We carry out a numerical study of the elastic deformations of the cross-reinforced and compound multilayer structures produced by winding of the unidirectional composite material (organic-plastic Kevlar 49/PR-286) on a thin-walled metal mandrel. We consider a two-stage static loading, when the temperature and internal pressure are combined. We analyze the influence of the winding angles on the elastic deformations, and reveal the packet structures and the schemes of reinforcement which improve stability of dimensions. Based on an example of a thin-walled pipeline with a running liquid, we prove that application of the composite structures of stable dimensions allows us to reduce significantly the level of forces and stresses in the pipes. The calculations were made applying the finite element method in displacements.