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In this article, the transverse vibrations of bi-directional functionally graded (FG) nanobeams subjected to a longitudinal magnetic field are investigated. The nonlocal elasticity theory is exploited to consider the small-scale effect. Material properties of FG nanobeam are assumed to vary continuously along the thickness and length with arbitrary function. The equations of motion and boundary conditions are obtained using Hamilton’s principle. These equations are solved by employing the generalized differential quadrature method. The effect of different parameters such as bi-directional FG parameters, size-dependent parameter, and value of the magnetic field on vibration are investigated. It is concluded that different factors as notated, play significant roles in the vibrational response of bi-directional FG nanobeams, and by using the magnetic field, the vibration frequencies of nanostructures can be controlled. Communicated by Jie Yang.

On vibration of bi-directional functionally graded nanobeams under magnetic field