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For the system of Fig. 5.12, the incidence table of Eq. (5.64) shows that the indices of the
stiffness matrix
k
1
for bar 1 are
a
and
b
, while for bar 2 the stiffness matrix indices are
b
and
c
. Summation of element stiffness matrix coefficients or submatrices of like subscripts
gives
k
aa
k
ab
K
aa
K
ab
K
ac
=
k
ba
k
bb
+
k
bb
k
bc
K
=
K
ba
K
bb
K
bc
(5.68)
k
cb
k
cc
K
ca
K
cb
K
cc
As can be imagined, numerous schemes have been devised for the efficient handling of
data during the assembly process. In some cases, all element stiffness matrices are generated
and stored in a background storage unit. They are called up as needed to create
K
in active
storage. In other cases, the information from each element is processed into the system
matrix as each element stiffness matrix is created, thus eliminating the use of background
storage.
5.3.5 Incorporation of Boundary Conditions, Reactions
The global stiffness matrix, like the element stiffness matrix, is singular. This singularity
can be illustrated using the structure of Fig. 5.12. The global stiffness can be expressed
as [Eq. (5.68)]
K
aa
K
ab
K
ac
V
a
V
b
V
c
P
a
P
b
P
c
=
K
ba
K
bb
K
bc
(5.69)
K
ca
K
cb
K
cc
K
V
=
P
If the structure is not constrained, the displacements can occur in two forms: elastic defor-
mation of the structure due to the applied loading and movement as a rigid body. Due to
this rigid body motion, two of the nodal displacements, say
V
a
and
V
c
,
can have arbitrary
values, while the third nodal displacement
depends on
V
a
and
V
c
. The stiffness rela-
tions of Eq. (5.69) form a set of simultaneous linear equations. From Cramer's rule, if the
equations do not have a unique solution, the determinant of the coefficient matrix
K
must
be zero,
K
is singular. A unique solution exists only after the structure is constrained, i.e.,
displacement boundary conditions are imposed, so that rigid body motion is prevented.
The introduction of boundary (support) conditions is r
e
adily portrayed by appropri-
ately partitioning the global equilibrium equations
KV
(
V
b
)
P
, although, in practice, matrix
rearrangement operations are avoided. Blind partitioning of the global equations would
normally lead to an increase in the bandwidth, which, as explained in Section 5.3.7, is
undesirable. To illustrate the partitioning approach, suppose the
di
splacement vector
V
contains both prescribed displacements, which will be denoted by
V
, and unknown nodal
displacements which will be designated by
V
y
. Typically, the prescribed displacements are
joint displacements that are zero. Often these displacements are referred to as constrained
DOF. The remaining unconstrain
ed
displacements are called the active DOF. Also, the force
vector can contain applied forces
P
and unk
no
wn nodal forces
P
v
,
i.e., the support reactions
corresponding to prescribed displacements
V
. Then the global equations can be rearranged
to achieve the partitioned form
=
K
11
V
V
y
P
P
K
12
=
(5.70)
K
21
K
22
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