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For the second element, which begins at node
b
and ends at node
c
,
k
bb
2
k
bc
k
2
=
(5.46b)
k
cb
k
cc
a
T
ka
is obtained in terms of the submatrices
The global stiffness matrix
K
formed using
K
=
k
i
jk
as
I
k
1
0
0
k
aa
k
ab
I
=
I
I
k
ba
k
bb
+
k
bb
k
bc
II
(5.47)
2
I
k
cb
k
cc
I
a
T
k
a
=
K
The process of forming an assembled matrix can be observed in Eq. (5.47). The number
of columns in the global connectivity matrix
a
is equal to the number of system DOF. Post-
multiplication of
k
by
a
places the coefficients of
k
in the proper columns of the assembled
system stiffness matrix, whereas premultiplication by
a
T
locates the coefficients of
k
in the
proper rows of the global stiffness matrix. The same process performed separately on each
element stiffness matrix expands it into its proper location in the global stiffness matrix.
In contrast to the unassembled stiffness equations which appear as
=
=
=
p
a
p
b
p
b
p
c
v
a
v
b
v
b
v
c
p
1
v
1
k
1
[
k
]
(5.48)
k
2
p
2
v
2
p
=
k
v
the assembled global stiffness equations are of the form [Eq. (5.47)]
k
1
p
a
p
b
+
V
a
V
b
V
c
k
aa
k
ab
V
a
V
b
V
c
0
=
=
p
b
(5.49)
k
ba
k
bb
+
k
bb
k
bc
k
2
p
c
k
cb
k
cc
0
KV
with
v
a
v
b
=
V
a
V
b
V
c
=
v
b
(5.50)
v
c
Inspection of the assembled stiffness matrix of Eq. (5.49) provides insight as to why a
stiffness matrix should be assembled using addition rather than the multiplication implied
by
K
a
T
ka
. In Eq. (5.49), it is apparent that all coefficients of
K
either are taken directly
from
k
1
or
k
2
, or, as in the case of the overlapping boxes, are the sum of
k
1
and
k
2
coeffi-
cients. The summation process shown in Eq. (5.49) can be programmed without much
difficulty.
Although this summation assembly process is considered in depth in Section 5.3.4, the
summation procedure is evident from Eq. (5.49). The assembled stiffness matrix
K
is formed
by summation of those element stiffness matrix coefficients with identical subscripts.
=
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