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The equations of equilibrium for the nodes were expressed in Eq. (5.52) in the form
p
1
p
2
p
M
P
1
P
2
P
N
[
b
∗
1
b
∗
2
b
∗
M
]
...
=
(5.120)
b
∗
p
=
P
where
N
is the number of nodes. The matrix
b
∗
contains information detailing which
element is connected to which node.
A comparison of Eqs. (5.38) and (5.52) indicates that
b
∗
=
a
T
(Eq. 5.56). This relationship
can be verified in general. Recall that from the principle of virtual work
M
v
T
p
v
iT
p
i
V
T
P
V
T
P
δ
=
δ
=
δ
=
δ
(5.121)
i
=
1
where
v
and
p
are the unassembled vectors of element displacements and forces. Substi-
tute Eq. (5.120) into the right-hand side of this expression and Eq. (5.35) into the left-hand
side, giving
V
T
b
∗
p
T
p
V
T
a
T
p
δ
=
δ(
aV
)
=
δ
(5.122)
so that
b
∗
=
This equality is an interesting relationship between the global equilibrium
and kinematic matrices.
a
T
.
bP
[Eq. (5.55)] is needed to superimpose the element
relationships. Recall that since
b
∗
is usually not a square matrix,
b
cannot be obtained by
inversion of
b
∗
.
The reciprocal relationship
p
=
However,
b
∗
is a square matrix for statically determinate structures, such
as a tree-like structure containing no meshes, i.e., closed branches.
For statically indeterminate systems, it is convenient to distinguish between a statically
deter
m
inate set of forces and the remaining
redundant
forces. Thus, the equilibrium relations
p
=
bP
will be split into two parts. Define
b
=
b
0
+
b
1
X
(5.123)
where
b
0
is obtained from a statically determinate system,
X
is formed of dimensionless
forces corresponding to those forces that are selected as redundants and comprises the
unknowns for the force method, and
b
1
is the equilibrium state derived for unit conditions
associated with the redundants of
X
.
The equilibrium conditions then appear as
p
=
(
b
0
+
b
1
X
)
P
=
p
0
+
p
x
(5.124)
with
p
0
=
b
0
P
and
p
x
=
b
1
XP
=
b
1
P
x
The introduction of
P
x
=
XP
permits Eq. (5.124) to be expressed as
p
=
b
0
P
+
b
1
P
x
(5.125)
where
P
x
now represents the unknown redundant forces of the force method.
Equation (5.123) provides us with the procedure for establishing
b
0
and
b
1
If
P
x
is set
equal to zero,
b
0
can be calculated column by column as the elements of
P
are set equal
.
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