An innovative formulation for buckling analysis of nano-tapered Timoshenko beams with axially varying materials

This paper presents a novel and low-cost formula based on the first-order shear deformation theory and Eringen's nonlocal elasticity theory for the stability analysis of tapered Timoshenko nanobeams with axially varying materials. The coupled governing differential equations of the problem, involving both the transverse displacements and rotations, stem from the energy method. Based on a mathematical manipulation, the system of equilibrium equations is converted to a novel single fifth-order differential equation with variable coefficients in terms of the vertical deflection, which is solved numerically to obtain the axial buckling load. The accuracy of the proposed formulation is first verified against the available literature, with the additional advantage related to its reduced computational effort, compared to other formulations. A systematic investigation is, thus, performed to check for the influence of the non-local parameter, power-law index, tapering ratio and length-to-thickness aspect ratio on the linear buckling strength of simply supported functionally graded nano-tapered Timoshenko beams. Due to the generality of the derived formula, it can be adjusted for the optimal design of Timoshenko nanobeams with favorable axial changes in material properties as well as the geometrical features.