Following the study of axisymmetric deformations of annular membranes under a normal surface load on the basis of the Foppl-Hencky small-finite-deflection theory, which has been completed in recent works, the corresponding methods are here generalized as to apply to finite-deformation problems of the Simmonds-Libai simplified Reissner equations of finite rotations. The question of uniqueness of positive regular solutions is solved in general, and an integralequation technique is developed which yields existence and nonexistence results for an extensive range of the boundary data where previous investigations failed. In contrast to the Foppl theory, some of the boundary conditions are nonlinear, and hence an extension to the finite-rotation case is not straightforward. Moreover there are significant differences between the small-and the finite-rotation theory.

A simplified version of Reissner's theory of thin shells of revolution suffering small strains but arbitrarily large deflections and rotations reduces, when specialized to axi-symmetric deformations of annular membranes under a vertical surface load, to a nonlinear ordinary differential equation which is free of Poisson'ratio. Within this framework the questions of existence and non-existence of non-negative solutions of the associated stress and displacement boundary value problem are brought to a final answer. Progress in this direction was made in an earlier study (Grabmüller & Pirner 1987). In this paper a continuous monotone curve is constructed which effects a subdivision of the respective ranges of boundary data into complementary domains of existence and non-existence of strictly positive solutions