Determination of the Upper Limit of the Bearing Capacity of Axisymmetric Reinforced Shallow Shells in Contact with an Incompressible Fluid

An axisymmetric problem is formulated for determining the upper (kinematic) limit of the bearing capacity of spherical shallow shells of annular shape in plan, the internal openings of which are closed by rigid inserts. Such compound structures are in contact with an incompressible fluid. The shells are reinforced with fibers along spiral trajectories symmetrical with respect to the meridian, as well as along meridional and/or circumferential directions. The materials of the composition components are assumed to be rigid-plastic and have different yield strengths under tension and compression. Plastic flow in the phases of the composition is determined by piecewise linear flow conditions. A two-layer model of a thin-walled structure is used, the kinematics of which in the limit state is described by the relations of the classical theory of shallow shells. The extreme problem of determining the ultimate load is formulated on the basis of the application of the principle of virtual power. An unconventional discretization of this problem was carried out, the solution of which was obtained using methods of linear programming theory. The convergence of the numerical solution is tested and compared with exact solutions of similar problems for homogeneous isotropic plates. Good accuracy of the numerical solution is demonstrated. The influence of the reinforcement structure parameters, the magnitude of the shallow shell lift and boundary conditions on the value of the ultimate load is investigated. It is shown that for annular plates the best arrangement of fibers is in the radial direction, and for shallow shells the rational one is a meridional-circumferential structure with specially selected reinforcement densities. It has been demonstrated that with an increase in the lifting height of a shallow shell, its load-bearing capacity more than doubles compared to a plate of the same geometry in plan and the same thickness.