Comparison of Simple and Chebychev Polynomials in Rayleigh‐Ritz Analysis

Abstract

The purpose of this paper is to demonstrate the efficacy of using Chebychev polynomials in the Rayleigh‐Ritz method. For purposes of illustration, the problem of free torsional vibration and buckling of doubly symmetric thinwalled beams of open section of constant cross section subjected to an axial compressive static load and resting on a continuous elastic foundation is considered. Both simple polynomials as well as orthogonal functions are used as displacement functions in order to demonstrate the advantages of the latter over the former. Two sets of boundary conditions are treated: (1) Fixed‐fixed; and (2) fixed‐simply supported. Wherever possible, the functions are chosen so that the kinematic boundary conditions are satisfied. In the cases in which the functions do not satisfy all the kinematic boundary conditions, the penalty‐type approach is adopted. In this approach, appropriate springs with large stiffness coefficients are provided to simulate the kinematic boundary conditions. Numerical results for natural frequencies and buckling loads for various values of warping and elastic foundation parameters are obtained and compared with those obtained by other researchers. A good agreement is observed.