Civil Engineering Reference
In-Depth Information
In terms of the dimensionless coordinate
s
=
x
/
L
, we have
d
x
=
L
d
s
and
d
/
d
x
=
(1
/
L
)d
/
d
s
,
so the terms of the conductance matrix are expressed as
1
k
x
A
L
d
N
i
d
s
d
N
j
d
s
k
ij
=
d
s
i
,
j
=
1, 3
0
The derivatives of the interpolation functions are
d
N
1
d
s
=
4
s
−
3
d
N
2
d
s
=
4(1
−
2
s
)
d
N
3
d
s
=
4
s
−
1
Therefore, on substitution for the derivatives,
1
1
k
x
A
L
k
x
A
L
(4
s
−
3)
2
(16
s
2
k
11
=
d
s
=
−
24
s
+
9) d
s
0
0
16
s
3
3
+
9
s
1
k
x
A
L
7
k
x
A
3
L
−
12
s
2
=
0
=
Via mathematically identical procedures, the remaining terms of the conductance matrix
are found to be
8
k
x
A
3
L
k
12
=
k
21
=−
k
x
A
3
L
k
13
=
k
31
=
16
k
x
A
3
L
k
22
=
8
k
x
A
3
L
k
23
=
k
32
=−
7
k
x
A
3
L
k
33
=
A two-element model with node numbers is shown in Figure 7.1b. Substituting
numerical values, we obtain, for the aluminum half of the rod (element 1),
=
7
−
81
−
8 6
−
8
1
2
.
6389
−
3
.
0159
0
.
3770
k
(1)
=
200(
/
4)(0
.
006)
2
3(0
−
3
.
0159
6
.
0319
−
3
.
0159
.
5)
−
87
0
.
3770
−
3
.
0159
2
.
6389
