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2 Variational Problems with Fixed Boundaries
O
O
x
x
y
y
L
(a)
(b)
L
q
q
Fig. 2.10 Deflection curves for the beams with a uniform load
or they are abbreviated to
n i
k=0
(−1)
k d
k
dx k F y
(k)
i
= 0
(2.7.18)
Example 2.7.1 A simple elastic beam of length L is subjected to the uniform load
q, as shown in Fig. 2.10a, what kind of deflection curve does the beam take, the total
potential energy J of the system is minimum? If it is fixed-end beam, as shown in
Fig. 2.10b, while the other conditions are unchanged, then what is the result?
Solution Let the deflection curve be y = y(x), the flexural rigidity of the beam is
EI, according to material mechanics knowledge, the bending strain energy of the
beam is
J 1 =
1
2
L
0
E I y
dx
The potential energy of load caused by the deflection of the beam is
J 2 = −
L
0
qydx
The total potential energy of the system is
J = J 1 + J 2 =
1
2
L
0
(E I y
− 2qy)dx
When the two ends of the beam is simply supported, the boundary conditions are
y(0) = 0, y(L) = 0, y
(0) = 0, y
(L) = 0
The Euler-Poisson equation of the functional is