This paper proposes a new variational RVR-method for calculating the three-dimensional stressed-strained state of statically loaded shell elements of structures with holes of arbitrary shapes and sizes. The scientifically substantiated RVR method is based on the use of the Reissner variational principle, the Vekua method, the theory of R-functions by Rvachev, and the general equations of the spatial theory of elasticity. The use of mixed Reissner variational principle leads to an increase in the accuracy of solving boundary value problems due to the independent variation of the displacement vector and the stress tensor. The Vekua method makes it possible to replace the solution to a three-dimensional problem with a regular sequence of solutions to two-dimensional problems. The theory of R-functions at the analytical level takes into account the geometric information of boundary value problems, which is necessary for the construction of solution structures that accurately satisfy all boundary conditions. At the same time, the developed algorithm for bilateral integral accuracy assessment makes it possible to automate the search for such a number of approximations in which the process of convergence of solutions becomes stable. The possibilities of the RVR method are shown in numerous examples of solving boundary problems of calculating cylindrical shells with an elliptical hole when setting centrifugal loads according to a deformed scheme. Calculations according to the specified load scheme of the anisotropic cylinder lead (at certain values of the angular velocity of rotation) to a significant increase in stresses. Therefore, to obtain reliable results, it is necessary to set a centrifugal load that takes into account the change in the size of the body in the process of its deformation. The characteristic features of the proposed RVR-method, which can be used effectively in the manufacture of shell elements of structures in various branches of technology, are discussed.