Page 440 - Wind Energy Handbook

414 COMPONENT DESIGN

Substituting ó 1 ¼ E x (å 1 þ ı y å 2 )=(1 ı x ı y ), ó 2 ¼ E y (å 2 þ ı x å 1 )=(1 ı x ı y ) and ô ¼
G xy ª, where E x , E y and G xy are the longitudinal, transverse and shear moduli of the
laminate respectively (obtained by averaging the corresponding moduli of the
individual plies) and ı x and ı y are the effective Poisson’s ratios, the in-plane strain
energy becomes:
ðð
h 2 2 2
U 2 ¼ [E x å þ E y å þ 2E x ı y å 1 å 2 þ (1 ı x ı y )ª G xy ]r dŁ dx (7:32)
2
1
2(1 ı x ı y )
Substituting the expressions for å 1 , å 2 , and ª from Equation (7.31) and integrating
over the width of the panel, łr, and the length of one half wave, L=m, we obtain
2 " 2
E x h L mð C 2 E y n n
2
U 2 ¼ łr Æ þ â 2 þ2ı y Æâ
1 ı x ı y m L 8 E x º º
2 #
G xy n ł
þ (1 ı x ı y ) Æ þ â þ (7:33)
E 1 º nð

where º ¼ młr=L and the ratios Æ ¼ A=C and â ¼ B=C are yet to be determined.
The expression for the strain energy of curvature is derived as follows. Replacing
the angular coordinate Ł by the linear coordinate y (¼ rŁ), the bending energy
absorbed in an area dx:dy is:

!
2
2
@ w @ w
dU b ¼ 1 M x þ M y dx:dy
2 @x 2 @ y 2
where

2
2
@ w @ w
M x ¼ D x D xy
@x 2 @ y 2
and

2
2
@ w @ w
M y ¼ D y D xy
@ y 2 @x 2
for a specially orthotropic laminate, i.e. one in which the reinforcement in each layer
is either oriented at 08 or 908, or is bi-directional with the same amount of fibres at
þŁ8 and Ł8. D x and D y are the flexural rigidities of the laminate when flat, for
bending about the y-axis and x-axis respectively, and D xy is the ‘cross flexural
rigidity’ i.e. the moment per unit width about one axis generated by unit curvature
about the other. Hence