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columns, and define
V
=
elements and
V
S
V
]
T
[
V
i
V
j
...
with
as the column vector
formed by deleting
V
from
V
.
Similar definitions apply to
V
and
V
S
as subsets of
V
.
Equation (5.109) can be simplified
to
K V
V
=
K
−
1
V
−
(5.110)
K
so that Eq. (5.110) can be written as
=
K
−
1
Let
Y
Y V
V
=
−
V
(5.111)
Rearrange Eq. (5.111) into the following partitioned set of equations
V
V
S
V
V
S
Y
Y
S
V
=
−
(5.112)
where
Y
has been partitioned into
Y
and
Y
S
to correspond with the rearrangement of
V
into
V
and
V
S
.
From the first
equations of Eq. (5.112)
V
=
V
−
Y V
(5.113)
or
V
=
(
+
Y
V
)
−
1
I
(5.114)
and from the lower portion of Eq. (5.112)
V
S
=
V
S
−
Y
S
V
(5.115)
Equations (5.114) and (5.115) constitute the reanalysis solution to the modified structure
problem. In Eq. (5.114), the response
V
of the modified system is expressed in terms of the
available response
V
of the original structure. The computational efficiency in analyzing
the modified structure stems from the fact that Eq. (5.114) is a system of
equations, rather
than a much greater number of equations that occur if the modified system of Eq. (5.107)
were to be solved.
5.4 Force Method
The force method, although it is used consistently for hand calculations of small problems,
does not enjoy much popularity as an approach for solving large-scale problems. This
method was the subject of intense investigation during the early evolution of computer au-
tomated structural analysis. If the currently available general purpose computer programs
are used as a measure of popularity, the displacement method completely overshadows
the force method. For reasons that will be delineated in this section, the force method is not
as easily automated for large-scale problems as the displacement method and the menu
of elements available for a force method analysis is quite limited. Variations on the force
method are still under development. See, for example, Gallagher (1993). The basis of the
force method is the principle of complementary virtual work. Recall from Chapter 2 that the
principle of complementary virtual work is equivalent to the global form of the kinematic
admissibility conditions. Hence, the force method is sometimes referred to as the
compatibil-
ity
method, as well as the
flexibility
or
influence coefficient
method. The derivation of the force
method equations follows closely the derivation (Section 5.3.1) of the displacement method
equations, since the two methods, as will be demonstrated later, are “dual” approaches.
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