On the Nonuniqueness of Solutions Obtained With Simplified Variational Principles

[+] Author and Article Information
H. Mang, P. Helnwein

Institute for Strength of Materials, Technical University of Vienna, Vienna, Austria

R. H. Gallagher

Aerospace and Mechanical Engineering Department, University of Arizona, Tucson, AZ 85721

J. Appl. Mech 63(3), 820-827 (Sep 01, 1996) (8 pages) doi:10.1115/1.2823368 History: Received April 11, 1995; Revised November 22, 1995; Online December 04, 2007


The attempt to satisfy subsidiary conditions in boundary value problems without additional independent unknowns in the form of Lagrange multipliers has led to the development of so-called “simplified variational principles.” They are based on using the Euler-Lagrange equations for the Lagrange multipliers to express the multipliers in terms of the original variables. It is shown that the conversion of a variational principle with subsidiary conditions to such a simplified variational principle may lead to the loss of uniqueness of the solution of a boundary value problem. A particularly simple form of the geometrically nonlinear theory of bending of beams is used as the vehicle for this proof. The development given in this paper is entirely analytical. Hence, the deficiencies of the investigated simplified variational principle are fundamental.

Copyright © 1996 by The American Society of Mechanical Engineers
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