A great number of experiments have been carried out to study in detail the influence of the angle of reinforcement on the axial and circular stiffness and on the strength of pipes made of organic- and carbon-plastics. The purpose was to determine mechanical characteristics of unidirectional reinforced ribbon, which is used in production of multilayer pipes by winding. The statistical tests have been performed on the pipes with the angles of winding ±25°, ±35°, ±45°, ±55°, ±65°, and 90°. The pipes were loaded by axial tensile (compressive) force and by internal pressure. The pipes with small angles of winding (±25°) under loading by axial force and the pipes with large angles of winding (±65°) under loading by internal pressure failed because of tearing of filaments (ribbons). The pipes with large angles of winding under axial force and the pipes with small angles of winding under internal pressure failed because of initial destruction of bonds with cracking of binder/ this type of failure is followed by load decrease (radical decrease of the modulus of elasticity). We present numerical values of modulus of elasticity, Poison’s ratio, and limiting stresses. It is proven that the values of modulus of elasticity of the tested organic- and carbon-plastics in compression and tension are of similar values of magnitude (they differ in 2.3 to 8.5%), and the values of the modulus of elasticity and the limiting stresses in the circular direction are higher than the modulus of elasticity and the limiting pressures in the axial direction in 1.25 to 2.56 times. The dependences of the modulus of elasticity and limiting stresses on the angle of winding are determined in the form of quadratic function. A substantial dependence of the characteristics of the composite materials on the production technology of the samples (illustrated by the discrepancies of the experimental results and the theoretically estimated ones) is revealed.