材料力学的结构稳定性英文

CHAPTER10

STABILITYOFCOMPRESSEDCOLUMNS1CHAPTER10STABILIZATIONOF COMPRESSIVECOLUMNS§10–1CONCEPTSOFSTABILITYOFCOMPRESSEDCOLUMNS§10–2EULER’SFORMULAOFTHECRITICALFORCEOFSLENDERCOMPRESSEDCOLUMNS§10–3CRITICALSTRESSOFCOMPRESSEDCOLUMNSASSTRESSEXCEEDSPROPORTIONALLIMIT

§10-4STABILITYCHECKANDREASONABLESECTIONOFCOMPRESSEDCOLUMNS2§10–1

CONCEPTSOFSTABILITYOFCOMPRESSEDCOLUMNSTheload-carryingcapacityofstructuremembers：①Strength②Rigidity③Stability

Somestructuremembersinengineeringhaveenoughstrengthandrigiditybuttheyareunabletoworksafelyandreliably.STABILITYOFCOMPRESSEDCOLUMN3PSTABILITYOFCOMPRESSEDCOLUMN41、Stableandinstableequilibrium：1).instableequilibrium

STABILITYOFCOMPRESSEDCOLUMN52).StableequilibriumSTABILITYOFCOMPRESSEDCOLUMN63).Stableandinstableequilibrium

STABILITYOFCOMPRESSEDCOLUMN72、Lossofstabilityandthecriticalpressureofthecolumn：1).Idealcompressivecolumns：thematerialisabsolutelyideal；theaxisisabsolutelystraight；thecompressiveforceisabsolutelyalongtheaxisofthecolumn.2).Stableandinstableequilibriumofcompressivecolumns：STABILITYOFCOMPRESSEDCOLUMNInstableequilibriumStableequilibrium83).lossofstabilityofcompressedcolumn：4).CriticalpressureofcompressedcolumnsStableequilibriumInstableequilibriumCriticalstateCriticalpressure:Pcr

correspondingpressureSTABILITYOFCOMPRESSEDCOLUMNintermediatestate9

§10–2

EULER’SFORMULAOFTHECRITICALPRESSUREOF SLENDERCOMPRESSEDCOLUMNS

1、Criticalpressureforthecolumnwithtwohingedends:①bendingmoment：②ApproximatedifferentialequationofthedeflectioncurvePPxyPMSTABILITYOFCOMPRESSEDCOLUMNSupposethepressurehasreachedthecriticalvalueandthecolumnhasbeenintinybendingstateasshowninthefigure.Starttodeterminethecriticalforcewiththedeflectivecurve.PxL10③Solutionofthedifferentialequation：④Determinetheintegralconstants：

ThecriticalforcePcr

isthesmallestpressureundertinybending，thereforeweonlytaken=1andthecolumnwillbendabouttheaxiswiththesmallestmomentofinertia.STABILITYOFCOMPRESSEDCOLUMNThatis112、Applicationrangeoftheformula：3、Euler’sformulaofthecriticalpressureforthecolumnwithotherendconditions:1).Idealcompressivecolumns；2).Inlinearelasticrange；3).Theendsofthecolumnaresupportedbyhinges.—Lengthcoefficient（orconstraintcoefficient）Euler’sformulaofthecriticalpressureforthecompressivecolumnwithtwohingedends

GeneralformofEuler’sformulaofthecriticalpressureSTABILITYOFCOMPRESSEDCOLUMN120.5lTable10–1Euler’sformulaoftheslendercompressivecolumnundervariousconstraintconditionsSupportsTwohingedendsOnefreeendandonehingedendTwofixedendsOnefixedendandonefreeendTwofixedendsbutoneofthemismovablelaterally.

TheshapeofthedeflectivecurveinlostofstabilityPcrABlEuler’sformulaofthecriticalforcePcr

Lengthcoefficientμμ=1μ0.7μ=0.5μ=2μ=1PcrABlPcrABl0.7lCCDC:InflectionpointC、D:Inflectionpoint0.5lPcrPcrl2llC:InflectionpointSTABILITYOFCOMPRESSEDCOLUMN13Solution：Thedeformationofthecolumnisshowninthefigure.Theapproximatedifferentialequationofitsdeflectioncurveis:Theboundaryconditionsare:

Example1

Trytodeducethecriticalforceofthefollowingslendercolumnbytheapproximatedifferentialequationofthedeflectivecurve.PLxPM0PM0PM0xPM0STABILITYOFCOMPRESSEDCOLUMNLet14Inordertodeterminetheminimumcriticalpressure“k”mustbetheminimumvalueexceptzero,thatis：Thereforethecriticalpressureis：=0.5STABILITYOFCOMPRESSEDCOLUMNand15③CriticalpressureofthecolumnExample2

Determinethecriticalpressureofthefollowingslendercompressive column.=1.0，Solution：①Abouttheaxis

y，

twoendsarehinged：=0.7，②Abouttheaxis

z，theleftendisfixedandtherightendishinged：yzL1L2yzhbxSTABILITYOFCOMPRESSEDCOLUMN16Example3

Determinethecriticalpressureofthefollowingslendercompressivecolumns.Knowing:L=0.5mandE=200GPa.Solution：Fig.(a)Fig.(b)Fig.(a)Fig.(b)5010PLPL(45456)Equal-leganglesteelyzSTABILITYOFCOMPRESSEDCOLUMN17

§10–3

CRITICALSTRESSOFCOMPRESSEDCOLUMNSASSTRESS EXCEEDSTHEPROPORTIONALLIMIT

1、Basicconcepts1).Criticalstress：Meanstressinthecrosssectionofthecompressivecolumninthecriticalstate.3).Flexibility：2).Criticalstressforslendercompressedcolumns：22lpsEcr=Thatis:STABILITYOFCOMPRESSEDCOLUMNRadiusofinertiaFlexibility(orslendernessratio)ofthecolumn184).Divisionoflargeflexibilitycolumn：2、Calculationofthecriticalstressforthecolumnswithmiddleorsmallflexibility1).Empiricalformulawiththelinearvariation①AsP<<S

：STABILITYOFCOMPRESSEDCOLUMNisoneofthemiddleflexibilitycolumn.ThecolumnwithItscriticalstressmaybedeterminedbytheempiricalformula.Thecolumnswithiscalledthelargeflexibilitycolumn(orslenderclumn).ThecriticalpressureofthiskindofcolumniscalculatedbyEuler’sformula.Thecolumnswithiscalledmiddleorsmallflexibilitycolumn.ItscriticalpressureofthiskindofcolumncannotbecalculatedbyEuler’sformula.19③To裂ta堆l崇fi邮gu企re购o数f话th胳e井cr炼it慈ic圆al丢s亦tr爹es麻s②A热sS<：bss=a-sl

PPEspl2

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