Approximate analytical solution for Bernoulli-Euler beams under different boundary conditions with non-linear Winkler type foundation

In this study, a powerful analytical method, known as Homotopy Analysis Method (HAM), is used to obtain an analytical solution to nonlinear ordinary deferential equations arising for Bernoulli-Euler beams with a non-linear foundation of the Winkler type. A comparison between the HAM solution and a solution obtained by a numerical method is made to show the accuracy of the method. It is shown that the present solution is valid for the whole domain of the solution and also for high nonlinear terms, where other methods such as the perturbation method fail to converge. The results clearly indicate that the convergence region can be controlled and adjusted by HAM. Finally, after validating the results, the effect of constant parameters on the deflection and slope for different boundary conditions is presented.