Transcription of Introduction to Nonlinear Analysis
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., ,"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"(\.)))
2 "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(~ESoOFENSINl:. ~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.)]
3 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?LimitLoad?PlasticZones?ResidualStre sses?YieldingwhereLoadsareApplied?CreepR esponse?PermanentDeflections?POSSIBLEANA LYSESP lasticPlasticanalysisanalysis(Smalldefor mations)(Largedeformations)Linearelastic analysisTransparency1-10 Determine:TotalStiffness;YieldLoadDeterm ine:SizesandShapesofPlasticZonesDetermin e:UltimateLoadCapacityTopicOne1-13 CLASSIFICATIONOFNONLINEARANALYSEST ransparency1-111)Materially- Nonlinear -On ly( ) Analysis : Displacementsareinfinitesimal.
4 Strainsareinfinitesimal. :/ ,..- ~L< Aslongastheyieldpointhasnotbeenreached,w ehavea )Largedisplacements/ largerotationsbutsmallstrains: Displacementsandrotationsarelarge. 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?+--+---+--+--+symmetryRThreekine maticformulationsareused: 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.
5 ,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~/"/"/"/"/"/"/"/"/.~~~"/"/"/"/"/"/"/" /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.
6 -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+.! ~vectorofexternallyappliednodalpointforc es(theseincludetheinertiaforcesindynamic Analysis ) ~vectorofnodalpointforcesequivalenttothe internalelementstressesTransparency1-33 ~iJ-ttv,..areknown. Wewanttoobtainthesolutioncorrespondingto timet+.
7 !1t(Le.,fortheloadsappliedattimet+.!1t). Forthispurpose,wesolveinstaticanalysistK .!1U= tu+.!1 UMoregenerally,wesolveusingTopicOne1-27 Transparency1-341-28 IntroductiontoNonlinearAnalysisSlide1-1 Slide1-2R~ ~ 'Ib/in2II=113O'y~ 'lb/in2E1= : Wj( )=ShPllC05 'EI~Slic'Plasl, (E'P,III,EI~slic PI~slicsmalld, .:,(.. 'P2V' ll:..::.---EI~sl;csm~1 Idisp.(E1_Slide1-30 80 60 40 2EI~ (E,Tl)o' 00 8 PRES~RADIALDlSPlAC~NTAT~.O-,ncheSStaticr esponseofperfect(5=0)shell0 80 60 40 2o0 8/E'PSlide0 61-4\ 4 PRES~PIPer0 2 RADIALDlSPlACEf'lENTAT~ O-,ncheSStaticresponseofimperfect(5= )shell1-30 IntroductiontoNonlinearAnalysisSlide1-5 PRESSlREPIPero0 20 40 6 RADIALDISPLACEMENTAT~=0inchesElastic-pla sticstaticbucklingbehavioroftheshellwith variouslevelsofinitialimperfectionSlide1 -68 MEANDISPLACEtvENT"642o:-:i' '0.""!2468 TIMETD ynamicresponseofperfect(~=0) (15= ) 'Slide5" 886 IoIEANDISPLACEr- (0= )shell1-32 IntroductiontoNonlinearAnalysisSlide1-90 DynamicunstableoDynamicstable0 40 20 30 4A~L1 TUDEOFIMPERFECTION'6 Effectofinitialimperfectionsontheelastic -plasticbucklingloadoftheshellMIT OpenCourseWare Resource: finite Element Procedures for Solids and Structures Klaus-J rgen Bathe The following may not correspond to a particular course on MIT OpenCourseWare, but has been provided by the author as an individual learning resource.)))
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