Mathematical Problems in Engineering
Volume 2008 (2008), Article ID 786520, 15 pages
Research Article

Constrained and Unconstrained Optimization Formulations for Structural Elements in Unilateral Contact with an Elastic Foundation

Ricardo A. M. Silveira,1 Wellington L. A. Pereira,1 and Paulo B. Gonçalves2

1Department of Civil Engineering, School of Mines, Federal University of Ouro Preto, Campus Universitário, Morro do Cruzeiro, Ouro Preto, MG 35400-000, Brazil
2Department of Civil Engineering, Catholic University, PUC-Rio, Rua Marquês de São Vicente 225, Gávea, Rio de Janeiro, RJ 22451-900, Brazil

Received 9 February 2007; Accepted 18 October 2007

Academic Editor: J. Richard Barber

Copyright © 2008 Ricardo A. M. Silveira et al. 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.


In this work, two numerical methodologies are proposed for the solution of unilateral contact problems between a structural member (beam or arch) and an elastic foundation. In the first approach, the finite element method is used to discretize the structure and elastic foundation and the contact problem is formulated as a constrained optimization problem. Only the original variables of the problem are used, subjected to inequality constraints, and the relevant equations are written as a linear complementary problem (LCP). The second approach is based on the Ritz method, where the coordinates defining the limits of the contact regions are considered as additional variables of the problem. The contact problem here is treated as an unconstrained optimum design problem. These proposed methodologies are then tested and compared using results from specific problems involving structures under unilateral contact constraints.