Civil Engineering Reference
In-Depth Information
3
CHAPTER
Truss Structures:
The Direct Stiffness
Method
3.1 INTRODUCTION
The simple line elements discussed in Chapter 2 introduced the concepts of
nodes, nodal displacements, and element stiffness matrices. In this chapter, cre-
ation of a finite element model of a mechanical system composed of any number
of elements is considered. The discussion is limited to
truss structures,
which we
define as structures composed of straight elastic members subjected to axial
forces only. Satisfaction of this restriction requires that all members of the truss
be bar elements and that the elements be connected by pin joints such that each
element is free to rotate about the joint. Although the bar element is inherently
one dimensional, it is quite effectively used in analyzing both two- and three-
dimensional trusses, as is shown.
The
global
coordinate system is the reference frame in which displace-
ments of the structure are expressed and usually chosen by convenience in con-
sideration of overall geometry. Considering the simple cantilever truss shown in
Figure 3.1a, it is logical to select the global
XY
axes as parallel to the predomi-
nant geometric “axes” of the truss as shown. If we examine the circled joint, for
example, redrawn in Figure 3.1b, we observe that five
element nodes
are physi-
cally connected at one
global node
and the
element x
axes do not coincide with
the
global X
axis. The physical connection and varying geometric orientation
of the elements lead to the following premises inherent to the finite element
method:
1.
The element nodal displacement of each connected element must be the
same as the displacement of the connection node in the global coordinate
system; the mathematical formulation, as will be seen, enforces this
requirement (displacement compatibility).
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