strength caracteristics of composites materials

7/28/2019 Strength Caracteristics of Composites Materials

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CONTRACTORREPORT

i

a : AEprovecar puciiceieast'I

1 9 9 6 0 5 0 3 0 2 5

NASA CR-224

HARACTERISTICSFOMPOSITEMATERIALSyStepbenW.TsainderContractNo.NAS7-215 yORPORATIONBeach,Calif.TIC QUALITYIK3FBCISSDI'

rERONAUTICSNDPACEDMINISTRATION WASHINGTON,..


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NASACR-224

STRENGTHCHARACTERISTICSOFCOMPOSITEMATERIALSByStephenW.sai

Distributionofthisreportsprovidedinthenterestofinformationexchange.esponsibilityforthecontentsresidesntheuthororrganizationthatpreparedit.PreparedunderContractNo.AS7-215by PHILCOCORPORATIONNewportBeach,alif.

forNATIONALAERONAUTICSANDSPACEADMINISTRATION

ForoleTjy~^k


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ABSTRACTThetrengthcharacteristicsfquasi-homogeneous, nonisotropic

materialsrederivedfrom generalizeddistortionalworkcriterion. Forunidirectionalomposites, thetrengthsovernedbyhexial, transverse,andheartrengths, andhengleffiberrientation.

Thetrengthofalaminatedcompositeonsistingflayersfuni-directionalcompositesependsnhetrength, thickness, andorientationofeachconstituentlayerandheemperatureatwhichhelaminatesured.Inheprocessflamination, thermalandmechanicalnteractionsrenducedwhichffectheesidualtressndheubsequenttressistributionunderexternaload.

Amethodoftrengthanalysisflaminatedcompositesselineatedusinglass-epoxycompositess examples. Thevalidityofhemethodsdemonstratedbyappropriatexperiments.

Commonlyncounteredmaterialconstantsndcoefficientsortressandtrengthnalysesorglass-epoxycompositesrelistednheAppendix.


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SECTION

CONTENTS

REFERENCESAPPENDIX. .

PAGE1NTRODUCTION

StructuralBehaviorfCompositeMaterials....ScopefPresentInvestigation2TRENGTHOFANISOTROPICMATERIALSMathematicalTheoryQuasi-homogeneousCompositesExperimentalResults3TRENGTHOFAMINATEDCOMPOSITES

MathematicalTheory 9Cross-plyComposites Angle-plyComposites J

4ONCLUSIONS 35759


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ILLUSTRATIONS Figure. ComparativeYieldSurfacesFigure2. CoordinateTransformationofStressFigure3. TensileTestSpecimens 4Figure4. StrengthofUnidirectionalComposites 6Figure5. StrengthofaTypicalCross-plyComposite7Figure . StrengthofCross-plyComposites 9Figure7. ThermalWarpingofaTwo-layerComposite0Figure . StrengthofAngle-plyComposites 1


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NOMENCLATURE A= In-planetiffnessmatrix, lb/in.A"~'" = Intermediaten-planematrix, in./lbA'' = In-planeompliancematrix, in./lbijBStiffnessouplingmatrix, lbB'""~ = Intermediateouplingmatrix, in.B11 = Complianceouplingmatrix, 1/lbCnisotropietiffnessmatrix, psiijD= Flexuraltiffnessmatrix, Ib-in.D'= Intermediateflexuralmatrix, lb-in.D'' = Flexuralompliancematrix, 1/lb-in.Eoung'smodulus, psiE,xialtiffness, psiH'~" = Intermediateouplingmatrix, in.hlatethickness, in.M.= Distributedbendingandtwisting)moments, lbTM.= Thermalmoment, lblM= EffectivemomentM.M.lmos 0, or

=ross-plyratiototalthicknessfoddlayersverhatofvenlayers)


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NOMENCLATUREContinued)N.Ntressesultant, lb/iiTN.N = Thermalforces, lb/in.l

N.N = Effectivetressesultant N. N.l lnsin or= totalnumberflayerspRatioofnormaltresses a~l,qRatioofheartress a /io 1 rRatioofnormaltrengthsX/YSSheartrengthfunidirectionalcomposite, psisSheartrengthatioX/sS-Anisotropieompliancematrix, 1/psiTTemperature, degreeFTCoordinatetransformationwithpositiveotationTCoordinatetransformationwithnegativeotationXAxialtrengthofunidirectionalomposite, psiYTransversetrengthofunidirectionalomposite, psiaiThermalxpansionmatrix, in./in./degreeF( Straincomponent, in./in.o(^In-planetrain, component, in./in.


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NOMENCLATUREContinued)= Fiberrientationorlaminationangle, degree

KCurvature, 1/in.l1 - v u 2 21

aStressomponents, psilT. = Sheartress, psivPoisson'satiov ^ - MajorPoisson'satio12vMinorPoisson'satioSUPERSCRIPTS+Positiveotationortensileproperty= Negativeotationorcompressivepropertykk-thlayernalaminatedcomposite-1InversematrixSUBSCRIPTSi} j = 1, 2, ... 6rx, y, zn3-dimensionalpace, or

= 1, 2, 6rx, y , sn2-dimensionalpace


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SECTIONINTRODUCTION

StructuralBehaviorofComposi teMaterialsThepurposefhepresentinvestigationisoestablisharational

basisfhedesignsfcompositematerialsortructuralapplications.Ultimately, materialsesigncanbeintegratedntotructuraldesignasnaddeddimension. Higherperformanceandowerostnmaterialsndstructurespplicationsanthereforebexpected.

Followingheesearchmethodoutlinedpreviously, ' thepresentprogramombineswotraditionalareasfresearchmaterialsndstructures. Thesewoareasreinkedbyamechanicalconstitutivequa-tion, theimplestformfwhichshegeneralizedHooke'saw. Themate-rialsesearchsoncernedwithhenfluencesfheonstituentmaterialsonheoefficientsfheonstitutivequation, whichnthisase, areheelasticmoduli. Thetructuresesearch, onheotherhand, isoncernedwithhegrossbehaviorfananisotropicmedium. Anintegratedstructuraldesigntakesntoaccount, inadditionohetraditionalvariationsnthick-nessesndhapes, thecontrollablemagnitudeanddirectionofmaterialpropertieshroughheelectionofproperonstituentmaterialsndtheirgeometricarrangement.

'Referencesrelistedathendofthiseport.


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Followingheframeworkjustdescribed,helasticmoduliofaniso-tropicaminatedcompositeswereeportedpreviously.2'3heappropriateconstitutivequationwas:

"N "

M =

"AI

B ~ " tiB D K (1)

Thisquation, ofcourse, includedhequasi-homogeneousorthotropicom-posite, whichrepresentedaunidirectionalcomposite, as pecialcase.ThematerialcoefficientsA, B, andDwerexpressedntermsfmaterialandgeometricparametersssociatedwithheconstituentmaterialsndhemethodoflamination. Thisnformationprovidedaationalbasisforhedesignofelastictiffnessesfananisotropiclaminatedcomposite. Thus,theinvestigationreportednReferences nd nvolvedbothtructuresresearch, inhestablishmentofEquation1)snappropriateonstitutiveequation, andmaterialsesearch, inhestablishmentofheparametersthatgovernhematerialcoefficientsfEquation(1).

Thepresentreportcovershetrengthcharacteristicfanisotropiclaminatedcomposites, whichagainncludesheuasi-homogeneousom-posite, as pecialcase. Unlikeheasefhelasticmoduli, thepresentreportoversnlyhetructuresspectoftrengths;hematerialsspectisobeinvestigatednhefuture. Theappropriateonstitutivequationforthetrengthcharacteristicssstablishednthiseport. Onlywhenthisinformationsvailable, canheareaofresearchfromhematerialstand-pointbedelineated. Guidelinesforheesignofcompositesfromhestrengthconsiderationcanbeerived.

ScopeofPresentnvestigationThepresentinvestigationsoncernedwithhetructuresspectof

thetrengthcharacteristicsfcompositematerials. Thetrengthofaquasi-homogeneousnisotropicompositeisfirststablished. Thenthe strengthofalaminatedcompositeonsistingoflayersfquasi-homogeneous


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compositesbondedtogetherisnvestigated.hevalidityofhetheoreticalpredictionssemonstratedbyusinglass-epoxyresincompositesstestspecimens.

Themainresultofthisnvestigationshatamoreealisticmethodofstrengthanalysishanheprevailingnettingnalysissbtained. Thestructuralbehaviorfcompositematerialssnowbetternderstood, andonecanusethesematerialswithhigherprecisionandgreateronfidence. A. stridesmadetowardheationaldesignofcompositematerials. Althoughmuchmoreanalysesnddatagenerationstillemainobedone, thepresentknowledgeoftiffnessesndtrengthsfcompositematerials, aseportedinReferences nd3, andnthiseport, respectively, ispproachinghelevelofknowledgepresentlyavailablenheseofisotropichomogeneousmaterials.


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SECTIONSTRENGTHOFANISOTROPICMATERIALS

MathematicalTheorySeveralstrengththeoriesofanisotropicmaterialsarefrequently

encounterednhestudyofcompositematerials. Hillpostulatedatheoryn19484andaterepeatedtnhisplasticitybook. Usinghisnotation, itisassumedthattheyieldconditionsaquadraticunctionofthestresscomponents

2f(a..) = F(ay - of + G( -a + H( - a+ r 2 + 2M T I + 2 r fyzxy (2)whereF, G , H, L, M , Narematerialcoefficientscharacteristicofhestateofanisotropy, andx, y, zareheaxesofmaterialsymmetrywhichareassumedtoexist. ThisyieldconditionsageneralizationofvonMises'conditionproposedn1913orsotropicmaterials. NotehatEquation2)

reducesovonMises'conditionwhenthematerialcoefficientsareequal.Beyondthisnecessarycondition, thereseemsobenofurtherrationale.Nevertheless, thisyieldconditionhasheadvantagesofbeingeasonableandreadilyusablenamathematicalheoryofstrengthbecausetsacon-tinuousunctionnthetresspace. Fordentificationpurposes, thiscon-ditionwillbecalledthedistortionalenergy condition.


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Marinproposed atrengththeoryquivalenttoEquation2), excepttherincipaltressomponentsweresedinsteadofthegeneraltresscomponents. Thesefprincipaltressess, infact, moreifficulttoapplytonisotropicmaterials, sincehexesfmaterialymmetry, theprincipaltress, andtheprincipaltrainare, ingeneral, notoincident.Thus, principaltressespereonothavemuchphysicalignificance.

Anothertrengthheoryfanisotropicmaterialsalledthe "inter-actionformula, sescribedby eriesfeportsytheForestProductsLaboratory ' ' andpparentlyndependentlyyAshkenazi. ThenteractionformulanHill'sotation' takesheollowingorm:

()m

y + yX Ya,,a7ay"z + (_ + zYZ T \ \2,2 (3)

,r\2oo \2z "x / x \ ,/zx1+ U? + ^lZ XX /RSinceheompositematerialfinterestsowsntheormfthinplates,tatefplanetresssssumed.henEquations2)nd3)anbeeduced, respectively:

x\2gxfyay \2 x /rxy \2IT) - XY + (l)MS J :14> TheheartrengthssedherereQ, R, SatherhanR, S, T, inorderopare ortemperature.


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o2oo. '2xx y /: _ y + utz.X /Y r (5 )= 1

Thedifferencebetweenheieldconditionofdistortionalenergy, thenter-actionformula, andvonMisess shownnFigure, assumingtensileandcompressivetrengthsfhematerialsrequal.

Forhepresentprogram, itsssumedthatheistortionalenergyconditionsalid. This, ofcourse, willbeubstantiatedexperimentallylaternthiseport. Itslsoassumed, forhepresent, thatfailurebyyieldingandbyultimatetrengthareynonymous. Thiswillbehowntobereasonableforlass-epoxycomposites, whichexhibitlinearlyelasticbehaviorpofailuretresswithlittleornoielding. Theworkcontained7 8 90intheForestProducteports ' ' andAskenazi hadwoestrictions:(1)oifferentiationwasmadebetweenhehomogeneousndlaminatedcom-posite, (Z ) sheartrengthwasottreatedasnndependenttrengthprop-erty. Inhepresentinvestigation, boththeseestrictionsreemoved.

Quas i -homogeneousCompos i t esThetrengthofquasi-homogeneousnisotropycompositeswas

reportedbyAzziandTsai.orheakeofcompleteness, thessentialpointsfthiseferencereepeatedhere.

Itshepurposeofthisectionodemonstratehowhedistortionalenergyconditioncanbeappliedoaquasi-homogeneousnisotropycompositesubjectedocombinedtresses. Onefhebasicssumptionsfthisondi-tionsthattherexistthreemutuallyperpendicularplanesfymmetrywithinheanisotropybody. Thismeansthathebodyseallyorthotropicratherhangenerallyanisotropicfromhepointofviewofstrength. Under


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Z2.Y

ANISOTROPICYIELDCONDITIONSX = 1 (vonMises)

X

INTERACTIONFORMULA

Figure. ComparativeYieldSurfaces


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thisssumption, theieldconditionmustbeppliedohetateoftressexpressednhecoordinateystemcoincidentwiththatofhematerialsymmetry. Thus, thetateftressmposedonabodymustbetransformedtoheoordinateystemofmaterialymmetryandhenheieldconditionapplied. Letx-ybehematerialymmetryaxes, and-2, theeferencecoordinateaxesfhexternallyappliedtresses, thesualtransformationequation inmatrixforms.

" " CTX

Oy

sL J

2mn2n2mn

2 2-mnn- ~ ala2abL J (6)wherem os 6, n in 0, andpositive 9shownnFigure2.

*> 1

Figure. CoordinateTransformationofStress


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Foronvenience, theollowingnotationsresed:p = Va\ q = C T 6/CT 1' r = X/Y S = X/S7)

SubstitutinghenotationsnEquations6)nd(7)ntoheieldconditionntheformofEquation(4), oneobtains:

T222 4 , , f.2 , . 2l 3 [1-p+pr +qsj m +2q 13 - r +p 1) mn+ [82 (p 2) r2 p l)2s2 - 1) -22S2J m2n2

(8)4-72\ 21a . T 2- 2 X 2 2 * 1tZq 3 - -(p-l)s mn + lp -p+r +q s n- (X/ax)2 = (rY/Cj)2 = (sS/ 2

Thisesultmaybeummarizedasollows: Foragivenanisotropicbodynreferenceoordinates 1-2, specifiedbyX, Yor), andS(or), withagivenorientationofhematerialymmetryaxes,d andsubjectedoom-binedtresses o a^orp)nd, (or), themagnitudeftheappliedstress o atfailure, canbedeterminedbyolvingEquation8)or a,.Alternatively, Equation(8)maybeegardedashetransformationequationforhetrengthofaquasi-homogeneousnisotropicmaterialubjectedocombinedtresses;.e. thetrengthcharacteristicss functionofhe orientationofheymmetryaxes,6.

Foruniaxialtension, p , thefailureconditions4+s2 1)m2n2 2n4 = (X/ .)2 9)

rrVr4X/2N22 4]o - X/ m +s - 1)m n + n 1/2 (10)10


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Thus, byperforminguniaxialtensiontestsnpecimenswithdifferentorientationsfhematerialymmetryaxes;.e., differentvaluesf & , onefindsirectlyhetransformationpropertyofstrength. Whatsquallyimportantsthathetrengthcharacteristicsofaquasi-homogeneousniso-tropicmaterialunderombinedtressesreimultaneouslyverified. ByasimpleubstitutionofEquation(6)nto9), whilemaintainingp , onerecovers, asxpected, theriginalyieldconditionshownnEquation(4).

Equation(8)anbeeducedootherimpleasesnatraight-forwardmanner. Forxample, theaseofhydrostaticpressureequiresp 1, q , fromwhichoneanshowthathemaximumpressuresqualtohetransversetrength, Y , andsndependentofherientation, v

Thecaseofaninternallypressurizedcylindricalhellsescribedbyp , q , fromwhichEquation(8)educeso

(4r2 1)m4+4r2 1 2)m2n2+r2 2)n4 = (X/)211)Forsotropicmaterial, itcanbehownthat

r = 1 = V~3~ 5whichagreeswithvonMises'ondition. Equation(11)henreduceso

1 X/V3and12)

C T2x/yr .issxwhichshewell-knownresultbetweenhemaximumhooptress a andhe 5uniaxialtrengthX.

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Theaseofpurehearanbederivedbyletting = C T? nEquation(6), andhenbyubstitutingtntoEquation(4),*neobtains

4m2n2r2+2)/s2 + (m2 2)2 = (S/of13)or

b S/Um2n2r2+)/S2 + (m2 2)2]14)Itisnterestingonotehat:

when6 = 0or0 = S 15)when6 = 45 = X/r2+] 1/2

= Y , ifr >> 116)= X/v3, ifr 1 (isotropy)

Inconclusion, itseenthathedistortionalnergyconditioncanbeeasilyappliedocasesfrequentlyncounterednhedesignanduseofaniso-tropicomposites. Thetrengthcharacteristicsnvolveheaxial, trans-verseandheartrengths, X, Y , andS , respectively, andherientationofthematerialymmetryaxes, Thistrengththeoryisuitedifferentfromthenettinganalysis, whichistillusedextensivelynhefilament-windingindustry. Thenaccuracy ofnettinganalysiss theoryordesigncriterionisfarlessamagingperehanhenfluenceofitsrroneousmplicationsonmanyrecentandevencurrentesearchprogramsnfilament-winding.

Equation(8)annotbeseddirectlyforthisasebecause cr1squalozero.ThissheheartrengthusednMarin'stheory. Its derivedquantity, aspposedoX, Y , andS , whichareheprincipaltrengths.

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Exper imenta lResultsInheprecedingubsection, theutilityfromhemathematicaltand-pointofyieldconditionasppliedoaquasi-homogeneousnisotropicom-

positehasbeenoutlined. Inthis subsection, experimentalresultswhichdemonstratehevalidityofheproposedtheoryoftrengthwillbeeported.

Thepecimenssedweremadeofunidirectionalglass-filamentspreimpregnatedwithpoxyresin. Thismaterials suppliedbyheU.S.PolymericCompanywithadesignationofE-787-NUF.* Thecuringycleinvolvednopreheat, 50psipressure, and300Ftemperaturefor hoursfollowedbylowcooling. Tensiletestpecimenswerecutfromheuredpanelssingwet-bladedmasonryaw. Astwasoundthatpecimensofuniformcross sectionhadaendencyofailunderhegripstowanglesoffiberorientation, adiamond-coatedrouterwassedohapepecimenswithaeducedtestection, indog-bone"fashion. Approximatepecimendimensionswereinnches): overallength, 8.00;verallwidth, 0.450;lengthoftestection, 2.50;widthoftestection, 0.180;hickness, 0.125.A3-inch-radiusirculararc, tangentohetestection, connectedhetestsectiontohemaximumndection. Additionally, aluminumabsacata-loguetem)werebondedohendsfhepecimensodistributeheoadsimposedbyhegrips. A.specialfixturewasevised: (1)oalignthetabswithhepecimensonsureapplicationofpureaxiaload, and(2)obecapableofmakingpo20ndividualpecimensimultaneously. Samplespecimens, beforeandaftertest, arehownnFigure.

Thevaluesfhexialandtransversenormaltrengths andYforthematerialemployedweredeterminedfromimpletensiontestsfpeci-menshavingfiberrientationsf nd "72oheirectionofappliedstress,respectively. TheheartrengthSwaseterminedfromheimpletorsiontestofafilament-woundthin-walledtorsionubehavingallcircumferentialwindings.;TheamematerialwassedomaketestpecimenseportednReference2.

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Figure. TensileTestSpecimens

1 4


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Toverifyhetheoreticalresults, specimenswerecutat5-degreeincrementsnheowerngleangeswheretrengthvariationsgreatest,andat5-degreeincrementsorhigherngles. ThetrengthsmeasuredforthesepecimenswerehencomparedwithresultsbtainedfromhetheoryevaluatedwithhecorrespondingvaluesforX, Y , andS. Thetheoreticalpredictionusingquation(10), andexperimentalresultsrehownnFigure4. Theesultsndicatethathevalidityofheproposedtheoryofstrengthsemonstrated, asmostmeasuredstrengthvaluesrenagree-mentwiththeoreticalpredictions. ThevaluesforX, Y , andSforheaseillustratedwere50, 4and ksi. Thelackofexcellentagreementatomeofhehighervaluesfmaybecausedbyincreasedsensitivityfhepeci-mendgesohehapingperationandheminutecrazingthatitometimesinduces. Thisensitivityincreaseswithhefiberrientation 6 ence,greatcaremustbexercisednhepreparationofpecimens.

AlsohownnFigure4shetheoreticallypredictedtiffnesssfunctionoffiberrientation, togetherwithexperimentalmeasurements. Thetheoreticalcurve, shownasheolidine, isomputedusingheusualtrans-formationequationofhetiffnessmatrix:

C'n = m4n2m2n2C n4C 4m2n266whereheollowingmoduli, sameashosenReference2, areused:

Cn = 7.97 0 psiC,2 = 0.66x0 psiC22 = 2.66x06psiC16 = C26 = C,, = 1.25 06psiDO

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COco LJ

I CO CO Q_ LUsOI- oJ 7 5 C J oCLoo

10

10070

40

20

n :l C3 h-co

X


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FromEquation10), oneanexaminehevariationofhetransforma-tionpropertyofcompositetrengthwithhebasictrengthcharacteristicsX, Y , andS. TheffectofYsignificantforargeanglesforientation,andheffectofaxialtrength, X, isignificantformallangles. Further,theheartrength, S , becomeshedominanttrengthcharacteristicorn-termediateanglesforientation. Thesenfluencesfachtrengthcharac-teristicmustbeakenntoonsiderationnanyattemptoimprovehestrengthfcompositematerialsavingrbitraryfiberorientationsoheappliedoad.

Itseasonableooncludethathepresentinvestigationofhestrengthofaquasi-homogeneousnisotropicompositendernytateofcombinedstressesanbepredictedwithaccuracy. Thetheoryhasbeendevelopedforhemostgeneralcasefplanetressnddiscussedndetail.Althoughhexperimentconfirmationwasimitedouniaxialension, astateofcombinedtressessctuallynducednheoordinateystemrepresentinghematerialymmetry. Itsssumedhathetensileandcompressivetrengthcharacteristicsrequal. Ifheyarenotqual, onecaneasilyintroduceayX+, X-, Y+, Y", whereheplusndminusuper-scriptsenotetensileandcompressivetrengths, respectively. Nocon-ceptualdifficultysxpectedforthismodification, asndicatedforxampleinReferences nd7.

Forheparticularpecimens, theheartrength, S, fallsbetweenthewonormaltrengths, XandY. Theatioofheheartrengthoverthetransversetrengthare.5orhepecimens. Thisvaluesnotmuchdifferentfrom /a"whichsheatioforisotropicmaterialsr compositematerialreinforcedbyphericalinclusions. Thepresentpecimenhaslowertransversetrengthhanheartrength. Thismpliesthathehearstrengthstaminimumfor45-degreefiberrientation, asanbeeenfromEquations14)nd(16)assuming+ Y"). Thissparticularlyinterestingnviewfhefactthathehearmodulusfommonorthotropicmaterials, whichncludehepresentpecimens, istamaximumat45-degreerientation. Thebehaviorfalaminatedcomposite, ontheotherhand, willbequitedifferentfromaquasi-homogeneousomposite, aswillbeeportednhenextections.

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SECTION3STRENGTHOFLAMINATED COMPOSITES

MathematicalTheoryThetrengthofaminatedanisotropiccompositessdependentonthe

thermomechanicalpropertiesoftheconstituentayersandthemethodoflam-ination, whichncludehehicknessandorientationofeachayer, thestack-ingsequence, cross-plyratio, helicalangle, theaminatingemperature, etc.Intheprocessofamination, twoourcesofinteractionarenduced. First,theresamechanicalnteractioncausedbythetransverseheterogeneityofthecomposite;.e. materialpropertiesvaryacrosshehicknessofthecomposite, andthecross-couplingofthe"16"and"26"componentsofthestiffnessmatrix. Asaresult, thestressacrosshecompositesnotuni-formandsdistributedaccordingoherelativestiffnessesoftheconstituentlayers. Second, theresathermalnteractioncausedbythedifferentialthermalexpansion(o rcontraction)betweenconstituentayers. Sincemostcompositesareaminatedatelevatedtemperatures, initialstressesareinducedftheserviceemperatureofthecompositesdifferentfromheam-inatingemperature. Takingntoaccountbothmechanicalandthermalinter-actions, thestrengthofaaminatedcompositecanbedescribedbyapiece-wiseinearstress-strainrelation. Discontinuouslopesnthiscurveoccurwhenoneormoreoftheconstituentayershavefailed. Theultimatestrengthofthecompositeseachedwhenalltheconstituentayershavefailed.Throughoutthisection, itsassumed, asbefore, thattheensileandcom-pressivepropertiesareequal, andyieldingandstrengtharesynonymous.

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Thetrengthanalysisorhepresentinvestigationsbasedonhe strength-of-materials'pproach. Thegeneralthermoelasticnalysisoflaminatedanisotropicompositessutlinedfirst. Onlyheproblemofhrinkagetresssreatedhere, althoughheanalysisspplicableothermaltressproblemsngeneral.

Forheakeofcompleteness, thebasiconstitutivequationofthermoelasticityandhessentialpointsfReference3reepeatedhere.

Itsssumedthatachconstituentlayerfhelaminatedcompositeisquasi-homogeneousndorthotropic, andsnhetate of-planetress.12 Usinghesualcontractednotations, thethree-dimensionalgeneralizedHooke'sawforanyconstituentlayeris:

e =..a. a.T,1 ij i i, j , 2, . . . 6 (17)Thisquationtatesthathetotaltrainsheumofmechanicaltrain(thefirstterm)ndfreethermaltrain(theecondterm). OnecaninvertEquation(17)ndobtain

a.C.(.a.T)1j J J (18)12 14Fornorthotropicayer, thetiffness andthermalxpansion matricesare:

C.ij

Cll1213C2223

C 33C44

C 55 '66

(19)

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a -l*10.i. 000 (20)

For tateflanetress, itsssumedthat:a3= a4= a5 (21)

Substitutingquations19), (20), and(21)nto18), . f_ =. and4c3--T=--(f-T)-L(,2-2T

(22)

(23)

SubstitutingEquation(23)nto18),rl = Cl 13 C13C32% l'C (fl-alT C12--T(f2-a2T> 33 33 (24)

C23C13, 32[C --(e -alT) C22-(


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15Intermsfngineeringonstants,C2

c223C22 " C^ 7 =V 2 7 ) C1 2 " = ^ 2 1El/ W*

where A 1 v vThequivalentonstitutivequationfor laminatedanisotropicom-

positeanbederivedsinghebasicssumptionofthenondeformablenor-malsfthetrengthfmaterials. Itsssumedthat

( = Ct+Ki 2 8 ) where, followinghenotationsnReference, i , 2, and.

Equation18), whenntegratedacrosshethicknessftheaminatedcomposite, becomes:

N.N. NTA.f B../c. 29)1 iij Jj J y'M.M.MT .f? D..K. 30>i iiJ Jj J '

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whererh/2(N., M.) = / a.1, z)z (31)1 x J-h/Z x

(NT, MT) f C.a.T1, z)dz (32)1 J-h/2 1J JJi/2 (33)(A.., B.., D..) / C.1,z, z zij ij ij' J_h/2 iJ

Equations29)nd(30)rehebasiconstitutivequationsoralam-inatedanisotropicomposite, takingntoaccountquivalentthermalloadings.

Thetresstanylocationacrosshethicknessfhecompositecan2bedeterminedasfollows:N A B "f

(34)LM B D K

Then, bymatrixinversion, A* J B " ~ N

(35).M _H* 1 D""_ K

N (36)

.H D M

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where A' = A"1B; = - A BH' = BA"1D' = DBA_1BA' = A" ~ " ~D"~-1HB' = H' ="~ DV_1 (37)D T = D

SubstitutingEquation(36)nto28),t. +K.li

=A.'. B.'.). +B!. Dl.) . (38)fromEquation(18), thetressomponentsforhek-thlayerare:

a (f.-(T)1J,(k)| (39)C '(A +B, , +B' +D!. .jkk' kk jk' r - ] (k)

Thisshemostgeneralxpressionofstressessfunctionsftressresultants, bendingmoments, andtemperature. Theamematerialcoeffi-cientsA', B'ndD 1, aseportednReference2ndalsotabulatednthe Appendixofthiseport, canbeusedforhethermaltressnalysis. Thissingleinkbetweenheisothermalandnonisothermalanalysesschievedbytreatingthermaleffectssquivalentmechanicaloads;.g. NTndMTnEquation(32).

Itcanbehownthatforuasi-homogeneousplates, B'=H'=;.e.noross-couplingexists. Inaddition,

A.. = C.h

D.. = C..h3/12 = A..h2/12 (40)

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Equation(39)anbeeducedoA.a . -'i h Ak(Nk+rr-Mk>-ajT

(41)=-(N. -^5.M.)C.a.Th i ,2'j j

usingheelationshipfA beingheinverseofAforquasi-homogeneousplates. Ifheplateislsoisotropic,

ipn~r C..O.T C,,a +17a_)ij j11 12 2' ' -vN. =N. NT=. -^-^-fdz 42>

T rh/2M.M. M =M. zzi 1 -vJ_h/2SubstitutingEquations42)nto(41), weobtainheameesultasEquation(12.2.7)fReference6.

Astatedbefore, thermaltressesrenducedwhenheperatingtemperaturefhecompositediffersromitsaminatingtemperature. Astypicalexample, itsssumedthathelaminatingtemperatures egreesaboveheperatingtemperaturewhichisssumedobeambient. Itsur-therassumedthatheero-stresstatexiststhelaminatingtemperaturewhichsowetashedatumtemperature. Theoperationtemperaturesthen-T. Foratraction-freeondition,

_ _T^2(N., M N.\ M.1) T /..a (1, z)z43)iiiih/2 ij J 25


32/100

FromEquation39)i 4k) ^k+zB] k )Nk+(Bik+ZDJ k )Mk+GtT (44)

Fornsotropieuasi-homogeneouslatenderniformtemperature,NTf MT=0-i/B' =0, C. A...Jkjj (45)

SubstitutingEquation45)n41)nd.(38), onebtains, asxpecteda. .NT C.a.T1!J J (46)e.Al.N. T1J J

Ifthetemperaturesinearcrosshehicknessfthesotropicquasi-homogeneouslate;.e.

T z (47)thenbyubstitutingEquation47)nto32), onebtains

TN. = 0, M.iEaah" 121 -v) (48)Hence, fromEquations41)nd38), onebtains, againasxpected,

a =_MTC.a.T1 h ,z ye. D..M. =aaz1j J (49)

TheesultsfEquations46)nd49)greewiththelementarytheory;.g.Equation9.5.66)fReference6.26


33/100

Thetrengthanalysisfalaminatedanisotropicompositesccom-plishedbyubstitutinghetressomponentsfhek-thconstituentlayer,calculatedfromquation(39), intohegeneralyieldconditionofEquation(8),ortsquivalentquation/when-^squaloero, e.g., Equation(13).Fromquation(8), themaximumy incombinationwithheparticularpandqthatachconstituentlayercanustain, canbebtained. Whenthismaxi-mumseached, failurenheparticularayerrlayerssonsideredohaveoccurred. Afterthisailure, theemaininglayers, whichhavenotfailed, willhaveocarryadditionalloads. Thishiftingfoadssccom-paniedbyapartialorompleteuncouplingfhemechanicalandthermalinteractionsmentionedabove. Thenetesultshatanewffectivetiffnessofhelaminatedcompositesownoperation. Thisewtiffness, asreflectednnewvaluesfA, B, andDmatricesfEquation(34), willcauseachangenhedistributionoftressesnachofheonstituentlayers stillintact. Theffectivetress-strainrelationofheompositeshangedandaknee"sxhibitedashelopefhetress-strainelationbecomesis-continuous. NewvaluesfA 1, B\ andD'matriceswhichareomputedfromtheevisedA, B, andD, mustnowbeusednEquation39)orheomputa-tionofhetresses. ThesenewtresseswillagainbeubstitutedntoheyieldconditionofEquation(8), fromwhichhenextlayerrlayersthatwouldfailcanbedetermined. Thisrocesss repeateduntilallhelayershavefailed.

Themathematicaldescriptionofheuncouplingfhemechanicalandthermalinteractionssotasyoascertain. Asnepossibility, crackstransverseohefiberswilldevelop, whichcauseadegradationofheffec-tivetiffnessndachangenhetressistributionnhecomposite. Anotherpossibilitys completeelaminationofhelaminate, therebyuncouplinghethermalandmechanicalinteractions. Thexactdescriptionofhedegradationprocessmustbetreatedforparticularlaminates, aswillbehownlater.

Theimportantpointntendedforthisectionsoillustratehexist-enceofmechanicalandthermalinteractionss directconsequenceoflam-ination. Internaltressesrenduced. Thesetressesxistnadditiono

27


34/100


35/100

a1) /rA +CA N2 " ( 21 11 22. 21' 1 + ' < 4V Ail+C22A2 1 > NT 1 + (C21 )A'l 2 +4zA2 2 > N2

(51).(m0i)+c i)u)T [l) (52)6

Inthebove, Equation29)wassed;.e.Nj x NJ" N2N^ N6 (53)

forhennerayer,a {Z)=cZ) (A!,Nt-()) (54)

Thisquation, whenxpanded, willbeheamesEquations50)hrough(52), exceptthatuperscript1)willbeeplacedbyuperscript2).

Usingheollowingxperimentallydeterminedmaterialpropertieswhichepresenttypicalunidirectionalglassilament-epoxyesincompos-ites,*neanvaluatethetressomponentsorhennerndouterayersintermsfthexialtressesultantN^ndtheaminationtemperature.

*TheameompositewhichwaseportedinSection2.

29


36/100

ClV2Z.97x106psiC12l2-66xQ6siC??...66x0 psiC6666-25xl06pxrl _ rl _ r2) _ r2) _U16 " ^26 " C16 " U26 "a[V =a - 3.5xlO-6/F = a = 11.4x10"6/oF6(55)

In 'athree-layern )ndm .2ross-plycomposite, oneancomputetheollowinguantitieswhichareneededforubstitutionntoEquations50)through(53). FromEquations33)nd(37),*

A .29 0"6n./lbA 0.03 0"6n./lb 56)A'22 .14x0"6n./lb

Thedetailedcalculationandometypicaldataforglass-epoxycompositesrehownnheAppendix.

30


37/100

FromEquation(32), assuming constantlaminationtemperature, oneancomputehequivalentthermalforcesndmoments:

N^=33. T lb-in.NJ=5. lb-in. 57)uj-MT=, asxpectedforthree-layercross-ply6

SubstitutingheomputedvaluesnEquations56)nd(57)ntohequationsforhetressomponents50)hrough(55), forheuterayers,

a1) .271 5.5a21] .12N1 16.0 58>

andforheinnerlayer,(2)1

a W .02N2 3.2T 59> a


38/100

Equation8)orheasef zerohear)ecomes, for 6-0degree(outerayer),

1 -p+p2r2 X/fl)2(60)22orY-oz+ ozKfor 0 0degreesinnerayer),

p2 +2 X/ax)2(61)2 22 vUsingheollowingxperimentallydeterminedtrengthvalueswhichepre-sent typicalunidirectionalglassilament-epoxyesinomposite,

AxialStrengthX = 150ksiTransverseStrength = Y =ksi 62)ShearStrengthS -sifromwhich, onebtainsrX/Y 7.5 (63)sX/S =5.0

SubstitutingEquations63)nd59)nto61),ndolvingheesultinguad-raticquationforN^,nebtainshetressesultantthatausesailurenthennerayer:

N1 . .33 64)TheameompositeseportednSection2.

32


39/100

Foracompositelaminateat270 F, orT 200,N, =400psi 65)

Forthatlaminatedatoomtemperature, orT ,N, =320psi 66)

Similarly, substitutingEquations63)nd(58)nto60), oneobtainshetressresultantthatcausesfailurenheuterlayer:

N, = 110 57.52 0002)1/ 67)For compositelaminatedat270 F, or 200,

N, =300psi 68)Forthatlaminatedatoomtemperature, or ,

N, =0,400psi 69)Comparingheesultsbove, onecanseethatheinnerlayerwillfailbeforetheuterlayer. Itslsohownthathefirstfailurewouldoccuratahigherstressfhelaminationtemperaturesmbient. FromEquation(59)tcanbeeenthatanelevatedlaminationtemperature(T egative)auses pre-tensionn whichshenormaltressransverseohefibers. ThiswillreducehemaximumN, atheknee."

33


40/100

Theffectivetiffnessfheaminatedcompositeupoheknee"simplytheeciprocalofA' (fornitythickness);.e. fromquation(56)heffec-tivetiffnesss.4x0 psi. Thus, thein-planetrainatheknee"s,usingN, =400fromEquation(65),

f 400/3.4x06 .1% 70)Thebehaviorofheross-plycompositeafterheknee"dependsnhedegreefuncouplingfhemechanicalandthermalinteractions. Animme-diatepossibilityshatcracksransverseohefibersredevelopednhe

(2)innerayer. ThisanbeescribedbylettingC\'fthennerayeremainconstantwhileheemainingomponentsredegraded"o verymallfractionofheirntactvalues, asistednEquation55). Theesultingmate-rialpropertiesfthisartiallydegradedompositeinnerayeregraded)becomenplacefEquation56), (58)nd59),

A'n - 0.75 0"6n./lbA'12 = 0.01 0"6n./lb 71)A22 = 0.4 0~6n./lbC T() =.00N1

(1)72)2 =.47Nj - 19.ai1>=

and

6

? ' =


41/100

Notethathethermalcouplingnhe 1-directionseducedoero. Buthethermalcouplingnhe-direction, as shownnEquation(72), isncreasedafterheegradation. Infact, thencreasesohigh(equalo9. )hat theuterlayersannotemainintactafterhenitialdegradation. Whatthismeansshatheuterayerswillalsodegrademmediately, thusausingcompleteuncouplingbetweenhelayers. Thereafter, onlyhencoupledouterlayersancarryheoad. Oneaneasilyolveforhexialoadhatapar-tiallydegradedcross-plycancarrybyubstitutinghetressomponentsfEquation72), intoheieldconditionofEquation60). ThemaximumNjturnsutobeonsiderablyowerhanhexistingtressf3400psi.

Afterwouccessivefailures, whichoccuralmostimultaneously, thelaminatedcompositebecomesompletelyuncoupledbothmechanicallyandthermally. Actualeparationamongonstituentlayershasbeenobserved. Inorderocharacterizehisompletelydegradedcomposite, itsssumedthatonlyhetiffnessarallelohefibersemain;.e. C andC aretheonlynonzerocomponents. (Inorderoavoidcomputationaldifficultiesnhematrixinversion, thetheromponentsreassumedobevanishinglymallbutnotzero. heesultingmaterialpropertiesfthisompletelydegradedcompositebecomenplacefEquations56), (58)nd(59),

A' = 0.77x0" in./lbA' = 0.15 0" in./lb

ThenlynonzerotressomponentsdueoN^s:o[V = 6.00N 75)

35


42/100

Thus, theffectivetiffnessfheompositeafterheknee"s1/hA', = 1.3x10 psi. Theltimatetrengthcanbeomputedasollows.ThetressnheouterlayersmmediatelybeforehedegradationofhennerlayersomputedfromEquation(58)singN, =400andT 200,

(1)cv ' = 618si = 600psi 76)Sincehemaximumtress er . canreachsqualoheaxialtrength,150,000psi, theuterlayersanbetressedanadditionalamountof150,000 00 149,400si. Usingquation(75), thisdditionaltressbeyondheknee"epresents tressesultantof49,400/6.00 4,900psi.ThenheltimatetressesultantN, isheumf24,900and3,400, whichis8,300psi. Thexperimentalmeasurementofheffectivetress-strainrelationofathree-layerross-plycompositeshownnFigure. Theagreementwithhetheoreticalpredictionsxcellentforthisase.

Itcanbetatedhataknee"doesxistanditsxistenceanbex-plainedntermsfhencouplingfhemechanicalandthermalinteractions.Ifhelaminationtemperaturesmbient, thenheknee"wouldoccur, fromEquation(66), atN, equalo5320psi, insteadof3400psi. Theesultantultimatetrengthofheomposite, however, turnsutobepracticallyhesameshatlaminatedat270 F.

Theonventionalnettingnalysisredictsheollowingtiffnessndstrength, basedontwo-thirdsfglassbyvolume, withglass stiffnessndstrengthof0.6x10 psiand400,000psi, respectively,

En = 10. x06x2/3x2/12 = 1.18xl06pi(77)

a1 = 400,000x2/3x2/12 = 44,000psiThesedataarealsohownnFigure. Itsnterestingonotethathemeasuredstrengthsnly68ercentofhatpredictedbynettingnalysis.

36


43/100

C O

J (PERCENT)

Figure5. StrengthofaTypicalCross-PlyComposite

37


44/100

Forhepurposefmorextensivexperimentalconfirmation, three-layerross-plycompositeswithdifferentcross-plyatiosweremadeandtested. Thetheoreticalpredictionsndhexperimentalresultsforbothheeffectivenitialandfinaltiffnessesbeforeandafterheknee,"espec-tively), andhetressevelstheknee"andheltimateoadarehownnFigure. Itsfairotatehathepresenttheoryseasonablyconfirmedexperimentally. Thecatterfdatacanbetracedpartlyohedifficultynmakingross-plytensilepecimens. Inheprocessfhapinghepecimensbyarouter, thelayerorientedtransverselyoheaxisfhedog-bone specimenssftendamaged.

Thepresenttheorynvolvesengthyarithmeticperations. PartofthisburdencanbeelievedbyusinghetablesistednheAppendix. TheinputdataarehoselistednEquation(55). Theompositemoduliandheequationsforhetressomponentsndhethermalforcesndmomentsrecomputedforwo-ndthree-layerompositeswithcross-plyatiosaryingfrom.2o4.0. Forachcross-plycomposite, twocaseswillbelisted:Case representslllayersntact;ndCase2, alllayersompletely"degraded." Withheaidofthesetables, thedataashownnEquations56)through(59), and(74)nd(75)anbeeaddirectly.

Inorderoemonstratehexistencefthermalforcesndmoments,atwo-layerross-plywithwoqualconstituentlayersm 1)wasaminatedat270 F. Attemperaturesowerhanhelaminationtemperature, theami-natedplatebecomes addle-shapedurface. Foraquareplatewithlength, thicknessh, clampedatonedgey ), as shownnFigure,

' '"As shownnReference2, two- andthree-layerlaminatedcompositesrepresentwoxtremeases, withallcompositeshavingargernumbersoflayersfallingnbetweenhextremes.

38


45/100

CO Q-oCO CO L LL

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M'0.4 CASE2ALLLAYERSDEGRADE:)2LAYERSNi2>

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2.2285.0000O.OOOO.57150. 0.

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0. 0.

APRIME(10-6IN./LB.)8.8637 -0.0000 0, -0.0000 0.2656 0. U. 0. 0000,0000

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THERMALFORCE(LB./IN./DFG.F.)Nl-T .7996N2-T9.5004N6-T' 0. THERMALMOMENT(LB./DEG.F.)

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z(IN. ) STRESSCOMPONENT COEF.FNl(1/IN.) COEF .F2(1/IN.) COEF,FN6(1/IN.) COEF.FMl(1/IN.SO. ) COEF.F2(1/IN.SO.)--AYER --

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STIFFNESSMATRIXC)(10*6La./IN.SO.)

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(10*6B./IN.)3.9000.OUOO.0.0000.9000.0. 0. 0.

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THERMALMOMENT(LB./DEG.F.)0 . 0 .0 . 0 ,( 1 . 0 .

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000

1- 00 0 0 07582o u u o - 0 . 0 0 0 01 2 . 3 0 7 70 .

0 ,0 ,0 0 0 . 0 0 0 0

z( IN . )

STRESSCOMPONENTCOEF.FNl(1/IN.) COtt.OFN2(1/IN,)

COEF.OFN6 COEF.OFM l(1/IN.) (1/IN.SO.) COEF.O FM2(1/IN.50.) COEF.FM6(1/IN.SO.)COEF.F(L8/IN.SQ

- -.AYER --0.5000 SIGMA 1

26

2 . 0 0 0 00 . 0 0 0 00 .

- 0 . 0 0 0 00 . 0 0 0 00 .

0 .0 ,1 , 0 0 0 0

-6.8571- 0 . 0 0 0 00 .

0 . 1) 0 0 0-0.00000.

0. 0. -6.0000

0.0000-0.00000.

-0.2500 SIGMA 156

2 . 0 0 0 0O.OOOO0 .- 0 . 0 0 0 00 . 0 0 0 00 .

0 .0 ,1 , 0 0 0 0--_AYEK

-3.4286-o.oooo0. o.oooo-0.00000.

0. 0. -3.O000

0.0000-0.00000.

-0.2500 SIGMA 1?6

0 . 0 0 0 0- 0 . 0 0 0 00 .O.OUOO2 . 0 0 0 0 0 .

0 .0 .1 , 0 0 0 0-0.00000.00000 .

-0.0000-24.000L

0.0. 0. -3.0000

-0.00000.00000,

0.2500 SIGMA 126

0 . 0 0 0 0- 0 . 0 0 0 00 .0 . 0 0 0 02 . 0 0 0 00 .

0 .0 .1.0900 - --AYER

0.0000-o.oooo0 .0.000024.00000.

0. 0. 3.0000

-0.00000.00000.

0.2500 SIGMA 12S

2 . 0 0 0 00 . 0 0 0 00 .- 0 . 0 0 0 00 . 0 0 0 00 .

0 ,0 ,1 , 0 0 0 03.42860 . 0 0 0 00 .

-0.0000o.oooo0.

0. 0. 3.0000

0.0000-0.00000.

0.5000 SIGMA 126

2 . 0 0 0 00 . 0 0 0 00 .- 0 . 0 0 0 00 . 0 0 0 00 .

0 ,0 .1 . 0 0 0 0 6.85710 . 0 0 0 00 .

o.oooo0.00000.

0. 0. 6.0000

0.0000-0.00000,

77


83/100

H2.0 CASE ALLLAYERSDEGRADED!2LAYERSN = 2)

STIFFNESSMATRIXCl (10*6Lb./IN.SQ. )

ODDLAYERS THERMALEXPANSIONMATRIXALPHA)(IN./IN./EG.F. )7.8000o.oooo0.

. o o o o

.0000ALPHA .5000ALPHA2 1.4000ALPHA6=0.

STIFFNESSMATRIXC)(10*6LB./IN.SO.)

EVENLAYERS THERMALEXPANSIONMATRIXALPHA)(IN./IN./DEG.F.)0.0000o . o ooo 0.00007.6000 0.0.

0.OtlOOALPHA 1.4000ALPHA .5000ALPHA 0.

(10*66./IN.) (10-6N./LS.) ARIME(10-6IN./LBi) THERMALFORCE(LB./IN./DEG.F.)5.2003 0.0000 0..1923 0. 0 0 01 10.0000 2.5957 0.O.OOOO 0.36470. ) . 0.0000 0. 0.

0.3365 -0.0300 0.-0.00 0 0 5.0018 0.0. 0. 0000.0 000

Nl-T 18.2009N2-T .0991N6-T 0.

8PRIME(10-6/L3.) THERMALMOMENT(LB./OEG.F.)

0.8666 0 . 0 .0 . 0.6666 3 .0 . 0 . 0.

0.1666 0.0000 0.-0.0000 -0.3333 0.

0.8652.0000,0.000013.8509.0. 0. 0, Ml-T -3.0332M2-T 3.0332M6-T 0. -0.1666-0.000U0.

0.00UU.0.3333.0. 0.

(10*6L8.IN.) (10*6LB.IN. )DPRIME(10-6/L3.IN.)

0.3370 0.0000o.naoo o,3i3o0. 1 .

0.1926.000O.0,0000.0241.0. 0. 0.0000

5.1915 -0.0000 0.-O.OOOO 1.5506 0,u. 0. 0000.000

7(IN. ) STRESSCOMPONENTCOEF.FNl

(1/IN.)COEF.F2(1/IN,) COEF.FN6(1/IN.) COEF.FMl COEF.FM2 COEF.FM6(1/IN.SQ.) (1/lN.SO.) (1/IN.SO.) EF .FEMP.8/IN.SO/F.)

LAYER0.5000 SIGMA 12

6-0.7496-0.00000.

0.00000 . 0U000.

1 1 .0.1.0000

-13.4986-0.0000u .o.ooooo.ooooa.

0.0.-6.OOOO-o.oooo-o.oooo0.

0.1667 SIGMA 126

3.7495O.OOOO0.

-0.00000.00000.

0.0.1.0000--AYER2--

13.49860.0000u .0.0 0000.00000.

0.0.2.0004

0.0000-o.oooo0.

0.1667 SIGMA 126

o.oooo-0.00000.

0 . 0 U 0 021.U0460.

0.0.1.0000

0.0000u.oooo0.

-0.0000-54.01030.

0.0.2.0004

-o.oooo0.00020.

0.5000 SIGMA 126

0.00000.00000.

-0.0000-15.00410.

0.0.1.0000

0.0000- o . o o o a0.0.000054.01040.

0.0.6.0000

-0.0000-0.00010.


84/100

MOSS-PLY M2.0ASE2ALLLAYERSDEORAOEO)3AYERSN3>

--DDLAYERS--STIFFNESSMATRIXC> HERMALEXPANSIONMATRIXALPHA)(1(1.6B./IN.SO.) IN./IN./DEG.F.)

7.800(1 0.0000 3. LPHA1 3.50000.0000 0.0000 3. LPHA 1.40000. -0. O.OOOO LPHA 0.

--VENLAYERS--STIrFNtSSMATRIXC) HERMALXPANSIONMATRIXA.PHO(10-6LB./IN.SO.) IN./IN./DEG.F.)

0.0000 0.0000 0. LPHA1 1.4000U.0000 7.BO0O 0. LPHA2 3.5000( I . 0. 0.0000 LPHA6sO.

A A PRIME THERMALFORCE(10-6LB./IN.) (10-6IN./LB.)10-6IN./LB,) (LB./IN./DEG.F.>5.1999 0.0000 0, 0,1923 -0.0000 0..1923 -0.0000 0. Nl-T 8.19960.0000 2.6001 0. -0.0000 0.3846 0.0.0000 0.384 0, N2-T 9.10020. 0. 0.0000 0, 0. 0000.0000. 0. OOOO.OOOO N6-T 0.

B B PRIME THEH-4ALMOMENT(10.6IN.) (10*0N,)10-61/LB.) (LB./DEG.F.)

0. 0. 0. 0. 0. 0.. 0. 0, Ml-T 0.00000. 0. 0. 0, 0. 0.. 0. 0. M2-T 0.00000. 0. 0. 0, 0. 0.. 0. 0, M6-T O.H.(10*0N,)0. 0. 0.0. 0. 0.0. . .D O PRIME(10*6LB.IN.) (10*6LB.IN.)10-61/LB.IN.)0.6259 0.0000 0. 0.6259 0.0000 0..5V76 -0.0000 0, 0.0000 0.C241 0. 0.0000 0.0241 0.0.0000 41.5359 0, 0. 0. 0,0000 0, 0. 0.0000. 0. 0000,0005

Z STRESS COEF.FNl COEF.F2OEF.FN6 COEF.FMl COEF.FM2 COEF.FM6 COEF.FEMP.(IN.) COMPONENT (1/IN.) (1/IN,)1/IN.) (1/IN.SO.) (1/IN.SO.) (1/IN.SO.) (LB/IN.SO/F.)

--AYER --O.5000 SIGHA 12

61.50000.00000.

-0.00000.00000.

0.0,i . o o o o -6.2308-0.00000.0.00000.00000.

0.0.-6.OOOOo.oooo-0.00000.

-0.1667 SIGMA 126

1.50000.00000.

-0.0000o.oooo0.

0,0,1,0000--LAYER --

-2.0770-0.00000.

0.0000.0.00000.

0.0.-2.O00C

o.oooo-o.oooo0.

-0.1667 SIGN 126

0.0000-0.00000.

o.oooo2.99990.

0,0,l.oooo

-0.00000.00000.

"0.0000- 51 . 9 9 7 7

0.0.0.-2.0000

-o.oooo0.00000.

0.1667 SIGNA 126

0.0000-0.00000.

0.00002.99990.

0,0,1.0000

--LAYER --

U.0000-o.oooo0.0.000053.99770.

0.0.2.0000

-0.0000o.oooo0.

0.1667 SIGMA 125

1.50000.00000.

-0.0000o.oooo0.

0, 0,1.0000

2.0770o.oooo0.-0.0000o.oooo0.

0.0.2.OOOO

o.oooo-o.oooo0.

0.5000 SIGMA 126

1.50000.00000.

-0.00000.00000.

0, 0, 1,0000

6.23080.00000.

o.oooo0.00000.

0.0.6.OOOO

o.oooo-0.00000.

7 9


85/100

CROSS-PLY

STIFFNESSMATRIX c i(10*6LB./IN.SO.)

1=4.0 CASE2ALLLAYERSDEGRADEO)2LAYERSN=2)

".5000.0300

O.OOOO0.0000

STIFFNESSMATRIXCl(10*6LB./IN.SO.)

0.0000Q . 0 0 0 00.

0.00007.80000.

ODDLAYERS

EVEVLAYERS

THERMALEXPANSIONMATRIXALPHA)(IN./IN./DEG.F.)ALPHAALPHAALPHA

3.500011.4000

0 .

THERMALEXPANSIONMATRIXALPHA)(IN./IN./DEG.F.)ALPHA 1.4000ALPHA2=.5000ALPHA6=0.

(10*6LB./IN. )

(10*6\l . )

(10-6IN./LS.)6.2400 o.booo 0. 0.1603 -0.00000.0000 1.5600 0. -0.0300 0.64100. 0. 0. 0000 0. 0. . 0. 30.00

(10*0N, 0,1300 0.00000.0300 -0.4000

0. 0.

APRIME(10-6N,/LB.)0.1903 -0.0300 0,

-O.OOOO 31.4388 0. 0. 0, 0000,0000

BPRIME(10-6/LB.)

U . 3 u 0 5.0300.0.000076.9194.0. 0. 0.

THERMALFORCE(LU./IN./DEG.F. )Nl-T 1.8400M2-T .4600NO-T 0. THEHMALMOMENT(LB./OEG.F. )

Ml-T 2.1340M2-T .1S40M6-T 0.

(10*0N, )-0.1000-0,0000

O.OOOO0.4000 0 .

0 .

(10*6L8.IN.)0.3952.0000.0.0000.2549.0 . . .

(10*6B.IN.>0.3 32 60.000B0.

0.00000.0052

DPRIME(10-61/LB.I^.)

3.0048 -O.OOOO 0, -0.000092.2987 0.

O . 0. 0000,0005

z(IN.) STRESSCOMPONENT

COEF,Fl(1/IN. )

COEF.FN2(1/IN,) COEF.FN6(1/IN.)

LAYER

COEF.Fl(1/1N.SQ.)

COEF.FMzIl/IN.SO.)

COEF.F6(1/IN.SO.) COEF.FEMP.(LB/IN.SD/F.)

0.5000 SIGMA 1 ?6

0.31230.00000.

-0.00000.00010.

0 ,0. 1.0000

-9.3750-0.00000.

0.0000-0.0002P.0. 0. -6.O0O0

-0.0000-0.00000.

0.3000 SIGMA 1?6

2.16750.0003n .

-0.00000.00000.

0. 0 .1,0000--AYER2--

9.37530.0003O .

0.0000- o . o o o o0 .

0. 0. 3.600C

0.0000-0.00000.

0.3000 SIGMA 1?6

O.OOOO-0.00030.

0.000064.9970

O .0. 0 .1.0000

O.OOOOO.0003O .

-O.OOOO-149.99260.

0. 0. 3.6OO0

-0.00000.00050.

0.5000 SIGMA 126

0.00030.00000 .

-0.0000-54.9973

0 .0. 0 .1.0000

U.0O00-U.00030.

0 .0 00 0149.99340.

0. 0. 6.U00C

-0.0000-0.00040.

80


86/100

M = 4. 0 CASE2(ALLLAYERSDEGRADES)3LAYERS(N=3)

STIFFNESSMATRIX( C )(10*6LB./IN.SO.)7.8000 0.000u.OOOO 0.0000. 0.

- -OODLAYERS-- THERNAtEXPANSIONMATRIX(ALPHA)(IN./IN./DEC.F. )ALPHA 3.5000ALPHA - 1 1.4O00ALPHA6=0.

VENLAYERSSTIFFNESSMATRIX(C)( 1 ( 1 * 6LB./IN.SO. )0 . 0 0 0 0 D . 0 0 0 0U.OOOO 7.9000j . .

THERMALEXPANSIONMATRIX(ALPHA)(IN./IN./BEG.F. )ALPHA1 1 1 . 4 0 0 0ALPHA2 .5000ALPHA6 .

(10*6L9./IN.)6.2395. 0 0 0 0.0 . 0 0 0 0.5605.0 . . 0 . 0 0 0 0 (10-6IN./LB.)0 , 1 5 0 3- 0 . 0 0 0 00 . .00 0 0 0 ..6408 0 .0 0 0 U . 0 0 0 0

APRIME(10-6IN./LB.)U.1603 - 0 . 0 0 0 0 0 .- 0 . 0 0 0 0 0.6408 0 ,u . 0 . 0 0 0 0 . o nu o

BPRIME( 1 ( 1 - 61/L8.)

THERMALFORCE(LB./IN./DEG.F. )Nl-T 21.8361 N2-T .4619N6-T 0 .THERMALMOMENT(LB./OEG.F.)

- 0 . 0 0 0 1..0.. 0 0 0 1.0 . 0 . 0 . 0 . 0 0 0 0- 0 . 0 0 0 0 0. 0 0 10 .- 0 . 0 0 0 0 0 .0 . 0 . 0.OU00 0 . 0 0 0 0O.OUDO - 0 . 0 06 7 0 .0 . Ml-T -0.O002 M2-T . 0 0 0 2M6-T 0 .0.0000 0.00000,0000 o.oouo0. 0.

(10-6LB.IN.10.6448. 0 0 0 0.0 . 0 0 0 0.005?.0 . 0 . 0 . 0 0 0 0 (10*6LB.IN. )0,6448. 0 0 0 0.0 . 0 3 0 0. 0 0 5 2.0 , 0 . 0 . 0 0 0 0

DPRIME(10-61/LB.IN.).5509 - 0 . 0 0 0 0 0 ,.00 0 0192 .1 0 2 9 0 .0 . 0 0 0 0 ,0 0 0 5

z (IN. ) STRESSCOMPONENT COEF.Fl(1/IN. ) COEF.FN2(1/IN,) COEF.FN6(1/IN.) COEI- .FMl(1/IN.SO.)COEF.F2(1/lN.SO.)

COEF.FM6(1/IN.SQ.)

COEF.FEMP(LB/IN.SO/F.)--AYER1-

0.5000 SIGMA 126

1.25010.00000.

-0.00000.00000.

0, 0. 1.0000

-6.0483-0.00000 .

0.0000- 0 . 0 0 0 10.

0. 0. -6.UO0C0.0000-0.00000.

0.1000 5IGM 126

1.25010.00000.

-0.00000.00000.

0, 0, 1,0000--LAYER2--

-1.2096-0.00000.

0.0000-0,00000.

0. 0 .-1.200C

0.0000-0.00000.

0.1000 SIGMA 126

0.0000-0.00000.

0.00005.00350.

0. 0. 1.0000

-0.00000.00000.

0.0000-149.89270.

0. 0. -1.2000

-0.00000.00000.

0.1001 SIGMA 126

0.0000-0.00000.

0.00004.99300.

0, 0. 1.0000--LAYEH3-"

o.cooo-0.00000.

U.OOOO149.6927

0. 0. 0. 1.2009

-0.00"00.00000.

0.1001 SIGMA 126

1.25010.00000.

-0.00000.00000.

0. 0. 1.0000

1.2106o.oooc0. -U.OOOO0.00000.

0. 0. 1.2009

o.oooo-o.oooo0.

0.5000 SIGMA 126

1.25020.00000.

-0.00000.00000.

0. 0. 1,0000

6.0485U.OOOOu .

-0.00000 . U 0 0 10.

0. 0. 6.U000

0.0000-o.oooo0.


87/100


88/100

ANGLE-PLY 5DEGREES CASE ALLLAYERSINTACT!3LAYERS(N-31OD DLAYERS

STIFFNESSMATRIX (C )110*6LB./IN.SC.7.8930 0.6962 -0.41400.6962 2.6630 -O.C4710.4140 -0.0471 1.2830

EVENLAYERSSTIFFNESSMATRIX IC)110*6LB./IN.SC 1

7.8930 0.6962 0.41400.6962 2.6630 0.C4710.4140 0.0471 1.2830

THERMALEXPANSIONMATRIXIAIPHA1IIN./IN./CEG.F.)ALPHAALPHAALPHA6

3.560011.34000.6859

THERHALEXPANSIONMATRIXIALPHAIUN./IN./BEG.F.IALPHAALPHA2ALPHA6

3.560011.34C0-0.6859

110*6LB./IN.)7.8930 0.6962 -0.13790.6962 2.6630 -0.0157-0.1379 -0.C157 1.2830

110*6IN.C. 0.0. 0.0. 0.

110*6LB.IN.)0.6577 0.0580 -0.03190.0580 0.2219 -0.0036-0.0319 -0.C036 0.1069

110-6IN./LB.I0.12990.03390.0136

-0.0339C.38440.0011B110*0IN.)

0.0136O.OCII0.7809

0.0.0.C.C .C.

H(1C*0IN.)

0 .0.0.

0.0 .0.C.c . c .

0.0 .0.

Do(10*6LB.IN)0.6577 C.0580 -0.03190.0580 C.2219 -0.0C36-0.0319 -0.0036 0.1069

APRIME(10-6IN./LB)0.1299 -0.03390.0339 0.38440.0136 0.0011

0.0136C.OOll0.78098PRIME110-61/LB.

0 . 0.0. c.0. 0.0.0.C.

CPRIME110-61/LB.IN.)1.5783 -0.4051-0.4051 4.61270.4578 0.0357

0.45780.03579.4911

THERMALFORCEILB./IN./DEG.F.)Nl -T >35.710CN2 T *32.6446N6 T -0 3758THERMALMOMENT(L6./0EG.F.)1 - T .-0 0000M2-T -0 0000M6-I 0.

I(IN. STRESSCOMPONENT

SIGMA26

SIGMA2

SIGMA2

SIGMA2

SIGMA2

SIGMA2

COEF.OFNl(1/IN.)

0.9963-0.0004-0.0348

COEF.OFN2I1/IN.)

0.9963-0.0004-0.0348

1.0075O.0O090.06961.0075O.00C90.0696

0.9963-0.0004-0.03480.9963-0.0004-0.0348

00030000002700030000

0.00061.00010.00540.00061.00010.0054

00030000002700030000

COEF.OFN6(1/IN.)LAYER

-0.2156-0.02450.9962-0.2156-0.02450.9962LAYER 0.43100.04901.00750.43100.04901.0075LAYER

-0.2156-0.02450.9962-0.2156-0.02450.9962

CCEF.OFMl(1/IN.SO

-5.99300.00080.0235

-1.99810.00030.0078

-2.0612-0.0069-0.20362.06120.00690.2036

1.9981-0.0003-0.00785.9930-0.0006-C.0235

COEF.OFM2ll/IN.SO.)

0.0005-5.99990.00180.0002-2.00040.0006

-0.0048-2.0009-0.0159004800090159

-0.0C022.0004-0.O0O6-0.00055.9999-0.0018

CCEF.OFM6ll/IN.SO.) COEF.OFTEMP(LB/IN.SO/F.)

0.14560.0166-5.9929

-0.0621-0.0071-0.5780

0.04850.C055-1.9980-0.0621-0.0071-0.5780

-1.2615-0.1435-2.06180.12420.01411.1556

1.26150.14352.06180.12420.01411.1556

-0.0485-0.00551.9980-0.0621-0.0071-0.5780

-0.1456-0.01665.9929-0.0621-0.0071-0.5780


89/100

ANGLE-PLY THETA0DEGREES CASE LLAYERSNTACTI2LAYERSN>2)ODDLAYERS--STIFFNESSMATRIXC) HERMALXPANSIONMATRIXALPHA!

110*6LB./IN.SCI IN./IN./DEC.F.)ALPHA .7382ALPHA =1.1620ALPHA6- 1.3510

VE N LAYERS STIFFNESS MATRIX ICI hERMAL EXPANSION MATRIX (ALPHA)(106 LB./IN.SCI N/IN/CEG.F.7.6800.7893.7969 LPHA 1 - 3.7382C.7893.6900.1093 LPHA 2 11.1620C.7989.1C93.3760 LPHA 6 " -1.3510768C0 0.7893 -0 798907893 2.69C0 -0 10930 7989 -0.1093 1 3760A

(1C6P./IN 1 A(10-6N./LB 1 APRIPE(1C-6N./LB.)7.68000.7893C.

0.78932.69COC.

e1 1C*6N.>

001 3760

0.-0 .0

11


90/100

768C0 0 7693 -0.7989C 7893 269C0 -0.1093C 7989 -0 1C93 1.3760

ANGLE-PLY IHETA' 1 0DEGREES CASE ALLAYERSNTACT]3LAYERSN=3IODDLAYERS

STIFFNESSMATRIXCl HERMALXPANSIONMATRIXALPHA)(1C6LB./IN.SCI IN./IN./DEG.F.)

ALPHA .7382ALPHA2 1.1620ALPHA6' 1.3510

VE N LAYERS --STIFFNESS MATRIX (Cl HERMAL EXPANSION MATRIX (ALPHA)(1C6 LB./INSCI I./I./EGF.ALPHA .7382ALPHA 1.1620ALPHA 6 -1.3510

THERMAL FORCE (10*6 LB./INI10-6 IN/BI1C-6 IN/LB.ILB/IN/DEGF.)7.6800 0.7893 -0.2662.1351 -C.0393 0.0251.1351 -C.0393 0.0251l-T " 36.44020.7893.69C0 -0.03640.0393.3833.0C250.0393.3833.0C252-T 32.8287

-0.2662 -C.0364.3760.0251.0025.7317.C251.0025.73176-T . -0.78227 6800 0.7693 0.7989C 7893 2.6900 0.10930 7989 0.1093 1.376C APRIME(1C-6N./LB )00C 1351 -C.03930393 0.38330251 0.00258PRIME

(1C-6/LB.

00C

1

02510C257 3 1 7

000

0. 0. C.

000

THERMAL fCMENT(10*6 INI10+0 INI1C-6 1/LBILB/DEC.F.)

C.......l-T * -O.COOC0.......2-T = -O.OOOC0.......-fc-T * 0.0000H (1C0 IN)C..C..C..c (106LB.IN.1 D'10*6LB.IN 1 CPRIME(1C-6/LB.IN . ]C.64000.06580.0616 CCC C658 -02242 -0C084 0 C616CC641147 0.64000.C658-0.C616 0.0658C.2242-C.0084 000 06160C84 -01147 0 6947 -0.46434643 4.6CC58768 0.0868 0.87680.08889.1988

I(IN. 1

STRESSCOMPONENTCOEF.FNl

(t/IN. 1CCEF.FN2

11/IN.)COEF.OFN6 CCEF.OFMl

(1/IN.1 11/IN.SC.1CCEF.FM2(1/IN.SO.)

CCEF.FM6(1/IN.SCI COEF.OFTEMP(LB/IN.SC/F.I--LAYER

-0.5C00 SIGMA 1 26

0.9866-0.0018-0.0691

-0.C0140.9998

-0.0070-0.3896-0.05330.9864

-5.97400.C036C.0463

0.0026-5.99960.0C49

0 .2 7 2 30 . C 3 7 3-5.97 3 7

-0 .2 2 68-0 .0 3 1 0 -1.1 7 2 3

-0.1667 SIGMA 1 26

0.9866-0.0018-0.0691

-0.00140.9998-0.0070

-0.3898-0.05330.9864LAYER

-1.9917C.C012C.0161

0.0C09-2.0003C.0C16

0.C9080 . C 1 2 4-1.9916-0 .2 2 68-0 .0 3 1 0 -1.1 7 2 3

-0.1667 SIGMA 1 26

1.02670.00370.1382

0.00271.00040.0140

0.77930.10661.0271

-2.2253-0.0308-0.4184

-0.0228-2.0C35-0.0424

-2.3593-0 .32 2 8^2 .2 2 8 40.45350.0620 2.3440

0.1667 SIGMA 126

1.02670.00370.1362

0.00271.00040.0140

0.77930.10661.0271LAYER

2.22530.03080.4184

0.02282.00350.0424

2.3593C. 3 2 262 . 2 2 840.4535O.06202.3440

0.1667 SIGMA 1 26

0.9866-0.0018-0.0691

-0.00140.9998-0.0070

-0.3898-0.05330.9864

1.9917-C.0012-0.0161

-0.00092.0C03-0.0016

-0.C9C8-0 .C 1 2 41.9916-0.2268- 0 . 0 3 1 0-1.1 7 2 3

0.5C0C SIGMA 1 26

0.9866-0.0018-0.0691

-0.00140.9998-0.C070

-0.3898-0.05330.9864

5.9740-C.C036-0.0483

-0.00265.9996-0.0049

- 0 . 2 7 2 3-0 .C 3 7 35.9737 - 0 . 2 26 8-0 .0 3 1 0-1.1 7 2 3

85


91/100

ANGLE-PLY IHETA . 15 DEGREES CSE 1 ULL LAYERS INTACT)2AVERS (N-2100 LAYERS SIIFFNESS MATRIX Id HERMAL EXPANSION MATRIX (ALPHA)(1C6 LB./INSCI IN/IN/DEG.F.I

7.3420 0.9320 -1.1290 LPHA 1 .0292C.9320 2.7430 -0.1993 LPHA 2 10.8700-1.1290 -0.1993 1.5190 LPHA 6 1.9750~VEN LAYERS SIIFFNESS MATRIX ICI HERHAL EXPANSION MATRIX (ALPHA)(1C>6 LB./INSO.I 1N/IN/0EG.F.I

7.3*20 0.9320 1.1290 LPHA I - 4.02920.9320 2.7430 0.1993 LPHA 2 " 10.87001.1290 0.1993 1.5190 LPHA 6 -1.9750

THERMAL FORCE I106 LB./IN)10-6 IN/B.I1C-6 IN/LB.ILB./IN/EG.F.)N-I7.4835N2-T3.1780N6-I 0.THERMAL MOMENTILB/OEGF.IMl-T .0000M2-T .0000M6-T 0.9288

7.34200.932C0.0.93202.7430C .

BI1C6IN.

0 .0 .1.5190

0.0.0.2822C.0 .0.C498

0.28220.04980.

c(10(6LB0.611t C.0777 C.0.0777 C.2266 0.

A(10-tIN./LB 1 APRIME(1C-6IN./L6.I

0.14230.04840.-C.0484C.3810C.

B(1C0IN.)

0 .0.0.65830-00

15470466 -0.0466 0 .0.3812 0.C. C.7205BPRIME10-61/LB.I

0.0 .0.1856C .0.-C.0328

HI1C


92/100

ANGLE-PLY 15DEGREES CASE ALLAYERSINIACtl3LAYERSN"1>

STIFFNESSKATRIXCI110*6IB./IN.SO.I

7.3*200.9320-1.1290

0.93202.7*30-0.1993

-1.1290-0.19931.5190

STIFFNESSMATRIXC)(106LB./IN.SO.I

7.3*20.9320.12900.9320.7*30.19931.1290 0.1993 1.5190 OCDLAYERSEVENLAYERS THERMALEXPANSIONMATRIXALPHA)IIN./1N./0EG.F.)ALPHA ..0292ALPHA2>0.8700ALPHA6 1.9750THERMALEXPANSIONMATRIXALPHA IIIN./IN./OEG.F.)ALPHA *.0292ALPHA2 0.8700ALPHA6-1.9750

(10*6LB./IN.)7.3*2C 0.9320 -0.37620.9320 2.7*30 -0.066*-0.3762 -0.C66* 1.5190

(10-6N./LB.I0.144C -0.0*61 0.0336-0.0*81 0.3810 0.00*70.0336 0.00*7 0.6668

APRICE(10-6N./LB.I

0.1**0 -0.0*81 0.0336-0.0*81 0.3810 0.00*70.0336 0.00*7 0.6668

THERMALFORCEILB./IN./DEG.F.INl-T 37.4835N2-T 3.1780N6-T -1.2379

B110*6N.) B110*0N.)

BPRIME(10-61/LB1 THERMALILB./OEGCMENT.F.)0.0.0.

C.C.0.

0.00

0. 0.0.

C.C. C.

0.0 .0.0.0 .0.

C.0.0.C.0.0.

Ml-T M2-T-M6-T -0-00

. oooc. ooco.0000H

!1C0N.I0.0.0.

C. c .c .0.0.0.

CI106LB.IN.) 10*6LB.IN. CPRIME(10-61/LB.IN.)

000.6118.0777.0871

CC

-C0777 -02286 -0C154 0

087101541266

0.61180.0777-0.0871C.0777 -C.2286 --C.0154

0 .00.

oe7i .015* -0.1266 1 .8796 -0.55625562 4.575*2259 0.1731

1.22590.17318.7646

Z(IN. 1STRESSCOMPONENT COEF.OFNl ll/IN.)

CCEF.OFN2ll/IN.) COET.OF (ll/IN.) CCEF.OFMlll/IN.SO COEF.CFM2ll/IN.SO COEf.OFM6ll/IN.SO COEF.FTEMP(LB/IN.SC/F.ILAYER -

-0.5C0C SIGMA 126

0.9747-0.0045-0.1020

-C.C0360.9994-C.0144

-O.5O20-0.08860.97*1

-5.9*87C.00910.07*5

0.0072-5.99870.0105

0.36670.0647-5.947*

-0.4**1-0.078*-1.7931

-0.1667 SIGMA 126

0.9747-0.00*5-0.1020-0.00360.9994-0.01*4

-0.5020-0.08860.9741~LAYER2--1.9833C.0030C.0248

0.002*-2.00000.00350.12230.0216. -1.9829

-0.4441-0.0784-1.7931

-0.1667 SIGMA 126

1.05050.00890.20*00.00711.00130.0288

1.00370.17721.0518-2.4447-0.0784-C.6457

-0.0627-2.0115-0.0912-3.1768-0.5608-2.*55B

0.88790.15671.5952

0.1667 SIGMA 126

1.05050.00890.20*C0.00711.C0130.0288

1.00370.17721.05ULAYER -2.44*70.078*C.6457

0.06272.01150.09123.17680.56C82.4558

0.89790.15671.5952

0.1667 SIGMA 126

0.9747-0.0045-0.102C

-0.00360.9994-0.0144-0.5020-0.08860.9741

1.9833-0.C030-0.C248-0.00242.0C0C-0.0035

-0.1223-C.C2161.9829-0.4441-0.0784-1.7931

0.5000 SIGMA 126

0.9747-0.0045-0.102C

-0.0O360.9994-0.C14*-0.5C20-0.08860.9741

5.9487-0.0091-0.C745-O.OC725.9987-0.0105

-0.3667-C.C6475.9474-0.4441-0.0784-1.7931

87


93/100

30DEGREES CASE (ILLAYERSINTACT)2LAYERS(N"2I

STIFFNESSPATRIXCOI 10*6LB. /IN.SCIOD DLAYERS THERMALEXPANSIONMATRIX(ALPHA)IIN./1N./DEG.F.)

5.83*0l.*690-1.6150

l.*6903.1780-0.6852

-1.6150-0.68522.0550

ALPHAALPHA2ALPHA6

5.7509.4250S.*20

STIFFNESSCATRIX(Cl 110*6LB./IN.SO.IEVENLAYERS THERMALEXPANSIONMATRIX(ALPHA)(IN./1N./0EG.F.)

5.83*0l.*6901.6150l.*6903.17800.6852

1.61500.68522.C550ALPHA .4750ALPHA2 9.4250ALPHA6-3.4201

UC6La./IN.I (10-6IN./LB.) APRIME110-6IN./LB.) THERMALFORCEILB./IN./OEG.F.)5.83*0.*690.l.*690.1780.0. C. 2.0550 0.19*0 -C.0897 0.-0.0897 0.3561 0.0 . C . 0.*866 0.2220 -0.O786 0 .-0.0786 0.3605 0.0. 0. 0.5886 Nl-T 40.2619N2-T 35.6515N6-T '0.0000

110*6IN.0. C .t. C .0.4037 0.1713

0.*037C.17130.

(1C0IN.0 .0 .0.1965

C.c.-C.083*H(1C0IN.

-0.0630-0.02*80.

0 .0.0.0630C.c.C.02*8

0.19650.083*0.

BPRIME10-61/LB.IC. 0. -0.44470. 0. -0.1752-0.4447 -0.1752 0.

THERMALMOMENT(LB./OEG.F.IMl-T-.0000M2-T.0000M6-T- 2.0676

(10.S LB.IN. ) 10*6LB.IN. 110-61/LB.IN.).4862.1224 C .0 .C .

122* 026*8 0C 1712

0.*C680.08880 .C.C888C.2506C.

0 .00 .

2 .-01*16 0 .663B -0.94379437 4.3255

0.0.0 .7.0631

I1IN . 1 STRESSCOPPCNENT COEF.OFNl11/IN.I CCEF.OFN2ll/IN.I COEF.OF (ll/IN.I COEF.OFMl(1/IN.SC.) COEF.CFM2(1/IN.SO.I COEF.OFM6(1/IN.SC.1

~LAYER --0.500C SIGPA 1

26

0.B2C*-0.07620.1523-O.C70 70.97000.0600

0.*7530.20170.790*-6.3591-0.15240.9139

-0.1414-6.06000.35992.15171.2099-6.4191

0. SIGCA 126

1.17960.0762-0.30*60.07071.0300-0.1200

-0.9506-0.*0331.2096LAYER -0.71820.3047-0.9139

0.28290.1200-0.3599-2.8517-1.20990.(382

0 . SIGMA 126

1.17960.07620.30*60.07071.03000.1200

0.9506O.*0331.2096-0.7182-C.3047-0.9139

-0.2829-0.1200-0.3599-2.8517-1.2099-0.B3B2

0.5CO0 S1GPA 126

0.820*-0.0762-0.1523-0.07070.9700-0.0600

-0.4753-0.20170.79046.3591C.15240.9139

0.14146.06000.35992.85171.20996.4191

COEF.OFTEMP.(LB/IN.SC/F.)

-J.8540-1.63523.26943.B5401.6352-6.53B7

3.85*01.63526.5387-3.8540-1.6352-1.2694


94/100

ANCLE-PLV THETA 30DEGREES CASE CALLLAYERSINTACT!3LAYERS(N"3)STIFFNESSMATRIX(Cl(10*6LR./IN.SQ.I

5.83*0 l.*690 -1.6150l.*690 3.1780 -0.6852-1.6150 -0.6652 2.C550STIFFNESSMATRIX(Cl (10*6LB./IN.SO.I

5.83*0 l.*69C 1.6150l.*690 3.1780 0.68521.6150 0.6852 2.C550

OCOLAYERS--

EVENLAYERS

THERMALEXPANSIONMATRIX(ALPHA)(IN./IN./OEG.F.IALPHAALPHA2 ALPHA6

5.*T509.*2503.*208

THERMALEXPANSIONMATRIX(ALPHA)(IN./IN./DEC.F.)ALPHAALPHA2ALPHA6

5.*7509.*250-3.*208

(10*6LB./IN.)5.83*C 1.4690 -C.5381l.*690 3.1780 -0.2283-0.5381 -C.2283 2.0550

C.0.C.0.0.0.

A(10-6IN./LB1APRIME(10-6IN./LB) THERMALFORCE(LB./IN./DEC.F.

0 .0 .0 .

1S7508830*19-C.0883C.35670.0165

8I1C0IN.)

0.0*190.01650.*99*0.1975-0.08830.0*19

-0.08830.35670.01656PRIME1C-61/LB.

000

) 0*190165*99*

Nl-T-*0.2619N2-T-35.6515N6-T.-2.7557THERMALMOMENT(L8./0EG.F.I

000

0.C.C.0.0.0.

0.0.0.0.0.0.

000

Ml-I -0.0000M2-T--O.OCCCM6-T O.OCOO

110*0IN.)0.0 .0.

0.C.c.

110*6LB.IN. )0.*862 0.122* -0.12*60.122* 0.26*8 -0.0529-0.12*6 -C.C529 0.1712

I 10*6LB.IN. I0.*862 C.122* -0.12*60.122* 0.2648 -0.0529-0.1246 -C.0529 0.1712

CPRIME110-61/LB.IN.)2.7238 -0.9201 1.6979-0.9201 4.33*8 C.66871.6979 0.6687 7.2813

IUN.) STRESSCOMPONENT COEF.OFNl (1/IN.I CCEF.OFN2[1/IN.I COEF.OFN6(1/IN.I COEF.OFMl(l/IN.SO.) COEF.CFM21l/IN.SO.)COEF.OFM6(1/IN.SC.) COEF.OFTE M(L8/IN.SC/F.

LAYER 0.5C0C SIGMA 1

26

0.95*9-0.0191-0.172*-0.C1780.9925-0.0679

-0.5576-0.22820.9*75-5.898*0.0*310.1397

0.0*00-5.98300.0550

0.*3360.18*9-5.881*

-0.9687-o.*uo-3.6995

0.1667 SIGMA 126

0.95*9-0i0191-0.172*-0.01780.9925-0.0679

-0.5578-0.22820.9*75LAYER2---1.96650.01**0.0*66

0.0135-1.99*70.01650.1*550.0616-1.9609

-0.9687-0.4110-3.6995

0.1667 SIGMA 1261.09C20.03830.5**6

0.03551.01510.13571.07550.45621.10)5

-2.88C7-0.3735-1.209B-0.5*67-2.1*75-0.*765

-5.7753-1.6017-5.C2781.93690.62167.3968

0.1667 SIGMA 1261.09020.05830.3**6

0.03551.01510.13571.07550.*5621.1053LAYER

2.88070.37551.20980.5*672.1*750.*765

5.77551.60175.02761.93690.62187.3966

0.1667 SIGMA 1 26

0.95*9-0.0191-0.172*-0.01780.9925-0.0679

-0.5578-0.22820.9*731.9665-0.01**-0.0*66

-0.01331.99*7-0.0183-0.1*55-0.C6161.9609

-0.9687-0.M10-3.69950.5C00 SIGMA 12

60.95*9-0.0191-0.172*

-0.01780.9925-0.0679-0.5378-0.22820.9*73

5.898*-0.0*51-C.1397-0.0*005.9830-0.0550

-0.*358-0.16*95.881*-0.9687-0.4110-3.6995

89


95/100

ANGLE-PLY 45 DEGREES CASE 1 (ALL LAVERS INTACT!2 LAYERS (N2ISTIFFNESS MATRIX (C)(106 LB./INSCI

4.23E0.737C1.32B01.7370.23601.3280-1.32801.3280.3230 OD D LAYERS THERMAL EXPANSION MATRIX (ALPHA)II./I./EG.F.IALPHAALPHAALPHA 6 7.45C07.45003.95C0STIFFNESS MATRIX ICI(10*6 L8./INS0.I EVENLAYERS THERMALEXPANSIONMATRIXIALPHAIIIN./IN./DEG.F.I4.2380 1.7370 1 32801.7370 4.2380 1 32801.3280 1.3280 2 3230 ALPHAALPHAALPhA6 7.45007.4500-3.9500

I10*6LB./IN. I4.2380.7370.1.7370.2380.0. 0 . 2.32300.0.0.3320

C..3320C ..3320C.3320 0 . 110-6[N./LB.I0.2836 -C.1162 0 .-0.1162 C.2636 0 .0 . C . 0.4305110*0IN.0 .0.0.1429 C.C.-C.1429H(IOCIN. -0.0556-0.05560.10.0.0.C556

C.C.C.05560.14290.14290 .

APMIMEI1C-6IN./LB.I THERMALFORCEILE./IN./OEC.F.0.3C33 -0.C965 00.0965 0.3033 00. 0. 0 5318

Nl-T 39.2681N2-T 39.2681N6-T 0 .8PRIME110-61/LB.I THERMALMOMENTILP../CEG.F.I

0. 0 . -C.0. C . -0.0.3546 -0.3546 0 .35463546 Ml-T 0.0000M2-T 0.M6-T 2.6528

I I06LB.IN.)0.3532.1447.0.1447.3532.C. C. 0.1936 110*6 LB.INI 0.3057.0973.0.0973.3057.0... C PRIME110-6 1/LB.IN.)3.6397-1.15840. -1.1584.3.6397.0..3821

IIN. STRESSCOMPONENT COEF. OF NlI1/INI CCEF. DF N2 I1/INI COEF. OF N6 ll/INI CCEF. CF Ml I1/1NSC.I CCEF. CF M2 ll/INSO COEF. OF M6 ll/INSCI COEF. CF TEMP ILE/INSC/F.I

SIGMA 1 2SIGMA 1 2

SIGMA26 SIGMA 1 2

0.8823-0.11770.13731.11770.1177-0.2746

1.11770.11770.27460.8823-0.1177-0.1373

-0.11770.88230.13730.11771.1177-0.2746

0.11771.11770.2746-0.11770.8823-0.1373

LAYER 1 0.35310.35310.7645-0.7063-0.70631.2355LAYER 2 0.70630.70631.2355

-0.3531-0.35310.7645

.2355.2355.8238

.4709.4709.8238

-0.47C9-0.47C9-C.82386.2355C.2355C.8238

-0.2355-6.23550.82380.47090.4709-0.8238

-0.4709-0.4709-0.82380.23556.23550.8238

2.11892.1189-6.4709-2.1189-2.11890.9419

-2.1189-2.1189-C.94192.11892.11896.47C9

-3.6254-3.62544.22783.62543.6254-8.4556

3.62543.62548.4556-3.6254-3.6254-4.2278

90


96/100

ANCLE-PLY 5DEGREES CASE (ALLLAYERSINTACT)3LAYERS(NO)ODDLAYERSSTIFFNESSMATRIX I C II106L8./IN.S0 1

*2380 1.7370 -1 32801 7370 4.2380 -1 32801 3280 -1.3280 2 3230EVENLAYERSTIFFNESSMATRIX IC)(10*6LB./IN.SO.

*23P0 1.737C 1 32801 7370 4.2380 1 32801 3280 1.3280 2 3230

THERMALEXPANSIONMATRIXALPHA)I IN./IN./OEG.F.)ALPHAALPHA2ALPHA6

7.45007.45003.9500

THERMALEXPANSIONMATRIX(ALPHA)IIN./IN./OEG.F.)ALPHA ALPHA2ALPHA6

7.45007.45003.9500

[10*6LB./IN.) (10-6IN./LB.) APRIME(10-6IN./LB.) TFERMALFORCEILB./IN./OEG.F.)4.238C 1.7370 -C.4425 0. 1.737C 4.2380 -0.4425 -0.-0.4425 -C.4425 2.3230 0.

286C0.1138.03281138.2860.03280328 C.0328 0.4430 0.2860 -0.1138 0.0328-0.1138 0.2860 0.03280.0328 0.0328 0.4430 Nl-T-39.2681N2-T39.2681N6-T -3.5357B

( 1C*60.c .0.

C. C.C.

UCtO IN)C.C.C.

0.0.0.

8 PRIME110-6 1/LB

0.0.0.C.0.0.

THERMALMOMENT(L8./CEG.F.IMl-T -0.0000M2-T -0.0000Mt-T -0.0000

(10*6LB. IN. ) 110*6LB.IN.I PRIME61/LB.IN.I0.3532 0.1447 -0.10250.1447 C.3532 -0.1025-0.1025 -0.1025 0.1936

0.3532 C.1447 -0.1C250.1447 C.3532 -0.1C25-0.1C25 -C.1025 0.19363.6829 -1.1152-1.1152 3.68291.3591 1.3591

1.35911.35916.6045

Z( IN . I

STRESSCOMPONENT

SIGMA2

26

SIGMA26

SIGMA2

SIGMA26

COEF.OFNl(t/IN.I

0.971C-0.029C-0.1525O.9710-0.0290-0.1525

1.05810.05810.30491.05810.05810.3049

0.9710-0.029C-0.15250.9710-0.029C-0.1525

CCEF.OFN2(I/IN.)

-0.02900.9710-0.1525-0.02900.9710-0.1525

0.05811.05810.30490.05811.05810.3049

-0.02900.9710-0.1525-0.02900.9710-0.1525

COEF.OFN611/IN.)LAYER

-0.3923-0.39230.9419-0.3923-0.39230.9419LAYER 0.78430.78431.11620.78430.78431.1162LAYER

-0.3923-0.39230.9419-0.3923-0.39230.9419

COEF.OFMlI I/IN.SO

-5.93310.06690.1264-1.97810.02230.0421

-2.5799-0.5795-1.09472.57990.57951.0947

1.9781-0.0223-0.0421 5.9331-0.0669-0.1264

COEF.CFM2ll/IN.SO.I COEF.OFM6(1/IN.SC.) COEF.OFTEMP(LB/IN.SO/F.)

0.0669-5.93310.1264 0.32500.3250-5.8662-0.8945-0.8945-4.6961

0.0223-1.97810.04210.10840.1084-1.9558

-0.8945-0.8945-4.6961

-0.5795-2.5799-1.0947-2.8158-2.8158-3.1593

1.78851.78859.38940.57952.57991.0947

2.81582.81583.15931.78851.78859.3894

-0.02231.9781-0.0421-0.1084-0.10841.9558

-0.8945-0.8945-4.6961-0.06695.9331-0.1264

-0.3250-0.32505.8662-0.8945-0.8945-4.6961

91


97/100

ANGLE-PLY THETA 60DEGREES CASE (LILAYERSINTACT]2AYERS(N"2I

DD LAYERS STIFFNESS MATRIX (Cl HERMAL EXPANSION MATRIX (ALPHA)(106 LB./INSCI IN/IN/OEG.F.)3.1780 1.4690 -0.6853 LPHA 1 9.42501.4690 58330 -1.6150 LPHA 2 54750-0.6853 -1.6150 2.C550 LPHA 6 - 3.420

VE N LAYERS STIFFNESS MATRIX Id HERMAL EXPANSION MATRIX (ALPHA)(106 L8./1NSC.) IN/IN/OEG.F.)

3.1780 1.4690 0.6853 LPHA 1 9.42501.4690 58330 1.6150 LPHA 2 547500.6853 1.6150 2.C550 LPHA 6 -3.420

A(IC6LB./IN.) A110-6IN./LB ) APRIME110-6IN./LB.) THERMALFORCE(LB./IN./OEG.F.)31 .0 .

17804690 1.4690 0.5.6330 0.C. 2.0550eI10


98/100

ANGLE-PLY THETA 60DEGREES CSE ALLLAYERSNTACTI3LAYERSN>3)ODDLAYERS--STIFFNESSMATRIXC) HERMALXPANSIONMATRIXALPHAI110*6LB./IN.SCI IN./IN./OEG.F.I

ALPHA 9.4250ALPHA2-.4750ALPHA6 3.4208

~EVENLAYERSSTIFFNESSMATRIXCl HERMALEXPANSIONMATRIXALPHAI110*6LB./IN.SO.I IN./IN./DEC.F.I

ALPHA .4250ALPHA2.4750ALPHA6 -3.4208

3.1780 1.4690 -0.68531.4690 5.8330 -1.6150C.6853 -1.6150 2.C550

3.1780 1.4690 0.68531.4690 5.8330 1.61500.6853 1.6150 2.C550

I1C*6LB./IN.I3.178C 1.4690 -0.22831.469C 5.8330 -0.5381-0.2283 -C.5381 2.0550

I1C*6N.)0. C..0. 0..0. 0.. A 110-6N./LB 1 1 APRICE0-6N./LB 1 THERMALFORCEILB./IN./OEG.F.I0.0.0.356708830165 -C.0883C.1975C.0419 000016504194994 0-00 356708830165 -0.06830.19750.C419 0.0165C.04190.4994 Nl-T-35.6512N2-T40.2564N6-T-2.756CI1C0N.) BPRIME10-6/LB. THERMALMOMENT(LB./DEG.F.I0.0.0. c .c .c .H*I1C0N.I 0 .0 .0 . 000 0. 0. 0. 000 Ml-T -O.OCOCM2-T>-0.0000M6-T O.OCOC0.0.0. C.c .0. 0 .0 .0 .

C110*6LB.IN ) CPRIME110-6/LB.IN.0.0.

26481224

C.12240.4861 -0. -0. 05291246 4-0 33499203 -0.92032.7244 c 1 66886982110*6LB.IN.I

0.2646 C.1224 -C.05290.1224 C.4861 -0.1246-0.0529 -C.1246 0.1712 -0.0529 -C.1246 0.1712 0.6688 1.6982 7.2816

I1 IN.I

STRESSCOMPONENT CHEF.OFNl ll/IN.I CCEF.OFN211/IN.I COEF.F611/IN.1 CCEF.OFMlll/IN.SO.I CCEF.CFM2ll/IN.SO.I COEF.OFM6I1/IN.SQ.1 COEF.CFIE(LA/IN.SQ/FLAYER

0.500C SIGMA 126

0.9925-0.0178-0.0679

-0.01920.9549-0.1724-0.2282-0.53780.9473

-5.98300.0400C.05500.0431

-5.89840.13970.18490.4358-5.C814

-0.4111-0.9687-3.6993

0.1667 SIGMA 126

0.9925-0.0178-0.0679-0.0192ff.9549-0.1724

-0.2282-0.53780.9473LAYER~

-1.9947C.01330.01830.0144

-1.96650.04660.06170.1453-1.9609

-0.4111-0.9667-3.6993

0.1667 SIGMA 126

1.01510.03550.1357

0.03831.09030.34470.45631.07531.1053

-2.1476-C.3466-0.4766-0.3736-2.8609-1.2101

-1.6020-3.7754-3.C281

0.B2191.93687.3965

0.1667 SIGMA 126

1.01510.03550.13570.03831.09030.3447

0.45631.07531.1053LAYER~2.1476C.34680.4766

0.37362.88091.2101

1.60203.77543.02610.82191.93667.3965

0.1667 SIGMA 126

0.9925-0.0178-0.0679

-0.01920.9549-0.1724

-0.2282-0.53780.9473

1.9947-0.0133-C.C183

-0.01441.9665-0.0466-0.0617-0.14531.9609

-0.4111-0.9667-3.69930.5C0C SIGMA 1

26

0.9925-0.0178-0.06 79-0.01920.9549-0.1724

-0.2282-0.53780.94735.9830

-C.04C0-C.0550-0.04315.8984-0.1397

-0.1849-0.43585.8814

-0.4111-0.9687-3.6993

93


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ANCLE-PLY THETA75DEGREES CASE (ALLLAYERSINTACT)2LAYERS(N-2)ODDLAYERSSTIFFNESSMATRIX(Cl HERMALEXPANSIONMATRIX(ALPHA!(106L8./IN.S0.I IN./IN./DEC.F.I

2.7*30 0.9321 -0.1993 LPHA 10.8700C.S321 7.3*20 -1.1290 LPHA *.0292-0.1993 -1.1290 1.5190 LPHA6 1.9750

EVENLAYERSSTIFFNESSMATRIXIC) HERMALEXPANSIONMATRIX1ALPHA)(10*6LB./IN.SO.I IN./IN./DEG.F.IALPHA 10.8700ALPHA2*.0292ALPHA6--1.97502.7*30 0.9321 0.19930.9321 7.3*20 1.12900.1993 1.1290 1.5190

A110*6LB./IN.) A110-6IN./LB ) APRIME(10-6IN./LB.)2.7*300.93210.

C.9321 0.7.3*20 0.0. 1.5190et i c eIN.)

0-00

381C0*8* -0.0*8*0.1*23C .B(10*0IN.)

0006583

0-00

38120*66 -0.0*66 0.0.15*7 C.0. 0.7205BPRIME10-61/LB.)

0.0.0.0*980 . 0.0*980. 0.28220.2822 0.

0 .0 .-0.C3280.c .-0.1858

-0-00

00530378 0 .0 .-0 0*610. -0.0*610. -0.3265-0.3265 C.

THERMALFORCE(Le./IN./CEG.F.)Nl-T 33.17!*N2-T-37.48*5Nt-T 0.THERMALMOMENT(LB./OEG.F.)Ml-T-.0000M2-T .OOOOM6-T- 0.9288

0 .0 .0.0053C.0.0.0378

0.03280.18580.c110*6LB.IN.) 10*6LB. N ) 0PRIME110-61/LB

0.2286 C.C777 0.0.0777 0.6118 0. 0.22690.068* C.068*0.559* 0 .0. *.5750 -0.5595-0.5595 1.8561 0 .0.8.6*62I STRESS COEF.DFNl CCEF.OFN2 COEF.OFN6 COEF.OFMl COEF.OFM2OEF.OFM6OEF.OFTEMP.(IN.I COMPONENT (WIN.I ll/IN.I ll/IN.I (1/IN.SC.) (1/IN.SO.)1/IN.SC.)LB/IN.SO/F.I

-0.0325.*3080.2859-6.18*3.**C*1.61970.*9606.1889.*52B0.06510.*308.28590.36872.**0*.6197-0.4960.37782.9056-0.06510.43C8.2859-0.36872.4404.6197-0.49600.3778.90560.0325.43C80.28596.1843.44041.61970.4960.18891.4528STRESSCOMPONENT COEF.OFNlll/IN.I CCEF.OFN211/IN.I COEF.OFN6ll/IN.) COEF.OF(1/IN.SC.LAYER SIGMA 126 0.9977-0.01300.0117 -0.01630.90780.0827 0.07180.40670.9055 -6.0046-0.02600.0700SIGMA 126 1.00230.013C-0.0233 0.01631.0922-0.1653 -0.1436-0.81351.0945LAYER 0.0092C.0520-C.0700SIGMA 126 1.00230.013C0.0233 0.01631.09220.1653 0.14360.81351.0945 -0.0092-0.0520-0.07COSIGMA 126 0.9977-0.0130-0.0117 -0.01630.9078-0.0827 -0.0718-0.40670.9055 6.00460.02600.0700

94


100/100

2 . 7 4 3 0 0 . 9 3 2 1 -0 .1993C.9321 7 . 3 4 2 0 -1.1 2 90ci993 -1.1 2 90 1.5190

ANGLE-PLYHETA = 75 DEGREESAS E 1 (ALI LAVERS INTACT)3 LAYERS IN=3I-- OD D LAYERS --STIFFNESS KATRIX IC) HERMAL EXPANSION MATRIX (ALPHA)

(10.6 LB./INSCI IN/lN/DEG.F.lALPHA 0.8700ALPHA.0292ALPHA 6 1.9750

EVEN LAYERS STIFFNESS MATRIX (Cl HERMAL EXPANSION MATRIX (ALPHA)(10.6 LB./INSCI IN/IN/DEC.F.I

2.7430.9321.1993 LPHA 1 10.87000.9321.3420.1290 LPHA 2 = 4.02920.1993.1290.5190 LPhA 6 -1.9750THERMAL FORCE

(1C6 LB./IN 10-6 IN/B.)IC-6 IN/LB.)LB./IN/DEO.F.I2.7430.9321 -0.0664.3810 -C.0481.0047.381C -0.0481.0047-T . 33.17840.9321.3420 -0.37620.0431.1440.03360.0481.1440.03362-T 37.4845C664 -0.3762.5190.0C47.0336.6668.CC47.C336.66686-T -1.2379APRIME(IC-6IN./LB )000 381C - 0 . 0 4 8 10481 0 . 1 4 4 0CC47 0 . C 3 3 6

BPRIME(10-61/LB.

000

1

0 0 4 70 3 3 66668

000a .0.0.

00c

E PRIMCHERMAL MOMENT[[0.6NI10*0 IN)10-6 1/LB.IL8./0EG.F.)0......l-T = -0.0000o'".......2-T . -O.OOCCo'.......6-T O .OOOC 0. C..0. C..o. c.L 0 PRICE(10*6 LB.IN) (10.6 LB.I.I10-6 1/LB.INIC.2286 C.C777 -0.C154 0.2286 C.07770.0154.5754 -C.5562 C.173C0.0777 C.611B -0.0871 0.0777 C.61180.08710.5562 1.B796 1.2259-0.0154 -C.0671 0.1266 -0.0154C.0871.1266 0.1730 1.2259 B.7646

(IN.STRESS COEF.OFNl CCEF.OFN2 COEF.OFN6 CCEF.OFM1 CCEF.CFM2 CCEF.OFM6 COEF.OFTEMP.COMPONENT (1/1N.I (1/1N.) (1/IN.I (1/IN.SCI (1/IN.SO.) (1/IN.SO.I (LBV IN.SO/F.I

--LAYER 0 5C0U SIGMA 126

0.9994-0 .0 0 36-0 .0 1 44-0.C0450 . 9 7 4 7-0 .1 0 2 0

-0.0886-0 .50 2 00.9741

-5.99870 . 0 0 7 2C. 0 1 C5

0.0C91 -5.94870 . 0 7 4 5

0.C6470.3667 -5.9474

-0 .0 78 4-0.4441-1.7931

0 1667 SIGMA 1260.9994-0 .0 0 36-0 .0 1 44

-0 .C 0 450 . 9 7 4 7-C .1 0 2 0-0 .0 886-0 .50 2 00.9741LAYER

-2.00C0 0 . 0 0 2 4C.0O350 . 0 0 3 0 -1.98330 . 0 2 4 8

0 .C 2 160 . 1 2 2 3-1.98 2 9- 0 . 0 7 84-0.4441 -1.7931

0 1667 SIGMA 1261 . 0 0 1 30 . 0 0 7 10 . 0 2 88

0 . 0 0 891.05050 .2 0 4 00 . 1 7 7 21 . 0 0 3 71.0518

-2 .0 1 15-0.0627 -0 .0 91 1-0 .0 78 4-2.4447 -0.6457

-0.5608-3.1 768-2.45560.1567 0 . 88 7 93.5852

0 1667 SIGMA 1261 . 0 0 1 30 . 0 0 7 10 . 0 2 88

0 . 0 0 891 . 0 5 050 . 2 0 4 00 . 1 7 7 21 . 0 0 3 71.0518

2 . 0 1 1 50 . 06 2 7C.0911 0 . 0 7 842.4447 0.6457

0.56083.17682.455B0.1567 0.88793.5852

strength caracteristics of composites materials

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