MODELLLING ALUMINIUM ALLOY PLATES

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1.MODELLLINGALUMINIUMALLOYPLATES1.1ProblemDefinitionTheplatesanalyzedinthischapteraremadeofaluminiumalloy.ThematerialpropertiesareshowinTable2.1.Young’smodulusE11(GPa)TensilestrengthσT(MPa)Poisson’sratioν12MaximumstrainAluminiumAlloy804000.3310%Table2.1MaterialPropertiesofaluminiumalloy.ThedimensionsofthetwoaluminiumplatesareshowninFigure2.1below.Theyare150mmlong,30mmwideand2mmthick.Oneoftheplateshasnocentralhole;anotherplatehasa5mmdiametercentralhole.Figure2.1Dimensionsofaluminiumplates. TensileloadsontheplatesareshowninFigure2.2.Figure2.2Tensileloadsonaluminiumplates.Theproblemistocalculatethestrengthofthenotchedlaminateplate.1.1AnalyticalCalculationsHolesinplatesmaycausestressconcentrations.Tocalculatethestrengthofanotchedplate,thestressconcentrationaroundtheholeshouldbeunderstood.InBoresi,APandSchmidt,RJ‘sbook[17],thestressdistributionaroundaholeinaninfiniteisotropicplateisgiven.Theresearchshowsthatthelargeststressesoccurattwonearestpointstotheedgeofthehole,andifastraightlinegoesthroughthesetwopoints,thelinemayperpendiculartotheloadstressdirection.Thestressattheedgeoftheholeisgivenbyequation2.1.2.1Inthisequation,σθθisthestressofapointattheedgeofthehole,σistheloadingstress,asshowninFigure2.3.Whenθequalsnπ/2,σθθequals3σ,whichisthelargeststress. Figure2.3Stressdistributionattheholeedge.Sothestressdistributiononthelineperpendiculartotheloadingdirectionshouldbestudied.InTan’sbook[9],theauthorquotedfromapreviousworks[18]andillustratedthestressdistributionofaisotropicplatealongthisline,whichcanbeobtainedbyequation2.2.KT∞=σyσ=1+12Rx2+12Rx42.2WhereKT∞isstressconcentrationfactor,σyistheconcentratedstress,andσistheloadingstress.Ristheradiusofthehole,andxistherangefromthecalculatedpointtotheholecentre.Forafinitewidthplate,thefinite-widthcorrection(FWC)factorshouldbeconsidered[9].ForisotropicmaterialtheFWCcanbecalculatedinequation2.3. 2.3WhereKT∞isthestressconcentrationfactorforaninfiniteplateandKTisstressconcentrationfactorforafinitewidthplate;aisthediameterofthecircularholeandWisthewidthoftheplate.Nowthestrengthofthenotchedaluminiumplatecanbecalculated.Ifusemaximumstressfailurecriterion.Thelargeststressattheholeedgewillbe:2.4Thestressconcentrationfactoris:2.5AndtheFWCis:2.6Hencethestrengthofthenotchedplateis:2.71.1ModellingUn-notchedAluminiumPlateSincethetwoplatesaresymmetrical,theirfiniteelementmodelscanbecreatedas1/4sizeoftheoriginalplates.Thissimplificationbringstwobenefits.First,ifelementsizeisfixed,thesimplifiedmodelsmayhavemuchlesselementsthanoriginalsizemodels,whichmayreducetimeandmemoryrequirements.Second,moreelementswhichhavesmallersizecanbeputintoasamearea,whichmayincreasetheaccuracyofthemodel. Theun-notchedaluminiumalloyplateismodelledinANSYSas75mmlong,15mmwidthand2mmthick.TheelementtypeisPLANE182,whichisa2-D4-nodeelasticelement.ThemodelisshowninFigure2.4.Figure2.4Un-notchedaluminiumalloyplatemodelThereare64nodesinthismodel.Themodelismappedmeshedtogetgoodstressdistribution.Symmetricboundaryconditionisappliedonthelinegoesthroughnode1and2(line1)andthelinegoesthroughnode2and17(line2).Theloadisadisplacementloadpointstominusxaxisdirectiononthelinegoesthroughnode1and20(line4).TheloadandsolvingprocedureisdefinedinmacroLoadunal.mac,andtheprocessingflowofthemacroisshowninFigure2.5.ThemacroiscodedinAPDL. Figure2.5FlowchartofLoadunal.mac.Theprocessofcheckingtheplatefailurestatusandreplacingthefailedmaterialisdonebyplatefcc.mac.TheprocedureofthismacroisshowninFigure2.6. Figure2.6Flowchartofplatefcc.mac.ThedataobtainedfromthismodeisshowninFigure2.7.Figure2.7Stress/Straincurveoftheun-notchedaluminiumplate. AsshownintheFigure2.7,theANSYSprogramsuccessfullyobtainedthemaximumstressoftheplate.Butduetousingmaximumstressfailurecriterionthestressfallstozeroaftertheloadstepwhichitculminatestothehighestpoint.ThestressdistributionatthefailuremomentisshowninFigure2.8.Figure2.8Thestressdistributionatthefailuremoment.1.1ModellingNotchedAluminiumPlateThemodelofnotchedaluminiumplateisshowninisshowninFigure2.9.Comparetotheun-notchedmodel,thismodelhasa1/4circularcorner.Togetaccurateresults,smallermeshsizeisappliedintheareaneartothenotch,andthewholemodelismappedmeshed. Figure2.9Notchedaluminiumalloyplatemodel.NodesonleftsideoftheplateareshowninFigure2.10.Figure2.10Nodesontheleftsideoftheplate. Inthismodel,theloadwillbethesamedisplacementloadasintheun-notchedcase.Theloadstresswillbecalculatedthroughnode1,40,70,71,72,73,74,75,76.ThecorneroftheplateisshowinFigure2.11.Theprogramwillusemaximumstressfailurecriteriontocheckthestressesonnode287.Oncethenode287fails,thematerialpropertieswillbechanged.Inotherwords,node287isselectedasfailurecheckingpoint.Figure2.11Corneroftheplate.Thecalculationprocedureissimilartotheprocedureofun-notchedmodel.ThemaincodeisinLoadnotchal.mac.TheprogramflowchartisshowninFigure2.12. Figure2.12FlowchartofLoadnotchal.mac.Theplatefcc.machasalsobeenchanged;thenewflowchartforitisshowninFigure2.13. Figure2.13Newflowchartofplatefcc.mac.ThestressandstraindataobtainedfromthismodelisshowninFigure2.14.Figure2.14Stress/Straincurveofthenotchedaluminiumplate Thelargeststressis0.149263GPa,whichislargerthantheanalyticalresult.Themainreasonforthiserrormaybetheloadstepofthissolutionissettoolarge.ThestressdistributionontheplateisshowninFigure2.15.Figure2.15Stressdistributionsonthenotchedaluminiumplate1.1ResultAnalysisTheresultsarecomparedinTable2.2.Theoreticalstrength(GPa)StrengthbyANSYS(GPa)Errorrate(%)Un-notchedplate0.40.40Notchedplate0.13142660.14926313.57Table2.2Resultcomparison. Fortheun-notchedplate,thestrengthobtainedfromANSYSfixeswellwiththegivenvalue.Butforthenotchedplate,thestrengthcalculatedbyANSYSislargerthantheresultfromanalyticalcalculation.Themainreasonfortheerrormightbetheloadstepissettoolarge.1.1ConclusionsThischaptershowsthestrengthcalculationprocedureofun-notchedandnotchedaluminiumalloyplates.Bothanalyticalmethodandfiniteelementmethodareadopted.InspiteoftheANSYSresults’smallerrorinnotchedplatecalculation,theANSYSmacrosworkcorrectlyandcanbemodifiedtoanalyselaminatestructures.