Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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



On the nonunique equilibrium states of a shallow arch subjected to a uniform lateral load

Authors: Arnold D. Kerr and Lotfi El-Bayoumy
Journal: Quart. Appl. Math. 28 (1970), 399-409
DOI: https://doi.org/10.1090/qam/99783
MathSciNet review: QAM99783
Full-text PDF

Abstract | References | Additional Information

Abstract: Using the exact solutiom of a shallow elastic arch clamped at both ends and subjected to a uniform lateral load, it is shown that, in addition to the equilibrium states usually discussed in the literature, also other equilibrium states, which correspond to higher modes of deformation, do exist. It is shown that there exist only a finite number of equilibrium branches which correspond to these higher modes, the number depending upon the shallowness of the arch. A number of graphs are presented which demonstrate these findings. The paper concludes with the proof that all states of equilibrium which correspond to the branches of higher order are unstable.

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

  • [1] S. Timoshenko, Buckling of flat curved bars and slightly curved panels, J. Appl. Mech. 2, A17-A20 (1935)
  • [2] K. O. Friedrichs and J. J. Stoker, Seminar on nonlinear elasticity, Section IX, New York University, 1947/48
  • [3] Y. C. Fung and A. Kaplan, Buckling of low arches and curved beams of small curvature, Technical Note 2840, NACA (Currently NASA) Washington, D. C., 1952.
  • [4] M. S. Kornishin and Kh. M. Mushtari, The instability of an infinitely long shallow cylindrical strip subjected to a normal uniform pressure, Izv. Kazansk. Fil. Akad. Nauk USSR Ser. Fiz.-Mat. Tehn. Nauk 7, 36-50 (1955) (Russian) See also: Nonlinear theory of thin elastic shells, Chapter XI, by Kh. M. Mushtari and K. Z. Galimov, Tatknigoizdat, Kazan, 1957; English transl., NASA-TT-62, 1961.
  • [5] A. S. Vol′mir, Flexible plates and shells, Gosudarstv. Izdat. Tehn.-Teor. Lit., Moscow, 1956 (Russian). MR 0090250
  • [6] A. Gelsvik and S. Bodner, Energy criterion and snap buckling of arches, Proc. ASCE, Journal of the Engineering Mechanics Division, EM5, Oct. 1962, pp. 87-134.
  • [7] L. H. Schreyer and E. F Masur, Buckling of shallow arches, Proc. ASCE, Journal of the Engineering Mechanics Division, EM4, August 1966, pp. 1-19
  • [8] A. D. Kerr and M. T. Soifer, The linearization of the prebuckling state and its effect on the determined buckling loads, J. Appl. Mech. Dec. 1969, 775-783
  • [9] Herbert B. Keller and Antoinette W. Wolfe, On the nonunique equilibrium states and buckling mechanism of spherical shells, J. Soc. Indust. Appl. Math. 13 (1965), 674–705. MR 0183174
  • [10] A. D. Kerr and L. El-Bayoumy, Equilibrium and stability of a shallow arch subjected to a uniform lateral load, New York University Report No. AA-68-44, November 1968
  • [11] W. T. Koiter, The energy criterion of stability for continuous elastic bodies. I, II, Nederl. Akad. Wetensch. Proc. Ser. B 68 (1965), 178–189, 190–202. MR 0198772

Additional Information

DOI: https://doi.org/10.1090/qam/99783
Article copyright: © Copyright 1970 American Mathematical Society

American Mathematical Society