Engineering Basic Mechanics I Statics 工程基礎力學Ⅰ 靜力學 課件 Chapter 14 Stability of Columns

StaticsStaticsofdeformablebodyChapter14

StabilityofColumns14.1Introduction14.2Criticalcompressiveforceofslendercolumnwithpinsupport14.3Criticalcompressiveforceofslendercolumnwithotherconstraints14.4Diagramofcriticalstress14.5CalculationofstabilityofcolumnsContents14.1IntroductionTheproblemofwhetherthecolumncanmaintaintheoriginallinearbalancestateiscalledthestabilityproblemofthecolumn.disturbanceforce(c)crF<FcrF=FcrF>FTheprocessofthelinearequilibriumofthecolumnfromstabletounstableiscalledbuckling.TheextremevalueFcrofpressureF,isalsoknownasthecriticalpressureorcriticalforce.(b)(a)disturbanceforcedisturbanceforceOtherformsofstabilityproblemsAthin-walledcylindersubjectedtouniformexternalcompressiveforcebecomesellipticalduetoinstability.Abeamofnarrowrectangularsectionbendslaterallyduetoinstability.F14.2CriticalcompressiveforceofslendercolumnwithpinsupportylFFxxvBytakingthecartesiancoordinatesystemshown,thedeflectionofanycrosssectionatadistancexfromtheoriginisv,andthebendingmomentis（a）whereFisanabsolutevalue.theapproximatedifferentialequationforthedeflectioncurveofcolumnis

（b）let，andthenequation(b)canbewrittenas（c）Thegeneralsolutionofthisdifferentialequationis:

（d）Boundaryconditions:

a.Bysubstitutingitintoequation(d),wegetb=0，（e）

b.v(l)=0Similarly,wecangetv(0)=0

（f）Fortheequation

(e),Ifa=0,Theremustbev=0.Thatmeansthedeflectionatallpointsontheaxisofthecolumnisequaltozero.Thiscontradictsthefactthatthecolumnremainsinequilibriuminaslightlybentstate.So,wegetsin(kl)=0.Thevalueofklthatsatisfiesthisconditionshouldbe

or（g）

whennandFiszero,itismeaningless.Whenn=1,thevalueofFistheminimum.So,FcroftheslendercolumnisThisistheformulaforcalculatingthecriticalcompressiveforceofaslendercolumnwithpinsupportatbothends,knownasEulerequation.TheFcrisproportionaltotheminimumbendingrigidityEIofthecolumnandinverselyproportionaltothesquareofthebarlengthl.Thismeansthattheslimmerthebaris,thesmallerthecriticalcompressiveforceis,andthemoreunstablethebaris.UndertheactionofFcr,wehavek=π/l.Substitutionitintoequation(e),weget

（h）Theaboveequationshowsthatthedeflectioncurveofaslendercolumnwithpinsupportatbothendsisahalf-wavesinecurve.Ifwemakex=l/2,andsubstituteitintoequation(h),wegetWhereaisthedeflectionofthemidcrosssectionofcolumn.Thereisnodefinitevalueofa.Thisisbecauseinderivingtheequationforthedeflectioncurveofthecolumn,theapproximatedifferentialequationforthebendingdeflectioncurveofthecolumnisbasis.Byexactdifferentialequationforthedeflectioncurve,WecanobtainthedeterminedvalueandthetheoreticalrelationshipbetweenthemaximumdeflectionandpressureF,asshowninthecurveOABinthefigure.Foractualmembersundercompression,duetofactorssuchasheterogeneityofmaterial,initialcurvature,orslighteccentricityduringloading,microbendingdeformationactuallyoccursbeforethepressureFreachesthecriticalpressureFcr,whichcanberepresentedbythecurveODinthefigure.FcrF14.3CriticalcompressiveforceofslendercolumnwithotherconstraintsForslendercolumnswithotherconstrainedsituations,

ThisisthegeneralformofEulerequation.

μisthelengthfactorofthecolumnunderdifferentconstraints.μlistheequivalentlength.Oneendfree，oneendfixed:

＝2.0Oneendhinged，oneendfixed:

＝0.7Bothendfixed:

＝0.5Bothendhinged:

＝1.0ExampleAcastironcolumnofcircularcrosssectionwithonefixedendandonefreeend.Ithasalengthl=3m,adiameterd=0.2mandamodulusofelasticityE=120GPa.CalculatethecriticalcompressiveforceofthecolumnfromEuler'sformula.Solution:CheckTable15-1forthelengthfactorμ=2,andthenthemomentofinertiaofthesectionisTherefore,thecriticalcompressiveforceis14.4Diagramofcriticalstress

1.Criticalstressandslendernessratiocriticalstressσcr

(a)Introducingtheradiusofgyration,weget

(b)CitatefollowingmarkTheequation(b)forthecriticalstresscanbewrittenas

whereλisadimensionlessquantity,calledtheslendernessratioorflexibilityofthecolumn.2.ApplicabilityofEuler'sformulatheconditionsfortheapplicabilityofEuler'sformulaare（c）Citatefollowingmark

Theconditions(c)fortheapplicationofEuler'sformulacanbewrittenas

Thistypeofbariscalledlargeflexibilityorslendercolumn.isonlyrelatedtothemechanicalpropertiesofthematerial,anddifferentmaterialshavedifferentvalues.TakingQ235low-carbonsteelasanexample，wehave，，andthen

ThisindicatesthattheEulerequationcanonlybeappliedtocompressionrodsmadeofQ235steelwhentheflexibility.3.CriticalcompressiveforceabovetheproportionallimitForthecommoncolumninengineering,TheircriticalstressexceedstheproportionallimitandcannotbecalculatedbyEulerequation.Weoftencalculationcriticalstressbytheempiricalformulaestablishedonexperiments,

whereaandbaretheconstantsrelatedtothemechanicalpropertiesofthematerial.Forcolumnsmadeofplasticmaterials:

or（d）theminimumvalueofslendernessratio:

Forcolumnsmadeofbrittlematerials:

Therefore,theempiricalformula(14-18)isapplicableunderλs<λ≤λp(orλb<λ≤λp).Thistypeofbariscalledmedium-flexibilitycolumnormedium-lengthcolumn.(Strengthissue)Whenλ≤λs

or≤λb,thecolumniscalledsmall-flexibilitycolumnorshortcolumn.Becauseitscriticalstresswillreachorexceedtheyieldlimitofthematerial,materialwillbedamagedduetoinsufficientstrengthwithoutbuckling,Forshortcolumnsmadeofplasticmaterials,iftheyarestillformallytreatedasastabilityproblem,theyieldstressσs

shouldbeusedasthecriticalstress.Insummary4.DiagramofcriticalstressByplottingtherelationshipbetweenthecriticalstressandtheslendernessratioofthecolumnwithinthethreeslendernessratiorangesinthecartesiancoordinatesystem(),wecangetthecriticalstressdiagramofthecolumn.sABCDOlplslssps2Eplcrs=2crabl=-scrs=ss5.Parabolicformulaanditsstressgeneraldiagram

Whenthecriticalstressexceedstheproportionallimit,weexpresstheparabolarelationshipbetweenthecriticalstressσcrandslendernessratioasfollows

China'ssteelstructurecodeprovidesaparabolicformulaestablishedbyourownexperiments:ForthecommonlyusedstructuralA2andA3steels,manganese16steelExampleAbarincompression,fixedatbothends,madeofA3steel,withacross-sectionalareaof32×102mm2.Calculatethecriticalloadwhenthecrosssectionisrectangularandcircular,respectively.F3msolution（1）rectangularsection

Minimumradiusofgyrationofthecrosssection:b2bSlendernessratioofthecolumn:

Thecolumnisaslendercolumn.F3mb2b（2）circularsectionby

weget

radiusofgyrationofthecrosssection:F3mb2bSlendernessratioofthecolumn:

Thecolumnisamedium-lengthcolumn.F3mb2bDiscussionthecriticalcompressiveforceisgreaterforcircularsectionsthanforrectangularsections,i.e.circularsectionsaremoreresistanttoinstabilitythanrectangularsections.tocalculatethecriticalcompressiveforce,itisnecessarytofirstcalculatetheslendernessratioofthecolumn,andthentoselecttheappropriateformula.Thestabilityconditionofthecolumnis

orexpressedusingsafetyfactorforstability

wherenwistheworkingsafetyfactorforstabilityofthecolumnand[nw]istheallowablesafetyfactorforstability.14.5CalculationofstabilityofcolumnsReferencevaluesof[nw]forseveralsteelcolumnsarelistedbelow.Columnsinmetalstructures---[nw]=1.8～3.0Screwsformachinetools---[nw]=2.5～4.0Tappetforlowspeedengines---[nw]=4～6Pistonbarforgrindingmachinecylinders-[nw]=4～6Liftingspirals---[nw]=3.5～5Note:Whenthereisapartialsectionweakeningofthebar,suchasoilholes,screwholes,etc.,asthecriticalcompressiveforceofthebarisdeterminedbythebendingdeformation,thelocalsectionweakeninghaslittleeffectonthevalueofthecriticalcompressiveforceandcanbeneglectedinthestabilitycalculation.Allcross-sectionalareasandminimummomentsofinertiaarecalculatedfortheunweakenedcrosssection.Strengthcheckmustbecarriedoutforcolumnwithpartialsectionwea