1CHAPTER10STABILITYOFCOMPRESSEDCOLUMNS2CHAPTER10STABILIZATIONOFCOMPRESSIVECOLUMNS§10–1CONCEPTSOFSTABILITYOFCOMPRESSEDCOLUMNS§10–2EULER’SFORMULAOFTHECRITICALFORCEOFSLENDERCOMPRESSEDCOLUMNS§10–3CRITICALSTRESSOFCOMPRESSEDCOLUMNSASSTRESSEXCEEDSPROPORTIONALLIMIT§10-4STABILITYCHECKANDREASONABLESECTIONOFCOMPRESSEDCOLUMNS3§10–1CONCEPTSOFSTABILITYOFCOMPRESSEDCOLUMNSTheload-carryingcapacityofstructuremembers：①Strength②Rigidity③StabilitySomestructuremembersinengineeringhaveenoughstrengthandrigiditybuttheyareunabletoworksafelyandreliably.STABILITYOFCOMPRESSEDCOLUMN4PSTABILITYOFCOMPRESSEDCOLUMN51、Stableandinstableequilibrium：1).instableequilibriumSTABILITYOFCOMPRESSEDCOLUMN62).StableequilibriumSTABILITYOFCOMPRESSEDCOLUMN73).StableandinstableequilibriumSTABILITYOFCOMPRESSEDCOLUMN82、Lossofstabilityandthecriticalpressureofthecolumn：1).Idealcompressivecolumns：thematerialisabsolutelyideal；theaxisisabsolutelystraight；thecompressiveforceisabsolutelyalongtheaxisofthecolumn.2).Stableandinstableequilibriumofcompressivecolumns：STABILITYOFCOMPRESSEDCOLUMNInstableequilibriumStableequilibrium93).lossofstabilityofcompressedcolumn：4).CriticalpressureofcompressedcolumnsStableequilibriumInstableequilibriumCriticalstateCriticalpressure:PcrcorrespondingpressureSTABILITYOFCOMPRESSEDCOLUMNintermediatestate10§10–2EULER’’SFORMULAOFTHECRITICALPRESSUREOFSLENDERCOMPRESSEDCOLUMNS1、Criticalpressureforthecolumnwithtwohingedends:①bendingmoment：②ApproximatedifferentialequationofthedeflectioncurvePPxyPMSTABILITYOFCOMPRESSEDCOLUMNSupposethepressurehasreachedthecriticalvalueandthecolumnhasbeenintinybendingstateasshowninthefigure.Starttodeterminethecriticalforcewiththedeflectivecurve.PxL11③Solutionofthedifferentialequation：④Determinetheintegralconstants：ThecriticalforcePcristhesmallestpressureundertinybending，thereforeweonlytaken=1andthecolumnwillbendabouttheaxiswiththesmallestmomentofinertia.STABILITYOFCOMPRESSEDCOLUMNThatis122、Applicationrangeoftheformula：3、Euler’’sformulaofthecriticalpressureforthecolumnwithotherendconditions:1).Idealcompressivecolumns；2).Inlinearelasticrange；3).Theendsofthecolumnaresupportedbyhinges.—Lengthcoefficient（orconstraintcoefficient）Euler’’sformulaofthecriticalpressureforthecompressivecolumnwithtwohingedendsGeneralformofEuler’sformulaofthecriticalpressureSTABILITYOFCOMPRESSEDCOLUMN130.5lTable10–1Euler’’sformulaoftheslendercompressivecolumnundervariousconstraintconditionsSupportsTwohingedendsOnefreeendandonehingedendTwofixedendsOnefixedendandonefreeendTwofixedendsbutoneofthemismovablelaterally.TheshapeofthedeflectivecurveinlostofstabilityPcrABlEuler’’sformulaofthecriticalforcePcrLengthcoefficientμμ=1μ0.7μ=0.5μ=2μ=1PcrABlPcrABl0.7lCCDC:InflectionpointC、D:Inflectionpoint0.5lPcrPcrl2llC:InflectionpointSTABILITYOFCOMPRESSEDCOLUMN14Solution：Thedeformationofthecolumnisshowninthefigure.Theapproximatedifferentialequationofitsdeflectioncurveis:Theboundaryconditionsare:Example1Trytodeducethecriticalforceofthefollowingslendercolumnbytheapproximatedifferentialequationofthedeflectivecurve.PLxPM0PM0PM0xPM0STABILITYOFCOMPRESSEDCOLUMNLet15Inordertodeterminetheminimumcriticalpressure“k”mustbetheminimumvalueexceptzero,thatis：Thereforethecriticalpressureis：=0.5STABILITYOFCOMPRESSEDCOLUMNand16③CriticalpressureofthecolumnExample2Determinethecriticalpressureofthefollowingslendercompressivecolumn.=1.0，Solution：①Abouttheaxisy，twoendsarehinged：=0.7，②Abouttheaxisz，theleftendisfixedandtherightendishinged：yzL1L2yzhbxSTABILITYOFCOMPRESSEDCOLUMN17Example3Determinethecriticalpressureofthefollowingslendercompressivecolumns.Knowing:L=0.5mandE=200GPa.Solution：Fig.(a)Fig.(b)Fig.(a)Fig.(b)5010PLPL(45456)Equal-leganglesteelyzSTABILITYOFCOMPRESSEDCOLUMN18§10–3CRITICALSTRESSOFCOMPRESSEDCOLUMNSASSTRESSEXCEEDSTHEPROPORTIONALLIMIT1、Basicconcepts1).Criticalstress：Meanstressinthecrosssectionofthecompressivecolumninthecriticalstate.3).Flexibility：2).Criticalstressforslendercompressedcolumns：22lpsEcr=Thatis:STABILITYOFCOMPRESSEDCOLUMNRadiusofinertiaFlexibility(orslendernessratio)ofthecolumn194).Divisionoflargeflexibilitycolumn：2、Calculationofthecriticalstressforthecolumnswithmiddleorsmallflexibility1).Empiricalformulawiththelinearvariation①AsP<<S：STABILITYOFCOMPRESSEDCOLUMNisoneofthemiddleflexibilitycolumn.ThecolumnwithItscriticalstressmaybedeterminedbytheempiricalformula.Thecolumnswithiscalledthelargeflexibilitycolumn(orslenderclumn).ThecriticalpressureofthiskindofcolumniscalculatedbyEuler’sformula.Thecolumnswithiscalledmiddleorsmallflexibilitycolumn.ItscriticalpressureofthiskindofcolumncannotbecalculatedbyEuler’sformula.20③Totalfigureofthecriticalstress②AsS<：bss=a-sl

PPEspl2

=STABILITYOFCOMPRESSEDCOLUMNThecriticalstressofthiscolumnisitsyieldlimit.isoneofthesmallflexibilityThecolumnwithcolumn.

212)EmpiricalformulawiththeparabolicvariationThecommonforminthestructuralengineeringis：①AsP<<s：②Ass<：STABILITYOFCOMPRESSEDCOLUMNForA3steel,A5steeland16Mnsteel,AsThisformulaisusedtodeterminethecriticalstress.22Solution：Foronepieceofanglesteel：Aftertwopiecesofanglesteeliscombinedasshowninthefigure,ThereforethecriticalpressureshouldbedeterminedbytheempiricalformulayzSTABILITYOFCOMPRESSEDCOLUMNExample4Acompressivecolumnismadefromtwoequal-leganglesteelof56568.Itstwoendsarebothhinged.ItslengthisL=1.5m.ThepressureisP=150kN.TheanglesteelisA3steel.TrytodeterminethecriticalpressureandthesafetycoefficientofstabilitynstbyEuler’’sformulaortheempiricalformulawiththeparabolicvariation.withtheparabolicvariation.23SafetycoefficientSTABILITYOFCOMPRESSEDCOLUMN24§10–4STABILITYCHECKANDREASONABLESECTIONSOFCOMPRESSEDCOLUMNS1、Permissiblestressforthecompressedcolumnsinstabilization:1).Determinethepermissiblestressbythemethodofthesafetycoefficient:2).Determinethepermissiblestressbythemethodofthereductioncoefficient:2、Stabilityconditionofthecompressedcolumns:STABILITYOFCOMPRESSEDCOLUMNReductioncoefficient.Itisrelatedtomaterialpropertiesandtheflexibilityofcolumns,and.25Example5ThecolumnABinthesimplecraneshowninthefigureismadefromcirculardeal.ItslengthisL=6manditsdiameterisd=0.3m.[]=11MPa.Trytodeterminethepermissiblepressureofthecolumn.Solution：Applythemethodofreductioncoefficient.①MaximumflexibilityInplanexy，=1.0Inplanezy，=2.0T1ABWT2xyzOSTABILITYOFCOMPRESSEDCOLUMN26②Determinethereductioncoefficient③DeterminethepermissiblepressureSTABILITYOFCOMPRESSEDCOLUMNForwoodencolumn:as273、Reasonablesectionofthecolumn:Itisreasonable.STABILITYOFCOMPRESSEDCOLUMN保國寺大殿的拼柱形式SakyamniwoodenpagodaYingzhouwoodenpagodaItwasbuiltin1056withthedouble-sleevestructure.Thepagodaplaneisoctagon.Itunderwentthelevel8earthquakeof1305.28Solution：Foreachchannelsteelofnumber10,thecentroidisatthepointC1.Aftertwochannelsteelsarecombined，PLz0yy1z1C1aSTABILITYOFCOMPRESSEDCOLUMNExample6Theuprightcolumnshowninthefigureismadefromtwochannelsteels.ItslengthisL=6m.KnowingthematerialisA3steel,E=200GPaand.Itslowerendisfixedanditsupperendissupportedbyasphericalhinge.Howmuchisthevalueofawhenthecriticalpressureoftheuprightcolumnisthemaximum?Thatisitisreasonableasso29Determinethecriticalforce：Itisalargeflexibilityrod.ThecriticalforceisdeterminedbyEuler’’sformula.STABILITYOFCOMPRESSEDCOLUMN30