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
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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
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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
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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.
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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.
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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
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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.
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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."
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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
? ' =
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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)
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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.
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C O
J (PERCENT)
Figure5. StrengthofaTypicalCross-PlyComposite
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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.
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CO Q-oCO CO L LL
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M'0.4 CASE2ALLLAYERSDEGRADE:)2LAYERSNi2>
STIFFNESSHATRIXC) (10*6d./IN.SO.)
'.8000 0.0000o.oooo o.oooo0..STIFFNESSMATRIXC)(10*6L8./IN.SO.)o.oooou.oooo 0.00007.80000. --ODULAYERS"-EVENLAYERS THERMALEXPANSIONMATRIXALPHA)(IN./IN./DEG.F.)ALPHA 3.5000A.PHA =1.40 0 0ALPHA .THERMALXPANSIONMATRIXALPHA)(IN./IN./DEG.F.)ALPHA =1.4000ALPHA .5000ALPHA6= 0. (106LB./IN. )
2.2285.0000O.OOOO.57150. 0.
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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*6LB./IN.SO.I
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ALPHA .5000ALPHA2 1.40001.PH6=0.
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0. 0, 1.0000
-24.0003-0.00030.
0.0000-0.03000.
0. 0. -6.0003
-0.0000-0.00300.
0. SI3M 126
8.00000.00030.
-0.30000.00000.
0. 0. 1.0000--IAYCR2
24.00000.0000I ) .
-O.OOOO-0 .(130 0 0.
0. 0. 0.
0.0001-o.oooo0.
0. SIGNA 126
0.0033-0.03000.
0.00006.00000.
0. 0. 1.0000
0.00000.00000.
- (' . 0 0 0 0-24.00000.
0. 0. 0.
-o.oooo0.00010.
0.5000 5I3MA 126
0.00030.00000.
-0.0000-4.00000.
0. 0. 1.0000
0.0033-0.00330.
0.000024,00000.
o. 0. 6.0003
-o.oooo-o.oooo0.
76
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M1.0 CASE LLLAYERSDEGRADED)3LAYEHSN.3)
STIrFNtSSMATRIXCl(10-6B./IN.SO.)
7 .0000.00000.0000.0000..STIFFNESSMATRIXCl(10*6L./IN.SQ.)U.0OOO0.00000. 0.00007.80000. ODDLAYERS--EVENLAYERS THERMALEXPANSIONMATRIXA^PHA)(IN./IN./OEG.F.)ALPHA .5000ALPHA =1.4000ALPHA = 0. THERHALEXPANSIONMATRIXA^PHA)(IN./IN./DEG.F.)A.PHA 1.000ALPHA =.5000ALPHA6=0.
(10*6B./IN.)3.9000.OUOO.0.0000.9000.0. 0. 0.
(10-6N,/LB.)0,2564 -O.OUOO .
-0.000 0.2564 0. 0. 0. 0000.000
0.0.0.
(10-0N.)0. 0.
APRIME(10-6N./LB THERMALFORCE(LB./IN./DEG.F. )0.2564 - 0 . 0 0 0 0 0 .U . 0 0 ( 1 0 0.2564 0 ,0 . 0 . 0 0 0 0 0 0 0 0
Nl-T 13.6500N2-T 13.6500N6-T 0 .f aPRIME(10-61/LB.)
THERMALMOMENT(LB./DEG.F.)0 . 0 .0 . 0 ,( 1 . 0 .
Ml-T 0 .M-T 0 .M6-T 0 .
(10*6LB.IN. ) (10*6LB.IN.)DPRIME(10-61/LB.IN.)
0.5688O.00000 . 000000813
5 00000.55880.00000.
0 . 0 0 0 00 . 0 B 1 30 .
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
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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.
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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
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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
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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.
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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