Civil Engineering Reference
In-Depth Information
Figure 6.24b, a three-node triangular element is shown having nodal coordinates
(
r
i
,
z
i
)
. In the axisymmetric case, the field variable is discretized as
3
(
r
,
z
)
=
N
i
(
r
,
z
)
i
(6.94)
i
=
1
where the interpolation functions
N
i
(
r
,
z
)
must satisfy the usual nodal conditions.
Noting that the nodal conditions are satisfied by the interpolation functions
defined by Equation 6.37 if we simply substitute
r
for
x
and
z
for
y
, the inter-
polation functions for the axisymmetric triangular element are immediately
obtained. Similarly, the interpolation functions in terms of area coordinates are
also applicable.
Since, by definition of an axisymmetric problem, the problem, therefore its
solution, is independent of the circumferential coordinate
, so must be the inter-
polation functions. Consequently, any two-dimensional element and associated
interpolation functions can be used for axisymmetric elements. What is the dif-
ference? The axisymmetric element is physically three dimensional. As depicted
in Figure 6.25, the triangular axisymmetric element is actually a prism of revo-
lution. The “nodes” are circles about the axis of revolution of the body, and the
nodal conditions are satisfied at every point along the circumference defined by
the node of a two-dimensional element. Although we use a triangular element
for illustration, we reiterate that any two-dimensional element can be used to
formulate an axisymmetric element.
As is shown in subsequent chapters in terms of specific axisymmetric
problems, integration of various functions of the interpolation functions over the
volume are required for element formulation. Symbolically, such integrals are
represented as
F
(
r
,
,
z
)
=
f
(
r
,
,
z
)d
V
=
f
(
r
,
,
z
)
r
d
r
d
d
z
(6.95)
V
z
r
Figure 6.25
A three-dimensional
representation of an axisymmetric
element based on a three-node
triangular element.
