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Principle of Stationery Potential Energy
Among all admissible displaced configurations of a conservative system those that satisfy
equations of equilibrium make the potential energy stationery with respect to small
admissible variations of displacement.
If the stationery condition is a relative minimum,
the equilibrium state is stable.
Example
:
For the beam shown, using an admissible displacement function
(
)(
)
x
L
x
a
x
v
−
=
1
, use
the principle of stationery potential energy to find the maximum deflection of the beam.
P
x
y,v
Solution:
The potential energy of the system is given by
W

U
=
Π
The strain energy in the beam is
dV
2
1
U
x
x
V
∈
=
∫
σ
Since
I
My
x
=
and correspondingly
EI
My
x
=
∈
,
∫
=
V
dV
EI
My
I
My
2
1
∫
=
V
dV
EI
y
M
2
2
2
2
1
Now given,
2
2
dx
v
d
I
E
M
=
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View Full Document∫
=
V
dV
dx
v
d
EI
y
I
E
U
2
2
2
2
2
2
2
2
1
∫
V
dV
y
dx
v
d
E
2
2
2
2
2
1
∫
∫
=
dx
dA
y
dx
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 Fall '09
 Law

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