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Applied Calculus of Variations for Engineers, 2nd Edition
book

Applied Calculus of Variations for Engineers, 2nd Edition

by Louis Komzsik
June 2014
Intermediate to advanced content levelIntermediate to advanced
233 pages
5h 42m
English
CRC Press
Content preview from Applied Calculus of Variations for Engineers, 2nd Edition
172 Applied calculus of variations for engineers
where y is the distance from the neutral plane and dG is the weight of the
cross-section. Using the unit length weight of the beam, we obtain the mo-
ment as
dM = ywdx = M (x)dx.
The total work of bending will be obtained by integrating along the length of
the beam:
W =
dM =
L
0
M(x)dx = w
L
0
ydx,
since the unit weight is constant. We are now ready to state the equilibrium
of the beam
E
s
= W
as a variational problem of the form
I(y)=
L
0
(E
s
(x) M (x))dx = extremum,
or
I(y)=
L
0
(
1
2
EI
1
r
2
wy)dx = extremum.
Since the radius of curvature is reciprocal of the second derivative of the bent
curve of the beam,
r =
1
y

(x)
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Publisher Resources

ISBN: 9781482253597