Introduction to Finite Element Analysis and Design, 2nd Edition

Chapter 2Weighted Residual Methods for One‐Dimensional Problems

In the previous chapter, we developed the finite element equations for the truss element using the direct stiffness method. However, this method becomes impractical when element formulations are complicated or when multidimensional problems are considered. Thus, we need to develop a more systematic approach to constructing the finite element equations for general engineering problems. In fact, the finite element method can be applied to any engineering problem that is governed by a differential equation. There are two other methods of deriving the finite element equations: weighted residual method and energy method. In the first part of this chapter, we will consider ordinary differential equations that occur commonly in engineering problems, and we will derive the corresponding finite element equations through the weighted residual method, in particular using the Galerkin method. Energy methods are alternative methods that are very powerful and amenable for approximating solutions when solving structures that are more realistic. In the second part of this chapter, we will use the principle of minimum potential energy to derive finite element equations of discrete systems and uniaxial bars.

2.1 EXACT VS. APPROXIMATE SOLUTION

2.1.1 Exact Solution

Many engineering problems such as the deformation of a beam and heat conduction in a solid can be described using a differential equation. The differential equation ...