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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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