The implementation of the theory of multicomponent dry friction in some engineering problems of the contact interaction of composite shells and rough rigid support planes is proposed. The main attention is paid to the construction of analytical models of combined dry friction, taking into account the anisotropy of the dry friction coefficients and the real distribution of normal and tangential contact stresses. These models can be used for a more detailed study of transient rolling modes of pneumatics, characterized by simultaneous sliding and rotation. The quasi-static distribution of the contact pressure is based on the solution of the contact problem for the spherical composite shell ant the absolutely rigid plane. The governing equation of S.A.Ambartsumyan for a transversally isotropic elastic shell is used as a background, the superposition principle, and the transient function for a shell. Such a function is a normal translation as a solution of the problem for a shell loaded by the unit concentrated force. This problem is solved by expanding the unknowns in series with respect to Legendre polynomials and their derivatives. After construction of the transient function the contact problem is reduced to the integral equation for the contact pressure on the basis of the superposition principle. The integral equation is then reduced to the algebraic one for the expansion factors of the contact pressure using the orthogonality of Legendre polynomials. Finally, after the truncation of the series and by discretization along the meridian coordinate the obtained problem is reduced to the algebraic system for the expansion factors of the contact pressure.