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ManufacturingProcesses-MFRG315-Mohr'sCircle

1Introduction

Intwodimensionalstressanalysis planestress thereisofteninformationavailableaboutdirect sigmaxandsigmay andshearstresses tauxy actingonaparticularplaneandeither

thevaluesofthedirectandshearstressesactingonaplaneinclinedatsomeangletothefirstplanearerequired,oralternativelythevaluesofthe2principalstresses sigma1andsigma2

andtheinclinationoftheplaneonwhichtheyactarerequired.

Thistransformationcalculationcanbecarriedoutusingequations,butitwasnoticedbyMohrthatthesetransformationequationscanbecombinedtogivetheequationofacircle.Thisenablesthetransformationtobecarriedoutgraphically andtheequationsneednotberemembered .Usingthegraphicalconstructionalsomakesiteasiertovisualisealternative

scenarios.

InaMohrstresscircle,directstressesareonthehorizontalaxisandshearstressesareontheverticalaxis.RotationsofdirectioninaMohrcirclearedoublethoseintherealstressdistribution.Tensilestressesandashearstresstendingtocauseclockwiserotationaredeemedtobepositive.Compressivestressesandshearstressestendingtocausecounter-clockwiserotationaredeemedtobenegative.

Manymetalformingoperationsinvolvea3dimensionalstateofstresssothisisconsideredbelow.

2Mohr'sCircleforThreeDimensionalStatesofStress AgeneralapproachtoMohr'scirclein3dimensionsiscomplicated,howeverthisishardlyeverrequiredasinvirtuallyallmetalformingproblemsweknowthedirectionsofatleast1of the3principalstresses.

Ifthedirectionof1principalstressisknown,saysigma3,thenwecanimaginelookingback

alongthedirectionofsigma3totheplaneonwhichitacts asthisisaprincipalplanethere

arenoshearstressesactingonit andforthestressesactinginthisplanea2dimensionalMohr'scirclecanbeconstructedtodeterminethevaluesanddirectionsofthetwoprincipalstressesactinginit.

Giventhe3principalstresses,threecirclescanbeplotted,oneforeachprincipalplane.Wecandeterminethestressstatewithinaprincipalplaneaswerotateabouttheprincipalstressdirectionnormaltotheplane.Itisnotpossibleusingthisapproachtoconsidersimultaneousrotationsabout2ormoreprincipalaxes.

Itcanbestatedthat:Thestressesonanyplaneatanyrotation,whenplottedinthe3dimensionalMohr'scirclediagramwillberepresentedbyapointeitherononeofthe3circles,orwithinthe shaded areabetweenthelargestand2smallercircles.


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Themaximumshearstressisgivenbytheradiusofthelargestcircle.

SomesketchesofMohrcircleforcommonstressstatesareshownbelow:


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DeterminingtheStressStateonAnotherPlane Afurthergraphicalconstructionenablesthestateofstressonanyplanewhosenormalisinclinedatangles'alpha','beta'and'gamma'tothesigma1,sigma2andsigma3directions

respectivelytobedetermined. Withdirectioncosinestotheprincipalaxesdirections:l,mandn,respectively-Forinformationaboutdirectioncosines .

http://www.tech.plym.ac.uk/sme/mfrg315/dircos.gif


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Thefullprocedureis:

1.Drawthedirectstress horizontal andtheshearstress vertical axes.2.Markthethreeprincipalstressesonthehorizontalaxis,sigma1greaterthan

sigma2greaterthansigma3.

3.Drawthe3circles4.Drawthelinesigma1-A-Binclinedatanglealphafromtheverticalthroughsigma1.5.Drawthelinessigma2-Candsigma2-Dwithinclinationsofbeta,eithersideofthe

verticalthroughsigma2.

6.WithcentresC13andC23drawarcsDCandAB

7.ThedirectandshearstressatParegivenbyitscoordinates.

Anexamplecalculationisshowninthediagrambelow:

The3circlesaredrawnforthreeprincipalstresses:

sigma1=120MPa

sigma2=80MPa

sigma3=20MPa.

Thisgivesamaximumshearstressof50MPa.

Theconstructionshowsthedeterminationofdirectandshearstressesonaplaneat45oto

thesigma1directionand60otothesigma2direction point'P'onthediagram .Fromthescale


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diagramthedirectstressvaluevalueisabout87MPaandtheshearstressvalueisabout39MPa seebelow .

Thiscanbecheckedusingtheequations:

Firstdeterminethethirddirectioncosine:n= 1- l2+m2 0.5

n= 1- 0.70712+0.52 0.5=0.5

sigma=sigmall2+sigma2m

2+sigma3n2

sigma=120 0.7071 2+80 0.5 2+20 0.5 2=85MPa

Theshearstressisgivenby:

Shearstress2= sigma1l 2+ sigma2m

2+sigma3n 2-sigma2

shearstress=7200+1600+100-7225=40.9MPa.

Formostpurposesareasonablycarefulgraphicalconstructionprovidesadequateaccuracy

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