河海大学土力学英文教案Chapter 2 Permeability of Soil文档格式.docx
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河海大学土力学英文教案Chapter 2 Permeability of Soil文档格式.docx
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2.2Darcy’sLaw(达西定律)
In1856,Darcypublishedasimpleequationforthedischargevelocityofwaterthroughsaturatedsoils,whichmaybeexpressedas
(3.3)
where
q=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.
2.3DeterminationofCoefficientofPermeability(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.
2.4SeepageandFlowNets(渗流和流网)
2.4.1Laplace’sequation
Laplacedifferentialequationofcontinuityisusedtodescribethetwo-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)
2.4.2FlowNets
Thecontinuityequation[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)
2.5.1Effectivestressprinciple
Effectivestress(’)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-waterpressureandeffectivestressunderhydrostaticcondition
Thetotalstressatdepthzis
(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)
(3.14b)
2.5.4Criticalhydraulicgradient(临界水力梯度)
Atboilingorquickconditionunderacriticalhydraulicgradient(ic),theweightofthesoilwillbebalancedbytheupwardseepageforcesuchthattheeffectivestressbecomeszero.Fromequation(3.14b)
(3.15)
whereeisvoidratioandnisporosity.
Table2.1Typicalvaluesofcoefficientsofpermeability(k)
Soiltype
k(cm/s)
Gravel(砾石)
>
10-1
Mixtureofgravelandsand(砾石与砂混合物)
10-3–10-1
Finesand(细砂)
10-5–10-3
Mixtureofsand,siltandclay(砂,粉土与粘土混合物)
10-7–10-5
Clay(粘土)
<
10-7
Keywords
Clay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.2.1Setupofconstantheadpermeabilitytest
Fig.2.2Setupoffallingheadpermeabilitytest
Fig.2.3Seepagethroughasoilelement
Chapter3Stressesinsoil(土体应力)
3.1Introduction
Soilmasscanbearloadingsfroms
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