The analysis of a cylindrical shell with cut-outs, which are reinforced to reduce the stress concentrations, is of importance in many engineering fields, particularly in aircraft structures. The most common assumptions made are that the curvature effect is negligible and that the reinforcement is a compact flexible bead. In the analysis of holes in fuselage shells the neglect of the effect of curvature may lead to serious errors as shown theoretically by Lurie and Vandyke and experimentally by Houghton and Rothwell. In some cases the bending stresses are of the same order as the membrane stresses which themselves may be much higher than the flat plate solution. Lurie obtained a perturbation solution for an unreinforced hole by expanding in powers of curvature parameter β, the results being valid for small values of β. Vandyke extended this solution up to β=4 by using a collocation procedure. In fact, if the collocation method is used it seems possible to obtain a convergent series solution for any β, the only limitations being the shallowness of the shell and the restriction that the deflections are not large.