Civil Engineering Reference
In-Depth Information
2
CHAPTER
Stiffness Matrices, Spring
and Bar Elements
2.1 INTRODUCTION
The primary characteristics of a finite element are embodied in the element
stiffness matrix
. For a structural finite element, the stiffness matrix contains the
geometric and material behavior information that indicates the resistance of
the element to deformation when subjected to loading. Such deformation may
include axial, bending, shear, and torsional effects. For finite elements used in
nonstructural analyses, such as fluid flow and heat transfer, the term
stiffness
matrix
is also used, since the matrix represents the resistance of the element to
change when subjected to external influences.
This chapter develops the finite element characteristics of two relatively
simple, one-dimensional structural elements, a linearly elastic spring and an elas-
tic tension-compression member. These are selected as introductory elements be-
cause the behavior of each is relatively well-known from the commonly studied
engineering subjects of statics and strength of materials. Thus, the “bridge” to the
finite element method is not obscured by theories new to the engineering student.
Rather, we build on known engineering principles to introduce finite element
concepts. The linear spring and the tension-compression member (hereafter re-
ferred to as a
bar
element and also known in the finite element literature as a
spar,
link,
or
truss
element) are also used to introduce the concept of
interpolation
functions.
As mentioned briefly in Chapter 1, the basic premise of the finite ele-
ment method is to describe the continuous variation of the field variable (in this
chapter, physical displacement) in terms of discrete values at the finite element
nodes. In the interior of a finite element, as well as along the boundaries (applic-
able to two- and three-dimensional problems), the field variable is described via
interpolation functions (Chapter 6) that must satisfy prescribed conditions.
Finite element analysis is based, dependent on the type of problem, on sev-
eral mathematic/physical principles. In the present introduction to the method,
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