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5 Variational Problems of Conditional Extrema
w(0) = 0, w
(0) = 0
(5.1)
The conditions at variable endpoint B are
w(x 1 ) = H, w
(x 1 ) = 0
( 5 . 2 )
The total energy of the beam is
U =
x 1
0
1
2
E I w
2
− qw
dx − q H(L − x 1 )
(5.3)
where,
x 1
0
1
2
E I w
2 dx is the bending deformation energy of AB,
x 1
0 qwdx is the
work done by the uniformly distributed load q of AB, q H(L − x 1 ) is the work done
by q of BC. This is a mixed type functional that belongs to the expression (5.4.7),
the Euler-Poisson equation is
F w −
d
dx
F w +
d
2
dx 2 F w = 0
( 5 . 4 )
that is
F w −
d
2
dx 2 F w = −q + E I
d
4
w
dx 4 = 0
(5.5)
According to the endpoint condition (5.4.9), the transversality condition of the
variable endpoint is
F − w
F w −
d
dx
F w
− w
F w + Φ x 1
x=x 1
δx 1 +
F w −
d
dx
F w + Φ w 1
x=x 1
δw 1 + F w | x=x 1 δw
1 = 0
(5.6)
Since the beam maintains invariable horizontal direction at the variable endpoint,
there is δw 1 = δw
1 = 0, when δx 1 is arbitrary, its coefficient should be zero, there is
(F − w
F w + Φ x 1 )
x=x 1
= 0
(5.7)
or
1
2
E I w
2
− qw − w
(E I w
) + q H
x=x 1
= 0
(5.8)
Note that w| x=x 1 = H , Eq. (8) becomes
1
2
E I w
2
x=x 1
= 0 or w
x=x 1
= 0
(5.9)