Burton T. (et. al.) Wind energy Handbook

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blade, but with differing phase angles. The blade root out-of-plane bending

moments due to the ﬁrst component will be in equilibrium, and will apply a

‘dishing’ moment to the hub which will result in tensile stresses in the front and

compression stresses in the rear. These stresses will be uniaxial for a two bladed

rotor, and biaxial for a three-bladed rotor.

The ﬂuctuations in out-of-plane blade root bending moment due to wind shear,

shaft tilt and yaw misalignment will often be approximately sinusoidal, with a

frequency equal to the rotational frequenc y. Using Equations (5.118), it is easily

shown that, for a sinusoidally varying blade root bending moment with amplitude

, the amplitude of the resulting shaft bending moment is 1:5M

for a three-

bladed machine and 2M

for a rigid hub two-bladed machine.

In the case of wind shear conforming to a power law, the loading on a horizontal

blade is always greater than the average of the loadings on blades pointing

vertically upwards a nd downwards, so the loading departs signiﬁcantly from

sinusoidal. The shaft bending moment ﬂuctuations due to wind shear with a 0.2

Blade 1

Blade 3

Blade 2

Figure 5.40 Shaft Bending Moments, with Rotating Axis System Referred to Blade 1

HUB AND LOW-SPEED SHAFT LOADING 295

exponent are compared in Figure 5.41 for two- and three-bladed rigid hub machines

operating at 30 r.p.m. in a hub-height wind speed of 12 m=s. The ratio of moment

ranges is still close to 2:1:5.

5.10.3 Stochastic aerodynamic loads

The out-of-plane blade root bending moments arising from stochastic loads on the

rotor will result in both a ﬂuctuating hub ‘dishing’ moment (see above) and

ﬂuctuating shaft bending moments. For a two-bladed, rigid hub rotor, the shaft

moment is equal to the differen ce between the two out-of-plane blade root bending

moments, or teeter moment, the standard deviation of which is given by Equation

(5.102). Similarly, the standard deviation of the mean of these two moments (i.e.,

the ‘dishing’ moment) is given by Equation (5.103).

The derivation of the standard deviation of the shaft moment for a three-bladed

machine is at ﬁrst sight more complicated, as the integ ration has to be carried out

over three blades instead of two. However, if the shaft moment about an axis

parallel to one of the blades, M

(Figure 5.35), is chosen, the contribution of

loading on that blade disappears, and the expression for the shaft moment standard

deviation becomes:



Mzs

˜

dÆ



R

, r

,0)c(r

)c(r

)

ﬃﬃﬃ

jjr

jdr

(5:119)

where the limits of the integrations refer to the other two blades. k

, r

,0) is

given by Equation (5.51), with  set equal to zero when r

and r

are radii to points

on the same blade, and replace d by 2= 3whenr

and r

are radii to points on

different blades. Note that, compared with the two-bladed case, the cross correla-

-50

-40

-30

-20

-10

100

110

120

130

140

150

160

170

180

190

200

210

220

230

240

250

260

270

280

290

300

310

320

330

340

350

360

Azimuth of blade 1 (degrees)

Shaft bending moment about axis perpendicular to blade 1

Blade 1 out-of-

plane root bending moment

Shaft bending moment: three bladed machine

Shaft bending moment: two bladed rigid hub machine

Rotor diameter = 40 m

Rotational speed = 30 r.p.m.

Hub-height wind speed = 12 m/s

Shear exponent = 0.2

Figure 5.41 Shaft Bending Moment Fluctuation due to Wind Shear

296

DESIGN LOADS FOR HORIZONTAL-AXIS WIND TURBINES

tion function, k

, r

, 0) will be increased when r

and r

relate to different blades,

because of the reduced separation between the two blade elements resulting from

the 1208 degree angle between the blades. Equation (5.119) can be rewritten in terms

of the normalized cross correlation function, r

, r

,0)¼ k

, r

,0)=

,as

follows:



Mzs

¼ 

r

dÆ



R

, r

,0)c(r

)c(r

)

ﬃﬃﬃ

jjr

jdr

(5:119a)

In the case of 40 m diameter turbines with TR blades operating in wind with a

turbulence length scale of 73.5 m, the standard deviation of shaft moment due to

stochastic loading for a three-bladed machine is 82 percent of that for a two-bladed,

ﬁxed hub machine rotating at the same speed. This ratio would rise to

ﬃﬃﬃ

=2 if the

effect on the cross correlation function of the 1208 blade spacing were ignored. It is

worth noting that, for a three-bladed machine, the standard deviation of the shaft

moment M

due to stochastic loading is the same as that of M

By analogy with the derivation of the shaft moment above, the standard deviation

of the hub ‘dishing’ moment for a three-bladed machine due to stochastic loading is

given by:



r

dÆ



R

, r

,0)c(r

)c(r

)

ﬃﬃﬃ

(5:120)

where the integrations are carried out over two blades only, and the cross correla-

tion function is modiﬁed as before to account for the 1208 angle between the blades.

It can be shown that



MZS

þ 



My1

(5:121)

5.10.4 Gravity loading

An important component of shaft loading is the cyclic cantilever bending moment

due to rotor weight, which usually has a dominant effect on shaft fatigue design. As

an illustration, a rotor consisting of three TR blades, each weighing 2 tonnes, and a

5 tonne hub cantilevered 0.85 m beyond the shaft main bearing, will produce a

maximum shaft gravity moment of about 90 kNm. This compares with a shaft

moment range due to wind shear of 70 kNm for a hub height wind of 12 m=s and a

shear exponent of 0.2, and a shaft moment standard deviation of 50 kNm due to

turbulence, taking a turbulence intensity of 20 percent and the same hub-height

mean wind speed. Note that the shaft moment due to wind shear relieves that due

to gravity, so it would be wise to adopt a smaller wind shear exponent for shaft

fatigue calculations.

HUB AND LOW-SPEED SHAFT LOADING 297

5.11 Nacelle Loading

5.11.1 Loadings from rotor

The previous section considered the moments applied to the shaft by the rotor hub

using an axis system rotating with the shaft. In addition to these moments, the shaft

also experiences an axial load due to rotor thrust, and radial forces arising from

differential blade edgewise loadings and any out-of-balance centrifugal force.

In order to calculate loadings on the elements of the nacelle structure, it is ﬁrst

necessary to transform the shaft loads (or the constituent blade loads) deﬁned in

terms of the rotating axis system into nacelle loads expressed in terms of a ﬁxed

axis system. Here the conventional system in which the x-axis is dow nwind, the y-

axis is horizontal to starboard and the z-axis is vertically upwards will be adopted

(Figure C2). Thus the moments acting on the nacelle about the y- and z-axes as a

result of the blade root out-of-plane bending moments are as follows for a three

bladed machine with shaft tilt :

¼ M

cos ł þ M

cos(ł  1208) þ M

cos(ł  2408)(5:122)

¼ (M

sin ł þ M

sin(ł  1208) þ M

sin(ł  2408)) cos  (5: 123)

where ł is the azimuth of blade 1 (see Figure 5.42).

Blade 1

Blade 3

sinψ

Shaft tilt η

cosψ

sinψ

cosψ

Figure 5.42 Components of Blade 1 Root Out-of-plane Moment about Fixed Axes Set

298

DESIGN LOADS FOR HORIZONTAL-AXIS WIND TURBINES

It is instructive to compare the moments acting on the nacelle due to determi-

nistic loading for three-bladed and two-bladed machines. The ﬂuctuations of out-

of-plane root bending moment due to wind shear and yaw misalignment are

approximately proportional to the cosine of blade azimuth for an unstalled blade.

Substituting M

¼ M

cos ł, M

¼ M

cos(ł  2=N) etc into Equations (5.122)

and (5.123) yields M

¼ 1:5M

and M

¼ 0 for a three-bladed machine,

whereas the corresponding results for a rigid hub two-bladed machine are

¼ M

(1 þ cos 2ł) and M

¼ M

sin 2ł cos . Thus the moments on the

nacelle are constant for a three-bladed machine, but continually ﬂuctuating with

amplitude M

for a rigid hub two-bladed machine. Parallel results are obtained

for M

¼ M

sin ł which approximates to the out-of-plane root bending moment

due to shaft tilt – again for an unstalled blade. The full comparison is given in

Table 5.6 below.

It is clear that the moments acting on the nacelle due to deterministic loading are

much more benign for a three-bladed rotor than for a two-bladed rotor with rigid

hub.

In the case of three-bladed machines, the standar d deviation of shaft bending

moment due to stochastic rotor loading is independent of the rotating axis chosen

(Section 5.10.3), so the standard deviation of the resulting moment on the nacelle

will take the same value about both the nacelle y- and z-axes.

5.11.2 Cladding loads

Except in the case of sideways wind loading, cladding loads are not usually of great

signiﬁcance. They may be calculated according to the rules given in standard wind

loading codes. For sideways wind loading, a drag factor of 1.2 will generally be

found to be conservative.

Table 5.6 Comparison Between Nacelle Moments due to Deterministic Loads for Two- and

Three-bladed Machines

Nacelle moments resulting from

out-of-plane blade root bending

moment ﬂuctuations due to wind

shear and yaw misalignment

approximated by: M

¼ M

cos ł,

¼ M

cos (ł  2=N) etc

Nacelle moments resulting from

out-of-plane blade root bending

moment ﬂuctuations due to shaft

tilt approximated by: M

¼ M

sinł,

¼ M

sin(ł  2=N) etc

Nacelle nodding

moment, M

Nacelle yaw

moment, M

Nacelle nodding

moment, M

Nacelle yaw moment,

Three-bladed

machine

1:5M

Zero Zero 1:5M

cos 

Two-bladed,

rigid hub

machine

(1 þ cos 2ł) M

sin 2ł cos  M

sin 2ł M

(1  cos 2ł)cos 

NACELLE LOADING 299

5.12 Tower Loading

5.12.1 Extreme loads

As noted in Section 5.4.1, it is customary to base the calculation of extreme loads on

a non-operational turbine on the 50 year return 3 s gust. Several loading conﬁgura-

tions may need to be considered, and the critical load case for the tower base will

generally differ from that for the tower top. In addition, it is necessary to investigate

the extreme operational load cases, as these can sometimes govern instead if the tip

speed is high in relation to the design gust speed.

In the case of non-operational, stall-regulated machines, the critical case for the

tower base occurs when the wind is blowing from the front and inducing max imum

drag loading on the blades. By contrast, sideways wind loading to produce maxi-

mum lift on a blade pointing vertically upwards or rear wind loading on the rotor

with one blade shielded by the tower will produce the maximum tower top bending

moment.

One of the beneﬁts of pitch-regulation of three-bladed machines is that blade

feathering at shut-down considerably reduces non-operational rotor loading. The

critical conﬁguration as far as tower base bending moment is concerned is sideways

wind loading, with two of the blades inclined at 308 to the vertical. The horizontal

component of the loading on these blades is cos

308 of the loading on a vertical

blade, so that the total rotor loading is only 43.3 percent (¼ 100:

ﬃﬃﬃ

=4%) of the

maximum experien ced by a stall-regulated machine.

The cases of sideways wind loading on a wind turbine referred to above can only

arise if the yaw drive is disable d by grid loss for sufﬁcient time for a 908 wind

direction change to take place. In many areas, the level of grid security will be high

enough for this possibility to be ruled out, so that sideways wind loading need not

be considered if the yaw drive is program med to remain operational in high winds.

The critical non-operational load case for a three-bladed, pitch-regulated machine

then occurs when the wind is from the front with a 158 to 208 yaw error. The rotor

load is maximum when one of the blades is vertical, and is similar in mag nitude to

loading produced by a sideways wind. However, as the load resu lts from blade lift

rather than drag, it is at right angles to the loading on the tower, so the total

moment at the tower base is signiﬁcantly less.

Information on the drag factors appropriate for cylindrical and lattice towers is to

be found in Eurocode 1, part 2–4 (1997), and in national codes such as BS 8100

(1986) or DS 410 (1983). The drag factor for a cylindrical tower is typically 0.6–0.7.

Rotor loading is generally the dominating component of tower base moment for

stall-regulated mach ines, but with pitch-regulated machines the contributions of

tower loading and rotor loading are often of similar magnitude.

5.12.2 Dy namic response to extreme loads

Just as in the case of the single, stationary cantilevered blade considered in Section

5.6.3, the quasistatic bending moments in the tower calculated for the extreme gust

300 DESIGN LOADS FOR HORIZONTAL-AXIS WIND TURBINES

speed will be augmented by inertial moments resulting from the excitation of

resonant tower oscillations by turbulence. As before, it is convenient to express this

augmentation in terms of a dynamic factor, Q

, deﬁned as the ratio of the peak

moment over a 10 min period, including resonant excitation of the tower, to the

peak quasistatic moment over the same period. Thus

Max

e50



1þ2Æ

dA:Q

(5:124)

where U

e50

is the 50 year return gust speed at hub height, z is height above ground,

H is the hub height, C

is the force factor (lift or drag) for the element under

consideration, Æ is the shear exponent, taken as 0.11 in IEC 61400-1, and

1 þ g 2





ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

SMB



2

)º

1 þ g





ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

SMB

(5:17)

The integral sign

signiﬁes that the integral is to be undertaken over each blade

and the tower.

The derivation of Equa tion (5.17) is explained in Section 5.6.3 and the Appendix

in relation to a cantilevered blade. The essentially similar procedure for a tower

supporting a rotor and nacelle is as follows.

(1) Calculate the resonant size reduction factor, K

), which reﬂects the effect of

the lack of correlation of the wind ﬂuctuations at the tower natural frequency

along the blades and tower. Adopting an exponential expression for the normal-

ized co-spectrum as before, Equation (A5.25) becomes:

) ¼

þþ

exp[Csn

=U]C

c(r)c(r9)

(r)

(r9)dr dr9

c(r)

(r)dr



(5:125)

where the integral sign

denotes integration over the blades and the tower, r

and r9 denote radius in the case of the blades and depth below the hub in the

case of the tower, s deno tes the separation between the elements dr and dr9, C

is the relevant force coefﬁcient, c(r) denotes chord in the case of the blades and

diameter in the case of the tower, and 

(r) denotes the tower ﬁrst mode shape.

This expression can be considerably simpliﬁed by setting 

(r) to unity for the

rotor and ignoring the tower loading contribution entirely. This is not unreason-

able, as only loading near the top of the tower is of signiﬁcance, and this does

not add much to the spatial extent of the loaded area.

TOWER LOADING 301

(2) Calculate the damping logarithmic decremen t, , for the tower ﬁrst mo de. The

aerodynamic component is given by



¼ 2

¼ 2

(r)

(r)dr

(5:126)

where m

is the generalized mass of the tower, nacelle and rotor (including the

contribution of rotor inertia) with respect to the ﬁrst mode given by Equation

(5.110), and n

is the tower natural frequency in Hz. For a stall-regulated

machine facing the wind, the rotor contribution to aerodynamic damping is

simply r

=2m

where A

is the rotor area.

(3) Calculate the standard deviation of resonant nacelle displacement according to



¼ 2





ﬃﬃﬃﬃﬃﬃ

2

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

)

p ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

)

(5:6)

(4) Calculate the ratio º

, which relates the ratio of the standard deviation of

resonant tower base moment to the mean value to the corresponding ratio for

nacelle displacement as follows:



¼ º



(5:7)

If 

(r) is set to unity for the rotor, º

is given by:

m(z)

(z)z dzC

U(z)



d(z)

(z)dz

()

U(z)



d(z)

()

(5:127)

where z is the height up the tower measu red from the base, and H is the hub

height. If the loading on the tower is relatively small, this approximates to

m(z)

(z)z dz

(5:127a)

which is close to unity because the tower head mass dominates the integral.

(5) Calculate the size reduction factor for the root bending moment quasistatic or

background response, K

SMB

, which reﬂects the lack of correlation of the wind

ﬂuctuations along the blades and tower. K

SMB

may be derived from a similar

expression to that for the resonant size reduction factor given in Equation

(5.125), but with the exponential function modiﬁed to exp[s=0: 3L

302 DESIGN LOADS FOR HORIZONTAL-AXIS WIND TURBINES

(6) Calculate the peak factors for the combined (i.e., resonant plus quasistatic) and

quasistatic responses in terms of the respective zero up-crossing frequencies. (In

estimating the zero up-crossing frequency of the quasistatic response, the blade

area should be replaced by the rotor area in Equation (A5.57).

(7) Substitute the parameter values derived in Steps (1)–(6) into Equation (5.17) to

obtain the dynamic factor, Q

The procedure is illustrated by the following example.

Example 5.3: Estimate the dynamic factor, Q

, for the extreme tower base moment

for a stationary 40 m diameter three-bladed stall-regulated turbine.

Data:

Hub height, H ¼ 35 m

Tower ﬁrst mode natural frequency, n

¼ 1:16 Hz

Generalized mass of tower, nacelle and rotor,

¼ 28 000 kg

Rotor area, A

¼ 23:9m

Blade drag fact or ¼ 1:3

Tower top diameter ¼ 2m,

base diameter ¼ 3:5m

Tower drag fact or ¼ 0:6

50 year return 10 min mean wind speed at hub

height,

U ¼ 50 m=s

Air density,

r ¼ 1:225 kg= m

Roughness length, z

¼ 0:01 m

Turbulence intensity at hub height, 

=U ¼ 1=ln(H=z

) ¼ 0:1225 (based on Euro-

code 1, Part 2.4 (1997) for z

¼ 0:01)

The non-dimensional power spectral density of longitudinal wind turbulence,

(n), is calculated at the tower ﬁrst mode natural frequency according to the Kaimal

power spectrum deﬁned in Eurocode 1 (see Equation (A5.8) in Appendix) as 0.0425.

The decay constant, C, in the exponential expression for the normalized co-spectrum

in Equation (5.125) is taken as 9.2. The calculation of Q

procedes as follows.

Size reduction factor for resonant response – Equation (5.125) with tower loading

ignored and 

(r) taken as unity over the rotor 0.166

Aerodynamic damping logarithmic decrement, 

– rotor contribution ,  UC

=2m

0.079

– tower contribution, 

d(z)

(z)dz =2m

0.007

Structural damping logarithmic decrement, 

0.02

Damping logarithmic decrement, total, 

þ 

0.106

Ratio of standard deviation of resonant tower base moment to mean value –

Equations (5.6), (5.7) and (5.127)



¼ 2





ﬃﬃﬃﬃﬃﬃ

2

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

)

p ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

)

¼ 2 3 0:1225 3 6:823

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

0:0425

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

0:166

3 1:02 0 :1432

TOWER LOADING 303

Size reduction factor for quasistatic or background response, K

SMB

– see Step (5)

above 0.837

Ratio of standard deviation of tower base moment quasistatic response to mean

value



¼ 2



ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

SMB

¼ 2 3 0:1225 3

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

0:837

0:2241

Ratio of standard deviation of total tower base moment response to mean value



ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ









ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

0:2241

þ 0:1432

0:2660

Zero up-crossing frequency of quasistatic response, n

– see Step (6) above 0.31 Hz

Zero up-crossing frequency of total tower base moment respons e,  – Equation

(A5.54) 0.68 Hz

Peak factor, g, based on  – Equation (A5.42) 3.63

Ratio of extreme tower base moment to mean value

max

¼ 1 þ g





¼ 1 þ 3:63(0:2660) 1:966

Peak factor, g

, based on n

– Equation (A5.42) 3.41

Ratio of quasista tic component of extreme tower base moment to mean value

¼ 1 þ g





¼ 1 þ 3:41 (0: 2241) 1:764

Dynamic factor, Q

¼ 1:966=1:764 – Equation (5.17) 1.115

It is apparent from Equation (5.17) that a key parameter determining the dyn amic

factor is the damping value. If the rotor contribution to aerodynamic da mping were

not available, the damping in the above example would be reduced by a factor of

four, which would increase the dynamic factor to 1.41. This is of relevance to non-

operational pitch-regulated machines facing into the wind, because if the lift

loading on a blade is near the maximum, the aerodynamic damping may be near

zero or even negative.

5.12.3 Operational loads due to steady wind (deterministic

component)

Tower fore-and-aft bending moments result from rotor thrust loading and rotor

moments. The moments acting on the nacelle due to deterministic rotor loads have

already been described in Section 5.11.1. Although the thrust loads on individual

blades vary considerably with azimuth as a result of yaw misalignment, shaft tilt or

wind shear, the ﬂuctuations on different blades balance each other, so that the total

304 DESIGN LOADS FOR HORIZONTAL-AXIS WIND TURBINES