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FIGURE 5.33
Equilibrium states b 1 for the redundants.
Now that f , b 0 , and b 1 have been established, P x and X can be computed using Eqs. (5.129)
and (5.130), respectively. Vector P x contains the redundant forces as noted in (5), and matrix
X isa2
2 matrix.
Before introducing specific calculations, it may be of interest to observe how the nodal
force equations [Eq. (5.129) or (5.130)] can be derived directly without reference to the
principle of complementary virtual work. Equation (5.129) is based on the compatibility
conditions, which require that all end displacements of the various elements joined at a
particular node must be equal to the value of the displacement of the node. That is, v
×
=
aV
.
Since a =
b T , where a is defined by V
a v , the compatibility can be expressed as
=
= b 0 + (
T v
b T v
b 0 v
X T b 1 v
V
=
b 1 X
)
=
+
(9)
The compatibility conditions that we must impose across the in-span supports, the locations
of the redundant reactions, are the continuity of the slopes, i.e.,
1
b
2
b
2
c
3
c
θ
= θ
and
θ
= θ
.
From
(1) and (8), these conditions are
b 1 v
0
=
(10)
Then, with v
=
fp and p
= (
b 0 +
b 1 X
)
P , (10) can be written as
b 1 v
b 1 fp
b 1 f
0
=
=
=
(
b 0
+
b 1 X
)
P
(11)
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