Eigenvalue sensitivity in the stability analysis of Beck's column with a concentrated mass at the free end

Abstract

The stability of a uniform cantilever column carrying a concentrated tip mass and subjected to a follower force at the free end is investigated by using the finite element method and the technique of eigenvalue sensitivity. In the present finite element formulation, the shape functions, which contain a shear deformation parameter, are developed from the static deflection of a Timoshenko beam, and the effects of both transverse shear deformation and rotatory inertia are included. The advantage of such a formulation is that if a suitable value of the shear deformation parameter is given, a specific beam model is then simulated without too much additional work to modify the original formulation. In order to obtain promptly the critical flutter load, a simple and direct method which utilizes the eigenvalue sensitivity with respect to the follower force is proposed instead of the conventional trial-and-error technique. Numerical results demonstrate the excellent agreement of the present solutions and the rapid rate of convergence of the iterative procedure. They also show that the critical flutter load is more affected by the effect of transverse shear deformation than by that of rotatory inertia.