Simple Stresses in Machine PartsSimpleStressesinMachineParts

VedatTemizAssistant Professor ofMachine Design

Introduction Inengineeringpractice,themachinepartsaresubjectedto

variousforceswhichmaybeduetoeitheroneormoreofthefollowing:1) Energy transmitted,2) Weight ofmachine,3) Frictional resistances,4) Inertiaofreciprocatingparts,5) Changeoftemperature,and6) Lackofbalanceofmovingparts.

Load

It is defined as any external force acting upon a machine part Itisdefinedasanyexternalforceactinguponamachinepart.Thefollowingfourtypesoftheload areimportantfromthesubjectpointofview:j p

1.Deadorsteadyload.Aloadissaidtobeadeadorsteadyload,whenitdoesnotchangeinmagnitude or direction.

2.Liveorvariableload.Aloadissaidtobealiveorvariableload,whenitchangescontinually.

3.Suddenlyappliedorshockloads.Aloadissaidtobeasuddenlyappliedorshockload,whenitissuddenlyappliedor

dremoved. 4.Impactload.Aloadissaidtobeanimpactload,whenitis

applied with some initial velocityappliedwithsomeinitialvelocity.

Stress Whensomeexternalsystemofforcesorloadsactonabody,

the internal forces (equal and opposite) are set up at varioustheinternalforces(equalandopposite)aresetupatvarioussectionsofthebody,whichresisttheexternalforces.Thisinternalforceperunitareaatanysectionofthebodyisp y yknownasunitstressorsimplyastress.Itisdenoted byaGreeklettersigma().Mathematically,

AF= Forceorloadactingonabody

C ti l f th b dA CrosssectionalareaofthebodyInS.I.units,thestressisusuallyexpressedinPascal(Pa)suchthat1Pa=1N/m2

Strain

When a system of forces or loads act on a body it undergoes Whenasystemofforcesorloadsactonabody,itundergoessomedeformation.Thisdeformationperunitlengthisknownasunitstrainorsimplyastrain.ItisdenotedbyaGreekletterp y yepsilon().Mathematically

Ch i l th f th b d

ll=

Changeinlengthofthebody

Originallengthofthebody

TensileStress and Strain

When a body is subjected to two equal and opposite axial WhenabodyissubjectedtotwoequalandoppositeaxialpullsF(alsocalledtensileload)asshowninFig.(a),thenthestressinducedatanysectionofthebodyisknownastensiley f ystressasshowninFig.(b).

A little consideration will show that due to theAlittleconsiderationwillshowthatduetothetensileload,therewillbeadecreaseincrosssectionalareaandanincreaseinlengthofthebody The ratio of the increase in length to thebody.Theratiooftheincreaseinlengthtotheoriginallengthisknownastensilestrain.

TensileStress and Strain

Tensile stress Tensilestress

F AxialtensileforceactingonthebodyAt

=Crosssectionalareaofthebody

Tensile strain

lt

= Increase inlengthlt

Original length

Compressive Stress and Strainp

When a body is subjected to two equal and opposite axial WhenabodyissubjectedtotwoequalandoppositeaxialpushesF(alsocalledcompressiveload)asshowninFig.(a),thenthestressinducedatanysectionofthebodyisknownasy ycompressivestressasshowninFig.(b).

A little consideration will show that due to theAlittleconsiderationwillshowthatduetothecompressiveload,therewillbeanincreaseincrosssectionalareaandadecreaseinlengthofthebody.The ratio of the decrease in length to the originalTheratioofthedecreaseinlengthtotheoriginallengthisknownascompressivestrain.

Compressive Stress and Strainp

Compressive stress Compressive stress

F AxialcompressiveforceactingonthebodyAc

=Crosssectionalareaofthebody

Compressive strain

lc

= Decrease inlengthlc

Original length

Young'sModulusorModulusofElasticity

Hooke's law* states that when a material is loaded within Hooke slaw*statesthatwhenamaterialisloadedwithinelasticlimit,thestressisdirectlyproportional to strain

whereEisaconstantofproportionalityknownasYoung'smodulusormodulusofelasticity.In S.I.units,itisusuallyexpressedinGPai.e.GN/m2 orkN/mm2.ItmaybenotedthatHooke'slawholdsgoodfor tension as well as compressionfortensionaswellascompression.

lFE lAlFE ==

*ItisnamedafterRobertHooke,whofirstestablisheditbyexperimentsin1678.

V l f E f th lValuesofE forthecommonlyusedengineeringmaterialsg g

Shear Stress and Strain

When a body is subjected to two equal and opposite forces Whenabodyissubjectedtotwoequalandoppositeforcesactingtangentiallyacrosstheresistingsection,asaresultofwhichthebodytendstoshearoffthesection,thenthestressy ,inducediscalledshearstress.

Shear Stress and Strain

Thecorrespondingstrainisknownasshearstrainanditisp gmeasuredbythe angulardeformationaccompanyingtheshearstress.Theshearstressandshearstrainaredenotedbythe

k l ( ) d hi () i l h i llGreekletterstau()andphi()respectively.Mathematically,

2

4dFF

AF === 22

4ddA

ShearModulusorModulusofRi iditRigidity

It has been found experimentally that within the elastic limit Ithasbeenfoundexperimentallythatwithintheelasticlimit,theshearstressisdirectlyproportionaltoshearstrain.Mathematicallyy

G/orGor == Shear stress Constantof

proportionality knownShear strain

proportionality,knownasshearmodulusormodulusofrigidity

ValuesofG forthecommonlyused materialsusedmaterials

Bearing Stressg Alocalised compressivestressatthesurfaceofcontact

betweentwomembersofamachinepart,thatarerelativelyatrestisknownasbearingstressorcrushingstress.Thebearing stress is taken into account in the design of rivetedbearingstressistakenintoaccountinthedesignofrivetedjoints,cotterjoints,knucklejoints,etc.

ldF

AFpb ==

average bearingpressure

Radialloadonthejournal

Di f h j l

Lengthofthejournalincontact

Diameter ofthe journal

Stressstrain Diagramg

In designing various parts of a Indesigningvariouspartsofamachine,itisnecessarytoknowhowthematerialwillfunctioninservice For this certainservice.Forthis,certaincharacteristicsorpropertiesofthematerialshouldbeknown.The

h l l dmechanicalpropertiesmostlyusedinmechanicalengineeringpracticearecommonlydeterminedfromastandardtensiletest.Thistestconsistsofgraduallyloadingastandardspecimenofamaterialpandnotingthecorrespondingvaluesofloadandelongationuntilthe specimen fracturesthespecimenfractures.

Stressstrain Diagramg 1.Proportionallimit. Thediagram shows

that from point O to A is a straight linethatfrompointOtoAisastraightline,whichrepresentsthatthestressisproportionaltostrain.Itisobvious,thatH k ' l h ld d t i t A dHooke'slawholdsgooduptopointAanditisknownasproportionallimit.

2.Elasticlimit.ItmaybenotedthateveniftheloadisincreasedbeyondpointAuptothepointB,thematerialwillregainitsshapeandsizewhentheloadisremoved.s ape a d s e e t e oad s e o edThismeansthatthematerialhaselasticpropertiesuptothepointB.Thispointisknown as elastic limitknownaselasticlimit.

Note: Sincetheabovetwolimitsareveryclosetoeachother,therefore,forallpracticalpurposesthesearetakentobeequal.

Stressstrain Diagramg 3.Yieldpoint.Ifthematerialisstressed

beyond point B the plastic stage willbeyondpointB,theplasticstagewillreachi.e.ontheremovaloftheload,thematerialwillnotbeabletorecoveritsi i l i d h At thi i t thoriginal sizeand shape. Atthispoint,the

materialyieldsbeforetheloadandthereisanappreciablestrainwithoutanyincreaseinstress The stress corresponding to yield pointstress.Thestresscorrespondingtoyieldpointisknownasyieldpointstress.

4.Ultimatestress.AtD,thespecimenregainst th d hi h l f tsomestrengthandhighervaluesofstresses

arerequiredforhigherstrains,thanthosebetweenAandD.Thestress(orload)goesonincreasing till the point E is reached At E theincreasingtillthepoint Eisreached. AtE,thestress,whichattainsitsmaximumvalueisknownasultimatestress.It isdefinedasthelargest stress obtained by dividing the largestlargeststressobtainedbydividingthelargestvalueoftheloadreachedinatesttotheoriginalcrosssectionalareaofthetestpiece.

Stressstrain Diagramg 5.Breakingstress.Afterthespecimenhas

reached the ultimate stress a neck isreachedtheultimatestress,aneckisformed,whichdecreasesthecrosssectionalareaofthespecimen,asshowni Fi (b) A littl id ti ill hinFig.(b). Alittleconsiderationwillshowthatthestress(orload)necessarytobreakawaythespecimen,islessthanthemaximumstress.Thestressis,therefore,reduceduntilthespecimenbreaksawayatpointF.ThestresscorrespondingtopointFisknownasbreakingstress.

6.Percentagereductioninarea.

A = Original cross sectional area andA=Originalcrosssectionalarea,anda=Crosssectionalareaattheneck.

Stressstrain Diagramg

7 P t l ti It i th 7.Percentage elongation. Itisthepercentageincreaseinthestandardgaugelength(i.e.originallength)obtainedbymeasuringthefracturedspecimenafterbringingthebrokenpartstogether.

l=Gaugelengthororiginallength,andL=Lengthofspecimenafterfractureorfinallength.

Note : The percentage elongation gives aNote :Thepercentageelongationgivesameasureofductilityofthemetalundertest.

Working Stressg

Whendesigningmachineparts,itisdesirabletokeepthestresslowerthanthemaximumorultimatestressatwhichfailure of the material takes place This stress is known as thefailureofthematerialtakesplace.Thisstressisknownastheworkingstressordesignstress.Itisalsoknownas safeorallowablestress.

Note: Byfailureitisnotmeantactualbreakingofthematerial.Somemachinepartsid t f il h th h l ti d f ti t i th d tharesaidtofailwhenthey haveplasticdeformationsetinthem,andtheynomore

performtheirfunctionsatisfactory.

Factor ofSafetyy Itisdefined,ingeneral,asthe ratioofthemaximumstressto

the working stress Mathematicallytheworkingstress.Mathematically,

Incaseofductilematerialse.g.mildsteel,wheretheyieldpointisclearlydefined,thefactorofsafetyisbasedupontheyieldpointstress.Insuchcases,

Incaseofbrittlematerialse.g.castiron,theyieldpointisnotwelldefinedasforductilematerials.Therefore,thefactorofsafetyforbrittlematerialsisbasedonultimatestress., y

SelectionofFactorofSafetyy Theselectionofaproperfactorofsafetytobeusedindesigninganymachine

componentdependsuponanumberofconsiderations,suchasthematerial,modeofmanufacture,typeofstress,generalserviceconditionsandshapeoftheparts.Beforeselectingaproperfactorofsafety,adesignengineershouldconsiderthefollowingpoints:g p

1) Thereliabilityofthepropertiesofthematerialandchangeofthesepropertiesduringservice;

) h li bili f l d f li i f h l2) Thereliabilityoftestresultsandaccuracyofapplicationoftheseresultstoactualmachineparts ;

3) Thereliabilityofappliedload;) y pp ;4) Thecertaintyastoexactmodeoffailure;5) Theextentofsimplifyingassumptions;6) Theextentoflocalised stresses;7) Theextentofinitialstressessetupduringmanufacture;8) The extent of loss of life if failure occurs ; and8) Theextentoflossoflifeiffailureoccurs;and9) Theextentoflossofpropertyiffailureoccurs.

Valuesoffactorofsafety

StressesduetoChangeinT t Th l StTemperatureThermalStresses

Wheneverthereissomeincreaseordecreaseinthetemperatureofabody,itcausesthebodytoexpandorcontract.Alittleconsiderationwillshowthatifthebodyisallowed to expand or contract freely with the rise or fall ofallowedtoexpandorcontractfreely,withtheriseorfallofthetemperature,nostressesareinducedinthebody.But,ifthedeformationofthebodyisprevented,somestressesarethe deformation of the body is prevented, some stresses areinducedinthebody.Suchstressesareknownasthermalstresses. Iftheendsofthebodyarefixedtorigid

Increaseordecreaseinlengthy g

supports,sothatitsexpansionisprevented,thencompressivestraininducedinthebody

tll tll =

l=Originallengthofthebody,

tl

tlll

c ===

t=Riseorfalloftemperature,and=Coefficientofthermalexpansion

Thermal stress tEEcth ==

Linear and Lateral Strain

Alittleconsiderationwillshowthatduetotensileforce,thelengthofthebarincreasesbyanamountlandthediameterdecreasesbyanamountd.Similarly,ifthebarissubjectedtoacompressiveforce,thelengthofbarwilly, f j p , gdecreasewhichwillbefollowedbyincreaseindiameter.

It is thus obvious, that every direct stress is accompanied by a strain in its ownItisthusobvious,thateverydirectstressisaccompaniedbyastraininitsowndirectionwhichisknownaslinearstrainandanoppositekindofstrainineverydirection,atrightanglestoit,is known aslateral strain.

Poisson's Ratio

Ithasbeenfoundexperimentallythatwhenabodyisstressedwithinelasticlimit,thelateralstrainbearsaconstantratiotothelinearstrain,Mathematically,

ThisconstantisknownasPoisson'sratioandisdenotedby1/mor.

Values of Poissons ratio for commonlyValuesofPoisson sratioforcommonlyusedmaterials

Volumetric Strain

When a body is subjected to a system of forces it undergoes Whenabodyissubjectedtoasystemofforces,itundergoessomechangesinitsdimensions.Inotherwords,thevolumeofthebodyischanged.Theratioofthechangeinvolumetothey g goriginalvolumeisknownasvolumetricstrain.Mathematically,volumetricstrain, V/V= V/Vv =

V=Change involume,and V=Original volumeVolumetricstrainofarectangularbodysubjectedtoanaxialforceisgivenas

== Vv 21 =Linear strain. mVvVolumetricstrainofarectangularbodysubjectedtothreemutuallyperpendicularforcesisgivenbyg y

wherex,y andz arethestrainsinthedirectionsxaxis,yaxisandzaxisrespectively

Bulk Modulus

When a body is subjected to three mutually perpendicular Whenabodyissubjectedtothreemutuallyperpendicularstresses,ofequalintensity,thentheratioofthedirectstresstothecorrespondingvolumetricstrainisknownasbulkp gmodulus.

ItisusuallydenotedbyK.Mathematically,bulkmodulus,

Relation Between Bulk Modulus and Youngs ModulusRelationBetweenBulkModulusandYoung sModulus

Thebulkmodulus(K)andYoung'smodulus(E)arerelatedbythefollowingl irelation,

R l ti B t Y M d l d M d l f Ri iditRelationBetweenYoungsModulusandModulusofRigidityTheYoung'smodulus(E)andmodulusofrigidity(G)arerelatedbythefollowingrelationrelation

Impact StresspThestressproducedinthememberduetothefallingloadisknownasimpactstress.a g oad s o as pact st ess.ConsiderabarcarryingaloadWataheighthandfallingonthecollarprovidedatthelowerend as shown in Figend,asshowninFig.

Weknowthatenergygainedbythesysteminthe form of strain energytheformofstrainenergy

d t ti l l t b th i htandpotentialenergylostbytheweight

Sincetheenergygainedbythesystemisequaltothepotentialenergylostbytheweight,therefore

Impact Stressp

F thi d ti ti fi d th tFromthisquadraticequation,wefindthat

Note :Whenh=0,theni =2W/A.Thismeansthatthestressinthebarwhentheloadinappliedsuddenlyisdouble ofthestressinducedduetograduallyappliedload.

Torsional Shear Stress Whenamachinememberissubjectedtotheactionoftwoequaland

opposite couples acting in parallel planes (or torque or twisting moment)oppositecouplesactinginparallelplanes(ortorqueortwistingmoment),thenthemachinememberissaidtobesubjectedtotorsion.Thestresssetupbytorsionisknownastorsional shearstress.Itiszeroatthecentroidalaxisandmaximumattheoutersurface.

= Torsional shearstressinducedattheoutersurfaceoftheshaftormaximumshear stress,r= Radiusoftheshaft,T= Torque or twisting moment,J=Secondmomentofareaofthesectionaboutitspolaraxisorpolarmomentofinertia,G= Modulusofrigidityfortheshaftmaterial,l= Lengthoftheshaft,and= Angleoftwistinradiansonalengthl.

Torsional Shear Stress

Theequation(i)isknownastorsionequation.Itisbasedonthefollowingassumptions:1.Thematerialoftheshaftisuniformthroughout.2.Thetwistalongthelengthoftheshaftisuniform.3 The normal crosssections of the shaft which were plane and circular3.Thenormalcrosssectionsoftheshaft,whichwereplaneandcircularbeforetwist,remainplaneandcircularaftertwist.4.Alldiametersofthenormalcrosssectionwhichwerestraightbeforetwist,remainstraightwiththeirmagnitudeunchanged,aftertwist.5.Themaximumshearstressinducedintheshaftduetothetwistingmoment does not exceed its elastic limit valuemomentdoesnotexceedits elastic limitvalue.

Polarmomentofinertia

For a solid shaft of diameter (d) the polar moment of inertia Forasolidshaftofdiameter(d),thepolarmomentofinertia,

Incaseofahollowshaftwithexternaldiameter(do)andinternaldiameter(di),thepolarmomentofinertia,

Note:Theexpression(G J)iscalledtorsional rigidityoftheshaft.

ShaftsinSeriesandParallel Whentwoshaftsofdifferent

diametersareconnectedtogethertoformoneshaft,itisthenknownascompositeshaft.Ifthedrivingtorqueisappliedatoneendandq pptheresistingtorqueattheotherend,thentheshaftsaresaidtobeconnected in series as shown inconnectedinseriesasshowninFig.Insuchcases,eachshafttransmitsthesametorqueandthetotal angle of twist is equal to the

Iftheshaftsaremadeofthesamematerial,thenG1 =G2 =G.totalangleoftwistisequaltothe

sumoftheangleoftwistsofthetwoshafts.Mathematically,total

l f i

1 2

angleoftwist,

ShaftsinSeriesandParallel

Whenthedrivingtorque(T)isg q ( )appliedatthejunctionofthetwoshafts,andtheresistingtorquesT1and T2 at the other ends of theandT2 attheotherendsoftheshafts,thentheshaftsaresaidtobeconnectedinparallel,asshownin Fig In such cases the angle ofinFig.Insuchcases,theangleoftwistissameforboththeshafts,

Iftheshaftsaremadeofthesamematerial,thenG1 =G2 =G.1 2

BendingStressinStraightBeamsg gConsiderastraightbeamsubjectedtoabendingmomentMasshowninFig.below.Thefollowing assumptions are usually made while deriving the bending formulafollowingassumptionsareusuallymadewhilederivingthebendingformula.1.Thematerialofthebeamisperfectlyhomogeneous(i.e.ofthesamematerialthroughout)andisotropic(i.e.ofequalelasticpropertiesinalldirections).2 The material of the beam obeys Hookes law2.ThematerialofthebeamobeysHooke slaw.3.Thetransversesections(i.e.BCorGH)whichwereplanebeforebending,remainplaneafter bending also.4. Each layer of the beam is free to expand or contract, independently, of the layer, above or4.Eachlayerofthebeamisfreetoexpandorcontract,independently,ofthelayer,aboveorbelow it.5.TheYoungsmodulus(E)isthesameintensionandcompression.6.Theloadsareappliedintheplaneofbending.pp p g

BendingStressinStraightBeamsg gAlittleconsiderationwillshowthatwhenabeamissubjectedtothebendingmoment,thefibres on the upper side of the beam will be shortened due to compression and those on thefibres ontheuppersideofthebeamwillbeshortenedduetocompressionandthoseonthelowersidewillbeelongatedduetotension.Itmaybeseenthatsomewherebetweenthetopandbottomfibres thereisasurfaceatwhichthefibres areneithershortenednorlengthened.Suchasurfaceiscalledneutralsurface.Theintersectionoftheneutralsurfacewithanynormalf ycrosssectionofthebeamisknownas neutralaxis.Thebendingequationisgivenby

M=Bendingmomentactingatthegivensection, = Bending stress Bending stressI=Momentofinertiaofthecrosssectionabouttheneutralaxis,y=Distancefromtheneutralaxistotheextremefibre,E=Youngsmodulusofthematerialofthebeam,andg f f ,R=Radiusofcurvatureofthebeam.

Bending Stress in Straight BeamsBendingStressinStraightBeams

Rearrangingaboveequationgives

TheratioI/yisknownassectionmodulusandisdenotedbyZ.

N tNotes:1. Theneutralaxis ofa sectionalwayspassesthroughitscentroid.2. Incaseof symmetricalsectionssuchascircular,squareorrectangular,the neutral axis passes through its geometrical centre3. Incaseof unsymmetrical sectionssuchasLsectionorTsection,theneutralaxisdoesnotpassthroughitsgeometricalcentre.Insuchcases,firstofallthecentroid ofthesectioniscalculatedandthenthedistanceoftheextremefibres forbothloweranduppersideofthesection isobtained.obtained.

Propertiesofcommonlyusedcrosssections

Propertiesofcommonlyusedcrosssections

Propertiesofcommonlyusedcrosssections

Shear Stresses inBeams Itisgenerallyassumedthatnoshearforceisactingonthesection.But,in

actual practice when a beam is loaded the shear force at a section alwaysactualpractice,whenabeamisloaded,theshearforceatasectionalwayscomesintoplayalongwiththebendingmoment.Ithasbeenobservedthattheeffectoftheshearstress,ascomparedtothebendingstress,is

it li ibl d i f t h i t B t ti th hquitenegligibleandisofnotmuchimportance.But,sometimes,theshearstressatasectionisofmuchimportanceinthedesign.Itmaybenotedthattheshearstressinabeamisnotuniformlydistributedoverthecrosssectionbutvariesfromzeroattheouterfibres toamaximumattheneutralsurfaceasshowninFigs.

Shear Stresses inBeams Theshearstressatanysectionactsinaplaneatrightangleto

h l f h b di d i l i i btheplaneofthebendingstressanditsvalueisgivenby.

F =Verticalshearforceactingonthesection,I=Momentofinertiaofthesectionabouttheneutralaxis,b =Widthofthesectionunderconsideration,f ,A =Areaofthebeamaboveneutralaxis,andy =DistancebetweentheC.G.oftheareaandtheneutral axisneutralaxis.

Shear Stresses inBeams Forabeamofrectangularsection,asshowninFig.,theshear

di f l i i i bstressatadistanceyfromneutralaxisisgivenby

Maximum shear stress(substituting y=h/2)(substitutingy h/2)

Shear Stresses inBeams ForabeamofcircularsectionasshowninFig.,theshear

di f l i i i bstressatadistanceyfromneutralaxisisgivenby

Maximum shear stress(Substituting y=d/2)

Shear Stresses inBeams ForabeamofIsectionasshowninFig.,themaximumshear

h l i d i i bstressoccursattheneutralaxisandisgivenby

Shearstressatthejointofthewebandtheflange

Shearstressatthejunctionofthetopofthewebandbottomoftheflange

State ofstressatapointp

Thestressesonthehiddenfaces becomeequalandoppositetothoseontheopposingvisiblefaces.Thus,ingeneral,a completestateofstressisdefinedbyninestresscomponents,xx ,,yy ,,zz,,xyxy ,, xzxz,,yxyx ,,yzyz,,zxzx ,,andand zyzy ..

Mohrs Circle for Plane Stress

A very common state of stress occurs when the stresses on Averycommonstateofstressoccurswhenthestressesononesurfacearezero.

When this occurs the state of stress is called plane stressWhenthisoccursthestateofstressiscalledplanestress. Supposethedx dy dz elementofFigure iscutbyanoblique

planewithanormalnat anarbitraryanglep y g counterclockwisefromthexaxis.

Mohrs Circle for Plane Stress2

By summing the forcescausedb ll hbyallthestresscomponentstozero,thestressesandare found to bearefoundtobe

These equations are called theThese equations arecalledtheplanestresstransformationequations

Mohrs Circle for Plane Stress3DifferentiatingEq.onthe left withrespecttoandsettingtheresultequaltozero gives

This equation definestwoparticularvaluesfortheangle2p,oneofwhichpdefines themaximumnormalstress1 andtheother,theminimumnormalstress2.Thesetwo stressesarecalledtheprincipalstressesprincipalstresses,andtheircorrespondingp pp p , p gdirections,theprincipalprincipal directionsdirections.

substitute

weseethat=0,meaningthatthesurfacescontaining principal, g f g p pstresseshavezeroshearstresses

Mohrs Circle for Plane Stress4DifferentiatingEq.onthe left withrespecttoandsettingtheresultequaltozero gives

This Equationdefinesthetwovaluesof2s atwhichtheshearstress reachesan extremevalue.Theanglebetweenthesurfacescontainingthemaximumshearstresses is90.

substitute

This equation tellsusthatthetwosurfacescontainingthemaximumshearstressesalsocontainequalnormalstressesof(x +y)/2.

Mohrs Circle for Plane Stress5 Formulasforthetwoprincipalstressescanbeobtainedby

substitutingthe angle2p fromEq.inEq.g g p f q q

Principal stresses

Inasimilarmannerthetwoextremevalueshearstresses

Mohrs Circle Shear Convention Thisconventionis

followedindrawingMohrscircle:

Shearstressestendingtorotatetheelementclockwise(cw)areplottedabovethe axisaxis.

Shearstressestendingtorotatetheelement

l kcounterclockwise(ccw)areplottedbelow the axis.

TheoriesofFailureUnderStaticLoad

Sincethesepropertiesareusuallydeterminedfromsimpletensionori t t th f di ti f il i b bj t d t i i lcompressiontests,therefore,predictingfailureinmemberssubjectedtouniaxial

stressisbothsimpleandstraightforward.Buttheproblemofpredictingthefailurestressesformemberssubjectedtobiaxialortriaxialstressesismuchmorecomplicated.Infact,theproblemissocomplicatedthatalargenumberofdifferenttheorieshavebeenformulated.Theprincipaltheoriesoffailureforamembersubjectedtobiaxialstressareasfollows:

1.Maximumprincipal(ornormal)stresstheory(alsoknownasRankines theory). 2.Maximumshearstresstheory(alsoknownasGuestsorTrescas theory). 3.Maximumprincipal(ornormal)straintheory(alsoknownasSaintVenant

theory). 4. Maximum strain energy theory (also known as Haighs theory).4.Maximumstrainenergytheory(alsoknownasHaigh s theory). 5.Maximumdistortionenergytheory(alsoknownasHencky andVonMises

theory).

MaximumPrincipalorNormalSt Th (R ki Th )StressTheory(Rankines Theory)

Accordingtothistheory,thefailureoryieldingoccursatapointinab h th i i i l l t i bi i l tmemberwhenthemaximumprincipalornormalstressinabiaxialstress

systemreachesthelimitingstrengthofthematerialinasimple tensiontest. Sincethelimitingstrengthforductilematerialsisyieldpointstressandforbrittlematerials(whichdonothavewelldefinedyieldpoint)thelimitingstrengthisultimatestress,thereforeaccordingtotheabovetheory,takingfactorofsafety(F.S.)intoconsideration,themaximumprincipalornormalstress(t1)inabiaxialstresssystemisgivenby

yt =Yieldpointstressintensionasdeterminedfromsimpletensiontest,andu =Ultimate stress.

Sincethemaximumprincipalornormalstresstheoryisbasedonfailureintensionorcompressionandignoresthepossibilityoffailureduetoshearingstress,thereforeitisnot

d f d til t i lusedforductilematerials.However,forbrittlematerialswhicharerelativelystronginshearbutweakintensionorcompression,thistheoryisgenerallyused.

MaximumShearStressTheory(G t T Th )(GuestsorTrescas Theory)

According to this theory the failure or yielding occurs at a point in a Accordingtothistheory,thefailureoryieldingoccursatapointinamemberwhenthemaximumshearstressinabiaxialstresssystemreachesavalueequaltotheshearstressatyieldpointinasimple tensiontest Mathematicallytest.Mathematically,

max =Maximumshearstressinabiaxialstresssystem,max y ,yt =Shearstressatyieldpointasdeterminedfromsimpletensiontest,andF.S.=Factor ofsafety.

Sincetheshearstressatyieldpointinasimpletensiontestisequaltoonehalftheyieldstressin tension therefore the equationmay be written asintension,thereforetheequationmaybewrittenas

MaximumPrincipalStrainTh (S i t V t Th )Theory(SaintVenants Theory)

According to this theory the failure or yielding occurs at a point in a Accordingtothistheory,thefailureoryieldingoccursatapointinamemberwhenthemaximumprincipal(ornormal)straininabiaxialstresssystemreachesthelimitingvalueofstrain(i.e.strainatyieldpoint)asdetermined from a simple tensile test The maximum principal (or normal)determinedfromasimpletensiletest.Themaximumprincipal(ornormal)straininabiaxialstresssystemisgivenby

According to theAccordingtotheabovetheory

t1 andt2 =MaximumandminimumprincipalFromequation,wemaywritethat

t1 t2stressesinabiaxialstresssystem,=Strainatyieldpointasdeterminedfromsimpletensiontest,1/m=Poissons ratio,E=Youngs modulus,andF.S.=Factor ofsafety.

Thistheoryisnotused,ingeneral,becauseitonlygivesreliableresultsinparticularcases.

MaximumStrainEnergyTheory(H i h Th )(Haighs Theory)

Accordingtothistheory,thefailureoryieldingoccursatapointinamemberwhenthestrainenergyperunitvolumeinabiaxialstresssystemreachesthelimitingstrainenergy(i.e.strainenergyattheyieldpoint)perunitvolumeasdeterminedfromsimpletensiontest.p

Weknowthatstrainenergyperunitvolumeinabiaxialstresssystem,

Limitingstrainenergyperunitvolumeforyieldingasdeterminedfromsimpletensiontest,

Accordingtotheabovetheory,U1 =U2.

Thistheorymaybeusedforductilematerials.

MaximumDistortionEnergyTheory(H k d V Mi Th )(Hencky andVonMises Theory)

According to this theory the failure or yielding occurs at a point in a Accordingtothistheory,thefailureoryieldingoccursatapointinamemberwhenthedistortionstrainenergy(alsocalledshearstrainenergy)perunitvolumeinabiaxialstresssystemreachesthelimitingdistortionenergy (i e distortion energy at yield point) per unit volume as determinedenergy(i.e.distortionenergyatyieldpoint)perunitvolumeasdeterminedfromasimpletensiontest.Mathematically,themaximumdistortionenergytheoryforyieldingisexpressedas

Thistheoryismostlyusedforductilematerialsinplaceofmaximumstrainenergy theoryenergytheory.

Note: Themaximumdistortionenergyisthedifferencebetweenthetotali d h i d ifstrainenergyandthestrainenergydueto uniform stress.

StressesinaThinCylindricalShelld t I t l PduetoanInternalPressure

The analysis of stresses induced in a thin cylindrical shell are Theanalysisofstressesinducedinathincylindricalshellaremadeonthefollowingassumptions:

1) Theeffectofcurvatureofthecylinderwallisneglected.2) The tensile stresses are uniformly distributed over the2) Thetensilestressesareuniformlydistributedoverthe

sectionofthewalls.3) The effect of the restraining action of the heads at the end3) Theeffectoftherestrainingactionoftheheadsattheend

ofthepressurevesselisneglected.

StressesinaThinCylindricalShelld t I t l PduetoanInternalPressure

Whenathincylindricalshellissubjectedtoaninternalpressure,itislikelyto fail in the following two ways:tofailinthefollowingtwo ways:

1.Itmayfailalongthelongitudinalsection(i.e.circumferentially)splittingthecylinderintotwotroughs,asshowninFig.(a).

2. Itmayfailacrossthetransversesection(i.e.longitudinally)splittingthecylinderintotwocylindricalshells,asshowninFig.(b).

Thus the wall of a cylindrical shell subjected to an internal pressure has toThusthewallofacylindricalshellsubjectedtoaninternalpressurehastowithstandtensilestressesofthefollowingtwotypes:(a)Circumferentialorhoopstress,and(b) Longitudinalstress.

Circumferential or Hoop Stressp

Consider a thin cylindrical shell subjected to an internal pressure as shown ConsiderathincylindricalshellsubjectedtoaninternalpressureasshowninFig.(a)and(b).Atensilestressactinginadirectiontangentialtothecircumferenceiscalledcircumferentialorhoopstress.Inotherwords,itisa tensile stress on longitudinal section (or on the cylindrical walls)atensilestressonlongitudinalsection(oronthecylindricalwalls).

p=Intensityofinternalpressure,d=Internaldiameterofthecylindricalshell,l=Lengthofthecylindricalshell,t=Thicknessofthecylindricalshell,andt1 =Circumferentialorhoopstressforthematerialofthecylindrical shell.

Circumferential or Hoop Stressp

Weknowthatthetotalforceactingonalongitudinalsection(i.e.alongthediameterXX)oftheshell

andthetotalresistingforceactingonthecylinderwalls

...(oftwo sections)

Fromthesetwoequations,wehave

Longitudinal Stressg

Consider a closed thin cylindrical shell subjected to an internal pressure as ConsideraclosedthincylindricalshellsubjectedtoaninternalpressureasshowninFig.(a)and(b).Atensilestressactinginthedirectionoftheaxisiscalledlongitudinalstress.Inotherwords,itisatensilestressactingonthetransverse or circumferential section Y Y (or on the ends of the vessel)transverseorcircumferentialsectionYY(orontheendsofthevessel).

Longitudinal Stressg

Let t2 =Longitudinal stress.Inthiscase,thetotalforceactingonthetransversesection( l )(i.e.alongYY)

and totalresisting force

F th t tiFromthesetwoequations,wehave