Vibration analysis of Timoshenko beam-columns on two-parameter elastic foundations

This paper presents a finite element technique for determining the vibration characteristics of a uniform Timoshenko beam-column supported on a two-parameter elastic foundation. The beam-column is discretized into a number of simple elements with four degrees of freedom each. The effects of axial force, foundation stiffness parameters, transverse shear deformation and rotatory inertia are incorporated into a finite element model. The matrix equation governing the free vibrations of the beam-column on the elastic foundation is derived from Hamilton's principle. The numerical results for the natural frequencies and the associated mode shapes of the classical Euler-Bernoulli and Timoshenko beam-columns on the elastic foundation are presented and compared with the exact or available solutions, wherever possible. It is shown that the present technique provides a unified approach for the vibration analysis of beam-columns with any end conditions, resting on the elastic foundation.