Thin-shelled elements made of fibered composites are widely used in different branches of modern industry. Thus, a composite material and a workpiece are produced simultaneously as part of one manufacturing procedure. Until recently, reinforcement was mostly provided by linear fibers. Such reinforcement method cannot be used for constructions with large gradients of stress and deformation fields in areas of holes and adapters. In this case, it is necessary to design constructions with special curvilinear structures of reinforcement. In the paper, the problem of rational reinforcement with curvilinear fibers sets is formulated in polar coordinates for an axially symmetric ring-shaped lamel. Discovery of curvilinear structures is performed by the structural model determined in terms of a non-homogeneous planar linear problem of elasticity. Effects of structural parameters for ultimate loading conditions are studied. Examples of ring-shaped lamel reinforcements with different kinds of spiral sets are presented. A family of spiral structures is obtained by plotting of trajectories isogonal to provided trajectories.