Civil Engineering Reference
In-Depth Information
Observing that
r
is independent of
z
, Equation 7.96 becomes
k
∂
∂
r
∂
r
∂
Qr
N
i
d
r
d
z
T
∂
∂
∂
T
2
+
+
=
0
r
r
z
∂
z
A
(
e
)
,
M
(7.97)
As in previous developments, we invoke the chain rule of differentiation as, for
example,
∂
∂
i
=
1,
...
rN
i
∂
r
∂
r
∂
T
∂
N
i
∂
∂
T
r
∂
T
∂
∂
N
i
∂
N
i
∂
∂
T
∂
=
+
⇒
r
r
r
∂
r
r
r
r
r
rN
i
∂
∂
∂
T
r
∂
T
∂
∂
N
i
∂
=
−
i
=
1,
...
,
M
(7.98)
r
∂
r
r
r
Noting that Equation 7.98 is also applicable to the
z
variable, the residual equa-
tions represented by Equation 7.97 can be written as
k
∂
∂
rN
i
∂
rN
i
∂
d
r
d
z
T
∂
∂
∂
T
∂
2
+
+
2
QN
i
r
d
r
d
z
r
r
z
z
A
(
e
)
A
(
E
)
k
∂
r
d
r
d
z
T
∂
∂
N
i
∂
r
+
∂
T
∂
N
i
∂
=
2
i
=
1,
...
,
M
(7.99)
r
∂
z
z
A
(
e
)
The first integrand on the left side of Equation 7.99 is a perfect differential in two
dimensions, and the Green-Gauss theorem can be applied to obtain
k
∂
n
z
rN
i
d
S
T
∂
k
∂
T
2
n
r
+
+
2
QN
i
r
d
r
d
z
r
∂
z
S
(
e
)
A
(
e
)
k
∂
r
d
r
d
z
T
∂
∂
N
i
∂
r
+
∂
T
∂
∂
N
i
∂
=
2
i
=
1,
...
,
M
(7.100)
r
z
z
A
(
e
)
where
S
is the boundary (periphery) of the element and
n
r
and
n
z
are the radial
and axial components of the outward unit vector normal to the boundary. Apply-
ing Fourier's law in cylindrical coordinates,
k
∂
T
∂
q
r
=−
r
(7.101)
k
∂
T
∂
q
z
=−
z
and noting the analogy with Equation 7.33, we rewrite Equation 7.100 as
∂
r
d
r
d
z
k
T
∂
N
i
∂
r
+
∂
T
∂
∂
N
i
∂
2
∂
r
z
z
A
(
e
)
=
2
QN
i
r
d
r
d
z
−
2
q
s
n
s
N
i
r
d
S
i
=
1,
...
,
M
(7.102)
A
(
e
)
S
(
e
)
