Nanocomposite materials are produced by using of nanotubes, the most significant structural elements of nanomaterials used in nanotechnologic applications. In the reinforcement (in the fibers) of the structure of composite materials, the appearance of the self-balancing stresses results from the initial curvature, caused by either structural reasons or technological processes. Because of exceeding safety limits of material caused by high magnitude self-balancing stresses, investigating the mechanical behaviors of the material theoretically, both under tensile and compression in the direction of strengthening (fiber) is essential for the engineering. Unlike the literature, in this study, composite materials containing triple-walled nanotube are investigated in the scope of the piecewise homogeneous body model by using of geometric non-linear exact equations of the 3-D theory of elasticity. The normal stress analysis, on the outermost surface of the carbon nanotube and the matrix intersection, is investigated under various external effects. Nanotube is first formed as having a small local curvature. Van der Waals forces existing between the carbon nanotube walls are taken into consideration.