Engineering Basic Mechanics I Statics 工程基础力学Ⅰ 静力学 课件 Chapter 11 State of Stress and Theories of Strength

StaticsStaticsofdeformablebodyChapter11

StateofStressandTheoriesofStrength11.1Stateofstress11.2Analysisofplanestress11.3Mohr’scircleforplanestress11.4Triaxialstressstate11.5GeneralizedHooke'slawandvolumetricstrain11.6Strainenergydensityoftriaxialstress 11.7TheoriesofstrengthContents1.ThestateofstressofapointThestateofstressofapointinvariousorientationsiscalledthestressstate.

Thestudyofthestressstateatonepointiscalledstressanalysis.Thepurposeofthestressanalysisistodeterminethemostdangerouspositionanddirectionofthemember,andtoprovideabasisforanalyzingthestrengthofthemember.11.1Stateofstress2.originalelement,principalelementInordertostudythestressstateatapoint,anelement(infinitesimalnormalhexahedron)canbetakenoutaroundthepoint.2.Thestressconditionsontwomutuallyparallelsidesarethesame3.Itrepresentsthestressesinthreemutuallyperpendiculardirectionsatthecorrespondingpoint1.Thestressdistributiononeachsideoftheelementisuniform.characteristicsAnelementwherethestressesonallthreemutuallyperpendicularsurfacesareknownquantitiesiscalledaoriginalelement.AsshowninFig.(a).Thenormalstressesontheleftandrightsidesoftheelement()areknowntobeσx=P/Aandthereisnoshearstress.Thestressontherestofthefacesiszero.So,theelementistheoriginalelementofpointA.APPaxsxsA(a)(a)a=0°xsxsAA2xsmaxta=45°2xsTheelementwitharoundpointAinFigure(b)isalsotheoriginalelement.AeMAA(b)ta=45°maxs=tmins=-ta=0°Ingeneral,therearethreestressesoneachfaceoftheoriginalelement:anormalstressandtwoshearstresses.Theelementthathavenoshearstressonallsidesandonlynormalstress(Includingthecasewherethenormalstressis0)iscalledtheprincipalelement.Itcanbeprovedthataroundanypointwithinthememberaprincipalelementcanbealwaysfound.Thenormalstressoneachfaceoftheprincipalelementiscalledtheprincipalstress.Theplaneofprincipalstressiscalledtheprincipalplane.Itcanalsobedefinedasfollows:theplanewheretheshearstressisequaltozeroiscalledtheprincipalplane,andthenormalstressintheprincipalplaneiscalledtheprincipalstress.3.ClassificationofstressstatesTherearethreepairsofprincipalstressesonthesixfacesoftheprincipalelementatonepoint,whichareusuallydenotedasσ1,σ2,σ3.Theiralgebraicmagnitudeisσ1≥σ2≥σ3.

Accordingtothenumberofprincipalstresses(notzero),thestressstatescanbeclassifiedintothreecategories.s1s3s21.UniaxialstressstateOnlyoneprincipalstressisnotzero.2.BiaxialstressstateTwoprincipalstressarenotzero.3.Triaxialstressstate

Allthreeprincipalstressesarenotzero.4.Exampleofcomplexstressstate

1.Biaxialstressstatethestressstateofaboilerorothercylindricalvessel()

mnmnl(a)nn(b)Calculationofaxialstress:thetotalpressureactingonthebottomofthecylinderatbothendsalongthecylinderaxisisThentheaxialstressismnABCDs¢s¢s¢¢s¢¢mnl(a)ps¢nn(b)3.TriaxialstressstateInaballbearing,thestressstateatthecontactpointbetweentheballandtheouterringisatriaxialstressstate.s1s3s2(a)(b)PFig.6-4Ontheplanewiththexaxisasthenormaldirection,thenormalstressisdenotedbyσx.Theshearstressisdenotedbyτxy，thesubscriptxofτxyindicatesthatthedirectionoftheshearstressisnormaltothexaxis,andthesubscriptyindicatesthatthedirectionoftheshearstressisparalleltotheyaxis.11.2AnalysisofplanestressyxxsxsysysxytxytyxtyxtThesignofthenormalstress:thetensilenormalstressandtheclockwiseshearstressarepositive.1、StressesontheinclinedsectionsyxxsxsysysxytxytyxtyxtyxysxsxytyxtabcdafenaanaeafysyxtxytxstasatfeasindAadA(a)(b)(c)(d)Accordingtotheequilibriumoftriangularprism,theequilibriumequationofthenormalndirectionandthetangenttdirectioncanbewrittenasaanaeafysyxtxytxstasatfeasindAadA(c)(d)Thesearetheequationsforcalculatingthestressesinanyinclinedsectionoftheelement.2、PrincipalstressandprincipalplaneConsidertheextremevalueofthenormalstressinobliquesection.Takingthederivativeoffollowingequationtoα,wegetwhen

,weobtain

thentherearetwovaluesthatsatisfytheaboveequation:

and,thenwecanget

.sothemaximumshearstressandminimumshearstress:

xoaminsmaxs3、Maximum(minimum)shearstressandcorrespondingplaneThesamemethodcanbeusedtodeterminethemaximumshearstressandminimumshearstressoftheelementandtherotationangleofthecorrespondingplane.differentiateThenweget

Fromaboveequation,twoanglesα1with90°differencecanalsobedetermined,Themaximumshearstressandtheminimumshearstressplanesareperpendiculartoeachother.solvefromaboveequationtofindsin2α1

andcos2α2,substituteintofollowingequation,themaximumshearstressandminimumshearstresscanbeobtainedas

Comparingfollowingtwoequationswecanseethat（e）Theplanewherethemaximumandminimumshearstressesarelocatedisatanangleof45°totheprincipalplane.fromandExample1

Theoriginalelementatapointisshowninfigure.Trytocalculate:theprincipalstressandtheprincipalplaneorientation;themaximumandminimumshearstresses,andtheorientationoftheplanesinwhichtheyarelocated.solutionfrompicturewecanget

(1)Lettheangleoftheprincipalplanebeα0,then(a)x100MPa40MPa20MPaor(b)22.5°(a)x(b)116.6MPa22.5°3.4MPaTheprincipalstressintheprincipalplaneatangleα0=-22.5∘(clockwisedirection)withthex-axisisσmax，whiletheprincipalstressontheplaneofα0=-122.5∘isσmin.ThemaximumandminimumnormalstressesareTherefore,theprincipalstressis100MPa40MPa20MPa(2)

Theangleoftheplanewherethemaximumshearstressislocatedis

themaximumandminimumshearstressesarecorrespondingnormalstressis(a)(c)xx22.5°100MPa40MPa20MPa56.6MPa60MPaExample2

Abeamundertransverse-forcebendingisshowninFig.(a).AfterfindingthebendingmomentMandshearforceFQinthecrosssectionm-n,thenormalandshearstressesatapointAofthecrosssectionareσ=-70MPaandτ=50MPa,respectively.TrytodeterminetheprincipalstressatpointAandtheorientationoftheprincipalplane,anddiscussthestressstateatotherpointsonthesamesection.alm(a)Amnst(b)nsolution：TheelementofpointAisenlargedandshowninFig.(c)Accordingtosetverticalupwardasthepositivedirectionofx-axis,wegetx1s3s(c)70MPaxyt27.5°or50MPaThereforethestressintheα0=27.5°planeisσmax,whilethestressintheα0=117.5°planeisσmin，themaximumandminimumnormalstressesare：theprincipalstressisx1s3s(c)xyt27.5°70MPa50MPa1.Mohr’scircle1.Mohr'scircleequation

11.3Mohr’scircleforplanestressSquaretheleftandrightsidesoftheabovetwoequations,andthenaddthetwoequationstogethertogetThiscircleiscalledtheStresscircle,orMohr’scircle.

Equation(a)iscalledtheMohr’scircleequation.（a）Center2.DrawMohr’scircle0aBCADyxtxytosyxxytysyxxytys3.ProcessofproofThecenterCofthiscircleisontheσaxisandithorizontalcoordinateisBCADxytxytosradius2.ApplicationofMohr’scircle1.determinethestressinanyinclinedsectionofanelement.yxxytnaatasys(a)AsshowninFig.(a),lettheanglefromthexaxistothenormalnofanyinclinedsectionbethecounterclockwiseangleα.yxnaxaxytatasys(a)BCADxytxytosFSothecoordinatesofthepointEare:(b)oBCAFDE1G1A1BD¢2Gs2sysxs1s2a02axytyxttTheresultisexactlythesameastheresultcalculatedbyequation.(b)oBCAFDE11A1BD¢2Gs2sysxs1s2a02axytyxttG2.determinetheprincipalstressandthelocationoftheprincipalplaneA1andB1onthestresscirclehavemaximumandminimumtransversecoordinates,respectively.Sincetheverticalcoordinatesofbothpointsarezero,thetransversecoordinatesofthesetwopointsaretheprincipalstressintheprincipalplane.(b)oBCAFDE11A1BD¢2Gs2sysxs1s2a02axytyxttG3.determinethemaximumshearstressandcorrespondingplaneMakeaverticalradius，theverticalcoordinatesofthepointsG1andG2arethemaximumandminimumshearstressesrespectively.SowehaveBecausetheabsolutevaluesofτmaxandτminareequaltotheradiusoftheMohr’scircle,wecanget(b)oBCAFDE11A1BD¢2Gs2sysxs1s2a02axytyxttGExample4IntheoriginalelementshowninFig.(a),ithavetheσx=80MPa,σy=-40MPaandτxy=-60MPa.TrytofindtheprincipalstressesandthelocationoftheprincipalplaneusingtheMohr’scirclemethod.solution：yx0a1s3sC1B1AD¢02aD1s3s0204060a80MPscale(a)(b)a60MPa40MPa80MPostAttheselectedscale：

C1Bo1AD¢02aDst1s3s(b)Example5Analyzenormalstressσαandtheshearstressτα

insectiondeusingtheMohr’scirclemethod.

solution：Selectaappropriatescale010515a20MPa40MPyedxasat60°120°cBEtosas(a)(b)atWecanget

010515a20MPa40MPyedxasat60°120°tocBEsas(a)(b)atExample

6:Intwoplanespassingthroughapoint,thestressesareshowninthefollowingFigure(stressesinMPa).Trytofindtheprincipalstressandtheprincipalplane(expressedinelements).150°4595Solvingtheaboveequationtogethergives150°459575°105°then：

150°4595yx0a1s

11.4Triaxialstressstate

bdac1s2s3s3s(b)1B1A1Cost1s3s2sD1G2s1s3s(a)abcdThepointsrepresentingthestressesinanycrosssectionoftheelementintheσoτcartesiancoordinatesystemmustlieonthecircumferenceofthethreestresscirclesandwithintheshadedareaenclosedbythem.st

Itcanbeseenthatthemaximumandminimumnormalstressesinthetriaxialstressstatearethemaximumandminimumprincipalstresses,respectively,

（6-10）

（6-11）the

maximumshearstressshouldbe

（6-12）

withinthe45°sectionformedbyand.01BD1A1C1Gst1s3s2sExample7

Apointisinatriaxialstressstate,anditselementisshowninthefigurebelow.Finditsprincipalstressandmaximumshearstress.solution：Oneprincipalplaneandtheprincipalstressonthatfaceareknown.Theothertwoprincipalstressescanbefoundinasimilarwaytobiaxialstressstate.a20MPAs=a60CMPs=a40MPt=AtBC(a)a40MPt=a20MP(b)t1s2s3ss(60,0)Co(c)TaketwopointsA(-20,40),B(0,-40)intheσoτcartesiancoordinatesystem.WithABasthediameteroftheMohr’scircle,thetwointersectionpointsoftheMohr’scircleontheσaxisaretheothertwoprincipalstresses,whichare31MPaand-51MPa,respectivelya20MPAs=a60CMPs=a40MPt=AtBC(a)A(-20,40)(0,-40)Ba40MPt=a20MP(b)Sincetheprincipalstressis60Mpa,andaccordingto,wehaveThenmaketwootherMohr’scircles.ThemaximumshearstressisontheMohr’scircleofthelargestdiameter,wegeta20MPAs=a60CMPs=a40MPt=AtBC(a)a40MPt=a20MP(b)t1s3ss(60,0)Co(c)A(-20,40)(0,-40)B2smaxt11.5GeneralizedHooke'slawandvolumetricstrain

1.GeneralizedHooke'slawinitsgeneralformInuniaxialtensileorcompressivedeformation,therelationshipbetweenstressandstraininthelinearelasticrangeis

or（a）Thisisthetension(pressure)Hooke'slaw.Transversestrainε'canbeexpressedas（b）TherelationshipbetweenshearstressandshearstrainobeysHooke'slawinshearwhentheshearstressdoesnotexceedtheshearproportionallimit

（c）Thenormalstrainatapointforanisotropicmaterialisonlyrelatedtothenormalstressatthatpoint,butnottotheshearstress.Atthesametime,theshearstrainatthatpointisonlyrelatedtotheshearstress.Thesetwotypesofrelationshipsareinvestigatedseparately.yxzysxszsoxytxztxztyxtyztstresstensorFirstdiscusstherelationshipbetweenthenormalstrainεxinthex-directionandthenormalstressσx,σy

andσz.Thenormalstraininthex-directioncausedbyσxaloneisThenormalstrainscausedbyσyandσz

inthex-directionareyxzysxszsoxytxytxztxztyxtyzt

Thenormalstraininthex-directioncanbeobtainedbysummingtheabove3equations

（d）Similarly,wecangetthenormalstraininyandzdirection.Finally,weget

Hooke'slawinshear:ThisisthegeneralizedHooke'slawinitsgeneralform.andWhenallfacesoftheelementareintheprincipalplane,Letthex、yandzaxisdirectionbethesameasσ1，σ2，σ3direction.Then

thegeneralizedHooke'slawbecomes1s2s3sabcwhichisthegeneralizedHooke'slawexpressedintermsofprincipalstresses.2、SimplificationoftheplanarcaseIntheplanestress,substitutingσz=τyz=τzx=0intoequation(11-29)andequation(11-30),thenon-zerostraincomponentisobtained.Usingstraintoexpressstress,aboveequationbecomeswhichareHooke'slawfortheplanestressstate.3.Volumetricstrain

AsshowninFig.11-17,letthelengthsofthethreeedgesofa、b

andcbeforedeformationbedx、dyanddz,thenthevolumeoftheelementis

Ifthenormalstrainsofthethreeprismaticedgesofa、b

andcafterdeformationareε1、ε2

andε3,thevolumeoftheelementbecomesExpandingtheaboveequationandnotingthatthehigher-orderdifferentialisnegligibleinthecaseofsmalldeformations,wehave

Therefore,thechangeperunitvolumeis

Substitutingequation(11-31)intotheaboveequation,weget

symbols:thebulkmodulusofelasticity

theaverageprincipalstressThenweobtaintheequationforthevolumeHooke'slaw:

11.6Strainenergydensityoftriaxialstress1.Strainenergydensity

Inuniaxialtensionorcompression,thestrainenergyisnumericallyequaltotheworkdonebytheexternalforce.ByusingtheHooke'slaw,theformulaofelasticstrainenergyis（a）

Inthecomplexstressstate,thestrainenergyoftheobjectisstillnumericallyequaltotheworkdonebytheexternalforce.Inthelinearelasticcase,therelationshipbetweeneachprincipalstressanditscorrespondingprincipalstrainisstilllinear.Thestrainenergydensitycorrespondingtoeachprincipalstresscanbecalculatedaccordingtoequation(a).ThestrainenergydensityinthetriaxialstressstateiscalculatedasSubstitutingintotheaboveequation,weget（b）2、Volumechangeenergydensityanddistortionenergydensity

Theprincipalstrainscorrespondingtoσ1、σ2

andσ3

areε1、ε2、ε3.AndthevolumestrainisΘ.Thedeformationenergydensityucanbeconsideredasconsistingoftwocomponents:

（1）Thestrainenergydensityutstoredduetovolumechangeiscalledvolumechangeenergydensity.Volumechangemeansthateachedgeoftheelementisdeformedequally,andtheoriginalshapeismaintainedafterthedeformation,butthevolumechanges.（2）Thestrainenergydensitystoredwhenthevolumeremainsthesamebuttheshapeoftheelementischangediscalleddistortionenergydensity.

Ifthethreeprincipalstressesarereplacedbytheaveragestress,thevolumestrainatthistimeisequaltothevolumestrainΘoftheoriginalelement.Thus,thestrainenergydensityofthiscaseisequaltout

oftheoriginalelement,so

（d）BytheHooke'slawSubstitutingitintoequation(d),weget

（e）SubstituteIntoequation（c）,wegetsimplify

Example8

Thepicturebelowshowsanelementinapureshearstressstatewithisotropicmaterial.TrytoprovethatthefollowingrelationshipexistsbetweenthethreematerialconstantsE,Gandμ.cabxyodtSolution：Letthelengthofeachsideoftheelementbedx,dyandt.Theshearforceoftheleftandrightfacesoftheelementareτtdy.Themisalignmentoftheleftandrightsidesoftheelementduetoshearisγdx.ThestoredstrainenergydUis（a）DividetheaboveequationbythevolumetdxdytoobtainthestrainenergyTheprincipalstressesontheelementinthepureshearstressstateareTheprincipalelementisalsoshowninrightfigure.cabxyodt45°1s3s（b）（c’）substitute

intotheequationbelowthenweget

（d）because，wehave（e）so11.7Theoriesofstrength1.Formsofmaterialfailure

(1)Plasticyielding(plasticflow)

Mildsteel

(2)BrittlefractureCastiron2、Theoriesofstrength

Strengthconditionsinaxialtensionorcompression

Theallowablestress[σ]isobtainedbydividingσbbythefactorofsafety.

Inengineering,mostofthedangerouspointsofthemembersareinthecomplexstressstate.Itisbelievedthatnomatterwhatstressstate,aslongasthesameformoffailureoccurs,thecausesoffailureandthefactorsleadingtofailurearethesame.Inthisway,itispossibletousetheresultsofsimpleexperimentstopredictthefailurebehaviorofmaterialsundervariousstressesandtoestablishfailurecriteria.Suchhypothesisiscalledthetheoriesofstrength.3、FourcommonlyusedstrengththeoriesSincetherearetwoformsofmaterialdamage,strengththeoriesarealsodividedintotwocategories：Oneistoexplainthebrittlefractureofmaterials;Theotheristoexplaintheplasticyielddamageofmaterials.1.Maximumtensilestresscriterion（firsttheoryofstrength）Themaximumtensilestressσ1isthemainfactorcausingthebrittlefractureofthematerial.Whetheritisauniaxialstressstateoracomplexstressstate,thematerialwillfractureifthemaximumtensilestressσ1

reachesalimitvalue.Thislimitisthestrengthlimitσbofthematerialmeasuredduringtheaxialtensiletest.fracturecriterion:strengthcondition:Thistheorybasicallyreflectscorrectlythepropertiesofsomebrittlematerials.Itcanbeusedforcertainbrittlemetalssubjectedtotensilestresses,suchascastiron.2.Maximumtensilestraincriterion(secondtheoryofstrength)

Thistheorybelievesthatthemaximumelongationnormalstrainisthemainfactorcausingbrittlefractureofthematerial.Whetheritisauniaxialstressstateoracomplexstressstate,thematerialwillfractureiftheelongationnormalstrainε1reachesthemaximumelongationnormalstrainε0.

Inthecaseofanaxiallystretchedmaterial,itisassumedthattheultimatenormalstrainofthematerialcanstillbecalculatedbyHooke'slawuntilfractureoccurs，Incomplexstressstates,fracturedamageoccurswhenthemaximumelongationlinestrainε1reachesε0.Theconditionsunderwhichfracturedamagethusoccursare（b）BythegeneralizedHook'slawthereis

Substitutingitintoequation(b),thefracturecriterionofthematerialcanbeobtainedas（c）Divideσbbythefactorofsafetytoobtaintheallowablestress[σ]andthestrengthconditionestablishedbythesecondtheoryofstrengthis

Whenbrittlematerialssuchasstoneorconcretearemainlysubjectedtocompressivestress,thistheoryisbasicallyconsistentwiththeexperimentalresults.Thistheoryisonlyconsistentwiththedamagelawofafewbrittlematerialsinsomecases,andcannotbeusedtodescribethegeneraldamagelawofbrittlematerials.3.Maximumshearstresstheory(thirdtheoryofstrength)Thistheorybelievesthatthemaximumshearstressτisthemainfactorcausingplasticyieldingdamageofthematerial.Whetheritisauniaxialstressstateoracomplexstressstate,thematerialwillundergoplasticyieldingdamageifthemaximumshearstressτmaxreachesthelimitvalueτ0

ofthematerial.

Inthecaseofaxialtension,whenthetensilestressinthecrosssectionreachestheyieldstressσsofthematerial,themaximumshearstressoccursintheinclinedsectionatanangleof45°totheaxisisTheplasticyieldcriterionis

（d）Inthecomplexstressstate,themaximumshearstressis

Thustheplasticyieldcriterioncorrespondingtothethirdtheoriesofstrengthis

ThisisalsoknownastheyieldingcriterionofTresca.σisdividedbythefactorofsafetytoobtaintheallowablestress[σ].Thestrengthconditionestablishedbythethirdtheoriesofstrengthis

orThistheorycanbetterexplainthephenomenonofplasticyieldinginplasticmaterials.Theshortcomingofthistheoryisthatitdoesnottakeintoaccounttheprincipalstressσ2(orrather,theeffectofotherprincipalshearstresses)andisonlyapplicabletomaterialswiththesametensileandcompressiveyieldstress.4.Maximumdistortionene