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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
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