Appendix: All Solutions to the Problems
791
2.69 The total potential energy of the linear bending of a clamped orthotropic circular
plate is given by the following expression
J [w] =
1
2
a
0
D 11 rw
2
+ 2D 12 w
w
+ D 22
w
2
r
− 2 f w
dr
where, r is radial coordinate; a is the radius of the plate; f is the distributed transverse
load; D i j are the plate stiffnesses. Determine the governing differential equation of
the plate.
Solution The Euler equation of the functional is
d
2
dr 2 (D 11 rw
) −
d
dr
(D 12 w
) +
d
2
dr 2 (D 12 w
) −
d
dr
D 22
1
r
w
− f = 0
or
d
2
dr 2 (D 11 rw
) −
d
dr
D 22
1
r
w
− f = 0
If D i j are constants, then
D 11
d
2
dr 2 (rw
) − D 22
d
dr
1
r
w
− f = 0
2.70 Find the extremal curve of the functional J [y] =
x 1
x 0
(2x y + y
2
)dx.
Solution Let the integrand F = 2x y + y
2 , substituting it into the Euler–Poisson
equation, we get
2x − 2y
(6)
= 0
or
y
(6)
= x
Integrating the above equation six times, we get
y =
x
7
7!
+ c 1 x
5
+ c 2 x
4
+ c 3 x
3
+ c 4 x
2
+ c 5 x + c 6
2.71 Find the extremal curve of the functional J [y] =
1
−1
yy
+
y
2
2
dx, the
boundary conditions are y(−1) = y(0) = y(1) = 0, y
(−1) = y
(0) = y
(1) = 1.