Analytical Solutions for Functionally Graded Beams under Arbitrary Distributed Loads via the Symplectic Approach

A novel symplectic approach is employed in the analysis of homogenous and functionally graded beams subjected to arbitrary tractions on the lateral surfaces. Two models of functionally graded beams are heterogeneous in the sense that the material properties are exponential functions of the length and thickness, respectively. Within the symplectic framework, the method of separation of variables alone with the eigenfunction expansion technique is adopted to obtain the exact analysis of displacement and stress fields. The complete solutions include homogenous solutions with coefficients to be determined by two end boundary conditions and particular solutions satisfying the lateral boundary conditions. Two examples are presented for functionally graded beams to demonstrate the effects of material inhomogeneity. The efficiency and accuracy of the symplectic analysis are shown through numerical results.