Bending, Buckling and Free Vibration of an Axially Loaded Timoshenko Beam with Transition Parameter: Direction of Axial Force

This paper aims to investigate the bending, buckling and free vibration problems of an axially loaded Timoshenko beam in a systematic manner. A recently-developed unified model is adopted, where a transition parameter characterizing the direction of the axial force during the deformation is introduced. For beams with various boundary conditions, closed-form expressions of deflections and critical buckling loadings are obtained in an analytical fashion and frequency equations are derived. It is highlighted that the inherent relations of the buckling loadings for the Timoshenko and Euler-Bernoulli beams are explicitly extracted. Numerical calculations are carried out to check the validity of the present solutions and to quantify the effects of the transition parameter. Significant effects of the transition parameter are revealed on the bending, buckling and free vibration of the axially loaded beam. The present analysis is of significance to axially-loaded beam-like mechanical system, especially for determination of the direction of axially force.