Page 441 - Wind Energy Handbook

BLADES 415
0 1
2 2 2 2 2 ! 2
dU b ¼ 1@ D x @ w þ 2D xy @ w @ w þ D y @ w A dx dy (7:34)
2
2 @x 2 @x @ y 2 @ y 2

The twisting energy absorbed in an area dx dy is:
2
@ w
1
dU t ¼ (M xy þ M yx ) dx dy
2 @x@ y
where
" ð h=2 # 2
2
M xy ¼ 2 G xy (z):z dz @ w
h=2 @x@ y

in which z is the distance measured from the mid-plane of the laminate, G xy (z)is
the in-plane shear modulus at that distance and h is the laminate thickness. Denot-
ing the torsional rigidity,
" #
ð
h=2
2
G xy (z):z dz
h=2
by D T , then
! 2
2
@ w
1
dU t ¼ :4D T dx dy (7:35)
2 @x@ y
The total strain energy of curvature over the width of the panel and the length of
one half wave is found by substituting the out-of-plane deflection given by Equa-
tion (7.27) in Equations (7.34) and (7.35) and integrating over this area, which gives:

4 " 4 #
2
2
C łrL mð n D y n D xy D T
U 1 ¼ U b þ U t ¼ D x 1 þ þ 2 þ 4 (7:36)
8 m L º D x º D x D x
The energy absorbed by the panel during buckling as a result of in-plane strains
and out-of-plane curvature is equal to the work done by the critical axial load as the
panel shortens. The shortening of the panel over one half wave length is given by
ð L=m 2 2
1 @w dx ¼ ð C 2 m sin 2 nðŁ (7:37)
2 ł
0 @x 4 L
so the work done by the axial force of N x per unit width over the panel width is
ð 2 m
T 1 ¼ C 2 łrN x (7:38)
8 L