Civil Engineering Reference
In-Depth Information
Exact
Four elements
Exact
0.25
0.5
0.75
1.0
1
2
3
4
x
L
Number of elements
(a)
(b)
Figure 1.4
(a) Displacement at
x
L
for tapered cylinder in tension of Figure 1.3. (b) Comparison of the exact solution
and the four-element solution for a tapered cylinder in tension.
On the other hand, if we plot displacement as a function of position along the
length of the cylinder, we can observe convergence as well as the approximate
nature of the finite element solutions. Figure 1.4b depicts the exact strength of
materials solution and the displacement solution for the four-element models.
We note that the displacement variation in each element is a linear approximation
to the true nonlinear solution. The linear variation is directly attributable to the
fact that the interpolation functions for a bar element are linear. Second, we note
that, as the mesh is refined, the displacement solution converges to the nonlinear
solution at
every point
in the solution domain.
The previous paragraph discussed convergence of the displacement of the
tapered cylinder. As will be seen in Chapter 2, displacement is the primary field
variable in structural problems. In most structural problems, however, we are
interested primarily in stresses induced by specified loadings. The stresses must
be computed via the appropriate stress-strain relations, and the strain compo-
nents are derived from the displacement field solution. Hence, strains and
stresses are referred to as
derived
variables. For example, if we plot the element
stresses for the tapered cylinder example just cited for the exact solution as well
as the finite element solutions for two- and four-element models as depicted in
Figure 1.5, we observe that the stresses are constant in each element and repre-
sent a
discontinuous
solution of the problem in terms of stresses and strains. We
also note that, as the number of elements increases, the jump discontinuities in
stress decrease in magnitude. This phenomenon is characteristic of the finite ele-
ment method. The formulation of the finite element method for a given problem
is such that the primary field variable is continuous from element to element but
