- This paper presents a new semi-analytical method for modeling rectangular plates with variable thickness and cutouts. The plate thickness is represented as a finite sum of multiplications of one-dimensional functions. The plate deflections are also assumed in the similar separable form and the variational extended Kantorovich method is applied. In order to enhance the accuracy of the solution, a multi-term formulation of the extended Kantorovich method is developed. It is shown that this representation is very general and it allows the description of a complex variation of the thickness including step thickness changes and cutouts. It is demonstrated that this approach avoids singularities at the cutout areas and it does not require assembly of predefined trial functions or computational domains satisfying plate geometry and boundary conditions. The presented method is applied for the free vibration analysis of rectangular plates with various rectangular cutouts and variable thickness. The accuracy and convergence of the solution is studied through comparisons with other semi-analytical methods (where applicable) and the results of finite element analysis.