河海大学土力学英文教案Chapter-2-Permeability-of-Soil

Chapter2PermeabilityofSoil2.1 IntroductionSoilsareassemblagesofsolidparticleswithinterconnectedvoidsthroughwhichwatercanflowfromapointofhighenergytoapointoflowenergy.Thestudyoftheflowofwaterthroughporoussoilmediaisimportantinsoilmechanics.Thisisnecessaryforestimatingthequantityofundergroundseepageundervarioushydraulicconditions;forinvestigatingproblemsinvolvingthepumpingofwaterforundergroundconstructions;andformakingstabilityanalysesofearthdamsandearth-retainingstructuresthataresubjectedtoseepageforces.(i) HydrostaticGroundwaterCondition(静水条件)Pore-waterpressure(u,孔隙水压力)atadepthzbelowthewatertableisdefinedas (3.1)wherewisunitweightofthewater(水的重度)andzisdepth(深度)ofthewaterbelowgroundwatertable.TotalHead(总水头)Thetotalhead(h)isrelatedtothepore-waterpressure(u)andelevationhead(z)bythefollowingequation: (3.2)zofequation(3.2)isdefinedwithrespecttoachosendatum.GroundwaterFlowWaterwillflowthroughthesoil(sayfrompointAtoB)ifhydraulicgradient(oradifferenceintotalhead)existsbetweenpointsAandB.2.2 Darcy’sLaw(达西定律)In1856,Darcypublishedasimpleequationforthedischargevelocityofwaterthroughsaturatedsoils,whichmaybeexpressedas (3.3)whereq=flowrate(m3/s),渗流量i=hydraulicgradient水力梯度h=totalheaddifference(m)水头差L=lengthofflow(m)渗径长度k=coefficientofpermeability(m/s)渗透系数A=cross-sectionalareaofthespecimen(m2)横截面积Equation(3.3)impliesthattheflowrate(q)bearsalinearrelationshiptothehydraulicgradient(i).However,anon-linearrelationshipbetweenqandiisfoundforclay[叁考:土力学p.69图3-3].Forpracticalreason,thefollowingrelationshipbetweenqandiisassumedforclay: (3.3a)whereibisstartinghydraulicgradient(起始水力梯度).Forgravellysoil(砾土),linearrelationshipbetweenqandiappearsatsmallhydraulicgradientonly.Ashydraulicgradientincreasesbeyondacriticalvalue,therelationshipbecomesnon-linearastheflowbecomesturbulent(紊流)[叁考:土力学p.69图3-3].Darcy’slawisvalidforlaminarflow(层流)conditionwhereReynoldsnumbersissmallerthanorequalto1.Reynoldsnumbers(雷诺数)isdefinedasfollows: (3.4)whereisdensityofwater(水的密度),visvelocityofwater(流速),isviscosityofwater(水的粘滞系数)anddisaveragediameterofsoilgrains(土粒子平均粒径).As=1g/cm3,v=0.25cm/s,=0.0131g/scmattemperatureof10C,discalculatedbyequation(3.4)ThusDarcy’slawcanbeappliedtosoilswhicharefinerthancoarsesand.2.3 DeterminationofCoefficientofPermeability(渗透系数的测定)2.3.1Constantheadtest(常水头测试)AtypicalarrangementoftheconstantheadpermeabilitytestisshowninFig.2.1.Inthesetup,thetotalwaterheadisalwaysmaintainedconstantduringtheperiodofthetest.Theconstantheadtestissuitableforcoarse-grainedsoils.Thecoefficientofpermeabilityiscalculatedbythefollowingequation: (3.5)whereVisvolumeofwatercollected,Lislengthofthespecimen,Aiscross-sectionalareaofthespecimen,histotalheaddifferenceandtisdurationofthetest.2.3.2FallingHeadTest(变水头测试)AtypicalarrangementofthefallingheadpermeabilitytestisshowninFig.2.2.Itissuitableforfine-grainedsoils.Therateofflowofwaterthroughthespecimenatanytimetisgivenby (3.6a)whereVisvolumeofwatercollected,Lislengthofthespecimen,A1iscross-sectionalareaofthespecimen,A2iscross-sectionalareaofthestandpipeandhistotalheaddifference.Re-arrangingequation(3.6a)into (3.6b)Integratingequation(3.6b) (3.6c)ThevaluesofkfordifferenttypesofsoilaretypicallywithintherangesshowninTable2.1.2.4 SeepageandFlowNets(渗流和流网)2.4.1Laplace’sequationLaplacedifferentialequationofcontinuityisusedtodescribethetwo-dimensionalsteadyflowconditionforagivenpointinthesoilmass.LetusconsiderasoilelementasshowninFig.2.3.Theelementhasdimensionofdxanddzinxandzdirection,respectively.Waterflowsthroughtheelementduetoahydraulicheaddifferencebetweenupstreamanddownstreamside.Letvxandvzbethedischargevelocityinthexandzdirection,respectively.Theratesofflowofwaterintotheelementinthexandzdirectionare (3.7a) (3.7b)Theratesofflowofwateroutoftheelementinthexandzdirectionare (3.7c) (3.7d)Assumingwaterisincompressibleandnovolumechangeinthesoilmassoccurs,thetotalrateofinflowshouldbeequaltothetotalrateofoutflow (3.7e)UsingDarcy’slaw,vxandvzcanbeexpressedas (3.7f) (3.7g)Substitutingequations(3.7f)and(3.7g)intoequation(3.7e) (3.7h)Ifthesoilisisotropickx=kz,equation(3.7h)becomes (3.7i)2.4.2FlowNetsThecontinuityequation[equation(3.7i)]inisotropicmediumrepresentstwoorthogonalfamiliesofcurves–thatis,theflowlines(流线)andequipotentiallines(等势线).Aflowlineisalinealongwhichawaterparticlewilltravelfromupstreamtothedownstreamsideinthepermeablesoilmedium.Anequipotentiallineisalinealongwhichthetotalhead(总水头)atallpointsisthesame.Ifpiezometers(测压计)areplacedatdifferentpointsalonganequipotentialline,theheightofwaterwillrisetothesameelevationinallofthem.Acombinationofanumberofflowlinesandequipotentiallinesiscalledaflownet.Tocompletethegraphicalconstructionofaflownet,theflowandequipotentiallinesaredrawninsuchawaythat(i)theequipotentiallinesintersecttheflowlinesatrightangle,(ii)theflowelementsareapproximatesquares,(iii)theupstreamanddownstreamsurfacesofthepermeablelayerareequipotentiallinesand(iv)theboundaryoftheimpervouslayerisaflowline.Basedontheflownet,thetotalflowrate(q)canbeestimatedby (3.8)whereHisthetotalheaddifferencebetweentheupstreamanddownstreamsides,NfisthenumberofflowchannelsandNdisthenumberofequipotentialdrops. Inaddition,thepore-waterpressure(u)atagivenpointAdeterminedfromtheflownetis (3.9a) (3.9b)whereHistotalheaddifference,Ndistotalnumberofequipotentialdrops,ndisnumberofequipotentialdropsatpointA,htistotalheadatpointAandheiselevationheadatpointA.2.5Effectivestress(有效应力)2.5.1EffectivestressprincipleEffectivestress(’)isdefinedas (3.10a)whereNsisthesumofinter-contactforcesandAisthecross-sectionalareaofthesoilmassunderconsideration.Totalstress()isdefinedas (3.10b)whereAsiscross-sectionalareaofthesoilmassoccupiedbysolid-to-solidcontacts.AstheratioAs/Aissmallandcanbeneglectedforpressurerangesencounteredinpracticalproblems.Thus,equation(3.10b)becomes (3.11)2.5.2Pore-waterpressureandeffectivestressunderhydrostaticconditionThetotalstressatdepthzis (3.12a)Thepore-waterpressureatdepthzis (3.12b)Theeffectivestressatdepthzis (3.12c)wheresatissaturatedunitweightofsoil,wisunitweightofwaterand’issubmergedunitweightofsoil.2.5.3Pore-waterpressureandeffectivestressunderseepage(i) Downwardseepage Thepore-waterpressureatdepthzisreducedbecauseofthedownwardflowofwater (3.13a)wherehisthehydraulicheaddifferencebetweengroundsurfaceanddepthz.Theeffectivestressatdepthzbecomes (3.13b)(ii) Upwardseepage Thepore-waterpressureatdepthzisincreasedbecauseoftheupwardflowofwater (3.14a)wherehisthehydraulicheaddifferencebetweengroundsurfaceanddepthz.Theeffectivestressatdepthzbecomes (3.14b)2.5.4Criticalhydraulicgradient(临界水力梯度)Atboilingorquickconditionunderacriticalhydraulicgradient(ic),theweightofthesoilwillbebalancedbytheupwardseepageforcesuchthattheeffectivestressbecomeszero.Fromequation(3.14b) (3.15)whereeisvoidratioandnisporosity.Table2.1Typicalvaluesofcoefficientsofpermeability(k)Soiltypek(cm/s)Gravel(砾石)>10-1Mixtureofgravelandsand(砾石与砂混合物)10-3–10-1Finesand(细砂)10-5–10-3Mixtureofsand,siltandclay(砂,粉土与粘土混合物)10-7–10-5Clay(粘土)<10-7KeywordsClay粘土Constantheadtest常水头测试Cross-sectionalarea横截面积Darcy’sLaw达西定律Density密度Diameterofsoilgrains土粒子粒径Dischargevelocity渗透速度EffectiveStress有效应力Equipotentialline等势线Fallingheadtest变水头测试Flowline流线Flownet流网Flowrate渗流量Gravel砾石HydraulicHead水头Hydraulicheaddifference水头差Hydraulicgradient水力梯度Breakinghydraulicgradient起始水力梯度Criticalhydraulicgradient临界水力梯度Hydrostaticcondition静水条件Laminarflow层流LaplaceEquation拉普拉斯方程Mixture混合物Permeability渗透性Coefficientofpermeability渗透系数Pore-waterpressure孔隙水压力Porosity孔隙率Reynoldsnumbers雷诺数Sand砂土Seepage渗流SeepageForce渗流力Silt粉土TotalStress总应力Turbulentflow紊流Viscosity粘滞系数Voidratio孔隙比Fig.2.1SetupofconstantheadpermeabilitytestFig.2.2SetupoffallingheadpermeabilitytestFig.2.3SeepagethroughasoilelementChapter3Stressesinsoil(土体应力)3.1 IntroductionSoilmasscanbearloadingsfromstructuresaswellasitsself-weight.Stressesareproducedduetotheseloadings.Thestressesareusedtoevaluatethedeformationofthestructuresfoundedonthesoilmassandthestabilityofthesoilmass.Thischapterpresentsthemethodstodeterminetheselfweightandadditionalstressesactedonasoilmass.3.2 In-situstresses(原位应力)3.2.1Effectiveverticalstress(竖向应力)Theeffectiveverticalstress(’z)iscalculatedfromtheunitweightofthesoil(土的重度),.(i)Nogroundwatertable(地下水位)exists[叁考:土力学p.39图2-1] (3.1)where(kN/m3)isunitweightofthesoilandz(m)isdepthofthesoil.(ii)Groundwatertablelocatesatthegroundsurface (3.2)where’(kN/m3)issubmergedunitweightofthesoil(土的浮重度)andw(9.81kN/m3)isunitweightofwater(水的浮重度).(iii)Ifthegroundconsistsofndifferentlayersofstratum(土层) (3.3)whereiandziareunitweightanddepthofthesoiloftheithstratum,respectively.3.2.2Effectivehorizontalstress(侧向应力)Theeffectivehorizontalstress(’h)ismuchmoredifficulttobeevaluated.Normally,thehorizontalstressisexpressedasafunctionverticalstressbythefollowingexpression: (3.4)whereKoiscoefficientofearthpressureatrest(静止侧压力系数).TypicalvaluesofKoarelistedinTable3.1.3.3 Contactpressuresatthebottomofthefoundation(基底压力)3.3.1FlexibleandrigidfoundationsThepressuredistributionbeneathafoundationdependsonthetypeofload,theflexuralrigidityofthefoundationandthetypeofmaterialonwhichitisresting.Aflexiblefoundation(柔性根底)haszerostiffnessbutarigidfoundation(刚性根底)hasinfinitestiffness.Auniformlyloadedflexiblefoundations(e.g.oiltankandearthdam)restingonanelasticmaterialsuchassaturatedclaywilltakeasaggingprofileandthecontactpressureisalsouniformlydistributed[叁考:土力学p.41图2-4].Arigidfoundation(e.g.concretedam,boxfoundation)restingonsaturatedclaywillundergouniformsettlementandthecontactpressuredistributionlookslikeasaddle,i.e.largeattheedgebutsmallinthemiddle[叁考:土力学p.42图2-5].Ifthesizeofthefoundationisnottoolargeandtheappliedloadissmall,lineardistributionofthecontactpressurecanbeused.3.3.2Pressuredistributionatthebottomoftherigidfoundation(i)Verticalloadatthecenterofthefoundation(中心竖向荷载) (3.5)whereFvisthesumoftheappliedverticalload(竖向荷载)andtheweightofthefoundationandbackfill(根底和回填土的总重),ListhelengthofthefoundationandBisbreadthofthefoundation.(ii)Eccentricandverticalload(偏心竖向荷载) (3.6)whereeistheeccentricityofthetotalverticalload.Fore<L/6,atrapezoidalpressuredistribution,Fore=L/6,atriangularpressuredistribution,Fore>L/6,pmin<0,partofthefoundationisseparatedfromthesoilandthestressredistributestoatriangularpressuredistribution[叁考:土力学p.43图2-7e]withpmaxequalsto (3.7)(iii)Netpressureatthebottomofthefoundation(基底净压力)Beforetheconstructionofafoundation,thesoilexhibitsanoverburdenpressureduetoitsselfweight.Duringconstruction,soilisexcavatedtotheleveloffoundation.Thefinal(ornet)pressureexertedonthefoundationistheappliedpressureminustheweightoftheexcavatedsoil. (3.8)wherezoisdepthoffoundation.3.4 Stressincreaseduetosurfaceload(地基中的附加应力)3.4.1Stressesduetoaverticalpointload(竖向集中力作用下附加应力)Boussinesq(1883)solvedtheproblemofstressesproducedatanypointinahomogeneous(均质),elastic(弹性)andisotropic(各向同性)mediumastheresultofapointloadappliedonthesurfaceofasemi-infinitehalfspace(半无限空间体).Boussinesq’ssolutionforstressesatapointM[叁考:土力学p.47图2-10]duetothepointloadFis (3.9a) (3.9b) (3.9c) (3.9d) (3.9e) (3.9f) (3.9g) (3.9h)whereisPoisson’sratio.Equation(3.9c)canbere-arrangedasfollows: (3.9i)where (3.9j)Kisafunctionofr/z.Byusingtheprincipleofsuperposition(叠加原理),theverticalstressproducedatpointMduetoseveralpointloadsFi(i=1,2,…n)[叁考:土力学p.47图2-12]canbededucedfromequation(3.9j)andpresentedasfollows: (3.9k)3.4.2Stressincreasein3-dimensionalproblems(空间条件下的附加应力)(i) Verticalstressbeneaththecornerofaverticalanduniformlyloaded(竖直均布荷载)rectangularfoundation Letthenetpressureontherectangularareabeequaltop[叁考:土力学p.48图2-13].Thetotalloadontheelementaryareaispdxdy.Theverticalstress(dz)atpointMduetotheloadonthiselementaryareacanbeobtainedfromequation(3.9c) (3.10a)TheincreaseofstressatMduetotheentirerectangularloadedareacanbefoundbyintegratingequation(3.10a) (3.10b)wherem=L/Bandn=z/B,Lislengthoftherectangle,Bisbreadthoftherectangle,zisdepthofpointAfromthebottomofthefoundation.Equation(3.10b)canbere-writtenasfollows: (3.10c)KsisafunctionofmandnorL.Bandz.ValuesofKscanbeobtainedfrom土力学p.52表2-2.(ii) VerticalstressbeneaththecenterofaverticalanduniformlyloadedcircularfoundationLetthenetpressureonthecircularfoundationofradiusrobeequaltop[叁考:土力学p.52图2-18].Thetotalloadontheelementaryareaisprdrd.Theverticalstress(dz)atpointAbelowthecenterduetotheloadonthiselementaryareacanbeobtainedfromequation(3.9c)TheincreaseofstressatAduetotheentirecircularareacanbefoundbyintegratingdz (3.11)Krisafunctionofz/r0andvaluesofKraresummarizedin土力学p.55表2-5.3.4.3Stressincreasein2-dimensionalproblems(平面问题条件下的附加应力)(i) Stressesduetoalineload(线荷载)Thefigure[叁考:土力学p.56图2-19]showsalineloadofinfinitelength(ydirection)havingapressurep/unitlengthonthesurfaceofasemi-infinitesoilmass.Thetotalloadontheelementarylengthispdy.Theverticalstress(dz),horizontalstress(dx)andshearstress(xz)atpointAduetotheloadonthiselementarylengthcanbeobtainedfromequations(3.9c),(3.9a)and(3.9e),respectively.TheincreaseofstressesatMduetotheentirelineloadcanbefoundbyintegratingdz,dxandxz, (3.12a) (3.12b) (3.12c)where (3.12d) (3.12e)Thisisaplanestressproblemandfromtheoryofelasticity (3.12f)(ii) Verticalstressbeneathaverticalanduniformlyloadedstripfoundation(条形根底竖直均布荷载)Equation(3.12a)derivedfortheverticalstressincreasesastheresultofalineloadcanbeusedtodeterminethestressincreasesatapointduetoastriploadofwidthB[叁考:土力学p.56图2-20].Letthepressureactingonthefoundationbepnandconsideranelementarystripofofwidthd.Theloadperunitlengthofthisstripispnd.Thiselementarystripcanbetreatedasalineload.Theverticalstressincrease(dz)iscalculatedfromequation(3.12a)byreplacingpbypnd.andxby(x-).TheincreaseofstressesatAduetotheentirestriploadcanbefoundbyintegratingdz, (3.13a)where (3.13b)wherem=x/Bandn=z/BandvaluesofKzsaretabulatedin土力学p.59表2-6.Table3.1TypicalvaluesofKoSoiltypeKo(静止侧压力系数)Loosesands(松砂)0.40–0.45Densesands(密砂)0.45–0.50Well-compactedfills(压实填土)0.8–1.5Normallyconsolidatedclays(正常固结粘土)0.5–0.6Overconsolidatedclays(超固结粘土)1.0–4.0KeywordsClay粘土 Normallyconsolidatedclay正常固结粘土Overconsolidatedclay超固结粘土Coefficientofearthpressureatrest(Ko)静止侧压力系数Earthdam土坝Eccentric偏心Elastic弹性Fill填土 Backfill回填土Well-compactedfill压实填土Foundation根底 Bottomoffoundation基底 Flexiblefoundation柔性根底Rigidfoundation刚性根底Stripfoundation条形根底Groundwatertable地下水位Homogeneous均质In-situstresses原位应力Isotropic各向同性Load荷载 Horizontalload水平荷载Inclinedload倾斜荷载Lineload线荷载Triangularload三角形分布荷载Uniformload均布荷载Verticalload竖向荷载Verticalpointload竖向集中力Moment力矩Pressure压力 Netpressure净压力Principleofsuperposition叠加原理Right-angledtrapezoidalload直角梯形分布荷载Sand砂土 Loosesand松砂Densesand密砂Semi-infinitehalfspace半无限空间体Stratum土层Stress应力Horizontalstress侧向应力Stressincrease附加应力Verticalstress竖向应力Two-dimensionalproblem平面问题Unitweightofsoil土的重度Submergedunitweightofsoil土的浮重度Chapter4CompressibilityandConsolidationofSoil[土的压缩与固结]4.1 IntroductionTheprocessofconsolidation[固结]isoftenconfusedwiththeprocessofcompaction[压实].Compactionincreasesthedensityofanunsaturatedsoilbyreducingthevolumeofairinthevoids(seeFig.4.1).However,consolidationisatime-relatedprocessofincreasingthedensityofasaturatedsoilbydrainingsomeofthewateroutofthevoids(seeFig.4.1).Consolidationisgenerallyrelatedtofine-grainedsoils[幼粒土]suchassiltsandclays.Coarse-grainedsoils[粗粒土],suchassandsandgravels,alsoundergoconsolidationbutatamuchfasterrateduetotheirhighpermeability.Saturatedclaysconsolidateatamuchslowerrateduetotheirlowpermeability.Consolidationtheoryisrequiredforthepredictionofboththemagnitudeandtherateofconsolidationsettlements[沉降量与沉降速度]toensuretheserviceabilityofstructuresfoundedonacompressiblesoillayer.4.2 Asimpleone-dimensionalconsolidationmodel[单向固结模型]Sincewatercanflowoutofasaturatedsoilinanydirection,theprocessofconsolidationisessentiallythree-dimensional[三维].However,inmostfieldsituations,waterwillnotbeabletoflowoutofthesoilbyflowinghorizontallybecauseofthevastexpanseofthesoilinhorizontaldirection.Therefore,thedirectionofflowofwaterisprimarilyvertical[竖向]orone-dimensional.Asaresult,thesoillayerundergoesone-dimensional(1-D)consolidationsettlement[单向固结沉降]intheverticaldirection.Fig.4.2showsasimplemodelfor1-Dconsolidation.Thespring[弹簧]isanalogoustothesoilskeleton[土骨架].Thestifferthespring,thelessitwillcompress.Therefore,astiffsoilwillundergolesscompressionthanasoftsoil.Thestiffnessofasoil[土的硬度]influencesthemagnitudeofitsconsolidationsettlement.Thevalve[阀门]openingsizeisanalogoustothepermeabilityofthesoil[土的渗透性].Thesmallertheopening,thelongeritwilltakeforthewatertoflowoutanddissipateitspressure.Therefore,consolidationofafine-grainedsoiltakeslongertocompletethanthatofacoarse-grainedsoil.Permeabilityofasoilinfluencestherateofitsconsolidation.4.3 One-dimensionalconsolidationtest[单向固结试验] Theone-dimensional(1-D)consolidationtestisperformedinanoedometer[固结仪].AnoedometerisshowninFig.4.3.Thesoilspecimen[土样本]isinaformofadisc(normally20mminheightand80mmindiameter),whichisconfinedinasteelringandimmersedinawaterbath.Averticalload[竖向荷载]isappliedtocompressthespecimenandwaterispermittedtodrainawaythroughporousstones[透水石]placedatthetopandbottomofthespecimen.4.3.1Time-relatedconsolidation Foreachverticalloadincrement[竖向荷载增量],theverticalsettlement[竖向沉降]ofthesoilspecimenisrecordedusingadialgauge[百分表].Fig.4.4showsthetimerelationshipsofverticalsettlement,verticaltotalstress,excesspore-waterpressure[超孔隙水压力]andverticaleffectivestress.Initially,100%oftheverticalloadistakenbyporewaterbecause,duetolowpermeabilityofthesoilspecimen,theporewaterisunabletoflowoutofthevoidsquickly.Therefore,thereisverylittlesettlementofthesoilspecimenimmediatelyafterplacingtheverticalload.Thesettlementofsoilispossibleonlywhenthereisanincreaseineffectivestress,whichinturnrequiresthatthevoidratioofthesoilbereducedbytheexpulsionofporewater.Afterafewseconds,theporewaterbeginstoflowoutofthevoids.Thisresultsinadecreaseinexcessivepore-waterpressureandvoidratioofthesoilspecimenandanincreaseineffectivestress.Finally,allexcesspore-waterpressuredissipatesandtheverticaleffectivestressequalstotheverticaltotalstress.4.3.2CompressionCurve[压缩曲线]Severalincrementsofverticalstressareappliedinanoedometertest.Foreachincrement,thefinalsettlementofthesoilspecimenisrecorded.Theendpointsfromnumberofloadingandunloadingincrements[荷载与卸载增量]ofanoedometertestmaybeplottedasaconventionalstress-straincurve[应力-应变曲线]asshowninFig.4.5.Theincrementofverticalstrain[竖向应变增量](v)foreachloadingincrementisgivenby (4.1)wherehisthefinalsettlementfortheloadingincrement(i.e.thechangeinspecimenheight)andH0istheinitialspecimenheight[样本初始高度]beforeloadingincrementisapplied.(i) e:’vcurveAsthesoilspecimenisnotallowedtodeforminhorizontaldirection,thereisonlychangeinvoidratio.Thus,vcanbeexpressedintermsofthevoidratio[孔隙比](e)by (4.2)whereeisthechangeinvoidratioduetotheloadingincrementande0istheinitialvoidratioofthespecimen[初始孔隙比]beforeloadingincrementisapplied.Thetestresultsareplottedintermsofverticaleffectivestress(’v)andvoidratio(e)andshowninFig.4.6.Thecoefficientofvolumecompressibility[体积压缩系数](mv)isdefinedastheratioofchangeinvolumetricstrain(vol)overchangeinverticaleffectivestress(’v)andshownasfollows: (4.3)Theunitformvism2/kNanditsvaluedependsonthestressrangeoverwhichitiscalculated.Example:ConsiderthesecondloadingincrementshowninFig.4.7,(ii) e:log’vcurveThee:’vcurvebecomesalmostlinearif’visplottedonalogscaleasshowninFig.4.8.TheslopeoftheloadingcurveiscalledtheCompressionIndex[压缩指数](Cc)andisdimensionless.Itisdefinedas: (4.4)Thenegativesignisusedbecausethevoidratiodecreaseswhentheeffectivestressisincreased.TheslopeoftheunloadingcurveiscalledtheExpansionIndex[回弹指数](Ce)andiscalculatedusingthesameprocedure.(iii) OthercompressibilityparametersOedometricmodulus[侧限压缩模量](Es)isdefinedastheratioofverticaleffectivestress(’z)oververticalstrain(z)andshownasfollows: (4.5)FromgeneralisedHooke’slaw[广义虎克定律],elasticstrains[弹性应变]inx,yandzdirectionsareexpressedasfollows: (4.6a) (4.6b) (4.6c)whereEisYoung’smodulus[弹性模量].In1-Dconsolidation,x=y=0,thusEquations(4.6b)and(4.6c)become (4.6d) (4.6e)SubstituteEquation(4.6e)intoEquation(4.6d) (4.6f)Sincex=y=zK0,whereK0isthecoefficientofearthpressureatrest[静止土压力系数],therefore (4.6g)Inone-dimensionalconsolidation,substituteEquations(4.5)and(4.6f)intoEquation(4.6c)andre-arrangedtothefollowingexpression: (4.6h)4.4 Effectofstresshistory[应力历史]Fig.4.9showsthestresshistoryofaclaydeposit.Duetothemeltingoficeandgrounderosion,thepresentverticaleffectivestress(’v)issmallerthanthemaximumpasteffectiveverticalstress(’vmax).Overconsolidationratio[超固结比](OCR)isdefinedastheratioof’vmaxover’v: (4.7)’vmaxisalsocalledthepre-consolidationpressure[前期固结压力](’c).Asoilthathasneverexperiencedaverticaleffectivestressthatwasgreaterthanitspresentverticaleffectivestressiscalledanormallyconsolidated[正常固结土](NC)soil.TheOCRforanNCsoilisequalto1.MostNCsoilshavefairlylowshearstrength.Asoilthathasexperiencedaverticaleffectivestressthatwasgreaterthanitspresentverticaleffectivestressiscalledanoverconsolidated[超固结土](OC)soil.TheOCRforanOCsoilisgreaterthan1.MostOCsoilshavefairlyhighshearstrength.TheOCRcannothaveavaluelessthan1.The1-DconsolidationofanOCsoilisshowninFig.4.10.Theslopeofthecompressioncurveisfairlyflatuntilaverticaleffectivestressequaltothepre-consolidationpressure(’c)isreached.Beyondthispoint,theslopeofcompressioncurvebecomessteeper,i.e.thesoilbecomesmorecompressible.Infact,’cislikeayieldstress[屈服应力]forsoil.Thisfactcanbeappreciatedbyrotatingthecurveby90°inanti-clockwisedirection(seeFig.4.11).Doesn’tthiscurveresembletheextensioncurveofametalrod?4.5 Settlement[沉降]Ingeneral,thesettlementcausedinsoilduetoloadingmaybedividedintothreecategories: (4.8)whereStistotalsettlement,Siisimmediatesettlement[瞬时沉降],Scisprimaryconsolidation[主固结沉降]andSsissecondaryconsolidation[次固结沉降].Immediatesettlementisduetotheelasticdeformation[弹性变形]ofsaturatedsoilswithoutanychangeinthewatercontent.Thereisnovolumechangeofthesoilbutlateraldeformation.Immediatesettlementforfoundationsrestingonelasticmaterialcanbederivedfromthetheoryofelasticity[弹性理论]andexpressedas (4.9)wherepisthepressureatthebottomofthefoundation[基底压力],biswidthofthefoundation[根底的宽度](=diameterofcircularfoundation),EisYoung’smodulus[弹性模量]andIisinfluencefactor[影响系数].ValuesofIisfoundin土力学p.94表4-1. Primaryconsolidationisthesettlementduetothegradualdissipationoftheporewaterfromthesaturatedsoils.Thesecondaryconsolidationisthesettlementwithtimeunderconstanteffectivestress[有效应力不变下的沉降].Primaryconsolidationiscalculatedfromthefollowingtwo