Nonlinear large deflection problems of beams with gradient: A biparametric perturbation method

For beams with gradient, due to the combined influences introduced by loads and gradient, the first derivative item in Euler-Bernoulli equation can not be neglected thus making the solution of the problem be a nonlinear large deflection one. In this paper, we use a new perturbation method with two small parameters, one describes the loads effect and another describes the geometrical nature of the problem, to solve the nonlinear large deflection problem of beams with gradient under the two different boundary conditions. We derive the first and second order approximate analytical solution of the deflection, the rotation and the arc length of the beam, as well as the internal forces of the beam at the end. The results indicate that the choice of two independent parameters may describe comprehensively the nonlinear effects caused by loads and gradient, which enables the approximate solution to be precise enough to be used for the analysis of large-deflection beam with gradient. (C) 2013 Elsevier Inc. All rights reserved.