Civil Engineering Reference
In-Depth Information
or
x
2
x
2
x
2
d
2
T
d
x
2
A
k
x
N
i
(
x
)d
x
−
hP
T
(
x
)
N
i
(
x
)d
x
+
A
QN
i
(
x
)d
x
x
1
x
1
x
1
x
2
+
hPT
a
N
i
(
x
)d
x
=
0
i
=
1, 2
(7.7)
x
1
Integrating the first term by parts and rearranging,
x
2
x
2
d
N
i
d
x
d
T
d
x
k
x
A
d
x
+
hP
T
(
x
)
N
i
(
x
)d
x
x
1
x
1
x
2
x
2
x
2
d
T
d
x
=
A
QN
i
(
x
)d
x
+
hPT
a
N
i
(
x
)d
x
+
k
x
AN
i
(
x
)
i
=
1, 2
(7.8)
x
1
x
1
x
1
Substituting for
T
(
x
)
from Equation 7.5 yields
d
N
1
d
x
T
2
d
x
x
2
x
2
d
N
i
d
x
d
N
2
d
x
k
x
A
T
1
+
+
hP
N
i
(
x
)[
N
1
(
x
)
T
1
+
N
2
(
x
)
T
2
]d
x
x
1
x
1
x
2
x
2
x
2
d
T
d
x
=
A
QN
i
(
x
)d
x
+
hPT
a
N
i
(
x
)d
x
+
k
x
AN
i
(
x
)
i
=
1, 2
(7.9)
x
1
x
1
x
1
The two equations represented by Equation 7.9 are conveniently combined into
a matrix form by rewriting Equation 7.5 as
N
2
]
T
1
T
(
x
)
=
[
N
1
=
[
N
]
{
T
}
(7.10)
T
2
and substituting to obtain
d
N
d
x
T
d
N
d
x
x
2
x
2
[
N
]
T
k
x
A
{
T
}
d
x
+
hP
[
N
]
{
T
}
d
x
x
1
x
1
x
2
x
2
x
2
d
T
d
x
Q
[
N
]
T
[
N
]
T
k
x
A
[
N
]
T
=
A
d
x
+
hPT
a
d
x
+
(7.11)
x
1
x
1
x
1
Equation 7.11 is in the desired finite element form:
k
(
e
)
{
}=
f
(
e
Q
+
f
(
e
h
+
f
(
e
g
T
(7.12)
