Abstract: In order to quantify the effect of the dead weight (hereinafter called the distributed axial force) of the beam, the rod and the column on the static instability and the transverse vibration, the mechanical and mathematical models of the instability and the transverse vibration are established with consideration of the distributed axial force. The finite difference method, the Galerkin method and the numerical integration method are used to obtain the numerical results. The results show that: the natural frequency of the rod with the axial distributed force decreases with the increase of the rod length, and the first order natural frequency is zero when the rod is unstable; the axial distributed force has a small impact on the critical load of the shorter rod, but has a greater impact for a longer rod; the Galerkin method is not suitable for calculating the buckling critical load of the super long rod.