PlasticTheoryofBendingDescribesBendingabovetheyieldStressforelasticmaterials(MildSteel).

EngineeringMaterials

Contents

1. Introduction2. AssumptionsInThePlastic

Theory.3. TheMomentOfResistance

AtAPlasticHinge.4. MomentsOfResistanceFor

VariousCrosssections.5. CollapseLoads6. CombinedBendingAnd

DirectStress7. CollapseLoadsInPortal

Frames8. PageComments

Introduction

BendingBeyondTheYieldStress.MostEngineeringdesignisbasedonthe"ElasticTheoryofBending"andthemethodistocalculatethemaximumStresseswhichoccur,andtothenkeepthemwithintheworkingStressesinbothcompressionandTension.TheseworkingStressesarecalculatedfromtheYield(orultimate)StressandaFactorofSafety.ThisapproachisalittleunrealisticsinceMildSteelStructuresdonotfailwhentheedgeStressofanycrosssectionreachestheYieldpoint,andwillcontinuetowithstandtheloadaslongasthecentralcoreofthesectionremainswithintheElasticState.

PlasticBendingOfBeamsAstheloadonaparticularbeamisgraduallyincreased,thegreatestStresseswilloccurattheextremefibresofthe"weakest"section(Note:InsomeSteelswhentheelasticlimitisreachedthereisamarkedreductioninStressandinanycalculationsthelowerYieldStressistakenSeegraph).Theseouterfibresaresaidtobeintheplasticstate,andanyincreaseinloadingwillresultinaconsiderableincreaseinStrainandhencedeflectionatthatsectionoftheBeam.TherewillalsobearedistributionofStress.WithMildSteelthisincreaseinStraincantakeplacewithouttheStressrisingabovetheyieldpoint(i.e.anyStrainHardeningeffectscanbeneglectedandtheplasticStrainatyieldisintheorderof1020timestheElasticStrain).

ItcanthereforebeassumedthattheStressintheplasticregionisConstant.WhenthewholecrosssectionatanypointinastructurebecomesPlastic,nofurtherincreaseinthemomentofresistanceispossiblewithoutexcessiveStrain(equivalenttoanincreaseintheCurvatureatthatsection)andaplastichingehasbeendevelopedoneormoresuchhingesarerequiredforacompletecollapse.Thenumberdependsuponthetypeofstructureandwhetheritis,forexample,asimplysupportedbeam,abuiltinbeamorarigidframe.ThevalueoftheloadrequiredtoproducethisstateiscalledtheCollapseLoad,andtheratiooftheCollapseLoadtotheWorkingLoadiscalledtheLoadFactor.InplasticdesignthisfactorisusedinsteadofthenormalFactorofSafety.

AssumptionsInThePlasticTheory.TherequirementistocalculatetheBendingMomentneededtoformaPlastichingeinanyparticularcrosssection,andtodeterminethedistributionofBendingMomentalongthebeamattheCollapseLoad.Todothisitisnormaltomakethefollowingassumptions:

ThatthematerialexhibitsamarkedyieldandcanundergoconsiderableStrainatYieldwithoutanyfurtherincreaseinStress.IneffectthislimitsthetheorytoapplicationsusingMildSteelsasthematerialhasadropinStressatYield.Theloweryieldstressisusedincalculations.

TheYieldStressisthesameinTensionandCompression.

TransversecrosssectionsremainplanesothattheStrainisproportionaltothedistancefromtheNeutralAxis.However,inthePlasticregiontheStresswillremainConstantandisnotproportionaltotheStrain.

OnceaPlasticHingehasdevelopedatanycrosssection,theMomentofResistanceatthatpointwillremainConstantuntilthecollapseofthewholestructurehastakeneffect.ThiswillonlyhappenwhentherequirednumberofPlasticHingesatotherpointshavedeveloped.

TheMomentOfResistanceAtAPlasticHinge.ThediagramshowsthevariationsinStressandStraininabeamofsymmetricalcrosssectionsubjectedtoaworkingload.

(a)UsingtheformulafromtheSimpleTheoryofBending,themaximumworkingStressis .NotethattheStressandStrainareproportionaltothedistancefromtheNeutralAxis.

(b)TheloadhasbeenincreasedsothattheextremefibresYieldandthebeamisinapartialPlasticstate.NotethatisthelowerYieldStress.

(c)TheLoadisincreasedfurtheruntilafullyPlasticStateisobtained.Itisnowassumedthatthestress isuniformoverthewholecrosssection.Infactthisisnotstrictlytrue,sincetherewillbeaverysmallelasticregionaroundtheNeutralAxis(shownonthediagram)buttheeffectofthisonthevalueoftheMomentofResistanceisverysmallandcanbeneglected.

MomentsOfResistanceForVariousCrosssections.a)RectangularSection

IfthewidthisbandthedepthdthentheTotalloadsaboveandbelowtheNeutralAxisareboth eachactingatfromtheNeutralAxis.Hence,thePlasticMomentisgivenby:

(2)

ThiscanbecomparedwithaWorkingMomentof:

(3)

ThishasbeencalculatedfromtheElasticTheoryfromwhich:

(4)

AndZistheNormalSectionModulus.

Theratio iscalledtheShapeFactorSsinceitdependsonlyupontheShapeofthecrosssection.Forarectangularsection,fromequations(1)and(3):

(5)

Andfromequations(2)and(3)TheNormalFactorofSafetybasedontheinitialYieldequals:

(6)

Andfrom(1)and(2)

(7)

Notethattheequations(5)and(6)applytoanycrosssection.

b)Isection

TheShapefactorwillvaryslightlywiththeproportionsofflangetoweb.TheaveragevalueforSisabout1.15.

c)UnsymmetricalSections

IfAistheTotalAreaofacrosssection,thenitisclearthatforpurePlasticBendingthe"NeutralAxis"mustdividetheareainhalf.IftheCentroidsofthesehalvesare atadistanceapartof ,

then:

(8)

Example: 1 2 3

Butatfirstyield.

(9)

(10)

[imperial]

ExampleMomentsOfResistanceForRectangularCrosssectionsProblem

ASteelbarofrectangularcrosssection3in.by1.25in.isusedasasimplysupportedbeamoveraspanof48in.withacentralload.IftheyieldStressis18tons/sq.in.andthelongedgesofthesectionarevertical,(a)findtheloadwhenyieldingfirstoccurs.

Assumingthatafurtherincreaseinloadcausesyieldingtospreadinwardstowardstheneutralaxis,withthestressintheyieldedpartremainingat18tons/sq.in.,(b)findtheloadrequiredtocauseyieldingtoadepthof0.5in.atthetopandbottomofthesectionatmidspan.(c)Findalsothelengthofbeamoverwhichyieldinghasoccurred.(U.L.)Workings

If tonsistheloadatfirstyield,thenfromequation(3):

(1)

(2)

(3)

UnderahigherloadWthecentralsectionofthebeamisinapartiallyPlasticstate,thestressdistributionbeing

similartotheouter1/2in.oneachsideoftheneutralaxisbeingunderconstantstressof18tons/sq.in.withnodropofstressatyield.TheMomentofResistancecalculatedfromthestressdiagramis:

(4)

(5)

(6)

Usingequation(12).AtfirstyieldtheMomentofResistanceis andif

thisoccursatadistancexfromeitherendunderacentralloadWthen:

(7)

Thelengthofbeamoverwhichyieldingoccurs

Solution

(a)

(b)

(c)Thelengthofbeamoverwhichyieldingoccurs

CollapseLoadsOncetheMomentsofResistanceataplastichingeinasectionofabeamhasbeenfounditisnecessarytodecidefromtheconditionsatthesupports,howmanyhingesarerequiredtocauseCollapse.Ifthereareanumberofpointsof"local"maximumbendingmomentsalongthebeam(Underworkingloadconditions),itisclearthatthefirstplastichingewilloccuratthenumericalmaximumpoint.Iffurtherplastichingesarerequiredforcollapse,thenthesewilloccuratthenextlowervaluechosenfromtheremaininglocalmaxima.Whensufficientplastichingeshavebeenformedtoconvertthestructureintoamechanism(i.e.thehingesareconsideredtobepinjoints),thencollapsewilloccur.

Thecaseofasinglebeamsupportedinthreedifferentwayswillnowbeexamined.

a)ASimplySupportedBeam.

Lettheloaddividethelengthintheratioofa:b.ThereisonlyonepointofMaximumBendingMoment(i.e.Wab/lundertheload)andthecollapseconditionswillbereachedwhenaPlasticHingeisformedatthispoint,

TheBendingMomentatthehingeis ,andhencethecollapseLoadisgivenby:

(11)

UsingEquation(#6)

(12)

But whereWistheworkingLoad.

Rearranging:

(13)

(14)

Thisisthesimpleresultwhichwillalwaysbeobtainedwhenonlyoneplastichingeisrequiredforcollapse.ForagivenmaterialandworkingstressitcanbeseenthattheloadfactorisgreaterbytheShapeFactor,thanthenormalfactorofsafetyusedonelasticdesign(thisconsidersthatfailureoccursatfirstyield).ItcanalsobeseenthatdifferentloadFactorswillbeobtainedfrom,say,rectangularandIsections,evenunderthesamesystemofloading.Alternatively,bybasingthedesignonaconstantloadfactortheworkingStressmaybevariedtosuittheparticularsection.

(15)

(16)

(17)

Theresultsobtainedfordistributedloadsandforsimplecantileversarealsoasinequations(29and(30).

b)ProppedCantilever

ThefollowingdiagramshowsacantileverwhichiscarryingacentralloadW,andwhichisproppedatthefreeendtothesameheightasthefixedend.

Theloadonthepropunderelasticconditionsis (Seepagesonthedeflectionofbeams).TheBendingMomentdiagramisshownimmediatelyunderneath.Therearelocalmaximaatthefixedendandundertheload,andagradualincreaseinloadwillcauseaplastichingetoformatthefixedendfirstasthecentralB.M.issomewhatless.However,duetothesupportatthefreeend,collapseconditionswillnotbereacheduntilasecondplastichingehasformedundertheload.AtthatpointtheB.M.'satthecentreandfixedendwillbethesameandnumericallyequalto .Thedistributionisshownonthelowerdiagram.NotethattheshapeoftheB.M.diagramatthecollapseconditionsisnotsimilartothatatWorkingconditions.ThisisduetotheredistributionofStressandStrainwhenaplastichingeisformed.Thevalueof isassumedtobethesameateachhinge.IfPistheloadonthepropatcollapse,thenequatingthenumericalvalueoftheB.M.'satthefixedendandthecentregives:

(18)

(19)

(20)

UnderworkingconditionsthemaximumBendingMomentis:

(21)

Usingequation(#27)

(22)

andfromequation(6)

(23)

Substitutingfromequation(#28)

(24)

HencetheLoadFactorLis:

(25)

Thisisanincreaseof9:8overasimplysupportedbeamunderthesameworkingconditions.

c)BuiltinBeamwithaUniformlydistributedLoad.

ForcollapsetooccurthreePlasticHingesmustbeformed,andastheloadingissymmetrical,thesewillbeateitherendandinthecentre.TheBendingMomentdiagramatcollapseisthenconstructedbymakingthevaluesatthesepointsequalto .

Byinspectionitcanbeseenthatthereactionsattheendsare andhenceatthecentre:

(26)

fromwhich:

(27)

Example: 1

Fromtheelastictheory:

(28)

Substitutinginequation(#34)

(29)

Substitutingfromequation(#35)

(30)

(31)

Forallcasesofbuiltinbeamsthecollapseloadisnotalwaysaffectedbyanysinkingofthesupportsorlackofrigidityatthefixedends,providedthattherigidityissufficienttoallowthefullyplasticmomenttodevelop.

[imperial]ExampleApplicationofCollapseLoadsProblem

A12in.by4in.BritishStandardBeamiscarriedoveraspanof20ft.andhasrigidlybuiltinends.Findthemaximumpointloadwhichcanbecarriedat8ft.fromoneendandthemaximumworkingstresswhichwillhavebeensetup.Workings

(8)

UnderElasticconditionsthemaximumBendingMomentwillbeattheendnearesttotheload.Hence:

(9)

Atcollapsehingesmustbeformedateachendandundertheloadanditcanbeseenthatthecollapseload isfoundbyequatingthenumericalvalueoftheBendingMoments.

(10)

(11)

Sincetheloadfactoris1.8xWorkingLoad.

(12)

fromequation(40)

(13)

(14)

Example: 1 2

(15)

ItisnowpossibletocalculatetheworkingStressfrom:

(16)

(17)

Solution

CombinedBendingAndDirectStressWhenaBeamorColumnissubjectedtoanaxialStressaswellasaBendingStress,theneutralaxiswillbedisplacedtoonesideofthecentroid.(a)inthefollowingdiagramshowsthevariationinworkingStress.AnincreaseinloadwillcausethestresstoreachtheYieldPointononesidefirstandthentospreadacrossthesectiontogiveafullyplasticstate(c).

Itcanbeseenthatthedisplacementoftheneutralaxisintheplasticstateisgivenbyh,where:

(32)

Wherebisthewidthofthesectionneartothecentroidaxis.Theplasticmomentofresistanceisgivenby:

(33)

(34)

(35)

Comparingthistoequation(6)itcanbeseenthattheplasticmomenthasbeenreducedbyatermdependingupontheAxialLoad,theLoadFactorandtheShapeofthesection.Thepermissibleworkingmomentforasinglehingeisthenobtainedfrom

bydividingbytheLoadFactorL.

(36)

[imperial]ExampleApplicationofCombinedBendingAndDirectStressProblem

A12in.by5in.BritishStandardBeamhastowithstandanaxialloadof10tons.Ifaloadfactorof1.8istobeapplied,determinethemaximumpermissibleBendingMoment.Workings

(18)

AtCollapsetheaxialloadis1.8X10=18tonswhichrequiresadepthofWebof .Thiswillbespacedequallyaaboutthecentroidi.e.1.705oneitherside(hofequation(#49)).

Thereductionin isgivenbytheproductofhalftheaxialloadandthedistancebetweenthecentresofareasofeachhalfload(i.e.GHCDJKintheabovediagram)

(19)

(20)

(21)

Notethatthereductionin inthiscaseisonlyabout %,whereasontheelastictheory,withaworkingstressof10tons/sq.in.,thepermissibleBendingStressisgivenby:

(22)

(23)

Andthereductionin duetotheexistenceoftheaxialloadis:

(24)

Solution

CollapseLoadsInPortalFramesInaframeworkwithrigidjoints,underanyappliedload,pointsofmaximumBendingMomentwilloccuratthejoints.AtcollapsesomeorallofthejointswillbecomePlasticHinges.

Thediagramshowsaportalframeofheighthandspanl.ItisunderacentralverticalloadVandahorizontalloadH.PlastichingesmayforminanycombinationatthepointsABCDandE.ItshouldbenotedthatifAandEarepinjointed,theywillrotateunderzeroBendingMoment.

Acollapseconditionisreachedwhensufficienthingesareformedtocreatea"mechanism".Theonlythreeformsofcollapsemechanismare:

BeamCollapsewithhingesatB,CandD.

SwayCollapsewithhingesatA,B,DandE

CombinedCollapsewithhingesatA,C,D,andE

Ifonelinkofthemechanismisgivenarotationof (undertheactionoftheplasticmoment ),thenthevalueofthecollapseloadcanbecalculatedbytheprincipleofwork,choosingtheleastloadforallthepossiblemechanisms.

Somestandardcasesareconsideredbelow.Toallowfordifferentsectionbeamstheplasticmomentswillbeindicatedasfollows:

forthestanchionsABandDE.

forthebeamBD

forthecornersBandD.

a)HingedBasePortal

VerticalLoadonly.ThesymmetricalBeamCollapsewillapply.Ajointrotationof willoccuratBandDwhichequatesto atC.IfitisassumedthatthewholeStraintakesplaceunderaconstantcollapseloadandneglectinganyelasticstrain,thentheworkdonebytheLoadis ,andtheenergydissipatedintheplastichingesis

.ButworkdonemustequalEnergydissipated.

HorizontalLoadonly.Thiswillproducea"SwayCollapsewithrotationsof occurringatBandD.EquatingtheWorkdonebytheloadandattheplastichinges:

(37)

Example: 1 2 3

LastModified:24Nov11@14:40PageRendered:2014021016:44:08

(38)

CombinedLoad.GenerallytherewillbenorotationatthepointBandcollapsewillbebyformingplastichingesatCandD.TheworkEnergyequationnowbecomes:

(39)

(40)

Itcanbeshownthatifthesectionisuniformthroughout,thenif ,collapsewilloccurbySway.TheCollapseloadisgivenbyequation(#62)

b)FixedBasePortal

VerticalLoadingonly.Thebeamcollapsesinthesamewayasthehingedbaseportalframe.Hence:

(41)

HorizontalLoadOnly.Swaycollapsenowrequirestheformationof4hingesandtheworkequationis:

(42)

(43)

CombinedLoading.Thecombinedcollapsemechanismgives:

(44)

(45)

Foraneconomicaldesignequation(#69)andeither(#67)or(#65)shouldbesatisfiedsimultaneously.

[imperial]ExampleCollapseLoadsInHingedBasePortalFramesProblem

Ifintheabovediagram obtainanexpressionforthehorizontalandverticalcollapseloadswhenthePlasticmoment isthesameforthebeamandthestanchions.Workings

Fromequation(#69)

(25)

(26)

(27)

Iftheothertwomodesofcollapsearecheckedthenitwillbefoundthattheybothrequirehighercollapseloadsandconsequentlythecombinedcollapsemechanismgivestheleastload.

Themosteconomicalsectionsforagivenloadarecalculatedbysatisfyingequations(#67)and(#69)simultaneously.Solution