Post-buckling analysis of a nonlinear beam with axial functionally graded material

The postbuckling analysis of a modified nonlinear beam composed of axial functionally graded material (FGM) is investigated by a canonical dual finite element method (CD-FEM). The governing equation of the axial FGM nonlinear beam is derived through a variational method. The CD-FEM is adopted to find the nonconvex postbuckling configurations of the beam according to Gao's triality theory. Using duality transition, the original potential energy functional becomes a functional of deformation and dual stress fields. By variation of the mixed complementary energy, the coupling equations are derived to find deformation and dual stress fields. In FEM, matrices of a beam element depend on the gradient of material property (elastic modulus). To obtain general forms of matrices of a beam element, the graded elastic modulus is approximated by piecewise linear functions with respect to axial position. Numerical examples are presented to show the effects of graded elasticity on the postbuckling configurations of the beam