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FIGURE 5.4
Incorporation of point occurrences into a transfer matrix solution.
due to the spring is upwards, while
P
of Fig. 5.2 was downwards. Thus,
+
−
1000
.
w
V
M
...
w
V
M
...
0
0100
.
0
010
.
k
0
=
(1)
0001
.
0
... ... ... ... . ...
0000
.
1
1
1
j
j
No distinction is made between the use of point and field matrices in the progressive
matrix multiplications of a transfer matrix solution. Thus, for example, for the beam of
Fig. 5.4, the state vector
z
f
appears as
z
x
=
L
=
U
5
U
4
U
d
U
3
U
2
U
b
U
1
z
f
=
z
a
(5.10)
5.1.2 Transfer Matrix Catalogue
Catalogues of transfer matrices for various structural elements with arbitrary loading are
available in many sources, e.g., Pestel and Leckie (1963), Pilkey (1994), Pilkey and Chang
(1978). One of the most useful transfer matrices, which was derived in Chapter 4 and is
displayed in Table 4.3, is that for the beam element. If extension, as well as bending, is to be
included, the transfer matrix is expanded as in Chapter 4, Eq. (4.123). Then, the state vector
is
NVM
]
T
(5.11)
where
u
is the axial displacement, and
N
is the axial force. Torsion is included in a similar
manner. Table 5.1 provides a variety of point matrices, including one necessary to transfer
the state variables across a corner.
z
=
[
u
wθ
5.1.3
Incorporation of Boundary Conditions
The transfer matrices developed thus far can be used to solve boundary value problems
for beams. Once the overall or global transfer matrix is formed as in Eq. (5.7), the boundary
conditions can be applied to find the unknown initial parameters of
z
a
. As mentioned
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