The subject of this investigation is to study the buckling of orthotropic cylindrical thin shells under torsion, which is a power function of time. The dynamic stability and compatibility equations are obtained first. These equations are subsequently reduced to a time dependent differential equation with variable coefficient by using Galerkin's method. Finally, the critical dynamic and static loading, the corresponding wave numbers, the dynamic factors, critical time and critical impulse are found analytically by applying the Ritz type variational method. Using those results, the effects of the variations of the power of time in the torsion load expression, of the loading parameter, the ratio of the Young's moduli and the ratio of the radius to thickness on the critical parameters are studied numerically. It is observed that these factors have appreciable effects on the critical parameters of the problem in the heading.