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We can treat the beam of Fig. 3.14c in a similar fashion. According to the reciprocal
theorem,
M
∗
P
·
w
|
x
=
0
,
due to
M
∗
=
·
θ
|
x
=
0
,
due to
P
x
2
L
2
where
θ
=−
d
w/
dx
=−
P
(
−
)/(
2
EI
).
Thus,
x
=
0
,
due to
P
=
M
∗
P
θ
M
∗
L
2
2
EI
w
|
x
=
0
,
due to
M
∗
=
(3)
In the case of the beam of Fig. 3.14d,
L
PL
4
8
EI
P
6
EI
(
x
3
3
L
2
x
2
L
2
P
w
|
x
=
0
,
due to
p
0
=
p
0
−
+
)
dx
=
p
0
(4)
0
or
p
0
L
4
8
EI
w
|
x
=
0
,
due to
p
0
=
(5)
An influence line is a response, such as a reaction force, produced by a unit applied
loading, suc
h a
s a force, as it traverses the member. Then, for any other value of the applied
loading, say
P
, the response at a poi
nt
in the system is obtained by multiplying the value
of the influence line at that point by
P
. As demonstrated by example in Chapter 3 of the
first edition of this topic, reciprocal theorems are useful in finding influence lines.
Problems
Principle of Virtual Work and Related Methods
3.1 Use the principle of virtual work to find the reactions in the beam of Fig. P3.1.
FIGURE P3.1
Beam with hinge.
Answer:
=
(
a
2
−
a
1
)
a
1
L
−
a
2
R
a
P,
R
b
=
P
−
R
a
,
M
b
=
P
a
2
a
2
3.2 Use the principle of virtual work to find the forces in the bars of the system shown in
Fig. P3.2 if the material is linearly elastic. Find the horizontal displacement of point
a
.
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