Civil Engineering Reference
In-Depth Information
we present several such principles applicable to finite element analysis. First, and
foremost, for spring and bar systems, we utilize the principle of static equilib-
rium but—and this is essential—we include
deformation
in the development;
that is, we are not dealing with rigid body mechanics. For extension of the finite
element method to more complicated elastic structural systems, we also state and
apply the first theorem of Castigliano [1] and the more widely used principle of
minimum potential energy [2]. Castigliano's first theorem, in the form presented,
may be new to the reader. The first theorem is the counterpart of Castigliano's
second theorem, which is more often encountered in the study of elementary
strength of materials [3]. Both theorems relate displacements and applied forces
to the equilibrium conditions of a mechanical system in terms of mechanical
energy. The use here of Castigliano's first theorem is for the distinct purpose of
introducing the concept of minimum potential energy without resort to the higher
mathematic principles of the calculus of variations, which is beyond the mathe-
matical level intended for this text.
2.2 LINEAR SPRING AS A FINITE ELEMENT
A linear elastic spring is a mechanical device capable of supporting axial loading
only and constructed such that, over a reasonable operating range (meaning ex-
tension or compression beyond undeformed length), the elongation or contrac-
tion of the spring is directly proportional to the applied axial load. The constant
of proportionality between deformation and load is referred to as the
spring con-
stant, spring rate,
or
spring stiffness
[4], generally denoted as
k
, and has units
of force per unit length. Formulation of the linear spring as a finite element is
accomplished with reference to Figure 2.1a. As an elastic spring supports axial
loading only, we select an
element coordinate system
(also known as a
local
co-
ordinate system) as an
x
axis oriented along the length of the spring, as shown.
The element coordinate system is embedded in the element and chosen, by geo-
metric convenience, for simplicity in describing element behavior. The element
k
u
1
u
2
1
f
1
f
2
x
1
2
Deflection,
u
2
u
1
k
(a)
(b)
Figure 2.1
(a) Linear spring element with nodes, nodal displacements, and nodal forces.
(b) Load-deflection curve.
