Research Papers

The Two-Dimensional Elasticity Solution for the Buckling of a Thick Orthotropic Ring Under External Pressure Loading

[+] Author and Article Information
Wooseok Ji

Research Fellow

Anthony M. Waas

Felix Pawlowski Collegiate Professor
e-mail: dcw@umich.edu
Department of Aerospace Engineering,
Composite Structures Laboratory,
University of Michigan,
Ann Arbor, MI 48109

1Corresponding author.

Manuscript received November 12, 2012; final manuscript received January 25, 2013; accepted manuscript posted February 14, 2013; published online August 22, 2013. Editor: Yonggang Huang.

J. Appl. Mech 81(1), 011005 (Aug 22, 2013) (12 pages) Paper No: JAM-12-1513; doi: 10.1115/1.4023682 History: Received November 12, 2012; Revised January 25, 2013; Accepted February 14, 2013

This paper is concerned with the 2D elasticity solution for the buckling of a thick orthotropic ring under external hydrostatic pressure loading. The bifurcation buckling problem is first formulated using two methods, distinguished by the manner in which the external work done by the pressure loading during the buckling transition is treated. In doing so, the correct buckling equations and associated traction boundary conditions are derived. The resulting sets of equations and associated boundary conditions are then cast in a weak form, amenable to a numerical solution using the finite element method. The necessity of using the correct pairs of energetically conjugate stress and strain measures for the buckling problem is pointed out. Errors in using the incorrect traction boundary condition and terms that influence the buckling load and that have been omitted in popular commercial codes are pointed out and their significance in influencing the buckling load is identified. Results from the present two-dimensional analysis to predict the critical pressure are compared with previous theoretical results. The formulation and results presented here can be used as the correct benchmark solution to establish the accuracy in computing the buckling load of thick orthotropic composite structures, of contemporary interest, due to the increased use of thick-walled composite shell type structures in diverse engineering applications.

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Brush, D. O., and Almroth, B. O., 1975, Buckling of Bars, Plates, and Shells. McGraw-Hill, New York.
Carrier, G. F., 1947, “On Buckling of Elastic Rings,” J. Math. Phys., 26(2), pp. 94–103.
Koiter, W. T., 1963, “Elastic Stability and Postbuckling Behavior,” Langer, R. E., ed., Proceedings of the Symposium on Nonlinear Problems, University of Wisconsin Press, pp. 257–275.
Thompson, J. M. T., 1969, “A General Theory for the Equilibrium and Stability of Discrete Conservative Systems,” Z. Angew. Math. Phys., 20(6), pp. 797–846. [CrossRef]
Budiansky, B., 1974, “Theory of Buckling and Postbuckling Behavior of Elastic Structures,” Adv. Appl. Mech., 14(2), pp. 1–65. [CrossRef]
Kardomateas, G. A., 1993, “Buckling of Thick Orthotropic Cylindrical-Shells Under External Pressure,” ASME J. Appl. Mech., 60(1), pp. 195–202. [CrossRef]
Novozhilov, V. V., 1953, Foundations of the Nonlinear Theory of Elasticity, Graylock, Rochester, NY.
Fu, L., and Waas, A. M., 1995, “Initial Postbuckling Behavior of Thick Rings Under Uniform External Hydrostatic Pressure,” ASME J. Appl. Mech., 62(2), pp. 338–345. [CrossRef]
Sewell, M. J., 1967, “On Configuration-Dependent Loading,” Arch. Ration. Mech. An., 23(5), pp. 327–351. [CrossRef]
Bodner, S. R., 1958, “On the Conservativeness of Various Distributed Force Systems,” J. Aeronaut. Sci., 25(2), pp. 132–133.
Bažant, Z. P., 1971, “A Correlation Study of Formulations of Incremental Deformation and Stability of Continuous Bodies,” ASME J. Appl. Mech., 38(4), pp. 919–928. [CrossRef]
Bažant, Z. P., and Cedolin, L., 1991, Stability of Structures: Elastic, Inelastic, Fracture and Damage Theories, Oxford University Press, New York.
Singer, J., and Babcock, C. D., 1970, “On Buckling of Rings Under Constant Directional and Centrally Directed Pressure,” ASME J. Appl. Mech., 37(1), pp. 215–218. [CrossRef]
Bower, A. F., 2010, Applied Mechanics of Solids, CRC Press, Boca Raton, FL.
Ji, W., and Waas, A. M., 2009, “2D Elastic Analysis of the Sandwich Panel Buckling Problem: Benchmark Solutions and Accurate Finite Element Formulations,” Z. Angew. Math. Phys., 61(5), pp. 897–917. [CrossRef]
Ji, W., and Waas, A. M., 2010, “Errors Caused by Non-Work-Conjugate Stress and Strain Measures and Necessary Corrections in Finite Element Programs,” ASME J. Appl. Mech., 77(4), p. 044504. [CrossRef]
Ji, W., and Waas, A. M., 2012, “Accurate Buckling Load Calculations of a Thick Orthotropic Sandwich Panel,” Compos. Sci. Tech., 72(10), pp. 1134–1139. [CrossRef]
Ji, W., Waas, A. M., and Bažant, Z. P., 2012, “On the Importance of Work-Conjugacy and Objective Stress Rates in Finite Deformation Incremental Finite Element Analysis,” ASME J. Appl. Mech. (accepted for publication).
Hibbitt, H. D., 1979, “Some Follower Forces and Load Stiffness,” Int. J. Numer. Meth. Eng., 14(6), pp. 937–941. [CrossRef]
Mang, H. A., 1980, “Symmetricability of Pressure Stiffness Matrices for Shells With Loaded Free Edges,” Int. J. Numer. Meth. Eng., 15(7), pp. 981–990. [CrossRef]
Schweizerhof, K., and Ramm, E., 1984, “Displacement Dependent Pressure Loads in Nonlinear Finite-Element Analyses,” Comput. Struct., 18(6), pp. 1099–1114. [CrossRef]
Simo, J. C., Taylor, R. L., and Wriggers, P., 1991, “A Note on Finite-Element Implementation of Pressure Boundary Loading,” Commun. Appl. Numer. M., 7(7), pp. 513–525. [CrossRef]
Zienkiewicz, O. C., and Taylor, R. L., 2000, The Finite Element Method, Butterworth-Heinemann, Oxford, Boston.
Kardomateas, G. A., 2000, “Effect of Normal Strains in Buckling of Thick Orthotropic Shells,” J. Aerosp. Eng., 13(3), pp. 85–91. [CrossRef]


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Fig. 1

Configuration of a ring under external pressure loading in xy-coordinate

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Fig. 2

Configurations during the buckling deformation with incremental variables

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Fig. 3

Stress vectors action on a infinitesimal tetrahedron in the current configuration. The reference (undeformed) configuration is also shown.

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Fig. 4

(a) Finite element modeling of a ring (b) deformed configuration of the ring under hydrostatic pressure loading

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Fig. 5

Critical pressure of a thick isotropic ring as a function of the ratio of the outer radius to the inner radius

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Fig. 6

Relative errors of predicted buckling loads from different formulations with respect to those from Formulation A

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Fig. 7

Critical pressure of an orthotropic ring with a variance of the thickness with a fixed inner radius

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Fig. 8

Relative errors of buckling loads of thick orthotropic rings obtained from different formulations with respect to those from Formulation A

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Fig. 10

Critical pressure of an orthotropic ring as a function of the modulus ratio Eθθ/Err

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Fig. 9

Critical pressure of an orthotropic ring as a function of the modulus ratio Eθθ/Grθ



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