Abstract An effective method for the free vibration of arbitrary plan-form shallow shells is proposed. The algorithm is based on the R-function theory and variational Ritz method. The effectiveness of the method offered is illustrated by examples of shallow shells of a complex plan form at different boundary conditions. Three types of curvatures are considered, which are the spherical, circularly cylindrical and of hyperbolic paraboloid shape. A number of the test problems were solved to check the veracity of the proposed method. The comparison between the obtained results and those available in the literature, was carried out.