science

Onbend-stretchformingofaluminumextrudedtubes*I:experiments

J.E.Miller,S.Kyriakides*,A.H.Bastard

ResearchCenterforMechanicsofSolids,Structures&Materials,WRW110,TheUniversityofTexasatAustin,

Austin,TX78712,USA

Received13September1999;receivedinrevisedform27April2000

Abstract

Tubularaluminumframepartsforautomotiveapplicationsarebestproducedbyextrusion.Thetubesarethencoldformedtotherequiredshapebyprestretching,pressurizingandbendingthemoverrigiddies.Tensionpreventsbucklingofthecompressedsideandsigni\"cantlyreducesthespringbackonunloading.Anunwantedbyproductoftheprocessisdistortionofthecrosssection.Ithasbeenfoundthatmodestlevelsofpressurecanreducethisdistortion.Theselectionoftheleveloftensionandpressureforoptimumformingispresentlyempirical.Thestudydiscussedhereinseekstodevelopascienti\"cbasisforoptimizingformingprocessessuchthatbucklingisavoidedanddistortionandspringbackareminimized.PartIdescribesacustombend-stretch-pressureformingfacilitydevelopedforthestudy.Thefacilityisoperatedbyonepneumaticandtwoservohydraulicclosed-loopsystems.Thisallowscomputercontroloftheprocess,anda!ordsselectableloadinghistories.Theplanarformingprocesswasmodeledbyapproxi-matingthetubeasanonlinearelastic}plasticbeamwhichcanundergolargerotations.Themodelwasshowncapableofreproducingaccuratelytheloadinghistoryexperiencedbydi!erentsectionsalongthelengthofthepartduringforming.Representativeresultsfromformingexperimentsinvolvingrectangularaluminumarepresented.Theresultsareusedtodiscussthee!ectoffriction,tensionandpressureonthecross-sectionaldistortion,springbackandnetelongationofthepart.PartIIpresentsamodelforestablishingthecross-sectionaldistortioninducedduringforming.Themodelisusedinconjunctionwithexperimentalresultstoestablishwaysofoptimizingtheprocess.PublishedbyElsevierScienceLtd.

Keywords:Bend-stretchforming;Aluminumtubes

*Correspondingauthor.Fax:00-1-512-471-5500.E-mailaddress:skk@mail.utexas.edu(S.Kyriakides).

0020-7403/01/$-seefrontmatterPublishedbyElsevierScienceLtd.PII:S0020-7403(00)00039-4

1284J.E.Milleretal./InternationalJournalofMechanicalSciences43(2001)1283}1317

1.Introduction

Themajorityofcurrentcarframesaremadebystampingsteelsheetsintotherequiredshapesandspotweldingthemtogether(unibodyconcept).Bycontrast,asigni\"cantnumberofmodernautomotiveframecomponentsarecoldformed,extrudedaluminumtubes.Suchtubesareassem-bledtoformaspaceframe;arevolutionarydesignconceptwhichhasbeenusedinseverallowproductionvolumeallaluminumcarssuchastheAudiA8[1}3],themorerecentA12[4],thePlymouthProwler[5,4],thebatteryoperatedEV1ofGM[6],theLamborginiP140[7]andothers.Themainadvantageofthespaceframeisareductioninweightandcorrespondingimprovementsinfuelconsumption.IntheA8,areductionofmorethan40%intheweightofthecarwaspossible.Inadditiontobeinglighter,thespaceframeresultsinasti!erautomobilewithimprovedcrushingperformanceduetothetubularnatureofthecomponents.Anotheradvantageofthespaceframedesignisthatithasfewerparts(by30%ormore*[8,1])whichreducesthecostofmanufacturing.Therecyclablenatureofaluminummakesitattractivewhenconsideredonthebasisoftotallifecycle[9,10].

TheframeoftheA8isshowninFig.1.Itiscomposedofaluminumextrudedtubesjoinedwithcastaluminumconnectingnodes.Thejointsarearc-weldedandtheskinsareattachedtotheframebyrivets.Beforethecarcouldbebuilt,appropriatealloysfortheframe,forthenodesandfortheskinshadtobedeveloped.Innovationswerealsonecessaryinadvancedextrusionprocesses,innewcoldformingmachinesandprocesses,inroboticarc-weldingtechniques,inroboticassemblingprocesses,etc.Typically,thetubesarestretchformed.Inthisprocess,theyareprestretched,internallypressurizedandbenttotherequiredshapesoverdies.

Fig.1.AudiA8frame(reprintedwithpermissionfromCarandDriver,November1996).

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Automatedassemblyrequiresunusuallytightdimensionaltolerancesontheformedtubes.Typicalcross-sectionaldimensionsare50}150mmandwallthicknessrangesbetween1and4mm.Thecross-sectionaldistortionsresultingfromformingmustbekeptwithin$0.5mmandshapedeviationsalongthelengthofthemembersmustbeequallysmall.Forsuchaluminumdesignstobecomeaseriousalternativetocurrentunibodysteeldesigns,theymustbeamenabletoautomatedmanufacturingprocessesforhighproductionspeedandlowercost.Forthesereasons,thespaceframemaynotbeeasilyadaptabletomassproducedcars.However,coldformedaluminumextrudedtubesremainacompetitiveoptionforuseinotherhybridframedesignconceptswheretubularcomponentsareincorporatedintothemoretraditionalunibodydesign(e.g.HondaInsight).Theextenttowhichhollowclosedsectionscanbebentislimitedbyvariousformsofbucklingwhich,forthematerialsandgeometriesofinteresthere,occurintheplasticrange.Thus,theonsetandevolutionofprevalentbucklingmodesareastrongfunctionofthesectiongeometryandthestress}strainbehaviorofthealloy.Onewayofdelayingtheonsetofsuchbendinginstabilitiesisapplicationoftensionduringforming.Tensionreduces,andcaneveneliminate,compressivestressesduetobending,andthusbucklingisavoided.Apracticalwayofaccomplishingthisistobendthepartoveracurvedmandrel.Theaxialtensionisreactedbycontactwiththemandrel(seereviewbyWelo[11]).Contactprovidesanadditionalconstraintandcanfurtherdelaytheonsetofbuckling.Asecondimportantbene\"tfromtheaxialtensionisreductionofspringback.Presentpracticeistoprestretchalltubularcomponentsintotheplasticrange.Thisguaranteesthatonunloadingthespringbackisminimal.

Anunwantedbyproductofbendingofthin-walledsectionsisdistortionoftheircrosssection[12].Forexample,inthecaseofacirculartube,thistakestheformofovalizationofthecrosssection[13,14];inthecaseofarectangularsection,bentinoneofthesection'sprincipaldirections,thecross-sectiondeformsinawaythatmaintainstherightanglesatthecorners[15].Unfortunately,tensionappliedduringbendingoveramandrelaggravatessuchcross-sectionaldistortionsbecausethenormalcompressivereactionforcefromthemandrelhasthee!ectof#atteningthesection[16,17].Becauseofthisproblem,stretchforminghasnottraditionallybeenfavoredforcoldformingtubularmembers.Inthecaseofautomotivetubularcomponents,themethodisattractivebecauseofthespeedwithwhichpartscanbemanufactured.Theprocesswasmodi\"edtoovercometheproblemofcross-sectionaldistortionbyinternallypressurizingthepart[18].Ithasbeenfoundthatmodestvaluesofinternalpressure(usuallycompressedairatlessthan3bar)canreducedistortionofthecrosssection.Becausethepartisalreadyplasticizedbyprestretching,theselectionoftheoptimummagnitudeofpressureiscrucial.Excessivepressureofevenafractionofabarcanresultinbulgingofthetubewhich,ofcourse,isalsounacceptable.Anotherwayofreducingdeformationofthecrosssectionistoaddlateralsupporttothewalls.Thisispresentlydonebytrialanderrorrequiringseveralexpensiveandtime-consumingdiemodi\"ca-tionsbeforetheoptimumdieforthepartisfound.

Thecurrentstateoftheartremainsempirical,requiringextensivetrialanderrortesting,andseveraldiemodi\"cationsbeforeanacceptableformingprocess(i.e.,dieshape,prestretchload,levelofinternalpressure)isestablishedforanewpart.Itisimportanttonotethatthecross-sectionalshapesofmanyofthecarcomponentsarerathercomplexandtheirdesignisoftendecidedbyestheticconcernsratherthanstructuralperformance.Anynewcardesignwillhavealargenumberofdi!erenttubesforwhichformingdiesmustbemade.Theempiricismofthepresentwayofdesigningthemanufacturingprocessisine$cient.

1286J.E.Milleretal./InternationalJournalofMechanicalSciences43(2001)1283}1317

Theobjectiveofthisstudywastodevelopascienti\"cbasisfordesigningaformingprocessforeachpartwhichensuresaccuracyinshapewithminimumdistortionofitscrosssection,isfreeofwrinklesandwithminimumorpredictablespringback.Thesegoalsarepursuedthroughacombi-nationofexperimentandanalysis.InPartI,we\"rstdescribeacustombend-stretch-pressureformingfacilitydevelopedforthestudy.Twoseriesofexperimentsinvolvingrectangularaluminumtubeswereperformedusingtheformingfacilitytodeterminehowtheappliedloadsinteracttodeterminethe\"nalshapeoftheformedtubes.InPartIIoftheseries,arelativelysimpletwo-dimensionalmodeloftheformingprocessispresentedwhichisabletoaccuratelycapturethecross-sectionaldistortioninducedbytheformingprocess.Themodelisvalidatedbycomparingthepredictedshapeswithcorrespondingexperimentalresults.Themodel,inconjunctionwithexperi-ments,isthenusedtoinvestigatealternativeloadinghistories.Finally,theinsightgainedisusedtoformulateamethodologyforselectingtheoptimumloadinghistoryforaparticularformedpart.

2.Experimental

Astretch-formingmachineofthetypeusedinindustrytocoldformtubesofvariouscrosssectionsisshownschematicallyinFig.2.Thepartisgrippedinopen-sidedjawsmountedontwolargehydraulicactuators.Thegripsalsosealtheends.Thetubeis\"rstpressurizedtotherequiredlevelandissubsequentlystretchedtoaplasticstressstate.Thepressureandaxialloadarethenkeptconstantwhilethehorizontaltablemovesforwardandthedieengagesthepartforcingitto

Fig.2.Industrialbend-stretchformingmachine.

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conformtotherequiredshape.Asthediemovesforward,theactuatorsrotatesoastoremainapproximatelytangentialtothetwoendsofthetube.Thepressureandtheaxialloadaresubsequentlyreleased,thedieisretracted,andthepartisremovedfromthemachine.Typicalformingcyclescurrentlyrangefrom30to60s.

Theformingmachinedescribedisprohibitivelylargeforlaboratorytesting.Inaddition,wewantedtoaddfeedbackandcomputercontroltoourfacility.Forthesereasons,weoptedtodevelopasmallercustomfacilityforthisexperimentalprogramwhichhadthedesired#exibilityandfeatures.

2.1.Custombend-stretch-pressuretestingfacility

Aphotographofthebend-stretch-pressuretestingfacilitydevelopedisshowninFig.3.Themajorcomponentsofthefacilityareidenti\"edinthescaleddrawingsinFig.4.Itconsistsofarigidreactionframe(length108inor2743mm)madeofstructuraltubing(8;4;0.5inor203;102;13mm).Theframeismountedhorizontallyon󰀂-framelegsequippedwithwheelsandlevelingmountsandisataheightof50in(1270mm)o!the#oor.Whiletheactualmachine(Fig.2)operatesinahorizontalplane,thisoneoperatesinaverticalplane.Thischangeremovedthebendingloadsduetogravityandeliminatedtheneedforlinearsupportslidesforthemovingdie.Bendingislimitedtojustoneplaneandtwistingofthespecimenisprecluded.(AnalternatelaboratorystretchformingfacilityisdescribedinRefs.[19,20].)

Fig.3.Custombend-stretch-pressuretestingfacility.

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Fig.4.Schematicoftestfacility:(a)frontview,initialcon\"guration;(b)topview;(c)frontview,deformedcon\"guration.

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Tensionisappliedtothetestspecimenviatwohydraulicactuatorswiththefollowingcharacter-istics:bore3.25in(83mm),roddiameter1.375in(35mm)androdlength12.5in(318mm).Theyareconnectedtotheframeviatrunnionmountswhichslideintospeciallydesignedbearingsmountedtothesidesofthefame.Thebearingsallowtheactuatorstorotatefreelyaboutpivotssothattheycanfollowthemotionoftheendsofthepartandremainapproximatelytangentialtoitduringtheformingprocess(cf.Figs.4aandc).Theappliedtensionismonitoredbyin-lineloadcells(20kipsor90kNcapacity)placedbetweeneachactuatorandpinconnection.TheextensionofeachactuatorismonitoredbyanLVDTdisplacementtransducer.

Severalcircularformingdieshavebeenfabricatedoutof3in(76mm)thicksolidsteelplate.Theyweremilledtoprecisionandground\"nishedusingNCmillingmachines.Theyhaveradiiintherangeof50}20in(1270}508mm).Thediesareconnectedtotwoverticalparallelactuators14in(356mm)apartwiththefollowingcharacteristics:3.25in(83mm)bore,1.75in(44mm)rodand26in(660mm)extension.Thetwoactuatorsaremechanicallyandhydraulicallyconnectedsotheydisplaceequally(seeFig.4a).Thedoubleactuatorarrangementwasadoptedbecauseitprovidesadditionaltorsionalandbendingrigiditytothesystemrequiredduetothelengthofrodswhenfullyextended.Thisallowedustoeliminatetheslidesusedintheactualsystem(seeFig.2).Lowpro\"le,highsti!nessloadcells(20kipsor90kNcapacity)areattachedbetweeneachactuatorandtheconnectingplate.Thepositionofthedieismonitoredbyamagnetostrictivedisplacementtransducer.TheloadanddisplacementcapacitiesofthefacilityarelistedinTable1.

Theresultspresentedherecomefromrectangularcross-sectionaltubeswithdimensionsof50;30;1.8mmcustommadebyAlcoaforthisresearchprogram(Al-C210-T4).Steelend-plugsarebondedtotheendsofthetubeandpin-connectedtotheactuatorrods.High-strengthHysol(EA9430)epoxywasusedoveranareaof8in󰀲(5160mm󰀲)sothatthestrengthofthebondedjointexceededthatofthetubes.Thepin-to-pinlengthofthespecimencanvary,but1000mmistypical.Byusingdi!erentend-plugs,thespecimencanbebenteitheraboutitswideornarrowside.Switchingtotubesofothershapesmainlyinvolvestheconstructionofnewpairsofcompatibleend-plugs.

Thetestspecimencanbepressurizedinternallybycompressedairor,alternatively,itcanbeevacuatedthroughportsinthetubeend-plugs(seeFig.5).Thepressureissetbyapneumaticservovalvewithacapacityof!1}10atm(bar)andmonitoredbyapressuretransducer.

Themostsigni\"cantimprovementinthetestingfacilityovercommercialunitsistheadditionoftheinstrumentationtoeachofthethreemodesofloadingandinclusionoffeedbackcontrol.Theseinturnallowtheexecutionofaformingtestbyacomputeranda!ordsomefreedomsinthechoiceoftheloadingpathfollowed.Thestretching(H)andforming(V)modesofloadingareconnectedto

Table1

Bend-stretch-pressuretestfacilitycapacitiesCapacities

Displacementin(mm)Loadkips(kN)Pressure(atm)

Stretchingmode(H)12.5(305)20(90)*

Formingmode(V)25(635)40(180)*

Pressuremode(P)**

!1}10

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Fig.5.Detailofconnectionbetweenspecimenandtestingfacility.

Fig.6.Controlsystem.

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ageneralpurposehydraulicpowerunit(3000psior207bar).Thesetwoloadingmodesarecontrolledviaclosed-loopservohydrauliccontrolsystemsshownschematicallyinFig.6.Theycanbeoperatedeitherunderloadordisplacementcontrol.Intheexperimentsthatfollow,theH-modewasoperatedunderloadcontrolwhiletheV-modewasoperatedunderdisplacementcontrol.TheP-modehasitsownfeedbackloopwithpressureasthefeedbacksignal.

Thedistortionofthecrosssectionofthetubeismonitoredbyacustom-builtbiaxialtransducershowninFig.7a,usuallymountedatthemid-spanofthespecimen.Thetransducermonitorssimultaneouslythemaximumde#ectionofthetopfaceoftherectangulartube(seeFig.7b)andthemaximumchangeinthewidthofthetube(bulgingseeFig.7c).Themovingpartsofthetransducercontactthespecimenviarollers.

Thetransducerrestsonthecirculardiewithwheelssothatbyinterruptingatestitcanberolledalongthedietorecordcross-sectionalde#ectionsalongthelengthofthespecimen.Thetransducer

Fig.7.Biaxialdistortiontransducer.

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positionalongthedieismonitoredasfollows.Athintapewithregularlyspacedblackandsilvermarkingsisbondedtooneedgeofthecurvedsurfaceofthedie.Anemitter/receiverphotodiodepairmountedonthetravelingtransducerproducesanelectricalpulsewheneveritencountersasilverband(seeFig.7a).Positionalongthedieisthenrelatedtothecountofpulsesfromthecenterofthespecimen.Typically,asignalisrecordedevery0.2in(5.1mm)onthecurvedsurfaceofthedie.(Theimportanceofthe\"nalshapeofformedpartsandanalternatemethodofmeasuringtheshapeisdiscussedinRef.[21].)

Thesignalsofone-loadcellandone-displacementtransducerfromeachoftheV-andH-modeareconditionedviaafour-unitMTS458.20controlconsole.Theconsolealsoenablescontrolofthetwomodesviathefeedbacksignalsofchoice.TheremainingtransducersignalsfromtheV-andH-modeareconditionedseparatelyasshownintheblockdiagraminFig.8.Thepressuretransducerandtheassociatedconditionerandcontrollerareanintegralpartofthepneumaticservovalve.Thesignalsfromthetwo-displacementtransducersinthebiaxialdistortiondeviceareconditionedasshowninFig.8.

Thecommandsignalsforthethreeclosedloadingloopsaregeneratedbyadataacquisi-tion/controlsystemoperatedbyaPC(200MHzPentiumPro).Twodataacquisitioncards,eachwith2D/Aoutputchannels,areusedtogeneratethecommandsignals(12bitresolution)fortheV-,H-,andP-modes.The$10Vsignalsaregeneratedatamaximumupdaterateof1MS/s.ThecontrolprogramusestheLabVIEWenvironmentandgeneratesthecommandsignalswhichallowachoiceofthetension-pressure-verticaldisplacementpaththatwillbefollowedinatest.

Concurrently,thedataacquisitioncardsrecordthesignalsfromthe11transducerspresentinthetestfacility.Eachofthecardsisequippedwith8di!erentialinputD/Achannelswhichoperatewith

Fig.8.Testfacilityinstrumentation.

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12-bitresolutionatamaximumsamplingrateof1.25MS/s.Whenall11transducersaremonitored,thesamplingrateforanindividualchannelisapproximately100kS/s.ThesignalsareprocessedthroughLabVIEWsoastohavereal-timedisplayofallofthevariables.Initsstandardoperationmode,thesystemcontinuouslymonitorsandrecordsthetransducersignals.Whenanaxialscanofthecross-sectionalde#ectionsisperformed,recordingofmeasurementsistriggeredbythevoltagepulsesgeneratedbythephotodiode.Forredundancyandsafety,severalofthekeyvariablesarealsomonitoredviadigitalvolt-metersandanalogplotters.2.2.Experimentalprocedure

FivetestspecimenswerecutfromeachofthetubesprovidedbyAlcoa.Thefollowinggeometricmeasurementsweremadeoneachusingamicrometer.Threemeasurementsofthetubeheight(h)andwidth(b)weremadeatthetwoendsandatthemiddleofthespecimen.Thethickness(t)wasmeasuredat12locations(threemeasurementsofeachofthefourwalls,identi\"edbysubscriptsi\"1,4)oneachendofthespecimen.ThemeanvaluesandthevariationsofeachofthevariablesarelistedinTable2.Thewallthicknesswasusuallynearlyconstantforthreeofthefourtubewalls.Thefourthwall(oneofthetwoshortsides)wasusuallyabout5%smaller.

Theexcessmaterialwasusedtoobtainthemechanicalpropertiesofeachlengthoftubeusedinthestudy.Uniaxialtensiontestswereperformedonstripscutfromthetubes.ThemeanvalueandthevariationofYoung'smodulus(E)andtheyieldstress(󰀊)ofthetubesusedinthisseriesoftests

󰀰

aregiveninTable3.Testsperformedusingstripsfromdi!erentsidewallsofthetubedidnotshowvariationinthematerialproperties.Tensiletestswerealsoperformedonspecimenscuttransversetothetubeaxis.Thedi!erencebetweentheaxialandtransverseyieldstresswasinsigni\"cantlysmall.Thetubescamefromthesamebatchthusdi!erencesinmechanicalpropertiesfromtubetotubewererelativelysmall.

Thesurfacesoftheend-plugsandthetubewerepreparedforbondingbysandingwithamedium-gritsandpaperandcleaningwith1}1}1trichloroethane.Theadhesivewascuredat1803F(823C)for1h.

Tominimizethee!ectoffriction,thesurfaceofthediewascoveredwithan8milTe#ontapewhilesimultaneouslythecontactingsurfaceofeachtestspecimenwascoatedinlightgrease.Thee!ectoffrictionwillbediscussedlaterinlightoftheresults.

Table2

Dimensionsofcrosssectionsoftubestested

tin(mm)󰀱

MeanS.D.Min.Max.

0.0717(1.82)

0.37;10\\󰀳(9.40;10\\󰀳)0.0711(1.81)0.0728(1.85)

tin(mm)󰀲

0.0687(1.74)

0.25;10\\󰀳(6.35;10\\󰀳)0.0681(1.73)0.0693(1.76)

tin(mm)󰀳

0.0717(1.82)

0.40;10\\󰀳(10.2;10\\󰀳)0.0707(1.80)0.0727(1.85)

tin(mm)󰀴

0.0713(1.81)

0.30;10\\󰀳(7.62;10\\󰀳)0.0702(1.78)0.0718(1.82)

tin(mm)0.0709(1.80)

0.25;10\\󰀳(6.35;10\\󰀳)0.0701(1.78)0.0713(1.81)

bin(mm)1.968(50.0)

0.73;10\\󰀳(18.5;10\\󰀳)1.967(50.0)1.970(50.0)

hin(mm)1.180(30.0)

0.71;10\\󰀳(18.0;10\\󰀳)1.178(29.9)1.181(30.0)

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Table3

Mechanicalpropertiesoftubestested

Emsi(GPa)

MeanMin.Max.

9.6(66.2)9.4(64.8)10.0(68.9)

󰀊ksi(MPa)󰀰15.2(105)14.9(103)15.5(107)

Atypicalformingtestwascomprisedofthestepslistedbelow.Testsinwhichwedeviatedfromthisprocedurewillbedescribedanddiscussedinthesectiononexperimentalresults.

(i)Testsbeganwiththediefullyretracted(Fig.4a).TheH-modeactuatorswereadjustedtothecorrectpositiontoallowthetestspecimento\"tbetweenthepinconnections.Theywereheldinahorizontalpositionbyelasticbandswhilethespecimenwasattachedbyinsertingthepins.(ii)Theorderandmagnitudeoftheloadswerepresetintothecontrolprogram.Thespecimen

was\"rststretchedaxiallytoachosenvalueoftension.Atthispoint,theelasticbandssupportingtheH-modeactuatorswereremovedandthecross-sectionaldeformationtrans-ducerwasinstalled.

(iii)Theinternalpressurewasthenrampedtothedesiredlevelwhilethetensionwasheld

constant.

(iv)Motionofthediewasthencommencedengagingandformingthespecimen.Duringthe

formingprocess,thetensionandpressureloadswerekept\"xed.Asthediemoves,theH-modeactuatorsextendandrotatesothattheyremainapproximatelytangentialtotheendsofthedeformingspecimen.Thediewasstoppedwhenitwasincontactwithmostofthespecimen(usually82%,seephotographinFig.3).

(v)Thebiaxialcross-sectionalmonitoringtransducerwasparkedatthespecimenmid-span

throughoutthetest.Inseveralofthetests,thediemotionwasperiodicallystoppedandanaxialscanofthecross-sectionaldistortionwasperformedonthepartofthetubeincontactwiththedie.

(vi)Atthecompletionofforming,theinternalpressurewasallowedtoreturntotheambient

value,thetensionwasreleasedandthespecimenremovedfromthemachine.

(vii)Theformedpartwasevaluatedbymeasuringitscurvature,#angesaggingandmaximum

widthat1inintervalsalongthelength.3.Experimentalresults

Twoseriesofformingexperimentswereperformedusingthecustombend-stretch-pressuretestingfacilityandproceduredescribedintheprevioussection.Thegoaloftheseexperimentswastodeterminehowthe\"nalcurvature,cross-sectionalshapeandoverallextensionwerea!ectedbytheformingprocess,andinparticular,themagnitudeoftheappliedloads.Tubeswereformedeitheraboutthetallortheshortsideofthecrosssection.Fortheexperimentsdiscussedhere,the

dieradiuswas20in(508mm).Thepressure,tensionandtheloadinghistorywerevariedsu$cientlytoestablishtrends.Allspecimenswereapproximately36.5in(927mm)long(2¸)whichincludesthe1.5in(38mm)endsectionsbondedtothesolidinserts.Inallexperimentsreportedhere,thediewas

\"1.3in/min(33mm/min).Thisrelativelyslowformingratewasmovedataconstantvelocityofd

adheredtoinordertoenablecompleteacquisitionofthesignalsfromallvariablesdiscussedearlier.Theexperimentwasusuallyterminatedwhenapproximately82%oftheinitiallengthofthetubewasincontactwiththedie.Inotherwords,justbeforethepluggedendscameintocontactwiththedietopreventfailureofthebonds.

TheformingprocessisillustratedinFig.9throughasequenceofphotographsofthetubeasitprogressivelyconformstotheshapeofthedie.Thistubewasbentwiththelongsidesofitscrosssectionvertical.Atension¹\"4000lbf(18kN*i.e.62%oftheyieldtension,¹)wasapplied

󰀱󰀰

\"rstandwassubsequentlyheldconstantduringforming.Nointernalpressure(orvacuum)wasappliedinthiscase.Fig.10ashowsaplotoftheverticalforceseenbythedie(F)normalizedby

J

2¹versusthediedisplacement(d).Therelationshipisnearlylinear.(NotethatF/2¹represents󰀱J󰀱

thesineofthecurrentangletheaxisofthetensionactuatorsmakewiththehorizontaldirection.)Markedontheforce}displacementresponsearethelocationscorrespondingtothenumberedphotographsinFig.9.Theinitialstraightcon\"gurationisnotvisiblefromtheobservationangleused.Bendingisinitiallylocalizedatthemid-spanwherethetube\"rstcomesintocontactwiththemandrel.Afterthispointhasconformedtothemandrel'scurvature,theactivezonemovesawayfromthemid-span.Asthediemovesvertically,thedeformingtubeemergesfrombehindthehorizontalsupportbeamofthedevice.Incon\"guration

terminatedatcon\"guration

FanddseeninFig.10a.ThetestwasJ

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Fig.9.Sequenceoftubedeformedcon\"gurationsduringtheformingprocess.

pre-stretchedtoatension¹\"0.95¹andthenpressurizedinternallyto2.3bar(P).The

󰀱󰀰

tensionandpressurewerethenheldconstantduringforming.Fig.11ashowsaplotoftheverticalforceseenbythedie(F/2¹)versusthediedisplacement.Therelationshipisagainnearly

J󰀱

linear.

J.E.Milleretal./InternationalJournalofMechanicalSciences43(2001)1283}13171297

Fig.10.(a)Verticalforceandcontactlengthduringforming;(b)calculatedtubedeformedcon\"gurations.

Thecrosssectionofthedeformingtubedistortsasitcomesintocontactwiththedie.Thedistortionispartlyduetothebending[15,12]butalsoduetothecompressivereactionoftheaxialtensileforcebythedie[16].Indeed,aswewillsee,tensioncanbeasigni\"cantcontributortodistortion.Forthesectionorientationusedinthistest,distortiontakestheformshowninFig.12bandcanbemitigatedbyarelativelysmallvalueofinternalpressure.Theextentofdistortionisde\"nedbythevariables󰀃(or󰀃H)and󰀁de\"nedinthesame\"gure.Thedistortionvariableswillbeexpressedasratiosofthetubemid-surfaceheight(h)andwidth(b),respectively(seeFig.12a).

󰀰󰀰

Becausethetubecrosssectionhasanaspectratioofapproximately1.62thevaluesoftheratioswillbedistinctlydi!erentinthetwotubeorientationsused.

1298J.E.Milleretal./InternationalJournalofMechanicalSciences43(2001)1283}1317

Fig.11.Responseofatubebentinthehorizontalmode:(a)verticalforce;(b)distortionatmid-span.

Theevolutionofdistortionatthemid-spanofthetubewasmonitoredbythebiaxialdistortiontransducerdiscussedinSection2.1.Themeasuredvaluesof󰀃and󰀁areplottedagainstthedisplacementofthedieinFig.11b.Initially,thetopsurfaceofthetubebulgesslightlyoutwardduetotheinternalpressure.Asthetubeisbentfurther,thetopsurfacebeginstosaginwardandreachesits\"nalvalue(󰀃/h+!5.7%)whenthediedisplacementisapproximately8in(203mm).The

󰀰

smalldiscontinuitiesonthetwotrajectoriescorrespondtothelocationswhenthetestwasinterruptedandanaxialscanofthecrosssectionofthepartofthetubeinconductwiththediewasperformed.Thesidewallsdistortinasimilarmannerbutmuchless(󰀁/bK1%).Indeed,this

󰀰

distortioniscoupledtothedistortionofthetopsurfaceinordertomaintaincompatibilityofrotationsatthecornersofthecrosssection.Duringtheinitialphasewhenthetopsurfaceisbulging,thetubewallsarebentinwardcausingthedevicetomeasurethepositionofthetubecorners.Afterthediehasdisplaced2in(51mm),thesidewallsbegintodeformoutwardandengagethedevice.Localdistortionceasesoncethislocationhasconformedtotheshapeofthedie.Inthistest,theformingwasstoppedperiodicallyandanaxialscanoftheportionofthetubeincontactwiththediewasperformedusingthebiaxialdistortiontransducer.Resultsfromfourscans

J.E.Milleretal./InternationalJournalofMechanicalSciences43(2001)1283}13171299

Fig.12.(a)Tubegeometricparameters;(b)de\"nitionofdistortionparameters;(c)stress-strainresponseoftubealuminumalloy.

takenattheverticaldiedisplacementsindicatedareshowninFig.13(sisameasureofdistance

\"

alongthecircularsurfaceofthedie).Thedistortionissymmetricaboutthemid-spanandincreasesslightlyawayfromthecenter.Onceagain,distortionceasesataparticularcrosssectiononceithaslocallyconformedtotheshapeofthedie(i.e.bendingceases).Therelativemagnitudeofthetwomeasuresofdistortionremainssimilartothatseenatmid-span.

MeasurementsofthedistortionmadeafterthespecimenwasremovedfromthetestfacilityarepresentedinFig.14a(sisameasureofdistancealongtheinnersurfaceofthedeformedtube).Here󰀃*representsthesumofthedistortionofthetopandbottomsurfacesofthetube,measuredindividually,relativetothecorners(seeFig.12b).(Notethatthecontributionto󰀃*fromthebottomsurfaceofthetubeisrathersmall.)Thisremovesthecontributionfromthechangeinhdue

󰀰

tothePoissone!ectwhichisincludedinthevariable󰀃measuredbythebiaxialtransducerduringtheexperiment.Asaresult,thismoreaccuraterepresentationofdistortionislessthanthatseeninFigs.11band13a.Asimilarsymmetricdistributionisobservedwiththeminimumdistortionoccurringatthemid-spanandincreasingasthedistancefromthecenterincreases.

1300J.E.Milleretal./InternationalJournalofMechanicalSciences43(2001)1283}1317

Fig.13.Axialscansofdistortionatdi!erentdiedisplacements.

Aswewillsee,thispatternofthedistortionatmid-spanbeingsmallerthanintherestofthetubeispersistentinallresultsandforbothtubeorientations.Oneofthereasonsforthisisthefactthatthecentralpartinitiallybendswithashearpointforcewhichstartsatzeroandgrowstocertainvaluewhenthesectionhasconformedtotheradiusofthedie(seeFig.27b).Asecondreasonisthatthedeformingsectorofthetubeissupportedonbothsidesbyundeformedsectionswhichaddtoitssti!ness.Bycontrast,forsectionsawayfromthemid-span,thelocalshearisatthefullydevelopedvaluewhichtendstogrowslightlyasthecontactlengthincreases.Inaddition,nowthedeformingsectorseesadeformedsectorononesideandanundeformedsectorontheotherwhichmakesitabitmorecompliant.

The\"nalradiusofcurvature(󰀉)ofthetubewasmeasuredalongitslengthandiscomparedto

D

theradiusofthedie(󰀉)inFig.14b.Thetubesprangbackapproximately4.9%.Thespringbackisrelativelyuniformalongthelengthwithaslightincreaseclosetotheends.Thisispartlyduetothein#uenceofthesolidinsertplugattheendsofthetestspecimen.

Wenowconsideratestinwhichthecrosssectionisorientedsothatthelongsidesarevertical.Thetestspecimenwasalsoprestretchedtoatension¹\"0.95¹butnointernalpressurewas

󰀱󰀰

J.E.Milleretal./InternationalJournalofMechanicalSciences43(2001)1283}13171301

Fig.14.Finalshapeofatubeformedinthehorizontalmode:(a)distortion;(b)springback.

appliedbecause,withthetubeinthisorientation,internalpressureonlyexacerbatesthebendinginduceddistortions.TheverticalforceseenbythedieisplottedagainstitsdisplacementinFig.15a.Therelationshipisagainnearlylinear.Thedistortionvariables󰀃and󰀁measuredatthemid-spanareplottedinFig.15bagainstthediedisplacement.Thedistortionsincreasemonotonicallytovaluesof󰀃/h+!4.76%and󰀁/b+6.62%achievedatadiedisplacementofapproximately8in

󰀰󰀰

(203mm).Subsequently,thedistortionsdonotchange.Inpractice,bothvaluesaretoohighsothatthispartwouldberenderedareject.

Againaxialscansofthedistortion,intheportionofthetubeincontactwiththedie,weretakenduringperiodicpausesofthemotionofthedie.FoursuchscansofthetwodistortionvariablesareshowninFig.16.Asinthepreviouscase,thedistortionpro\"leshavedepressionsaroundthemid-spanandgrowslightlytowardstheends.Weobserveoncemorethat,atanygivenaxialposition,thedistortionreachescertainvaluesataparticulardiepositionandsubsequentlyremainsunchanged.Clearly,forthisorientationofthecrosssection,thedistortionofthelongersidewallsismorepronounced.

1302J.E.Milleretal./InternationalJournalofMechanicalSciences43(2001)1283}1317

Fig.15.Responseofatubebentintheverticalmode:(a)verticalforce;(b)distortionatmid-span.

Fig.17showshowthedistortionvariable󰀁variedalongthelengthoftheformedtubeafteritwasremovedfromthemachine.ThedistributionalongthelengthissimilartothatinFig.16b(includingthesmallasymmetryaboutthemid-span).Themaximumvalueisapproximately9.8%.The\"nallocalradiusofcurvatureofthetubeiscomparedtotheradiusofthedieinthesame\"gure.Thetubesprangbackapproximately4.34%.Onceagain,thespringbackisrelativelyuniformalongthelengthwithaslightincreaseclosetotheends.3.2.Ewectoffriction

Frictioncanplayaroleintheformingprocess,especiallyifthetensionischangedoncethetubeisincontactwiththedie.Frictionale!ectswereminimizedbycoveringthesurfaceofthediewithathinTe#ontapeandsimultaneouslylubricatingthesurfaceofthespecimenwhichcontactsthedie.Theadequacyofthisschemewasevaluatedinspecialtestsinwhichtheaxialstrainsatdi!erentpositionsalongthelengthofthetubewerecompared.ResultsfromsuchatestareshowninFig.18.

Fig.16.Axialscansofdistortionatdi!erentdiedisplacements.

Thetubeisformedwiththelongsidesvertical.Pairsofstraingageswerebondedoneachsidewallofthetube(seeinsetinFig.18a)atthemid-spanand7in(178mm,S/¸\"0.39)awayfromitataheightof0.5in(12.7mm)abovethesurfaceofthedie.Thetubewasprestretchedto¹\"0.77¹

󰀱󰀰

andevacuatedtoapressureof!0.48bar.Fig.18ashowsaplotoftheappliedaxialforceversustheaveragestrain(󰀅)ateachofthetwolocations.Attheendofforming,thestrainsatthetwo

Q

locationsdi!erby3%.Oncethetubecameintocontactwiththediethetensionwasincreasedto¹\"0.98¹whilethevacuumwasheldatthesamelevel.(Numericvariablesubscriptsrefertothe󰀵󰀰

genericloadinghistoryshownschematicallyinFig.19.Inthiscase,nopostpressurewasappliedthuspoints

1304J.E.Milleretal./InternationalJournalofMechanicalSciences43(2001)1283}1317

Fig.17.Finalshapeofatubeformedinverticalmode.

Subsequently,󰀅remainsunchangeduntilthetensionisincreasedattheendofforming.Thebeam

1

modeloftheformingprocessyieldsthatthemid-spanconformstotheshapeofthedieatd\"3.68inwhichcompareswellwiththetimethemaximumdistortionisrecorded.Bycontrast,inFig.11wesawthatinthatexperimentthedistortionvariablesreachedtheirfullvaluesatadiedisplacementofapproximately8in(203mm).Althoughmostofthedistortionisinducedduringthetimethesectioninquestionisbent,asmalladditionalincreasetakesplacesubsequentlyastheneighboringsectionsbendintheprocessreducingtheirsupportofthecrosssectionatmid-span.Theevolutionof󰀅atS/¸\"0.39issigni\"cantlydi!erent.Thissectionbendsgraduallydueto

1

beamactionuptoadiedisplacementofapproximately12in(305mm).Atthistime,thestrainexperiencesasharpincreaseasthetubelocallyconformstothecurvatureofthedie.Thesteady-statevalueisreachedwhend\"15in(380mm).Asmentionedabove,thesteady-statevaluesatthetwolocationsdi!erbyapproximately3%.

The\"naldistortion(󰀁)distributionalongthelengthofthetubeissimilartothatinFig.17butwithasomewhatreducedamplitudeduetotheinternalpressure.Thetubewasfoundtohavesprangbackapproximately5.05%.Onceagain,thespringbackwasrelativelyuniformalongthelength.

3.3.Ewectoftension

Tensionplaystwoimportantrolesinbend-stretchformingoftubes.First,itisusedtoeliminate(orreduce)compressionontheconcavesideofthepartandthuspreventwrinklingorlocalbuckling[22]fromoccurring.Inthepresentsetting,theaxialtensionisreactedbycontactwiththedie,anadditionalconstraintwhichcanfurtherdelaybuckling.Asecondroleplayedbyaxialtensionisreductionofspringback[23].Presentpracticeistoprestretchtubularpartsintotheplasticrange[24,19,20,25].Thisguaranteesthatonunloadingthespringbackisminimal.Thee!ect

J.E.Milleretal./InternationalJournalofMechanicalSciences43(2001)1283}13171305

Fig.18.Evolutionofaxialstrainduringformingprocessincludingpost-tension.

Fig.19.De\"nitionofloadinghistory.

1306J.E.Milleretal./InternationalJournalofMechanicalSciences43(2001)1283}1317

Fig.20.E!ectoftensionontubebentinhorizontalmode:(a)distortion;(b)springback.

oftensiononspringbackisillustratedinFig.20bfortubesbentwiththeirlongsideshorizontalandinFig.21fortubesbentwiththelongsidesvertical.The\"rstsetshowsthe\"nalradiusofcurvaturealongthelengthoffourtubesformedattensionsof¹/¹\"0.54,0.65,0.95and1.08.Asthe

󰀱󰀰

tensionincreases,thespringbackisreducedfromapproximately7.3%toapproximately4.5%.Thee!ectoftensiononspringbackfortheothertubeorientation(Fig.21)issimilar.Inbothsetsofexperiments,theshapeisquiteuniformalongthelengthalthoughatendencytobecomemoreuniformathighertensionsisseeninFig.20b.

Anunwantedbyproductoftensionisthatitaggravatescross-sectionaldistortionasthenormalcompressivereactionforcefromthemandreltendsto#attenthesection[16,11,24].ThisisillustratedinFig.20aforthreetubesformedwiththelongsideshorizontal.Aninternalpressureof2.28barwasappliedwhilethetensionhadvaluesof¹/¹\"0.65,0.95and1.08.Theplot

󰀱󰀰

comparesthedistortionpro\"les(󰀃*)alongthelength(measuredafterunloading)forthethreecases.Aswasthecaseforthetubesdiscussedearlier,thedistortionpro\"lesexhibitdepressionsaroundthemid-spanwhichgrowwithtension.Awayfromthemid-span,thedistortionclearlygrowswithtension(bynearlyafactoroftwobetweenthelowestandhighestvaluesof¹).

󰀱

J.E.Milleretal./InternationalJournalofMechanicalSciences43(2001)1283}13171307

Fig.21.E!ectoftensiononspringbackoftubebentinverticalmode.

Itisinterestingtopointoutthat,fortheverticalorientationofthecrosssection,thebottom#angebecomesneutrallystressedatatensionofapproximately1.28¹whilefortheother

󰀰

orientationatapproximately0.94¹.Buckling,ofcourse,requiressigni\"cantcompressiontotake

󰀰

place;thus,inviewofthedetrimentale!ectoftensionondistortion,forthistube,thecurrentpracticeofprestressingtotheyieldtensionisexcessiveforbothorientationsofthecrosssection.AlternativeswillbediscussedinPartIIwiththehelpofnumericalresultssupportedbyadditionalexperimentalresults.3.4.Ewectofpressure

Thee!ectofthebend-stretchprocessonthecrosssectionbecomesmorepronouncedifwecomparethe\"naldistortionpro\"lesoftwotubesformedatthesametensionwithandwithoutinternalpressure.Fig.22ashowssuchacomparisonof󰀃*/hpro\"lesfortwotubesformedat

󰀰

¹/¹\"0.95;onehadnointernalpressure,whileforthesecond,aconstantpressureof󰀱󰀰

P\"2.28barwasapplied.Inthepresenceofpressure,theaveragedistortionwasreducedfrom󰀲

!15.2to!3.1%.(ThedramaticimprovementbecomesevenmoreevidentinthecomparisonofphotographsofthetwocrosssectioninFig.3ofPartII.)Thespringbackwasmildlyincreasedbythepressurefrom3.9to4.9%asevidencedinFig.22b.

Becauseofthisdetrimentale!ectoftensiononcross-sectionaldistortion,untilrecently,stretchformingwasnotfavoredforcoldformingtubularmembers.Atthesametime,themethodisattractivebecauseofthehighspeedwithwhichpartscanbemanufactured.Itwasfoundthatmodestvaluesofinternalpressure(usuallycompressedairatlessthan3bar)canhelpreducedistortionofthecrosssection[18].Becausethepartisalreadyplasticizedbypre-stretching,theselectionoftheoptimummagnitudeofpressureiscriticalasexcessivepressurecanresultinbulgingofthetube.ThiswillbefurtherdiscussedinPartIIinthelightofanalyticalparametricstudies.

1308J.E.Milleretal./InternationalJournalofMechanicalSciences43(2001)1283}1317

Fig.22.E!ectofpressureontubebentinhorizontalmode:(a)distortion;(b)springback.

Inthecourseofthisstudy,itwasobservedthatinternalpressureisnotsuitableforformingtubesofallcrosssections.Infact,whenformingtubeswithtallsections,internalpressurecanaggravatethedistortion.Insuchcases,vacuumwasfoundtobebene\"cial.ThisisillustratedinFig.23fortubesbentwiththetallsidesverticalatatensionof¹/¹\"0.62.The\"rstonewasbentat

󰀱󰀰

atmosphericpressurewhilethesecondhadavacuumof0.97barappliedtoit.Thee!ectofthevacuumonthe\"naldistortionvariable󰀁isquitedramaticasitisreducedfrom11.4to1.05%(Fig.23a;seealsopicturesofcrosssectionsinFig.7ofPartII).Inaddition,vacuumdecreasedthespringbackfrom6.4to5.7%(Fig.23b).AdditionalresultsforloadinghistorieswhichincludevacuumwillbediscussedinPartII.4.Summary

InthisPartIofthistwo-partseriesofpapers,amodelbend-stretch-pressuretestingfacilityhasbeendescribed.Thefacilityisoperatedbyonepneumaticandtwoservohydraulicsystems.Itdi!ers

J.E.Milleretal./InternationalJournalofMechanicalSciences43(2001)1283}13171309

Fig.23.E!ectofvacuumontubebentinverticalmode:(a)distortion;(b)springback.

fromcomparablecommercialmachinesinthateachofthethreemodesofloadingisoperatedunderfeedbackcontrol.Thisallowscomputercontroloftheprocess,anda!ords#exibilityintheloadinghistorythroughwhichthepartistaken.Anadditionalfeatureofthetestingfacilityisabiaxialtransducerwhichmonitorscross-sectionalchangesofthepartduringforming.Theplanarformingprocesswasmodeledbyapproximatingthetubeasanonlinearelastic}plasticbeamwhichcanundergolargerotations.Themodelwasshowncapableofreproducingaccuratelytheloadinghistoryexperiencedbydi!erentsectionsalongthelengthofthepartduringtheformingprocess.Representativeresultsfromformingexperimentsinvolvingrectangularaluminumtubesrepre-sentativeofthoseusedinautomotiveapplicationswerepresented.Theresultswereusedtodiscussthee!ectoffriction,tensionandpressureonthecross-sectionaldistortion,springbackandnetelongationofthepart.Therepeatabilityoftheexperimentalresultswasingeneralfoundtobegood.ThemajorconclusionsfromtheexperimentsarelistedintheintroductionofPartIIofthisstudythatfollows.There,arelativelysimple,two-dimensionalmodelispresentedwhichiscapableofpredictingthecross-sectionaldistortionresultingfrombend-stretch-pressureforming.Resultsfromthemodel,supportedwithadditionalexperimentalresults,willbeusedtoillustratestrengths

1310J.E.Milleretal./InternationalJournalofMechanicalSciences43(2001)1283}1317

andweaknessesofcustomaryandalternateforminghistories.TheconclusionsdrawnfromthestudyasawholeappearattheendofPartII.

Acknowledgements

Theauthorsacknowledgewiththanksthe\"nancialsupportoftheNationalScienceFoundationthroughgrantsDMI-9522774andDMI-9734947inconjunctionwithAlcoathroughtheGOALIprogram.SpecialthanksgotoF.PourboghratwhocoordinatedthecooperationwithAlcoaduringtheperiodof1996}1998.Any\"ndings,conclusionsandrecommendationsexpressedhereinarethoseoftheauthorsanddonotnecessarilyre#ectthoseofthesponsors.

AppendixA.NumericalsimulationofformingprocessA.1.Problemformulation

Numericalsimulationsofthestretchformingprocessasperformedinourmodelfacilitywereconductedthroughanonlinearbeamanalysis(seealsoRef.[16]).Thedieisassumedtoberigidandtohavearadius󰀉.Thebeam,withaninitiallengthof2¸,isassumedtodeformsymmetricallyaboutthemid-spanwithoutcross-sectionaldistortion(seeFig.24a).AnendloadFisappliedatBinsuchawaythatitslineofactionalwayspassesthroughpointCrepresentingthepivotofthehorizontalactuator.Duringtheformingprocess,aportionofthetube,sectionOA,comesintosmoothcontactwiththedie.TheremainingsuspendedsectionABisanalyzedthroughthefollowingsmallstrain,largede#ectionbeamequations[26]:

dy

\"(1#e)cos󰀇,dS

dy

\"(1#e)sin󰀇,dSd󰀇

\"󰀈,dSdH

\"0,dSd<

\"0,dS

dM

\"(1#e)(Hsin󰀇!(A.1)

J.E.Milleretal./InternationalJournalofMechanicalSciences43(2001)1283}13171311

Fig.24.Problemgeometry.

Here(x,y)arethecoordinatesofpointsonthemid-surfaceofthebeam,󰀇istheanglebetweenthemid-surfaceofthebeamandthex-axis,󰀈isthecurvatureandethestrainofthemid-surfacewhileSandsaretheundeformedanddeformedlengthsofthemid-surface.HandVareforcesde\"nedin

1312J.E.Milleretal./InternationalJournalofMechanicalSciences43(2001)1283}1317

Fig.24bandMisthemomentactingonthebeamcrosssection.TheaxialforceTandshearforceQcanberelatedtoHandVasfollows:

¹\"Hcos󰀇#hh

󰀅\"e#󰀆󰀈,!)󰀆).

22

(A.3)(A.2)

Thematerialstress}strainrelationshipisbasedonamultilinearrepresentationoftheresponse

measuredinuniaxialtensiletestsonstripscutfromthetubestested.TandMaregivenby

¹\"

󰁁󰁁

Thefollowingincrementalrelationshipsrelate(¹,M)and(e,󰀈):

󰁐

󰀊dAandM\"

󰁐

󰀊󰀆dA.(A.4)

󰁐󰀃󰀄

󰁐

¹M\"󰁁

󰁁

Thefollowingboundaryconditionscompletetheformulation:

x\"󰀉sin󰀇,y\"󰀉(1!cos󰀇),󰀇\"s/󰀉,M\"M(󰀉,e),(A.6)󰁁󰁁󰁁󰁁󰁁󰁁󰁁󰁁󰁁M\"0,F\"(H󰀲#<󰀲.

Theproblemissolvedincrementallyinthreephases.Initially,thetensionisincrementeduntilthedesiredvalueisreached.Subsequently,thetensionisheldconstant.Inthesecondphase,thecurvatureatthemid-spanisincrementeduntilitreachesthecurvatureofthedie.Inthethirdphase,thelengthofthetubeincontactwiththedie(s)isprescribedincrementally.Ineachphase,the

󰁁

resultanttwo-pointboundaryvalueproblemisdiscretizedthrougha\"nitedi!erenceschemeandsolvedthroughNewton'smethodusingtheIMSLpackageBVPFDroutine(seeRefs.[27,28]).Aftereachincrement,thepositionofthedieiscalculatedthrough

<(b!x)

.d\"y# H

A.2.Typicalresults

Theperformanceofthemodelwillbeillustratedthroughanexampleofatubeformedwiththelongsideshorizontalatatensionof¹/¹\"1.08andaninternalpressureof2.28bar(thebeam

󰀰

modeldoesnotaccountforthee!ectofpressure).Inthiscase,anadditionalfeaturewasaddedtotheexperimentalsetupwhichenabledustomonitorthelengthoftubeincontactwiththedie.

(A.7)

󰀁d󰀊dAd󰀅d󰀊

󰀆dAd󰀅

󰁐󰁐

󰁁󰁁

d󰀊

󰀆dAd󰀅

d󰀊

󰀆󰀲dAd󰀅

󰀂󰀃󰀄

e󰀈.(A.5)

J.E.Milleretal./InternationalJournalofMechanicalSciences43(2001)1283}13171313

Aspecialresistancepaper,whichhasregularlyspaced,continuous,conductivelinesofsilverdepositedononeside,isbondedtothedie.Thetubeismountedtotheformingmachineintheusualmanner,buttheresistancepaperandthetubeareconnectedformingtwoconstantcurrentcircuitsasshowninFig.25a.Thecircuitsarecalibratedpriortotheexperiment.Asthetubedeformsandcomesintocontactwiththedie,thecontactlengthssandsaredeterminedfrom

󰁁󰀱󰁁󰀲

thechangesinvoltageineachofthetwocircuits(technique\"rstusedinRef.[16]).

ThecontactlengthsrecordedareplottedasafunctionofthediedisplacementinFig.25b.Contactlengthisseentobesymmetricaboutthemid-span,indicatingthatthebeamdoesnotslideonceinplace.Thecontactlengthsexhibitaninitialnonlinearityduringwhichtheneighborhoodofthemid-spanconformstothecircularshapeofthedie.Subsequently,contactlengthincreasesnearlylinearlywiththediedisplacement.

IncludedinFig.25barepredictionsofthecontactlength(s)fromthenonlinearbeammodel.Thecalculatedcontactlengthremainszerountilthemid-spanconformstothecurvatureofthedieatadisplacementof2.75in(70mm).Subsequently,thecontactlengthgrowsessentiallyinthesamemannerasrecordedintheexperiment.Thesti!erinitialresponseexhibitedbythemodelispartlyduetothemathematicallysharppointcontactandpointreactioninherenttobeamtheory,butalsoduetothefactthatthepresenceoftheinternalpressurewasneglectedinthebeammodel.(Thelongwallsexhibitedaninitialmildbulgingduethepressurewhichmadetheinitialcontactproblemthreedimensional.)

Fig.25.(a)Contactlengthmeasurementsystem;(b)contactlengthevolutionduringforming.

1314J.E.Milleretal./InternationalJournalofMechanicalSciences43(2001)1283}1317

Fig.26.Calculateddeformedbeamcon\"gurations.

Anotherviewoftheevolutionofcontactbetweenthetubeandthedieisshowninasetof10calculatedbeam/diedeformedcon\"gurationsinFig.26.Themainfeaturesofthecon\"gurationsaresimilartothoseoftheexperimentdiscussedinSection3.

Figure27ashowsacomparisonbetweenthemeasuredandcalculateddieforce-displacementresponse.Thepredictionsareinverygoodagreementwiththemeasuredresults.Avariablethoughttoplayaroleincross-sectionaldistortionisthelocalshearforceintheneighborhoodofthelifto!pointofthebeamfromthedie.Inthebeammodel,thisisidealizedasapointreactionforce(QinFig.24).Figure27bshowshowitsvalueevolveswiththediedisplacement.Initially,

󰁁

asthemid-spangraduallybendsandcomesintocontactwiththedie,Qgrowslinearly.At

󰁁

adiedisplacementof2.75in(70mm),themid-spanhasconformedtothediecurvatureandthepointoflifto!startstomoveoutwards.Qisseentoremainessentiallyunchangeduntilfull

󰁁

contactisachieved.(Notethatforlowervaluesoftension,Qgrowstosomedegreewithcontact

󰁁

length.)

Abyproductofthisformingprocessisanetelongationofthepart.ThisisillustratedinFig.28awhichshowsaplotoftheevolutionofaxialstrain(e,strainofmid-surfaceofbeam)withthedie

J.E.Milleretal./InternationalJournalofMechanicalSciences43(2001)1283}13171315

Fig.27.(a)Verticalforceduringforming;(b)evolutionofcontactshearforceduringforming.

displacementatalocationofS/¸\"0.38.Thebeamdevelopsaninitialstrainof0.9%duetothepretension.Thepointmonitoredremainsinthesuspendedpartofthebeamuntild\"12.2in(310mm).Thesuspendedsectionbendsgraduallyinducingtheslowgrowthintheaxialstrainseeninthe\"gure.Asitcomesclosertobeincontactwiththedie,thelocalcurvaturegrowsfasterandtheaxialstrainexperiencesasuddensurge.Oncecontactisachieved,theelongationceasestogrow.Again,becauseoftheassumptionsinherenttobeamtheory,contactisamathemat-icallysharpeventresultinginthesharpdiscontinuityinslopeinthevalueofthestrainseeninthe\"gure.

AnalternativeviewofthiscouplingbetweentheaxialstrainandthebendingdeformationisshowninFig.28bwherethestrainvariableeatthemid-span(S/¸\"0)andatS/¸\"0.38isplottedagainstthecurvature󰀈atthecorrespondinglocations.Interestingly,therelationshipbetweeneand󰀈atthetwopointsisessentiallyidentical.Theaxialstraingrowsessentiallylinearlywith󰀈andstopsgrowingoncebendingdeformationceases.

1316J.E.Milleretal./InternationalJournalofMechanicalSciences43(2001)1283}1317

Fig.28.CalculatedaxialstrainatS/¸\"0and0.38duringforming.

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