Loads, Deflections, and Strains

Loads

WISDEM estimates the ultimate loads by running a steady-state CCBlade simulation at rated pitch and rotor speed values and at a wind speed corresponding to the peak of the three-sigma gust for the extreme turbulence model. This approach to estimate loads is known to be somewhat over-conservative, but it is capable of capturing the relative trends and it is suitable to run iterative optimization loop on standard hardware in just a few minutes, offering to the designer the chance of a wide exploration of the solution space.

The blade loads computed with such setup are used to compute blade deflections, strains, buckling margins, and are finally summed together to estimate overall loads at the rotor hub.

Deflections

The loads computed with CCBlade are applied to the beam model of Frame3dd, which returns the overall blade deflections. From those, the tip deflections are exctracted.

Strains

RotorSE estimates the strains in the mid-point of the spar caps and of the trailing edge reinforcements. The strains are computed combining the sectional properties of the wind turbine blade generated by PreComp with the flapwise and edgewise loads computed in RotorSE. The strains are computed with the formula from [Han08]

(1)\[\epsilon(x,y) = \frac{M_1}{[EI]_1} y - \frac{M_2}{[EI]_2} x + \frac{N}{[EA]}\]

Buckling

A panel buckling calculation is added to augment the sectional analysis. The constitutive equations for a laminate sequence can be expressed as

(2)\[\begin{split}\left[\begin{matrix} N \\ M \end{matrix}\right] = \left[\begin{matrix} A & B \\ B & D \end{matrix}\right] \left[\begin{matrix} \epsilon^0 \\ k \end{matrix}\right]\end{split}\]

where N and M are the average forces and moments of the laminate per unit length, and \(\epsilon^0\) and \(k\) are the mid-plane strains and curvature (see [Hal92]). The D matrix is a \(3 \times 3\) matrix of the form (while wind turbine blade cross-sections are not always precisely specially orthotropic they are well approximate as such).

\[\begin{split}\left[ \begin{array}{ccc} D_{11} & D_{12} & 0 \\ D_{12} & D_{22} & 0 \\ 0 & 0 & D_{66} \end{array} \right]\end{split}\]

The critical buckling load for long (length greater than twice the width) simply supported panels at a given section is estimated as [Joh94]

\[N_{cr} = 2 \left(\frac{\pi}{w}\right)^2 \left[ \sqrt{D_{11} D_{22}} + D_{12} + 2 D_{66}\right]\]

where \(w\) is the panel width. If we denote the matrix in the constitutive equation (Equation (2)) as \(S\) and its inverse as \(S^*\), then \(\epsilon_{zz} \approx S^*_{11}N_z\). This expression ignores laminate shear and bending moment effects (the latter would be zero for a symmetric laminate), a good approximation for slender turbine blades. At the same time, an effective smeared modulus of elasticity can be computed by integrating across the laminate stack

\[E_{zz} = \frac{1}{\epsilon_{zz} h} \int_{-h/2}^{h/2} \sigma_{zz} dh = \frac{N_z}{ \epsilon_{zz} h}\]

where \(N_z\) in this equation is the average force per unit length of the laminate. Combining these equations yields an estimate for the effective axial modulus of elasticity

\[E_{zz} = \frac{1}{S^*_{11} h}\]

The critical strain can then be computed as

\[\epsilon_b = - \frac{N_{cr}}{h \ E_{zz}}\]

where the negative sign accounts for the fact that the strain is compressive in buckling.

Bibliography

[Hal92]

John C. Halpin. Primer on Composite Materials Analysis. Technomic, 2nd edition, 1992.

[Han08]

Martin O. L. Hansen. Aerodynamics of Wind Turbines. Earthscan, 2nd edition, 2008.

[Joh94]

Alastair Johnson. Handbook of Polymer Composites for Engineers, chapter Structural Component Design Techniques. Woodhead Publishing, 1994.