Hierarchical theories for the free vibration analysis of functionally graded beams

This work addresses a free vibration analysis of functionally graded beams via several axiomatic refined theories. The material properties of the beam are assumed to vary continuously on the cross-section according to a power law distribution in terms of the volume fraction of the material constituents. Young’s modulus, Poisson’s ratio and density can vary along one or two dimensions all together or independently. The three-dimensional kinematic field is derived in a compact form as a generic N-order polynomial approximation. The governing differential equations and the boundary conditions are derived by variationally imposing the equilibrium via the Principle of Virtual Displacements. They are written in terms of a fundamental nucleo that does not depend upon the approximation order. A Navier-type, closed form solution is adopted. Higher-order displacements-based theories that account for non-classical effects are formulated. Classical beam models, such as Euler–Bernoulli’s and Timoshenko’s, are obtained as particular cases. Bending, torsion and axial modes are investigated. Slender as well as short beams are considered. Numerical results highlight the effect of different material distributions on natural frequencies and mode shapes and the accuracy of the proposed models.