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FIGURE 5.12
A three-node, two-element framework.
node. This matrix would be populated with null or unit values. Such a matrix is termed a
Boolean
1
matrix.
To illustrate the compatibility conditions between the element and system displacements
for a structure, consider the three-node, two-element system of Fig. 5.12. The compatibility
conditions at nodes
a
,
b
, and
c
are
v
a
=
V
a
v
b
=
v
b
=
V
b
(5.36)
v
c
=
V
c
or, in the notation of Eq. (5.35),
=
v
a
v
b
v
b
v
c
=
I
v
1
V
a
V
b
V
c
I
I
(5.37)
v
2
I
v
=
a
V
where
I
is the unit matrix. Note that these compatibility conditions serve to transform a
displacement vector
v
containing displacements at the same node from different elements,
i.e.,
v
b
and
v
b
,
into the nodal displacement vector
V
.
If Eq. (5.32) were written in matrix rather than in summation form, it would appear as
M
v
iT
k
i
v
i
p
io
v
T
1
δ
(
−
)
=
δ
(
kv
−
p
)
=
0
(5.38)
i
=
where
v
as defined in Eq. (5.35) is an unassembled displacement vector,
[
p
10
p
20
p
M
0
]
T
p
=
···
(5.39)
1
George Boole (1815-1864) was a self-taught British mathematician. Among his works was a brief, influential
topic written in 1847,
The Mathematical Analysis of Logic
, wherein he asserted that logic should be grounded in
mathematics, rather than be associated with metaphysics. He founded symbolic logic with the topic
An Investi-
gation of the Laws of Thought
, which appeared in 1854. At the time he was a professor of mathematics in Queen's
College, Cork.
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