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Trefftz's method is not universally applicable, since solutions are not always available
for the governing differential equations. Also, the method leads to algebraic equations that
may be less banded than those of some other methods.
7.5
Trial Function Methods for Solids Divided into Elements: The Method
of Finite Elements
Thus far in this chapter, we have utilized trial functions that apply for the whole body.
The selection of appropriate trial solutions for complicated bodies, e.g., two- and three-
dimensional configurations, is not simple. Furthermore, for a formulation in terms of
displacements, the chosen trial solution is usually expected to approximate forces (e.g.,
moments or stresses), as well as the displacements. This goal is often not easy to achieve.
As a further complication, the use of what would appear to be an adequate trial solu-
tion can lead to numerical instabilities. These difficulties can usually be circumvented
by subdividing the body into elements and utilizing a separate trial solution in each ele-
ment. This is, of course, the procedure that forms the basis of the finite element method, the
powerful computational technology that was considered in-depth in the previous chapter.
Basically, Ritz's method is employed to generate stiffness matrices for the elements. And in
certain instances, weighted-residual methods can be used, provided that interelement con-
ditions are properly taken into account. Then the techniques discussed in Chapters 5 and
6 for assembling a system stiffness matrix are utilized to create a set of global equilibrium
equations.
7.5.1 Analog Solutions of Differential Equations
Substantial generic finite element software has been developed for the solution of vari-
ous problems, especially for structural mechanics problems. Although the finite element
method is fundamentally a variationally based technique, as shown in Chapter 2, differ-
ential equations can often be shown to be equivalent to variational forms. Since many
different kinds of problems are described by the same kind of differential equations, it is
often not necessary to create a computer program for each problem. For example, the differ-
ential equation for the one-dimensional static heat distribution and that for the extension of
straight bars have the same form except that the coefficients of the derivatives of the depen-
dent variables and the nonhomogenous terms have different physical meanings. Problems
which have this characteristic are said to be analogical . A wide variety of problems can be
solved using computer software suitable for analogical problems. The primary concern for
this solution method is the interpretation and exchange of the coefficients of the derivatives
and the nonhomogeneous terms between the analogical problems. Special attention to the
boundary conditions is often necessary.
EXAMPLE 7.12 The Analogy Between the Heat Equation and the Extension of a Straight Bar
The governing differential equation for one-dimensional heat transfer under stationary
conditions is
K d 2 T
dx 2 =
Q
(1)
with boundary conditions, where T is the temperature, K is the thermal conductivity, and Q
is the heat source. The differential equation for the extension of a straight bar is (Chapter 1,
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