Civil Engineering Reference
In-Depth Information
more general finite element formulations, the computed displacements are also
substituted into the strain relations to obtain element strains, and the strains are,
in turn, substituted into the applicable stress-strain equations to obtain element
stress values.
EXAMPLE 2.2
Figure 2.4a depicts a system of three linearly elastic springs supporting three equal
weights
W
suspended in a vertical plane. Treating the springs as finite elements, deter-
mine the vertical displacement of each weight.
■
Solution
To treat this as a finite element problem, we assign node and element numbers as shown
in Figure 2.4b and ignore, for the moment, that displacement
U
1
is known to be zero by
the fixed support constraint. Per Equation 2.6, the stiffness matrix of each element is
(preprocessing)
3
k
k
(1)
=
−
3
k
−
3
k
3
k
2
k
k
(2)
=
−
2
k
−
2
k
2
k
k
−
k
−
kk
k
(3)
=
1
3
k
U
1
3
k
1
W
2
U
2
2
k
2
k
2
W
3
U
3
k
k
3
4
W
U
4
(b)
(a)
Figure 2.4
Example 2.2: elastic
spring supporting weights.
