Civil Engineering Reference
In-Depth Information
d
d
z
r
Figure 6.26
Differential volume
in cylindrical coordinates.
where
V
is the volume of an element and
d
V
d
z
is the differential vol-
ume depicted in Figure 6.26. For axial symmetry, the integrand is independent of
the circumferential coordinate
=
r
d
r
d
, so the integration indicated in Equation 6.95
becomes
(6.96)
F
(
r
,
,
z
)
=
F
(
r
,
z
)
=
2
f
(
r
,
z
)
r
d
r
d
z
A
Equation 6.96 shows that the integration operations required for formulation of
axisymmetric elements are distinctly different from those of two-dimensional
elements, even though the interpolation functions are essentially identical. As
stated, we show applications of axisymmetric elements in subsequent chapters.
Also,
any
two-dimensional element can be readily converted to an axisymmetric
element, provided the true three-dimensional nature of the element is taken into
account when element characteristic matrices are formulated.
EXAMPLE 6.5
In following chapters, we show that integrals of the form
N
i
N
j
d
V
V
where
N
i
,
N
j
are interpolation functions and
V
is element volume, must be evaluated in
formulation of element matrices. Evaluate the integral with
i
=
1,
j
=
2
for an axisym-
metric element based on the three-node triangle using the area coordinates as the interpo-
lation functions.
■
Solution
For the axisymmetric element, we use Equation 6.96 to write
N
i
N
j
d
V
=
2
N
i
N
j
r
d
r
d
z
=
2
L
i
L
j
r
d
r
d
z
V
A
A
where
A
is the element area. Owing to the presence of the variable
r
in the integrand, the
integration formula, Equation 6.49, cannot be applied directly. However, we can express
r
in terms of the nodal coordinates
r
1
,
r
2
,
r
3
and the area coordinates as
r
=
L
1
r
1
+
L
2
r
2
+
L
3
r
3
