Large amplitude free vibration analysis of Timoshenko beams using a relatively simple finite element formulation

Free vibration analysis of uniform isotropic Timoshenko beams with geometric nonlinearity is investigated through a relatively simple finite element formulation, applicable to homogenous cubic nonlinear temporal equation (homogenous Duffing equation). Geometric nonlinearity is considered using von-Karman strain displacement relations. The finite element formulation begins with the assumption of the simple harmonic motion and is subsequently corrected using the harmonic balance method. Empirical formulas for the non-linear to linear radian frequency ratios, for the boundary conditions considered, are presented using the least square fit from the solutions of the same obtained for various central amplitude ratios. Numerical results using the empirical formulas compare very well with the results available from the literature for the classical boundary conditions such as the hinged–hinged, clamped–clamped and clamped–hinged beams. Numerical results for the beams with non-classical boundary conditions such as the hinged-guided and clamped-guided, hitherto not studied, are also presented.