Long waves of the form of loss of stability of a compressed cylindrical shell in the Timoshenko-Lorentz solution

Using the form of loss of stability proposed by S.P. Timoshenko and R. Lorenz, the problem of stability of a compressed cylindrical shell in a linear formulation is considered. The expressions for bending deformations of the median surface take into account longitudinal displacements. An expression for the critical load is obtained and, based on the analysis of the obtained dependence, a dimensionless half-wave length of the loss of stability, at which the critical load has a minimum, is assigned to each number of transverse half-waves. With a decrease in the number of transverse half-waves, the length of the longitudinal half-wave increases, and the critical load decreases. The critical load for the multi-wave form is always less than the critical load for the axisymmetric form of loss of stability, that is, less than the load corresponding to the axisymmetric form of loss of stability; and with a decrease in relative thickness, it tends to the value determined by the Timoshenko-Lorentz formula. The relations between three mutually perpendicular displacements of the points of the median surface are determined. The marginal transitions showed a coincidence with the corresponding analytical models.