Stochastic bifurcation in the theory of flexure of spherical shells and circular membranes (Q1087047)

The capacity of rigidity clamped membranes and open shallow spherical shells of circular outline that are in equilibrium under the action of a radical stress, given uniformly on the contour, and transverse loads distributed radially along the surface to form a field with a quasi- Gaussian probability measure to retain shape is investigated. It is assumed that the behaviour of the membranes and shells is described by von Kármán equations taken in a radial approximation. The following method is used. A generalization of the probability density, a probability functional (PF) induced by the probability measure of the load and the operator of the problem is constructed in the space of possible solutions of the initial boundary value problem. The times of a substantial change in the shape or an abrubt decrease in the shell (and membrane) carrying capacity are related to the first bifurcation of the PF modes with respect to the growth of the compressive force. The application of this method starts with the derivation of the equations for the PF extremals in the space of weighted derivatives of the deflection function with respect to the dimensionless variable radius. Within the framework of the Galerkin method, solutions of the designated equation are determined. Simple relationships are determined that relate the radial stresses to the statistical characteristics of the transverse load field at the time of bifurcation of these solutions. It is shown that up to the time of the first bifurcation PF has just one extremal, a trivial mode for the membranes but a non-trivial mode for the shells. Then by starting with the time mentioned the membrane PF reaches maxima on the extremals bifurcating from the trivial, while the shell PF acquires a new maximum (in addition to the existing maximum) on still another non-trivial extremal. Results are presented of a computation of the compressive forces of the first bifurcation of the PF modes in the case of transverse loads with a small correlation scale.