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requiring a major effort to achieve a solution. Often substructuring can be employed to
avoid excessive computational cost. This technique is especially suitable for structures in
which parts of the system (substructures) with the same number and types of elements
are repeated. Then the stiffness matrix of a repeated part needs to be evaluated only once.
Substructuring also helps one to retain mechanical insight into the behavior of complicated
systems despite a very large number of unknowns.
The first step in substructuring is to divide the large structure into several smaller parts
referred to as “substructures.” Then discretize each substructure into elements, obtain the
element stiffness matrices and loading vectors, and assemble the element matrices into a
set of global stiffness equations for each substructure. At the boundaries where the sub-
structures connect, there will be common nodes called
exterior nodes
. The substructures
can be viewed as being formed of elements with many interior nodes along with some
exterior nodes. The global stiffness matrix and the loading vector of the complete struc-
ture is assembled from the stiffness matrices and loading vectors of the substructures, with
the displacements of the exterior nodes as unknowns. The degrees of freedom of the int
e
-
rior nodes are
condensed out
of the stiffness equations for each substructure. Let
K
i
i
and
P
be the stiffness matrix and applied loading vector for the
i
th substructure,
V
i
1
an
d
V
i
2
be
the displacements of the exterior nodes and interior nodes, respectively, and
P
i
1
and
i
2
P
be the loading vectors associated with
V
i
1
and
V
i
2
. The stiffness equations for this sub-
structure can be expressed as
K
i
11
V
i
1
V
i
2
i
1
P
K
i
22
=
(5.75)
K
i
21
K
i
22
i
2
P
i
=
P
K
i
V
i
In expanded form,
i
1
K
i
11
V
i
1
+
K
i
12
V
i
2
=
P
(5.76)
i
2
K
i
21
V
i
1
+
K
i
22
V
i
2
=
P
Solve the second equation of Eq. (5.76) for
V
i
2
,
giving
V
i
2
=
K
i
22
−
1
−
K
i
22
−
1
i
2
P
K
i
21
V
i
1
(5.77)
Substitute the expression of
V
i
2
into the first equation of Eq. (5.76), giving
P
i
K
i
V
i
=
(5.78)
i
2
. Equation (5.78)
contains only the displacements of the exterior nodes as the displacements of the interior
nodes have been condensed out. This process is called
static condensation
.
The stiffness equations in the form of Eq. (5.78) for the substructures can be assembled
into the global stiffness equations for the whole structure, with the displacement vector
formed of the displacements of the exterior nodes. After these displacements are obtained,
they can be substituted into the substructure stiffness equations, which can then be used to
compute the displacements of the interior nodes.
i
1
=
P
P
=
K
i
11
−
K
i
12
(
K
i
22
)
−
1
K
i
21
,
V
i
=
V
i
1
,
and
P
i
−
K
i
12
(
K
i
22
)
−
1
with
K
i
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