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International Journal of Mathematics and Mathematical Sciences
Volume 16, Issue 1, Pages 193-198

On the existence of equilibrium states of an elastic beam on a nonlinear foundation

1Department of Mathematics, University of Wisconsin-Eau Claire, Eau Claire 54702, WI, USA
2Department of Mathematics, Michigan State University, East Lansing 48824, MI, USA

Received 23 September 1991; Revised 6 June 1992

Copyright © 1993 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


This paper concerns the existence and uniqueness of equilibrium states of a beam-column with hinged ends which is acted upon by axial compression and lateral forces and is in contact with a semi-infinite medium acting as a foundation. The problem is formulated as a fourth-order nonlinear boundary value problem in which the source of the nonlinearity comes from the lateral constraint (the foundation). Treating the equation of equilibrium as a nonlinear eigenvalue problem we prove the existence of a pair of eigenvalue/eigenfunction for each arbitrary prescribed energy level. Treating the equilibrium equation as a nonlinear boundary value problem we prove the existence and uniqueness of solution for a certain range of the acting axial compression force.