By using decomposition of hoop and radial components of the displacement vector to the trigonometrical and generalized power series, we obtained the new exact analytical solutions to problems of the equilibrium state of thick-walled heavy anisotropic central-and axial-symmetric bodies fixed on their external surfaces and subject to uniform internal lateral pressure. The estimation of an initial strength of cylindrical and spherical reinforced concrete monolithic mine working supports for a long-term storage of gas and oil is carried out on the basis of a multicriteria approach taking into account real damage mechanisms. They include damage from tension or compression in radial, hoop and axial directions, and from transversal and antiplane shear) of anisotropic axial- and central-symmetric bodies.