Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X

   
 
 

 

Buckling of cylindrical shells with small curvature


Author: John Mallet-Paret
Journal: Quart. Appl. Math. 35 (1977), 383-400
MSC: Primary 73.35
DOI: https://doi.org/10.1090/qam/478917
MathSciNet review: 478917
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We consider the bifurcation buckling of a rectangular plate with an imperfection of magnitude $ \alpha $ under an applied lateral force of magnitude $ \lambda $. The analysis allows the parameters $ \left( {\lambda ,\alpha } \right)$ to vary independently in a neighborhood of some $ \left( {{\lambda _0}, 0} \right)$, and describes all buckled states of small magnitude. If the plate is represented by the domain $ \left( {0, \sqrt 2 } \right) \times \left( {0, 1} \right)$ in $ {R^2}$, then the lateral force is applied to the edges $ x = 0$, $ \sqrt 2 $, and the imperfection is a small vertical displacement of the form $ z = \left( {\alpha /2} \right){y^2}\left( {\sigma x + \tau } \right)$, where $ \sigma $ and $ \tau $ are fixed. Roughly, then, the plate has a small curvature in the $ y$-direction, of magnitude $ \alpha \left( {\sigma x + \tau } \right)$.


References [Enhancements On Off] (What's this?)

  • [1] L. Bauer and E. Reiss, Nonlinear buckling of rectangular plates, SIAM J. 13, 603-626 (1965)
  • [2] L. Bauer, E. Reiss and H. Keller, Multiple eigenvalues lead to secondary bifurcations, SIAM Review 17, 101-122 (1975) MR 0378546
  • [3] M. Berger, On von Kármán's equations and the buckling of a thin elastic plate: I. The clamped plate, Comm. Pure Appl. Math. 20, 687-719 (1967) MR 0221808
  • [4] M. Berger and P. Fife, On von Kármán's equations and the buckling of a thin elastic plate: II. Plate with general edge conditions, Comm. Pure Appl. Math. 21, 227-241 (1968) MR 0229978
  • [5] S. N. Chow, J. Hale and J. Mallet-Paret, Applications of generic bifurcation I, Arch. Rat. Mech. Anal. 59, 159-188 (1975) MR 0390852
  • [6] S. N. Chow, J. Hale and J. Mallet-Paret, Applications of generic bifurcation II, Arch. Rat. Mech. Anal. 62, 209-235 (1975) MR 0415673
  • [7] M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Func. Anal. 8, 321-340 (1971) MR 0288640
  • [8] J. P. Keener, Perturbed bifurcation theory at multiple eigenvalues, Arch. Rat. Mech. Anal. 56, 348-366 (1974) MR 0355710
  • [9] J. P. Keener, Secondary bifurcation and perturbed bifurcation theory, preprint.
  • [10] J. Keener and H. Keller, Perturbed bifurcation theory, Arch. Rat. Mech. Anal. 50, 159-175 (1973) MR 0336479
  • [11] G. H. Knightly and D. Sather, On nonuniqueness of solutions of the von Kármán equations, Arch. Rat. Mech. Anal. 36, 65-78 (1970) MR 0261835
  • [12] G. H. Knightly and D. Sather, Nonlinear buckled states of rectangular plates, Arch. Rat. Mech. Anal. 54, 356-372 (1974) MR 0349106
  • [13] G. H. Knightly, Some mathematical problems from plate and shell theory, in Proceedings of the Michigan State University Conference, Nonlinear Functional Analysis and Differential Equations, Marcel Dekker, Inc., New York, 1976 MR 0489242
  • [14] L. D. Landau and E. M. Lifshitz, Theory of elasticity, Pergamon Press, 1970 MR 884707
  • [15] S. E. List, Generic bifurcation with application to the von Kármán equations, thesis, Brown University, Providence, R. I., 1976
  • [16] B. Matkowsky and L. Putnick, Multiple buckled states of rectangular plates, Int. J. Nonlinear Mech. 9, 89-103 (1974)
  • [17] D. H. Sattinger, Group representation theory and branch points of nonlinear functional equations, SIAM J. Math. Anal. 8, 179-201 (1977) MR 0438383
  • [18] D. H. Sattinger, Group representation theory, bifurcation theory and pattern formation, preprint. MR 493378

Similar Articles

Retrieve articles in Quarterly of Applied Mathematics with MSC: 73.35

Retrieve articles in all journals with MSC: 73.35


Additional Information

DOI: https://doi.org/10.1090/qam/478917
Article copyright: © Copyright 1977 American Mathematical Society

American Mathematical Society