The Analysis of Bending of an Elastic Beam Resting on a Nonlinear Winkler Foundation with the Galerkin Method

Elastic beams resting on an elastic foundation are frequently encountered in civil, mechanical, aeronautical, and other engineering disciplines, and the analysis of static and dynamic deflections is one of the essential requirements related to various applications. The Galerkin method is a classical mathematical method for solving differential equations without a closed-form solution with a wide range of applications in engineering and scientific fields. In this study, a demonstration is presented to solve the nonlinear differential equation by transforming it into a series of nonlinear algebraic equations with the Galerkin method for asymptotic solutions in series, and the nonlinear deformation of beams resting on the nonlinear foundation is successfully solved as an example. The approximate solutions based on trigonometric functions are utilized, and the nonlinear algebraic equations are solved both numerically and iteratively. Although widely used in linear problems, it is worth reminding that the Galerkin method also provides an effective approach in dealing with increasingly complex nonlinear equations in practical applications with the aid of powerful tools for symbolic manipulation of nonlinear algebraic equations.