1、go:EiE$o 4IIJm 5NATIONAL.4DV1SORYCOMMITTEE-=FORAERONAUTICSTEiCHNICAI.30TENo.1347 .CRITICALCOMBINATIONSOFSHEARANDLONGITUDINALDCTSTRESSFORLONGPLATESWITHTRANSVERSECURVATUREByS. B. Batdorf,MurrySchildcrout,andManuelSteinLangleyMemorialAeronauticalLaboratoryLangleyField,Va.WashingtonJune1947 AFMDC -TECHN
2、!C?.LLBSARYAFL 2fjII13/9. qq74/-_,._._ _ _ - .Provided by IHS Not for ResaleNo reproduction or networking permitted without license from IHS-,-,-NATIONAL!ADVISORYcoMMITTEETEc3NmLNom?0a71F AERONAUTICS1347CR3?EWALCOMBINATIONSOFSHEANDIONGITUDINALDIRE(XE91?RESSF LQNGlZATESW13HTRANSVY3RSECURVATUREByS.B.B
3、atdorf,MurrySchildcrout,andManuelSteinSUMMARYA theoreticalsolutionispresentedforthebuckling6tressesoflongplateswithtransversecurvatureloaded.inshearandlongi-tudinaldirectstress.Thetheoreticalcritical-6tressccxabinations.forplateshavingeitherstiplysupportedorclampededgesaregiveninfigures andtablesand
4、a cgmparisismade%thaprevioustheo-reticalsolutionforsimplysupportedplates.InthecompressionrangetheoreticalcurvesareunsuitableforuseindesignbecauselongplateswithsubstantialcurvatureloadedinaxialcompressionbuclcJeatstressesthataremuchlessthanthetheoreticalvaluesofcriticalstress.Aninvestigationwasthere-
5、foremadetodetenuinetheuodiffcaticmsrequiredtomakethetheo-reticalcurvescompatiblewiththeavailableeerimentaldataforplatesinaxialcompression.Interactioncurvesbaseduponthisinvestipjationareprovisionallyrecczmnendedforuse in design.Boththeoreticalagdsuggesteddesigncurvesme essentiallyparabolas,a circumst
6、ance%Mchpermitssimpleapproximateinteractionformulastobegiven., .“ ,.INTRODUCTION,. . .Theoreticalsolutionstoanumber”ofproblemsconcernedwiththedeterminationofthecriticalstresseswhichcauselongcurvedplatestobucklehavebeenpresentedinvariousinvestiti.ons.Inreferences1to3 shearaloneacticm,bothsimplysuppor
7、tedandclampedplatesisinvestigated)inreferencesk anddirectaxialc.oppression.aloneactingonbothsimplysupportedandclampedplatesiqinvestigated;andinreference6the-criticalmzibinations“ofshearand,tiectaxialstressforsimplysuppo:tedplatesonlyare8 .ven. :Thepresentpaperdealsth thedeterminationofthecombi-ttons
8、ofshearanddirectaxialstress.whichcameplateswitheitherProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-2 NACATNNoo1347simylysupptedorclampededgestobuckle(appendixA). Thepresentsolutionaswellasthesolutionsofreferences1 to 6 isbaseduponthesmall-deflectio
9、ntheoy.Ascurvedplatesloadedinaxialcompressionmaybuckleatastressmuchlessthanthetheo-reticalvalue,thetheoreticalinteractioncurvesofrefeience6andthepresentpapermustbemodifiedinthecompressionrangeforuseindesign.Aninvestigationwasthereforemadeofavailableeerimentaldataonthocriticalstressesoflongplateswith
10、transversecurva-tureloadedinaxialcompression(appendixB),andapproximateinter-actioncurvesincorporatingtheseresults.weredevelopedandareprovisionallyrecommendedfordesignposes. Theresultsoftheresentanatisiswe venIntheformofandformulas.bm,n,Jrtuv“wxYDEQSYMBOLSwidthofplateIntegersradiusofcurvatureofplatet
11、hicknessofplatedisplacementofpoint(x-)directiondisulacmentofpointonmedianonmedianerential(y-directicmdisplacementofpointonmediandirection;positiveoutward.-axialcoordinateofplate .tables,interactioncurves,1.a71surfaceofplateinaxialsurfaceofplateincircwn- surfaceofplateinradialcircumferentialcoordinat
12、eofateflexuralstiffnessofplateperunitlength()Et3,. 12(1-U2)YoungsmodulusofelasticitymathematicaloperatordefinedinappendixAProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NACA0. 1347 3zan, bnk.%(%thWJ*Q(%)thVmzwm.$=:?8PaxTcurvatureparametercoefficient
13、sofdeflectionfunctions .?,. k#%shear-stresscoefficientappearinginequation7=b2tdirect-axial-stresscoefficientappearinginequa-kx#D .tionax= b2tdiagonalelementinstabilitydeterminanttheoreticalshear-stressratio(ratioofshearstresspresenttotheoreticalcriticalshearstressinabsenceofotherstresses)empiricaldi
14、rect-axial-stressratio(ratioofdirectaxialstresspresenttoempiricalcriticaldirectthereforeanapproximatedesigncurveconsistingoftwoparts(asfndfcatedinfig.6) is sested.Onepart,applyingtothecompressionrange,istheparabolapassingthroughthepointscorrespondingtotheexperimentalcriticalcan-pressivestressandthet
15、heoreticalcriticalshearstress(obtainedfromfigs.5 and3, respectively).Theseconflpart,applyingtothetensionrange,isthetheoreticalcurvewhichisessentiallytheparabolaassingthroughthepointscomespondingtothetheoreticalProvided by IHSNot for ResaleNo reproduction or networking permitted without license from
16、IHS-,-,-6 # NACATNNO.1347criticalstressinpurecaqressionandpureshear(obtainedfromfigs.4 and3, respectively);. .IXTE!MOTIO?FORMULASThetheoreticalinteractioncurvefora long platewithtrans-verse curvatureloadedinshearandlongitudinal direct dress isveryneartheresultsaresubstantiallythesameastheresultsofre
17、ference6*Inordertodeterminethecriticalstresscoefficientsforthebucl.ingofalongcurvedplateloadedinaxialcompressionalone,equation(.410)issolvedbysettinke equal.tozero, In theresultantequationallthooff-diagonaltermsareequaltozero.Thesolutiontothisequationis .Provided by IHSNot for ResaleNo reproduction
18、or networking permitted without license from IHS-,-,-NACATNNo.1347 15For theminimumvalueofthestresscoefficientthatsatisfiesequa-tion(Alh),therelationshipMl=O t5nuwtbeatisfied.Thevalueof givenbyequation(thesecurvesgivethecompressive-lxzcld.ing-stresscoefficientsforactualcurvedplates,-.Athighvaluesof
19、Z thecurvesapproacha seriesofstraightlineswhichareparalleltothetheoreticalcurve.These-straightlinesarefunctionsof r/t andmaybeapproximatedbytheequa-tion= CZ whereC isafunctionofr/t expressedbytheequationC= 0.68- 0.000+. Thisezqressionfor C, plottedintfie 9,wasobtainedf?x?meerjmentalresultsgiven“figu
20、res-7and8. As Z decreasesandapproacheszero,theempiricalcurvesapproachthevalueof k . 4 whichisthetheoreticalcompressive-Ystresscoefficientforthebucklingofflatplateswithsimplysup-portededgesloadedinlongitudinalcompression.(Seecurvesforstmpl.ysupportedplatesinfig.4.) Theempiricalcurvesoffigures7and8may
21、thereforebeusedtodeterminethecompressivebucklingstressesofcurvedplatestithsimply supyortededges. .InordertodeterminethestressesthatcausecurvedplateswithclampededgestobucKLe,itisnecessarytomodifythectuwesoffigures7 and8. Thelongitudinalloadswhichcausebucld.ingarepracticallyindependentofedgerestrainta
22、tlargevaluesof Z.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-22 NACA TOo1347(See fi.k.)Flatplateswithclampededgesload.longitti.1willalsobuckleata stresswhichagreescloselywiththetheo-reticallypredictedvalue(reference1).Thecurvesoffigures7and8areth
23、ereforemodifiedforcurvedplateswithclampededgesbyfairingsmoothtransitioncurvesbetweenthetheoreticalvaluesatlowvaluesof z.values of thecurvatureparameterZ andtheempiricalestablishedforthebucklingofcurvedplatesathighvalueaTheresultsareshownasdashed.curvesinfigure7.Provided by IHSNot for ResaleNo reprod
24、uction or networking permitted without license from IHS-,-,-NAC!ATNNO.1347REFEmmEs231.Leggett, D,M.A.: TheElastic Stability of aBentRectangularPlateunderUniformShea-r.(Iondon),ser.A.,volt162,no, 908,Sept.2. Krcnm,A.: TheLimitof StabilityofaCurvedShearandAxialStresses.NACATMNo.898,3.Batdorf,S.B.,Schi
25、ldcrout,Murry,andStein,ShearStressofLongPlateswithTransverseNo.1346,1947sLonganaslightlyProc. Roy. SOC.1, 1937,pp. 62-83*Plate Stripunder1939.Manuel:CriticalCurvature.,NACATN4.Redshaw,S.C.:TheElasticStabilityofaCurvedPlateunderAxialThrusts.Jow:R.AS.,VO1O xKC, no- 330,Je 1938)p. 536.5. Leggett,D.M.A.
26、: TheBucklingofaLongCurvedPanelunderAxialCompression.Rga7157 - -5 3.71 .60 .64 - -6 2.65 “73 2161 a7175 - -7.03 .56 4 .98 .56 .98 -.- -30 -5 12.58 *10 11.94 u .92 a71121.1.73a7111 IJ.18 :E - -; 9;58a7112 9.24 a7112 - -10 ;.a7112 760 ,12 - -15m12 5,58 *12 - -8 3:98 .12 3*95”a7112 - -21 a7155 a7112 a7
27、153 ,la - -lco -10 21.,70042.$)38.834.3429.5024.1017.31,872.062.652.0399522.63.60*O1;01.01.01.01.01.01.003.0030003.003:003.00340*5636.w23.6017.31.868.6760.16505-38.722.43.60001.01.01a71O1a71OI.01.01003.003a71003.003.003.003Curved.plateswithclampedod.ges-501;77.09-602467*57*9712.789*598,88014,2310a71
28、468,947.20”5.062.73010031.411*W1.752*OO2.28243012.119.348.6.().1.26“1.671,802,082*352.60I I- -40.53.“-68.5560.0350.34-.”.-.-11*91ti”-. -.”.,-,-.-.-0.01.“ .-.-”.-.-. . . . -.003.003.003.-. ”-1.23. -.-.*-.“-.-.-.”-I?AT!1ONALADVEXXYCOMMITTEE3?ORAERONAUTICS.-.Provided by IHSNot for ResaleNo reproduction
29、 or networking permitted without license from IHS-,-,-NACATNNO,1347 27TABLE1 - ConcludedTHEORETICALCOMBINATIONSOFSHEKR-SSANDDTRECT-AXIKG-SZREEScomIcIENTsANDcomSPOND 8.65 2.50 8.= *, - -7 6.62 3.00 6.5z 2.95 - .-. .4.00 3.34 3.98 3.36 .-.- -:.14 0 3*75 - - - -30 -15 38.92 2.00 23.85 1.65 23.22 1.800
30、28.233.32 18.44 2.65 18.D 2.755 23.86 4.2Q 16.213.25 - -10 18.93 5.20 13.64 .4.20 .-.- -15 13.32 6.50 10.51 5.50 “.-. -18 9.50;.4$ 8.18 6.70 - -21493 4.72 8.35 - -22.39 0 9:44 -.- - - -1000 105 .25 92.5 ,54 - -&? 97*5 .22 78 .56 - -Wo $x3 .24 68.5 .62 62.5 1.4.1NATIONALADVISCE?YCOMMITTEEFORAERONAUI!
31、ICSProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-.1210820.LTension Compression_,+yyl: _- . 0 P3rabola I i-8 -4 0 4 .8 12 16 20 24kx.MATtOMAlADVISORYOmnrrm m EMmurlcsFigure 1.- Theoretical combinationsof stress coefficients for longplates withtransv
32、erse curvaturehavingsimply supportededges1 loadedinshearanddirectaxialstress.(Curvefor Z = O1obtainedfromreference7.)Eiz.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-242016122k,=%840,- -Tensionh q -12 -8 -4 0 4 8 12 16 20 24Figure2.-TheoreticalcombinationsofstresscoeHicientsforlongplateswithtransversecurvaturehavingclampededgesloadedinshearanddirectaxialstress.(CurveforZ =O obtainedfromreference7.). . ., IIProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-
