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Method of Weighted Residuals 5.1 INTRODUCTION Chapters 2, 3, and 4 introduced some of the basic concepts of the finite element method in terms of the so-called line elements. The linear elastic spring, the bar element and the flexure element are line elements because structural properties can be described in terms of a single spatial variable that identifies position along the longitudinal axis of the element. The displacement-force relations for the line elements are straightforward, as these relations are readily described using only the concepts of elementary strength of materials. To extend the method of finite element analysis to more general situations, particularly nonstructural applications, additional mathematical techniques are required. In this chapter, the method of weighted residuals is described in general and Galerkin’s method of weighted residuals [1] is emphasized as a tool for finite element formulation for essentially any field problem governed by a differential equation.
5.2 METHOD OF WEIGHTED RESIDUALS It is a basic fact that most practical problems in engineering are governed by differential equations. Owing to complexities of geometry and loading, rarely are exact solutions to the governing equations possible. Therefore, approximate techniques for solving differential equations are indispensable in engineering analysis. Indeed, the finite element method is such a technique. However, the finite element method is based on several other, more-fundamental, approximate techniques, one of which is discussed in detail in this section and subsequently applied to finite element formulation. The method of weighted residuals (MWR) is an approximate technique for solving boundary value problems that utilizes trial functions satisfying the 131
prescribed boundary conditions and an integral formulation to minimize error, in an average sense, over the problem domain. The general concept is described here in terms of the one-dimensional case but, as is shown in later chapters, extension to two and three dimensions is relatively straightforward. Given a differential equation of the general form D[y(x ), x ] = 0
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