Static and dynamic stability of an elastically restrained Beck column with an attached end mass

The static and dynamic stability of an elastically restrained Beck column with an attached end mass subjected simultaneously to gravity and follower axial forces is presented. The analytical solution takes into account the simultaneous effects of: (1) translational and rotational elastic restraints at the base support; (2) the uniform mass per unit length of the column (including any additional uniformly distributed mass) and rotary inertia of the column; (3) the rotary and translational inertias of the attached end mass; (4) the distance from the centroid of the attached end mass to the column free end; and (5) the shear deformation. The analytical eigenvalue solution can be used to study these effects on the static and dynamic stability of the Beck column. The analytical method and eigenvalue equation capture the static buckling (or divergence) as well as the dynamic (flutter) instability of cantilever columns elastically restrained at the base and subjected to any combinations of gravity and follower compressive axial forces applied at the free end. A parametric study is carried out on the effects of the translational and rotational inertias of the end mass, the distance from the centroid of the attached end mass to the column free end, and the rotational restraint on the static and dynamic stability of a perfectly clamped Beck column. Four comprehensive examples are presented to show the simplicity and effectiveness of the analytical method, and the obtained results compared with those obtained analytically and experimentally by others researchers.