1、COPYNATIONAL ADVISORY COMMITTEEFOR AERONAUTICSTECHNICAL NOTENo. 1341A SIMPLIFIED METHOD OF ELASTIC-STABILITYANALYSIS FOR THIN CYLINDRICAL SHEI_J_I - DONNELLS EQUATIONBy S. B. BatdorfLangley Memorial Aeronautical LaboratoryLangley Field, Va.WashingtonJune 1947Provided by IHSNot for ResaleNo reproduct
2、ion or networking permitted without license from IHS-,-,-Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NATIONAL ADVISORY CO_ITTEE FOR AERONAUT IC STECHNICAL NOTE NO. 15hl, ,|A SIMPLIFIED METHOD OF ELASTIC-STABILITYANALYSTS FOE THIN CYLINDRICAL SHEL
3、LSI - DONNELL tS EQUATIONBy S. B. BatdorfSUMMARYThe equation for the equilibrium of cylindricalshells introduced by Donneli in NACA Report No. 479 tofind the critical stresses of cylinders in torsion isapplied to find critical stresses for cylinders withsimply supported edges under other loading con
4、ditions.it is shown that by this method solutions may be obtainedvery easily and the results in each case may be expressedin terms of two nondimenslonal parameters, one dependenton the critical stress and the other essentially deter-mined by the geometry of the cylinder. The influence ofboundary con
5、ditions related to edge displacements in theshell median surface is dLscussed. The acciLracy of thesolutions found is established by comparing them withprevious theoretical solutions and with test results.The solution_ to a nt_iber of problems concerned withbuckling of cylinders with simply supporte
6、d edges on thebasi_ of a unified viewpoint are presented in a convenientform for practical use.INTR ODUCT IONThe recent emphasis on aircraft designed for veryhigh speed has resulted in a trend toward thicker skinand fewer stiffening _lements. As a result of thistrend, a larger fraction of the load i
7、s being carried bythe _kin and thus ability to predict accurately thebehavior of the skin under load has become more impor-tant. Accordingly, it was considered desirable to pro-vide the designer with mor_ information on the bucklingProvided by IHSNot for ResaleNo reproduction or networking permitted
8、 without license from IHS-,-,-of curved sheet than has been available in the past. Incarrying out a theoreticsl research program for this pur-pose, a method of analysis was developed which is believedto be s_mper to apply than those generally appearlni:_ inthe literature. The specific problems solve
9、d as a partof this research program are treated in dcta_l in othorpapers. The purpose of the present investigation, which isdiscussed in two papers,ls to present the method of analysisthat was developed to solve these problems. In the presentpaper the method is briefly outlined and applied to a mzmb
10、erof the simpler problems in the buckling of cylindricalshells. In reference i the method is generalized forapplication to more complicated problems.THEORET ICAI, BAC._(GROUNDIn most thcoretica! treatments of the buckling ofcylindrical shells (see refe ,_ _n,_e, _, 2 to 4) three simul-taneous partia
11、l differential equations have been used toexpress the relationship between the components of shellmedlan-surface displacemen_ u, v, and w in _.;hea:lal, circtui_ferential, and radial directions, recpec-tive!y. No general agreement has been reached, however,on just what these equations should be. In
12、1954 Donnell(refer_mce 5) pointed out that the differences in 1:hevarious sets of equations arose from the inclusion oromission of a num1_er of rel?_,ively unimportant terms(referred to in the present paper as higher-order terms),and proposed the use of simp.cr equations in which onlythe most essent
13、ial term,_; (first-order terms) “:ere retained.The omitted terms were shown to be small, and thus thesimplified equations to be appli_,able, if the cylindershave thin walls and if the square of the num_er of cir-cumfcrentia! waves is larFe compared with unity. Donnellfurther showed that the three si
14、m.01ifi 1 is no longer satisfied and appreciableerror is to be expected. Indeed it may be shown that forvery long cylinders when n = 2 Donnell,s equationgives !_D/r3 as the critical value of the appliedlateral oressure, whereas the accepted theoretical resultis 5D/r 5 (by use o!“ the formula given o
15、n p. _50 ofreference k). Tbe curves for n : 2 will probably notoften be needed, however, since they apply bnly whenF _, _ which in the case of thin cylinderscorresponds to a very large length-z_adius ratio, and ifneeded, the curves for n - 2 can be applied in con-junction with a correction factor 0
16、_a.Axial comoression.- The theory for the axial st_essat wb_ich a cylinder will buckle is given in appendix B,and the results are shown in figure 3. The ordinate isanalogous to, and the abscissa identical with,the corre-sponding coordinates used in figure i. Figure 3 showsthat for small values of Z,
17、 kx approaches the value i,which applies in the case of long flat plates in trans-verse compression with long edges simply supported(reference _). For large values of Z, the curvebecomos a straight line of slope i. This straight lineis expressed by the formulakx : “*Z = 0.70,9ZTr2Provided by IHSNot
18、for ResaleNo reproduction or networking permitted without license from IHS-,-,-NACATN No. 13J,1 9For any fixed value of r/t some value of Z alwaysexists above which L/r is so large that the cylinderfails as an i_ler strut rather than by buckling of thecylinder wal_s. Pin-ended Euler buckling of cyli
19、ndersis indicated in figure 3 by means of dashed curves.The result just given for the critical-stress coef-ficient for a cylinder in axial compression leads to thefollowing expression for the critical stress:X1 (1)3(i- rThe value given in equation (i) for the critical stressof a moderately long cyli
20、nder in axial compression byuse of Donnell,s equation is identical with the valuefound by a number of investigators using other equationsas starting points (references 2 to 4). In the case ofcylinders under axial compression the errors involvedin dropping the second-order terms are therefore con-clu
21、ded to be small.The buckling stresses given by equation (I) arenevertheless in serious disagreement with the bucklingstresses obtained by experiment (reference ii). For adiscussion of the degree of correlation that can be foundbetween theory and experiment for cylinders under axialcompression, see r
22、eference 12.Hydrostatic pressure on closed cylinders.- Whenclosed cylinders are subjected to external pressure, bothaxial and circumferential stress are present. The theoryfor buckling under these combined loads is given inappendix B. The results are shown in figure 4. Theordinate Cp used in this fi
23、gure is a nondimensionalmeasure of the pressure p defined as follows:prL 2Cp - _2DProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-I0 NACA TN No. 13151The coefficient Cp can be directly related to thecorresponding stress coefficients kx and ky. Bydefi
24、nitionOytL 2ky - n2Dand, according to the hoop-stress formula,Dr_Y =7It follows from the three preceding equations that Cpis ntunerically equal to ky. Similarly Cp can be shownto be numerically equal to 2k x.At low values of Z, Cp approaches the value 2,which implies that kx = i and ky = 2. Ti_at th
25、esevalues of k represent a critical combination of stressesfor an infinitely long fiat plate was shown in refer-ence 15. At large values of Z, the curve approachesthe curve given in figure i for buckling under lateralpressure alone and, like that curve, has Oranchesrepresenting buckling into two cir
26、cumferential waves.In figure 5 the computed values of the pressurecoefficient Cp at which the cylinder would buckle ifonly the axial pressure _ere acting and if only lateralpressure were acting are compared v_ith the results whenboth are acting because of hyckostatic pressure. Atlarge values of Z th
27、e circumferential stress at whichbuckling occurs under hydrostatic pressure is suostan-tially the same as it _ould be if no axal stress werepresent, as in the case of lateral pressure. The reasonthat the circumferential stress appears as the main factorin buckling at high values of Z presumably is t
28、hat atthese values of Z the axial stress required to producebuckling is many times the circumferential stress required,whereas under hydrostatic pressure the axial stressactually present is only one-half the circumferentialstress.Provided by IHSNot for ResaleNo reproduction or networking permitted w
29、ithout license from IHS-,-,-NACA TN No. 1341 !IIn figure 6 the curwc of figure )_ is compared withcurves re_resenting Sturms theoretical res1_!ts (refer-ence 6) and with a curve based on the follo_ing 1ormuladeveloped at the U S. _ocrimenta! ,_del Basin (refer-ence 14, equation (9):pi t_/2-,_,.42E )
30、- - - 0.45 jThis formula is an anproximation based on theoreticalresults obtained by von I_ises (reference 4_ P. _79) whichare identical with the results in the present paper forbuckling under hydrostatic pressure. Figure 6 shows thatSturms theoretical results (reference 6) are in reasonableagreemen
31、t with those of the present paper and that theformula from the U S. Exoerlmen_al Y_odel Basin practi-cally coi aoies v_ith the presen_ resclts exceot at verstlow values of Z.Test _esults from references 6 and 14 are includedin figure 6. The test data are in good agreement withthe theoretical resul_s
32、 except at low values of thecurvature oarame_er Z at which the theoIetical resultsare appreciably above those obtained experimentally. Apossible explanation _,f the screpancy between thetheoretical and experimental results at low curvature issuggested by the relative importance of axial and circum-f
33、erential stress in causing buckling. The axial stressbecomes important only at low values eY the curvatureparameter Z. It is known experimentally that bucklingunder axial stresses may occur far below the theoreticalvalue of the critical stress At low values of Zcylinders under hydrostatic pressure m
34、ay therefore beexpected to buckle well below the th_oretica! criticalload just as cylinders do under axial compression.Torsion.- The problem of the determ_ination of thebuckling stresses of cylinders in torsion was solved byDonneli (reference 5) who gave an approximate solutionof the equation of equ
35、ilibrium. ._ somewhat more accuratesolution of this equation is given in reference -5 TheProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-12 NACATN No. 13_,_lessential results are shown in figure 7 taken from refer-ence 15. _t low values of Z the buck
36、ling-stress coeffi-cient k s aooroaches, the value _,.3_ apnropriate, toinfinitely long fiat plabes loaded iu shear (reference 16).At higher values of Z the curve approac_les a straightline given by= .:,TParameters anpearing in buckling curves.- The factthat the buckling of a cylinder under axial co
37、mpression,lateral pressure, hydrostatic pressure, or torsioninvolves substantially the s_ue parsm_eters is not a mereProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NACA TN No. i_I 13coincidence but is a direct consequence of the differentialequation
38、. The differential equation implies that when therequirement of an integral number of circumferential wavesis removed the six variables L, r, t, E, K, and theload may be combined into two nondimensional parameters,one (kx, ky, k s , or Cp) describing the stress condi-tion, and the other (Z) essentia
39、lly determined by thegeometry. (See appendix C.) It is also shown in appendix Cthat the buckling of a curved rectangular plate of anygiven length-width ratio may be represented in terms ofthese parameters. The critical stress of a cylinder ora curved plate of given length-width ratio may thereforebe
40、 given by a single curve relating the two parametersprovided that the number of circtunferential waves may beregarded as continuously variable. This restrictionbecomes Lmportant at very large values of Z, for whichthe curves may split into a number of curves for cylindersof different values of r/t b
41、uckling into two circum-ferential waves.Except for hydrostatic pressure, each type of loadingconsidered results in a single uniform stress in thecylinder, and the nondimensional _rameter k describingthis stress is defined as follows in analogy to theparameter used in describing the buckling of a fla
42、t plate:(_ (or T)k -_2DL2tAs the radius of the cylinder increases toward infinity(the other dimensions remaining constant), the cylinderapproaches an infinitely long flat plate of the samethickness as the cylinder, having a width b equal tothe length L of the cylinder. Accordingly, as theradius appr
43、oaches infinity, the critical-stress coef-ficient k for the cylinder approaches the value of thecorresponding stress coefficient for an infinitely longflat plate under the appropriate loading condition.The other nondimensional parameter Z is definedby the equationProvided by IHSNot for ResaleNo repr
44、oduction or networking permitted without license from IHS-,-,-14 NACA 15hirt tIf the small correction due to Poissons ratio isneglected, a direct physical significance can be assignedto Z when its magnitude is small. The maximum distancefrom a slightly curved arc of length L and radius rto its Chord
45、 can be shown to be given by the expres-sion L2/Sr, which is called the “bulge“ by some writers(see references 9 and I0). Accordingly, in the case ofa curved strip of length L in the circumferential direc-tion, L2/_rt is the bulge divided by the thickness andis thus a nondimensional measure of the d
46、eviation fromflatness of the strip. As apolied to a short cylinder,L2/$rt is the deviation from flatness of a square panelof the cylinder, each side of “_vhich is equal to the lengthof the cylinder. For cylinders having a length greaterthan a few tenths of the di_neter, the parameter Zloses this sim
47、ple physical significance and is perhapsbest regarded as a nondLmensional measure of the lengthof the cylinder. Some indication of the variety ofcyliner shapes corresponding to a fixed value of Zis given in figure 9 Boundary conditio_ns. - When problems in the stabilityof cy_n-drical shells are solv
48、ed by the use of Donnellsequation, boundary conditions on u and v cannot beimposed _lirectly because only w appears in the equa-tions. The metl_od of solution, ho_ever, may in some casesimply boundary conditions on u or v. In appendix Dit is shown that for simply supported cylinders the methodused i
49、n the present paper (a solution using one or moreterms of a Fourier series satisfying the boundary condi-tions on w term by term) Implies that at both ends ofthe cylinder the circumferential displacement v iszero, but that the cylinder edges are free to warp inthe axial direction (u # 0). For a sim_oly supp
