Civil Engineering Reference
In-Depth Information
6.9 AXISYMMETRIC ELEMENTS
Many three-dimensional field problems in engineering exhibit symmetry about
an axis of rotation. Such problems, known as
axisymmetric problems,
can be
solved using two-dimensional finite elements, which are most conveniently de-
scribed in cylindrical
(
r
,
,
z
)
coordinates. The required conditions for a problem
to be axisymmetric are as follows:
1.
The problem domain must possess an axis of symmetry, which is
conventionally taken as the
z
axis; that is, the domain is geometrically
a solid of revolution.
2.
The boundary conditions are symmetric about the axis of revolution;
thus, all boundary conditions are independent of the circumferential
coordinate
.
3.
All loading conditions are symmetric about the axis of revolution; thus,
they are also independent of the circumferential coordinate.
In addition, the material properties must be symmetric about the axis of revolu-
tion. This condition is, of course, automatically satisfied for isotropic materials.
If these conditions are met, the field variable
is a function of radial and
axial (
r
,
z
) coordinates only and described mathematically by two-dimensional
governing equations.
Figure 6.24a depicts a cross section of an axisymmetric body assumed to be
the domain of an axisymmetric problem. The cross section could represent the
wall of a pressure vessel for stress or heat transfer analysis, an annular region
of fluid flow, or blast furnace for steel production, to name a few examples. In
(
r
3
,
z
3
)
z
z
(
r
2
,
z
2
)
r
r
(
r
1
,
z
1
)
(b)
(a)
Figure 6.24
(a) An axisymmetric body and cylindrical coordinates. (b) A three-
node triangle in cylindrical coordinates at an arbitrary value .
