Exact analysis of bi-directional functionally graded beams with arbitrary boundary conditions via the symplectic approach

Elasticity solutions for bi-directional functionally graded beams subjected to arbitrary lateral loads are conducted, with emphasis on the end effects. The material is considered macroscopically isotropic, with Young\'s modulus varying exponentially in both axial and thickness directions, while Poisson\'s ratio remaining constant. In order to obtain an exact analysis of stress and displacement fields, the symplectic analysis based on Hamiltonian state space approach is employed. The capability of the symplectic framework for exact analysis of bi-directional functionally graded beams has been validated by comparing numerical results with corresponding ones in open literature. Numerical results are provided to demonstrate the influences of the material gradations on localized stress distributions. Thus, the material properties of the bi-directional functionally graded beam can be tailored for the potential practical purpose by choosing suitable graded indices.