Free vibrational analysis of axially loaded bending-torsion coupled beams:a dynamic finite element

In this article, a dynamic finite element formulation for the free vibratio n analysis of axially loaded bending-torsion coupled beams is presented. Ba sed on the Euler-Bernoulli and St. Venant beam theories, the exact solution s of the differential equations governing the uncoupled vibrations of an ax ially loaded uniform beam are found. Then, employing these solutions as bas is functions, the analytical expressions for uncoupled bending and torsiona l dynamic shape functions are derived. Exploiting the principle of virtual work, together with the variable approximations based on the resulting shap e functions, leads to a single frequency dependent element matrix which has both mass and stiffness properties. The application of the theory is demon strated by an illustrative example of a bending-torsion coupled beam with c antilever end conditions, for which the influence of axial force on the nat ural frequencies is studied. The correctness of the theory is confirmed by the published results and numerical checks. (C) 2000 Elsevier Science Ltd. All rights reserved.