Civil Engineering Reference
In-Depth Information
6
CHAPTER
Interpolation Functions
for General Element
Formulation
6.1 INTRODUCTION
The structural elements introduced in the previous chapters were formulated on
the basis of known principles from elementary strength of materials theory. We
have also shown, by example, how Galerkin's method can be applied to a heat
conduction problem. This chapter examines the requirements for interpolation
functions in terms of solution accuracy and convergence of a finite element
analysis to the exact solution of a general field problem. Interpolation functions
for various common element shapes in one, two, and three dimensions are de-
veloped, and these functions are used to formulate finite element equations for
various types of physical problems in the remainder of the text.
With the exception of the beam element, all the interpolation functions dis-
cussed in this chapter are applicable to finite elements used to obtain solutions
to problems that are said to be
C
0
-continuous. This terminology means that,
across element boundaries, only the zeroth-order derivatives of the field
variable (i.e., the field variable itself) are continuous. On the other hand, the
beam element formulation is such that the element exhibits
C
1
-continuity, since
the first derivative of the transverse displacement (i.e., slope) is continuous
across element boundaries, as discussed previously and repeated later for em-
phasis. In general, in a problem having
C
n
-continuity, derivatives of the field
variable up to and including
n
th-order derivatives are continuous across ele-
ment boundaries.
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