Free Vibration of Axially Loaded Axial-Bending Coupled Timoshenko-Ehrenfest Beams

The free vibration analysis of an axially loaded Timoshenko-Ehrenfest beam coupled in axial and bending deformations is carried out in this paper for different boundary conditions. First, the governing differential equations of motion in free vibration are developed using Hamilton’s principle and then they are solved in closed algebraic form for axial displacement, bending displacement and bending rotation. The expressions for axial force, shear force and bending moment are also obtained in explicit algebraic form. Finally, by applying the boundary conditions, the natural frequencies and mode shapes of the axially loaded axial-bending coupled Timoshenko-Ehrenfest beam are computed for an illustrative example with clamped-free (C-F), pinned-pinned (P-P) and clamped-clamped (C-C) supports at the ends. The results are discussed, and some conclusions are drawn.