The dynamic stability of an elastic prismatic slender column with semirigid connections at both ends of identical stiffness and with sidesway between the two ends totally inhibited, subject to parametric axial loads including the combined effects of rotary inertia and external damping is investigated in a classical manner. Closed-form expressions that can be used to predict the dynamic instability regions of slender columns are developed by making use of Floquet’s theory. The proposed solution is capable of capturing the phenomena of stability of columns under periodic axial loads using a single column element. The proposed method and corresponding equations can be used to investigate the effects of damping, rotary inertia and semirigid connections on the stability analysis of slender columns under periodically varying axial loads. The effects produced by shear deformations along the span of the column as well as those produced by the axial inertia, the coupling between longitudinal and transverse deflections and the curvature are not taken into account. Sensitivity studies are presented in a companion paper that show the effects of rotary inertia, damping and semirigid connections on the dynamic stability of columns under parametric axial loads.