工学类地盘工学特论 14页

工学类地盘工学特论本文由defang168贡献ppt文档可能在WAP端浏览体验不佳。建议您优先选择TXT，或下载源文件到本机查看。地盤工学特論第10回Stress,Strain,ElasticityandPlasticity廣岡明彦ahirooka@civil.kyutech.ac.jphttp://geo.civil.kyutech.ac.jp12004/6/23地盤工学特論前回の講義のまとめ前回の講義のまとめ-1粘土の排水せん断強度–エネルギー補正dyτl=τ?σdx(σ1?σ3)l地盤工学特論dε1+2dε3??=(σ1?σ3)+σ3???dε1??22004/6/23前回の講義のまとめ前回の講義のまとめ-2排水せん断強度σ1f?σ3f=2cdcosφd+(σ1f+σ3f)sinφdφd≒φ′cd≒c′圧密非排水三軸試験で求めたφと圧密排水三軸試験で求めたφdとの比較32004/6/23地盤工学特論前回の講義のまとめ前回の講義のまとめ-3粘土の残留強度一面せん断あるいはリングせん断試験における応力～変位関係の特徴\n地盤工学特論42004/6/23前回の講義のまとめ前回の講義のまとめ-4Stress,Strain,ElasticityandPlasticitymodel:physicalrepresentationinthesenseofascalemodel?aconceptualideaoranumberofmathematicalequationsForSoilsmathematicalmodels:basedontheoriesofelasticity,plasticity,andfrictionCriticalStateModel地盤工学特論SchofieldandWroth,1968AtkinsonandBransby,197852004/6/23前回の講義のまとめ前回の講義のまとめ-5PlainStrainAxialSymmetrySpecialcase:Solidcylindersofsoilintriaxialtestsσ’2=σ’3δε2=δε3Soil:anaggregateofmineralgrainswithafluidfillingtheporespacesdry;porefluidisairalonesaturated;porefluidiswateralone地盤工学特論62004/6/23前回の講義のまとめ前回の講義のまとめ-6effectivestressσ’=σ–ucontrolssoilbehaviorTheprincipleofeffectivestress(statedbyTerzaghi1936)?Allmeasurableeffectsofachangeofstress,suchascompression,distortion,andachangeofshearingresistance,aredueexclusivelytochangesofeffectivestress.Mathematicalmodelsforsoilbehaviormustbewrittenintermsofeffectivestress,nottotalstress.地盤工学特論72004/6/231.3AnalysisofDeformationandStrain\nδσ’v,δσ’h:stressincrementδv,δh:smallchangeofdimensionsModeofdeformation.(a)Stressesonaplaneelement.(b)Continuousstraining.(c)Discontinuousslipping.(b)nodiscontinuitiesintheelement.thebehaviorineachandeverypartoftheelementisthesame.(c)alldeformationsareduetoverylargestrainwithintheslipzone.→SlipPlane(inthelimit,thezone:zerothickness)(b)and(c)oftenoccursimultaneously，but(b)or(c)usuallypredominates.82004/6/23地盤工学特論Incrementofplanedisplacement(δh)δεx=??x?(δv)δεz=??z1??(δv)?(δh)?δεxz=??+?2??x?z?(δv)?(δh)?+δγxz=2δεxz=????z???xtheengineer’sshearstrain地盤工学特論92004/6/23Incrementofstrain(a)incrementsofstraininaplaneelement(positive)(b)principalstraincounterclockwiseshearstrains:positiveP:pole地盤工学特論Mohr’scircleofplanestraincorrespondingtotheincrementsofstrainabove102004/6/23Planesofzerostrainincrement.ψ:anangleofdilation(δε1+δε3)sinψ=?(δε1?δε3)δεγ=δε1?δε3δεv=δε1+δε3εγ:maximumshearstrainεv:volumetricstrain(ifε2=0)\nP:pole地盤工学特論γ=45°－ψ/2theincrementofminorprincipalstrainisnegativeandtensile112004/6/23δεvsinψ=?δεγ,inthelimit,dεvsinψ=?dεγForthespecialcaseofaxialsymmetry(δε2=δε3)2δvδεs=(δε1?δε3),δεv=δε1+2δε3=?3vv=1+e:specificvolume11δγn=(δε1?δε3)cosψ22,inthelimit,δεvtanψ=?δγn地盤工学特論dεvtanψ=?dγn122004/6/23DirectionsofzerostrainIncrementδεn=δεvthesamevalueforδγn:theplaneofzeroAstrainincrementFromthegeometry,thetangenttothepointAmakesanangleψtotheδεaxis.P:poleβ=45°＋ψ/2地盤工学特論132004/6/23alldeformationsandstrainsarecontainedwithinslipline?discontinuousslippingδmδγn=?δyδnδεv=?δyδndntanψ==δmdm地盤工学特論\nAnalysisofdiscontinuousslipping142004/6/231.4AnalysisofStateofStressStateofstressintwodimensionsMohr’scircleofstresscorrespondingtothestateofstressaboveP:pole地盤工学特論1′(σ1′?σ3)′=2sinρ1′′15(σ1+σ3)2004/6/232stressparameterst’s’mobilizedangleofshearingresistance:ρ’1′′t′=(σ1?σ3)21′′s′=(σ1+σ3)2t′sinρ′=s′Forthespecialcaseofaxialsymmetry(σ’2=σ’3)1′′′′q′=(σ1?σ3),p′=(σ1+2σ3)3q′η=p′地盤工学特論162004/6/23theplanesfordiscontinuousslippingStressonplanesforwhichtangentsfromtheorigintouchtheMohr’scircle1′′′τn=(σ1?σ3)cosρ′21′=(σ1+σ3)cos2ρ′′′σn2′τntanρ′=′σnP:poleρ′α=45°+2172004/6/23地盤工学特論′τn=τnt′=tq′=q′σn=σn?us′=s?up′=p?uρ′≠ρη′≠η\nMohr’scirclesoftotalandeffectivestress地盤工学特論182004/6/231.5RelationshipsbetweenstressandstrainCoaxialitycondition:principalplainsofstressandprincipalplainsofstrainandstrainincrementcoincide（共軸）;notageneraltheoreticalrequirementofamaterialbehavior;agoodapproximationformaterialbehaviorUnitWorkδE:incrementofenergytransferredfromtheexternalforcesδE=δU+δWδU:apartofenergystoredandrecoveredonunloadingδW:apartofenergydissipatedinthematerial地盤工学特論δU=0δW=0→dissipative:ex.plasticmaterial→conservative:ex.elasticmaterial192004/6/23′′′δE=σ1?δε1+σ2?δε2+σ3?δε3←continuousstrain′′δE=σn?δεn+τn?δγn←discontinuousslipping{δε}:strainincrementvectors{δε}=[C′]{δσ′}{δσ′}:effectivestressincrementvectors[C′]:compliancematrixcontainingamaterial{δσ′}=[C′]{δε}1地盤工学特論←determinantof[C’]isnon-zero202004/6/23foraxialsymmetry\nforplanestrainδεs??δq′?′??=Cqp???δp′??δεv??δεγ??δt′?′??=[Cts]???δs′??δεv?[]IncrementalModel↓alternativeformulationwhichrelatesincrementsofstraintostates,ratherthanincrements,ofstress地盤工学特論212004/6/231.6ElasticandPlasticDeformationsMathematicalmodelformaterialbehavior:statesandincrementsofstress?statesandincrementsofstrainbasedonthetheoriesofelasticityandplasticityExaminethebehaviorofanidealsoil-likematerialsubjectedtoprincipaleffectivestressσa’,σb’andσc’?theprincipleofeffectivestress“Strainsdependsoneffective,nottotal,stresses.”地盤工学特論222004/6/23Stress-straincurvesforanidealsoil-likematerialforuniaxialcompression(a)StrainhardeningOY:linearlyelastic,strainsarefullyrecoveredforstatesbeyondY:irrecoverableplasticstrainsoccurY:yieldpoint′σa～εa′′σb=σcandεb=εcthestressisreducedfromG:linearlyelasticintherangeBGG:yieldpoint↓YieldstressincreaseswithplasticstrainF:failurepoint地盤工学特論StrainhardeningG→F:sufferingelasticandplasticstrains232004/6/23\nStress-straincurvesforanidealsoil-likematerialforuniaxialcompression(b)StrainsofteningOY:linearlyelastic,strainsarefullyrecoveredforstatesbeyondY:irrecoverableplasticstrainsoccurY:yieldpoint′σa～εa′′σb=σcandεb=εcthestressisreducedfromG:linearlyelasticintherangeBGG:yieldpoint↓YieldstressdecreaseswithplasticstrainStrainsoftening地盤工学特論G→F:sufferingelasticandplasticstrains242004/6/231.7Yielding,HardeningandPlasticFlowvariouscombinationsofσa’andσc’atyieldandatfailureforastrainhardeningmaterialYaYc:yieldcurveFaFc:failureenvelopeYaYc～FaFcgeometricallysimilarornotYaYc→GaGcStrainHardeningchangeofyeildcurve?Hardeninglaw25plasticstrain2004/6/23地盤工学特論yieldsurfaceplasticstrainThestateofthematerialcanbeonorwithintheyieldsurfacebutcannotlieoutsideit….forastrainhardeningmaterialforastrainsofteningmaterial?????yieldsurface地盤工学特論262004/6/23Hardeningmaterial\nloadingpathA→BA→CA→D：traversetheyieldsurfaceandplasticstrainsoccurunloadingpathA→E：traverseanelasticwallandplasticstrainsarezeropurelyelasticstrainsoccurPathsforloadingandunloading地盤工学特論OgGaGc:elasticwallunloadingpathA→E’Impossible!!!!272004/6/23SofteningmaterialPathsforloadingandunloading地盤工学特論282004/6/23Flowruleofplasticitytheory(a)Plasticpotential(b)Normalityconditiontheratioofincrementsofplasticstrainplasticitytheorythestateofstresscausingtheplasticdeformationthevectorofplasticstrainincrementflowrulethevectorofstressplasticpotential=yieldcurve?associatedflowrule29?normalitycondition2004/6/23地盤工学特論1.8IdealElasticBehavior()()δε=(1E′)[δσ′?ν′σ′?ν′σ′]δδδγ=(2E′)(1+ν′)δτE’：Young’smodulusν’：Poisson’sratio2)(1+ν′)δτδγ=(E′δγ=(2E′)(1+ν′)δτδεx=1E′[δσ′?ν′σ′?ν′σ′]δyδzxδεy=1E′[δσ′?ν′σ′?ν′σ′]δzδxyzzxyxyxyyzyzzxzxthebehaviorofisotropicelasticmaterial←Hooke’slawreversiblerecoverable→conservative地盤工学特論\n302004/6/23′δ′δε1=1E′[δσ1?ν′σ2?ν′σ3]δ′′δε2=1E′[δσ2?ν′σ3?ν′σ1]δ′δ′3312()()δε=(1E′)[δσ′?ν′σ′?ν′σ′]δδ2(1+ν′)1δεs=?δq′=?δq′3E′3G′K’：bulkmodulusν3(1?2′)1?δp′=?δp′G’：shearmodulusδεv=E′K′E′E′K′=G′=31地盤工学特論3(1?2′)ν2(1+ν′)2004/6/23forthespecialcaseofaxialsymmetry(σ’2=σ’3,δε2=δε3)ontheprincipalstress(strain)planeforplanestrain(δε2=0)1+ν′′[(1?ν′)δσ1′?ν′δσ3]δε1=E′1+ν′′[?ν′δσ1′+(1?ν′)δσ3]δε3=E′2(1+ν′)1δεγ=?δt′=?δt′E′G′psK’ps：bulkmodulusappropriateforplanestrainG’ps：shearmodulusappropriateforplanestrain地盤工学特論2(1+ν′)(1?2′)1νδεv=?δs′=?δs′E′K′ps1K′=K′+G′G′=G′psps3322004/6/23forthespecialcaseofaxialsymmetryδεs:dependentonlyonδq’incrementofshearstress→incrementofshearstrainδεv:dependentonlyonδp’incrementofnormalstress→incrementofvolumetricstrainforplanestrain\nδεγ:dependentonlyonδt’δεv:dependentonlyonδs’地盤工学特論332004/6/23δεv9(1?2′)δp′ν=?δεs2(1+ν′)δq′theratioδεvδεsisdependentontheratioδp′δq′plasticitytheory（1.7）therelationshipbetweentworatios←thevalueofPoisson’sratioindependentofYoung’smodulus地盤工学特論342004/6/231.9Elasto-PlasticBehaviorelasticityyieldinghardeningplasticflowgeneralstress-strainequationsδεs=δε+δεespsδεv=δε+δεevpvtotalstrains=elasticcomponent+plasticcomponent地盤工学特論352004/6/231δε=δq′3G′1eδεv=δp′K′eselasticδεsp=Fpδεvpvflowruleδε=FH?δq′+FG?δp′\npsδε=H?δq′+G?δp′traverseyieldsurfaceH,G:hardeningparametersdependontheshapeoftheyieldsurface地盤工学特論362004/6/231??δεs=?FH+??δq′+FG?δp′3G′??1??δεv=H?δq′+?G+??δp′K′??thecoaxialityconditionholdstheflowruleisassociatedHG=F1??δεs=?FH+??δq′+H?δp′3G′???H1?δεv=H?δq′+?+??δp′?FK′?地盤工学特論372004/6/231.10EquilibriumandCompatibilityacompletesolutionforstressesanddisplacementofasoilstructure←necessarytosatisfyconditionsofequilibriumandcompatibilityforplanestrainσz?τxz+=γ?z?x?σx?τxz=0+?x?zinthex:zplaneγ:theunitweightofthesoil地盤工学特論382004/6/23σ′?τ′?uxzz+=γ??z?xintermsofeffectivestresses?z?σ′?τ′?uxxz+=??x?z?xnoseepage;porepressurearehydrostaticu=γw?zu=0?x地盤工学特論σ′?τ′z+xz=(γ?γw)?x?z?σ′?τ′xxz+=0?x?z\n392004/6/23thecompatibilitycondition2εx?(δh)=?22?z?x?z23?εz?(δv)=?22?x?z?x33?γxz?(δv)?(δh)=??22?x?z?z?x?x?z3andhence地盤工学特論εx?εz?γxz+2?=02?z?x?x?z2240validfordrainedandundrainedloading2004/6/231.11RelationshipsbetweenStressandStrainforPlaneStrainthestress-strainrelationshipforplanestrain[][][]′=C′p+C′eCpspspspδεx??δσ′?x?????δεz?=C′?δσ′?psz?δγ??δτ′??xz??xz?[]:compliancematrix地盤工学特論FpsHps?δεγ???p?=?H?δεv??ps?Hps???δt′?Hps????δs′?Fps??412004/6/23更多资源xiti123.taobao.com更多资源xiti123.taobao.com初二语文初二英语初二数学初二物理初二政治初二历史初二地理初二生物地盤工学特論422004/6/231\n