Shell structures have high load-carrying capabilities compared to their own weight, while they are apt to be produced large deformations accompanied with unstable phenomena when loads exceed a critical level. So it is important to investigate their behavior under various environmental conditions, particularly, dynamic loading conditions, because of insufficiency of existing researches. The dynamic problems of conical shells have been little analyzed except ones for the case of lateral pressure loading. In the present paper, on the basis of nonlinear shell theory, the axisymmetric dynamic behavior of elastic conical shells clamped along edges subjected to an axial step load is analyzed, where the effect of the mass at the loading end is taken into account. The basic equations derived by the authors are solved numerically, and the influence of the mass at the loading end upon the dynamic behavior is clarified. The mechanism of snapping is discussed in detail, and it is shown that the snap-through phenomena of the shell are classified into four types depending on magnitudes of the applied load and end-mass.