Abstract: The dynamic partial differential equations of the stepped column structure were established for the top of the overrun support column subjected to axial loads, which were transformed into second-order ordinary differential Mathieu-type parametric equations, thus numerically calculating dynamic stability of the cotumn by Bolotin method. The safety margin equation of column structure is established and the boundaries of the dynamically unstable region were successfully solved. The effects of different length ratio of live column to cylinder, different elastic modulus of column and different diameter ratio of live column to cylinder on the dynamic stability of column were analyzed. The results show that the stability of the column decreases with the increase of the length ratio of live column to cylinder. The stability of the column is improved with the increase of the elastic modulus of the column and the diameter ratio of the live column to the cylinder. On this basis, the non-probabilistic reliability analysis of the column structure is carried out, and the non-probabilistic reliability of the column structure is obtained, that is, the non-probabilistic reliability of the column structure is opposite to the increase or decrease of the ratio of the length of the column and the cylinder, and same to the increase or decrease of the ratio of the column elastic modulus and the ratio of the live column and the cylinder diameter.