A Novel Approximate Analytical Method for Nonlinear Vibration Analysis of Euler–Bernoulli and Rayleigh Beams on the Nonlinear Elastic Foundation

In this study, a new analytical method called Laplace iteration method (LIM) is used to investigate nonlinear vibration behavior of Euler–Bernoulli and Rayleigh beam on nonlinear elastic foundation with three different combinations of edge conditions (simply supported, clamped–clamped and simply supported–clamped) subjected to axial load. Closed form solutions for natural frequencies, beam deflection, critical buckling and post-buckling loads are also presented. Furthermore, the effects of vibration amplitude, slenderness ratio, linear and nonlinear elastic coefficients of foundation and axial load on the nonlinear frequency and buckling load are discussed. Finally, results obtained by the proposed method are compared with those available in literature. The results showed that in comparison with other approximate analytical solutions, LIM leads to solutions that are in good agreement with respect to existing references which are valid for a wide range of vibration amplitudes meanwhile one or two iterations are enough for obtaining the results.