Civil Engineering Reference
In-Depth Information
Comparison of the results of parts b and c reveals that the two element solution
exhibits an error of only about 1 percent in comparison to the exact solution from
strength of materials theory. Figure 2.7e shows the displacement variation along the
length for the three solutions. It is extremely important to note, however, that the
computed axial stress for the finite element solutions varies significantly from that of
the exact solution. The axial stress for the two-element solution is shown in Fig-
ure 2.7f, along with the calculated stress from the exact solution. Note particularly
the discontinuity of calculated stress values for the two elements at the connecting
node. This is typical of the derived, or secondary, variables, such as stress and strain,
as computed in the finite element method. As more and more smaller elements are
used in the model, the values of such discontinuities decrease, indicating solution
convergence. In structural analyses, the finite element user is most often more inter-
ested in stresses than displacements, hence it is essential that convergence of the
secondary variables be monitored.
2.4 STRAIN ENERGY, CASTIGLIANO'S
FIRST THEOREM
When external forces are applied to a body, the mechanical work done by those
forces is converted, in general, into a combination of kinetic and potential ener-
gies. In the case of an elastic body constrained to prevent motion, all the work
is stored in the body as elastic potential energy, which is also commonly
referred to as
strain energy.
Here, strain energy is denoted
U
e
and mechanical
work
W
. From elementary statics, the mechanical work performed by a force
F
as its point of application moves along a path from position 1 to position 2 is
defined as
2
F
=
ยท
(2.37)
W
d
r
1
where
d
z k
(2.38)
is a differential vector along the path of motion. In Cartesian coordinates, work
is given by
d
xi
d
y j
=
+
+
d
r
y
2
x
2
z
2
=
+
+
(2.39)
W
F
x
d
x
F
y
d
y
F
z
d
z
x
1
y
1
z
1
where
F
x
,
F
y
,
and
F
z
are the Cartesian components of the force vector.
For linearly elastic deformations, deflection is directly proportional to ap-
plied force as, for example, depicted in Figure 2.8 for a linear spring. The slope
of the force-deflection line is the spring constant such that
F
. Therefore,
the work required to deform such a spring by an arbitrary amount
0
from its
=
k
