On the forms of loss of stability of a cylindrical shell under an external side pressure

Abstract

Two statements of the problem of the static loss of stability (through flexural-shearing and non-classical beam flexural modes) of a cylindrical shell made out of an isotropic material under the action of an external pressure on its side surface are considered. The first statement corresponds to the introduction of an assumption concerning the invariability of the direction of the action of the load when the shell changes from the unperturbed equilibrium state to a perturbed equilibrium state and the second statement corresponds to the action of a “tracking” hydrostatic pressure on the shell. The linearized equations of the theory of stability obtained earlier are used with a refinement of the terms in them associated with the action of the “tracking” external pressure and which are written in the linear approximation in the two versions. Exact analytical solutions are obtained for all the formulated problems. The solutions from them describing the beam flexural buckling modes (BMs) correspond to a hinged mounting of the ends of the shell, and they are obtained both on the basis of simplified equations, formulated in the momentless approximation, as well as the unsimplified (moment) equations. It is shown that their realization is only possible in the case of long shells under the action of a “non-tracking” side pressure with a fully explicable loss of stability mechanism. Under the action of a “tracking” pressure, the solutions for beam flexural BMs apparently only exists on account of the asymmetry with respect to the circumferential coordinate of the “loading” terms that occur in the neutral equilibrium equations used when the exact expression for the vector of the unit normal to the deformed middle surface of the shell is replaced by approximate linearized expressions for the vector of the normal that does not have a unit length. The critical values of the pressure corresponding to such beam flexural BMs are considerably greater than the critical pressure corresponding to the “shell” flexural BMs of the shell, and they are therefore of no practical interest whatsoever and only of purely theoretical interest.