The concern of this article is nonlinear elastic membrane theory with special emphasis on axisymmetric deformations of curved annular membranes subjected to a partially vanishing vertical surface load and with prescribed radial stresses on the edges. The nonlinear finiterotation theory of E. Reissner is employed, assuming linear stress-strain relations. This leads to consider a single second order ODE for the derivation of the principal stresses in the membrane. Analysis previously developed for a determination of that boundary data which corresponds to nonnegative radial and circumferential stress components is extended to the case, where parts of the membrane adjacent to the inner edge remain unloaded. The curved membrane is shown to flatten out on such parts which seems not yet to be observed.Mathematics Subject Classification (1991). 73C05, 73K15, 73H99, 34B15.