Lower bound estimate for buckling in axially graded cantilever rods

Functionally graded beams, bars and rods have been gaining a relevant consideration in engineering practice and research, taking into account variations of the material properties either in the transverse or in the longitudinal direction. Yet existing literature dealing with analytical study of structural instability for an arbitrary material longitudinal variation is still limited. In this paper, a variational approach to the buckling in axially graded cantilevers is developed within Euler-Bernoulli beam theory. Considering the large deflection static behavior and interpreting the first variation of the corresponding Action integral as a weak form of the associated Euler-Lagrange equation, the problem of analytically finding a lower bound estimate for the buckling load is investigated and solved for arbitrary variations of mechanical properties within an imposed condition on the maximal deflection of the free end. In particular, two examples of widely used material gradient forms have been considered and their lower bound buckling forces have been estimated in a closed form, compared with numerical results from literature developed within the linearized version of governing equations and validated using nonlinear finite element forecasts, showing promising results in terms of buckling prediction.