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Figure 10.2 Example of computed loads using FLEX4 for a mean wind speed of 11m/s
The tilt rotor moment in the main shaft, as shown in Figure 10.1, tries to tilt the nacelle over the tower. The yaw rotor moment tries to turn the nacelle on the tower. Sometimes the two bending moments at the root of the tower are also stated. Figure 10.2 shows an example of computed loads using the aero- elastic code FLEX4 at a mean wind speed of 11m/s. The wind speed at hub height and the power output are seen on the first graphs. The last four graphs show the corresponding time histories of the flapwise and edgewise bending moments and the tilt rotor and yaw rotor moments. The effect of gravity is clearly seen on the edgewise bending moment as a dominant sinusoidal variation, upon which is superimposed some small high frequency signal stemming from atmospheric turbulence. The flapwise bending is mostly influenced by the aerodynamic loads that vary with the turbulent wind field, and this signal is therefore more stochastic.
References
IEC 61400-1 (2004) ‘Wind turbines. Part 1: Design requirements’, CD, edition 3, second revision, IEC TC88-MT1
Sørensen, J. N. (1983) ‘Beregning af banekurver og kastelængder for afrevne
vindmøllevinger’ (in Danish), AFM83-06, Department of Fluid Mechanics (AFM), Technical University of Denmark (DTU)
11
Beam Theory for the Wind Turbine Blade
This section describes how a blade, whose outer contour is designed from aerodynamical considerations, is built to be sufficiently strong and stiff. In the past materials like wood, steel, aluminium, glass-fibre-reinforced plastics (GRPs) and carbon fibre reinforced plastics (CFRPs) have been used. The choice depends on many parameters such as strength, weight, stiffness, price and, very important for wind turbines, fatigue properties. The majority of wind turbine blades today are built using GRPs, and therefore a short description of a manufacturing process using this material is given. A negative mould for the upper part (suction side) and lower part (pressure side) of the blade is made. A thin film of so-called gelcoat is first laid in the moulds. The gelcoat gives a smooth white finish to the blades and therefore it is not necessary to paint the blades afterwards. Then a number of glass fibre mats are laid in. On each mat a layer of epoxy or polyester is rolled on to bind the mats into a hard matrix of fibres. The number of mats gives the thickness of the shell; typically a thin shell is made around the leading and trailing edges and a thick shell is made in the middle of the aerofoil. A section of such a blade is shown in Figure 11.1.
To make the blade stronger and stiffer, so-called webs are glued on between the two shells before they are glued together. To make the trailing edge stiffer, foam panels can also be glued on before assembling the upper and lower parts. Because such a construction consists of different layers, it is often called a sandwich construction; a sketch is given in Figure 11.2 which can readily be compared to the real section in Figure 11.1. It is seen that the thick layer of mats and epoxy in the middle of the skin and the webs form a box-like structure. For structural analysis, the box-like structure, which is the most important structural part of the blade, acts like a main beam on which a thin skin is glued defining the geometry of the blade. Fixing a thin skin on a main beam (so the skin is not carrying loads, but merely gives an outer aerodynamic shape) is an alternative way that is sometimes used to construct a blade.
A blade can thus be modelled as a beam, and when the stiffnesses EIand GIv at different spanwise stations are computed, simple beam theory can be applied to compute the stresses and deflections of that blade. E is the modulus of elasticity, Gis the modulus of elasticity for shear and Idenotes different moments of inertia. In the next section a more elaborate explanation is given for the moments of inertia and it is shown how the stiffnesses can be computed for a wind turbine blade such as the one shown in Figure 11.1.
The simple beam theory described here is found in almost any basic book on mechanics of materials (for example Gere and Timoshenko, 1972).
Further, it is outlined how to compute the important structural parameters shown in Figure 11.3. Values of these parameters are necessary to compute the deflection of a blade for a given load or as input to a dynamic simulation using an aeroelastic code.
Figure 11.1Section of an actual blade
Figure 11.2Schematic drawing of a section of a blade
EI1 – bending stiffness about first principal axis;
EI2 – bending stiffness about second principal axis;
GIv – torsional stiffness;
XE – the distance of the point of elasticity from the reference point;
Xm – the distance of the centre of mass from the reference point;
Xs – the distance of the shear centre from the reference point;
– the twist of the aerofoil section measured relative to the tip chord line;
v – angle between chord line and first principal axis;
+v – angle between tip chord line and first principal axis;
The point of elasticity is defined as the point where a normal force (out of the plane) will not give rise to a bending of the beam. The shear centre is the point where an in-plane force will not rotate the aerofoil. If the beam is bent about one of the principal axes, the beam will only bend about this axis. As will be seen later, it is convenient to use the principal axes when calculating the blade deflection.
Before continuing, some necessary definitions must be introduced. The following quantities are defined in terms of the reference coordinate system (XR,YR) in Figure 11.4:
• Longitudinal stiffness: [EA] = ∫AEdA.
• Moment of stiffness about the axis XR: [ESXR] = ∫AEYRdA.
• Moment of stiffness about the axis YR: [ESYR] = ∫AEXRdA.
Beam Theory for the Wind Turbine Blade | 109
Figure 11.3Section of a blade showing the main structural parameters
• Moment of stiffness inertia about the axis XR: [EIXR] = ∫AEYR 2d A.
• Moment of stiffness inertia about the axis YR: [EIYR] = ∫AEXR2d A.
• Moment of centrifugal stiffness: [EDXYR] = ∫AEXRYRd A.
From these definitions, the point of elasticity PE= (XE,YE) can be computed in the reference system (XR,YR) as:
[ESY R]
XE= ––––– (11.1)
[EA]
and:
[ESXR]
YE= ––––– (11.2)
[EA]
For E and ρ constant the point (XE,YE) equals the centre of mass for the section, where ρdenotes the density of the material used. Now the moments of stiffness inertia and the moment of centrifugal stiffness are moved to the coordinate system (X’,Y’), which is parallel to the reference system (XR,YR) and has its origin in the point of elasticity, using the formulaes:
[EIX’] = ∫AE(Y’)2dA= [EIXR] –YE2[EA] (11.3)
Figure 11.4 Section of a blade
[EIY’] = ∫AE(X’)2dA= [EIYR] –X2E[EA] (11.4) [EDX’Y’] = ∫AEX’Y’dA= [EDXYR] –XEYE[EA]. (11.5) It is now possible to compute the angle between X’and the first principal axis and the bending stiffness about the principal axes. The second principal axis is perpendicular to the first principal axis:
2 [EDX’Y’]
= 1–2 tan–1 ––––––––––– (11.6)
[EIY’] – [EIX’]
[EI1] = [EIX’] – [EDX’Y’]tan (11.7)
[EI2] = [EIY’] [EDX’Y’]tan (11.8)
The stress σ(x,y) in the cross-section from the bending moments about the two principal axes MXand MYand the normal force Nis found from:
σ(x,y) =E(x,y)ε(x,y), (11.9)
where the strain εis computed from:
M1 M2 N
ε(x,y) = –––– y– –––– x+ –––– . (11.10)
[EI1] [EI2] [EA]
σ, ε and N are positive for tension and negative for compression. The bending moments M1and M2and the normal force Nare computed from the loading of the blade, as is shown later.
The main structural data are now determined. Since a wind turbine blade is very stiff in torsion, the torsional deflection is normally not considered. A complete description of how to compute the shear centre and the torsional rigidity is, however, given in Øye (1978). An example of results from Øye (1988), for the 30m blade used at the 2MW Tjæreborg wind turbine, is listed in Table 11.1.
Table 11.1 shows that the position of the first principal axis, described by the angle +v, varies with the radius r. It is also seen that the position of the first principal axis is almost identical with the chord line since the angle v is small for most of the blade.
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