Civil Engineering Reference
In-Depth Information
Evaluating the first term at the limits as indicated, substituting Equation 5.60
into the second term, and rearranging, Equation 5.58 results in the two
equations
d
N
1
d
x
T
2
d
x
x
1
x
2
x
2
d
N
1
d
x
d
N
2
d
x
k
x
A
d
T
dx
k
x
A
T
1
+
=
A
QN
1
d
x
−
(5.63)
x
1
x
1
d
N
1
d
x
T
2
d
x
x
2
x
2
x
2
d
N
2
d
x
d
N
2
d
x
k
x
A
d
T
d
x
k
x
A
T
1
+
=
A
QN
2
d
x
+
(5.64)
x
1
x
1
Equations 5.63 and 5.64 are of the form
[
k
]
{
T
}={
f
Q
}+{
f
g
}
(5.65)
where [
k
] is the element conductance (“stiffness”) matrix having terms defined
by
x
2
d
N
l
d
x
d
N
m
d
x
k
lm
=
k
x
A
d
x
l
,
m
=
1, 2
(5.66)
x
1
The first term on the right-hand side of Equation 5.65 is the nodal “force” vector
arising from internal heat generation with values defined by
x
2
f
Q
1
=
A
QN
1
d
x
x
1
(5.67)
x
2
f
Q
2
=
A
QN
2
d
x
x
1
and vector {
f
g
} represents the gradient boundary conditions at the element
nodes. Performing the integrations indicated in Equation 5.66 gives the conduc-
tance matrix as
1
k
x
A
L
−
1
[
k
]
=
(5.68)
−
11
while for constant internal heat generation
Q
, Equation 5.67 results in the nodal
vector
QAL
2
QAL
2
{
f
Q
}=
(5.69)
