Exact spectral elements for follower tension buckling by power series

The spectral dynamic stiffness method using exact solutions of the governing equations as shape functions has been popular for vibration and dynamic stability analyses of framed structures consisting of uniform members. Since non-uniform members do not generally have closed form solutions, special cases only have been considered. However, exact solutions are still possible for generally non-uniform members using power series. The paper studies the exact dynamic stability of columns with distributed axial force by power series. Both uniform and distributed, compression and tension, and conservative and non-conservative axial forces are considered. Interaction diagrams of various kinds of axial loads on the natural frequencies including different intensities of the distributed loads and degree of tangency are given. Follower tension buckling is reported for the first time. It is found that the power series outperforms the dynamic stiffness method in terms of versatility in applications and numerical stability at the very low and high ends of the frequency spectrum.