Page 439 - Wind Energy Handbook

BLADES 413

7.1.10 Design against buckling

The stress at which a slender plate element without imperfections buckles under
compression loading is known as the critical buckling stress. The derivation of the
critical buckling stresses for thin-walled curved panels bounded by stiffeners,
which typically form the blade load-bearing structure, is relatively straightforward
when the panel material is isotropic and solutions are provided in Timoshenko and
Gere (1961). These do not apply to composite materials such as the GFRP and wood
laminates commonly used in blade manufacture, however, as these are anisotropic,
but solutions can be derived for a symmetric laminate using the energy method, as
outlined below.
Consider a long cylindrical panel of length L and radius r, supported along two
generators and subtending an angle ł at the cylinder axis, which is axially loaded
in compression. If it deflects to form n half-waves around the circumference
between supports and m half-waves along its length, then its out-of-plane deflection
can be written as:

nðŁ mðx
w ¼ C sin sin (7:27)
ł L

where Ł and x are the co-ordinates of the deflected point with respect to one of the
long edges and one end respectively. In the absence of in-plane direct strains in the
plate, this out-of-plane deflected profile will result in circumferential deflections
Cł nðŁ mðx
v 0 ¼ cos sin (7:28)
nð ł L

These deflections will result in in-plane shear stresses, which reach a maximum at
the corners of each rectangular buckled panel. In practice, additional in-plane
deflections occur to moderate these shear stresses, as follows:

nðŁ mðx
u ¼ A sin cos in the axial direction
ł L
nðŁ mðx
v ¼ B cos sin in the circumferential direction (7:29)
ł L

The in-plane strain energy is calculated as
ðð
1
U 2 ¼ h (ó 1 å 1 þ ó 2 å 2 þ ôª) r dŁ dx (7:30)
2
with the suffices 1 and 2 denoting the axial and circumferential directions respec-
tively, so that

@u @v @u @(v 0 þ v)
å 1 ¼ , å 2 ¼ , ª ¼ þ (7:31)
@x r@Ł r@Ł @x