Technical Brief

Technical Brief: Knockdown Factor for the Buckling of Spherical Shells Containing Large-Amplitude Geometric Defects

[+] Author and Article Information
Francisco López Jiménez, Joel Marthelot

Department of Civil and Environmental Engineering,
Massachusetts Institute of Technology,
Cambridge, MA 02139

Anna Lee

Department of Mechanical Engineering,
Massachusetts Institute of Technology,
Cambridge, MA 02139

John W. Hutchinson

School of Engineering and Applied Sciences,
Harvard University,
Cambridge, MA 02138

Pedro M. Reis

Department of Civil and Environmental Engineering,
Massachusetts Institute of Technology,
Cambridge, MA 02139;
Department of Mechanical Engineering,
Massachusetts Institute of Technology,
Cambridge, MA 02139

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received November 28, 2016; final manuscript received December 26, 2016; published online January 24, 2017. Editor: Yonggang Huang.

J. Appl. Mech 84(3), 034501 (Jan 24, 2017) (4 pages) Paper No: JAM-16-1580; doi: 10.1115/1.4035665 History: Received November 28, 2016; Revised December 26, 2016

We explore the effect of precisely defined geometric imperfections on the buckling load of spherical shells under external pressure loading, using finite-element analysis that was previously validated through precision experiments. Our numerical simulations focus on the limit of large amplitude defects and reveal a lower bound that depends solely on the shell radius to thickness ratio and the angular width of the defect. It is shown that, in the large amplitude limit, the buckling load depends on an single geometric parameter, even for shells of moderate radius to thickness ratio. Moreover, numerical results on the knockdown factor are fitted to an empirical, albeit general, functional form that may be used as a robust design guideline for the critical buckling conditions of pressurized spherical shells.

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Elishakoff, I. , 2014, Resolution of the Twentieth Century Conundrum in Elastic Stability, World Scientific, Singapore.
von Kármán, T. , and Tsien, H.-S. , 1939, “ The Buckling of Spherical Shells by External Pressure,” J. Aeronaut. Sci., 7(2), pp. 43–50. [CrossRef]
Tsien, H.-S. , 1942, “ A Theory for the Buckling of Thin Shells,” J. Aeronaut. Sci., 9(10), pp. 373–384.
Koiter, W. T. , 1945, “ Over de Stabiliteit van het Elastisch Evenwicht,” Ph.D. thesis, TU Delft, Delft University of Technology, Delft, Netherlands.
Bijlaard, P. P. , 1960, “ Elastic Instability of a Cylindrical Shell Under Arbitrary Circumferential Variation of Axial Stress,” J. Aerosp. Sci., 27(11), pp. 854–859.
Kobayashi, S. , 1968, “ The Influence of the Boundary Conditions on the Buckling Load of Cylindrical Shells Under Axial Compression,” J. Jpn. Soc. Aeronaut. Eng., 16(170), pp. 74–82. [CrossRef]
Almroth, B. O. , 1966, “ Influence of Edge Conditions on the Stability of Axially Compressed Cylindrical Shells,” AIAA J., 4(1), pp. 134–140. [CrossRef]
Hutchinson, J. W. , and Koiter, W. T. , 1970, “ Postbuckling Theory,” ASME Appl. Mech. Rev., 23(12), pp. 1353–1366.
Budiansky, B. , and Hutchinson, J. W. , 1972, “ Buckling of Circular Cylindrical Shells Under Axial Compression,” Contributions to the Theory of Aircraft Structures, Delft University Press, Delft, The Netherlands, pp. 239–259.
Babcock, C. D. , 1983, “ Shell Stability,” ASME J. Appl. Mech., 50(4b), pp. 935–940. [CrossRef]
Seide, P. , Weingarten, V. , and Peterson, J. P. , 1968, “ Buckling of Thin-Walled Circular Cylinders,” Report No. NASA SP-8007.
Samuelson, L. Å. , and Eggwertz, S. , 1992, Shell Stability Handbook, Elsevier Applied Science, New York.
Hilburger, M. W. , Nemeth, M. P. , and Starnes, J. H. , 2006, “ Shell Buckling Design Criteria Based on Manufacturing Imperfection Signatures,” AIAA J., 44(3), pp. 654–663. [CrossRef]
Hilburger, M. W. , 2012, “ Developing the Next Generation Shell Buckling Design Factors and Technologies,” AIAA Paper No. 2012-1686.
Castro, S. G. P. , Zimmermann, R. , Arbelo, M. A. , Khakimova, R. , Hilburger, M. W. , and Degenhardt, R. , 2014, “ Geometric Imperfections and Lower-Bound Methods Used to Calculate Knock-Down Factors for Axially Compressed Composite Cylindrical Shells,” Thin-Walled Struct., 74, pp. 118–132. [CrossRef]
Lee, A. , Brun, P.-T. , Marthelot, J. , Balestra, G. , Gallaire, F. , and Reis, P. M. , 2016, “ Fabrication of Slender Elastic Shells by the Coating of Curved Surfaces,” Nat. Commun., 7, p. 11155. [CrossRef] [PubMed]
Lee, A. , López Jiménez, F. , Marthelot, J. , Hutchinson, J. W. , and Reis, P. M. , 2016, “ The Geometric Role of Precisely Engineered Imperfections on the Critical Buckling Load of Spherical Elastic Shells,” ASME J. Appl. Mech., 83(11), p. 111005. [CrossRef]
Hutchinson, J. , 2016, “ Buckling of Spherical Shells Revisited,” Proc. R. Soc. A, 472(2195), p. 20160577. [CrossRef]
Starlinger, A. , Rammerstorfer, F. G. , and Auli, W. , 1988, “ Buckling and Postbuckling Behavior of Thin Ribbed and Unribbed Spherical Shells Under External Pressure (Beulen und Nachbeulverhalten von Duennen Verrippten und Unverrippten Kugelschalen Unter Aussendruck),” Z. Angew. Math. Mech., 68(4), pp. 257–260.
Riks, E. , 1979, “ An Incremental Approach to the Solution of Snapping and Buckling Problems,” Int. J. Solids Struct., 15(7), pp. 529–551. [CrossRef]
Zoelly, R. , 1915, “ Ueber ein Knickungsproblem an der Kugelschale,” Ph.D. thesis, ETH Zürich, Zürich, Switzerland.
Koga, T. , and Hoff, N. J. , 1969, “ The Axisymmetric Buckling of Initially Imperfect Complete Spherical Shells,” Int. J. Solids Struct., 5(7), pp. 679–697. [CrossRef]
Kaplan, A. , and Fung, Y. C. , 1954, “ A Nonlinear Theory of Bending and Buckling of Thin Elastic Shallow Spherical Shells,” NACA Technical Note 3212.
Homewood, R. H. , Brine, A. C. , and Johnson, A. E., Jr ., 1961, “ Experimental Investigation of the Buckling Instability of Monocoque Shells,” Exp. Mech., 1(3), pp. 88–96. [CrossRef]
Seaman, L. , 1962, “ The Nature of Buckling in Thin Spherical Shells,” Ph.D. thesis, Massachusetts Institute of Technology, Cambridge, MA.
Krenzke, M. A. , and Kiernan, T. J. , 1963, “ Elastic Stability of Near-Perfect Shallow Spherical Shells,” AIAA J., 1(12), pp. 2855–2857. [CrossRef]


Grahic Jump Location
Fig. 2

(a) Plateau knockdown factor, 〈κd〉plateau, as a function of the geometric parameter, λ, for different values of the dimensionless radius, η. The solid line corresponds to the fit of the data to Eq. (4), using the numerical values for η = 1000. Inset: representative example of κd versus δ¯, for λ = 2.5 and η = 100. (b) Plateau knockdown factor, 〈κd〉plateau, as a function of the defect angle, β0, for different values of the dimensionless radius, η.

Grahic Jump Location
Fig. 1

(a) Contour plot of the knockdown factor, κd, for different values of the normalized radius, η, and defect angle, β0. The solid, dashed, and dashed-dotted lines correspond to instances of constant geometric parameter λ (see legend). (b) Knockdown factor, κd, versus normalized radius, η, normalized by κd (η = 1000), for different values of λ and δ¯.



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