The Unified Approach to the Beam Deflection and Torsion Problem

Abstract

There are several useful approximate methods for the solutions of the engineering problems. Some of them, Rayleigh-Ritz method, finite difference method and finite element method, are introduced, compared each other and unified in one problem by selecting possible advantages over the others in certain domain or at the boundary conditions to be satisfied.

A. Rayleigh-Ritz Method One of the useful approximate methods coming from variational considerations is the Rayleigh-Ritz method. In the Rayleigh-Ritz analysis, the first step is to assume a solution. Suppose ve seek a function y(x,y) which is to extreraize a functional I(y). Assume that y(x,y) can be approximated by a linear combination of suitably chosen linearly independent coordinate functions of the form here the constant coefficients C₁ C₂, ..... Cn are to be found so as to extremize a functional I(y) [$] • When the values of the coefficients are thus determined, they are called Ritz coefficients.

Usually the coordinate functions are chosen so that this expression satisfies the specified boundary conditions for any choice of the constants C₁, C₂ , ..... C_n . This is 12 n known as admissible functions. If the selection of the coordinate functions are made carefully, we can reach very good approximations to the exact solution even though the small number of coordinate functions are used. Besides this restriction on the satisfaction of the specified boundary conditions, the coordinate functions ǿ₁ are largely arbitrary. After the suitable choice of the coordinate functions for the approximate solutions, substitution of expression (1-1) into the functional I(y) and performance of integration results in The value of the functional becomes a function of n unknowns C₁, C₂, ... C_n. The partial differentiation of I with respect to C's gives enough simultaneous equations for the coefficients.