A method for solving problems on the limit equilibrium of reinforced shells of revolution

Rigid-plastic reinforced shells of revolution with a piecewise linear condition of plasticity are considered. It is shown that, in solving problems on their limit equilibrium, the application of linearized yield surfaces or the approximation of derivatives by finite differences restricts the set of possible solutions. In this paper, an asymptotic method for solving the problems by constructing a convergent sequence of solutions is offered. Each of these solutions is constructed numerically, and to approximate the derivatives, special finite differences coordinated with suppositions of the theory of thin shells are used. A feature of this method is that, with piecewise smooth yield surfaces, it is not necessary to determine a sequence of various plastic states, because the approximating yield surfaces are constructed during solution of the problem. Shells of revolution with positive and negative Gaussian curvatures and compound constructions of shells with various structures of reinforcement are examined. It is shown that the junction boundaries of rigid and plastic regions and the sequence of realization of plastic hinges greatly depend on the accuracy of approximation of the surfaces. With these approximations tending to the true yield surface, the sizes of the rigid regions decrease, and the range of structural and geometrical parameters of the shells grows when the yield state is reached through out their span. It is noted that, for closed constructions of shells reinforced only with spiral fibers at placement angles less that 55°, all possible mechanisms of plastic flow correspond to the direction of operating forces, whereas for other reinforcement structures, mechanisms of plastic flow with the opposite direction of velocities are possible.