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to zero except for the element corresponding to the column being calculated. This entry
should be set equal to one. Then this column of
b
0
is equal to
p
T
h
us, to set up
b
0
, a unit
value is employed one at a time for each of the applied loads of
P
, while the redundant
forces
P
x
are set equal to zero. The forces
p
(which are now equal to a column of
b
0
) can then
be computed using equilibrium alone, since the structure under this loading is statically
determinate.
Matrix
b
1
can be computed in a similar manner. This time, set the applied loads
P
equal
to zero, and give the redundants
P
x
unit values one by one. Those redundants not assuming
a unit value are set to zero. This leads to the equilibrium states
b
1
.
=
p
, column by column.
In theory, this procedure for determining
b
0
and
b
1
applies to any problem. In practice,
however, it is often difficult to establish
b
0
and
b
1
because it is not easy to determine the
equilibrium states automatically.
Insertion of the equilibrium conditions
p
bP
in the form of Eq. (5.125) into the principle
of complementary virtual work expression will provide equations in terms of the system
nodal forces. Thus,
=
M
p
iT
v
i
v
i
p
T
p
T
1
δ
(
−
)
=
δ
(
v
−
v
)
=
δ
(
fp
−
v
)
i
=
P
x
b
1
[
f
=
δ
(
b
0
P
+
b
1
P
x
)
−
v
]
=
0
(5.126)
where
δ
p
=
δ
p
0
+
δ
p
x
=
δ
p
x
=
b
1
δ
P
x
f
1
=
f
2
diagonal [
f
i
] is an unassembled
global flexibility matrix
f
=
.
.
.
f
M
=
[
v
1
v
2
...
v
M
]
T
.
v
,
v
are unassembled displacement vectors, e.g.,
v
Define
b
1
fb
1
F
=
(5.127)
as the assembled system flexibility matrix and
b
1
(
=−
−
)
V
fb
0
P
v
(5.128)
as the assembled applied displacement vector, so that Eq. (5.126) becomes
P
x
(
δ
FP
x
−
V
)
=
0
or
=
FP
x
V
(5.129)
which is a set of algebraic equations for the un
kn
own nodal forces. These equations repre-
sent the global statement of compatibility
V
V
for a
ll
node
s
of the system.
If there are no non-ze
ro
prescribed displacements
v
, i.e.,
v
=
=
0, then from Eqs. (5.128)
and (5.129) with
P
x
=
X P
, the nodal force equations become
V
=
FX
(5.130)
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