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FIGURE P3.22
In-plane motion of circular bar.
3.23 For the tapered cantilever beam (Fig. P3.23) the moment of inertia varies as
I
=
)
−
1
(
c
1
x
+
c
2
where
c
1
and
c
2
are constants. Derive a formula for the deflection of the
free end.
FIGURE P3.23
Miscellaneous Problems
3.24 Show how to find the reactions in the continuous beam of Fig. P3.24a.
Hint:
Choose
R
a
,R
b
,
...
,R
n
, the reactions at supports at
a, b,
...
,n
, as redundant
forces and use the relations
∂
U
i
U
i
U
i
∂
,
∂
R
a
=
0
,
R
b
=
0
,
...
R
n
=
0
∂
∂
∂
3.25 Indicate how to find the forces in the springs of the beam of Fig. P3.24b.
Hint:
Instead of being equal to zero as in Problem 3.24, the derivative of the com-
plementary strain energy with respect to a reaction
R
s
is now equal to
−
/
R
s
k
s
, i.e.,
∂
,n
. The negative sign indicates that the displace-
ment in the spring is opposite in sense to the reaction on the beam.
U
i
/∂
R
s
=−
R
s
/
k
s
,
s
=
a, b,
...
3.26 Compute the displacement and reaction at the left end of the beam of Fig. P3.26. How
should this problem be approached if the spring does not fit properly?
p
0
L
4
/
8
Answer:
R
=
3
,
w
=
R
/
k
x
=
0
L
3
EI
/
k
+
/
If the spring is
V
s
units too long (replace
V
s
by
−
V
s
if it is too short) before
p
0
is
applied, then force the beam in place and use
∂
U
i
∂
R
k
R
=
V
s
−
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