
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)