Masud - A FEM Formulation for Multi-layered Shells

8/12/2019 Masud - A FEM Formulation for Multi-layered Shells

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A FiniteElementFormulationofMulti-layeredShellsfortheAnalysisofLaminatedComposites

by

ArifMasud1

DepartmentofCivil&MaterialsEngineeringTheUniversityofIllinoisofChicagoChicago,Illinois60607-7023

and

MohammadPanahandeh 2BerkeleyAppliedScience&Engineering,Inc.

San Francisco,CA94103

rx3Submittedto:

Computers&StructuresJMSffflUOllONTATEMENTAppsrovdoraWtaotoxMj

1AssistantProfessorofMechanics&Materials.orrespondingauthor.2Principal;BerkeleyAppliedScience&Engineering,Inc.DtlCQUALITYINSPECTED1


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AbstractThisaperresentsmulti-layered/multi-directorndhear-deformableinitele-

mentormulationfshellsorhenalysisfcompositeaminates.heisplacementieldisassumedcontinuousacrossthefiniteelementlayersthroughthecompositethickness.Therotationields,owever,ayer-wiseontinuousndsssumediscontinuouscrossheselayers.hiskinematichypothesisesultsnndependentheardeformationfthedirectorassociatedwithachndividualayerndhusllowshewarpingftheompositeross-section.heresultingstrainfieldisdiscontinuouscrossthedifferentmaterialsets,herebycreatingtheprovisionthattheinter-laminartransversestressescomputedfromtheconstitu-tiveequationscanbecontinuous.Numericalresultsrepresentedtoshowtheperformanceofthemethod.


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ContentsAbstract

1 .ntroduction .2.AssumptionsintheLayer-wiseShearDeformableShellTheory3.KinematicDescriptionofMulti-layeredShells

3.1Kinematicsinthecontextoffiniteelementmethod4.GeometricDescriptionofMulti-layeredShells5.ConstitutiveRelations6.heVariationalFramework7.NumericalExamples 48.Conclusions 8

Acknowledgement 9References 9


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A.MasudandM.Panahandehfiniteelementformulationforcomposites1.ntroductionInthelasttwodecades,ompositeshavefoundincreasingapplicationinmanyengineer-

ingtructures.ecentdvancesnheechnologiesfmanufacturingndmaterialsaveenhancedthecurrentapplicationofcompositematerialsfrombeingusedassecondarystruc-turalelementstobecomingprimaryload-carryingstructural components.Duetotheinherentinhomogeneityandanisotropyofthematerials,analysisof thesecompositestructuresimposesnewchallengesonengineers1 0,1 ].

Plateandshellstructuresmadeof laminatedcompositematerialshaveoftenbeenmodeledasanequivalentsinglelayerusingclassicallaminatetheory(C.L.T)5,8,2,3,6]inwhichthehicknesstressomponentsregnored.C.L.T.s directxtensionfclassicalplatetheorynwhichhewellnownKirchhoff-Loveinematicypothesissssumednforced.Thistheoryisadequatewhenthethicknesstosideorradiusratio)ssmall.owever,lami-natedplatesandshellsmadeofadvancedfilamentarycompositematerialsaresusceptibletothicknessffects,ecauseheirffectiveransversemodulireignificantlymallomparedtotheeffectiveelasticmodulusalongthefiberdirection.Furthermore,theclassicaltheoryofplates,whichassumesthatthenormalstothemid-planebeforedeformationremainstraightandnormaltotheplaneafterdeformation,underpredictsdeflectionsandoverpredictsnatu-ralfrequenciesandbucklingloads.Thesediscrepanciesariseduetotheneglectoftransverseshearstrains.TherangeofapplicabilityoftheC.L.T.solutionhasbeenwellestablishedfo rlaminatedlatlates1 0,2,0] .InrderovercomeheeficienciesnC.L.T.,efinedlaminatetheoriesavebeenproposed3,0,1 ,8 ,9,0,1 ,4].Theseresinglelayertheoriesinwhichthetransverseshearstressesaretakenintoaccount.Theyprovideimprovedglobalesponsestimatesoreflections,ibrationrequenciesnducklingoadsfmod-eratelyhickompositeswhenomparedohelassicalaminateheory. AMindlinypefirst-ordertransversesheardeformationtheoryS.D.T.)wasfirstdevelopedbyWhitneyandPagano24]ormulti-layeredanisotropicplates,ndbyDongandTso6 ]ormulti-layered


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A.MasudandM.Panahandehfiniteelementformulationforcompositesanisotropicshells.BothapproachesC.L.T.ndS.D.T.)onsideralllayersasoneequivalentsinglenisotropicayer,husheyannotmodelthewarpingofcross-sections,hats,hein-planedistortionftheeformedormalueoransverseheartresses.urthermore,theassumptionofanon-deformablenormalresultsinincompatibleshearingstressesbetweenadjacentlayers.helaterapproachalsorequirestheintroductionofanarbitraryshearcor-rectionactorwhichependsnheaminationparametersorobtainingccurateesults.Itswellstablishedhatuch heorysdequateoredictnlyhegrossehaviorflaminates.

ThexactnalyseserformedyPagano1 2,3,6]nheompositeiatplatesaveindicatedthathein-planedistortionofthedeformednormaldependsnotnlyonthelam-inatehickness,butlsonherientationndheegreeforthotropyfthendividuallayers.hereforehehypothesisfnon-deformableormals,whilecceptableorsotropicplatesndhells,sftenquiteunacceptableormulti-layerednisotropicplatesndhellsthatave argeatiofYoung'smodulusohearmodulus,venfheyreelativelythin4,8,9,0] .hus ransversesheardeformationheorywhichalsoaccountsforthewarpingofthedeformednormalisequiredorccurateredictionfthelasticehavior(deflections,thicknessdistributionof thein-planedisplacements,naturalfrequencies,etc.)fmulti-layeredanisotropicplatesandshells.

Inviewoftheseissuesavariationallysoundtheorythataccountsforthe3-Deffects,allowsthicknessvariation,andpermitsthewarpingofthedeformednormal,isrequiredfo rarefinedanalysisfthickndhinomposites.nhispaperweavendeavoredoddressheseissuesnddevelopanewfiniteelementformulationfo rlaminatedcomposites.Theassumeddisplacementieldsontinuoushroughthehickness,whiletheotationieldsayer-wisecontinuousin-D )ndaneiscontinuouscrossheinitelementayers.hisis-placementfieldfulfills priorithegeometriccontinuityconditionsbetweencontiguouslayers.Furthermore,theassumeddisplacementfieldiscapableofmodelingthein-planedistortionof


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A.MasudandM.Panahandehfiniteelementformulationforcompositesthedeformednormal,withoutncreasingtheorderofthepartialdifferentialequationswithrespectohefirst-orderransversesheardeformationtheory.nthisormulation,tmost,onlyfirstderivativesofdisplacementandrotationfieldsappearinthevariationalequations.ThepracticalconsequenceofthisfactisthatonlyCcontinuityoffiniteelementfunctionsisrequiredwhichisreadilysatisfiedbytheLagrangefamilyof elements7].Becauseofthe3-Dfeaturesintheformulation,tcanaccuratelymodeltheinter-laminarconditionsandpredictthe-Ddgeffects.inally,ikeheingle-layerheareformableheories,heproposedformulationprovidesflexibilityinthespecificationoftheboundaryconditions7].

Asummaryofthepaperisasfollows.Section2presentstheassumptionsinherentintheproposedmodel.hegeometricndkinematicypothesisfthemulti-layeredhellshataccommodate,oanappropriatedegreeofapproximation,heeffectsftransversalwarpingofthecross-sectionduetosheardeformationaswellasfibercompressibilityarepresentedinSections3and4,respectively.Section5talksaboutthedevelopmentofconstitutivematricesfo rcomposite laminates.Theunderlyingvariationalframeworkandtheensuing finiteelementformulationftheproblemrepresentednSection.ection presentsomeumericalexamplestodemonstratetherangeofapplicabilityandaccuracyof theproposedtheory,ndconclusionsaredrawninSection8.2.AssumptionsnheLayer-wiseShearDeformableShellTheory1 .hedomainftsofthefollowingspecialform

f t=\ x ,yz)e7 e 3 |z< 6 Y ]I)T= X> (0 ^y il G A(cn2\ (1)whereA stheareaofthereferencesurfacefo rlayerI,t sthethicknessoflayerIandTisthetotalthicknessof thecompositeshell.

2.hedisplacementfieldisssumedtotakethefollowingformU2\x,y,z)=uW(x,y,z)+ZU9W(x,y) a1,22)


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A.MasudandM.Panahandehfiniteelementformulationforcompositeswhereu\x,y,z)arehen-planeranslationsnayer,9l,'(x,y)areheoutfplanedirectorrotationsforthereferencesurfacessociatedwithlayer,andZ s functionthatstablisheshepositionfapointwithespectoheeferenceurfacenayer.Thesecondtermontherighthandsideof(2)canbeviewedasanenhancementtothein-planecomponentsofthedisplacementfieldinlayerIwithrespecttothereferencesurfaceassociatedwiththeayer.onsequentlyhisnhancedieldbecomesdenticallyeronthereferencesurfaces.

3.hedisplacementfieldinthethicknessdirectionisassumedtobeafunctionofz,.e.,Uil)(x,y,z)=uf{x,y,z) '3)

therebyproducingthroughthethicknessstrainswhichresultinthicknessvariationintheshell.onsequenceftheboveelaxationshat33 .e.,weootnvokeheplanestresshypothesis.

Remarks:1 .heontinuumequirementnheisplacementieldthenterfaceetweenthand

(I+)stboundedayersecessitatesheatisfactionftheontactonditions;aUa+l fo ra=1,2),ndU^ l)=U^ l+1).heseconditionsreinherentlysatisfiedviathedefinitionoftheassumeddisplacementfieldasdefinedin2)nd3) .

2.heinter-laminarstresscontinuityrequiresthesatisfactionofthefollowingconditionsattheinterfaceofIthand(l+l)stlayers;r =r ) fo ra=1,2),andT =T +1),where7 - ^ 3denotestheinter-laminarstressesfo rlayerI. tisimportanttonotethattheassumeddisplacementfield2)esultsinindependentnon-normalcross-sectionalrotationsineachfiniteelementlayerwhichisinaccordancewiththeMindlinkinematicassumption.Acon-sequenceistheindependentsheardeformationof thedirectorineachlayer,thusallowingthewarpingofthecompositecross-section.hereforetheresultingstrainfieldisdiscon-tinuousacrossdifferentmaterialsetsandcreatestheprovisionof stresscontinuityacrosstheinterfaces.Consequently,thetractioncontinuityequationsareinherentlysatisfied.


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A.MasudandM.Panahandehfiniteelementformulationforcomposites3.KinematicDescriptionof Multi-layeredShellsFigure hows ompositeaminatewithm'umberfmaterialayers.Forasefpresentation,nFig.,m=).onsequently,heotalhicknessTsividedntoWelementsthroughthethickness.Eachlayerofelementsisassociatedwithareferencesurfacewhich,orasefdiscussion,sssumedoeoincidentwithheowerurfacefthatlayer.tsmportantootehatheirectorotationaandhelope3)Q3shedisplacementield3projectedntoheeferenceurfaceorheayer)renotecessarilyequalndhusransverseheartrainsreccommodated.hissoeontrastedwiththeclassicallaminationtheoriesinwhich8a 3,a -Withineachlayerthedirectorrotatesby \generatingheartrain7^].onsequently,nheeformedonfiguration,ode(seeFig.)movesoocation2'.ow ,hedirectornthesecondlayerrotatesbynngle0a+l\generatingheartrainy Thiskinematicssepeatednsuccessiveayersnd,duetothecontinuityofthedisplacementieldinthethicknessdirection,weobtainthenewlocationsofnodalpoints(thatconstitutethelayerwisedirectors)asl',2','and4Thisnewlocationofpointsproducesahigher-ordervariationofthein-planedisplacementfieldthroughthehickness.ccordingly,rbitrarypolynomialxpressionsreotequiredomodelhehigher-orderin-planevariationofdisplacementthroughthethicknessofthecomposite.3.1Kinematicsofanindividuallayerintheontextofthefinitelementmethod

ThedisplacementfieldofeachlayerIintheshellisdefinedbythefollowingrelationsuW( ,rj,0(l)&V )U ( ,7 7 ,C ) 4)(0( )=- a( )7 )(


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A.MasudandM.Panahandeh Afiniteelementformulationforcomposites

nimnllll'imll

Fig.. Shellkinematics.Variationoftransverseshearstrainsthroughthethickness.4= ^w>i i(i+cK+8

whereu )+andu arethetranslationsoftheupperandlowersurfacesoflayerl,^ l)sthedisplacementofapointonthereferencesurfaceoflayerI,U{1)sthe'directordisplacement'fo rayer,Naepresentswo-dimensionalhapeunctionssociatedwithodea',nenarethenumberofelementnodes,ndZ sthethicknessfunctiondefinedas

z?)(o=|(i+o)+-i-)2iI ) (9 )wherezil)+\(l-C )|Z||ndZ?{l+C )l^ll-ere< * >=


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A.MasudandM.Panahandehfiniteelementformulationforcompositesstretchasshownbyanadditivedecomposition:#=(2 +-2')e2/whereu{and.(*. ? 414)

a=l

a=lXil)(C)=ZP(C)X (nosum)1 6)


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A.MasudandM.Panahandeh Afiniteelementformulationforcomposites

Fig. . Mesoandmacrostructureofthecomposite.

taiit

=constant

Fig. . Geometricdescriptionofanindividuallayer.wherexshepositionectorfagenericpointnayer,xshepositionectorfapointnheeferenceurfaceorayer,X(Z)shepositionectorf genericointrelativetoxwhatdefinesthedirectorthroughthepointfo rlayerI, incomputationalshellliterature,Xisreferredtoasfiberdirection),xi)=Z /\\Z \\isaunitvectoremanatingfromnode'a'inthedirectordirection,andZ^}isthethicknessfunctionassociatedwithnode'a'sdefinedin9).hisfunctionisdefinedbythelocationofthereferencesurface.asthepositionvectorofnodalpointa'nlayerIandisdefinedas

4)| i-c)*F+| i+ c)*if)+ (17)


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A.MasudandM.Panahandehfiniteelementformulationforcomposites5.ConstitutiveRelationsMostcompositesaremadeofarepeatedsequenceoflaminatesthat havethesamematerialpropertiesbutareorientedat+9and-6degreeswithrespecttotheextensiondirection.LetC'W epresentheinearonstitutivematrixornndividualaminatewithegardotsmutuallyperpendicularplanesfelasticsymmetry.hisconstitutivematrixfo r laminatecanbeprojectedfromitsmutuallyperpendicularplanesofelasticsymmetryontotheglobalcompositecoordinatesystemviaatransformationmatrixQ^>whichisafunctionoftheangle8.ThetransformedconstitutivematrixC sobtainedas

CT(0=QW)C{i)Qd) 1 8 )6.heVariationalFramework

WetartwithheHu-Washizuariationalrinciplehatsesheisplacement,trainandtressieldssndependentariablesndhusrovideshemostgeneralrameworktoformulateheroblemnderivetsweakorm.eenotehetrainieldy andtsvariationbya.imilarly,wedenotehetressieldyrandtsvariationby.inceoboundaryonditionsrepecifiednhetrainrtressields,heolutionpacendhespaceofadmissiblevariationscoincidefo rbothstrainsandstresses.

5:={e|[L2(n)rdKd+1)/2}T:={


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A.MasudandM.Panahandehfiniteelementformulationforcomposites0field(2)-(3)resultsinanenhancedstrain fieldbecauselayerwiserotationsresultinenhancingtheheartrainomponents.Wewritehetrainields =Vsu+e,whereVsushesymmetriccompatiblestrainandiistheenhancedstrain.Thecorrespondingstrainvariationisa=Vsa;+,whereVu>isthecompatibleandistheenhancedstrainvariation.Sincenoboundaryconditionsreprescribedontheaugmentedstrainsswell,bothspaces.e.,iandconincide.

:={|ee[L2(fi)p''(n a +1)/2}Weanowubstitutehisnhancedtrainieldn1 9)obtainnnhancedtrain

versionoftheHu-Washizuformulation.nHW(e(0),< r ,u)=/(V3u+i(9)) C Vsu+e{9))du-r i


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A.MasudandM.Panahandehfiniteelementformulationforcomposites1g Yg- TU '*-1,h:ThIT'*-1,findu,9}Ssuchthatfo rall{w,V}V n(u,0;u>,)= fV au+&{9)):C:(Vau+i{0))dn- ufdu- uhd

(24)whereVs(-)isthesymmetricgradientoperator,/denotesthebodyforcevectordefinedover0, epresentsheprescribedractionsnormalndhear)ve rboundaryTh,andgaretheprescribedisplacementsndotationsefinedverboundaryTg.ereTg=TuUTewithreingtheboundarywithprescribeddisplacementsandTebeingtheboundarywithprescribedrotations.

Nowonsider shellonsistingof'm'generallydirectionalbutperfectlybondedayers,andeferenceurfacefeachlayer< z)iscretizedntorwei'initelements.ete =^e(0x[-t/2,t/2](l),fi(0=Ur=T'e('andfi=in'where e(istheelementdomaininlayerI.Wecanwritetheprincipleofvirtualwork(24)forthelayeredbutperfectlybondedanisotropiccompositelaminatesas

fi(, 0;, t f )[J ( V so ,+())(Z C > (V'u+*(*) )d f t- / WWfMdSlY {l)h{l)dT25 )

Let7=Vn3-0betheshearstrainvector,where3sthedisplacementfieldprojectedontoheeferenceurfaceorayer.imilarly,etK=Vs9beheurvature,whichsalsoefinedlayerwise.Weanwriteheenergyunctional25 )orheayerednisotropiclaminatessn(,;a,^)El/ (WCffW>+7(I)C ? >7(i)+*WC)d f l

[=1n')_ fWdti-E/0 (/) f c (l)dr)

where7Wnd


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A.MasudandM.Panahandehfiniteelementformulationforcomposites2surfaceforlayer I.Theassumeddisplacementfieldin(2)and(3)leadstoamodifiedfunctionalII,definedasn(u,0;u; )=f[f eWC7(Z )dAjr((JnwAW

Tl,d+f*WCil) KdA- wf d-Y[ Wh^dT I (27)7^(0awzlJrWwhereJn(l)( )duisavolumeintegralndJA(l)()dArepresentsanareaintegral.Weddthefollowingboundaryconditionstotheenergyfunctional(27)sua=ua on TUa a=1 ,228 )u3=u3 on TU3 29)0a=a on ra a=1 ,2 30)whereTu=TUaU TU3epresentsheboundarywheretranslationboundaryconditionsreapplied,ndTga orrespondstotheboundarywithprescribedrotations,u^uajandoarep-resenttheprescribeddisplacementsandrotations,respectively.naddition,weprescribethetractionsas

T n=T on rT 31 )qn= on Tq 32)m-n=h on Tm 33)

randqrepresentthenormalndsheartractions,respectively,whilerhrepresentsthepre-scribedmoments.hisleadstoadecompositionofThasTh=rTurjUrmrrurrepresentstheoundarywhereractionboundaryonditionsrepplied,ndm orrespondsoheportionofboundarywithappliedmoments.

Thestrainvectorsfortheproposedcompositeelementare.(011

K{1) >oo / ) 22 [ ' ~ 1 (l ) f -2 2e33723 * W -12 e(/)ze12


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A.MasudandM.Panahandeh Afiniteelementformulationforcomposites 1 3where,7,/cnd epresenthen-plane,hear,endingndhrough-thicknesstrains,respectively.Wecancombinethein-planestrainswiththrough-thicknessstrainstoyield

=4T3 5 )Thestressescorrespondingtothestrainsare

11~22T33 r12

9= =(011

, 02 2 (0 12

(36)

wherer epresentsthecombinedin-planeandthethrough-thicknessstresses,ndqandmrepresenttheshearandbendingstresses,respectively.

Theelementstiffnessmatricesandforcevectorsemanatingfromtheweakform,thatareobtainediaGalerkinpproximationndntroductionfelementhapeunctionsanewrittenas m

1= 1whereAj*J shefiniteelementssemblyoperator.helementstiffnessmatrixcanex -plicitlybewrittenask(l )=f gfl)Cl)(D dn+f 7WCf7(0A+ * < * >CfWAin)/A>>vAWmodifiedmembrane shear bending (38)whereC$,Cl)andcjare themodified-membrane,shearandbendingconstitutivematricesforlayerI,respectively.Therighthandsideforcevectorcanbewrittenas

= E{AS-(jf^ ( I )/( I )* > + 4,I )h dr } (39)wherethefirsttermistothebodyforcevectorandthesecondtermisthetractionvector.Remark:Wehaveusedselectivereducedintegrationconceptsfo rtheevaluationof theelementstiffnessmatrices.


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A.MasudandM.Panahandehfiniteelementformulationforcomposites4 7.NumericalExamples7.1Freedgeboundary-valueproblem. [45,-45]sTheirstumericalimulations prismaticymmetricaminateavingraction-free

edgesatY=aandsurfacesZ=h,andissubjectedtostrainincrementonlyonitsendsatX=constant.Eachlayeriscomposedofuni-directionalfiber-reinforcedmaterialsuchthatthefiberdirectionisdefinedbyitsangle8withrespecttotheX-axis.TheelasticpropertiesusednthisimulationaretakenfromPagano12],..helaminateconsistsoffouruni-directionalfibrouscompositelayers,twowithaxisofelasticsymmetryat+45andtwoat 45tothelongitudinallaminate axis.SeeFig..)%straininoppositedirectionsisappliedatX=L,whileitisrestrainedtomoveintheaxial,lateralandthicknessdirectionsatX=0.ThephysicaldimensionsorhenumericalsimulationareX=60 ,Y=20 ,Z=2.5,with2elementsinXdirection,40elementsintheYdirectionand4elementsthroughthethickness.Forachinitelementayerhroughhehickness,heeferenceurfacesssumedoecoincidentwiththebottomsurfaceofthelayer.

Figure howshexialdisplacementdistributionthetopreesurfacefthesectioncuttX ,ndheesultsreomparedwithMoirest l.ndPaganot l.12].Figure6presentstheinter-laminarshearstressatmaterialinterfaceandtheircorrespondingvaluesfromReddy20].Figure7showsthemajorstresscomponents


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A.MasudandM.Panahandeh Afiniteelementformulationforcomposites 1 5

4hI

Figure4.Laminategeometryandthecoordinatesystem.0.00500.00400.00300.00200.00100.0000

-0.0010-0.0020-0.0030-0.0040-0.0050

, resentmodels Moirest. lV'a

aganot. d 's: /\- \ \

- \\

i-1-0.500.400.300.200.10 0.00 0.100.20 0.300.40 0.50NormalizedwidthFigure5 .Axialdisplacementdistributionatthetopfreesurface.

7.2 Freedgeboundary-valueproblemwithacylindricalhole. [45,-45]sThecompositelaminateconsideredinthissimulationhasthesamephysicaldimensions,

materialropertiesndoadingonditionssnhereviousase.oweverheresentlaminateas nitiameterylindricalolet0,0,0)withtsxisoincidentwithheZaxisseeFig.).heatiofthediameterftheoleohewidthftheaminates0.1.Traction-freeboundaryconditionsareappliedonthecylindricalsurfaceofthehole.The


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A.MasudandM.Panahandeh Afiniteelementformulationforcomposites 1 630-i resentmodel20- \\ .N.Reddy

~ IO - y -,CS 2 0. V>-- -10- *\ \

-20--30- V -40.

-0.500.400.300.200.10 0.00 0.10 0.20 0.30 0.40 0.50Normalizedwidth

Figure6.nterlaminarshearstressatmaterialinterface.252150I 100

w 9 50-5 0-100

22E22ZXSXS- rrr-ggragSPg,

tSSSSSSs-SBS- BWBWE2Z tggnn~ -gTT77g

Cl2Pipest.l

-0.500.400.300.200.100.00 0.10 0.20 0.30 0.40 0.50Normalizedwidth

Figure7.Majorstresscomponentsinthetop+45aminate.laminateisconstrainedtomoveintheaxial,lateralandtransversedirectionsbyappropriatelyconstrainingthenodesalongthesymmetrylinesatX=0.

Figure0presentshemajorstressomponentsan,ayz, ~ \zat-sectionuttX=0inheopayerwith+45egreeslyorientation.heseesultsavebeenomparedwithcomplete3Danisotropiccalculationsdonewithameshwhichistwiceasrefinedasthepresentmesh.ntheregionawayfromthefreeedgeboundaryandthetractionfreehole,theratioof


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A.MasudandM.Panahandeh Afiniteelementformulationforcomposites 1 7

'.0.50-0.40-0.30-0.20-0.10 0.00 0.10 0.20 0.30 0.40 0.50Normalizedwidth

Figure8 .Minorstresscomponentsinthetop+45laminate.anandc r12greescloselywiththatoftheprecedingnumericalsimulation.Figure1howstheminortressomponentswhichlsohow goodgreementwithheDnisotropicsimulation.

4 h I

Figure9.Laminategeometrywithacylindricalhole.


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A.MasudandM.Panahandeh Afiniteelementformulationforcomposites 1 840035 030 0

,- >50200

3t/J 1 501 00500

-5 0

C\2| -13

..H.D-Anisotropicg i- =s55a -a a a HS 3 - a SaH -

_g...B -g...a--ga a a g a a.-a- 'sgasg' .-0.00 0.05 0.10 0.150.20 0.25 0.30 0.35 0.40 0.45 0.50

NormalizedwidthFigure0.Majorstresscomponentsinthetop+45laminate.

72 2T3 3 23

aD-Anisotropic

n -5& 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50

Normalizedwidth

Figure1 .Minorstresscomponentsnthetop+45laminate.8.Conclusions

Inhisaperweaveresented initelementormulationwhichsuitableorheanalysisfmulti-layered/multi-directorandshear-deformablecompositelaminates.hege-ometricdescriptionemployedfo rthecompositeshellsfindsitsrootsintheso-calleddegener-atedshellapproach.Asetofkinematichypothesisisintroducedtoaccommodatetheeffectsoftransversewarpingftherossectionueoheareformationlongwithiberom -


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A.MasudandM.Panahandehfiniteelementformulationforcomposites9possibilitythatresultsfromtransversenormalstresses.Thekinematicsareindividuallyandindependentlyrepresentedfo reachlayerandarecoupledviatheconditionsofdisplacementcontinuitybetweencontiguouslayers.Therotationfieldishoweverlayer-wisecontinuousandisassumeddiscontinuousacrossthelayers.Transversalwarpingofthecompositecross-sectionisdescribedbyindividuallayerdirectorswhichsimultaneouslyrotatendstretch.hisre-sultsniscontinuoustrainieldscrossheifferentmaterialets,herebyreatingheprovisionfstressontinuitycrosshematerialnterfaces.inceheormulationesolvesal ltrains,lltressesreomputedhroughhehree-dimensionalonstitutivequationsandtheusual'zeronormalstress'shellhypothesisisnotemployed.ThevariationalequationcontainsonlythefirstderivativesofdisplacementandrotationfieldsthatrequirejusttheCcontinuityffinitelementunctions7].ndisplacementormulationfplatesndhells,specialttentionneedsoegiventoransverseshearandmembranetermsopreventheoccurrenceofmeshlocking.Wehaveemployedtheselective/reducedintegrationtechniquetoavoidthisproblem.Numericalresultsarepresentedthatdemonstratethegoodperformanceoftheproposedformulation.

AcknowledgementsTheuthorswishohankProfessorsThomas.R.HughesndRobert.ayloror

helpfulcomments.ThisresearchwassupportedbytheAirForceOfficeofScientificResearchunderContractF49620-94-C-0084,projectdirectorDr.WalterF.ones.

References[ 1 ].Argyrisnd.Tenek,Anfficientndocking-freelatnisotropicplatendhell

triangularelement ,orn-put.MethodsinAppl.Mech.Engrg.,118,3-119,994.[ 2 ]W.S.BurtonandA.K.Noor,Assessmentofcomputationalmodelsforsandwichpanels

andshells ,Comput.MethodsinAppl.Mech.Engrg.,124,25-151995.[ 3 ].ByunandR.K.Kapania,Predictionofinterlaminarstressesinlaminatedplatesusing

globalorthogonalinterpolationpolynomials ,AIAAJournal,30,No1 1 ,November1992.


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Masud - A FEM Formulation for Multi-layered Shells

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