Exploration of the high order theory for functionally graded beams based on Legendre's polynomial expansion

A high order theory for functionally graded (FG) beams based on expansion of the two dimensional (2-D) equations of elasticity for functionally graded materials (FGMs) into Legendre's polynomials series has been developed. The 2-D equations of elasticity have been expanded into Legendre's polynomials series in terms of a thickness coordinate. In the same way functions that describe functionally graded relations also has been expanded. Thereby all equations of elasticity including Hook's law have been transformed to corresponding equations for coefficients of Legendre's polynomials expansion. Then system of differential equations in term of displacements and boundary conditions for the coefficients of Legendre's polynomials expansion coefficients has been obtained. Cases of the first and second approximations have been considered in more detail. For the obtained boundary-value problems solution, a finite element method (FEM) has been used and numerical calculations have been done with MATHEMATICA, MATLAB and COMSOL Multiphysics software. Numerical results are presented and discussed.