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is an unassembled vector of elements loads, and
=
k
1
k
2
diagonal [
k
i
]
k
=
(5.40)
.
.
.
k
M
is an unassembled global stiffness matrix.
If
v
aV
of Eq. (5.35) is inserted in Eq. (5.38), the principle of virtual work expression
would then be written in terms of the system nodal displacements. Thus,
=
v
T
V
T
a
T
V
T
a
T
kaV
a
T
p
δ
(
kv
−
p
)
=
δ
(
kaV
−
p
)
=
δ
(
−
)
=
0
(5.41)
Define
a
T
ka
K
=
(5.42a)
as the assembled system stiffness matrix, and
a
T
p
P
=
(5.42b)
as the assembled applied load vector, so that Eq. (5.41), the principle of virtual work,
becomes
V
T
δ
(
−
)
=
KV
P
0
(5.43)
or for arbitrary nodal displacements
P
(5.44)
which
is
a set of algebraic equations for the unknown nodal displacements. The load
vector
P
also includes loads applied directly at the nodes.
In terms of the system
no
dal forces
P
, the principle of virtual work can be expressed as
−
(δ
KV
=
+
δ
)
=
δ
V
T
(
−
)
=
W
i
W
e
P
P
0
,
so that
V
T
V
T
−
(δ
+
δ
)
=
δ
(
−
)
=
δ
(
−
)
=
W
i
W
e
P
P
KV
P
0
(5.45)
Since the pr
in
ciple of virtual work is equivalent to the conditions of equilib
ri
um, it follows
that
KV
=
=
P
. The matrices
are “assembled” in the sense that the duplications occurring in
v
[e.g., in Eq. (5.37) where
v
b
=
P
is an expression of the global statement of equilibrium,
P
v
b
] are removed by invoking the compatibility conditions. Thus, the vector
v
of
unassembled unknown displacements is replaced in the governing equations of equili-
brium by the vector
V
of
assembled unknown displacements.
The relations
KV
P
are equilibrium equations that form the core of the displacement
method and which can be solved for the system nodal displacements. These displacements
can then be used in computing forces, stresses, and other displacements.
The topological information for the assembly of the stiffness matrix is contained in the
connectivity matrix
a
. The congruent transformation
a
T
ka
=
K
is rarely used in practice
to form
K
. The matrix
a
of this “assembly by multiplication” may contain many zero
coefficients. In practice, this assembly of
K
as a congruent transformation is avoided in
favor of an “as
se
mbly
b
y addition” procedure. Thus, the assembled matrix relationships
K
=
a
T
p
are more of conceptual than practical value.
The assembly of
K
by an addition or superposition technique is illustrated using the
two-element, three-node system of Fig. 5.12. For element number 1, which spans be-
tween global nodes
a
and
b
, the stiffness matrix can be written in terms of submatrices as
[Eq. (5.19)]
a
T
ka
and
P
=
=
k
aa
1
k
ab
k
1
=
(5.46a)
k
ba
k
bb
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