This paper is devoted to application of the variational Ritz method joined with the R-functions theory to investigate linear and geometrically nonlinear free vibrations of the functionally graded shallow shells/plates resting on elastic foundation. Mathematical statement is carried out in framework of the first order shear deformation theory of the shell (FSDT). The elastic foundation is defined by two parameters: Winkler and Pasternak. The effective material properties in the thickness direction are modeled by means of power and sigmoid laws. Analytical expressions of the matrix elements, which calculate forces and moments of the shells/plates for both types of law are presented. Proposed method consists of several steps, which include solution of the linear vibration problem, solution of the problem that is similar to the theory of elasticity problem. Ater that an initial system with partial derivatives is reduced to nonlinear ordinary differential equation due to application of the procedure by Bubnov-Galerkin. Obtained equation is solved by Runge–Kutta method. The developed approach is tested on a lot of problems and illustrated on shallow shells/ plates with complex plan shapes resting on elastic foundation.