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Section 5.6
Solid Mechanics Part I
Kelly
208
5.6 The Principle of Minimum Potential Energy
The
principle of minimum potential energy
follows directly from the principle of
virtual work (for elastic materials).
5.6.1
The Principle of Minimum Potential Energy
Consider again the example given in the last section; in particular rewrite Eqn. 5.5.15 as
0
2
2
2
2
2
2
1
1
1
=
⎭
⎬
⎫
⎩
⎨
⎧
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
+
−
B
B
u
L
A
E
L
A
E
Pu
δ
(5.6.1)
The quantity inside the curly brackets is defined to be the
total potential energy
of the
system,
Π
, and the equation states that the variation of
Π
is zero – that this quantity does
not vary when a virtual displacement is imposed:
0
=
Π
(5.6.2)
The total potential energy as a function of displacement
u
is sketched in Fig. 5.6.1.
With
reference to the figure, Eqn. 5.6.2 can be interpreted as follows: the total potential energy
attains a stationary value (maximum or minimum) at the
actual
displacement (
1
u
); for
example,
0
≠
Π
for an incorrect displacement
2
u
.
Thus the solution for displacement
can be obtained by finding a stationary value of the total potential energy.
Indeed, it can
be seen that the quantity inside the curly brackets in Fig. 5.6.1 attains a minimum for the
solution already derived, Eqn. 5.5.17.
Figure 5.6.1: the total potential energy of a system
To generalise, define the “potential energy” of the applied loads to be
ext
W
V
−
=
so that
V
U
+
=
Π
(5.6.3)
The external loads must be conservative, precluding for example any sliding frictional
loading.
Taking the total potential energy to be a function of displacement
u
, one has
0
)
(
=
Π
=
Π
u
du
u
d
(5.6.4)
)
(
u
Π
1
u
1
u
2
u
2
u
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View Full Document Section 5.6
Solid Mechanics Part I
Kelly
209
Thus of all possible displacements
u
satisfying the loading and boundary conditions, the
actual displacement is that which gives rise to a stationary point
0
/
=
Π
du
d
and the
problem reduces to finding a stationary value of the total potential energy
V
U
+
=
Π
.
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This note was uploaded on 01/20/2012 for the course ENGINEERIN 1 taught by Professor Staff during the Fall '11 term at Auckland.
 Fall '11
 Staff

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