We study stresses in an orthotropic material of a composite shell of revolution under the axisymmetric tangential (membrane) loading. Winding of one family of fibers ± f makes the shell. We consider a loading by inner pressure and reinforcement along the geodesic lines. The diagrams of the stresses s1 and s2 along the fibers and in the transversal direction and the shear stress t12 versus the radius of revolution are plotted for the case of a balanced segment. We determine the maximum values of s2 and t12 and corresponding values of the radiuses of the shell. The failure of the matrix is determined on the basis of the quadratic criterion of strength. When a tape of moderate width reinforces the shell, the angle of reinforcement of a fiber f depends not only on the radius of the shell but also on the location of the fiber in the tape as well as on the derivative of the equation of the contour. The diagrams of the stresses along the fibers and in the transversal direction and the shear stresses versus radius are constructed for the cases of two fibers located near the edge of the tape and for the middle fiber of the tape. We conclude that the maximum stresses are the same as in the case of a shell wounded by a fiber. .