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and, with
=
L
/
2
,
2
p
0
L
2
15
4
p
0
5
2
p
0
L
5
4
p
0
V
a
=
=
M
a
=−
=−
(11)
15
,V,
and
M
are calculated using Eq. (5.6) and can be printed out at designated locations. For
example, the displacements and forces at nodes
b
and
c
are given by
Since
w
=
0
,
θ
=
0 and
M
a
,V
a
are given by (11),
z
a
is now known. The variables
w
,
θ
a
a
U
1
z
a
z
b
=
z
x
=
L
/
2
=
(12)
U
2
U
1
z
a
=
z
c
=
z
x
=
L
=
Uz
a
Between nodes, results are computed by adjusting the coordinate in the transfer matrix for
that element. If, for example, the state variables are sought at the midpoint of the second
element, then
U
2
U
1
z
x
=
L
/
2
+
(
L
/
2
)/
2
=
((
L
/
2
)/
2
)
(
L
/
2
)
z
a
4
240
EI
17
p
0
7
p
0
L
4
3840
EI
17
p
0
L
3
3072
EI
−
7
p
0
0
0
3
384
EI
−
U
2
2
U
1
=
()
4
p
0
5
=
3
p
0
40
p
0
=
3
p
0
L
80
p
0
L
2
40
1
(13)
2
2
4
p
0
−
15
1
10
1
5.1.5 Some Computational Considerations
In Chapter 4 it was demonstrated that a transfer matrix can be transformed into a stiff-
ness matrix. Much of the present chapter deals with the displacement method of structural
analysis in which element stiffness matrices are assembled into a global stiffness matrix
and the resulting system of equations is solved for the displacements. This displacement
method of analysis is probably the most reasonable approach for solving structural me-
chanics problems for which element transfer matrices are available for the elements, that
is, rather than preparing a computer program to implement the progressive matrix multi-
plication which characterizes the transfer matrix method, it is usually better to transform
the transfer matrices into stiffness matrices and utilize the displacement method. After the
displacements at the nodes are computed and the forces at the nodes are determined using
the stiffness matrices, the transfer matrices can be used to print out the displacements and
forces along the member.
Numerical Difficulties
The use of the pure transfer matrix method, with its progressive multiplications of element
transfer matrices as it proceeds along the member, tends to encounter numerical difficulties.
It should not be surprising that serious numerical difficulties would arise for chain-like
structures of such a nature that an occurrence, e.g., an applied loading, at one location
has little effect on the response at a distant location. There are actually several causes
of numerical difficulties and several seemingly effective corrective measures that can be
taken. Detailed discussions of the sources of the numerical problems and techniques for
overcoming them are given in Horner and Pilkey (1978), Marguerre and Uhrig (1964), and
Pestel and Leckie (1963).
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