Dynamic Analysis of a Timoshenko Beam Using a Finite Element Approach

摘要

The finite element method was used in this investigation in order to determine the static and dynamic behavior of non-uniform Timoshenko beams subjected to various boundary conditions. When using this approach, the general displacement functions are based on the cubic polynomial expansion of the three principal quantities as degrees of freedom: transverse displacement, rotation of the cross section and shear deformation. The stiffness and mass matrices, however, are derived from expressions of the strain and kinetic energy which are based on the assumptions of curved, slightly curved and straight beams. In addition, a formula was obtained from the general equation for a slightly curved beam and as a result, frequency equations for a straight beam were obtained for ten common types, taking into account all the forced and natural boundary conditions. Good agreement was achieved by comparing the results of these models with exact solutions, as well as with other numerical methods for straight and tapered beams. In order to test the rate of convergence, the natural frequencies of typical cases were calculated as a function of the number of elements. Results showed that the model converged rapidly and required only a small number of elements to achieve good results. Finally, the present analysis demonstrated the reliability of the model chosen by including the transverse shear and rotary inertia effects at the higher modes.