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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
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