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H.A. Mang, P. Helnwein:
"On the Nonuniqueness of Solutions of Boundary Value Problems of the Theory of Elasticity on the Basis of Simplified Variational Principles";
in: "Proceedings of the First National Congress on Computational Mechanics of the Greek Association of Computational Mechanics", University of Patras Press, 1992, Vol. I, 201 - 211.

By Simplified Varational Principles (SVP) modifications of variational principles with subsidiary conditions are meant. They are characterized by replacing Lagrange multipliers by the Euler equations providing the physical interpretation of the multipliers. In particular, principles of stationary value of the potential energy of beams subjected to bending, characterized by the relaxation of geometric boundary conditions, are considered. Geometric nonlinearity is taken into account within the framework of a theory of small displacements and moderately large rotations. In the analytical part of the investigation it is shown that depending on the degree of relaxation of geometric boundary conditions the solution, by mistake, may be nonunique. In the numerical part it is demonstrated by means of the finite element method (FEM) that irrespective of the degree of relaxation of geometric boundary conditions and of the number of finite elements, the solution, by mistake, is not unique for all values of a dimensionless load parameter.

Keywords: finite element, simplified variational principle, nonuniqueness, theory of elasticity