均布荷载作用下功能梯度悬臂梁弯曲问题的解析解

采用弹性力学半逆解法,假设所有材料常数沿梁厚度方向按同一函数规律变化,求得了功能梯度悬臂梁在均布载荷作用下的解析解.该解退化到各向同性均匀弹性情况时与已有的理论解相一致.对弹性模量按指数函数梯度变化的算例进行了分析.所得到的解对任意梯度函数均成立,可作为数值解以及简化理论的检验依据.

A GENERAL SOLUTION OF A CLAMPED FUNCTIONALLY GRADED CANTILEVER-BEAM UNDER UNIFORM LOADING

This paper studies the bending problem of a clamped functionally graded cantilever-beam under uniform loading.The problem was firstly treated as a plane stress case and the basic equations of the orthotropic elastic materials were given.Assuming that the mechanical properties of the material have the same dependence on the thickness-coordinate,we obtained a general solution of the beam subjected to a uniform loading at the upper surface by using a semi-inverse method.Degenerate results for the special case of an isotropic homogeneous elastic beam are coincided well with the existing analytical solutions.Some numerical examples are also given by assuming an exponential gradient function.This solution is valid for arbitrary gradient functions,so it could serve as a benchmark result to assess other approximate methodologies or as a basis for establishing simplified functionally grade beam theories.