Transcription of Introduction to Nonlinear Analysis - MIT OpenCourseWare
1 Contents:Textbook:Examples:Reference:Top ic1 IntroductiontoNonlinearAnalysis Introductiontothecourse Theimportanceofnonlinearanalysis Fourillustrativefilmsdepictingactualandp otentialnonlinearanalysisapplications Generalrecommendationsfornonlinearanalys is Modelingofproblems Classificationofnonlinearanalyses Exampleanalysisofabracket,smallandlarged eformations,elasto-plasticresponse Twocomputer-plottedanimations-elasto-pla sticlargedeformationresponseofaplatewith ahole-largedisplacementresponseofadiamon d-shapedframe Thebasicapproachofanincrementalsolution Timeasavariableinstaticanddynamicsolutio ns Thebasicincremental/iterativeequations , , , ,T.
2 , ,"OnFiniteElementLargeDisplacementandEla stic-PlasticDynamicAnalysisofShellStruct ures,"Computers&Structures,12,309-318, 'fS1S ( '\MECHANICS FINliEEll::t-"\ENTD15 CR~.,..'"2A-TIONS NLAt-'\ "R11\-\MS SOfTWA'QE(ONS''t>E:'RAT\DN~Nt::-CONCENTR ATEON~- M&Tl-\o1: TI-\Po-TA'e&5 ENE'RALL'IA? !'"10t>E'RtJ'E"(\.. "PRACTICAL?ROCEblA~E:S~IHE1Hc>'t>sIItA-\ Al'fORARENOw~E(OMINbANINTEG1</\l'flPt'RI0r CAl:>leAfSO~TWAKET opicOne1-3'R~IEFOVERVI ~Ol=CouRSE 6fOMElRICANt>,"" S"1AT,c:AN~:b'fAlAM1C SOLIAllONS. EASICr~ll' >LE:,,;)ANbTIi~ <;EWILLEEOFINIERES,TINMAN'IE;KAN(~ESoOFE NSINl.))))
3 ~IN(;11-\" \ 0lAiTIi'EWO'RL\::>Markerboard1-1IN,HISLE :CTUR~WE"bl'>[\)\> <;$ol'\EINTRO"t:>l,\ ,Vlf\V G'l:.A'PItSAN!)SHoW<;C'nE<;It01':.,t-10\ 1\ESWETH-ENCLASS\f' ~'RAI\}AL'-/'SESWE~ ;C::;THE\SASICA??'KOACI-\I:JFANINCKEMENT AL$OLlAi\ONW(bl\lEEXAtWLESM arkerboard1-21-4 IntroductiontoNonlinearAnalysisTranspare ncy1-1 Transparency1-2 FINITEELEMENTNONLINEARANALVSIS Nonlinearanalysisinengineeringmechanicsc anbeanart. Nonlinearanalysiscanbea frustration. Italwaysisa nonlinearanalysis: Collapseorbucklingofstructuresduetosudde noverloads Progressivedamagebehaviorduetolonglastin gsevereloads Forcertainstructures( ), non linearfiniteelementanalysisarefoundinalm ostallbranchesofengineering,mostnotablyi n, ,a (INTERACTIONAND)MUTUALENRICHMENTBESTAPPR OACH Usereliableandgenerallyapplicablefinitee lements.]
4 Withsuchmethods,wecanestablishmodelsthat weunderstand. Startwithsimplemodels(ofnature) "PHILOSOPHY"FORPERFORMINGANONLINEARANALY SISTOPERFORMANONLINEARANALYSIS Staywithrelativelysmallandreliablemodels . Performa linearanalysisfirst. Refinethemodelbyintroducingnonlinearitie sasdesired. \"",u",/NECESSARYFORTHEINTERPRETATIONOFR ESULTST hpicOne1-11 Transparency1-7 PROBLEMINNATUREMODELINGMODEL:Wemodelkine maticconditionsconstitutiverelationsboun daryconditionsloadsSOLVEINTERPRETATIONOF RESULTST ransparency1-81-12 IntroductiontoNonlinearAnalysisTranspare ncy1-9 ATYPICALNONLINEARPROBLEMM aterial:MildSteelPOSSIBLEQUESTIONS:Yield Load?
5 LimitLoad?PlasticZones?ResidualStresses? YieldingwhereLoadsareApplied?CreepRespon se?PermanentDeflections?POSSIBLEANALYSES P lasticPlasticanalysisanalysis(Smalldefor mations)(Largedeformations)Linearelastic analysisTransparency1-10 Determine:TotalStiffness;YieldLoadDeterm ine:SizesandShapesofPlasticZonesDetermin e:UltimateLoadCapacityTopicOne1-13 CLASSIFICATIONOFNONLINEARANALYSEST ransparency1-111)Materially- Nonlinear -On ly( ) Analysis : Displacementsareinfinitesimal. Strainsareinfinitesimal. :/ ,..- ~L< Aslongastheyieldpointhasnotbeenreached,w ehavea )Largedisplacements/ largerotationsbutsmallstrains: Displacementsandrotationsarelarge.
6 Strainsaresmall. :yy' a'T< Aslongasthedisplacementsareverysmall, )Largedisplacements,largerotations,large strains: Displacementsarelarge. Rotationsarelarge. Strainsarelarge. :DTransparency1-16x Thisisthemostgeneralformulationofa problem, )NonlinearitiesinboundaryconditionsConta ctproblems:'1)e--------A~~--Il-Gapd Contactproblemscanarisewithlargedisplace ments,largerotations,materiallynonlinear conditions,..Example:BracketanalysisAlld imensionsininchesElasto- :36elementmesh Allelementsarea-nodeisoparametricelement sLineof?
7 +--+---+--+--+symmetryRThreekinematicfor mulationsareused: Materially- Nonlinear -onlyanalysis(smalld isplacements/smallrotationsandsmallstrai ns) TotalLagrangianformulation(largedisplace ments/largerotationsandlargestrains) UpdatedLagrangianformulation(largedispla cements/largerotationsandlargestrains)Th picOne1-17 Transparency1-19 Transparency1 201-18 IntroductiontoNonlinearAnalysisTranspare ncy1-21 Transparency1-22 However, , Themateriallawusedinconjunc tionwiththetotalLagrangianformulationisa ctuallynotapplicabletolargestrainsituati ons(butonlytolargedispl.)
8 ,rotation/smallstrainconditions). Themateriallawusedinconjunc (Ibs) +----+----+--o12 Totaldeflectionbetweenpointsofloadapplic ation(in)Thedeformedmeshcorrespondingtoa loadlevelof12000 Ibsisshownbelow( ). ,, ,..,mesh~.==-1,..-.,..--r---r- '''''IJIIII\"Jr':,I/T" '.I-_-+-_-+-_-+_'rIIIII---+---+--IIII--- +---+-III_~s-deformedmeshTopicOne1-19 Transparency1-231-20 IntroductiontoNonlinearAnalysisComputerA nimationPlatewithholeTIME: 8 LOAD : 41 LOAD : 52 LOAD ,LOADMPAtTIME,13llLOADI325llllMPA~/"/"/" /"/"/"/""/"/"/"/"/"/"/vTIME3llllLOAD7511 00 MPA~/"/"/"/"/"/"/"/"/.
9 ~~~"/"/"/"/"/"/"/"/vTopicOne1-21 ComputerAnimationDiamondshapedframe1-22 IntroductiontoNonlinearAnalysisTranspare ncy1-24 Transparency1-25 THEBASICAPPROACHOFANINCREMENTALSOLUTION Weconsidera body(astructureorsolid)subjectedtoforcea nddisplacementboundaryconditionsthatarec hanging. ,weuseanincrementalapproach,measuredinlo adstepsortimestepsTopicOne1-23 Transparency1-26timeWhentheappliedforces anddisplacementsvary-slowly,meaningthatt hefrequenciesoftheloadsaremuchsmallertha nthenaturalfrequenciesofthestructure,weh avea staticanalysis.
10 -fast,meaningthatthefrequenciesoftheload sareintherangeofthenaturalfrequenciesoft hestructure,wehavea Timeisa pseudo-variable,onlydenotingtheloadlevel inNonlinearstaticanalysiswithtime independentmaterialpropertiesRun1at= ~-I-----+ :Transparency1-29 IdenticallythesameresultsareobtainedinRu n1andRun2 Timeisanactualvariable-indynamicanalysis -innonlinearstaticanalysiswithtime-depen dentmaterialproperties(creep)Nowdtmustbe chosencarefullywithrespecttothephysicsof theproblem, (ortime)step,weneedtosatisfythethreebasi crequirementsofmechanics: Equilibrium Compatibility 26 IntroductiontoNonlinearAnalysisTranspare ncy1-32 Weidealizethebodyasanassemblageoffinitee lementsandapplytheprincipleofvirtualwork totheunknownstateattimet+.
