An Approximate Solution of the Bending of Beams on a Nonlinear Elastic Foundation with the Galerkin Method

The analysis of beam deformation on elastic foundation is very important in engineering applications, and studies of beams on linear elastic foundations are abundant and accurate. For practical problems, it is always demanded that the influence of the nonlinear effect of the foundation on the analytical methods and results must be considered. This study treats the static bending problem of an elastic beam resting on the nonlinear elastic foundation by solving the nonlinear differential equations with the Galerkin method, converting the nonlinear differential equations to a system of nonlinear algebraic equations for approximate solutions. The nonlinear equations are solved with a series expansion of the deflection satisfying the boundary conditions, and coefficients of the series are obtained with usual techniques including the iterative method. The accuracy of the approximate solution with the Galerkin method is verified through examples from earlier studies. The procedure and results show that the Galerkin method is effective in solving static nonlinear differential equations in addition to the nonlinear vibrations with the extended Galerkin method in earlier studies.