Keywords: admissible functions, Rayleigh-Ritz method, plates.

This paper presents a procedure to build sets of admissible functions to be used in the Rayleigh-Ritz method (RRM) and a discussion on the characteristics of these functions. 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. 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. In the present paper, a very simple set of functions is built using an intuitive method and a close form solution of the terms to define the elastic stiffness, geometric stiffness, mass and penalty matrices to solve vibration and buckling problems can be obtained solving very simple integrals. Because the set of admissible functions does not produce ill-conditioning it also allows for the possibility of defining very complex boundary conditions on the structure and/or to define the inter-connection of elements using penalty functions. Each penalty function added to the structure eliminates a degree of freedom which also influences the accuracy of the solution. Thus, for cases that include a large number of constraints a larger number of terms in the set of admissible functions can be used to improve the accuracy of the solution. Another advantage of this approach is that the same set of functions can be used to solve problems of beams, as well as plates and shells of rectangular planform with any type of boundary conditions and to solve problems with two or more elements joined by penalty parameters representing artificial springs. The results included in this paper show that very accurate results can be obtained for all fifty five cases of plates with free, simply supported, guided and clamped boundary conditions along the edges of a thin plate.

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