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FIGURE 4.4
Calculation of flexibility coefficients.
Since
V
b
and
M
b
are to be the independent variables, the variables
V
a
and
M
a
(or
p
a
) will
be eliminated. From Eq. (4.17c),
−
10
−
p
a
p
b
U
−
1
pp
I
p
b
=
1
10
01
g
T
p
R
p
=
=
p
b
=
(4.26)
In summary,
v
R
=
gv
g
T
p
R
p
=
(4.27)
For a properly established
g
, these general relations are valid when other variables are
selected as the independent variables.
Virtual Work
Since rigid body movement does not contribute to the internal virtual work, the virtual
work expressed in terms of the complete force and displacement vectors,
p
and
v
, has
the same value as when it is expressed in terms of the vectors
p
R
and
v
R
from which the
rigid body displacements have been removed. Thus
p
R
δ
v
R
=
p
R
δ(
)
=
p
R
g
δ
=
p
T
δ
gv
v
v
(4.28)
Case 2
As a second possibility, suppose
M
a
and
M
b
are selected to be the independent vari-
ables. Physically, this corresponds to a beam segment with end moments (Fig. 4.5). Again,
the rigid-body displacements are to be suppressed. For this case, set
M
a
M
b
p
R
=
(4.29)
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