On fivefold coupled vibrations of Timoshenko thin-walled beams

The objective of the present paper is to analyze the motion of the Timoshenko thin-walled beam, with arbitrary open cross section, by means of an exact solution, and to study the influence of shear deformation over the natural frequencies. The effects of rotary inertia and warping stiffness are included in the present formulations. The five governing differential equations for coupled bending–torsional–shearing vibrations were performed using the principle of virtual displacements. The resulting coupling is referred to as fivefold coupled vibrations. In the case of a simply supported thin-walled beam, the closed-form solution for the natural frequencies of free harmonic vibrations was derived. The frequency equation, given in determinantal form, is expanded in an explicit analytical form, and then solved using the symbolic computing package Mathcad 2001 professional. In order to demonstrate the validity of this method the natural frequencies of asymmetric thin-walled beams having coupled deformation modes are evaluated and compared with analytical results analyzed by Vlasov theory.