Dynamic and buckling of functionally graded beams based on a homogenization theory

In this work, the free vibration and the stability problems of functionally graded beams are analysed via the Timoshenko theory through the Navier procedure and via an appropriated finite element (FE) approach. In particular, it is shown how the definition of homogenized/generalized displacements allows to uncouple boundary conditions, obtaining a remarkable advantage in terms of computational effort. Moreover, a unified expression capable to express the buckling load for different constrain conditions is discovered. The latter may be considered the natural extension of the Euler's one derived in the century XVIII. In order to verify the reliability of the proposed method, natural frequencies, buckling loads and static displacements of differently constrained beams are numerically evaluated.