Solution of contact problems for Gao beam and elastic foundation

This paper presents mathematical formulations and a solution for contact problems that concern the nonlinear beam published by Gao (Nonlinear elastic beam theory with application in contact problems and variational approaches, Mech Res Commun 1996; 23: 11-17) and an elastic foundation. The beam is subjected to a vertical and also axial loading. The elastic deformable foundation is considered at a distance under the beam. The contact is modeled as static, frictionless and using the normal compliance contact condition. In comparison with the usual contact problem formulations, which are based on variational inequalities, we are able to derive for our problem a nonlinear variational equation. Solution of this problem is realized by means of the so-called control variational method. The main idea of this method is to transform the given contact problem to an optimal control problem, which can provide the requested solution. Finally, some results including numerical examples are offered to illustrate the usefulness of the presented solution method.