Engineering Basic Mechanics I Statics 工程基礎力學Ⅰ 靜力學 課件 Chapter 8 Shear and Torsion

StaticsStaticsofdeformablebodyChapter8

ShearandTorsionContents8.1Theconceptofshear8.2Practicalcalculationofshearand

bearing8.3Theconceptoftorsion8.4Torqueandtorquediagram8.5Torsionofthin-walledcylinders8.6Stressanddeformationduringtorsionofcircularshafts 8.7Torsionalstrengthandrigidity SmallshearingmachineBoltedconnectionRivetedconnectionPinconnectionFlatkeyconnection8.1TheconceptofshearFFmn0FFFsinglesheardoubleshearBearingbearingstress

：pressureonthebearingsurfacebearingdeformation：deformationonthecontactsurfacebearingsurface：thecontactsurfaceFFBearingsurface8.2Practicalcalculationofshearandbearing1、Practicalcalculationofshear

FFQtTheshearstressτisuniformlydistributedontheshearsurface.SotheformulaofshearstressiswhereAistheareaoftheshearsurface.Thisshearstressisbasedonassumptionsandisnotthetrueshearstress,whichisusuallyreferredtothenominalshearstress.Whentheshearstressτontheshearplanereachesacertainvalue,theshearmemberwillbedamagedbyshear.Allowableshearstress

Thisistheshearstrengthcondition.Iftheshearultimatestressofthematerialisandnisthesafetyfactorthentheallowableshearstressofthematerial

isExperimentalresultsshowthattheshearultimatestrengthofthematerialhasanapproximateproportionalrelationshipwiththetensile(compressive)ultimatestrength.Plasticmaterials:Brittlematerials:Basedonthisrelationship,thevalueofthetensileallowablestress[σ]isoftenusedinengineeringtoestimatethevalueoftheshearallowablestress[τ].

Example1

ThepinconnectionstructureisshowninFigure.TheloadisknowntobeF=15kN.Thethicknessist=8mm,thediameterofthepinisd=20mmandthepinallowableshearstressis[τ]=30MPa.Checktheshearstrengthofthepin.0FFd1.5tttFmmnnFQFQmmnn2F2Fsolution:FromthesectionmethoditiseasytofindSowecangetThereforethepinmeetsthestrengthrequirements.Theshearstressreachestheultimatestressofthematerial,i.e.FBearingpressure：forceactingonthecontactsurfacebearingdeformation：deformationonthecontactsurfacebearingsurface：thecontactsurfacebearingstress：pressureonthebearingsurfaceFFwhereAjyisthebearingsurfacearea.Thisbearingstressisnotthetruestressandisusuallyreferredtoasthenominalbearingstress.Bearingsurface2、Practicalcalculationofbearing

Thecalculationoftheareaoftheextrudedsurfaceisdiscussedintwocasesasfollows：（１）Whenthecontactsurfaceisflat,theareaoftheextrudedsurfaceforcalculationistheactualcontactsurfacearea,i.e.

lhh2（2）Whenthecontactsurfaceisasemi-cylindricalsurface,theareaofthebearingsurfaceforcalculationisthediameterprojectionareaoftheactualcontactsurface.Inthisway,thenominalbearingstresscalculatedinaccordancewithequationandtheactualmaximumbearingstressareverysimilar.tdShearingsurfaceDiameterprojectionareaActualcontactareaTopreventbearingdamage,themaximumbearingstressshouldnotexceedtheallowablebearingstress[σjy]ofthematerial,i.e.

Thisisthebearingstrengthcondition.Theallowablebearingstressandtheallowabletensilestress[σ]arerelatedasfollows：Plasticmaterials:Brittlematerials：Ifthetwocontactingmembersareofdifferentmaterials,thecalculationshouldbemadeforthememberwiththeweakerbearingstrength.

Therearethreepossibledamagetoconnectionscommonlyusedinengineering:Oneisthatthememberisshearedalongtheshearsurface;Second,Thebearingsurfaceshowsobviousplasticdeformation,whichmakestheconnectingrodloose;Third,theconnectionplatemaybepulledoffbecausethecross-sectionisweakenedafterdrilling.3、Strengthcalculationofconnectionparts

Tomakefulluseofthematerial,theshearandbearingstressesshouldmeet:DiscussionAjointisshowninthefigure.Itisknownthattheplateandrivetareofthesamematerialandthatσbs=2[τ].Tomakefulluseofthematerial,therivetdiameterdshouldbe________Example2

ArivetedjointstructureisshowninFig(a)withaknownloadF=100kN,arivetdiameterd=16mm,anallowabletensilestress[σ]=160MPaforthesteelplate.Theallowableshearstressis[τ]=130MPafortherivetandtheallowablebearingstressis[σjy]=320MPafortheplateandrivet.Checkthestrengthofthestructure.Fd=16mmF=100kN(a)t=10mmt=10mmSolution

Therearethreepossibleformsofdamagetoarivetedjointstructure:damagetotherivetduetoshear;damagetotherivetorsteelplateduetobearing;anddamagetothesteelplateduetotension.(1)ChecktheshearstrengthoftherivetTheforceoneachrivetisTherefore,theshearforceontheshearplaneoftherivetis

Theshearstressintherivetisthus321123F4F4F4p4FFb=90mmFF=100kNt=10mmd=16mmFF1p2p10FF123F3214p4p4p4pb=90Fd=16F=100KNt=10t=10(2)Checkthebearingstrengthoftherivetthebearingforceoftherivet:thebearingstressis2314p34FF1123F3214p4p4p4p+FF=100kN(3)Checkthetensilestrengthofthesteelplate.Sectionmethodsection2-2：section3-3：

Insummary,theentirestructure

meetsthestrengthrequirements.Apairofcoupleswithequalmagnitudeandoppositedirectionisappliedattheendsoftherod.Thecoupleplaneisperpendiculartotheaxisoftherod.Anytwocrosssectionsoftherodrotaterelativetoeachotheraroundtheaxisofthebar.Thisformofdeformationoftherodiscalledtorsionaldeformation.

Accordingtothesectionmethod,whentorsionaldeformationoccurstotherod,theinternalforceonthecrosssectionisonlythemomentofthecouplelocatedontheface.Itiscalledtorque.8.3Theconceptoftorsion1、CalculationoftheexternalmomentofcoupleIfthepowerisexpressedinNk(kW)andtherotationalspeedisn(r/min)，themomentisM,wecanget

Note：TheunitofNk

iskW,andtheunitofnisr/min.WhenthepowerishorsepowerNH

(H.P,1horsepower=735.5W),theformulaforcalculatingtheexternalmomentofcoupleis

8.4Torqueandtorquediagram2、TorqueandtorquediagramAssumethatthecircularaxisisdividedintotwosectionsalongthesectionm-m,theequilibriumoftheleftsectionasfollowingSowegetwhereMnisthecombinedmomentofthedistributedinternalforcesystemofthetwopartsIandIIinteractingonthesectionm-m.Similarly,iftherightsectionisthesubjectofstudy,thetorqueMnonsectionm-mcanalsobefound.Itsvalueisstillm,butitssteeringisoppositeMMnnIIIMnnIInnIxMnMnM

Thesignofthetorquecanbespecifiedasfollows:thetorqueMnisexpressedasavectoraccordingtotheright-handspiralrule.Whenthedirectionofthevectoristhesameasthedirectionoftheouternormalofthesection,thetorqueMnispositive,andtheoppositeisnegative.Inthisway,Thetorqueonthecrosssectionm-mispositivebothforpartIandpartII.

AgraphicalrepresentationofthevariationoftorqueMninthedirectionoftheaxisiscalledatorquediagram.Torquediagramsaredrawninasimilarwaytoaxialforcediagrams.

Example3

OntheshaftshowninFigure,theactivewheelAisconnectedtotheprimemoverandthedrivenwheelsB,CandDareconnectedtothemachinetool.TheinputpowerofwheelAisknowntobeNA=50kW,theoutputsofwheelsB,CandDareNB=NC=15kWandND=20kW,respectively.Thespeedoftheshaftisn=300r/min.Trytofindthetorqueineachsectionoftheshaftanddrawatorquediagram.(a)AMBMCMDMBACDIIIIIIIIIIIICSolution(1)Calculatetheexternalmomentofcouple(2)CalculatetorqueSectionBC:cuttheshaftalongsectionI.Fromtheequilibriumequation,wegetBMnMIBMCMnMIIDMnMIIIBACDIIIIIIIIIIIIAMBMCMDMAnegativeresultindicatesthattheactualdirectionofthetorqueIisoppositetothedirectionset.ThetorqueoneachsectionwithinthesectionBCisconstant,sothetorquediagraminthissectionisahorizontalline（Fig.e）.SectionCA：ThereforeSectionAD：BMnMIBMCMnMIIDMnMIIIBACDIIIIIIIIIIIIAMBMCMDM+-(3)Makingtorquediagram

Ascanbeseenfromthegraph,themaximumtorqueoccursinthesectionCAwithanabsolutevalueofBMnMIBMCMnMIIDMnMIIIBACDIIIIIIIIIIIIAMBMCMDM

8.5Torsionofthin-walledcylinders

Inordertostudythestressanddeformationduringtorsionofacircularshaft,thetorsionofathin-walledcylinderisfirstdiscussedtounderstandthelawofshearstressandshearstrainandtherelationshipbetweenthem.1.Stressinthin-walledcylindersduringtorsionInthefigureabove,athin-walledcylinderofequalthicknessisshown.Afterapplyinganexternalmomentatbothends,thefollowingphenomenacanbeobserved:

(1)Theshape,sizeandspacingofthecircumferentiallinesonthesurfaceofthecylinderremainunchanged,andjustrotaterelativelyaroundtheaxis.(2)Eachlongitudinallineisinclinedatthesameangleγ,andcanstillbeapproximatedasastraightline.

(3)Tinyrectangleformedbythelongitudinalandcircumferentiallinesbecomesaparallelogram.tRjg1Therearenonormalstressesineachcrosssectionofthecylindertwisted,onlytheshearingstressesperpendiculartotheradius.Theshearstressisthesameateverypointalongthecircumferenceofthecross-section.2Theshearstressesareuniformlydistributedalongthewallthicknessdirection.3ItsdirectioncoincideswiththesteeringofthetorqueMninthecrosssection.MnMnabcddxnMjgRRdqItfollowsfromstaticsthatsoor

whereistheareaenclosedbythemidlineofthecylinderwallonthecrosssection.RRdAdqt(e)Letlandbethelengthandtherelativeangleoftwistatbothendsofthethin-walledcylinderrespectively.Wecangetthereforetheshearstrainisproportionaltothetorsionangle.jgcabdgg2.PureshearShearforceEquilibriumconditioncouplemoment

Astheelementisinequilibrium,inthetopandbottomsurfaceoftheelement,theremustalsobeshearstressτ’

yxzdxtt￠dytTheaboveequationshowsthatshearstressmustexistinpairswithequalvaluesonthetwoplanesperpendiculartoeachotherintheelement.Theshearstressesarebothperpendiculartotheintersectionofthetwoplanes.Thedirectionisofpointingtoordeviatingfromthisintersectionconsistently.Thisrelationshipisknownasthetheoremofcomplementaryshearingstresses.Asshowninthefigureonthetop,bottom,leftandrightfoursidesoftheunitbodyonlyshearstressandnopositivestressexist,thestressstateofunitbodyiscalledpureshearstate.yxzdxtt￠dyt3.Hook'sLawinshear

Theτ-γcurveforlowcarbonsteelisshowninabovepictureHook'sLawinshear

WhereGisaconstantofproportionality,knownastheshearmodulusofelasticity.Itisanindicatoroftheabilityofamaterialtoresistsheardeformation.Becauseγisdimensionless,Ghasthesameunitastheτ.TheG-valueofthesteelisabout80GPa.gttg0

"Hooke'slawintensionandcompression","Hooke'slawinshear"and"theoremofcomplementaryshearingstresses"arethefundamentaltheoremsofmaterialmechanics.ThetensilemodulusofelasticityE,theshearmodulusofelasticityGandthePoisson'sratioμarethreeelasticconstantsofamaterial.Forisotropicelasticmaterials,thefollowingrelationshipsexistbetweenthem.

Onlytwoofthethreeelasticconstantsareindependent.4.Energyofsheardeformation

whentheshearstressdoesnotexceedtheshearproportionallimitofthematerial,theangleφoftwistisproportionaltotheexternaltorqueM.TheworkdonebytheexternalmomentisEnergyofsheardeformationU，

Strainenergyperunitvolumeisthestrainenergydensityu.ThevalueofushouldbeequaltotheshearstrainenergyUdividedbythevolumeofthethin-walledcylinder.SoAccordingtoHook'sLawinshear,wecanget

8.6Stressanddeformationduringtorsionofcircularshafts Trainofthought:Geometricrelation(planesectionhypothesis)RelationshipbetweenshearstrainandrelativeangleoftwistPhysicalrelation(Hook'sLaw)RelationshipbetweenshearstressandrelativeangleoftwistStaticrelation(Thecombinedmomentofshearstressontheshaft,i.e.thetorqueonthecrosssection)Relativeangleoftwistexpressionandshearstressexpression1.Stressduringtorsionofacircularshaft1.GeometricrelationAmicro-sectionoflengthdxisinterceptedfromthecircularshaftAsmallrelativemisalignmentoftheabsideTheanglechangeγoftheoriginalrectangleonthesurfaceofthecircularshaftisTheshearstraininthecross-sectionatadistanceρfromthecenterofthecircleis（a）jxeMeMmndxmn2.PhysicalrelationWhentheshearstressdoesnotexceedtheshearproportionallimitofthematerial,theshearstressisproportionaltotheshearstrain,thatis,obeyingtheshearHooke'slaw

（b）Substituteequation(a)intoequation(b)tofindtheshearstressatthedistanceρfromtheaxisas

（c）Theaboveformulashowsthattheshearstressτρatanypointinthecrosssectionisproportionaltothedistanceρ.Theshearstressvariesalongtheradiusinalinearfashion,withzeroshearstressatthecentreofthecircleandthemaximumshearstressatpointsonthecircumferentialedge.

Accordingtothetheoremofcomplementaryshearingstresses,thedistributionofshearstressesalongtheradiusinthelongitudinalandtransversesectionsofthesolidcircularshaftisshownasfollows.rt3.Staticrelation

TakeamicroareadA，micro-shearforcesonthemicro-areadA：Correspondingmicro-momentstothecenterofthecircle：torque

(d)Substituteintoaboveequation,weget

dArtdAnMrOTheintegralintheaboveequationisaquantityrelatedtothegeometryanddimensionsofthecrosssection.Itiscalledthepolarmomentofinertiaofthecrosssection.(denotedas)

rtdAdAnMrOequation(d)canagainbewrittenasconsidering

weget

Thisistheformulaforcalculatingtheshearstressatanypointonthecrosssectionwhenthecircularshaftistwisted.

Accordingtoequation

Wecanknow,whenρ=R(i.e.ateachpointontheedgeofthecrosssection),theshearstresstakesitsmaximumvalue.

let,aboveequationcancanbewrittenas

WhereWniscalledthesectionmodulusoftorsion.4.TorsionaldeformationofcircularshaftThetorsionaldeformationofacircularshaftcanbeexpressedintermsoftherelativeangleoftwistoftwocrosssections.

Integratingbothsidesoftheaboveequationgivestherelativeangleoftwistofthetwosectionsseparatedbyl.

Foracircularshaftofequalcross-sectionmadeofthesamematerial(ItsGIPisaconstant)，Ifthetorquebetweenthetwocrosssections(distancel)isalsoconstant,thetorsionanglebetweenthetwosections

is

Thisistheformulaforcalculatingthetorsionaldeformationofacircularshaftofequalcrosssection.

isknownasthetorsionalrigidity.

ThesignoftheangleoftwistisspecifiedinthesamewayasthatoftorqueMnanditsunitisradian(rad).

Ifthetorqueortorsionalrigiditybetweentwocrosssectionsisvariable,therelativetorsionalanglesofthetwosectionsshouldbecalculatedbyintegratingthetorsionalanglesofeachsectioninaccordancewithequationandthensummingthemalgebraically.5.Polarmomentsofinertiaandsectionmodulusoftorsion

annularmicroarea：polarmomentofinertiaofcircularsection:

drRDmaxtmaxtmaxtsectionmodulusoftorsion:

whereDisthediameterofthecircularsection.ThedimensionofIpisthefourthpowerofthelengthandthedimensionofWnisthethirdpowerofthelength.rOdrRDmaxtmaxtmaxtforthehollowcircularshaft，

WhereDanddaretheouterandinnerdiametersofthehollowcircularsection,respectively.rOdrRDmaxtmaxtmaxt6、ApplicationconditionsofstressanddeformationformuladuringtorsionTheabovestressanddeformationequationsarederivedbasedontherigidplanehypothesis.Theseformulasareonlyapplicabletoisotropiccircularbars.Whenthecircularcrosssectionchangesslowlyalongtheaxis,itcanalsobeapproximatedbytheaboveformulae.IpandWnarealsochangingalongtheaxisatthesametime.

Onlywhen，aboveequationsarecorrect.8.7Torsion