Proof of convergence for a set of admissible functions for the Rayleigh–Ritz analysis of beams and plates and shells of rectangular planform

This work presents a discussion on the characteristics of sets of admissible functions to be used in the Rayleigh–Ritz method (RRM). Of particular interest are sets that can lead to converged results when penalty terms are added to model constraints and interconnection of elements in vibration and buckling problems of beams, as well as plates and shells of rectangular planform. The discussion includes the use of polynomials, trigonometric functions and a combination of both. In the past, several sets of admissible functions that have a limit on the number of terms that can be included in the solution without producing ill-conditioning were used. On the other hand, a combination of trigonometric and low order polynomials have been found to produce accurate results without ill-conditioning for any number of terms and any number of penalty parameters that can be accommodated by the computer memory.