This article presents formulation of third-order non-local beam theories based on Eringen's non-local continuum theory and the Reddy and Leung higher-order beam models for bending, buckling, and vibration of nanobeams. By applying the variational principle, an asymptotic governing differential equation of infinite-order strain gradients is derived and subsequently the governing equation of motion of the nanobeams is obtained. For practical applications, the Navier solutions of the asymptotic non-local model are presented and discussed. Bending, buckling, and vibration of nanobeams with simply supported boundary conditions are investigated. Numerical examples are presented to illustrate the corresponding effects of non-local scale parameter, rotary inertia and transverse shear deformation on deflections, buckling loads, and natural frequencies of nanobeams.
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