1、BUCKLING OF CYLINDRICAL SHELL END CLOSURES BY INTERNAL PRESSURE by G. A. Thrston und A. A. Holston, Jr. Prepared by MARTIN-MARIETTA CORPORATION Denver, Colo. for Langley Research Center NATIONAL AERONAUTICS AND SPACE ADMINISTRATION . WASHINGTON, D. C. . JULY 1966 Provided by IHSNot for ResaleNo repr
2、oduction or networking permitted without license from IHS-,-,-TECH LIBRARY KAFB, NM I 111111 #Ill lllll lllll II11 lllll II1 Ill 1111 0099533 NASA CR-540 BUCKLING OF CYLINDRICAL SHELL END CLOSURES BY INTERNAL PRESSURE By G. A. Thurston and A. A. Holston, Jr. Distribution of this report is provided i
3、n the interest of information exchange. Responsibility for the contents resides in the author or organization that prepared it. Prepared under Contract No. NAS l-4782 by MARTIN-MARIETTA CORPORATION Denver, Colo. 1 / ;,/ I :,. :.I,; /:, : I ! p-t _ ,” (4 /q,z , , 6 for Langley Research Center NATIONA
4、L AERONAUTICS AND SPACE ADMINISTRATION For sole by the Clearinghouse for Federal Scientific and Technical information Springfield, Virginia 22151 - Price $2.00 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-Provided by IHSNot for ResaleNo reproducti
5、on or networking permitted without license from IHS-,-,-CONTENTS CONTENTS . iii SUMMARY . 1 INTRODUCTION . 1 SYMBOLS . 3 THEORY 4 BOUNDARYCONDITIONS . 6 NUMERICALRESULTS 7 CONCLUSIONS . 10 APPENDIX - A;)D;T;oNAL TERMS . 11 REFERENCES 14 TABLES 1. NUMERICAL RESULTS FOR ELLIPTICAL CLOSURES SUBJECTED T
6、O INTERNAL PRESSURE . . . . . . . . . . . . . . . . . 2. NUMERICAL RESULTS FOR TORISPHERICAL CLOSURES SUBJECTED TO INTERNAL PRESSURE . . . . . . . . . . . . . . . . . FIGURES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. NOTATION FOR AXISYMMETRIC BOUNDARY CONDITIONS FOR TORISPHERICAL CLOSURE . . . . . . . . . . .
7、. . . . . STRESS DISTRIBUTION FOR TORISPHERICAL CLOSURE SUB- JECTED To INTERNAL PRESSURE (CYLINDER BOUNDARY CON- DITIONS), NOTATION FOR TORISPHERICAL CLOSURE . . . . . . . . . . NOTATION FOR ELLIPTICAL CLOSURE . . . . . . . . . . . STRESS DISTRIBUTION FOR TYPICAL ELLIPTICAL CLOSURE SUBJECTED 0 INTER
8、NAL PRESSURE (CLAMPED BOUNDARY CONDITIONS). . . . . . . . . COMPARISON WITHMECALL(REF. FOR TRISPHERICA;. * * CLOSURES SUBJECTED TO INTERNAL PRESSURE . . . . . . . COMPARISON.WITH ADACHI AND BENICEK (REF. 4) FOR TORISPHERICAL CLOSURES SUBJECTED TO INTERNAL PRESSURE. PRESENT RESULTS FOR TORISPHERICAL
9、CLOSURES SUBJECTED TO INTERNAL PRESSURE . . . . . . . . . . . . . . . . . PRESENT RESULTS FOR ELLIPTICAL CLOSURES SUBJECTED TO INTERNAL PRESSURE . . . . . . . . . . . . . . . . . . REGION OF STABILITY FOR ELLIPTICAL CLOSURES SUBJECTED TO INTERNAL PRESSURE . . . . . . . . . . . . . . . . . PAGE 15 16
10、 17 18 19 19 20 21 22 23 24 25 iii Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-BUCKLING OF CYLINDRICAL SHELL END CLOSURES BY INTERNAL PRESSURE By G.
11、 A. Thurston and A. A. Holston, Jr. Martin-Marietta Corporation SUMWRY A theoretical study was conducted on buckling of shallow end closures of cylindrical shells under internal pressure. The most important result from the analysis is that elliptical domes can be designed that do not buckle under in
12、ternalpressure al- though they are shallower than the fi:l elliptical domes in common use in aerospace vehicles. This indicates that decreasing the rise of elliptical domes could result in a weight savings because of shortening the structure between tanks and stages of missiles. Finite-deflection th
13、eory was used to compute the prebuckling stress distribution. This theory predicts that the rate of change of compressive circumferential stresses as a function of pressure decreases as the internal pressure increases. This nonlinear re- lationship between hoop stress and pressure results in compute
14、d bifurcation pressures for asymmetric wrinkling that are higher than buckling pressures from linear theory and predicts that some clos- ures do not buckle under any pressure. INTRODUCTION Torispherical and elliptical shells are commonly used as end closures for cylindrical pressure vessels. The “sq
15、uare root of two to one“ elliptical dome has become virtually sacrosanct for propellant tanks in certain aerospace vehicles. This ratio of cylinder radius to dome rise is derived from membrane theory by postulating that no circumferential compressive stress should appear in the dome due to internal
16、pressure. With no compressive stresses, there can be no problem of designing for buckling due to internal pressure. Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-This approach would appear to be conservative for two reasons. First, the cylinder and
17、 any skirts will support the edge of the dome so that membrane theory will not apply. This support re- sists the inward radial displacements that must accompany compres- sion and lowers the level of compressive stresses from that pre- dicted from membrane theory. Second, the dome will have enough st
18、iffness to prevent buckling if the compressive stresses are not too high. This report contains theoretical results that provide some insight into. pressure levels that can be expected to produce wrinkling in shallow domes. The torispherical dome consists of a spherical cap joined to a toroidal.segme
19、nt, joined in turn to the cylindrical pressure vessel. Galletly (ref. 1) warned that membrane theory is not adequate for predicting stresses in the toroidal portion of tori- spherical heads and proceeded to compute stresses based on linear bending theory. He noted the possibility of elastic buckling
20、 due to the compressive hoop stresses that can be developed. Mescall (ref. 2) calculated pressures that would produce asymmetric buck- ling modes in torispherical shells. He used linear bending theory to compute the prebuckling stress state based on an asymptotic solution by Clark (ref. 3) and a Ray
21、leigh-Ritz procedure to com- pute bifurcation pressures from a Donnell-type buckling theory. The present analysis goes a step further by computing the axisymmetric stresses from nonlinear finite-deflection theory and the buckling pressures from an improved theory. The results are compared with Mesca
22、lls data and with experimental buckling pres- sures reported by Adachi and Benicek (ref. 4). The compressive circumferential stresses predicted by the nonlinear theory are lower than those from linear theory, and the agreement between the computed and experimental buckling pressures is good. The ell
23、iptical end closure has apparently not been studied as extensively as the torispherical shell. The present study indicates that elliptical domes can be shallower than K, N; q cr Q: R C r 0 R t 0 U radius of spherical cap extensional stiffness, W(1 - v2) influence coefficients element of arc length o
24、f shell shell thickness axisymmetric horizontal stress resultant curvature of meridian of undeformed shell rise of torispherical closure minor axis of elliptical closure axisymmetric meridional stress couple number of circumferential waves of buckling mode axisymmetric stress resultants in meridiona
25、l and circumferential directions, respectively internal pressure critical internal pressure for asymmetrical buck- ling axisymmetric transverse stress resultant cylinder radius horizontal radius to generic point of the unde- formed shell toroidal radius axisymmetric radial deflection 3 Provided by I
26、HSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-axisymmetric rotation of shell normal Poissons ratio angle from centerline to normal of shell opening angle of spherical cap portion of tori- spherical closure differentiation with respect to the independent variabl
27、e x THEORY The differential equations for shells of revolution used in the analysis are derived in reference 5, and that derivation will not be repeated here. The main feature of the theory is computing the prebuckled state of stress from nonlinear finite-deflection equations rather than assuming th
28、at membrane theory is adequate. At certain critical loads, bifurcation occurs with asymmetric equilibrium positions existing infinitesimally near the axisymmetric equilib- rium state. The bifurcation points are determined from linear- ized theory. The same approach has been used recently by several
29、authors for special cases of shallow spherical caps (refs. 6 thru 8), cylinders under axial compression (refs. 9 and lo), and cones under external pressure (ref. 11). The analysis in reference 5 uses Reissners finite-deflection equations for.shells of revolution under axisymmetric loads as the basis
30、 for the prebuckled stress distribution. The linearized equations for computing the critical loads are derived from the nonlinear str.ain-displacement relations with lines of curvature coordinates listed by Sanders (ref. 12). Weinitschke (ref. 7) has an extensive discussion of the per- tinent equati
31、ons for the special case of the shallow spherical shell. The differential equations are written in operator nota- tion, which makes it easy to see the differences between the present approach and classical solution. Reference 5 contains the equations for a general shell of revolution. Provided by IH
32、SNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-The prebuckled solution requires solving a fourth order set of nonlinear ordinary differential equations. The bifurcation points are defined by nontrivial solutions of a set of homogen- eous eighth order linear parti
33、al differential equations. In references 6 and 11, the solutions of the homogeneous equations were found by successive approximations. When this iterative procedure was applied to the case of spherical caps under point load (ref. 5), it failed to converge. Convergence problems also appeared in the p
34、resent study of domes under in- ternal pressure. The exact cause of the lack of convergence was not ascertained, but both problems were characterized by tensile stresses existing along with the compressive stresses that cause buckling. The computer program for the numerical solution of the dif- fere
35、ntial equations was rewritten to solve these problems direct- ly by relating critical loads to the vanishing of a determinant. The change in the numerical solution involved shifting vectors from the right-hand side of a matrix equation and adding them to other vectors appearing on the left side to m
36、ake a set of linear homogeneous algebraic equations approximating the homogeneous differential equations. The additional terms are listed in the appendix. The com- puter program is written so that three separate shell theories can be checked by the solution, depending on terms retained in the vector
37、s. If only the terms in the appendix that are under- lined twice are added, the result is a Donnell-type theory where the expressions for changes of curvature during buckling contain only w, the component of displacement normal to the undeformed shell. If the terms in the appendix with a single unde
38、rline are also added, the result will be analogous to Flugges theory for cylinders (ref. 13) where the prebuckled stress resultants enter all three equilibrium equations. Finally, if the terms. in the appendix containing p 0 that are not underlined are added X along with the rest, the complete set o
39、f terms consistent with linearizing Sanders equations (ref. 12) are retained. The terms in B 0 X are difficult to explain on physical grounds, but they reflect the change in strain-displacement relations of the shell due to the prebuckling rotation p“ of the shell normal. X 5 Provided by IHSNot for
40、ResaleNo reproduction or networking permitted without license from IHS-,-,-BOUNDARY CONDITIONS The critical loads are dependent on the boundary conditions assumed for the dome in the prebuckled state and during buckling. The boundary conditions for the axisymnetric prebuckled solution were varied. S
41、ome of the runs assumed that the domes were at- tached to a cylinder of the same material and thickness as the dome. These solutions are denoted in the results as “cylinder boundary conditions.“ The limiting case of a stiffer support was obtained by using “clamped“ conditions with no radial deflec-
42、tion or edge rotation (fig. 1). The other limiting case of flexibility was obtained by using “membrane“ conditions with no edge transverse shear or moment, Q; = ME = 0 In the present study, the boundary conditions for asymmetric buckling modes were restricted to clamped edges; all three compo- nents
43、 of displacement for the asymmetric solution plus the edge rotation vanish at the boundaries. Since the prebuckled theory is nonlinear, the effect of pres- sure on the cylinder must be considered in computing its stiff- nesses. The same is true in matching the spherical part of the torispherical dom
44、e to the toroidal knuckle. Good approximations for the effect of internal pressure on the edge stiffness of cylinders and spheres have been published by Grossman (ref. 14) and Cline (ref. 15). Cline (ref. 15) computes influence coeffi- cients for a spherical cap from edge stresses added to the prima
45、ry stress state. His equations at the edge of the spherical cap can be written 0 U -cl5 - 2 cos PO + Cl2 M; + v) sin (p 0 p; = c12(H0 - F cos “) + C22 M; The deformations and stress resultants must be continuous at the edge juncture of the spherical cap and the toroidal knuckle. Therefore, the above
46、 equations become two boundary conditions for the numerical solution of Reissners equation for the toroidal 6 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-shell. The program accepts them by listing as input the coeffi- 0 cients of u , B 0 X Ho, 9,
47、 and the constant term from the equations written in the form 0 U 95 - Cl1 Ho - Cl2 ME = 2 k (1 - v) sin Go - Cl1 cos 0 1 BE - Cl2 Ho - C22 ME = - F Cl2 cos PO The influence coefficients cl1 cl2 and c22 are a func- tion of the pressure q and decrease in magnitude as the shell gets stiffer with incre
48、asing pressure. The influence coefficients are computed from Clines solution (ref. 15) in an IBM 1620 pro- gram that punches the proper input cards for the main IBM 7090 program. A similar set of influence coefficients is computed as a function of pressure for the cylinder (ref. 14) and these appear in the other two boundary conditions for the toroidal shell. NUMERICAL RESULTS The numerical buckling loads obtained in this study for tori- spherical and elliptical closures subjected to internal pressure are summarized in tables 1 and 2. These results are also plotted in
