A method is developed for finding the buckling loads of beams or plates over continuous rigid or elastic supports. These loads are found as the roots of characteristic equations tha t are developed in terms of the known characteristic modes of buckling and corresponding characteristic buckling loads of the beam or plate without the rigid or elastic supports. The method is an application of the direct method of the Calculus of Variations and consists of minimizing the potential energy of the system subject to the constraints tha t are the inner supports, rigid or elastic. The object of this paper is to develop a method whereby the buckling loads of beams or plates, having simply supported, fixed, or free boundaries, and subject to inner constraints in the form of intermediate rigid or elastic supports, may readily be calculated. These buckling loads are the roots of the characteristic equations tha t arise from the condition tha t the potential energy of the system shall be a minimum subject to the constraints. By developing the buckled shape of the system as a Generalized Fourier Series in terms of the characteristic (or eigen) functions of the unrestrained system, in many cases known or easily found, it is possible to express the potential energy in a simple and compact form from which there is no difficulty in deriving the characteristic equation. Few terms of these equations need be used to obtain reasonably accurate results in most problems. The beam has been previously solved, but the present treatment will be found to have advantages in some cases over the existing method. This is particularly evident if the beam is uniform. In the case of the plate results believed to be new are found.