We consider the problem of bending of a three-layer shallow shell resting on an elastic foundation under the action of a transverse load. It is assumed that the lower and upper layers are made of functionally graded materials and the filler is made of an isotropic material (metal or ceramic). For the mathematical modeling of the problem, we use the refined first-order Timoshenko-type theory of plates that takes into account the presence of bending strains. The elastic foundation of the shell is modeled by a Pasternaktype two-parameter model. The effective elastic properties of the functionally graded materials vary according to the power law. The proposed algorithm for solving the problems of bending is based on the application of the R -functions theory and the Ritz variational method. The developed software, which realizes the proposed approach, is verified by analyzing test problems posed for rectangular plates and shallow shells with different schemes of arrangement of the layers and various characteristics of the elastic foundation. The efficiency of the method is demonstrated by an example of a shell with hexagonal hole and circular notches on the sides. We consider various conditions of fastening of the hole and the outer contour of the shell. The influence of the gradient index and the characteristics of elastic foundation on the maximum value of deflection are analyzed. The obtained results are presented in the form of tables and plots. They are used to study functionally graded plates and shallow shells on an elastic foundation with complex shape in the plan.