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The desired transfer matrix in Sign Convention 2 notation is obtained from Eqs. (4.77) and
(4.78) as
.
k
−
1
k
−
1
ab
v
b
...
−
ab
k
aa
v
a
...
U
vv
U
v
p
v
a
=
=
...............
.........
(4.79)
.
k
bb
k
−
1
k
bb
k
−
1
p
b
p
a
U
p
v
U
pp
p
a
k
ba
−
ab
k
aa
ab
4.4.6
Inclusion of Axial and Torsional Motion
Combine the stiffness matrices of Eq. (4.12) and Example 4.1 to give
.
−
N
a
u
a
EA
EA
.
V
a
w
a
12
EI
6
EI
12
EI
6
EI
−
−
−
3
2
3
2
.
M
a
θ
a
6
EI
4
EI
6
EI
2
EI
−
2
2
...
=
...
...
...
...
...
...
...
(4.80)
.
EA
EA
N
b
−
u
b
.
12
EI
6
EI
12
EI
6
EI
V
b
−
w
b
3
2
3
2
.
6
EI
2
EI
6
EI
4
EI
M
b
θ
−
b
2
2
p
i
k
i
v
i
=
as the stiffness matrix for a beam undergoing bending and extension. The sign convention
of Fig. 4.1b (Sign Convention 2) was used in constructing this stiffness matrix.
In the case of a bar undergoing bending and torsion, the stiffness matrix in a nondimen-
sional displacement form is (Sign Convention 2)
V
a
M
a
M
ta
V
b
12
−
60
−
12
−
6
0
w
a
/
θ
a
φ
−
6
4
0
6
2
0
EI
J
∗
J
∗
0
0
00
−
a
=
M
b
M
tb
−
12
6
0
12
6
0
w
/
b
−
6
2
0
6
4
0
θ
b
φ
J
∗
00
J
∗
0
0
−
b
(4.81)
p
i
k
i
v
i
=
where
GJ
EI
=
J
J
∗
=
I
Note that this form differs from that of the stiffness matrix of Eq. (4.13) due to different
scaling.
2
(
1
+
ν)
4.5
Transfer Matrices
4.5.1
Determination of Transfer Matrices
There are many approaches for deriving transfer matrices such as the one leading to Eq.
(4.8c). Some of the techniques are described in the references for this chapter. One of these
techniques is the direct integration of the governing equations, either using the higher order
or the first order system of governing equations.
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