Chapter 15 (Hansen 2012)

15

Fatigue

From earlier chapters it is clear that the loads on a wind turbine vary con-stantly with time, giving rise to a possible breakdown due to accumulatedfatigue damage. In Madsen et al (1990) a recommended practice to estimatethe fatigue damage, and thus the lifetime of a wind turbine, is outlined. Thischapter provides a summary of this practice.

First, the loads must be obtained from either computations using anaeroelastic code or directly from measurements. For normal operation theloads are monitored for 10 minutes in each wind speed interval Vp <Vo <Vp+1.An example of such a time history for V10min = 11m/s and a turbulence intensityI = 0.1 is seen in Figure 10.2, which is a result of a simulation using theaeroelastic code FLEX. The turbulence intensity, I, is defined as σ/V10min,where σ is the standard deviation of the wind speed within the 10 minutetime series. Knowing the loads, the stresses at critical points on the windturbine are computed using equations (11.9) (Hook’s law) and (11.10). As aminimum it is recommended in Madsen et al (1990) to monitor the bladebending moments, the yaw and tilt rotor moments, the axial thrust, the torquein the main shaft, the bending moments of the tower, and the torsionalmoment in the tower. From each 10 minutes time history the stresses aresorted in a matrix, where the elements mij(Vp <Vo<Vp+1) denote the number ofcycles in the mean stress interval σm,i<σm<σm,i+1 and range interval σr,j <σr

<σr,j+1 for the wind speed interval Vp <Vo <Vp+1.Figure 15.1 sketches one cyclewith a mean stress value σm and a range σr; it is seen that the range is twicethe amplitude σa.

To count the number of cycles from an actual time series such as the oneshown in Figure 10.2, a technique called ‘rainflow counting’ is used (for acomplete description of this algorithm, see Madsen et al, 1990). Then,knowing the annual wind distribution hW(Vo), the probability, f, of the windspeed being in the interval Vp <Vo <Vp+1 is computed from equation (6.48). Theactual number of annual 10 minute periods where the wind speed is in thisinterval is 6·8760·f. The number of cycles per year, nij, in the mean stressinterval σm,i <σm <σm,i+1 and in the range interval σr,j <σr <σr,j+1 is found by

adding together the contributions from each wind speed interval:

nij =�N–1

p=1

mij(Vp <Vo <Vp+l)·6·8760·f (Vp <Vo <Vp+l), (15.1)

where N–1 is the number of wind speed intervals. The matrix M, withelements nij, is called the Markov matrix. A wind turbine also experiencesloads when starting and stopping and when running under abnormalconditions such as at high yaw angles. Before a lifetime analysis can beperformed, it is also necessary to estimate these loads with respect to theannual number of occurrences with a given mean stress interval σm,i <σm

<σm,i+1 and range interval σr,j <σr <σr,j+1 and add this to the Markov matrix nij.In so doing one has to distinguish between starting and stopping at high windspeed and low wind speed, since the loads are different in these two cases.IEC 61400 (2004) gives a complete list of the different load cases needed forcertification. The total number of cycles in the entire lifetime within the meanstress interval σm,i <σm <σm,i+1 and range interval σr,j <σr <σr,j+1 is:

ntot,if = T·nij (15.2)

where the lifetime T is measured in years. To estimate T, the Palmgren-Minerrule, equation (15.3), for cumulative damage during cyclic loading is used.This rule assumes that the ratio between the number of applied stress cycles,

158 | Aerodynamics of Wind Turbines

Figure 15.1 Definition of mean stress σm and range σr for one cycle

nij, with a given mean stress level σm,i and range σr,j, and the number of cycles,Nij, which with the same mean stress and range would lead to failure,constitutes the expended part of the useful fatigue life and that the sum ofthese ratios is thus the damage D. Thus the criteria for not failing is that D isless than 1:

�ntot,ij––––Nij

= D < 1 (15.3)

Combining equation (15.2) with equation (15.3) yields the followingequation to estimate the lifetime T:

T = 1––––– (15.4)�

nij––Nij

The number of cycles, Nij, leading to failure for a cyclic loading with a givenmean stress level σm,i and range σr,j for a given material is found in a so-calledS-N curve or Wöhler curve like the one sketched in Figure 15.2.

Figure 15.2 Sketch of an S-N curve

Fatigue | 159

Sometimes only the range σr, j is taken into account and the influence of themean stress level σm,i is ignored. Under this assumption the S-N curve can beapproximated by:

m

N = N *σr(N *)

––––––––σr(15.5)

where m is a material constant and σr(N*) is the stress range giving failure forN* cycles. According to DS 412 (1983), for steel m is approximately 4.0; forglass fibre m is approximately 8–12.

The damage D can be estimated using the Palmgren-Miner rule. Tocompare the contribution from the different wind speeds to the total fatiguedamage, an equivalent, σr,eq, load can be used. The equivalent load is definedas the cyclic load which, when applied neq times, gives the same fatiguedamage on the wind turbine as the real turbulent flow at the considered windspeed. Since the total damage, D, is known, the equivalent load can becalculated similarly to using equation (15.5) for the S-N curve:

D = neq—Neq

= ––– neq––––––––––

N* σr(N*)

–––––m

(15.6)σr,eq

⇓

σr,eq = σr (N *) N

*D––––neq

1/m

Figure 15.3 plots a time series of 300s of a flapwise bending moment; Figure15.4 shows the result of using rainflow counting on this signal; in otherwords, the number of cycles with a given range contained in the time series.

Figure 15.3 An example of a time history of a flapwise bending moment

160 | Aerodynamics of Wind Turbines

� �

� �

� �

Figure 15.4 Result of using rainflow counting on the time series from Figure 15.3

A time series consisting of a sequence of cyclic loads with increasing rangecan be made that, assuming the Palmgren-Miner rule is correct, will give thesame fatigue damage as the original time series. Such a time series is shownin Figure 15.5.

Figure 15.5 Sequence of cyclic loads with increasing range that gives the same fatiguedamage as the original time series

ReferencesDS 412 (1983) Dansk Ingeniørforenings norm for stålkonstruktioner (Danish standard for

steel constructions)IEC 61400-1 (2004) ‘Wind turbines. Part 1: Design requirements’, CD, edition 3, second

revision, IEC TC88-MT1Madsen, P. H., Dekker, J. W. M., Thor, S. E., McAnulty, K., Matthies, H. and Thresher, R. W.

(1990) ‘Expert group study on recommended practices for wind turbine testing andevaluation. 3: Fatigue loads’, IEA Wind Energy Conversion Systems, second edition

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