Civil Engineering Reference
In-Depth Information
H
(
x
,
y
,
z
)
are functions defined in a region of xyz space
(the element volume in
our context),
then
∂
d
V
A
F
x
+
∂
G
y
+
∂
H
∂
=
(
Fn
x
+
Gn
y
+
Hn
z
)d
A
(7.75)
∂
∂
z
V
where
A
is the surface area of the volume and
n
x
,
n
y
,
n
z
are the Cartesian com-
ponents of the outward unit normal vector of the surface area. This theorem is the
three-dimensional counterpart of integration by parts discussed earlier in this
chapter.
Invoking Fourier's law and comparing Equation 7.75 to the first term of
Equation 7.74, we have
A
−
(
q
x
n
x
+
q
y
n
y
+
q
z
n
z
)
N
i
d
A
+
QN
i
d
V
V
k
x
∂
d
Vi
T
∂
N
i
∂
k
y
∂
T
∂
N
i
∂
k
z
∂
T
∂
N
i
∂
=
x
+
y
+
=
1,
...
,
M
(7.76)
∂
x
∂
y
∂
z
z
V
Inserting the matrix form of Equation 7.71 and rearranging, we have
k
x
∂
[
N
]
∂
∂
N
i
∂
k
y
∂
[
N
]
∂
∂
N
i
∂
k
z
∂
[
N
]
∂
∂
N
i
∂
x
+
y
+
{
T
}
d
V
x
y
z
z
V
A
=
QN
i
d
V
−
(
q
x
n
x
+
q
y
n
y
+
q
z
n
z
)
N
i
d
Ai
=
1,
...
,
M
(7.77)
V
Equation 7.77 represents a system of
M
algebraic equations in the
M
unknown
nodal temperatures {
T
}. With the exception that convection effects are not in-
cluded here, Equation 7.77 is analogous to the two-dimensional case represented
by Equation 7.34. In matrix notation, the system of equations for the three-
dimensional element formulation is
k
x
∂
d
V
T
[
N
]
T
∂
[
N
]
T
∂
∂
[
N
]
∂
k
y
∂
[
N
]
∂
∂
[
N
]
∂
k
z
∂
∂
[
N
]
∂
+
+
{
T
}
x
x
y
y
z
z
V
A
Q
[
N
]
T
d
V
q
z
n
z
)[
N
]
T
d
A
=
−
(
q
x
n
x
+
q
y
n
y
+
(7.78)
V
and Equation 7.76 is in the desired form
k
(
e
)
T
(
e
)
=
f
(
e
Q
+
f
(
e
q
(7.79)
Comparing the last two equations, the element conductance (stiffness) matrix is
k
(
e
)
=
k
x
∂
d
V
T
[
N
]
T
∂
[
N
]
T
∂
∂
[
N
]
∂
k
y
∂
[
N
]
∂
∂
[
N
]
∂
k
z
∂
∂
[
N
]
∂
+
+
x
x
y
y
z
z
V
(7.80)
