Civil Engineering Reference
In-Depth Information
associated with nodes on
S
3
. To generalize, we rewrite Equation 7.41 as
f
(
e
g
=−
k
(
e
hS
{
}+
f
(
e
)
hS
T
(7.42)
where
k
(
e
hS
=
h
[
N
]
T
[
N
]
t
d
S
(7.43)
S
is the contribution to the element conductance matrix owing to convection on
portion
S
of the element boundary and
f
(
e
)
hS
=
hT
a
{
N
}
t
d
S
(7.44)
S
is the forcing function associated with convection on
S
.
Incorporating Equation 7.42 into Equation 7.36, we have
k
(
e
)
{
}=
f
(
e
Q
+
f
(
e
h
+
f
(
e
g
+
f
(
e
)
hS
T
(7.45)
where the element conductance matrix is now given by
k
x
∂
t
d
A
T
∂
k
y
∂
T
∂
k
(
e
)
=
N
N
N
∂
N
+
∂
x
∂
x
y
∂
y
A
2
h
h
[
N
]
T
[
N
]
T
+
[
N
] d
A
+
[
N
]
t
d
S
(7.46)
A
S
which now explicitly includes edge convection on portion(s)
S
of the element
boundary subjected to convection.
EXAMPLE 7.4
Determine the conductance matrix (excluding edge convection) for a four-node, rectan-
gular element having 0.5 in. thickness and equal sides of 1 in. The material has thermal
properties
k
x
4
3
=
20
Btu/(hr-ft-
◦
F
) and
h
=
50
Btu/(hr-ft
2
-
◦
F
).
=
k
y
s
r
■
Solution
The element with node numbers is as shown in Figure 7.9 and the interpolation functions,
Equation 6.56, are
1
2
Figure 7.9
Element
node numbering
for Example 7.4;
the length of each
edge is 1 in.
1
4
(1
−
r
)(1
−
s
)
N
1
(
r
,
s
)
=
1
4
(1
+
r
)(1
−
s
)
N
2
(
r
,
s
)
=
