The elastic shell buckling of a circular cylindrical shell caused by a briefly applied, intense pressure loading is considered.When such cylindrical shells as these produce the elastic shell buckling by the hydrostatic pressure collpse under the transient load, it may be thought that the elastic buckling of the shell occure at the early stage of the collapse. Such a elastic buckling pressure is able to calculated by the method of minimumization of the total potential which is the sum of the elastic strain energy in the shell, the kinetic energy of the shell and the work done by the external forces acting on the shell, that is, the equation of the motion on the shell is obtained from the first variation of the total potential, and the second variation determines the critical pressure at which the elastic instability of the shell will occure. The calculating result as follows, q_s =α⁴/ n ²+α²+1/12 (1- v ²) h ²/ a ² ( n ²+α²) - ( n ²+α²/2) q ₀+λ²/ k ²/ [ n ² (1-cos kt ) -1] +2/ ( n ²+α²) ² [ n ² { n ²+ (2+ v ) α²} cos kt -α² ( n ²- v α²]) where P_s =critical transient pressure (asumming that the load-time relation is square) P ₀=static pressure q ₀= P ₀ a / Eh n =number of lobes α=π a / b a =radius of cylinder b =length of cylinder h =thickness of cylinder k ²= Eg /γ a ² γ=density of shell material λ=ratio of the velocity of buckling deflection to the buckling deflection