Buckling of nonuniform and axially functionally graded nonlocal Timoshenko nanobeams on Winkler-Pasternak foundation

Buckling of axially functionally graded and nonuniform Timoshenko beams is examined based on the nonlocal Timoshenko beam theory. Small scale effect on the buckling is taken into account via the length scale parameter often referred as the nonlocal parameter. The material properties vary in the axial direction and the nanobeam is modelled as a nonuniform Timoshenko beam which rests on a Winkler-Pasternak foundation. Rayleigh quotients for the buckling load are derived for Euler-Bernoulli and Timoshenko beams. Chebyshev polynomials based on the Rayleigh-Ritz method is used to obtain the numerical solution of the problem for a combination of clamped, simply supported and free boundary conditions. Accuracy of the method is verified by comparing the buckling loads obtained by the present approach with those available in the literature. Numerical results for the problem are given in the form of contour plots to study the effect of nonlocal parameter, cross-sectional non-uniformity, axial grading and the boundary conditions on the buckling loads.