EN0175
11 / 21/ 06
Principle of minimum potential energy (continued)
The potential energy of a system is
∫
∫
∫
−
−
=
S
i
i
V
i
i
V
S
u
t
V
u
f
V
w
V
d
d
d
Principle of minimum potential energy states that for all kinematically admissible
, the actual
displacement field minimizes
.
i
u
V
Example 1:
g
ρ
x
L
We have shown in the beginning of the semester that the exact solution is:
(
)
x
L
x
E
g
u
−
=
2
ρ
.
Now we discuss how to solve the same problem by using the principle of minimum potential
energy.
Procedure:
1)
Pick any displacement such that
( )
( )
0
0
=
=
L
u
u
.
2)
Minimize
for the chosen parameters.
V
An obvious choice is
( )
(
)
( )
x
f
x
L
x
x
u
−
=
since this satisfies the clamped displacement
boundary conditions for any
. We can assume
( )
x
f
( )
x
f
to be a polynomial function.
( )
N
N
x
C
x
C
x
C
C
x
f
+
+
+
+
=
L
2
2
1
0
To minimize the potential energy
, we take
(
)
N
C
C
C
C
V
,
,
,
,
2
1
0
L
0
0
=
∂
∂
C
V
,
0
1
=
∂
∂
C
V
, …,
0
=
∂
∂
N
C
V
1

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