effect of a local axisymmetric imperfection on the buckling behavior of a circular cylindrical shell...

8/12/2019 Effect of a Local Axisymmetric Imperfection on the Buckling Behavior of a Circular Cylindrical Shell under Axial Compression.pdf

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Reprinted rom AIAAJOURNAL, Vol. 9,No. I, January1971,pp. 48-52 Copyright,1971,by the AmericanInstituteofAeronauticsand Astronautics,andreprintedby permissionof the copyrightowner

Effect of a Local xisymmetric Imperfection on the BucklingBehavior of a Circular Cylindrical Shell under

xial CompressionJ. W.HUTCHINSON*

Harvard University, Cambridge, Mass.AND

R.C.TENNYSONtAND D.B.MUGGERlDGE1University of Toronto, Toronto, Ontario, CanadaA combinedtheoretical andexperimentalinvestigationhasbeencarriedout on the effects

ofcertaintypesoflocalaxisymmetricimperfectionson thebucklingofcylindricalshellsunderaxialcompression. Bucklingloadshavebeencalculatedfora varietyof"dimple"imperfections. Results have been obtained for cons tant thickness shells with middle surfacevariationsfrom thegeometryofa perfectcylinderaswellasforshellswithlocalaxisymmetricthickness variations. Nonlinearprebucklingelrectsand edgecondit ionsare taken into ac count. In the experimentalprograma seriesofsevenphotoelasticplasticcircularcylindricalshellseachcontaininga localaxisymmetricdimplecentered t mid-lengthweretestedunderpureaxialcompression. Allcylinderswereconstructedby the spin-castingtechniqueandthe local imperfection wascut onboth theinneraudouter walls usinga hydraulic tracer-toolapparatusin conjuuctionwitha metaltemplate. Abroadrangeofimperfectionamplitudesandwavelengthswasinvestigated. Theexperimentalresultswerein goodagreementwiththe theoreticalpredictions.

Nomenclature IntroductionC [3(1 1 2)]112 T HE presenceofshapeimperfectionsincircularcylindricalE = modulusofelasticity shellssubjected toaxialcompressiveloadinghasnowbeenL shelllength recognized as the dominantfactor in reducing the buckling

= axialhalfwavelengthofdimple load significantly below the classical value. Although theI, = 11 [12(1 - 1'2)]-1I4(Rt)1I2; halfwavelengthof classicalaxi effects ofend const raint dolower bucklingloads, it has beensymmetricbucklingmode shown both theoretically' and experimentally- that theR = shellradiusmeasuredtothemiddlesurfacet = shellwallthickness clamped case, which is perhapsmost often encountered in1 averagewallthickness practiceparticularlyinthelaboratory,accountsfornominally1* averagewallthicknessmeasuredoveronlydimplearea a10% reduction. Asdemonstratedrecently.tauniformdisw radialdisplacemeut-c-positiveoutward tributionofaxisymmetricimperfectionshaving theform ofax axialcoordinatewithoriginat theshellmid-length simple trigonometric function drastically reduces the shell:i = 1I x/le bucklingload. In particularitwasfoundthattheloadreducZ = (1 - 1 2)1I2L2/Rt tionwasdependent bothon theimperfectionamplitudeandr = I Ie wavelength,andcould beaccuratelypredictedusing Koiter's/ = peakamplitudeof axisymmetric middlesurfaceimperfec extended theory. Other investigators!. have also demontionA seeEq. (1) stratedexperimentally thatimperfectionssignificantlyaffect>'e 2 Et 2/C; classicalbucklingloadofaninfinitelylongcylin- thebucklingbehaviorofcircularcylindricalshellsunderaxialdricalshellunderaxialcompression compression. Althoughimperfectiondistributionsare likely * buckling load (i.e., maximum support load of imperfect toberandominnature,it!softenobserved thatlocaldimplesshell) orshapeimperfectionsarepresentinthe shellstructure. Rep. maximumdeviation ofmediansurface/averagewallthick cently,theeffectsoflocaleutouts'-'andlocalcirculardimples'ness on thebucklingofcircular cylinders have been studied andI Poisson's ratio havebeenfoundtogiveriseto lowbucklingloads.Here, a numberoftheoreticaland experimentalresultsarepresentedontheeffectsoflocalaxisymmetricimperfectionson

thebucklingbehaviorof cylindrical shellsunder axial comPresentedasPaper70-103 at theAIAA SthAerospaceSciencesMeeting, New York, January 19-21, 1970; revision received pression. Shells with variousdimple forms are studied inMay4,1970. Thisworkwassupportedinpart,bytheNational cluding constant thickness shells with initialmiddle surface ResearohCouncilofCanada (NRCGrant,No. A-27R3), the Na deviations as well as shells with variable thickness and astional AeronauticsandSpaceAdministration [NASAGrantNo. sociatedmiddlesurface variations. 52-026-(011); alsoNASAGrantNGL22-007-012] andthe DivisionofEngineeringandApplied Physics,HarvardUniversity.* ProfessorofAppliedMechanics,DivisionofEngineeringand AsymptoticandNumericalPredictionsAppliedPhysics. MemberAIAA.t AssociateProfessor,InstituteforAerospaceStudies. Mem Amazigo andBudiansky"haveobtaineda generalformulabel'AIAA. for thebucklingloadofaninfinitelylongcylindricalshelluntResearchAssistant,InstituteforAerospaceStudies. der axial compressioncontaining an arbitrary, localized axi


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49ANUARY 1971 BUCKLING BEHAVIOR OF A CIRCULAR CYLINDRICAL SHELL

-L/2 L/2e

T I 'I - --i'-. -..----:.;*I- - - - 4 - - -L 1.--+---.;.' a I _--- - .J1-1 .....\ .I.-lR SHELL

. J d ~ t l ~ ~ : l ,'1.:51 1.wJ.d; e , FORALLOTHe:R VALUES OF "

Fig, 1 Circular cylindrical shell of constant thicknesswith inward axisymmetric dimple imperfection in shape.symmetric imperfection. Their formula is an asymptotic onewhich is valid for sufficiently small imperfections in much thesame way as is Keiter's formula for a sinusoidal axisymmetricimperfection.w. '

The initial axisymmetric imperfection Wo in the middle surface (assumed to have a continuous slope) of a cylinder of constant thickness t enters into the buckling formula through(1)

where f = 1rxllc and l, = 1r[12 1 - p2) 1-1/4 Rt) 1/ 2 is the halfwavelength of the classical axisymmetric buckling mode.The asymptotic formula of Ref. 9 for the ratio of the bucklingload (..* to the classical buckling load Ac is

(1 - A' 1A) 3/2 = [3C1(23/2) ]ILil (A' 1A) (2)where C [3(1 - p2) ]1/2.Of primary interest here is a "cosine dimple" defined by

Wo -0/2 1 + cos1rxllz ) (3)=0so that l is the half length of the dimple. In this case,

Li = [sin(1rI3)/(::l2 - l] o/t) (4)where :J = l./C When ::l 1, Eq. (4) reduces to

ILil = 21r 0/t) (5)The maximum reduction in the buckling load, for a given imperfection amplitude-thickness ratio o/t, occurs when ::l 0.8with Li about 5% larger than Eq. (5).Numerical calculations have also been carried out for finitelength shells with the cosine dimple, Eq. (3), symmetri callylocated with respect to the ends of the shell as shown in Fig. 1.The calculation procedure, which has been described in Ref. 3,takes into account end effects and nonlinear prebuckling deformations. The prebuckling problem and the reduced eigenvalue problem are both obtained without approximation froma Karman-Donnell-type shell theory. Resulting ordinarydifferential equations are cast in finite difference form andsolved by a Gaussian elimination method usually referred to asPotters' method. Clamped end conditions are chosen for allthe results presented here since they pertain most closely tothose in the testing program.

0.2 .. s.Ie -to. ZollOO FOR SHELLS All e0.1 0.2 o.s 0.4 0.5ISIt I

Fig. 2 The effect of an axisymmetrical dimple imperfect ion on the buckling of clamped cylindrical shells (cons t ant thickness) under axial compression. .

Predictions based on the asymptotic formula [Eq. (2)1and the numerical calculations are compared in Fig. 2 forl.llc 1. Only the absolute value of the imperfection amplitude enters into the asymptotic formula and thus the curveshown holds for inward and outward dimples. On the otherhand, the numerical results bring out some difference betweenthe buckling load for a bulge (curve A) and a pinch (curve Bwith the latter having the more degrading effect. Agreementbetween the asymptotic formula and the numerical results isquite good except, of course, for extremely small imperfectionamplitudes in which case end effects dominate the bucklingbehavior of finite leng th shells. The length parameter Zfor these shells was taken to be 300, so that for all practicalpurposes the buckling load is independent for values of Z inthis range and greater. (All calculations were made with 120finite difference stations over the half length of the shell andisolated results were checked using 250 stations.)Effects of local thickness variations are illustrated by theresults presented in Fig. 3. These results were also obtainednumerically in the manner described previously and detailedfurther in Ref. 3. For both the lowest and uppermost curvesthe thickness variation is given by

Lit = 0 1 + COS1rx/l,,) Ixl t=0 Ixl > t;

with either the inner or outer wall unperturbed as shown.Thus, this variation gives rise to exactly the same initial mid

1I sO.30.2

Z -zoo0 0 ' 02 o.s 0.504Sril

Fig. 3 A comparison of t be effects of different thicknessvariations on the buckling of clamped cylindrical shellsunder axial compression.


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50 HUTCHINSON, TENNYSON, AND MUGGERIDGE AIAA JOURNAL

Fig.4 Cylinder profile measuring apparatus.

dle surface variation as described by Eq. (3). I t is not surprising that a shell which has both an inward middle surfacevariation and a reduction in its thickness buckles at a loadconsiderably below a constant thickness shell with an identicalmiddle surface variation. Even a thickening of the shell, asindicated by the uppermost curve, results in a small reductionin the buckling load if olt is sufficiently small.Localized dimples in constant thickness cylindrical shellshave an effect which is somewhat less severe than a sinusoidalaxisymmetric imperfection. Almrotht has made the sameobservation on the basis of his studies of the effects of axisymmetric imperfections on cylindrical shells. Axisymmetricdimple studies of conical shells lead to similar conclusions.Pv'

Experiment Fabricat ion and Test ProcedureAll test cylinders were made by the spin-casting techniquesusing a photoelastic liquid epoxy plastic. Initial ly, a geo

metrically near-perfect circular cylindrical shell was cast andthe inner wall machined to the prescribed profile with an axisymmetric dimple centered at mid-length. Subsequently,the cylinder was removed from the form and thickness measurements were made at discrete points around the circumference at both ends of the shell. Machined aluminum endplates were then attached to the test cylinder to provide theclamped edge constraint. To obtain a constant thicknessshell, the outer surface was machined to the same profile as theinner wall by inserting into the cylinder a ring stiffener locatedat the dimple to reinforce the shell wall during the machiningstage. Each imperfection shape was constructed using ametal template containing the desired amplitude and wavelength in conjunction with a hydraulic tracer-tool apparatus.In order to determine the actual inner and outer shell wallprofile, each model was placed in a rotation apparatus (Fig.4) and axial contours were obtained using two low pressure,linear contacting displacement transducers, with their outputsrecorded on an x-y plotter. Figure 5 illustrates median surface profiles determined from two of the test cylinders. I t isquite evident that the dimples are axisymmetric i.e., littlecircumferential variation, and centered at mid-length. Acomparison of the dimple shape with the assumed wave formgiven by Eq. (3) is shown in Fig. 6 using a randomly selectedgenerator from each test cylinder. I t can be readily seen thatthe assumed and actual profiles are in excellent agreement. Asummary of the shell properties is contained in Table 1. Ofparticular significance is the difference in values between theaverage shell wall thickness [ and the average shell wall thickness measured only over the dimple area, denoted by rThis variation appreciably affects the values of X' IX fJ and {3which are used in the comparison with the exact model results,as will be discussed later.

The final stage of the test procedure involved the properalignment of the cylinder in an electrically driven, four-screwcompression machine. Uniformity of the applied stress waseasily checked by examining photoelastically the membranestress distribution around the circumference of the cylinderand finally, by noting the postbuckled configuration of thecylinder. Complete circumferential buckling centrally located either in the dimple (one tier) or bounding the dimple(two tiers) attested to the uniformity of the load distribution.To serve as a reference shell, a geometrically near-perfect

_ wJ-1L I : I I .~ . I /-- SHELL NO.011 2 0 L L ~. SHELLNO.02

il2 LL/2 ~ .. (/2il:'LL/2 SHELL NO. 03y -----: /2SHELL NO.04O L/2--V------ /2

t / : k . ~ 0 S L L N ~ 03


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51ANUARY 1971 BUCKLING BEHAVIOR OF A CIRCULAR CYLINDRICAL SHELLTable 1 Properties of shells containing an axisymmetric dimple

Shell INNo. R 1* L s i; IJ Irlle (A*IAe).xpD1 3.92 0.0211 0.0242 11.0 0.0070 0.550 0.289 1.02 0.473D2 3.92 0.0189 0.0188 11.0 0.0069 1.375 0.367 2.87 0.857D3 3.92 0.0194 0.0192 11.0 0.0075 0.834 0.391 1. 72 0.629D4 3.92 0.0183 0.0188 11.0 0.0062 0.306 0.330 0.64 0.449D5 3.92 0.0207 0.0215 11.0 0.0022 0.550 0.102 1.14 0.714D6 3.92 0.0224 0.0236 11.0 0.0173 0.550 0.734 1.03 0.345D7 3.92 0.0179 11.0 0.919E = 3.94 X 1 )6 PSIII = 0.40Note: are based on 1*

cylinder was tested in axial compression to determine the modulus of the elasticity of the epoxy plastic used in the test series,and at the same time, provide a measure of the load reduction due to the clamped edge constraint common to each shell. In total, seven cylinders of varying imperfection wavelengthand amplitude were studied (refer to Table 1), the results ofwhich are discussed in the next section.Discussion of Experimental Results

A comparison of shell buckling data for shells with inwarddimples and theoretical predictions from both the asymptoticformula [Eq. (2)] and the numerical calculations is made inFig. 7. The theoretical curves have been obtained usinglrlle = 1.06 which is the average of the corresponding valuesfor the three tes t specimens. As alluded to previously, theaverage wall thickness i , which was measured only over thedimple area defined by [z] Ir was used as the referencethickness in calculating the buckling loads. Justification ofthis step rests on the argument that buckling is a local phenomenon governed by the dimpled region. In any case, thediscrepancy between the average shell thickness I and thelocal average l* was less than five percent in every shell butone as listed in Table 1. All the test specimens faU in therange in which the numerical predictions are essentially independent of the shell length.The agreement between test and theory, particularly withthe numerical results, is very good, especially so consideringthe long history of discrepancy between test and theory forsuch structures. Equally good comparisons have recentlybeen obtained for the effects of sinusoidal imperfections."

1 .01 - - - - - - - - - - - - - - - -SHELL NIS lGl1 5A 06 07

O.

0L----l--....J..--....J....--....L---- 0.2 0.4 0.6 0.8 1.08 T

Fig. 7 Buckling load of a circular cylindrical shell vsaxisymmetric dimple imperfection amplitude for a particular value of imperfection wavelength.

Figure 8 displays plots of buckling loads for constant thickness shells with inward dimples over a range of values of theimperfection wavelength ratio lrlle for a constant value of theimperfection-thickness ra tio olt The value of olt 0.363was chosen as the average of the values for the three experimental points plotted. It is clear that the critical half wavelength occurs for Irlle f 0.8 and the associated buckling load isonly a few percent below the predicted value for lxlle = 1.The asymptotic formula shows the same general trend as thenumerical results. Namely, when the ratio lrlle is a factor 3,say, greater or less than unity, the effect of the imperfection ismuch less than at the critical value of lrlle. The experimentsshow this very clearly.Conclusions

Based on the local shell geometry in the region of the axisymmetric dimple, experimental buckling loads were found tobe in rather good agreement with numerical calculations usingan exact model formulation including the effects of a clampededge constraint and arbitrary imperfection amplitudes andwavelengths. The asymptotic formula of Amazigo and Budiansky specialized to the cosine dimple, gives quite reasonablepredictions for values of the imperfection-thickness ratio evenas large as 0.5. In the critical wavelength range (i.e., 0.5 ;lxlle :::; 1.5) it is conservative. In general, for the axisymmetric dimple profile considered, buckling load reductionsare not as severe as compared to a sinusoidal distribution extending over the entire cylinder length. However, it is quiteapparent that very small axisymmetric dimple imperfectionsdrastically reduce the buckling load of a circular cylindricalshell in axial compression.

1.01-------------,-------,

.-0.6

0.4 RESULTS

u -0.4 SHELL NI0.2 > 02ll/jl 0 .363 e 03il 04

0.5 1.0 1.5 2_0I /IeFig. 8 Buckling load of a circular cylindrical shell vsaxisymmetric dimple imperfection wavelength for a particular value of imperfection amplitude.

3.0


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HUTCHINSON, TENNYSON, AND2eferences

Almroth, B. 0., "Influence of Edge Conditions on the Sta-bili ty of Axially Compressed Cylindrical Shells," CR-161, Feb.1965, NASA; also AIAA Journal, Vol. 4, No.1, Jan. 1966, pp.134-140.! Tennyson, R. C., "Photoelastic Circular Cylinders in AxialCompression," STP 419, 1967, American Society for Testing andMaterials.3 Tennyson, R. C. and Muggeridge, D. B., "Buckling of Axisymmetric Imperfect Circular Cylindrical Shells under AxialCompression," AIAA Journal, Vol. 7, No. 11, Nov. 1969, pp.2127-2131.4 Babcock, C. D. and Sechler, E. E., The Effect of InitialImperfections on the Buckling Stress of Cylindrical Shells,"TN D-2005, NASA, July 1963.6 Arbocz, J. and Babcock, C. D., TheEffect of General Imperfections on the Buckling of Cylindrical Shells," [ ransactions of

ASME Ser. 36, E; Journal of Applied Mechanics, 1969, pp.28-38.6 Tennyson, R. C., The Effects of Unreinforced Circular Cut outs on the Buckling of Circular Cylindrical Shells under AxialCompression," Transactions of ASME ; Journal of Engineering forIndmtry, Nov. 1968.

MUGGERIDGE AIAA JOURNAL7 Brogan, F. and Almroth, B. 0., "Buckling of Cylinders withCutouts, AIAA Journal, Vol. 8, No.2, Feb. 1970, pp. 236-240.8 Babcock, C. D., The Influence of a Local Imperfection onthe Buckling Load of a Cylindrical Shell Under Axial Compression," GALCIT SM 68-4, Feb. 1968, California Institute of Technology, Pasadena, Calif.S Amazigo, J. C. and Budiansky, B., private communication,1969,Harvard Univ., Cambridge, Mass.

io Kolter, W. T., On the Stability of Elastic Equilibrium,"thesis,I945; Delft, H. J. Paris, Amsterdam; also NASA Technical Translation F-I0, 833, March 1967.Keiter , W. T., The Effect of Axisymmetric Imperfections onthe Buckling of Cylindrical Shells under Axial Compression,"Koninklijke N ederlandse Akademie van Wetenschappen, Proceed-ings B66, 1963,pp. 265-279.11 Almroth, B. 0., "Influences of Imperfections and Edge Restraint on the Buckling of Axially Compressed Cylinders,"CR-432, April 1966, NASA.13 Schiffner, K., "Untersuchung des Stabilitatsverhaltens dunnwandiger Kegelschalen unter axialsymmetrischer Belastung mittels einer niehtlinearen Schalentheorie," Jahrbuch 196. i derWGLR, pp.448-453.14 Arbocz, J., "Buckling of Conical Shells under Axial Com.preseion," CR-1162, 1968, NASA.

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