1、REPORT NO. 874A SIMPLIFIED METHOD OF ELASTIC-STABILITY ANALYSIS FOR THIN CYLINDRICAL SHELLSBy S. B. BATDORFSUMMARYThis paper develops a new method for determining the bucklingstresses of cylindrical shells under various loading conditions.For convenience of exposition, it is divided into two parts.I
2、n part I, the equation for the equilibrium of cylindricalshells introduced by Donnell in NACA Report No. _79 to findthe critical stresses of cylinders in torsion is applied to findcritical stresses for cylinders with simply supported edges underother loading conditions. It is shown that by this meth
3、od solu-tions may be obtained very easily and the results in each case maybe expressed in terms of two nondimensional parameters, onedependent on the critical stress and the other essentially deter-mined by the geometry of the cylinder. The influence of boundaryconditions related to edge displacemen
4、ts in the shell mediansurface is discussed. The accuracy of the solutions found isestablished by comparing them with previous theoretical solutionsand with test results. The solutions to a number of problemsconcerned with buckling of cylinders with simply supportededges on the basis of a unified vie
5、wpoint are presented in aconvenient form for practical use.In part II, a modified .form of Donnells equation for theequilibrium of thin cylindrical shells is derived which is equiv-aleat to Donnell s equation but has certain advantages in physicalinterpretation and in ease of solution, particularly
6、in the caseoJ shells having clamped edges. The solution of this modifiedequation by means of trigonometric series and its application toa number of problems concerned with the shear buckling stressesof cylindrical shells are discussed. The question of implicitboundary conditions also is considered.I
7、NTRODUCTIONThe recent emphasis on aircraft designed for very highspeed has resulted in a trend toward thicker skin and fewerstiffening elements. As a result of this trend, a larger fractionof the load is being carried by the skin and thus ability topredict accurately the behavior of the skin under l
8、oad hasbecome more important. Accordingly, it was considereddesirable to provide the designer with more information onthe buckling of curved sheet than has been available in thepast. In carrying out a theoretical research program for thispurpose, a method of analysis was developed which is be-lieved
9、 to be simpler to apply than those generally appearingin the literature. The specific problems solved as a part ofthis research program are treated in detail in other papers.The purpose of this paper, which is discussed in two parts,is to present the method of analysis that was developed tosolve the
10、se problems.883026 50 -20In part I, the stability of a stressed cylindrical shell isanalyzed in terms of Donnells equation, a partial differentialequation for the radial displacement w, which takes intoaccount the effects of the axial displacement u and the cir-cumferential displacement v. Part I sh
11、ows the manner inwhich this equation can be used to obtain relatively easysolutions to a number of problems concerning the stability ofcylindrical shells with simply supported edges. The resultsof the solution of this equation are shown to take on a simpleform by the use of the parameter k (similar
12、to the buckling-stress coefficients for flat plates) to represent the state ofstress in the shell and the parameter Z to represent thedimensions of the shell, where Z is defined by the followingequations:For a cylinder of length LL 2Z=741-_and for a curved panel of width b5 2wherer radius of curvatu
13、ret thickness of shellandg Poissons ratio for materialThe accuracy of Donnells equation is established by compari-sons of the results found by its use with the results found byother methods and by experiment.In the simplest method that has been found for solvingDonnells equation, the radial displace
14、ment w is representedby a trigonometric series expansion. This method can be usedto great advantage for cylinders or curved panels with simplysupported edges but leads to incorrect results when applieduncritically to cylinders or panels with clamped edges.In part II, an equation is derived which is
15、equivalent toDonnells equation but is adapted to solution for clamped aswell as simply supported edges by means of trigonometricseries. This modified equation retains the advantages ofDonnells equation in ease of solution and simplicity of re-sults: The solution of the modified equation by means of
16、theGalerkin method is explained, and the results obtained bythis approach in a number of problems concerned with theshear buckling stresses of cylindrical shells are given ingraphical form and discussed briefly. Boundary conditionsimplied by the method of solution of the modified equationare also di
17、scussed.285Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-286 REPORT NO. 874-NATIONAL ADVISORY COMMITTEE :FOR AERONAUTICSLQ, Q1,Q2zSYMBOLSa length of curved panel (longer dimension)b width of curved panel (shorter dimension)d diameter of cylinderi,
18、j, m,t integersn, p, ffJp laterM pressure, positive inwardr radius of cylindrical shellt thickness of cylindrical shellu displacement in axial (x-) direction of point onshell median surfacev displacement in circumferential (y-) direction ofpoint on shell median surfacew displacement in radial direct
19、ion of point on shellmedian surface; positive outwardx axial coordinatey circumferential coordinate7 _numencal coefficientsCmn_, amn j(rtL 2 rtb 2k, shear-stress coefficient DTr 2 for cylinder or _ forcurved panel or infinitely long curved strip)/(r#L 2k_ axial compressive-stress coefficient D_r2 fo
20、rzxtb 2cylinder or_ for curved panel or infinitelylong curved strip)k_ circumferential compressive-stress coefficientz#L 2 o-_tb2D_ 2 for cylinder or _ for curved panel or in-finitely long curved strip)%/ (p L2 Up hydrostatic-pressure coefficient D_r2 w0 amplitude of deflection function(Et3_D plate
21、flexural stiffness per unit length _2(_2) /E Youngs modulusF Airys stress function for the median-surfacestresses produced by the buckle deformationb2F_, stress in axial direction; b2F_, stress in eir-b2F shear stress)eumferential direction; -b_ylength of cylindermathematical operators(“curvature pa
22、rameter rt _/1_2 for cylinder orb2rt _/1-_ for curved panel or infinitely longcurved strip)ft L/X for cylinder or b/X for infinitely long curvedstripX half wave length of buckles; measured cireumfer-entially in cylinders and axially in infinitely longcurved stripsTTcr0“x0“yR_R_V 4_7G4V 8dimensionles
23、s axial coordinate (x/b)dimensionless circumferential coordinate (y/b)Poissons ratioapplied shear stresscritical shear stressapplied axial stress, positive for compressionapplied circumferential stress, positive for compres-sionshear-stress ratio; ratio of shear stress present tocritical shear stres
24、s when no other stress is actingaxial-compressive-stress ratio; ratio of direct axialstress present to critical compressive stress whenno other stress is acting/_2 52 2 b4 b4 b4 ,_operatortoperator _+_ operator _2+b7 / /operator _+_ inverse operator defined by equation(v -_(vy) = v_(v-_) =2)1. DONNE
25、LLS EQUATIONTHEORETICAL BACKGROUNDIn most theoretical treatments of the buckling of cylin-drical shells (see references 1 to 3) three simultaneous partialdifferential equations have been used to express the relation-ship between the components of shell median-surface dis-placement u, v, and w in the
26、 axial, circumferential, andradial directions, respectively. No general agreement hasbeen reached, however, on just what these equations shouldbe. In 1934 Donnell (reference 4) pointed out that thedifferences in the various sets of equations arose from theinclusion or omission Of a number of relativ
27、ely unimportantterms (referred to in the present paper as higher-orderterms), and proposed the use of simpler equations in whichonly the most essential terms (first-order terms) wereretained. The omitted terms were shown to be small, andthus the simplified equations to be applicable, if the cylinder
28、shave thin walls and if the square of the number of circum-ferential waves is large compared with unity. Donnellfurther showed that the three simplified equations can betransformed into a single eighth-order partial differentialequation in _o (see appendix A of the present paper) in whichthe effects
29、 of the displacements u and v are properly takeninto account; chis equation will hereinafter be referred to asDonnells equation.When higher-order terms are included in the three partialdifferential equations previously mentioned, the resultingtheoretical buckling stresses are usually very complicate
30、dProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-A SIMPLIFIED METHOD OF ELASTIC-STABILITY ANALYSIS FOR THIN CYLINDRICAL SHELLS 287functions of the cylinder dimensions and the elastic proper-ties of the material. A family of curves is ordinarily drawn
31、giving the critical stress as a function of the length-diameterratio for specified values of the radius-thickness ratio andfor given elastic properties (references 2, 3, and 5). Whenthe higher-order terms are omitted from the equations andthe requirements of an integral number of circumferentialwave
32、s is removed, new parameters can be introduced whichcombine the cylinder dimensions and material properties insuch a way that the results can be given in terms of a singlecurve. These parameters have been used, with slight varia-tions in detail, by Donnell, Kromm, Leggett, and Redshaw(references 4 a
33、nd 6 to 9). The omission of the higher-orderterms also greatly simplifies the calculations, and the calcula-tions are simplest if Donnells equation, rather than the setof three simultaneous equations, is employed. Donnellsequation, or an equivalent equation, may therefore bepresumed to be the most p
34、romising for use in solving hithertounsolved problems in the stability of cylindrical shells.In spite of the fact that it was introduced some time ago,Donnells equatiort has not achieved the wide acceptancefor use in the stability analysis of cylindrical shells whichit appears to merit. Some investi
35、gators have continued touse simultaneous differential equations in which higher-order terms appear, presumably on the assumption that theerrors arising hom neglect of these terms might be undesh-ably large. Others have dropped second-order terms but havecontinued to employ simultaneous equations, pr
36、obably inorder to specify directly edge-restraint conditions having todo with displacements in the axial and circumferentialdirections, which cannot be done with Donnells equation.The purposes of part I are to establish the accuracy of theequation by comparing the results found by the use ofDonnells
37、 equation with the results found by other methodsand with experimental results and to investigate the questionof boundary conditions on u and v. The additional purposeis achieved of presenting the solutions of a number ofproblems concerned with buclding of cylinders with simplysupported edges on the
38、 basis of a unified viewpoint and in aconvenient form for practical use.BUCKLING STRESSES OF CYLINDERS WITH SIMPLYSUPPORTED EDGESLateral pressure.-The theory for the lateral pressure(uniform external pressure applied to walls only) at whicha cylinder will buckle is given in appendix B in which it is
39、assumed that the lateral pressure causes the buckling byproducing a circumferential stress _y and that it affects thebuclding in no other way. The results are shown in alogarithmic plot in figure 1. The ordinate in this figure is“the stress coefficient k_ which appears in the flat-platebuckling equa
40、tion (see, for example, reference 3, p. 339)7r2DOy =/gY-.L_I0 3I0 _ii/0 /JI I I I I I I/0 I I I 1 1 I I I/2I I I I I l I II0sLzI I I I I I 041 I I I I I I Iii0FIGURE 1.-Critical circumferential-stress coefficients for cylinders with simply supported edges.Provided by IHSNot for ResaleNo reproduction
41、 or networking permitted without license from IHS-,-,-288 REPORT NO. 874-NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS(The discussion given in the section of the present paperentitled “Parameters Appearing in Buckling Curves“ showsthe relationship between a cylinder of length L and aninfinitely long f
42、lat plate of width b=L.) The abscissaL 2 ,_ /LX 2 rmay be regarded either as a measure of the curvature, or,for any given ratio of radius to thickness, as a measure of thelength-radius ratio of the cylinder. Figure 1 shows thatfor small curvature % approaches the value 4, which appliesin the case of
43、 simply supported long flat plates in longitudinalcompression (reference 3, p. 327). As the curvature param-eter Z increases, the stress coefficient Icy also increases.For large values of Z, the curve approaches a straight line ofslope 1/2. This straight line is expressed by the formulaky_- 1.04Z /_
44、As the length-radius ratio increases, for a given value ofr/t, the number of circumferential waves n diminishes. A1- though n must be an integer, the curves of figure 1 wereobtained on the assumption that n is free to vary continu-ously. Only small conservative errors are involved in thisassumption.
45、 Because n-_l corresponds merely to a lateraldisplacement of the entire circular cross section, the minimum/0 3uI0 z/0value of n is 2, which corresponds to deformation of thesection into an ellipse. This limitation on n results insplitting the curve of figure 1 into a number of curves fordifferent v
46、alues of r/t when Z becomes large. A cylinderhaving a value of r=20 buckles into an ellipse when L/r istabout 10, and the value of Lfr at which such budding occursincreases with increasing r/t.In figure 2 the curve of figure 1 is compared with resultsbased on more complicated calculations given in r
47、eference 3and in reference 5. At fairly large values of Z the resultsgiven in reference 3 and in reference 5 are in good agreementwith the results of the present paper. At small values of Zthe curve based on reference 3 (Timoshenko) is definitely toolow, because t% should approach the fiat-plate val
48、ue of 4 as Zapproaches zero. An interesting feature of the comparisonis that one calculation gives results below, and the othercalculation results above, those given herein. The test data,taken from reference 5, are in reasonable agreement withand show more scatter than the theoretical curves.In the
49、 case of cylinders so long that n-2, the requirementfor the validity of Donnells equation that n21 is nolonger satisfied and appreciable error is to be expected.Indeed it may be shown tha_ for very long cylinders whenn_-2 Donnells equation gives 4D/r 3 as the critical value ofthe applied lateral pressure, whereas the accepted theoreticalJSfurrn F _i:-IA_ o#80 _:_o5 “ .-_ -X,- _b “M_, “./ o_ “_/ (t_moshenko.“-500/00FIGURE 2.-Comparison of
