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TABLE 4.3
Transfer Matrix for a Beam Elements, Part a: (
Continued
)
Note: The column 5 definitions of Part b along with case 6 or 7 of Part c of this table apply for
any magnitude of
. However, usually the transfer matrix elements will then be complex
quantities, and the computer calculations may be quite difficult.
To use this general transfer matrix, follow the steps:
λ
,
ζ
,
η
1.
Calculate the three parameters
λ
,
ζ
,
and
η
. If shear deformation is not to be considered, set
1
/
k
s
GA
=
0
2.
Compare the magnitude of these parameters and look up the appropriate
e
i
functions in Parts
b and c of this table.
3.
Substitute these
e
i
expressions in the general transfer matrix above.
A second general, viable technique for deriving transfer matrices is based on the use
of the Laplace transform. The homogeneous form of Eq. (4.94) can be combined into a
single fourth-order equation
d
4
w
dx
4
+
(ζ
−
η)
d
2
w
dx
2
+
(λ
−
ζη)w
=
0
(4.109)
where
k
∗
)/
ζ
=
(
N
−
EI,
η
=
k
/(
k
s
GA
)
,
λ
=
k
/
EI
The Laplace transform of Eq. (4.109) is
[
s
4
s
2
w(
s
)
+
(ζ
−
η)
+
(λ
−
ζη)
]
s
3
s
2
w
(
w
(
)
+
w
(
)
+
(ζ
−
η)w
(
=
w(
0
)
+
0
)
+
s
0
0
0
)
+
(ζ
−
η)
s
w(
0
)
(4.110)
where
s
is the transform variable. The inverse transform gives
w(
)
=
[
e
1
(
)
+
(ζ
−
η)
e
3
(
)
w(
)
+
[
e
2
(
)
+
(ζ
−
η)
e
4
(
)
w
(
)
x
x
x
]
0
x
x
]
0
)w
(
)w
(
+
e
3
(
)
+
e
4
(
)
x
0
x
0
(4.111)
where
L
−
1
s
4
−
i
e
i
(
x
)
=
(4.112)
s
4
+
(ζ
−
η)
s
2
+
λ
−
ζη
The quantity
L
−
1
indicates the inverse Laplace transform. It follows from Eq. (4.112) that
several useful identities hold:
d
dx
e
i
+
1
(
e
i
(
x
)
=
x
)
i
=−
2
,
−
1
,
0
,
1
,
2
,
3
(4.113)
x
e
i
+
1
(
x
)
=
e
i
(
u
)
du
i
=
4
,
5
,
6
0
Take the first three derivatives of
w(
x
)
and arrange the results as
w(
x
)
e
1
+
(ζ
−
η)
e
3
e
2
+
(ζ
−
η)
e
4
e
3
e
4
w(
0
)
=
w
(
x
)
e
0
+
(ζ
−
η)
e
2
e
1
+
(ζ
−
η)
e
3
e
2
e
3
w
(
0
)
(4.114)
w
(
w
(
x
)
e
−
1
+
(ζ
−
η)
e
1
e
0
+
(ζ
−
η)
e
2
e
1
e
2
0
)
w
(
w
(
x
)
e
−
2
+
(ζ
−
η)
e
0
e
−
1
+
(ζ
−
η)
e
1
e
0
e
1
0
)
w
(
x
)
=
Q
(
x
)
w
(
0
)
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