We study the problems of applicability of methods of the perturbation theory to solution of elastic problems for rational reinforcement of the plane composite structures having a uniformly stressed reinforcement of constant cross-sectional area. Based on these methods, we obtain, for the first time, two-dimensional solutions for corresponding boundary value problems. We demonstrate the existence of several solutions of such problems under one and the same input data and a possibility to provide a goal control over a set of projects with equally-stressed reinforcement. We make a comparison of the exact and approximate solutions for simply-connected and doubly-connected structures.