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TORSIONOFMEMBERSWITHRECTANGULARCROSSSECTIONS

Preparedby:

JeremyElder

For:

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Dr.MehrdadNegaban

ENGM325H

April23rd,1999

Torsionofmembersisanimportanttopicinthemechanicsofelasticbodies.Wecancalculateshearstressesandtherotationanglescreatedbyappliedtorques.Thestudyofpuretorsionincircularbarsismuchsimplifiedbecauseofthehighdegreeofgeometryinthemember.Thestudyofpuretorsioninrectangularbarsismoreindepth.FromthisreportIhopetohelpcreateabetterunderstandingoftorsioninnoncircularmembers.

Circularmembers

Incircularmembers,centralgeometrygreatlyenhancestheunderstandingoftorsionseffects.Shearstressesareequivalentalongcircumfrentiallinesaboutthecenter(seeFigure1.1).Studyingpuretorsionincircularmembersandtheirgeometryresultsinequations(1.1)and(1.2).

Figure1.1Shearstressincircularmembers.Shearstressuniformdistributedalongcircumfrentiallines.

=(rT)/IpEq(1.1)Where:

isshearstress

ristheradiusdistancefromcenter

Tistheappliedtorque

Ipisthepolarmomentofinertia

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=Tl/GIpEq(1.2)Where:

isthechangeinangleoverthelengthofthebar

listhelengthofthebar

Gistheshearmodulus

Rectangularmembers

Theobservationsmadefortorsionofmemberswithcircularcrosssectionsdonotholdforthosewithnoncircularcrosssections:

1.Theshearstressisnotconstantatagivendistancefromtheaxisofrotation.Asaresult,sectionsperpendiculartotheaxisofthememberwarp,indicatingoutofplanedisplacement.

2.Thetheoryofelasticityshowsthattheshearstressatthecornersiszero.

3.Maximumshearstrainandstressarenotatthefarthestdistancefromtherotationalaxisofahomogeneousnoncircularmember.

Outofplanedisplacementsrequiresolutionsofthenoncirculartorsionproblemtouseawarpingfunction.Thewarpingfunctionmakesthesolutionoftheproblemmorecomplex.Wewillnotexploreithere,rathershowtheresultsfromthefirstproposedsolutiontothenoncirculartorsionproblem.

St.VenantsSolutions

St.VenantwasthewasthefirsttoaccuratelydescribetheshearstressdistributiononthecrosssectionofanoncircularmemberusingtheTheoryofElasticity:

Applicablestatementsfromthetheoryofelasticity:

themaximumshearstrainandstressoccuratthecenterlineofthelongsidesoftherectangularcrosssectiontheshearstrainandstressatthecornersandcenteroftherectangularcrosssectionarezerothestrainandstressvariationsonthecrosssectionareprimarilynonlinear

Thefiguresin2.1(a)and(b)showSt.Venantsresultingstresszonesfromrectangularandsquaretorsionmembers.

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(a)

Figure2.1(a)showstheresultingstresszonesinanrectangularelementofb/t=2,wherebisthelengthofthelongsideandtistheheight.Dashedlinesindicateareasofdepressionanddarklinesindicateareasofelevation.St.Venantcomputedthatastheratiooft/bapproaches1.4513fromhighervaluesthefourregionsdecomposeintoeightregionslikethatofthesquareelementshowninfigure2.1(b)whereb/t=1.

Thetheoryofelasticityhasbeenappliedtofindanalyticalsolutionsforthetorsionofrectangularelasticmembers.Equationsforshearstressandangleoftwistarestatedinequations(2.1)and(2.2).

Eq(2.1)

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Eq(2.2)b=lengthofthelongside

t=thickness,orwidthofshortside

,=parametersobtainedfromTable2.1

Table2.1

b/t 1.0 1.5 1.75 2.0 2.5 3.0 4 6 8 10

.208 .231 .239 .246 .258 .267 .282 .299 .307 .313 .333

.141 .198 .214 .229 .249 .263 .281 .299 .307 .313 .333

ElasticMembraneAnalogy

Thesolutiontothenoncirculartorsionproblemrequirestheavailabilityofawarpingfunction.Theelasticmembraneanalogyprovidesforamuchsimplerwaytofindthesolutiontoathisproblem.Prandtl,showedthattheLaplaceequationdescribingthetorsionofanelasticmemberisidenticaltothatusedtodescribethedeflectionofanelasticmembranesubjectedtoauniformpressure.

Theelasticmembraneanalogyisasfollows:

Consideratubewhichhasthesamecrosssectionalboundaryasthebar.Thenahypotheticalmembraneisstretchedoverthetubescrosssectionandinternalpressureisapplied.Thedeflectedshapeofthemembranehelpsustounderstandthestresspatterninthebarundertorsion.

Thefollowingconclusionsareusedtohelpunderstandtheanalogy:

1.Linesofequaldeflectiononthemembrane(contourlines)correspondtoshearingstresslinesofthetwistedbar.

2.Thedirectionofaparticularshearstressresultantatapointisatrightangletothemaximumslopeofthemembraneatthesamepoint.

3.Theslopeofthedeflectedmembraneatanypoint,withrespecttotheedgesupportplaneisproportionalinmagnitudetotheshearstressatthecorrespondingpointonthebarscrosssection.

4.Theappliedtorsiononthetwistedbarisproportionaltotwicethevolumeincludedbetweenthedeflectedmembraneandplanethroughthesupportingedges.

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Figure3.1Showstheresultingshapeofthemembraneanalogyappliedtoasquaremember.12,3=0(Source:

http://ae.msstate.edu/~masoud/Teaching/SA2/A6.5_more2.html)

Figure3.2showstheresultingshapewhenapplyingthemembraneanalogytoasquaremember.2=1,3=0(Source:

http://ae.msstate.edu/~masoud/Teaching/SA2/A6.5_more3.html)

Inengineeringpracticemostmaterialsarenottheideal,perfectmaterialswestudy.Inthesameway,studyingpuretorsioninbarsofcircularcrosssectionallowunderstandingoftheeffectsoftorqueonamember,butmanytimesareexclusivetothesituationswhichariseintherealworld.Studyingtorsioninnoncircularstructuresallowsneededcalculationstocomputetheeffectsofappliedtorque.

Bibliography

Anniversaryvolumeonappliedmechanics,dedicatedtoC.B.Bienzobysomeofhisfriendsandformerstudentsontheoccasionofhissixtyfifthbirthday,March2nd,1953.Publisher:Haarlem,H.Stam,1953.

Basler,K,andC.F.Kollbrunner.TorsioninStructures.NewYork:SpringerVerlag,1969.

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Heins,C.P.BendingandTorsionalDesigninStructuralMembers.Lexington,Massachusetts:LexingtonBooks,1975.

http://ae.msstate.edu/~masoud/Teaching/SA2/chA6list.html