Size-dependent buckling analysis of non-prismatic Timoshenko nanobeams made of FGMs rested on Winkler foundation

Abstract

In this article, the buckling behavior of tapered Timoshenko nanobeams made of axially functionally graded (AFG) materials resting on Winkler type elastic foundation is perused. It is supposed that material properties of the AFG nanobeam vary continuously along the beam's length according to the power-law distribution. The nonlocal elasticity theory of Eringen is employed to contemplate the small size effects. Based on the first-order shear deformation theory, the system of nonlocal equilibrium equations in terms of vertical and rotation displacements are derived using the principle of total potential energy. To acquire the nonlocal buckling loads, the differential quadrature method is used in the solution of the resulting coupled differential equations. Eventually, an exhaustive numerical example is carried out for simply supported end conditions to investigate the influences of significant parameters such as power-law index, tapering ratio, Winkler parameter, aspect ratio, and nonlocal parameter on the buckling capacity of AFG Timoshenko nanobeams with varying cross-section supported by uniform elastic foundation.

An analytical solution for stability analysis of unrestrained tapered thin-walled FML profile

“…Since the flanges and/or web are variable, all stiffness quantities of the composite beam are functions of the xcoordinate. In this regard, the solution of the resulting fourth-order differential equation in terms of the twist angle ( 25) is not straightforward and only analytical or numerical techniques such as Galerkin's or Rayleigh-Ritz methods [37][38][39], the finite difference method [40,41], the differential quadrature method [26,36,42,43], and the power series method [44][45][46] are feasible. In the present work, Galerkin's method as a highly accurate analytical methodology is used to solve Eq.…”