1、 .k)!+Jw%bmC2cm(33NATIONALADVISORYCOMMITTEE“i= _FORAERONAUTICS E.-TECHNICALNOTENo.1348CZUTICALSHJ3ARSTRESS03?CURVEDR33CTAFGULARANELSByS. B. Batdorf,ManuelStein,andMurrySchildcroutLangleyMemorialAeronauticalLaboratoryLangleyField,Va.WashingtonMay1947 AFM9C_,.-Provided by IHSNot for ResaleNo reproduct
2、ion or networking permitted without license from IHS-,-,-.r.*?mTIoNALADVISORYcOmml?EETEOENICMLNOTENO.FORKERONAUTTCS;348-CRITICAL6EEARSTKESSOFCURW3DRECTANGUIW?PAN?HS -ByS.B.3atdorf,ManuelStein,andMurqSchildoroutSUMMARYA solutionbaseduponsmall-cleflegtionthgoqispresented. forthecriticalshearstremofcur
3、vedrectangularpanelswithsimplysupportededgesCcmput8dcurveswhichcoverawiderangeofpaneldeneionsarepresented;thesecurvesarefound,tobeingoodagreementwithtestresults.Estimatedcurvesarealsogivenforpmelswithclampededges. -.INTRODUCTIONA seriesofpapershasbeenpreparedtoprovideinformationonthebucklingofmrveds
4、heet.Theyoblemtreatedinthepreeentpaper,whichisapartofthatseries,isthedeterminationofthecriticalshearstressofacylindricallycurvedrectangularpanel.Forpanelshavingsimplysupportededgesthisproblemissolvedtheoretically(seeappendixandcmnpwtedcurvesareprovidedforfindingthecriticalshearstress.Estimatedresult
5、sarealsogivenforcurvedrectangularp6melshavingclampededges.,EXiStiANDDISCUSSION ,ThecriticalshearstressTcr forcyIindricawcuryedweJ+isgivenbytheequation : /. ,- T ks , cr=b%.-where, ks critical-shetresscoefficient,establishedbygeometry”ofpanelandtypeofedgesupportProvided by IHSNot for ResaleNo reprodu
6、ction or networking permitted without license from IHS-,-,-2D flexumlstiffnessofpanelperunitlengthE Young?smodulusofelaeticltyP PoissonrsrRtiob axialorcircumferentialdimensionofpanel,whioheverissmaller(exoeptwherenoted)t thicknessofpanel. .Twochartsarepresented,oneforpanelshavingcurvedsideslongertha
7、nthestzaglrb.kidesendtheotherforstraightsideslongertkanthecurvedsides.Inchartstheoritioal-sheam+tresscoefficientk6agzinstaourvatureparameterZ definedbytheorwhere ,r mdiusofourvatureofpanelpauelshavingeaohoftheseisplottedequationNACA!ENNO.1348()Et312(1-pa),Panelstithslmp3ysuyportedeib ismeastiredaxia
8、llywiththereeultthatthevaluesof ks and Z aredefinedinthemannerappropriatetoacylinder.Infigure2,thecritioabshea=tressooeffiolentsaregivenforpanelslongintheaxialdireotion;b ismeasured.oimmuferentiallysothatthevaluetht is, ks Z weredefinedintemnsoftheaxialratherthanthecircumferentialdimensionofthepanel
9、(dimensionbl infig.3). Figure5 showsthecomparisonforpanelslongintheaxialdireotion.Thecylinderourve is replottedintemeofdimensionb2 infigure3 sothattheseineparem.etersareusedaswereusedfortheothercurves.Ineachoffigures4and5 thefirstthreepanelbucklingcurveswerecomputedendthefourthcurvewasest-ted. These
10、esttitk “ -curveswereobtainedbyusingtheknownlimitingresultsasguidesandbyextrapolatingthetrendsobservedinthecasesfromwhichcomputedresultswereavdlable.Penelswitholampededges.-Figures6amd7sziveestimatedtheoreticalcritioal-shear-stresscthisfactindicat6sthattheinitialeccen-tricitiesInthetestspecimensare-
11、11. ThethreeSpeoimm inwhichsnapbucklingdidnotcmur fellinthe-e of 6“aa% XT wstress;, ccmlpressivestressiny-direction;thatis, .Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-.w(w) =W(%Y)=o W(x, o) = W(x,b) = o 1V(o,y) = v(a,y) =O t(x,o) =U(x,%)= o ()7
12、ISubstitutionoftheml.ussof Q andw givenbyequations(4)and(),rcspecttvely,intoequation(6)leadstothefciU.owingsetofslgobraicequations:wherep=1,2,.; q=l,2, .; andthesummationincludesonlythosevaluesof m and n forwhichmtp and n*q areodd.Theconditionforanonvanishingsolionoftheseequationsisthevanishingofthe
13、determinantofthecoefficientsoftheunknowns. Thisinfinitedeterminantcanbefaotoredintotheproductoftwoinfinite,su%dcterminan:s$oneinwhichptq iseven,andoneinwhichpq isodd.Thevanishingofthesesubdetorminantsleadstothefolluwingdsterminantalequations.Theequationinwhichpq isevenis,.Provided by IHSNot for Resa
14、leNo reproduction or networking permitted without license from IHS-,-,-10pa,q=:i?, I?=3)1=:p=l, q=:, !l=F3? 1=:p=l, q=;-p=2,q=(a71a15a15a71 a11.,. .a31o04-5000NACATNNo.1348a524 a33 a17 a26 a35. . ,0000a71.*.4.*,*4.,. . .a71 a11a13a15 a15 a15a71 a15 a15= o (9-).Provided by IHSNot for ResaleNo reprodu
15、ction or networking permitted without license from IHS-,-,-NACA0. 1348andtheequationinwhichpq isoddis%2 “21IQ=l,Chowever,asthepreciselocationofall.thcuspsinvolvesaohibitiveamountoflaborwithoutmy significantincreaseinaccuracy,thecuspswerefalredoutinfigure2. Table1 presentstherelativemagnitudesoftheco
16、efficientsofthetermsusedinthesolutions,end.tablo2givesthocomputedstresscoefficients.SolutionforDanelshaviwcircumferential2eaRD=t eaterthaqaxiallenath.-Whenthecircumferenti4dimensionisgreaterthantheaxial-dimension,a andb canbeinterchangedinequation(5inordertoretainb astheshortedimension,asfollows:w=c
17、rsinsin (1.1) “m=ln.1Thecoordinatesystmusedisshowninfigure10(b).Thisproblcmissolved.inamamnel* similar to thatusedinsolvingtheprobleminvolvingaxialdimensiongreaterthencircumferentialdimension.ThesetofalgebraicequationsfortheunknownFouriercooffici.entmisnow+12pf$J4z21TCW+32+L12)2-J (M). =0(m2- thecur
18、vesweiiefairedinamannersimilatothatforthecurvesoffigure2.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-14 MICATN S$0a71 1348 .REFERENCES1.2,394*5*6*7*8.910.u.12a71Batdorf,SB.: ASimplifiedMethod.ofElesticStabilityAnalysisforThinCylindricfiShells.II-
19、ModifiedEquilibriumEquationmNACATNNo.1342,1947.Batdorf,S.B.,Stein,Manuel,andSchUdcrout,Murry:CriticalStressofThin-WalledCylindersinTorsion.NACATNNo. 1344,1947.Batdorf,S.B.,Schildcrout,Murry,endStein,Manuel$Critical.SheerStressofLongPlateswith!IhnsverseCurvature.NACATNNo.1346,1947.Smith,R.C.T.: TheBu
20、cklingofPlywoodPlatesinShear.Rep.SM51,CouncilforSci, ad M:* g 17.59100 33”551000 157:k 164.51.5 0 7.07 79977.12 8.031; 8.55 9*7530 14*30 15.38100 30.54 . 27.151000 136.6 3.29*72 1 6.6510 % 8.43I-2.48 14.292% 26.96 26.u1000 U7*3 u8.9l-+aTaA- . .I kar z Evendetemiuiat I odddeterment1.5 1 7*3710 10.38
21、;:g15.23 15.511% 32.24 30.73ItATION.ALADVISORYCOMWXTEEFCIRJ!XRONNI!XOSProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-K?Id10,1II I IIII1 I I I I 1tI I 1 I II1410 102 103+f+qZ- MhmwuMvmOaYamnnEK9o IFigure1.-Critical-shear-stresscoefficientsforsimplysu
22、pportedcurvedpanelshavingcircumferenhllengthgreaterthanaxiallength.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-.10IId10201 J-L 10 z=g Ioz 105. NKrlalmMWORYmlu9mEF01KsFigdre2.-Critical-shear-stresscoefficientsforsimplysupportedcurvedpanelshavingsx
23、iallengthgreaterthancircumferentiallendh.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-Fig.3 NACATN NO. 1348IIIStripI I PanelNATIONALADVISORYCOMMITTEE FORAERONAUTICSLFigure3.- Curvedrectangularpanel. Limitingcasesfor a completecylinderandaninfinite
24、lylongstripalsoshown.a71Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-.Id, /Rot puldcl-“/4J1-qS+Cme?pmdtqtog=1u Lb.+.Eg.3&im&J)Otpo-d-. kwm9,1 K)-m Figure4.-Criticsl-shezw-stresscoefficientsforsimplysupported”curvedpanelscomparedwiththoseforcylinde
25、rsandlongcurvedstrips(ksad Z aredefinedintermsofaxiallenhofpanel).wf$lsIProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-. ,.1,:I, 1/1.-L uII (Figure5.-Critical-shear-stresscoefficientsfosimplysupportedcurvedpanelscomparedwiththoseforcylindersandinfin
26、itelylongcurvedstfips(ksand Z aredefinedintermsofcircmn-ferentiallengthofpanel). )“11Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-.,.NACATNNO. 1348,3102k.$10I1- 1 1.0Fig. 6NATIONALMVISORYCGWIITIEI!FM AEMIAAUTICS1 I I IIIII1, I I IIIIIII 10 102z.2I
27、 I IIIII(103Figure6.- Estimatedtheoreticalcritical-shear-stresscoefficientsforcurvedpanelsWithclampededgesd havingcircumferentiallengthgreaterthanaxiallength.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-Fig. 7 NACATN NO. 1348.103.I0210It-L I I III1II MATtOftAlADVISORYCOMMITTEEFC4AERONAUTICSI 10 100Ta1.gb .I.0I.!3 2.0=!Figure10.-CoordinatesystemsusedintheoreticalSolutio&.g., , , ,Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-
