We state the problems of junction of smooth equally stressed reinforced structures in plane thermo-elastic fibrous constructions. We obtain a system of governing equations along with corresponding boundary conditions and conditions of joining for structural sub-elements and perform a qualitative analysis of the system of equations. We propose an iterative procedure for solving the problem stated. This method allows us to solve the problem of equally stressed reinforcing sequentially in each sub-region where the solution is continuous. We show that there may be cases when the conjugate problem has more than one solution. At that, the number of possible alternative solutions grows with the number of sub-regions of continuous solutions. Thus, we have an opportunity to control these solutions varying the number of reinforcing members in each sub-region. We obtain analytical and numeric solutions for conjugate problems. On these problems we demonstrate the opportunity for solving problems of equally stressed reinforcing in the cases of wide class of structural parameters.