機(jī)械設(shè)計(jì)柱屈曲

機(jī)械設(shè)計(jì)柱屈曲第一頁，共十五頁，編輯于2023年，星期六Allowablestressisnotexceeded:Afterthesedesigncalculations,maydiscoverthatthecolumnisunstableunderloadingandthatitsuddenlybecomessharplycurvedorbuckles.FailureofColumnsDeformationfallswithinspecifications:Thecomponentunderacyclicloadingshouldhaveafatiguelifehigherthanthespecified:N>NspecStaticStrengthStaticStiffnessFatigueStability2第二頁，共十五頁，編輯于2023年，星期六DifferentialEquationforPin-EndedBeamsConsideranaxiallyloadedbeam.Afterasmallperturbation,thesystemreachesanequilibriumconfiguration.SolutiontotheDifferentialEq.

BendingMomentDifferentialEquation

SolutionColumn3第三頁，共十五頁，編輯于2023年，星期六

Euler’sFormulaforPin-EndedBeams

SolutionBoundaryConditionA=0y=0:nobucklingA≠0,thenon-trivialsolutionforbuckling:orCriticalLoad:n=1

IfP<Pcr

A=0y=0:nobucklingIfPPcr

y≠0

:bucklingoccurs4第四頁，共十五頁，編輯于2023年，星期六ExtensionofEuler’sFormulatoCantileverBeamsAcolumnwithonefixedandonefreeend,willbehaveastheupper-halfofapin-connectedcolumn.ThecriticalloadingiscalculatedfromEuler’sformula,Canwegetthesameresultfromsolvingadifferentialequation?Yes,BUT5第五頁，共十五頁，編輯于2023年，星期六OtherBeamsk=0.5k=2k=0.7k=16第六頁，共十五頁，編輯于2023年，星期六CriticalStressforColumns:Euler’sEquationCriticalLoad

CriticalStress,Euler’sEquation

ForacircularcolumnwitharadiusofR,7第七頁，共十五頁，編輯于2023年，星期六E=200GPaLet’splotscrv.s.Le/rforE=200GPaandsy=250MPa.CriticalStressforColumns:Johnson’sFormulaTheportionoversyisuseless,sincethestaticstrengthneedsalsotobesatisfied.Itwouldappearthatifweusedthecombinecurveasthecriticalstresscurve,itwouldcovertheentirerangeofcompressiveproblemsfromtheshortesttothelongestcompressionmember.89sy(Le/r)1ysyseplel0.2%However,theaboveconclusionisnottrue,as(a)theendconditionsmaynotthesameastheidealendcondition,(b)Eisreducedwhensapproachessy,…Itturnsouttherealcurveforashortcolumnwilllookliketheblueoneshowninthefigure.Johnson'sFormulaforthebluecurve8第八頁，共十五頁，編輯于2023年，星期六CriticalStressforColumns:SummaryInsummary,thefollowingequationcanbeusedtocalculatethecriticalstressforcompressivememberswherescrSR1LongColumnsIntermediateColumnsDifferentcodes(ASME,ASCE,etc.)mayusedifferentequationsfordifferentmaterials,etc.Checkwithappropriatecodesandstandardswhenyoudesign.9第九頁，共十五頁，編輯于2023年，星期六scrsysy2CriticalStressforColumns:OtherFormulaInengineeringapplication,thefollowingequationmayalsobeusedwhere10第十頁，共十五頁，編輯于2023年，星期六Example1Thehydrauliccylindershownhasa3-inboreandistooperateatapressureof800psi.Withtheclevismountshown,thepistonrodshouldbesizedasacolumnwithbothendspinnedforanyplanebuckling.TherodismadeofforgedAISI1030steelwithoutfurtherheattreatment.Useasafetyfactornd=3todeterminethediameteroftherodifthecolumnlengthis60in.Repeatpart(a)butforacolumnlengthof18in.ProblemSolutionStrategyCalculatethecompressiveaxialforce.Assumealongcolumn,anduseEuler’sEquationtodesignCheckthe“l(fā)ongcolumn”assumption.Iftheassumptioniscorrect,stopandthatisthedesign.Iftheassumptioniswrong,usetheJohnsonFormula.Obtainmaterialproperties.11第十一頁，共十五頁，編輯于2023年，星期六SolutiontoEx.1(a)MaterialCompress-iveforce

AssumeEuler,k=1

CheckEuler

Radiusofgyration:Foracircularcolumn,

UsingEuler’sformulaisO.K.d=1.5inissatisfactory.12第十二頁，共十五頁，編輯于2023年，星期六SolutiontoEx.1(b)AssumeEuler,k=1CheckEuler

Johnson’sFormula

<End>d>0.861inUsed=0.875in13第十三頁，共十五頁，編輯于2023年，星期六EccentricLoading;TheSecantFormulaEccentricloadingisequivalenttoacentricloadandacouple.Bendingoccursforanynonzeroeccentricity.Questionofbucklingbecomeswhethertheresultingdeflectionisexcessive