数学Chapter02-结构可靠度理论基本概念课件

Chapter2BasicConceptsofStructuralReliabilityTheoryChapter2BasicConceptsof1Chapter2:BasicConceptsofStructuralReliabilityTheory2.2FailureProbabilityofStructures2.3ReliabilityIndexofStructures2.1DefinitionsofStructuralReliability2.5RelationshipbetweenReliabilityIndexandSafetyFactorContents2.4GeometricMeaningofReliabilityIndexChapter2:BasicConceptsofS22.1DefinitionsofStructuralReliabilityChapter2

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StructuralReliabilityTheory2.1DefinitionsofStructural32.1DefinitionsofStructuralReliability…1StructuralReliability

istheabilityofastructureorstructuralelementtofulfillthespecifiedperformancerequirementsundertheprescribedconditionsduringtheprescribedtime.2.1.1ReliabilityofStructures1.ThePrescribedTimetheprescribedtime=thedesignworkinglifethedesignworkinglife

thedesignreferenceperiodDesignworkinglifeisassumedperiodforwhichastructureorastructuralelementistobebeusedforitsintendedpurposewithoutmajorrepairbeingnecessary.Designreferenceperiodisachosenperiodoftimewhichisusedasabasisforassessingvaluesofvariableactions,time-dependentmaterialproperties,etc.2.1DefinitionsofStructural42.1DefinitionsofStructuralReliability…2Table2.1DesignWorkingLifeClassDesignworkinglife(years)Examples11to5Temporary225Replacementstructural

parts,e.g.gantrygirders,bearings350Buildingsandothercommonstructures,otherthanthoselistedbelow4100ormoreMonumentalbuildings,andotherspecialorimportantstructures.Largebridges2.1DefinitionsofStructural52.1DefinitionsofStructuralReliability…32.ThePrescribedConditionsthenormaldesignconditionthenormalconstructionconditionthenormaloperationconditionThehumanerrorsarenotconsidered3.TheSpecifiedPerformanceRequirementsUltimatelimitstaterequirementTheyshallwithstandextremeand/orfrequentlyrepeatedactionsoccurringduringtheirconstructionandanticipateduse.ServiceabilitylimitstaterequirementTheyshallperformadequatelyunderallexpectedactions.StructuralrobustnessrequirementTheyshallnotbedamagedbyeventslikeflood,landslip,fire,explosion,impact,earthquake,tornado,etc.StructuraldurabilityrequirementTheyshallremainfitforuseduringtheirdesignworkinglivesintheirenvironment,givenappropriatemaintenance.2.1DefinitionsofStructural62.1DefinitionsofStructuralReliability…4Alimitstate

isastatebeyondwhichastructureorpartofitnolongersatisfiesthedesignperformancerequirements.2.1.2LimitStatesofStructures1.DefinitionofLimitStateTheconceptofalimitstateisusedtohelpdefinefailureinthecontextofstructuralreliabilitytheory.Wecouldsaythatastructurefailsifitcannotperformitsintendedfunction,orviolationofalimitstate.Alimitstateisaboundarybetweendesiredandundesiredperformanceofastructure.Theboundarybetweendesiredandundesiredperformanceofastructureisoftenrepresentedmathematicallybyalimitstatefunctionorperformancefunction.2.1DefinitionsofStructural72.1DefinitionsofStructuralReliability…52.TypesofLimitState—Classification1Ultimatelimitstatesaremostlyrelatedtothelossofload-carryingcapacity.Examplesofmodesoffailureinthiscategoryinclude:(1)UltimateLimitState(ULS)ExceedingthemomentcarryingcapacityFormationofaplastichingeCrushingofconcreteincompressionShearfailureofthewebinasteelbeamLossoftheoverallstabilityBucklingofflangeBucklingofwebWeldruptureLossoffoundationload-carrying2.1DefinitionsofStructural82.1DefinitionsofStructuralReliability…6Serviceabilitylimitstatesaremostlyrelatedtogradualdeterioration,user’scomfort,ormaintenance.Theymayormaynotbedirectlyrelatedtostructuralintegrity.Examplesofmodesoffailureinthiscategoryinclude:(2)ServiceabilityLimitState(SLS)ExcessdeflectionExcessvibrationPermanentdeformationsCracking2.1DefinitionsofStructural92.1DefinitionsofStructuralReliability…7(3)CollapseLimitState(CLS)ULSisforthelargestload-carryingcapacityofintactstructures,whileCLSisforthatoflocaldamagedstructures.Failure-SafeorConditionalLimitState

Transformationofthelocaldamagedstructureintoamechanism.Thekeypartsofalocaldamagedstructurefailwhenthestrengthofmaterialisexceeded.Instabilityoftheelementsofalocaldamagedstructure(e.g.buckling).Lossofequilibriumofthelocaldamagedstructureorofapartofit,consideredasarigidbody.2.1DefinitionsofStructural102.1DefinitionsofStructuralReliability…8(3)CollapseLimitState(CLS)2.1DefinitionsofStructural112.1DefinitionsofStructuralReliability…9(3)CollapseLimitState(CLS)2.1DefinitionsofStructural122.1DefinitionsofStructuralReliability…10(3)CollapseLimitState(CLS)2.1DefinitionsofStructural132.1DefinitionsofStructuralReliability…11(3)CollapseLimitState(CLS)(March,2007)(Jan.,2002)2.1DefinitionsofStructural142.1DefinitionsofStructuralReliability…122.TypesofLimitState—Classification2Irreversiblelimitstateisalimitstatewhichwillremainpermanentlyexceededwhentheactionswhichcausedtheexcessareremoved.(1)IrreversibleLimitStateTheeffectofexceedinganultimatelimitstateisalmostalwaysirreversibleandthefirsttimethatthisoccursitcausesfailure.Inthecasesofpermanentlocaldamageorpermanentunacceptabledeformations,exceedingaserviceabilitylimitstateisirreversibleandthefirsttimethatthisoccursitcausesfailure.Reversiblelimitstateisalimitstatewhichwillnotbeexceededwhentheactionswhichcausedtheexcessareremoved.(2)ReversibleLimitStateInthecasesoftemporarylocaldamage,temporarylargedeformationsandvibrations,exceedingaserviceabilitylimitstateisreversible.2.1DefinitionsofStructural152.1DefinitionsofStructuralReliability…132.1.3LimitStateFunctions(PerformanceFunctions)1.FormofComprehensiveVariablesIngeneral,thefactorsinfluencingstructuralreliabilitycanbeputintotwokindsofcomprehensivevariables,thatis,loadeffectS(demand)andstructuralresistanceR(supply).Aperformancefunction,orlimitstatefunction,canbedefinedasforsafestateforlimitstateforfailurestateThelimitstateequationisdefinedas2.1DefinitionsofStructural162.1DefinitionsofStructuralReliability…14Eachlimitstatefunctionisassociatedwithaparticularlimitstate.Evenforaparticularlimitstate,itslimitstatefunctionsmaybedifferent.Differentlimitstatesmayhavedifferentlimitstatefunctions.Zisalsonamedsafetymarginormarginofsafety.PropertiesofLimitStateFunctions2.1DefinitionsofStructural172.1DefinitionsofStructuralReliability…152.FormofBasicVariablesIngeneral,theperformancefunctioncanbeafunctionofmanyvariables:loadcomponents,environmentinfluence,resistanceparameters,materialproperties,dimensions,analysisfactors,andsoon.Theaboverandomvariablesarecalledbasicrandomvariables,orbasicvariables.Thegeneralformofaperformancefunction:Thebasicvariablespaceisaspecifiedsetofbasicvariables.Itisalsocalledstatespace:Thegeneralformofalimitstateequation:forsafestateforlimitstateforfailurestate2.1DefinitionsofStructural182.1DefinitionsofStructuralReliability…162.1.4MeasuresofStructuralReliability1.Safety(Survival)ProbabilityofStructuresTheprobabilityoftheeventthatthestructureistobesafeisdefinedassafetyprobability,orsurvivalprobability.Mathematically,Lettheperformancefunctionofastructurebe,bethebasicrandomvariablesinfluencingstructuralfunctions.Obviously,Zisarandomvariable.AssumethatthePDFofZisTheprobabilityoftheeventthatthestructureistobeunsafeisdefinedasfailureprobability.Mathematically,2.FailureProbabilityofStructures2.1DefinitionsofStructural192.1DefinitionsofStructuralReliability…17Thereliabilityindexofstructuresisdefinedas3.ReliabilityIndexofStructuresRelationshipbetweenandwhere,istheinversestandardizednormaldistributionfunction.2.1DefinitionsofStructural202.2FailureProbabilityofStructuresChapter2

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StructuralReliabilityTheory2.2FailureProbabilityofSt212.2FailureProbabilityofStructures…12.2.1GeneralBasicVariables1.AssumptionsConsidertheperformancefunctionwhere,isan-dimensionrandomvector.ItisassumedthatthejointPDFofis2.FormulaThefailureprobabilityofstructureswithgeneralbasicvariables:2.2FailureProbabilityofSt222.2FailureProbabilityofStructures…2Intheaboveformula,iscalledasfailuredomainofstructures.3.CalculationMethodAnalyticalMethodSimulationMethodIntegralMethodPreciseAnalyticalMethod:FOSMSORMProbabilityInterferenceMethod(PIM)ApproximateAnalyticalMethodMonteCarloMethod(MCS)LatinHypercubeSamplingImportantSamplingFORM2.2FailureProbabilityofSt232.2FailureProbabilityofStructures…32.2.2TwoComprehensiveVariables1.Assumptions(1)Sisthetotalloadeffect,known:2.ProbabilityInterferenceMethod(PIM)Thelimitstatefunction:(2)Risthememberresistance,known:(3)RandSarestatisticalindependent,thatis,,,Thefailuredomain:failuredomainsafedomain2.2FailureProbabilityofSt242.2FailureProbabilityofStructures…42.ProbabilityInterferenceMethod(PIM)…Thefailureprobability:2.2FailureProbabilityofSt252.2FailureProbabilityofStructures…5Formula2ofPIMFirstintegrationfors,thenforrFormula1ofPIMFirstintegrationforr,thenfors2.2FailureProbabilityofSt262.2FailureProbabilityofStructures…63.PhysicalMeaningofProbabilityInterferenceTheprobabilityofloadeffectSbeinginspace:TheprobabilityofresistanceRbeinglessthanaspecificloadeffects:SinceR&Sarestatisticalindependent,theprobabilityofR&Ssimultaneouslyoccurringinspaceis:2.2FailureProbabilityofSt272.2FailureProbabilityofStructures…7Attention!Actually,ithasbeenverifiedthatthereexistsarelationshipasfollows:2.2FailureProbabilityofSt282.2FailureProbabilityofStructures…8Example2.1ConsidertheperformancefunctionofthesquaresectionstrengthofastructuralelementwhereRandSarerandomvariables.ThedistributionparametersofR&Sareshownbelow:RisanormalRVSisaexponentRVThePDFofSisCalculatethefailureprobabilityoftheelement.2.2FailureProbabilityofSt292.2FailureProbabilityofStructures…9Solution:Let,then2.2FailureProbabilityofSt302.2FailureProbabilityofStructures…10Ifweplacethepracticalvaluesof,,,intotheaboveformula,thenwewillobtainthefailureprobability:EndofExample2.12.2FailureProbabilityofSt312.3ReliabilityIndexofStructuresChapter2

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StructuralReliabilityTheory2.3ReliabilityIndexofStru322.3ReliabilityIndexofStructures…12.3.1R&SareIndependentNormalVariables1.AssumptionsConsidertheperformancefunctionwhere,arenormalrandomvariables.2.FormulaSinceR&Sareallnormalrandomvariables,thenweknowthatZisalsoanormalRV.Therefore,wehave2.3ReliabilityIndexofStru332.3ReliabilityIndexofStructures…2Thefailureprobabilityis:Let,then2.3ReliabilityIndexofStru342.3ReliabilityIndexofStructures…3Let,thenFormulaofReliabilityIndex.iscalledReliabilityIndex.>formatshorte>beta=1.0:0.5:5.0;>Pf=normcdf(-beta,0,1)MATLABPrograms>n=1:9>Pf=10.^(-n);>beta=-norminv(Pf)2.3ReliabilityIndexofStru352.3ReliabilityIndexofStructures…41.01.59×10-12.56.21×10-34.03.17×10-51.56.68×10-23.01.35×10-34.53.40×10-62.02.28×10-23.52.33×10-45.02.87×10-7Table2.2Relationshipbetweenand

10-11.2810-22.3310-33.0910-43.7110-54.2610-64.7510-75.1910-85.6210-95.992.3ReliabilityIndexofStru362.3ReliabilityIndexofStructures…52.3.2R&SareIndependentLognormalVariables1.AssumptionsConsidertheperformancefunctionwhere,arelognormalrandomvariables.2.Formula2.3ReliabilityIndexofStru372.3ReliabilityIndexofStructures…62.3ReliabilityIndexofStru38Example2.2Considertheperformancefunctionofastructuralelement,Calculatethereliabilityindexoftheelement.2.3ReliabilityIndexofStructures…7whereRandSarenormalrandomvariables.Solution:Example2.2Considertheperfor39Example2.3Considertheperformancefunctionofastructuralelement,Calculatethereliabilityindexoftheelement.2.3ReliabilityIndexofStructures…8whereRandSarelognormalrandomvariables.Solution:Example2.3Considertheperfor402.3ReliabilityIndexofStructures…9Ifweusetheapproximateformula,thenwewillhaveTherelativeerrorofthesetwomethodsis:2.3ReliabilityIndexofStru412.4GeometricMeaningofReliabilityIndexChapter2

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StructuralReliabilityTheory2.4GeometricMeaningofReli422.4GeometricMeaningofReliabilityIndex…12.4.1ReducedVariablesThestandardformsofthebasicvariablesR&Scanbeexpressedas:Thevariablesandarecalledreducedvariables.2.4GeometricMeaningofReli432.4GeometricMeaningofReliabilityIndex…2Wecanfindastraightlineequationinthespaceofreducedvariables:Theaboveequationcanbetransformedintoanormalequation:2.4GeometricMeaningofReli442.4GeometricMeaningofReliabilityIndex…3failuredomainsafedomain2.4.2GeometricMeaningofReliabilityIndexTheReliabilityIndexistheshortestdistancefromtheoriginofreducedvariablestothelimitstateequation.Thedefinitioncanbegeneralizedfornvariablespace.iscalledthedesignpoint.thecoordinatevaluesof2.4GeometricMeaningofReli452.5Relationshipbetween