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CONTRACTORREPORT

NASA CR-1457

to

m ^ > ORTRUCTURALSTABILITYNALYSISFSANDWICHLATESANDHELLSbyR. .Sullins,G .W.Smith,andE.E.pierPreparedbyGE NE RALD YNAMIC SCORPORATIONSa nDiego,Calif.EPARTMENT - * }CMuV.,MannedSpacecraftCenterg* jyp rWv .NATIONALERONAUTICSNDPACEDMINISTRATION WASHINGTON,.. DECEMBER96Irsft

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THISOCUMENTSESTQUALITYVAILABLE.HECOPYURNISHEDOTICCONTAINEDIGNIFICANTNUMBEROFPAGESWHICHDONOTEPRODUCEEGIBLY.

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NASACR-1457

MANUALFORSTRUCTURALSTABILITYANALYSISO FANDWICHPLATESANDSHELLS

ByR.T.Sullins,G.W .Smith,ndE.E.Spier

Distributionofthiseportisprovidedinthenterestofinformationxchange.esponsibilityforthecontentsresidesinheuthorororganizationhatpreparedit.PreparedunderContractNo.NAS9-8244byGENERALDYNAMICSCORPORATIONSanDiego,Calif.

forMannedSpacecraftCenterN A T I O N A LAERONAUTICSAN DSPACEADMINISTRATION

-Forsale4>yjlieClearinghouseforF&tWslScientific/andTe&tmicalnformptiahpringfie/d,VirginiaZT5 -krPride$3.00DTIGQUALITYINSPECTEDt

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FACESHEET

BRAZEALLOYORADHESIVELAYER

FACESHEET

RIBBONDIRECTION

HONEYCOMBSANDWICHCONSTRUCTION111

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ABSTRACTThebasicobjectiveofthistudywastodevelopandcompileamanualwhichwouldincludepracticalandup-to-datemethodsforanalyzingth etructuralstabilityofsandwichplatesndshellsfortypicalloadingconditionswhichmightbeencounteredinaerospacepplications/ ) > Themethodsproposedforusewouldincludeknownanalyticalapproachessmodifiedforcorrelationwithapplicabletestdata.Thedataprcsented-he-recoversecommendeddesignequationsndcurvesforawideangeofstructuralconfigurationsndloadingconditions^includ-in gcombinedloads. Inanumberofcases,ctualtestdatapointsarein-cludedonthedesigncurvestosubstantiatetheecommendationsmade.Forthosetemswherelittleornotestdataexiststh ebasicanalyticalapproachispresentedalongwiththenotationthatthisrepresentedth ebestavailable"dataandshouldbeusedwithsomeautionandjudgmentuntilsubstantiatedbytest.Thefollowingsubjectsreamongthosecoveredinthemanual:

LocalInstabilityGeneralInstabilityofFlatPanelsGeneralInstabilityofCircularCylindersGeneralInstabilityofTruncatedCircularConesGeneralInstabilityofDome-ShapedShellsInstabilityofSandwichShellSegmentsEffectsofCutoutsontheGeneralInstabilityofandwichShellsInelasticBehaviorofandwichPlatesandShells

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SectionTABLEOFCONTENTS

PageINTRODUCTION _11.1ENERAL "11.2AILUREMODES _4LOCALINSTABILITY _12.1NTRACELLULARBUCKLINGFaceDimpling)...-12.1.1andwichwithHoneycombCore -12.1.2andwichwithCorrugatedCore -82.2ACEWRINKLING "212.2.1andwichwithSolidorFoamCoreAntisymmetricWrinkling) _212.2.2andwichwithHoneycombCoreSymmetricWrinkling) _282.3HEARCRIMPING ~372.3.1asicPrinciples ~372.3.2esignEquations ~40GENERALNSTABILITYOFFLATPANELS-13.1ECTANGULARPLATES -13.1.1eneral ~13.1.2niaxialEdgewiseCompression -53.1.3dgewiseShear -253.1.4dgewiseBendingMoment -373.1.5therSingleLoadingConditions -463.1.6ombinedLoadingConditions -473.2IRCULARPLATES -733.2.1vailableSingleLoadingConditions-733.2.2vailableCombinedLoadingConditions-733.3LATESWITHCUTOUTS -743.3.1ramedCutouts -743.3.2nframedCutouts -75GENERALINSTABILITYOFCIRCULARCYLINDERS....-14.1ENERAL ~14.2XIALCOMPRESSION ~54.2.1asicPrinciples "54.2.2esignEquationsndCurves -17

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TABLEOFCONTENTS,Cont'd.Section age

4.3UREBENDING -254.3.1asicPrinciples -254.3.2esignEquationsndCurves -294 . 4XTERNAL LATERAL PRESSURE- 3 24.4.1asicPrinciples -324.4.2esignEquationsndCurves -424.5ORSION 4-474.5.1asicPrinciples -474.5.2esignEquationsndCurves -544.6RANSVERSESHEAR -624.6.1asicPrinciples -624.6.2esignEquationsndCurves -644.7OMBINEDLOADINGCONDITIONS-654.7.1eneral -654.7.2xialCompressionPlusBending.........-674.7.3xialCompressionPlusExternalLateralPressure -714.7.4xialCompressionPlusTorsion -894.7.5therLoadingCombinations -95

5ENERALNSTABILITYOFTRUNCATEDCIRCULARC ONES -15.1XIALCOMPRESSION -15.1.1asicPrinciples -15.1.2esignEquationsndCurves -45.2UREBENDING -65.2.1asicPrinciples -65.2.2esignEquationsndCurves -75.3XTERNALLATERALPRESSURE-95.3.1asicPrinciples -95.3.2esignEquationsndCurves -125.4ORSION -145.4.1asicPrinciples -145.4.2esignEquationsndCurves -165.5RANSVERSESHEAR -185.5.1asicPrinciples -185.5.2esignEquationsndCurves -205.6OMBINEDLOADINGCONDITIONS-215.6.1eneral -215.6.2xialCompressionPlusBending -23

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TABLEOFCONTENTS,Cont'd.Section Page

5.6.3niformExternalHydrostaticPressure. . . .5.6.4xialCompressionPlusTorsion5.6.5therLoadingCombinations.GENERALNSTABILITYOFDOME-SHAPEDSHELLS .6.1ENERAL6.2XTERNALPRESSURE6.2.1asicPrinciples .6.2.2esignEquationsndCurves6.3THERLOADINGCONDITIONSINSTABILITYOFANDWICH SHELLSEGMENTS .7.1YLINDRICALCURVEDPANELS7.1.1xialCompression7.1.2therLoadingConditions7.2THERPANELCONFIGURATIONSEFFECTSOFCUTOUTSONTHEGENERALNSTABILITYOFSANDWICHSHELLSINELASTICBEHAVIOROFANDWICHPLATESA NDSHELLS9.1INGLELOADINGCONDITIONS9.1.1asicPrinciples9.1.2esignEquations9.2OMBINEDLOADINGCONDITIONS9.2.1asicPrinciples9.2.2uggestedMethod 5-275-345-396-16-16-36-36-126-157-17-17-17-117-113-19-19-19-19-39-109-109-13

IX

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LISTOFFIGURESFigure age1.1-1ypicalSandwichConstruction -21.2-1ocalizedInstabilityModes -51.2-2ltimateFailuresPrecipitatedbyFaceWrinkling-5 1.2-3on-LocalizedInstabilityModes -62.1-1riticalStressesforIntracellularBucklingUnderUniaxialCompression -32.1-2efinitionofDimensions -52.1-3hartforDeterminationofCoreCellSizeSuchThatIntracellularBucklingWillNotOccur -62.1-4orrugationConfigurations -82.1-5ucklingModes -102.1-6ocalBucklingCoefficientforSingle-Truss-CoreSandwich -142.1-7ocalBucklingCoefficientforSingle-Truss-Core

Sandwich -152.1-8ocalBucklingCoefficientforSingle-Truss-CoreSandwich -162.1-9ocalBucklingCoefficientforDouble-Truss-CoreSandwich -172.1-10ocalBucklingCoefficientforDouble-Truss-CoreSandwich -182.1-11ocalBucklingCoefficientforDouble-Truss-CoreSandwich -192.1-12ocalBucklingCoefficientforTruss-CoreSandwich -202.2-1ypicalVariationofQvs.. -242.2-2omparisonofTheoryvsTestResultsforFaceWrinklinginSandwichConstructionsHavingSolidorFoamCores. -252.2-3arametersforDeterminationofFaceWrinklinginSandwichConstructionsHavingSolidorFoamCores ....-27

xi

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FigureLISTOFFIGURES,Cont'd.

Page2.2-4ypicalDesignCurvesorFaceWrinklinginSandwichConstructionsHavingHoneycombCores -302.2-5omparisonofTheoryvsTestResultsorFaceWrinklinginSandwichConstructionsHavingHoneycombCores2-312.2-6elationshipfK goHoneycombCorePropertiesFcEcandFacingWavinessParameter6/tc) -342.2-7 GraphsfEquation2.2-12)ortheWrinklingStressfFacingsnSandwichConstructionsHavingHoneycomb

Cores ~352.3-1niaxialCompression ~402.3-2ureShear ~413.1-1lasticPropertiesndDimensionalNotationsor

TypicalSandwichPanel -43.1-2Mfor SandwichPanelwithEndsndSidesSimplySupported,sotropieFacings,ndOrthotropicCore,

(R=0.40) -103.1-3 K MforSandwichPanelwithEndsndSidesSimplySupported,sotropieFacings,ndIsotropieCore,

(R=1.00) _113.1-4 K MforSandwichPanelwithEndsndSidesSimplySupported,sotropieFacings,ndOrthotropicCore,

(R=2.50) _123.1-5MforSandwichPanelwithEndsSimplySupportedandSidesClamped,sotropieFacings,ndOrthotropicCore,

(R=0.40) "133.1-6MorSandwichPanelwithEndsSimplySupportedandSidesClamped,sotropieFacings,ndIsotropieCore,(R=1.00) -143.1-7MforSandwichPanelwithEndsSimplySupportedandSidesClamped,sotropieFacings,ndOrthotropicCore,

(R=2.50) _15

Xll

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LISTOFIGURES,Cont'd.Figure Page3.1-8 KMorSandwichPanelwithEndsClampedandSidesSimplySupported,sotropieFacings,ndOrthotropicCore,(R=0.40) -163.1-9 KMforSandwichPanelwithEndsClampedandSidesSimplySupported,sotropieFacings,ndIsotropieCore,(R=1.00) -173.1-10 KMforSandwichPanelwithEndsClampedandSidesimplySupported,sotropieFacings,ndOrthotropicCore,(R=2.50) -183.1-11 KMorSandwichPanelwithEndsndSidesClamped,IsotropieFacings,ndOrthotropicCore,R=0.40)....-193.1-12 KMorSandwichPanelwithEndsandSidesClamped,IsotropieFacings,ndIsotropieCore,R=1.00)-203.1-13MforSandwichPanelwithEndsndSidesClamped,IsotropieFacings,ndOrthotropicCore,R=2.50) -213.1-14 KMforSimplySupportedSandwichPanelHav ingaCorrugatedCore. CoreCorrugationFlutesrePerpendiculartotheoa dDirection -223.1-15 KMorSimplySupportedSandwichPanelHavingaCorrugatedCore. CoreCorrugationFlutesreParalleltotheoa dDirection -233.1-16 KM0orSandwichPanelwithIsotropieFacingsnEdgewiseCompression -243.1-17MforaSandwichPanelwithAllEdgesSimplySupported,andanIsotropieCore,R=1.00) -2 93.1-18MforaSandwichPanelwithAllEdgesSimplySupported,andanOrthotropicCore,R=2.50) -303.1-19MforaSandwichPanelwithAllEdgesSimplySupported,andwithanOrthotropicCore,R=0.40) . . -313.1-20MforaSandwichPanelwithAllEdgesSimplySupported,IsotropieFacingsndCorrugatedCore.oreCorrugationFlutesreParalleltoSide -32

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LISTOFFIGURES,Cont'd.Figure age3.1-21Mor SandwichPanelwithAllEdgesSimplySupported,IsotropieFacingsndCorrugatedCore.oreCorrugation

FlutesreParalleltoSideb -333.1-22Mor SandwichPanelwithAllEdgesClamped,sotropieFacingsndIsotropieCore,R=1.00) -343.1-23Mfor SandwichPanelwithAllEdgesClamped,sotropieFacingsndOrthotropicCore,R=2.50) 3-353.1-24MforaSandwichPanelwithAllEdgesClamped,sotropieFacingsndOrthotropicCore,R=0.40) -363.1-25Mfor SimplySupportedSandwichPanelwithnIsotropieCore,(R=1.00) -423.1-26Mor SimplySupportedSandwichPanelwithanOrthotropicCore,R=2.50) -433.1-27Mor SimplySupportedSandwichPanelwithnOrthotropicCore,R=0.40) -443.1-28Mfor SimplySupportedSandwichPanelwithCorrugatedCore. CoreCorrugationFlutesParalleltoSide -453.1-29nteractionCurveor HoneycombCoreSandwichPanelSubjectedtoBiaxialCompression -59

InteractionCurveor HoneycombCoreSandwichPanelSubjectedtoBendingndCompression -60InteractionCurveor HoneycombCoreSandwichPanelSubjectedtoCompressionandShear -61InteractionCurveor HoneycombCoreSandwichPanelSubjectedtoBendingandShear -62BucklingCoefficientsorCorrugatedCoreSandwichPanelsinBiaxialCompressiona/b=1/2) -63

3.1-34ucklingCoefficientsorCorrugatedCoreSandwichPanelsinBiaxialCompressiona/b=1.0) -64

3.1-35ucklingCoefficientsorCorrugatedCoreSandwichPanelsinBiaxialCompressiona/b=2.0) -65xiv

3.1-303.1-31 3,1-323.1-33

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LISTOFFIGURES,Cont'd.Figure age3.1-36 BucklingCoefficientsorCorrugatedCoreSandwichPanelsUnderCombinedLongitudinalCompressionandShearwith

LongitudinalCorea/b=1/2)... -663.1-37 BucklingCoefficientsorCorrugatedCoreSandwichPanelsUnderCombinedLongitudinalCompressionandShearwith

LongitudinalCorea/b=1.0) -673.1-38 BucklingCoefficientsorCorrugatedCoreSandwichPanelsUnderCombinedLongitudinalCompressionandShearwith

LongitudinalCorea/b=2.0) -683.1-39 BucklingCoefficientsorCorrugatedCoreSandwichPanelsUnderCombinedLongitudinalCompressionandShearwithTransverseCorea/b=1/2) -693.1-40 BucklingCoefficientsorCorrugatedCoreSandwichPanelsUnderCombinedLongitudinalCompressionandShearwithTransverseCorea/b=1.0) -703.1-41 BucklingCoefficientsorCorrugatedCoreSandwichPanelsUnderCombinedLongitudinalCompressionandShearwithTransverseCorea/b=2.0) -713.1-42 BucklingCoefficientsorCorrugatedCoreSandwichPanelsUnderCombinedLongitudinalCompression,TransverseCompression,ndShearwithLongitudinalCore-724.1-1quilibriumPathsorAxiallyCompressedCircularCylinders -24.1-2ypicalEquilibriumPathsorCircularCylinders.....-44.2-1chematicRepresentationfRelationshipBetweenK^and

Vcor "74.2-2 Semi-LogarithmicPlotfyQvsR/t forIsotropieNon-Sandwich)CylindersUnderAxialCompression-94.2-3nock-DownFactorycorCircularSandwichCylindersSubjectedtoAxialCompression -104.2-4 ComparisonofProposedDesignCriterionAgainstTestDataforWeak-CoreCircularSandwichCylindersSub-jectedtoAxialCompression -12

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LISTOFFIGURES,Cont'd.Figure age4.2-5 ComparisonofProposedDesignCriterionAgainst TestResultorWeak-CoreCircularSandwichCylinder

SubjectedtoAxialCompression -134.2-6tressesnvolvedinnterpretationfTestData-154.2-7ucklingCoefficientforAxiallyCompressedCircular

SandwichCylinders ~194.2-8esignKnock-DownFactororCircularSandwichCylindersSubjectedtoAxialCompression -204.2-9ucklingCoefficientforShortSimply-SupportedSandwichCylindersSubjectedtoAxialCompression0=)-244.3-1nock-DownFactor orCircularSandwichCylinders

SubjectedtoPureBending -284.3-2esignKnock-DownFactor-^orCircularSandwichCylindersSubjectedtoPureBending -304.4-1ircularSandwichCylinderSubjectedtoExternalLateralPressure -324.4-2 SchematicRepresentationofLog-LogPlotofCpVersusL/RorCircularSandwichCylindersSubjectedtoExternalLateralPressure -364.4-3 BucklingCoefficientsporCircularSandwichCylindersSubjectedtoExternalLateralPressure;sotropieFacings;TransverseShearPropertiesfCoresotropieorOrtho-

tropic;Vp ~444.4-4 BucklingCoefficientsporCircularSandwichCylinders

SubjectedtoExternalLateralPressure;sotropieFacings;TransverseShearPropertiesfCoresotropieorOrtho-tropic;V =0.05 ~45

4.4-5 BucklingCoefficientsporCircularSandwichCylindersSubjectedtoExternalLateralPressure;sotropieFacings;TransverseShearPropertiesfCoresotropieorOrtho-tropic;Vp 0.10 ~46

4.5-1ircularSandwichCylinderSubjectedtoTorsion-47xvi

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LISTOFIGURES,Cont'd.Figure age4.5-2ypicalLog-LogPlotoftheBucklingCoefficientKgorCircularSandwichCylindersSubjectedtoTorsion-504.5-3ucklingCoefficientsforCircularSandwichCylindersSubjectedtoTorsion -564.5-4ucklingCoefficientsforCircularSandwichCylindersSubjectedtoTorsion -574.5-5ucklingCoefficientsforCircularSandwichCylindersSubjectedtoTorsion -584.5-6ucklingCoefficientsforCircularSandwichCylindersSubjectedtoTorsion -594.5-7ucklingCoefficientsforCircularSandwichCylindersSubjectedtoTorsion -604.5-8ucklingCoefficientsforCircularSandwichCylindersSubjectedtoTorsion -614.7-1ampleInteractionCurve -664.7-2esignInteractionCurveforCircularSandwichCylindersSubjectedtoAxialCompressionPlusBending -684.7-3esignInteractionCurveforCircularSandwichCylindersSubjectedtoAxialCompressionPlusBending-704.7-4ircularSandwichCylinderSubjectedtoAxialCompressionPlusExternalLateralPressure -714.7-5 TypicalInteractionCurvesforCircularSandwichCylindersSubjectedtoAxialCompressionPlusExternalLateralPressure -744.7-6nteractionCurvesforCircularSandwichCylindersSubjectedtoAxialCompressionPlusExternalLateralPressure.... 4-794.7-7nteractionCurvesforCircularSandwichCylindersSubjectedtoAxialCompressionPlusExternalLateralPressure ..-804.7-8nteractionCurveforCircularSandwichCylindersSubjectedtoAxialCompressionPlusExternalLateralPressure....-814.7-9nteractionCurvesforCircularSandwichCylindersSubjectedtoAxialCompressionPlusExternalLateralPressure....-82

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LISTOFIGURES,Cont'd.Figure age4.7-10nteractionCurvesforCircularSandwichCylindersSubjectedtoAxialCompressionPlusExternalLateralPressure....-834.7-11nteractionCurvesforCircularSandwichCylindersSubjectedtoAxialCompressionPlusExternalLateralPressure....-844.7-12nteractionCurvesorCircularSandwichCylindersSubjected

toAxialCompressionPlusExternalLateralPressure....-854.7-13nteractionCurvesforCircularSandwichCylindersSubjectedtoAxialCompressionPlusExternalLateralPressure....-864.7-14nteractionCurvesforCircularSandwichCylindersSubjectedtoAxialCompressionPlusExternalLateralPressure....-874.7-15nteractionCurvesforCircularSandwichCylindersSubjectedtoAxialCompressionPlusExternalLateralPressure....-884.7-16ircularSandwichCylinderSubjectedtoAxialCompressionPlusTorsion -894.7-17onditionalInteractionCurveforCircularSandwichCylindersSubjectedtoAxialCompressionPlusTorsion-934.7-18onservativenteractionCurveforCircularSandwichCylindersSubjectedtoAxialCompressionPlusTorsion-945.1-1mpiricalKnock-DownFactors -25.1-2runcatedSandwichConeSubjectedtoAxialCompression -45.2-1runcatedSandwichConeSubjectedtoPureBending-75.3-1runcatedConeSubjectedtoUniformExternalLateralPressure -95.3-2runcatedSandwichCone -1 25.4-1runcatedSandwichConeSubjectedtoTorsion-1 75.5-1runcatedConeSubjectedtoTransverseShear......-185.6-1ampleInteractionCurve -2 15.6-2esignInteractionCurveorTruncatedSandwichConesSubjectedtoAxialCompressionPlusBending-26

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LISTOFFIGURES,Cont'd.Figure age5.6-3runcatedConeSubjectedtoUniformExternalHydrostaticPressure 5-275.6-4runcatedSandwichCone -315.6-5runcatedConeSubjectedtoAxialCompressionPlusTorsion -345.6-6onditionalInteractionCurveforTruncatedSandwichConesSubjectedtoAxialCompressionPlusTorsion-385.6-7onservativeInteractionCurveforTruncatedSandwichConesSubjectedtoAxialCompressionPlusTorsion-3 95.6-8runcatedConeSubjectedtoAxialCompressionPlusBendingPlusTorsion -406.1-1tructuralDomehapes -16.2-1andwichDomeSubjectedtoExternalPressure-36.2-2chematicRepresentationofRelationshipBetweenK,,andVc. -76.2-3nock-DownFactory-,forSandwichDomesSubjectedtoUniformExternalPressure -96.2-4ucklingCoefficientforSandwichDomesSubjectedtoExternalPressure -1 37.1-1ylindricalPanelandAssociatedFlat-PlateConfiguration -27.1-2 SchematicLogarithmicPlotofSchapitzCriterionforNon-SandwichCylindricalSkinPanels -37.1-3 SchematicLogarithmicPlotofTestDataforCylindricalIsotropieNon-Sandwich)kinPanelsUnderAxial

Compression -67.1-4raphicalRepresentationof Equations7.1-5)through(7.1-8) -1 0

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LISTOFTABLESTable age2-1ummaryofDesignEquationsforLocalInstabilityModesofFailure -423-1ummaryofDesignEquationsforGeneralInstabilityofFlatSandwichPanels -764-1ummaryofDesignEquationsorInstabilityfCircular

Cylinders -985-1ummaryofDesignEquationsforInstabilityofTruncatedCircularCones -4 26-1 SummaryofDesignEquationsforInstabilityofDome-ShapedShells -167-1ummaryofDesignEquationsforInstabilityofCylindrical,CurvedPanels -1 29-1ecommendedPlasticityReductionFactorsforLocalInstabilityModes -79-2ecommendedPlasticityReductionFactorsfortheGeneralInstabilityofFlatSandwichPlates -89-3 RecommendedPlasticityReductionFactorsforth eGeneralInstabilityofCircularSandwichCylinders,runcatedCircularSandwichCones,ndAxisymmetricSandwichDomes -9

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LISTOFSYMBOLSaanellength,nches. Majorsemi-axisofanellipse,nches.axiallengthofacylindricalpanel,nches.Raengthof thelatpanelshowninFigure7.1-1,nches.banelwidth,nches. Minorsemi-axisofanellipse,nches.bircumferentialwidthofacylindricalpanel,nches.Rbitchofcorrugatedcore,nches.bidthoftheflatpanelshowninFigure7.1-1,nches.CengthparameterdefinedbyEquation(4.7-25),imensionless.CarameterdefinedbyEquations4.2-21)nd(6.2-19),imensionless.Cucklingcoefficientforsandwichcylindersubjectedtoexternallateralpressure,imensionless. Vi yn2Dendingstiffnessofsandwichwallorpanel=inch-lbs. (Eltl+E2t2)2Dhearstiffnessofsandwichwallorpanel =hG )/tb/inch.dotalthicknessfsandwichwallorpaneld +t+t,.nches.J L 4i CEoung'smodulus,psi.Eoung'smodulusof thecoreinthedirectionnormaltothefacings,psi.Eoung'smodulusffacing,psi.Eecantmodulusffacing,psi.sEangentmodulusoffacing,psi.xxi

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E1'E2e.lFV(Fv)crFc

Young'smoduliforfacings nd2espectively,psi.StrainintensitydefinedbyEquation(9.2-2),n./in.Transverseshearforce,bs.Criticaltransversehearforce,bs.Flatwisesandwichstrength(thelowerofflatwisecorecompressive,flatwisecoretensile,ndflatwisecore-to-facingbondstrengths),psi.

Gransverseshearmodulusfcore,psi.cGoreshearmodulusssociatedwiththeplaneperpendiculartothefacingsandparalleltosideaof panel,psi.Goreshearmodulusssociatedwiththeplaneperpendiculartothecb facingsandparalleltosidebof panel,psi.Glastichearmodulusffacing,psi.G..oreshearmodulusassociatedwiththeplaneperpendiculartotheij facingsndparalleltothedirectionof loading,si.Gecantshearmodulusoffacing,psi.sGoreshearmodulusssociatedwiththeplaneperpendiculartothexz facingsndparalleltothexisfrevolutionofacylinder,si.Goreshearmodulusssociatedwiththeplaneperpendiculartotheaxisofrevolutionofacylinder,si.histancebetweenmiddlesurfacesof thetwofacingsof asandwichconstruction,nches.Kucklingcoefficientforanisotropicnon-sandwich)latplate,dimensionless. Bucklingcoefficientforflatrectangularsandwichpanelunderedgewisecompression(Kc),dgewiseshear(K g),oredgewisebending(K.). K K +KKheoreticalflatpanelbucklingcoefficientwhichisdependenton facingstiffnessndpanelaspectratio,imensionless.xxii

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Kheoreticalflatpanelbucklingcoefficientwhichisdependenton Mandwichbendingandshearrigidities,anelaspectratio,ndappliedloading,imensionless.Kucklingcoefficientforsandwichcylindersunderaxialcompressioncndsandwichdomesunderexternalpressure,imenionless.K 'ucklingcoefficientforshortsandwichcylindersunderaxialcom-pression,imensionless.KarameterdefinedbyEquation(4.4-2),imensionless.PKucklingcoefficientforsandwichcylindersubjectedtotorsion,sX

cr

dimensionless.KarameterdefinedbyEquation(2.2-4),imensionless.6 kucklingcoefficient,imensionless.kucklingcoefficientassociatedwithcompressivestressctinginxhexdirection,imensionless.k'oadingcoefficientforappliedcompressivestresswhichisctinginthexdirection,imensionless.kucklingcoefficientassociatedwithcompressivestressctingintheydirection,imensionless.k'oadingcoefficientforappliedcompressivestresswhichisctingin^heydirection,imensionless.Lver-alllength,nches.Lffectivelength,nches,eMppliedbendingmoment,n-lbs.Mriticalbendingmoment,n-lbs.crM.S.arginofsafety,imensionless.Nriticalcompressiveunningload,bs/inch.Numberofcircumferentialfull-wavesinthebucklepattern,dimensionless. xxm

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pxialload,bs.p'quivalentxialloaddefinedbyEquation5.6-32),bs.Priticalxialload,bs.crcr(p mpiricallower-boundvalueorcriticalxialoadwhenactingEmpirical aione,bs.pxternalpressure,si.priticalvalueorexternalpressure,si.cr(p )xperimentalvalueorcriticalxternalpressure,si.rcrTest(p lassicaltheoreticalcriticalpressureoracylinderubjectedtoXCLxternalpressurectingonlynthendclosures,si.%q

*b(R b)CL

Externalpressurectingonlyontheateralurfaceofacylinder,psi.

(p lassicaltheoreticalcriticalpressureoracylinderubjectedtoyCLxternalpressurectingonlyntheateralurface,si.Qheelativeminimum,withespectto,fexpression2.2-2),dimensionless.QuantitydefinedbyEquation2.2-3),imensionless.RegreeofcorehearmodulusorthotropicityGca/Gcb'dimensionless. Radiustomiddleurface,nches.StressatiodefinedbyEquation4.7-9),imensionless.StressatiodefinedbyEquation4.7-5),imensionless.Road,tress,rpressureatiossdefinedinappropriateectionscfthishandbook,imensionless.(R tressatiossdefinedinappropriateectionsfthishandbook,CLimensionless.Rffectiveadius,nches,e xxiv

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R.tressorloadratiofortheparticulartypeof loadingassociatedwiththeubscripti,imensionless.R.tressorloadratiofortheparticulartypeof loadingassociatedwith3 thesubscriptj,imensionless.Radiustomiddlesurfaceatthelargeen dofatruncatedconicalshell,measuredperpendiculartotheaxisfrevolution,nches.Raxiumradiusfcurvatureformiddlesurfaceofadome-shapedshell,nches.Rressureatiosasdefinedinappropriateectionsfthishandbook,dimensionless.(R^)ressureratiodefinedbyEquation(4.7-15),imensionless.CL Roa dorstressatiossdefinedinappropriateectionsof thishandbook,imensionle s s.(R tressatiodefinedbyEquation(4.7-29),imensionless.sCL Radiustomiddlesurfaceatthesmallen dofatruncatedconicalshell,measuredperpendiculartotheaxisfrevolution,nches.Rtressorloadratiocorrespondingtothexdirection,imensionless.Rtressorloadratiocorrespondingtotheydirection,imensionless.Riddle-surfaceradiusofcurvaturein theplaneperpendiculartothemeridian,nches.rarameterdefinedbyEquation(4.2-37),imensionless.a. sellsizeofhoneycombcore,nches.Txternaltorque,n-lbs.Triticalexternaltorque,n-lbs.cr(Tcrmpiricallower-boundvalueforcriticaltorquewhenactingalone,Empirical in-lbs.thickness,nches. xx v

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tRtcnches*ftPtott1'U

V

VcVpVsVxzVyzw

wcZZ

TotalthicknessfthecylindricalpanelshowninFigure7.1-1,nches.Thicknessfcoremeasuredinthedirectionnormaltotheacings),Thicknessofasingleacing,nches.TotalthicknessftheflatpanelhowninFigure7.1-1,nches.Thicknessofmaterialfromwhichcorrugatedcoreisormed,nches.Thicknessesof theespectivefacingsfasandwichconstruction(thereisnopreferenceastowhichfacingisdenotedbytheubscript1or2) ,nches.

h2Sandwichtransversehearstiffness,efinedasUGc~hGc,lbs.perinch._ DBendingandshearrigidityparameterwhichisdefinedasV-

dimensionless.ParameterdefinedinSections4.2nd6.2,imensionless.ParameterdefinedbyEquation(4.4-4),imensionless.ParameterdefinedbyEquation(4.5-4),imensionless.ParameterdefinedbyEquation(4.7-13),imensionless.ParameterdefinedbyEquation(4.7-14),imensionless.BendingandshearrigidityparameterforflatsandwichpanelswithcorrugatedcorewhichisdefinedasW-dimensionless. b Gcb(Eiti+E2t2)Runningcompressionload,bs/inch.LengthparameterdefinedbyEquation(4.2-33),imensionless.LengthparameterdefinedbyEquation(4.5-3),imensionless.

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angleofrotationatappropriatejointincorrugated-coresandwichconstruction(seeFigure2.1-5),egrees. Vertexhalf-angleof conicalshell,egrees.ngleofrotationatappropriatejointincorrugated-coreandwichconstruction(seeFigure2.1-5),egrees.ynock-downfactor,imensionless. Ratio=aa imensionless.y xy.nock-downfactorassociatedwithgeneralinstabilityunderpurebending,dimensionless.ynock-downfactorassociatedwithgeneralinstabilityunderaxialcompression,imensionless.y.nock-downfactorassociatedwiththegeneralinstabilityofadome-shapedshellunderexternalpressure,imensionless.(7i)T nock-downfactordeterminedfromatestspecimensubjectedtotheloadingconditioncorrespondingtothesubscripti,imensionless.ynock-downfactorassociatedwithgeneralinstabilityofacylinderunderuniformexternallateralpressure,imensionless.ynock-downfactorassociatedwithgeneralinstabilityunderpuretorsion,imensionless.mplitudeofinitialwavinessinfacing,nches.ormalstrainin thexdirection,n/in.x 'Cormalstrainin theydirection,n/in.Chearstraininthexyplane,n/in.Carameterinvolvingthecorelasticmoduli,orethickness,ndbucklewavelength,imensionless.t]lasticityreductionfactor,imensionless.T ? T tlasticityreductionfactorcorrespondingtoanexperimentalcriticalstressvalue,imensionless.X 2(1-n. (1-pi.)orisotropicfacings,imensionless. Ratio= aT/CT imensionless.x xxvii

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Ve

RatioftransversehearmodulifcoreseeEquation4.2-1)],dimensionless.ActualPoisson'satiooffacing,imensionless.ElasticPoisson'satiooffacing,imensionless.

padiusfgyrationorhellwallfsandwichandnon-sandwichconstructionsph/2orandwichconstructionswhosetwofacingsreofequalthickness),nches.C Ttress,si.aeakcompressivestressdueolelytonappliedbendingmoment,bsi.( C T lassicaltheoreticalvalueorthecriticalpeakcompressivestressCLnder bendingmomentctinglone,si.alassicalvaluefcriticalstress,si.CLaniformcompressivestressdueolelytonappliedxialload,c psi.CT'ffectivecompressivestressdefinedbyEquation4.7-38),si.c( C T eakxialcompressivestressdueolelytoanppliedbendingcboment,si.( C T niformxialcompressivestressdueolelytonppliedaxialcoad,si.(C T lassicaltheoreticalvalueorthecriticaluniformcompressiveCLtressunderanaxialloadactingalone,si.c rriticalstress,si.crc rxperimentalcriticalstressobtainedfromaparticulartestpeci-crtesten,si.CT 'xperimentalcriticalstresswhichwouldhavebeenttainedhadtesthetestpecimenemainedelastic,si.xxvui

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er

a ,aer, er1crimp

HMMax

M IN predicted

Rwr

a.xPx)

< C T x > CLPx

Criticalvalueorthecompressivestressctinginthexdirection,psi.Compressivetressesinfacings and2,espectively,nthepres-enceofthecriticalloadingforgeneralinstability(theresnoprefer-enceastowhichfacingisdenotedbythesubscript or2) ,psi.Uniaxialcompressivestressatwhichshearcrimpingoccursinsand-wichconstructions,si.StressintensitydefinedbyEquation(9.2-1),si.Hoopmembranetress,psi.Meridionalmembranetress,psi.Maximumpossiblecriticalstresscorrespondingtoaparticularmaterial,psi.Minimumvalueofstressforthepost-bucklingequilibriumpath,psi.Predictedvalueforcriticalstress,psi.Criticalbucklingstressoraflatplate,psi.Criticalbucklingstressforacompletecylinder,psi.Facingwrinklingstress,si.Stressactinginthexdirection,psi. Uniformaxialcompressivestressduetoanappliedaxialload,si.EffectivecompressivestressdefinedbyEquation(4.7-37),si.Peakaxialcompressivestressduesolelytoanappliedbendingmoment,psi.Classicaltheoreticalvalueforcriticaluniformaxialcompressivestresswhenactingalone,psi.

stressdu eolelytoanappliedaxialStressctingintheydirection,psi.Uniformaxialcompressiveload,si.

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7hearstress,si.T ffectiveshearstressdefinedbyEquation(4.7-39),psi.(T)CL crT

TV

Classicaltheoreticalvalueforcriticaluniformshearstresswhenactingalone,psi.Triticalshearstress,psi.T'riticalshearstressforanequivalentcylindersubjectedtoancrppliedtorque,psi.Pureshearstress,ctingcoplanarwiththeacings,twhichshearcrimprimpingoccursinsandwichconstructions,si.Tniformshearstressduesolelytoanappliedtorque,si.T Peakshearstressdueolelytoanappliedtransverseshearforce,psi.$ngulardimensionofcorrugatedcoreseeFigure2.1-4),egrees.QuantitydefinedbyEquation(4.2-10),imensionless.> ngleofrotationatappropriatejointincorrugated-coresandwichconstruction(seeFigure2.1-5),egrees. ParameterdefinedbyEquation(4.4-3),imensionless.

xxx

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CONVERSIONOFU.S.CUSTOMARYUNITSTOTHEINTERNATIONALSYSTEMOFUNITS1(Reference: MIL-HDBK-23)

Quantity U.S.Customary ConversionUnitactor2 SIUnitDensityLengthStressPressureModuli laS lty (RigidityTemperature

in.lbm/(lbm/ft3ftin.psi(lb/in.2jlb/ft2psi(F+460)

327.68x10ilograms/meter (kg/m16.02ilograms/meter (kg/m0.3048etersm)0.0254etersm)

36.895x10ewtons/meter (N/m6.895xlO3ewtons/meter2N/m2)47.88ewtons/meter2N/m2)36.895x10ewtons/meter (N/m)5/9egreesKelvin(K)

kgcal/hrmChermalonductivity Btuin./hrft F 0.1240Prefixestondicatemultiplesfunitsresollows

Prefix MultiplegigaG) io9megaM) io6kilok) 103millim) io"3microy j) io"6

ThenternationalSystemofUnitsSystementernationalSI)]wasdoptedbytheEleventhGeneralConferencenWeightsndMeasures,aris,October1960,nResolutionNo.2.

'MultiplyvaluegiveninU.S.CustomaryUnitbyconversionfactortoobtainequivalentvalueinSIunit.xxxi

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1INTRODUCTION1.1 GENERALThishandbookpresentspracticalmethodsforth estructuralstabilityanalysisofsandwichplatesandshells. Theconfigurationsndloadingconditionsoveredherearethosewhicharelikelytobeencounteredinaerospacepplications. Basicequa-tions,esigncurves,ndcomparisonsoftheoryagainsttestdataareincluded.Forthepurposesofthishandbook,structuralsandwichisdefinedas layeredconstructionformedbybondingtwothinfacingstoacomparativelythickcoreasdepictedinFigure1.1-1. Thefacingsprovidepracticallyallofth eover-allbendingandin-planeextensionalrigiditytotheandwich. Theoreservestopositionth efacestlocationsemovedfromth eneutralaxis,rovidesvirtuallyallofthetrans-versehearrigidityoftheandwich,ndstabilizesth efacingsagainstlocalbuckling.Thusth etructuralsandwichconceptisquiteimilartothatofaconventionalbeam. Theandwichcoreplays rolewhichisnalogoustothatofthebeamwe bwhileth eandwichfacingsperformafunctionverymuchlikethatofthebeamflanges. TheprimarydifferencebetweenthesetwotypesofconstructionliesntheNumbersnbrackets nth etextdenoteeferenceslistedaten dofeachmajorsection1;;tc.).

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factthatthetransverseheardeflectionsreusuallysignificanttoth eandwichbehavior;whereas,orbeams,hesedeflectionsreonlyimportantforth epecialcaseofelativelyshort,eepbeams.

FACING

FACING

Figure1.1-1. TypicalSandwichConstructionTheandwichisnattractivetructuraldesignconceptsince,ytheproperchoiceofmaterialsndgeometry,onstructionshavinghighratiosofstiffness-to-weightcanbechieved. Sinceigidityisequiredtopreventstructuralinstability,hesandwichisparticularlywellsuitedtoapplicationswheretheloadingconditionsreconducivetobuckling.Theuseofsandwichconstructioninaerospacevehiclesscertainlynotarecentinnovation. TheBritishdeHavillandMosquitobomberofWorldWarIemployed

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structuralsandwichthroughouttheairframe. In thisase,hesandwichwasntheformofbirchfacesheetsbondedto balsawoodcore. Manyotherairplanes,nclud-in gtheB-58,B-70,F-lll,C-5A,tc.,avetakenadvantageofthehighstrength-to-weightationjoyedbysandwichconstruction. SpacevehicleapplicationsaveincludedtheApollospacecraft,heSpacecraftMAdapter(SLA) fairingsontheCentaurandotherlaunchvehicles,swellaspropellanttankbulkheads.Inviewoftheeverincreasingapplicationofstructuralsandwich,thasbecomedesir-abletoassembleahandbookwhichpresentslatestdesignandanalysiscriteriaforthestabilityofsuchconstruction. Thepracticingdesignerandstressnalystneedthisinformationinaformsuitableforeasy,apiduse. Thisdocumentismeanttofulfillthatneed. However,tshouldbekeptinmindthat,nmanyareas,llpracticalproblemsavenotyetbeenfullyresolvedandon ecanonlyemploywhatmightbere-ferredtoas "best-available"approach. Inthesecasestisdvisabletosupplementnumericalcomputationswithsuitabletesting. Suchareasofuncertaintyareidentifiedin thisandbookinthesectionsdealingwiththeappropriateconfigurationsndloadingconditions.

In thesectionstofollowadiscussionisgivenofthebasicprinciplesehindthedesignequationslongwithconclusionsderivedfromananalysisofavailabletestdata. Thisisfollowedbythedesignequationslongwithanylimitationson theiruse. Also,ofacilitatetheiruse,tableoftheseequationsndrestrictionsmmediatelyprecedesthelistofreferencesnSections,,,nd5sincethesesectionscoverawiderangeofloadingconditionsndconsiderations.

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1.2 FAILUREMODESStructuralnstabilityof andwichconstructioncanmanifestitselfnanumberofdifferentmodes. ThevariouspossibilitiesresdescribedbelowndashowninFigures1.2-1hrough1.2-3.IntracellularBucklingFaceDimpling)Thiss ocalizedmodefnstabilitywhichoccursnlywhentheoresnotcontinuous. AsdepictednFigure1.2-1,ntheegionsdirectlyboveoreellssuchashosef honeycombore),hefacingsbucklenplate-likefashionwiththeellwallsctingsdgeupports. Theprogressivegrowthofthesebucklesaneventuallyprecipitatehebucklingmodeidentifiedbelowsfacewrinkling.FaceWrinklingThiss localizedmodefnstabilitywhichmanifeststselfntheformofhortwavelengthsnthefacings,snotconfinedondividualcellsfcellular-typecores,ndnvolvesheransversenormaltofacings)trainingofthecorematerial. AshownnFigure1.2-1,nemustconsiderthepossibleoccurrenceofwrinkleswhichmaybeitherymmetricalorantisymmetricalwithespecttothemiddleurfacefheoriginalundeformedandwich. AshowninFigure1.2-2,finalfailurefromwrinklingwillusuallyesulteitherfromcrushingofthecore,tensileupturefhecore,rtensileupturefheore-to-facingbond. However,ifpropercaresxercisedntheelectionofthedhesiveystem,neaneason-ablyssumehattheensilebondtrengthwillexceedboththeensilendom-pressivetrengthsfheoreproper.

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A-IntracellularBuckling(FaceDimpling)

SYMMETRIC NTISYMMETRICB-FaceWrinkling

fC-ShearCrimping

Figure1.2-1. LocalizedInstabilityModes

A-CoreCrushing

B-TensileRuptureofBond C-TensileRuptureofCoreProperFigure1.2-2. UltimateFailuresPrecipitatedbyFaceWrinkling

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CO 0+->

r-4C D a u0Q

C O oio

4-4oJ2cd 4-03 e cdC D 0

7300) c o iS

cd 03

0(H i I>ojio rH o

CO

T30a CO i

c d0

O oX3s c o

f f l

0T3O> -4J CQ I4 xs0NrH l Icd Og

CO I

0r-(

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ShearCrimping hearcrimpingisoftenreferredtoas localmodeoffailurebutisctuallyaspecialformofgeneralinstabilityforwhichth ebucklewavelengthisveryshortdu etoalowtransversehearmodulusforth ecore. Thisphenomenonoccursquiteuddenlyandusuallycausesth eoretofailinshear;however,tmayalsocauseashearfailurentheore-to-facingbond. Crimpingwillsometimesoccurincaseswhereelativelylong-wavegeneralinstabilityfirstdevelops. Insuchinstancesth ecrimpappearsbecauseofseverelocaltransversehearstressestth eendsofbucklepatterns. Asth erimpdevelops,hegeneralbucklemaydis-appearandapost-testexaminationwouldthenleadtoanerroneousconclusionastoth emechanismwhichinitiatedfailure.GeneralInstability orconfigurationshavingnosupplementarystiffeningsuchasrings)exceptatth eboundaries,hegeneralinstabilitymodeisdepictedinFigure1.2-3A. Thephenomenoninvolvesover-allbendingofth eompositewallcoupledwithtransversenormaltofacings)sheardeformations. Usually,ransverseexten-sionalstrainsdonotplayasignificantrolenthisbehavior. Whereasntracellularbucklingandwrinklingarelocalizedphenomena,eneralinstabilityisfamoregrossnature. Except forthepecialcaseitedundertheidentificationShearCrimping",hewavelengthsssociatedwithgeneralinstabilityarenormallycon-siderablylargerthanthoseencounteredinintracellularbucklingandfacewrinkling.Forconfigurationsavingsupplementarystiffeningatlocationsotherthanthebound-aries,hetermgeneralinstabilitytakesnnewsignificancendreferenceisalsomadetoanadditionalmodedentifiedaspanelinstability. Forthisase,eneral

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instabilityissdefinedabovebutwithth eaddedprovisionthatth ebucklepatterninvolvesimultaneousadialdisplacementofbothth eandwichwallandth enter-mediatetiffeners. AshowninFigure1.2-3B,heppropriatehalf-wavelengthofth ebucklepatternmustthereforeexceedthespacingbetweenintermediatetiffeners.TheexampleusedinFigure1.2-3Bsthatofasandwichcylinderstiffenedbyaseriesofringswhichhavensufficientstiffnesstoenforcenodalpointsattheire-spectivelocations.

PanelInstabilityThismodeofinstabilityappliesonlytoconfigurationswhichhavesupplementarystiffeningatlocationsotherthantheboundaries. Figure1.2-3Cdepictsthismodebyagainusingth eexampleofasandwichcylinderstiffenedbyaseriesofings. However,nthisasetheingshaveufficientstiffnesstoenforcenodalpointsattheirrespectiveocations. Theingsexperiencenoadialdeforma-tion. Therefore,hehalf-wavelengthofthebucklepatterncannotexceedth espacingbetweenrings. Asnth ecaseofgeneralinstability,hismodenvolvesover-allbendingofthecompositewallcoupledwithtransversesheardeformations. Hereagain,ransverseextensionalstrainsdonotplayasignificantrolenthebehavior.

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2LOCALNSTABILITY2.1 INTRACELLULARBUCKLINGFaceDimpling)2.1.1 SandwichwithHoneycombCore2.1.1.1BasicPrinciplesFromapracticalviewpoint,ntracellularbucklingcanbeegardedasflat-platebehavior. Evenwherecurvaturespresent,snth ecasesofcylindersandspheres,th ehoneycombcorecellsizewillnormallybeufficientlysmalltojustifysuchanassumption. AsnotedfromReference2-1,hecriticalstressforflatplatescanbeexpressedinth eform

=12(1-,/) \T)2-1"1)where

ccr =Criticalcompressivetress,si.k =Coefficientwhichdependsonth eplategeometry,boundaryconditions,ndtypeofloading,imensionless.V =Plasticityreductionfactor,imensionless.

Ef =Young'smodulusorfacingmaterialinth ecaseofintra-cellularbuckling),psi.ve =ElasticPoisson'satiofo rfacingmaterialinthecaseofintracellularbuckling),imensionless.

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tf =ThicknessofplateFacingthicknessnthecaseofintra-cellularbuckling),nches.

s =Aselectedcharacteristicdimensionofth eplate,nches.Itisonvenientheretocombineeveralofth eonstantsnEquation2.1-1)toobtain

O - = a (|2.1-2)or

2.o ;cr-~Ve*? Ef =KM)2.1-3)Toapplytheseequationstoth ecaseofintracellularbuckling,tisonlynecessarytodefineth edimension s andestablishacorrespondingvaluefor. InReference2-2,Norristook s tobeequaltoth ehoneycomborecellsize. Byconvention,thisstakenequaltoth ediameterofthelargestcirclethatcanbeinscribedwithinth ecell. Basedonth enalysisoftestdata,NorristhenchoseK=2.0orth ecaseofuniaxialcompression. Thisprovides reasonablygoodfittoth etestresultsashowninFigure2.1-1whichwastakendirectlyfromReference2-2. ItshouldbenotedthatthechoiceofK=2.0doesnotprovidealowerboundtoth edata. Sixofth etestresultsfallsignificantlybelowth evaluespredictedbyth eecommendedformula. Thisituationcanbetoleratedsinceth edimplingofseveralcellsinahoneycombandwichconstructionwillnotleadtocatastrophicfailureoon gassufficientlylargenumberofcellsemainunbuckled. A sndicatedbyth ecatterin Figure2.1-1,neouldreasonablyexpectthemajorityofunbuckledcellstopossessconsiderablygreaterbucklingstrengthsthanwouldbendicatedbyth eproposeddesigncurve. Undertheseconditions,omeedistributionoftresswouldoccur

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0.1

0.01

0.001

0.00050

. ///TheoryforSimply /Upijurieuqua.xBJfld.US v/

' Pi/br(1-V)'?Ef/ J

in'f/>V(j, I/V u4/ Legend: " ~ DimplinginElasticRangehfoi TestDataforSpruceCorewithSingleCircularHole TestDataforHoneycombCore

*7 DimplingBeyondElasticRangeOTestDataforSpruceCorewithSingleCircularHoleDTestDataforHoneycombCore. /,/////// /01 1.0m

Figure2.1-1. CriticalStressesforIhtracellularBucklingUnderUniaxialCompression2-3

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sincecan

butth etructurecouldcontinuetoupportth eappliedload. Inaddition,tispointedou tthatth edimpledregionsetainsignificantpost-bucklingload-carryingcapability

theybehaveessentiallyasflatplates. Thisdoesnotmean,owever,haton epermitthedimplestogrowwithoutbound. Thepointcanbeeachedwherethese

deformationsprecipitatewrinklingandthisannotbetolerated.Itisalsoofimportancetonoteherethatmostofth etestdatashowninFigure2.1-1wereobtainedfromsandwichplateshavingasolidsprucecorethroughwhichasinglecircularholewasdrilledtorepresentacoreell. Itisquestionablethatsuchspecimenstrulysimulateth ecelledgeupportlikelytobeencounteredinpracticalhoneycombonfigurations. Onlythreedatapointswereobtainedforspeci-mensactuallyhavinghoneycombcoresnd,showninFigure2.1-1,hesepointslieinth elowerregionofthetotalbandofscatter.In viewofth eforegoingdiscussion,tisvidentthattheuseofEquation2.1-3togetherwithth eelectionofK=2.0 isertainlynotarigorouspproachtoth eanalysisfintracellularbuckling. However,ntilfurtherworkisccomplishedin thisrea,tisecommendedthatthiscriterionbeemployedas best-available",approximatedesigntool.

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2.1.1.2DesignEquationsndCurvesThefacingstresstwhichintracellularbucklingwilloccurunderuniaxialcompres-sionisgivenbyth efollowingsemi-empiricalformula:

V Ef /tf\s=2 ' )W (2.1-4)Thedimensions isth ediameterofth elargestcirclethatcanbenscribedwithinthecellshape. Forexample,nth ecasesofhexagonalandsquarecells, s ismeasuredashownbelow.

y \(i# S /.Figure2.1-2. DefinitionofDimension s

SolvingEquation2.1-4)for s givesheesults fl2 crl-*feT 7 Ef (2.1-5)

Thisquationmaybeusedtodeterminehemaximumpermissibleellsize.corre-spondingtoparticular facingmaterialsndhicknesses. Figure2.1-3presentsfamilyofplotsfEquation2.1-5)forelectedvaluesf tf rangingfrom tf0.001 to tf .100.Forelasticcases,se T ? . Wheneverthebehaviorsnelastic,hemethodsfSectionmustbemployed.

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2.00

1.000.800.60

0.400.30

W

0.20

O Q N 0.10

i. (0) 0.08

0) uo0.06

0.040.03

0.02

0.01

IM C O * c o 00 r-i

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Whenth efacingsaresubjectedtobiaxialcompression,tisecommendedthaton euseth einteractionformula

Rx y 2.1-6)where

[AppliedCompressiveoading]nSubscriptDirection J1 [CriticalCompressiveoadingwhen '* ~[actingalone)nSubscriptDirectionJThisstraight-linenteractionrelationshipisbasedonth einformationprovidedinReference2-1forsquareflatplates. Forcasesnvolvingshearingstresseswhicharecoplanarwithth efacings,tisecommendedthattheprincipalstressesfirstbecomputedandthatthesevaluesthenbeusedintheaboveinteractionequation. When-everon eofth eprincipalstressesstensileandthebehavioriselastic,heanalysisshouldbebasedontheassumptionthatthecompressiveprincipalstressisactingalone.

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2.1.2 SandwichWithCorrugatedCore2.1.2.1 BasicPrinciplesThisectiondealswithcorrugated-coreandwichconstructionswhosecrossectionsmaybedealizedshownnFigure2.1-4. Forcylinders,henlycasereatedhereshatwherehexisfheorrugationssparalleltohexisfevolution.Forflatplates,owever,heorrugationsanbeorientedneithertheongitudinalortransversedirections.

h-bf-H I h-bf-H ? *t (TYPICAL)o

^(TYPICAL)((TYPICAL)

Single-Truss ouble-TrussFigure.1-4. CorrugationConfigurations

Eachofhefollowingoadingonditionssonsidered:a.niaxialcompressionactingparalleltothexisfheorrugations.b.niaxialcompressionactingparalleltohefacingsbutnormaltoheaxisfheorrugations.c.iaxialcompressionesultingfromombinationsf a andb above.

ThedesigncurvespresentedherereakendirectlyfromReference2-3ndrebasedentirelyontheoreticalconsiderations. Noomparisonsremadegainsttest

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datatoconfirmthevalidityoftheseolutions. Untilsuchsubstantiationisobtained,theecommendeddesigncurvescanonlybeconsideredasabest-available"criterion.Itispointedout,owever,hattheredoesnotappeartobeanyreasontouspectthattestdatawoulddisagreewithth ecurves.

AlthoughReference2-3isdevotedolelytoflatplates,heesultsareconsideredtobeapplicabletoth ecylindricalconfigurationshowninFigure2.1-4sincethedimen-sionsbfwillusuallybesmallwithrespecttotheadius. Undersuchconditions,curvatureinfluenceswillbenegligible.Thetheoreticaldevelopmentincludesconsiderationofeachofth ebucklingmodesshowninFigure2.1-5. Bothofthefollowingpossibilitiesrecovered:

a.hefaceheetsreth eunstableelementsandareestrainedbyth ecore.b.hecoreistheunstableelementandisestrainedbythefaceheets.

Bucklingisassumedtobeaccompaniedbyrotationofthejointsbutwithnodeflectionofth ejoints. Theanglesbetweenth evariouselementsatanyon ejointaretakentoremainunchangedduringbuckling. Itislsoassumedthattheover-allsandwichdimensionsareufficientlylargesuchthaten deffectsarenegligible.

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ClompedJ

--l a mpe d

.supported

Single-Truss-Coreouble-Truss-Core(a,,nd] Jdenotenglesfotationsttheppropriateoints)

Figure2.1-5. BucklingModes

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2.1.2.2De signEquationsandCurvesThetheoreticalstressatwhichintracellular bucklingofth efacingsorbucklingofth ecorrugatedcorewilloccurisgivenbythefollowingformula:

J TT ZVE /tf\32(l-^e3)bfjkiir8?Ecr .8(r)2.1-8)

wherecrcr =Criticalcompressivetress,psi.

kj =oefficientwhichdependsuponth egeometryandloadingconditions,imensionless.V =Plasticityreductionfactor,imensionless.

E =Young'smodulusoffacingsndcore,psi.P e - lasticPoisson'satiooffacingsandcore,si.tf =Facingthickness,nches.

bf =PitchofcorrugatedcoreseeFigure2.1-4),nches.Theonlycaseconsideredhereisthatwherethetwofacingsareofth eamethicknessandtheentiresandwichconstructionfacingsndcore)ismadeofasinglematerial.Figures2.1-6through2.1-12givevaluesforkjoreachofth efollowingloadingcombinations:

1 ? ^he n.= 0kxhe ny0.5kxhe ny1.0

d. kyhe nx0Thecoefficientskxndkyaredefinedasfollows:

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12(1-i/e2)bf\skx r1 (AppliedCompressiveo-x)2.1-9)12(1-14?)/bfVkv h-) (AppliedCompressiveo-)2.1-10)Theubscriptx (forkndk')isusedtodentifycaseswheretheloadingisdirectedalongthexisofth ecorrugationsxirection). Theubscript (forkand')isusedtoidentifycaseswhereth eloadingisactinginth e yirectionwhichisparalleltothefacingsbutnormaltoth exisofth eorrugations. Forcombinationsahrough eparateplotsarefurnishedforsingle-truss-coreanddouble-truss-coreconfigurations. Forcombination singlefamilyofcurvescoversbotharrangementssinceallofthecorrespondingappliedloadistransferredthroughthefacings. ThedashedlinesnFigures2.1-6through2.1-11divideth echartsintotworegions. Abovethedashedlines,hefaceheetsaretheunstableelementsandarerestrainedbythecore. Belowthedashedlines,hecoreisunstablendisestrainedbythefaceheets.ToclarifythedesignchartsgiveninFigures2.1-6through2.1-12,hefollowingadditionaldefinitionsareprovided:

t0=Thicknessofmaterialfromwhichthecorrugationsareformed(seeFigure2.1-4),nches.f> ngleshowninFigure2.1-4,egrees.

Inaddition,heampleproblemgivenbelowshouldbehelpfultoth euserofthishandbook.

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Given: SampleroblemDataoringle-TrussCoreypeandwichanelE 0 0* psi0 016"f 700"ve 30f 020"5 ProportionalimitCT=0,000siy 6,300siCompression)Required: Find 0 "cr Assuming7 , onebtainsk, 2y-V)/V\ _ 1 2x6,300x910 ATOOYyT2TjEtf/ " 9.87lx0X106 \.020/7bt0016I T=iT=800Usinginearnterpolationbetweenaluesivennigures.1-7nd.1-8ne

obtains .68.Hence, theriticaltressnhe irectionparalleloheorrugationxis)s

kxr2t]Ev2nr /f\2acrx 12(l-ve2)lband, assuming , onebtains

2.68 .87x 1 0 06,.,.,crx 2910(l)=930siCompression)/ .020V rftThetressntensityo * jSeeection)anoweomputedsollows:iVax+cry2-T xOy T2=oV(59.3)2 (16.3)2-59.3x6.3) 3,100siSincehisalueselowheroportionalimit, thessumption salid.Inaseswherehe7 jaluexceedsheroportionalimit, themethodsfection9mustemployed.

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2.2 FACEWRINKLING2.2.1SandwichWithSolidorFoamCoreAntisymmetricWrinkling)2.2.1.1BasicPrinciplesTheproblemoffacewrinklinghasbeentreatedbymanyinvestigatorsdatingbackasfaras1940. ThemostimportantpublicationsnthisubjectareistedasReferences2-4through2-14. Forthepurposesofthishandbook,twasdecidedthattheesultsinReferences2-7nd2-9wouldbehemostuseful. Theatterppliesnlytoand-wichconfigurationswhichhaveolidorfoamcores. Thedevelopmenttherencludesconsiderationofboththeymmetricndantisymmetricmodeslongwiththenfluencesfromnitialwavinessfthefacings. Itispointedoutthat,whentheoresufficientlythick,hewrinklepatternsfthetwofacingswillbendependentofachotherandthesamecriticalloadsobtainedfortheymmetricndantisymmetricmodes. However,forandwicheshavingthinnercores,hecoretrainsntroducedbyonefacinginflu-encethewavepatternintheotherfacing. Undertheseonditions,twasfoundthatsandwicheshavingsolidorfoamcoresanbexpectedtowrinklentisymmetrically.Thefollowinggoverningequationwasderivedtopredictthisformofwrinklingforisotropicfacingsubjectedtouniaxialcompression:

*7EfEf.Gr,

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Ec =Young'smodulusfheorenthedirectionnormaltohefacings,si.Gc =Corehearmodulusssociatedwiththeplaneperpendiculartohefacingsndparalleltohedirectionoftheppliedoad,si.ve=ElasticPoisson'satioffacings,imensionless.

ThequantityQ isheelativeminimum,withespectto fhexpression30q

16q cosh- \ 11 1inh+/

, , cosh \1 6.4Ksr~r =-)o*\ninh+5/ (2.2-2)where

tfGc[ (l-^e2) 13r? EfEcGc

K g SECtcFc(2.2-3)

(2.2-4)and

Parameternvolvingtheorelasticmoduli,orehickness,ndbucklewavelength,imensionless.tc =Thicknessfore,nches.tf =Thicknessffacing,nches. =Amplitudefnitialwavinessnfacing,nches.

Fc =Flatwiseandwichtrengththeowerofflatwiseoreompressive,flatwiseoreensile,ndflatwisecore-to-facingbondtrengths),psi.

Thenitialwavinessplaysnmportantolenthewrinklingphenomenonincetcausesransversefacingdeflectionsodevelopevenwhentheppliedoadingsverysmall. Asheoadncreases,hesedeflectionsgrowtsteadilyncreasingatesndleadtoransverseensileorcompressivefailurefthecoreortensileupturefthe

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core-to-facingbond. Thesefailuresoccur,fcourse,tloadvaluesbelowth epre-dictionsfromclassicaltheoryinwhichitisassumedthatth efacingsarenitiallyperfectKg ).TheesultsfromReference2-9canbesummarizedinth eformofEquation2.2-1)accompaniedbyplotsofQvsqwithgasparameter. AfamilyofsuchcurvesisgiveninReference2-9andtheyareofth egeneralshapeshowninFigure2.2-1.Thelimitingvaluesestablishedbyth estraightline0AcorrespondtothehearcrimpingmodeoffailureseeSection2.3). Allotherpointsonthecurvesareforantisymmetricwrinkling. Inactualpractice,urvesofthistypedonotprovetobeveryhelpfulsinceth eKgaluesappropriatetoparticularstructuresrearelyknown. Therefore,nordertoprovideapracticalmeansforth epredictionoffacewrinklinginsandwichconstructionshavingsolidorfoamcores,thasbecomeom-mon practicetoselectasingleconservativelower-boundQasedonavailabletestdata. Thispproachisfollowedhere. ElastictestdataselectedfromReference2-9areplottedinFigure2.2-2fromwhichth evalueQ=0.50hasbeenselectedasasafedesignvalue. AdditionaldataaregiveninReference2-6whicharenotshownherebutleadtoth esamevalueforalower-bound. Thissnconformancewithth eobservationmadebyPlantemainReference2-15thatthevalue .50asoftenbeenrecommendedforpracticaldesignpurposes. However,incemuchofth eexistingtestdatawereobtainedfromspecimensthatwerenotveryrepresentativeofconfigurationsikelytobeencounteredinrealistictructures,theselectionofQ=0.50canonlybeegardedasabest-available"approach. In viewoftheuncertainties

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KS0K8ConstantK8 Constant

Figure2.2-1. TypicalVariationof s. qamendedthatfortheverificationoffinaldesigns,wrinklingtests

whicharetrulyepresentativeofth eactualconfiguration.involved,tisecorbeperformedonspecimensThemethodpresentedhereforthepredictionofwrinklingshouldonlybeegardedasanapproximateguideline.

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i .m j tl

:__:i-ic a >O .bD II _ | | _ i-t,W .Q _linr 'T 3 ~:""~":.D cR4 -J _ :- y> ' *r -sri'

| |l_01 bU . 3 C O > C D rtH IB C O > C O r ? iHo< 1 > g Jt +J H A

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2.2.1.2DesignEquationsandCurvesThefollowingequationmaybeusedtocomputeth eapproximateuniaxialcompressivestressatwhichfacewrinklingwilloccurinsandwichconstructionshavingsolidorfoamcores:

ffwr " Z E fEcGcLl -^e3) (2.2-5)Incaseswhereth eamplitudeofinitialwavinesssknown,necanusethecurvesofFigure2.2-3toestablishQ . Wheneversuchinformationisunavailable,tisecom-mendedthatth evalueQ=0.50eusedtoobtainalower-boundprediction.Forelasticcases,seTj =. Wheneverth ebehaviorisnelastic,hemethodsofSection mustbeemployed.Whenth efacingsresubjectedtobiaxialcompression,tisecommendedthaton eusetheinteractionformula

Rx y 2.2-6)whereRi AppliedCompressiveoadinginSubscriptDirection,CriticalCompressiveoadingwhenactingalone)in SubscriptDirection (2.2-7)

andth eydirectioncorrespondstothedirectionofmaximumcompression. Thisinter-actionrelationshipisbasedonth einformationprovidedinReference2-1forrectangularflatplateshavingverylargeaspectratios. Forcasesinvolvingshearingstresseswhicharecoplanarwiththefacings,tisecommendedthatth eprincipalstressesfirstbecomputedandthatthesevaluesthenbeusedintheaboveinteractionequation. When-everon eofth eprincipalstressessensilendth ebehavioriselastic,heanalysisshouldbebasedontheassumptionthatth ecompressiveprincipalstressisctingalone.

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1 1 00 o / foo"II CO C O " j 1 ,/ / t-I j / ill 00 / CO / / / 1 > * / / / / j I to/ / / / 1 1

/// // CO CO / / / / // in/ / ' //

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2.2.2 SandwichWithHoneycombCoreSymmetricWrinkling)2.2.2.1BasicPrinciplesAsnotednSection2.2.1.1,heesultsfReference2-9applynlytoandwichcon-figurationswhichhaveolidorfoamcores. However,hebasictheoryofhateportisapablefxtensiontoconstructionshavinghoneycomboresndhissccomplishedinReference2-7. Thextensionschievedbyncorporatingonditionswhichecog-nizehatthehoneycomborelasticmodulintheplaneparalleltothefacingsreverymallnomparisonwiththeorelasticmodulinhedirectionnormaltohefacings. Fullconsiderationwasgivenoboththeymmetricndntisymmetricwrin-klingmodeslongwiththenfluencesfromnitialwavinessfhefacings. However,inthisasetwasfoundhat,xceptfortheegioncontrolledbyhearcrimpinglo wq),symmetricwrinklingdevelopststressevelswhichareowerthanthosetwhichtheantisymmetricmodewilloccur. Basedonthisobservation,hedevelopmentofRefer-ence2-7esultednthefollowingequationforthepredictionofwrinklingforsotropicfacingsnandwichconstructionshavinghoneycomboresndubjectedtouniaxialcompression:

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and< Tw =Facingwrinklingstress,si.

Ec =Young'smodulusofthecoreinth edirectionnormaltothefacings,psi.tfhicknessoffacing,nches.r \lasticityreductionfactor,imensionless.

E foung'smodulusoffacing,si.tchicknessofcore,nches.

Smplitudeofinitialwavinessnfacing,nches.Fc =Flatwiseandwichstrengththelowerofflatwisecorecompres-sive,latwisecoretensile,ndflatwisecore-to-facingbond

strengths),psi.Equation2.2-8)canbeusedtoplotafamilyofdesigncurvesofth eformshownin Figure2.2-4. Itshouldbenotedthatth ecurveforKg isanupper-boundclassi-calvaluewhichisbasedontheassumptionthatthefacingsreinitiallyperfect. ThisparticularcurveagreesverycloselywiththefollowingsymmetricalwrinklingequationrecentlyobtainedbyBarteldsandMayers2-14]

I*LOETTJ < ' * >2 -2-10)ComparisonofEquations2.2-8)and2.2-10)showsthat,wheng ,heformergivesriticalstresseswhichareapproximatelypercentlessthanthoseobtainedbyBarteldsandMayers2-14].

< rw .86

Numbersinbrackets[]inthetextdenoteeferenceslistedaten dofeachmajorsection1;;etc.)

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'wr'7Ef

K 50-K 5 =Constant

K=Constant

EftcyFigure2.2-4. TypicalDesignCurvesforFaceWrinklinginSandwichConstructionsHavingHoneycombCores

Inactualpractice,urvesofthetypeshowninFigure2.2-4donotprovetobeveryhelpfulsinceth eK , valuesppropriatetoparticularstructuresarearelyknown.Therefore,nordertoprovidepracticalmeansforth epredictionoffacewrinklinginsandwichconstructionshavinghoneycombores,lower-boundapproachistakeninthishandbook. Forthispurpose,estdataselectedfromReferences2-7and2-10areplottedinFigure2.2-5. A llofth epecimensfromReference2-7failedwithintheelasticange. Severalofthesefailuresoccurredbymeansofshearcrimpingandthesedatawerediscarded. FortheemainingtestseportedinReference2-7,hreedatapointsreplottedinFigure2.2-5foreachgroupofnominallyidenticalspecimens.Onepointisplottedforthemaximumtestvalueforthegroup,nepointforthemini-mum,ndon epoint forth everage. ThedatafromReference2-10wereelectedin asimilarmannerwithseveraladdedestrictions. Anumberofthesepecimenswrinkledunderhighlyinelasticconditions. Sinceathercrudeplasticityreduction

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03> o0)

8C O aa o

U i-i

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factors (77=E^/Ef)wereusednthedataeduction,twasdecidedoplotdatanlyforthosepecimenswhichwrinkledtstressevelswhere (E^/Ef) ?0.85. Inddi-tion,manyfheestspecimensfReference2-10hadverypoorcore-to-facingbondsasmeasuredbyflatwiseensiletrengths. Itwashereforedecidedoplotdataonlyforthosepecimenswhoseflatwiseensiletrengthsweretleastequaltoheflatwisecompressivetrengths. Adhesiveechnologyhasnowdvancedohepointwhere,withpropercare,neanusuallyelectandhesiveystemwhichatisfiesuchaequire-ment.

BasedntheplotofFigure2.2-5,heelationship/Ecf\w=- 33Ui~H (7?Ef)2 -2_11)

hasbeenelectedhereoprovideafedesignvalues. Thismplieshataknock-downfactorofpproximately.4spplicableohisv/rinklingphenomenon. Obviously,thissnotaigorouspproachtoheproblemndtwouldbedvisableobasehedesignequationonamuchwiderelectionfestdatafpecimenswhichwererulyrepresentativefcontemporarypracticaldesigns. Therefore,Equation2.2-11)anonlybeegardeds best-available"approachandtisecommendedthat,orveri-ficationoffinaldesigns,wrinklingtestsbeperformednpecimenshatactuallydup-licateheelectedandwichonfiguration. Themethodpresentedherehouldonlyberegardedsnapproximateguideline.

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2.2.2.2DesignEquationsandCurvesTh efollowingequationmaybeusedtocomputetheapproximateuniaxialcompressivestresstwhichfacewrinklingwilloccurinsandwichconstructionshavinghoneycombcores:

where

/Ec% \* -82[wrJ7 ? E f > O - =2.2-12)wr.64K(5 EC-Cc

Incaseswherethemplitudeofinitialwavinesssnown,necaneitherusetheseequationsorthecurvesgiveninFigures2.2-6and2.2-7toestablishthewrinklingstress. BothofthesefiguresaretakendirectlyfromMIL-HDBK-232-16]. When-everthenitialwavinesssunknown,tisecommendedthatth efollowingequationbeusedtoobtainalower-boundprediction:

a.wr - S 3 vwd(? E f )2 - 2 _ 1 4 )Forelasticcases,se . Wheneverthebehaviorisinelastic,hemethodsofSectionmustbeemployed.Whenth efacingsareubjectedtobiaxialcompression,tisrecommendedthaton eusetheinteractionformula

^x ^y^ y 2.2-15)where

AppliedCompressiveLoadinginSubscriptDirectionRi CriticalCompressiveoadingwhenactingalone)in 'SubscriptDirection2-33

(2.2-16)

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andtheydirectioncorrespondsothedirectionofmaximumcompression. Thisinteractionrelationshipisbasedonth enformationprovidedinReference2-1forrectangularflatplateshavingverylargeaspectratios. Forcasesnvolvingshearingstresseswhicharecoplanarwiththefacings,tisecommendedthatth eprincipalstressesfirstbecomputedandthatthesevaluesthenbeusedintheaboveinteractionequation. Wheneveron eofth eprincipalstressesstensileandth ebehavioriselastic,theanalysisshouldbebasedonth essumptionthatthecompressiveprincipalstressisactingalone.

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2.3 S H E A RRIMPING2.3.1BasicPrinciplesTounderstandhephenomenonofhearrimping,nemustkeepnmindhatthismodeffailuresimply imitingasefgeneralinstability. Thequationsforpredictinghearcrimpingemergefromgeneralnstabilitytheorywhenthenalyticaltreatmentextendsntoheegionofowhearmodulifortheore. Forexample,hetheoreticalderivationofReference2-17,seformulatednSection4.2.1.1fthishandbook,ieldsheesultthat,whenthetwofacingsrefheamematerial,hearcrimpingwilloccurnaxiallycompressedandwichcylinderswhenever

Vc 2 2.3-1)where

Vc 2.3-2)"crimph/Ms

=7Efi ^V(t1+ts)2'3-3> - h8"crimp (t 1+3)tc xz2.3-4)

V Plasticityeductionfactor,imensionless.Ef =Young'smodulusffacings,si.h =Distancebetweenmiddleurfacesffacings,nches.R =Radiusomiddleurfacefcylindricalsandwich,nches.

ta .ndgThicknessesfhefacingsTheresnopreferencesowhichfacingsdenotedbytheubscript 1r.),nches.Ve= ElasticPoisson'satiooffacings,imensionless.

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tc =Thicknessofcore,nches.G v . 7 =Coreshearmodulusassociatedwithth eplaneperpendiculartothe'xz facingsndorientedintheaxialdirection,psi.

Thecriticalstressanbedeterminedfromtheequationo -cr ccr0 2.3-5)

and,he ntheInequality2.3-1)holdstrue,ccanbecomputedasfollows:K = 2.3-6)C Vc

Hence,rimpo \^cr op ao crimp2-3_7>

Therefore,whenth etwofacingsaremadeofth eamematerial,hefollowingequationcanbewrittenforth eriticalstressforshearcrimpinginacircularsandwichcylinderunderaxialcompression:

a - - a -G2.3-8)cr-crimp {i,+l^)tc xzAnequivalentresultcanbeobtainedfromReference2-18forsandwichcylinderssub-jectedtouniformexternallateralpressure.hatis,wherethetw ofacingsaremadeofth esamematerial,necanwrite

hs

wherecr crimp tl+U dt0 yZ2'3"9>

G, =Corehearmodulusssociatedwithth eplaneperpendiculartoth eaxisfrevolution,si.

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Inaddition,hedevelopmentofReference2-19leadson etothefollowingformulaforcircularsandwichcylindersunderpuretorsionandhavingbothfacingsmadeofthesamematerial:

h8rcrTcrimp_t +tt v GxzGyz2s0)Itshouldbenotedthat,lthoughEquations2.3-8)through2.3-10)werederivedforsandwichcylinders,llofthesefinalexpressionsrendependentofcurvature. Thus,theseequationshaveageneralapplicabilitywhichisotlimitedtothecylindricalcon-figuration.

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2.3.2DesignEquationsThefollowingequationsmaybeusedtoomputehefacingstressestwhichhearcrimpingwilloccurnsandwichconstructionshavingbothfacingsmadeftheamematerial:

a. ForuniaxialcompressionactingcoplanarwiththefacingsseeFigure2.3-1),use

whereo r - CMcrimp (h ta)tc ^ (2.3-11)

Gij = Corehearmodulusssociatedwiththeplaneperpendicularothefacingsndparalleltohedirectionofoading,si.cr , si

o-,si a- , sio- ,si

Figure.3-1. UniaxialCompression2-40

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b. Forpureshearactingcoplanarwiththefacings(seeFigure2.3-2),useh2Tcrimp t + tGxzGyz (2.3-12)

Figure2.3-2. PureShearTheforegoingequationsarevalidregardlessof theoveralldimensionsofthestructure,inaddition,oknock-downfactorsarerequiredsinceshearcrimpingisinsensitivetoinitialimperfections. Thepredictionsfromtheseequationswillbesomewhatconserva-tiveincetheirderivationsneglectbendingof thefacingsbouttheirownmiddlesur-faces. Althoughsuchbendingisofnegligibleimportancetomostsandwichbucklingphenomena,nthecaseofshearcrimpingthisinfluencecanbeconsiderable.Furthermentionoftheshearcrimpingmodeoffailureismadeinthevarioussectionsongeneralinstabilityincludedinthishandbook.

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REFERENCES

2-1erard,GeorgeandBecker,Herbert,HandbookofStructuralStability,PartI-BucklingofFlatPlates,"NACATechnicalNote781,uly1957.2-2orris,C.B.,Short-ColumnCompressiveStrengthofSandwichConstruc-

tionsasAffectedbySizeofCellsofHoneycombCoreMaterials,"U.S.ForestServiceResearchNote,PL-026,anuary1964.

2-3nderson,M .S.,LocalInstabilityoftheElementsofaTruss-CoreSandwichPlate,"NASATechnicalReportR-30,959.

2-4ough,C.S.,Elam,C.F.,ndeBruyne,N .A.,TheStabilizationofaThinSheetByaContinuousSupportingMedium,"JournaloftheRoyalAero-nauticalSociety,anuary1940.

2-5illiams,D.,eggett,D.M.A.,ndHopkins,H.G.,FlatSandwichPanelsUnderCompressiveEndLoads,"RoyalAircraftEstablishmentReportN o.A.D.174,une1941.

2-6off,N..ndMautner,S.E.,"TheBucklingofSandwichTypePanels,"JournaloftheAeronauticalSciences,Vol.2,No.,uly1945.2-7orris,C.B.,Boiler,K.H.,ndVoss,A .W.,"WrinklingoftheFacings

ofSandwichConstructionSubjectedtoEdgewiseCompression,"FPLReportNo.810-A,une1953.

2-8usuff,.,TheoryofWrinklinginSandwichConstruction,"ournalof theRoyalAeronauticalSociety,Vol.9,anuary1955 .2-9orris,C.B.,Ericksen,W .S.,March,H.W.,mith,C.B.,ndBoiler,K.H.,WrinklingoftheFacingsof SandwichConstructionsSubjectedto

EdgewiseCompression,"FPLReportNo.810,March1956.2-45

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2-10 Jenkinson,.M .ndKuenzi,E.W.,WrinklingoftheFacingsofAluminumandStainlessSteelSandwichSubjectedtoEdgewiseCompression,"FPLReportN o.171,December1959.

2-11 Yusuff,S.,FaceWrinklingandCoreStrengthinSandwichConstruction,"JournaloftheRoyalAeronauticalSociety,Vol.4,March1960.

2-12 Harris,..ndCrisman,W .C,Face-WrinklingModeofBucklingofSandwichPanels,"roceedingsoftheAmericanSocietyofCivilEngineers,Journalof theEngineeringMechanicsDivision,une1965.

2-13 Benson,.S.ndMayers,J.,GeneralInstabilityandFaceWrinklingofSandwichPlates-UnifiedTheoryandApplications,"AIAAPaperNo.6-138PresentedinNew York,New York,anuary1966.

2-14 Barteids,.ndMayers,.,UnifiedTheoryfortheBendingandBucklingofSandwichShellsApplicationtoAxiallyCompressedCircularCylindricalShells,"DepartmentofAeronauticsndAstronautics,tanfordUniversityReportN o.UDAARN o.87,November1966.

2-15 Plantema,.J.,andwichConstruction,ohnWiley& Sons,nc.,New York,Copyright966.

2-16 U..Departmentof Defense, StructuralSandwichComposites,MIL-HDBK-23,30December968.

2-17 Zahn,..ndKuenzi,E.W.,ClassicalBucklingofCylindersofSandwichConstructioninAxialCompression-OrthotropicCores,"U ..ForestServiceResearchNoteFPL-018,November1963.

2-18 Kuenzi,E.W.,ohannan,B.,ndStevens,.H.,BucklingCoefficientsforSandwichCylindersofFiniteLengthUnderUniformExternalLateralPressure,"U.S.ForestServiceResearchNoteFPL-0104,eptember1965.

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2-19 March,.W .ndKuenzi,E.W.,Bucklingof SandwichCylindersinTorsion,"FPLReportN o.840,anuary1958.

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3GENERALINSTABILITYOFLATPANELS3.1 RECTANGULARPLATES3.1.1 GeneralAspreviouslynoted,nefthepotentialmodesffailureforsandwichpanelsisthatofgeneralinstability. Thisoccurswhenthepanelbecomeselasticallyunstableundertheapplicationofcertain typesof in-planeloads. Further,tshouldbenoted,heloadswhicharecriticalforinstabilitymayormaynotbeofsuchmagnitudeastocauseafailurefthebasicmaterials.Theflat,ectangularsandwichpanelrepresentsthatconfigurationforwhichthevastmajorityoffabricationandtestdatahasbeenaccumulatedoverthepastdecade. Thisisprobablydu etothefactthatthisconfigurationwasbestadaptedtothetructuralneedsforanumberofapplicationsndthatitrepresentedtheminimuminfabricationproblemsndcostsasfarasthistypefconstructionisconcerned. Bythesametoken,nalyticalsolutionshavebeendevelopedforawiderangefloadingapplicationsforflatpanels,ndanappreciablemountoftestingforcorrelationwiththesesolu-tionshasbeenaccomplished.Asaconsequenceofthispast work,tisnowpossibletoemploythenalyticalsolu-tionsforflatpanels,sgiveninMIL-HDBK-23,3-1],withahighdegreefcon-fidence. ThisviewisupportedbyrecommendationsgiveninReferences3-2through3-7,nclusive,orbasicpaneldesign. Therefore,withthisbackgroundinmind,he

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bucklingcoefficients,,whichwillbegiveninthisectionforthevariousplateloadingconditionswillbethosetakenfromtheapplicableectionsfReference3-1,withno"knockdown"factortobeappliedtothem.Th edevelopmentofplatebucklingcoefficientsforsandwichconstructionrequirestheconsiderationofanumberoffactors,omere: 1)thedegreeforthotropicityofthefaceplates,)th euseftheameorofdissimilarmaterialsforthefaceplatesn d,3)thedegreeforthotropicityofthecorematerial. Thegeneralequationsgivenintheollowingsectionsaccountforthesepossibilities;however,hecurveshowingKas functionof(a/b),,hetypefloadingandedgeupportconditionswillassumetheusefisotropicfaceplatematerialsincethisislargelytypicalofaerospacevehicledesignpractices.Inallcases,hefinaldesignofth eandwichpanelmustcomplywiththefollowingfourbasicdesignprinciples,Reference3-1;

a.hesandwichfacingsshallbetleastthickenoughtowithstandthehosendesignstressesunderthepplicationoftheultimatedesignloads.b.hecorehallbethickenoughandhavesufficientshearrigidityandstrengthsothatover-allsandwichbuckling,xcessivedeflection,ndshearfailurewillnotoccurunderthedesignloads.c.hecorehallhavehighenoughmoduliof elasticity,ndthesandwichshallhavegreatenoughflatwisetensilendcompressivetrengthsuch

thatwrinklingofeitherfacing willnotoccurunderthedesignloads.d.fthecoreisacellularhoneycomborconstructedofcorrugatedmaterialanddimplingof thefacingsisnotpermissible,hecellsizeorcorrugationspacingshallbemallenoughsothatdimplingofeitherfacingintothecorepaceswillnotoccurunderthedesignloads.

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Otherrequirementsincludetheuseofmoduliofelasticityandstressvaluesepre-sentativeofthosevalueswhichprevailundertheconditionsfuse. Also,wherethestressesarebeyondtheproportionallimit,heappropriateeducedmodulusfelas-ticityshouldbeused.Th efollowingsectionsonspecifictypesfpanelloadsdefinetheappropriateequationsforeachparticularsituationanddiscussusefullimitsndotherconsiderations,sapplicable. Asummarytable,Table3-1),istingthepanelinstabilityequationsgivenin thevariouspartsfthissection,longwithadefinitionof terms,quationlimitationsifany,ndreferencesfortheppropriatebucklingcurvesimmediatelyprecedesthelistofreferencestofacilitateuseofthemanualforspecificproblemsolution.Figure3.1-1showselasticpropertiesanddimensionsforthetypicalsandwichpanelunderconsiderationinthissection.

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h-bf-H I K -bfH^t (TYPICAL)

0TYPICAL)SINGLE-TRUSS

0(TYPICAL) fDOUBLE-TRUSS

CORRUGATEDCORECONFIGURATIONSIM 11

CORE- r n yv c

HONEYCOMBCORE

Figure3.1-1. ElasticPropertiesndDimensionalNotationsforaTypicalSandwichPanel3-4

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3.1.2 UniaxialEdgewiseCompression3.1.2.1asicrinciplesTh ebucklingcoefficientequationsndcurvesgivenhereforuniaxialedgewisecom-pressionarethoseoriginallydevelopedbyEricksenandMarch,[3-8],ndarein-cludedinth eMIL-HDBK-23documents,ssuedsincethen. Thebasicprinciplesndassumptionsemployedinthedevelopmentofthesegeneralinstabilityequationsarenotedinthereferencesandarenotrepeatedhereexceptwhererequiredtolimittheirusebecauseofth eoriginalrestrictionsimposed.Thebasicequationsforcalculationof theallowableandwichpaneledgewisecom-pressionloadsaregiveninthefollowingsection. Curvesforpanelbucklingcoeffi-cientsforpanelshavingisotropicfaceplatesandbothorthotropicandisotropiccoresforvariouspaneledgesupportconditionsfollowtheequations.3.1.2.2esignEquationsandCurvesAspreviouslynoted,heequationspresentedinthissectionarethosedevelopedby EricksenandMarch,ndpresentedinMIL-HDBK-23,swellasinotherdocuments.Supportingdatasuchaspertinentassumptionsanddefinitionof termsarelsoin-cludedalongwiththeequations.SandwichPanelsWithHoneycombCoresOn eofthebasicssumptionsusedinthedesignandanalysisfsandwichpanelsisthatthefaceplatescarrytheinplaneloadsappliedandthatthecoreprovidesthatshearsupporttotheaceplatesrequiredforthemtoactasunitinpreventingearly

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individualbuckling. Fromthis,heedgewisecompressioncapabilityofth epanelisgivenbythefollowingequations,whicharetakenfromSection5.3,Reference3-1:

N = 7 T 3/b8)(K)(D) 3.1-1)crwhereDistheandwichbendingstiffness. Solvingthisequationfortheacingstres-sesgivesthefollowing:

(E(t1)(E^t2),FK iLl l s*3.1-2)*ci,2 ff* (Et.+E )2 (b2) X

Forequalfacings:Tc(bp

where

KEF =-JM--13.1-3)

K =ucklingcoefficient=K +K seedefinitionsinfollowingMwork).E' (E 'E')~=effectivemodulusofelasticityfororthotropicbfacings.X - Mb)

=Poisson'satioasmeasuredparalleltotheubscriptdirection.f,l,2 =subscriptsdenotingfacings.

h,b =seeFigure3.1-1.Sincethebucklingcoefficientcurvestobepresentedhererebeinglimitedtothecaseofisotropicfaceplates,whichisepresentativefthelargemajorityofstructuralsandwichapplications,heffectedequationsgivenpreviouslyareevisedbelowforthisituation.

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Forisotropicfacings:E' =E' =.' J.E.;ndu = M= M aii1'ai %i Miwherer \ . plasticitycorrectionfactor(seeSection9.0).

Asnotedabovethebucklingcoefficientforthepanelunderthisloadingconditionisgivenby theequation

KKF+KMwhere

_ (EjV+ E&^pk+Ek)KF^ETMETyPSLO3-1-4> KM =K j^forthecasewhereV=0seeFigure3.1-16)]3.1-5)o

ValuesfKparegenerallyquitemallrelativetoK^,hus safefirstapproximationistoassumeitisequaltozerountilafinalpanelcheckismade. Onthisbasis,= MmaybeUSedt0develoPinitialfaceplatendcorethicknessesforthepanel.K ^.is theoreticalcoefficientwhichisdependenton theandwichpanelbendingandshearrigiditiesandpanelaspectratio. Otherfactorswhichinfluencethemagnitudeofthiscoefficientincludethepaneledgeupportconditionsndtheorthotropicityoftheore. Adiscussionoftheseconsiderationslongwithdevelopmentof thequationsforcalculationofthiscoefficientaregiveninReferences3-1and3-8. Thismanualdoesnotproposetorepeattheseequationshere;however,hecurvesshowninFigures3.1-2through3.1-15givevaluesofK^asafunctionofedgesupportcondition,panelaspectratio,ndthebending-shearrigidityparameter,whichisdefinedasfollows

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whichfurthercanbewrittenas:7 T 2t E^E^v=_3.1-7)Xb2Gc(E(t1+E^t 3)

7 T 3t E'tV= (forequal-facings)3.1-7a)2X b2G v cwhereUisandwichshearstiffness;Gcisthecorehearmodulusssociatedwiththeaxesparalleltodirectionofloading(alsoparalleltopanelsideflengtha)ndper-pendiculartotheplanefthepanel.AnindicationoftheinfluencendimportancefthecorehearmodulusmaybebtainedfrominspectionoftheaboveequationsforVandtheurvesgivingvaluesfK ^.givenlater. HoldingalltermsonstantexceptG nincreaseinitsvalueeducesthevalueofVtobeusedwiththebucklingcoefficientcurves,hiseducedvaluethencallsforanincreasedvaluefK..SandwichPanelsWithCorrugatedCoreTheequationsandformulaspreviouslygivenareforsandwichpanelswithhoneycombcores;however,heymaybeadaptedtocovertheasefpanelswithcorrugatedcoresbymeansfthefollowingmodifications:

a.ortheasewheretheorrugationflutesreorientednormaltothedirec-tionoftheloadapplication,hehearmodulusinthedirectionparalleltotheflutes,cb,sveryhighwithrespecttothehearmodulusparalleltothedirectionofloading, hus,hepreviouscurvesmaybeusedbylettingGcb= ondR=Gc?Gcb=0.b.orthecasewherethecorrugationflutesareparalleltothedirectionofloading,hecorrugationsmaybeassumedtocarryloadinadirectpro-portiontotheirareaandelasticmodulus. TheparameterVforthisaseiseplacedbytheparameterW ,whichisdefinedas

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T T 3tc(E )(E )Xb3Gcb(E +EcW"\wn T?'+ xT?'+ 3.1-8)

Or,orequalfacings,W=2tcEfY2Xb2Gcb3.1-8a)

Valuesf1 asafunctionof(b/a),=(GcG ),ndV,rW ,regivenforvariousedgeupportconditionsinFigures3.1-2through3.1-15,withFigures3.1-14and3.1-15representingthecasefpanelshavingcorrugatedcores.Figure3.1-16givesvaluesofKsafunctionof panelaspectratioandedgeupportconditionsforuseindeterminingvaluesfCnorderthatfinalvaluesforKmaybeobtainedforspecificdesigns.Thecurvesndequationsjustgivenmaybeusedindevelopingapaneldesigninaddi-tiontocheckingtheadequacyofanexistingdesign;however,hisis slowiterativeprocess. As consequence,hismanualrecommendstheusefthedesign-proceduresapproachdescribedinReference3-1sinceitwasspecificallydevelopedtoexpeditethenewdesignprocess.

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0.2 0.4 0.6 0 a

1.0 0.8 0.6 0.4 0.2bIT

11 HQii 0

1

t2.5GcEND 11\1

0_0.05*JL_100.200.150.30

Figure3.1-2. KMforaSandwichPanelwithEndsndSidesSimplySupported,"IsotropieFacings,ndOrthotropicCore,R=0.40)3-10

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1412

10

*M

1b 2u

M

c

I1

C G GcENDill1 b*-|Vn

0.10 ,.~ -o0 f l0.60FO RV>1.00I-i 1.00

0.2 0.4 0.6a

0.8 1.0 0.8 0.6 0.4 0.2b"a

0.200.40

Figure3.1-3. K^orSandwichPanelwithEndsndSidesSimplySupported,IsotropieFacings,ndIsotropieCore,R=1.00)

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a ba

Figure3.1-4. KMforSandwichPanelwithEndsndSidesSimplySupported,IsotropieFacings,ndOrthotropicCore,R=2.50)

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0.2 0.4 0.6 0.8 1.0 0a

0.6 0.4 0.2a

Figure3.1-5. 1%forSandwichPanelwithEndsimplySupportedandSidesClamped,sotropieFacings,ndOrthotropicCore,R=0.40)

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j1 'i

iow Qii eGcENDt1

0.2 0.4 0.6 0.8 1.0 0.8 0.6 0.4 0.2a b

" a

Figure3.1-6. KMforSandwichPanelwithEndsSimplySupportedandSidesClamped,sotropieFacings,ndIsotropieCore,R=1.00)

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1.001.875

Figure3.1-7. KMforSandwichPanelwithEndsSimplySupportedandSidesClamped,sotropieFacings,ndOrthotropicCore,R =2.50)

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0.050.150.30

a b_a

Figure3.1-8. KMforandwichPanelwithEndsClampedandSidesSimplySupported,sotropieFacings,ndOrthotropicCore,R=0.40)

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a a

Figure3.1-9. KMforSandwichPanelwithEndsClampedandSidesSimplySupported,sotropieFacings,andIsotropieCore,R=1.00)

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Figure -10. KforSandwichPanelwithEndsClampedandSidesSimplyupported,IsotropieFacings,ndOrthotropicCore,R-2.50)

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* M

Figure3.1-11. KMforSandwichPanelwithEndsndSidesClamped,sotropieFacings,ndOrthotropicCore,R=0.4)~3-19

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Figure3.1-12.%forSandwichPanelwithEndsndSidesClamped,sotropieFacings,ndIsotropieCore,R=1.00)

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,

c0.4GENDIII'H- V

0.100.200.300.601.001.875

Figure3.1-13. KMorandwichanelwithEndsndidesClamped,sotropieFacings,ndOrthotropicCore,R-2.50)

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Figure3.1-14. forSimplySupportedSandwichPanelHavingaCorrugatedCore. CoreCorrugationFlutesrePerpendiculartotheLoadDirection3-22

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14

12

J O

* M

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0.6 0.8 1.0

Figure3.1-16. KM forSa ndwichPanelwithIsotropieFacingsin~EdgewiseCompression

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3.1.3 EdgewiseShear3.1.3.1asicPrinciplesAsnotedearlierinSection3.1.1,ufficientanalysis,esign,ndtestingofflatsand-wichpanelshasbeenaccomplishedtodemonstratethedequacyof theanalyticalapproachespresentlyinuse. Thus,hepanelbucklingcoefficientequationsndcurvesgiveninthefollowingparagraphsforedgewiseheararethosetakenfromtheMIL-HDBK-23documentspresentlyinuse. TheseequationswereriginallydevelopedbyKuenziandEricksen[3-13]ndemploytheamegeneralassumptionsasthosedescribedinSection3.1.1. Specificlimitationsrrestrictionsntheuseftheseequationswillbenotedwheretheseequireonsideration.Th ebasicequationsforuseincalculationofthellowableandwichpaneledgewiseshearloadsaregiveninthefollowingsectionalongwithapplicablebackgrounddataandassumptions. Designcurvesandbucklingcoefficientsforpanelshavingisotropicfaceplatesandbothortho-tropicandisotropiccoresforbothsimplysupportedandclampededgeconditionsfollowtheequations.3.1.3.2esignEquationsandCurvesThedesignequationspresentedhereretakenfromReference3-1and3-13. Support-ingdataanddesignconstraintsarealsonotedanddiscussedasrequired.Th eedgewiseshearloadcarryingcapabilityofasandwichpanelisgivenby thefollow-in gequation:

Np,=

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where N =criticaledgewisehearload,bperinchscr(3.1-10)

D =sandwichbendingstiffnessSolvingthisforth eacingstressesgivesthefollowingequation:

(Ejt1)(E^)(h8)E^2Fsi)2= \ (Ek+ Eya(b^X

Or,orequalfacings7 T 2Kh^Ej

F = _ 3.1-10a)s(b2)Xwhere E' istheeffectivemodulusfelasticityoffacingatstresss=7 7 E r\ lasticitycorrectionfactorSection9.0)X -H 2M i=M s=Poisson'satiooffacingsh=istancebetweenfacingcentroidsFigure3.1-1)b =anelwidth(sa)Figure3.1-1)

K =K +K (Note: ThesetermsdifferfromthosefSection3.1.2)sMwhere

Or,orequalfacings(E+EME+EKMK = 3.1-11)

*F2(Et t,)h2K = ' 3.1-lla)H Fhs

KM =aluefK forV=W=03-26

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TheequationdefiningthevaluefK.isquitecomplexandinvolved,beingdependentonpanelaspectratio,a/b),henumberofhalf-waves,n),ortheminimumenergybucklepattern,ndthepanelbendingandshearrigidityparameter,V,rW). Thismanualproposestofollowgeneralpracticein theliteraturendprovidecurvesnlyforthedefinitionofthisbucklingcoefficient. Thoseinterestedinthebasicequationanditsdevelopmentwillf indthisnReference3-13.ValuesofK aregiveninFigures3.1-17through3.1-24asafunctionofthepanelaspectratioandtheparameterV,rW ,orvariouspaneledgesupportconditions.Thesefigurescoverpanelswithisotropicfaceplatesandbothisotropicandorthotropiccore,ncludingpanelsusingcorrugatedflutesforcores. Valuesfthebucklingcoeffi-cient,Mmayalsobeobtainedfromthesameetoffigures.TheequationsdefiningtheparametersVandWarethesameasthosegiveninthepreviousectionforedgewisecompression;however,heyareepeatedbelowtofacilitatetheiruse. Theequationnumberspreviouslyassignedtothemareetainedbelow

(E&)(E )(r)tcV=CEft+E^t2)b " )Gca-17*

V=T 2tE't/2Xb2Gequalfacings)3.1-7a)C II3,Forasandwichpanelwithacorrugatedcoreinwhichthecorrugationflutesareparalleltotheedgeflengtha,heparameterVisreplacedbytheparameterWwhichisde-finedasfollows: 3-27

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T?tc(Eft)(E^)xb2Gcb(Ejt1+E;y= -77wTT3-1-8)Or,orequalfacings

W=T^ tE't72\b8G 3.1-8a)C ffDIncheckingaparticulardesignforthecriticalbucklingstress,gcrigures3.1-17through3.1-21shouldbeusedforthosepanelshavingalledgessimplysupported.CurvesofK^forsandwichpanelshavingalledgesclampedaregiveninFigures3.1-22through3.1-24. Thesecurvesmaybeinterpolatedinordertoobtainthebucklingcoefficientsforothervaluesfcoreorthotropicity,R=Gca/Gcb>ndinter-mediatevaluesfVorW .ItshouldbenotedthatiftheesultingvaluefF isabovetheproportionallimitscrvalue,hevalueofE'hallbeaneffectivevaluebasedonthatstresslevel,ndthiseffectivevaluehallbeusedincomputingthevaluefV,Equation(3.1-7)r3.l-7a)orW ,Equation(3.1-8)r3.1-8a),sthecasemaybe. ThisameeffectivevalueforE'hallalsobeusedinEquation(3.1-10),r(3 .l-10a)whencalculatingthecriti-calpanelbucklingstress. Thus,everalinterationswillberequiredtoestablishtheactualvaluefF inthosecaseswhereitexceedstheproportionallimit,scrTheequationsndcurvesjustgivenmaybeusedinthedevelopmentofpaneldesignsaswellasincheckinganexistingdesign;however,swastheaseforuniaxialcom-pression,hisis lengthyiterativeprocess. Thus,hismanualre