This paper considers the doubly periodic problem of elasticity for anisotropic solids containing regular sets of thin branched inclusions. A coupling principle for continua of different dimension is utilized for modeling of thin inhomogeneities and the boundary element technique is adopted for numerical solution of the problem. The branches of the inclusion can interact both inside the representative volume element and at the interface of neighbor representative elements. A particular example of the elastic medium reinforced by a doubly periodic set of I-beams is considered. Stress intensity and stress concentration inside and outside thin inclusions are determined. The dependence of the effective mechanical properties of the reinforced composite material on the volume fraction of the filament and its rigidity is obtained.