Effect of Bracing on Linear Free Vibration Characteristics of Thin-Walled Beams with Open Cross Section

The paper presents an analysis of the coupled vibration of beams with arbitrary thin-walled open cross section, braced with identical transversal header beams uniformly distributed along their length. The explicit form of analytic solution is derived by directly solving the governing differential equations of motion. The development is based on Vlasov theory which includes the effect of flexural-torsion coupling, the constrained torsion warping, and rotary inertia. The governing differential equations for coupled bending-torsional vibrations are performed using the principle of virtual displacements. In the case of simply supported beam, exact explicit expressions are derived to predict the natural frequencies and the corresponding mode shapes. The frequency equation, given in determinantal form, is expanded in an explicit analytical form, and then solved using the symbolic computing package Mathcad. The expressions are concise and very simple and as such convenient to be used by a practicing engineer who does not need to go into detail of thin-walled beam theory. Also, the use of explicit expressions gives significant savings in computing time compared with the alternative numerical methods [finite-element method (FEM), finite strip method, differential transform method, etc.]. To demonstrate the validity of this method the natural frequencies of braced thin-walled beams, having coupled deformation modes, are evaluated and compared with FEM.