twisting moment calc

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twisting moment calculations

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  • ShaftsShafts

    VedatTemizAssistant Professor ofMachine Design

  • Introduction Ashaftisarotatingmachineelementwhichisusedto

    transmitpowerfromoneplacetoanother. Thepowerisdeliveredtotheshaftbysometangentialforce

    andtheresultanttorque(ortwistingmoment)setupwithintheshaftpermitsthepowertobetransferredtovariousmachines linked up to the shaft In order to transfer themachineslinkeduptotheshaft.Inordertotransferthepowerfromoneshafttoanother,thevariousmemberssuchaspulleys,gearsetc.,aremountedonit.Thesemembersp y , g ,alongwiththeforcesexerteduponthemcausestheshafttobending.

    Inotherwords,wemaysaythatashaftisusedforthetransmissionoftorqueandbendingmoment.Thevarious

    b t d th h ft b f kmembersaremountedontheshaftbymeansofkeysorsplines.

  • Introduction Notes Theshaftsareusuallycylindrical,butmaybesquareorcross

    shapedinsection.Theyaresolidincrosssectionbutsometimeshollowshaftsarealsoused.

    Anaxle, thoughsimilarinshapetotheshaft,isastationaryhi l t d i d f th t i i f b dimachineelementandisusedforthetransmissionofbending

    momentonly.Itsimplyactsasasupportforsomerotatingbody such as hoisting drum a car wheel or a rope sheavebodysuchashoistingdrum,acarwheeloraropesheave.

    Aspindleisashortshaftthatimpartsmotioneithertoacuttingtool(e.g.drillpressspindles)ortoaworkpiece(e.g.g ( g p p ) p ( glathespindles)

  • Material Used for Shafts

    The material used for shafts should have the following Thematerialusedforshaftsshouldhavethefollowingproperties:

    1.Itshouldhavehighstrength. 2 It should have good machinability 2.Itshouldhavegoodmachinability. 3.Itshouldhavelownotchsensitivityfactor. 4 It should have good heat treatment properties 4.Itshouldhavegoodheattreatmentproperties. 5.Itshouldhavehighwearresistantproperties.

  • Manufacturing ofShaftsg

    Shafts are generally manufactured by hot rolling and Shaftsaregenerallymanufacturedbyhotrollingandfinishedtosizebycolddrawingorturningandgrindinggrinding.

    Thecoldrolledshaftsarestrongerthanhotrolledh f b i h hi h id lshaftsbutwithhigherresidualstresses.

    Theresidualstressesmaycausedistortionoftheshaftwhenitismachined,especiallywhenslotsorkeywaysarecut.

    Shaftsoflargerdiameterareusuallyforgedandturnedtosizeinalathe.

  • Types ofShaftsyp

    1 Transmission shafts These shafts transmit power between 1.Transmissionshafts.Theseshaftstransmitpowerbetweenthesourceandthemachinesabsorbingpower.Thecountershafts,lineshafts,overheadshaftsandallfactoryshaftsare, , ytransmissionshafts.Sincetheseshaftscarrymachinepartssuchaspulleys,gearsetc.,thereforetheyaresubjectedtobendinginadditiontotwisting.

    2.Machineshafts.Theseshaftsformanintegralpartofthemachineitself.Thecrankshaftis anexampleofmachineshaft.

  • StandardSizesofTransmissionShafts

    Th t d d i f t i i h ft Thestandardsizesoftransmissionshaftsare: 25mmto60mmwith5mmsteps;60mmto110mmwith10mmsteps;110mmto140mm with 15 mm steps ; and 140 mm to 500mmwith15mmsteps;and140mmto500mmwith20mmsteps.h d d l h f h h f 6 Thestandardlengthoftheshaftsare5m,6mand7m.

  • Stresses inShafts

    The following stresses are induced in the shafts : Thefollowingstressesareinducedintheshafts: 1.Shearstressesduetothetransmissionoftorque(i d i l l d)(i.e.duetotorsional load).

    2.Bendingstresses(tensileorcompressive)duetotheforcesactinguponmachineelementslikegears,pulleysetc.aswellasduetotheweightoftheshaftitself.

    3.Stressesduetocombinedtorsional andbendinggloads

  • MaximumPermissibleWorkingStressesfor Transmission ShaftsforTransmissionShafts

    AccordingtoAmericanSocietyofMechanicalEngineers (ASME) code for the design of transmissionEngineers(ASME)codeforthedesignoftransmissionshafts,themaximumpermissibleworkingstressesintension or compression may be taken astensionorcompressionmaybetaken as

    (a)112MPa forshaftswithoutallowanceforkeywayskeyways.

    (b)84MPa forshaftswithallowanceforkeyways.

  • MaximumPermissibleWorkingStressesfor Transmission ShaftsforTransmissionShafts

    For shafts purchased under definite physical Forshaftspurchasedunderdefinitephysicalspecifications,thepermissibletensilestress(t)maybe taken as 60 per cent of the elastic limit in tensionbetakenas60percentoftheelasticlimitintension(el),butnotmorethan36percentoftheultimatetensile strength ( ) In other words the permissibletensilestrength(u).Inotherwords,thepermissibletensilestress,

    0 6 0 36 hi h i l t =0.6el or0.36u,whicheverisless.

  • MaximumPermissibleWorkingStressesfor Transmission ShaftsforTransmissionShafts

    The maximum permissible shear stress may be taken as Themaximumpermissibleshearstressmaybetakenas (a)56MPa forshaftswithoutallowanceforkeyways. (b) 42 MPa for shafts with allowance for keyways (b)42MPa forshaftswithallowanceforkeyways. Forshaftspurchasedunderdefinitephysicalspecifications,

    the permissible shear stress () may be taken as 30 per cent ofthepermissibleshearstress()maybetakenas30percentoftheelasticlimitintension(el)butnotmorethan18percentoftheultimatetensilestrength(u).Inotherwords,theupermissibleshearstress,

    =0.3el or0.18u,whicheverisless.

  • Design ofShaftsg

    The shafts may be designed on the basis of Theshaftsmaybedesignedonthebasisof 1.Strength,and2.Rigidityandstiffness. In designing shafts on the basis of strength the following Indesigningshaftsonthebasisofstrength,thefollowing

    casesmaybeconsidered: (a) Shafts subjected to twisting moment or torque only (a)Shaftssubjectedtotwistingmomentortorqueonly, (b)Shaftssubjectedtobendingmomentonly, (c) Shafts subjected to combined twisting and bending (c)Shaftssubjectedtocombinedtwistingandbending

    moments,and (d) Shafts subjected to axial loads in addition to combined(d)Shaftssubjectedtoaxialloadsinadditiontocombined

    torsional andbendingloads.

  • ShaftsSubjectedtoTwistingM t O lMomentOnly

    Whentheshaftissubjectedtoatwistingmoment(ortorque)only,thenth di t f th h ft b bt i d b i th t i tithediameteroftheshaftmaybeobtainedbyusingthetorsionequation.Weknowthat

    T=Twistingmoment(ortorque)actingupontheshaft,J=Polarmomentofinertiaoftheshaftabouttheaxisofrotation,=Torsional shearstress,andr=Distancefromneutralaxistotheoutermostfibre=d/2;wheredisthediameteroftheshaft.

    Weknowthatforroundsolidshaft,polarmomentofinertia,

    Finalequation

  • ShaftsSubjectedtoTwistingM t O lMomentOnly

    Wealsoknowthatforhollowshaft,polarmomentofinertia,

    do anddi =Outsideandinsidediameteroftheshaft,andr=do /2.

    Substituting these values in shear stress equationSubstitutingthesevaluesinshearstressequation

  • ShaftsSubjectedtoBendingM t O l (A l )MomentOnly(Axles)

    When the shaft is subjected to a bending moment only then Whentheshaftissubjectedtoabendingmomentonly,thenthemaximumstress(tensileorcompressive)isgivenbythebendingequation.Weknowthatg q

    M B di tM=Bending moment,I=Momentofinertiaofcrosssectionalareaoftheshaftabouttheaxis ofrotation,b =Bending stress,andy = Distance from neutral axis to the outer most fibrey=Distancefromneutralaxistotheoutermostfibre.

    Weknowthatforaroundsolidshaft,momentofinertia,

    Finalequationq

    Fromthisequation,diameterofthesolidshaft(d)maybeobtained.

  • ShaftsSubjectedtoCombinedTwistingMoment and Bending MomentMomentandBendingMoment

    When the shaft is subjected to combined twisting moment Whentheshaftissubjectedtocombinedtwistingmomentandbendingmoment,thentheshaftmustbedesignedonthebasisofthetwomomentssimultaneously.y

    Varioustheorieshavebeensuggestedtoaccountfortheelasticfailureofthematerialswhentheyaresubjectedtovarioustypesofcombinedstresses.

    Thefollowingtwotheoriesareimportantfromthesubjectpointofview:

    1.MaximumshearstresstheoryorGuest'stheory.Itisusedf d il i l h ild lforductilematerialssuchasmildsteel.

    2.MaximumnormalstresstheoryorRankines theory.Itisused for brittle materials such as cast ironusedforbrittlematerialssuchascast iron.

  • ShaftsSubjectedtoCombinedTwistingMoment and Bending MomentMomentandBendingMoment

    Let Let =Shearstressinducedduetotwistingmoment,and = Bending stress (tensile or compressive) induced due to b =Bendingstress(tensileorcompressive)induceddueto

    bendingmoment. According to maximum shear stress theory the maximum Accordingtomaximumshearstresstheory,themaximum

    shearstressintheshaft,

    Substituting the values of and Substitutingthevaluesofb and

  • ShaftsSubjectedtoCombinedTwistingMoment and Bending MomentMomentandBendingMoment

    The expression The expression

    isknownasequivalenttwistingmomentandisdenotedbyTe.The equivalent twisting moment may be defined as that twistingTheequivalenttwistingmomentmaybedefinedasthattwistingmoment,whichwhenactingalone,producesthesameshearstress()astheactualtwistingmoment.Bylimitingthemaximum( ) g y gshearstress(max)equaltotheallowableshearstress()forthematerial,thefirstequationmaybewrittenas

    Fromthisexpression,diameteroftheshaft(d)maybeevaluated.

  • ShaftsSubjectedtoCombinedTwistingMoment and Bending MomentMomentandBendingMoment

    Now according to maximum normal stress theory the Nowaccordingtomaximumnormalstresstheory,themaximumnormalstressintheshaft,

  • ShaftsSubjectedtoCombinedTwistingMoment and Bending MomentMomentandBendingMoment

    The expression The expression

    is known as equivalent bending moment and is denoted by M TheisknownasequivalentbendingmomentandisdenotedbyMe.Theequivalentbendingmomentmaybedefinedasthatmomentwhichwhenactingaloneproducesthesametensileorcompressivestress(b)astheactualbendingmoment.Bylimitingthemaximumnormalstress[b(max)]equaltotheallowablebendingstress(b),thenthefi t ti b ittfirstequationmay bewritten as

    Fromthisexpression,diameteroftheshaft(d)maybep , ( ) yevaluated.

  • ShaftsSubje