1、4ifaacJ.,.“1I,“-1 FORAERONAUTICS.TECHNICAL NOTE 9726COMPRESSIVEAND TORSIONALBUCKLINGOF THIN-WALL CYLINDERS“INYIELDREGIONByGeorgeGerardNewYorkUniversityWashingtonAugust19565 . s. . . . .-, - . . . . . . . . . . - . . . . . . . _J -IProvided by IHSNot for ResaleNo reproduction or networking permitted
2、without license from IHS-,-,-NATIONALADVISORYCOMIZTEE!rEcHNIcmmcl?l3726COMPRESSIVEANDTORSIONALBUCKUNGOFTEIN-WALGCYEDUMMINYIEIDREGIONByGeorgeGerardExJmARY”Basedonassumptionswhichham ledtothebestagreemexrtbetweentheoryandtestdataoninelasticbucklingoffl.atplates,ageneralsetofequilibriumdifferentialequa
3、tionsfortheplasticbucklingofcylind-ers hasbeenderived.Theseequationshavebeenusedtoobtnsolu-tionsforthecompressiveandtorsionalbucklingoflongcylindersintheyieldregion.Testdataarepresentedwhichindicatesatisfactoryagreementwith .thetheoreticalplasticity-reductionfactorsinmst cases.Whereadifferenceinresu
4、ltsexists,testdatasxeinsubstantiallybetteragree-mentwiththeresultsobtainedbyuseofthemaximum-shearlawratherthantheoctahedral-shearlawtotransformaxial.stress-straindatatoshearstress-straindata.INTRODUCTIONTJlel.asticCcapressiveBuckmlg.ofFlatPlatesThestateofknowledgeupto1936concerninginelasticbucklingo
5、fplatesandshellshasbeensumarizedbyThnoshenkoinreference1. Themaineffortswereconcernedwithattemptstomodifythevariousbend5ng-momenttermsoftheequilibriumdifferentialegpationsbytheuseofsuitableplasticitycoefficientsdeterminedfrcmexzbnentaldataoncolumns. Althoughsuchsemiempirical.effotismetwithareasonabl
6、edegreeofsuccess,thetheoretics.determinationofplasticity-reductionfactorsforflatplateshasbeenachievedwithinrecentyearsastheresultofthedevelopmentofinelastic-bucklingtheory.Becausesuchdevelopmentsarerecentandformthebackgroundfortheinelastic-bucklingtheoryforshellsdevelodherein,thefollowingdiscussionc
7、oncerningtheassump-tionssudresultsofthevarioustheoriesispresentedinsomedetailDifferentinvestigatorshaveuseddifferingassumptionsinthedevelopmentoftheirtheories.Themajorassmionsuuderl.yingeachofthesetheories= giveninthefol.lm *bk. . . .- ._ _ . - - . , . . . . _ - . _Provided by IHSNot for ResaleNo re
8、production or networking permitted without license from IHS-,-,-2 NACATN3726Investigator stress-strainlaw PlasticitylawBucklingmodelBijlaard Incrementaland octahedral Nostrain(ref.2) deformationtyps, Shesx reversalv instantaneousDeformationp, octahedral7=Y3) Strainv=0.5 Shesx reversalEandelmanandPrs
9、gerIncrementaltype,(ref.4) octahedralStrainv instmtaneous Sh!em? reversalStowell Deformationpe, octahedral Nostrain(refs.5ticregion,thehWS are obtained:(A4)Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NACATN3726.19(A5)(A6)lhelastic-BucklingConside
10、rationsAlloftheforegoingassumptionsformthebasisforsolutionofplasticiproblemsingeneral.Forthespecificproblemofinehsticbuckling,itisnecessarytomakeanadditionalassumptionconcerningthestressdistributionattheinstantofbuckling.lh?omthestandpointofclassicalstabilitytheory,theequilibriumdifferentialequation
11、sareformulatedonthebasisthatatthebucklingloadanexchangeofstableequilibriumconfigurationoccursbetweenthestraightformandtheslihtlybentform.Sincetheloadremainscon-stantduringthisexchange,a strainreversalmustoccurontheconvexside,and,therefore,thebucklingmodelleadingtothereduced-modulusconceptforcolumusi
12、scorrecttheoretically.Practiticolumnsandplatesinvariably.containinitial.imperfec-tionsandthereforeaxialloadingandbendingproceedsimultaneously.Sinceinthepresenceofrelativelylargeaxialcompressivestressesthebendingstresses generslldysmal!d.,nostrainreversalwouldbeexpectedtooccurandtheincrementalbending
13、stressesintheinelasticrangearegivenbythetangent-modulusmodel.However,thebentformistheonlystableconfigurationinthiscaseandthereforeuseofequi-Ubrim equationsbasedonperfectcolumns,plates,orshellsisclearlyunjustified.Partiallytoremuvethisdifficulty,StoweSlhasassumedthatthestraightformoftheplateorcolumni
14、sstableuntilbucklingoccurs(ref.5).Atbuckling,infinitesimal.bendingisassumedtoproceedsimultaneouslywithacorrespondinginfinitesimalincreaseinsxiiiloadingsothattheplateisnotsubjettedtoa strainreversalandremainsinelastic.Againthismodelposesanessentialdifficultysinceclassicalstabilitytheoryisbasedontheas
15、sumptionthattheaxiallosdingramdllsconstantduringthebucklingprocess.lhaendixB,inwhichtheequilibriumequationsareconsidered,anattemptismadetoremovethisdifficultybyShOWiIlgthat the infin-itesimalincreaseinloadassociatedwiththeno-strain-reversalmodelcontributeshigherordertermsthanthosegenerallyconsidered
16、intheeqyilibrimequation.Thisisbyvirtueoftpefactthattheaxialloadsaremul.tiplJedbyfirstorsecondderivativesofthedisplacementsand. . . -_ _ _ . . - . Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-20 IIACA!r!N3726thereforeproductsoftheincremental.loadin
17、creaseandthesederiva-tivesresultinsecond-orderterms. IncrementalForcesandMoments .Nhenbucld.ingoccurs,thedisplacementsvaryslightlyfromtheirvaluesbeforebuckling.Theresultingstrainvariationsarisepsrtlyfranvariationsofmiddle-surfacestrainsmd partlybecauseofbendingstrains.Theseresultingvariationsofstres
18、seshavebeenconsideredbyIlyushin(ref.3)smdStowell(ref.5).Usingtheassumptionthatnopartoftheplateisunloaded,%owellhasderivedthevariationsofthemomentsduringthebum process.- -iatio tithefiu-surfaceforcescanbederiveddirectlyfranthiswork.Whenthevariationsoftheforcesandmomentsprimes(),thefollowingrelationsa
19、pplytobuckling:I?x= 1!B 161i-(1/2)Ae2-% =BJ%2 + /A21% -=BxY1! 363-(1/2)A51e1-% = -DAl% -I-(1/2)Au-MY= -DA21+ (1/2)Xl-Mw 1! - (1/2)A31Xl-=- 3X3 equations(A7)to(A12),G1 aaredenotedbyplasticplates11(WA3262durin$(A7) (A8)“ (A9)(no)(Ku)WM32* (Au)E2 aremiddle-acenormalstrainvariations 63 isthemiddle-surfa
20、ceshearstrainvariation; .xlti=thechangesin- -is*_efitwist. Furthermore,theplasticitycoefficientsaredefinedasfoows: u. . -. . Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-MICATN3726 21. whereAl=1- (cLux2/.a%+law=ola#2Raxa2v =ax?27(C5)(c6)(C7)Byperf
21、ormingtheoperationapx onequation(C7)andthenusingegpa-tion(C5),a singleequilibriumequationin w isobtained:(c8)A solutiontoequation(c8)canbewritteninthefollowingform:w =wmsin(/A) (C9)whereh= Z/mUponsubstitutingtheappropriatederivativesofequation(C9)intoequation(c8)andusingthedefinitionsof D, B,andAl g
22、ivenbyequations(A15),(A14),and(Cl),respectively,andtherelationNx= awt,thefollowingnontrivialsolutionisobtained:()Est2 3%2+ % A2cl?=- 9 4+4EnR +% 4 4.%3(Clo).- .-. . z - ._Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-28 NACATN3726Thecylinderisconsi
23、defidtobelong,sothatmanywavesformalongthelengthandthereforeUa canbeconsideredasacontinuousfunctionof X. Byminimizingequation(C1O)length,l/2-()2E t%=S SREBcancorrespondinghalfwavelengthofthebuckleswithrespecttothewave(Cll)Fortheelasticcase,thecorrespondingsolutionsare:“=i+ - .2)l-124i-A=t)l/2l+ -Bycm
24、psringthecoefficientswhichappearinequations(Cll)(c12)(C13) -(C14)(C13),itcanbeobservedthatthecoefficientinequation(C1.1)beobtainedbysubstitutingaVS2ueof1/2for Ve inequation(C13).Thus,thefollowi.ngrelationscanbewrittenwhichsreexactintheelasticandfullyplasticrangesandresultinanexcellentdegreeofapproxi
25、mationintheinelasticrange:(C15)(c16) -,.- .Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NACATN3726 29APPENDIXDTORSIOBUCHJNGOFALONG -Foralongcircularcylindersubjectedtotorsionalmcmentaattheends,ax= =O. Thevalueof ui whichappearsinequations(A13)isgi
26、venbyequation(Al).Forthiscase,(ZJ_= (3)% (Dl)TheplasticitycoefficientsreducetoConsemntly,thefOllowinga%ax2A15=3=0 Itheequilibriumeressions:JA3=Eequations(eqs.(B5)to(B7)reduceto,J?3R:?32+(2+A3)a% , aW o4 -Rm+2R ax=a% .0W RhxaO(D2)(D3)(D4)(D5)(D6). - . . . -Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-
