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208
Finite Element Analysis and Design
21. Consider the tapered bar in Problem 16.
Use the RayleighRitz method to solve the
same
problem.
Assume
the
displacement
in
the
form
of
2
12
( )
(
1)(
)
u x
x
c x
c x
.
Solution:
The assumed displacements satisfy the essential BCs
u
(0) =
u
(1) = 0.
The assumed
displacements take the form:
2
( )
(
)
u x
x
c x
c x
Then, the strain is given by
2
(2
1)
(3
2 )
xx
du
c
x
c
x
x
dx
The strain energy in the bar becomes
1
2
2
0
( )
(3
2 )
2
E
U
A x c
x
c
x
x
dx
where
A
(
x
) is the area of crosssection defined as
22
( )
0.05 (1
0.8 )
A x
x
The potential energy of the distributed load
f
= 10,000 N/m is
11
2
00
( )
10,000
(
)
V
fu x dx
x
c x
c x dx
The total potential energy
1
2
1
2
1
2
( , )
( , )
( , )
c c
U c c
V c c
.
The principle of minimum
potential energy requires
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This note was uploaded on 08/22/2011 for the course EML 4500 taught by Professor Staff during the Fall '08 term at University of Florida.
 Fall '08
 Staff
 Finite Element Analysis

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