Size dependent free vibration analysis of 2D-functionally graded curved nanobeam by meshless method

Abstract

The free vibration of two-directional functionally graded (2D-FG) thick curved nanobeam with concentrated mass is investigated for various boundary conditions. Hamilton’s principle is employed to develop the governing equations of the 2D-FG curved nanobeam using the first-order shear deformation theory (FSDT). The small scale effect is captured by the nonlocal elasticity theory of Eringen. The nanobeam is functionally graded in the transverse and circumferential directions of the curved nanobeam. The governing equations of the system are obtained as three coupled partial differential equations with non-constant coefficients. An appropriate meshless formulation is developed to discretize the governing equations based on local weak formulation and radial basis function. The presented meshless method is employed to study the free vibration of the one-directional and two-directional functional graded curved nanobeams. Simple-simple, free-free, clamped-clamped, and clamped-free edge conditions are investigated. Moreover, an analytical solution is developed for simple-simple transversely 1D-FG curved nanobeam. In the numerical results, the natural frequencies and mode shapes of shadow and deep curved nanobeam are studied and the influence of effective parameters such as nonlocal parameter, FG power indexes, opening angle, edge conditions, and concentrated mass are investigated. It is seen that shear locking effect is eliminated in present meshless formulation.