An improved approach for the fatigue calculation of rotor blades based on sector loads

Abstract

In the design and calculation of rotor blades for wind turbines, a sector representation of the loads is often used. The investigation proposed in this paper indicates that, depending on the loading direction, the evaluated sector and the considered location on the rotor blade, the sector approach can result in a significant overestimation of the acting stresses. This translates in a higher calculated fatigue stress exposure (up to 101% in the considered example). However, under particular conditions, the stresses and therefore the stress exposure are correctly assessed. Based on the observation, an improvement of the current method is suggested. The improved method, called invariant-sector method, relies on the appropriate selection of the sector(s) to be evaluated. The method allows for an accuracy equal to the one of the direct method but with a computational effort that is orders of magnitude lower.

Keywords

Rotor blade sector loads fatigue Markov matrices range-mean matrices time series

Introduction

In the last decades, the wind energy field underwent a strong development. Pushed by the growing attention for the climate change issues and by subsidies from policy makers, the installed capacity has grown exponentially (Global Wind Energy Council, 2018). Driven by the growing demands, the technology and the products experienced a significant development, resulting in more efficient, larger, quieter and cheaper turbines. The reduction in the public subsidies, growing difficulties in established markets and the introduction of new players have resulted in an increase of the competition in the field and in a reduction of the profit margins for the turbine manufacturers (Vieira et al., 2019). The levelized cost of energy (LCEO) is expected to further significantly decrease in the coming years, with the reduction of turbine cost playing an important role (Wiser et al., 2016). To react to the market pressure, the turbine developers need to introduce innovative solutions and improve their design methods in order to achieve better performing design that are reliable yet structurally efficient.

In modern blade designs, fatigue aspects are dimensioning for many critical parts of the rotor blade. Due to the good fatigue behavior of composite materials (Mandell et al., 2002), a small reduction of the fatigue stress exposure would allow for a significant increase of the fatigue life.¹ A greater turbine life allows for amortizing the turbine costs over a longer period of time and has a positive influence on the LCOE. The current market trend is toward longer design lives, with the first design certified for 40 years operation being introduced (TüV Nord, 2019). Therefore, less conservative fatigue design methods can allow for longer design lives without needing to reinforce the structure.

At the same time, a greater reliability needs to be guaranteed in order to avoid expensive repairs in the field. These would negatively impact either the operational cost (OPEX) of the turbine operator of reduce the margins of the turbine developer.

It is clear that the capability of accurately estimating the fatigue life of a rotor blade is of paramount importance. Particularly, the method should be both reliable (conservative) in order to guarantee a safe design life and accurate to approach the design limit and therefore reduce costs.

This work introduces first the sector representation of the loads as used for rotor blade design (sec. 2). The effect of the sector projection on the calculated stress (sec. 3) and on the accuracy of the resulting fatigue stress exposure (sec. 4.2) is investigated. Based on the results of the investigation, an improvement of the current method is suggested (sec. 5).

Loads for the design of rotor blades

During their operational life, rotor blades are subjected to various loads (Manwell et al., 2010). These are mostly due to aerodynamic loads, gravitational loads and inertial loads. These loads are varying in time, both because of the rotation of the blade and the un-stationary wind conditions.

The loads acting on a rotor blade are generally calculated based on the Blade Element Momentum (BEM) theory (Lissaman, 1979). The blade is represented as beam and the aerodynamic characteristics are defined by the profile polars at various stations of the blade; further corrections are used for for example, tip and hub losses. Applying a time-domain approach, the response of the structure is time integrated based on several initial conditions corresponding to different operational and environmental conditions. The conditions to be investigated are defined in certification standards (International Electrotechnical Commission, 2005). Several commercial tools are used in the industry and have been shown to have a good agreement with experimental data (Zierath et al., 2016).

Research is concentrating on developing alternative methods based on metamodeling (Huchet et al., 2019), spectral approaches (Ragan and Manuel, 2007; Tibaldi et al., 2016), load case filtering (Stieng and Muskulus, 2019) and simplified time domain analysis (Schløer et al., 2016). These methods aim at reducing the computational effort while maintaining an acceptable accuracy and would be suitable for turbine optimization applications.

The load calculation returns the six load components (forces and moments) for the beam representation of the blade. The loads are evaluated at a discrete number of locations (load stations) along the blade. These loads vary both along the blade length and in time. Due to the small time discretization and the stochastic nature of the loads, this results in millions of load distributions. It is therefore needed to reduce such loads to a representative but manageable subset. The most common approaches for both static and fatigue loads are reported in the following. Although the focus of the work is on fatigue loads, the sector projection for static loads is first introduced as it is propaedeutic for the description of the fatigue loads.

Please note that, in this paper, we consider a coordinate system having Z directed in the blade radial direction (toward the tip) and Y directed along the chord toward the trailing edge (TE). X is therefore directed toward the suction side (SS) of the blade cross section. The coordinate system is shown in Figure 1.

Figure 1.

Used coordinate system.

Static loads

When considering static load envelopes, two main approaches are used: min/max loads and sector loads.

In the first approach, the loads for each load station are defined by the minimum and maximum envelope for all points in time for each of the six load components separately. The corresponding five remaining load components at the relevant point of time are reported together with the value for the enveloped component. The collection over all load stations results in 12 (six maxima and six minima) load envelopes. At this stage the time information is lost and therefore load envelopes of different load components are not consistent to each other (e.g. M_{y max} will not match the line integral of F_{x max}).

Another side effect of the loss of time information is the possibility of jumps in secondary load components over blade length; even changes of sign are common. This makes application of min/max loads in numerical blade modelling difficult and might cause unrealistic stress concentrations and stability problems. A possible solution consists in splitting the load case into multiple sub-load cases with smooth secondary load components²; this often results, however, in a great number of design loads.

A more detailed and advanced representation are the so called sector loads. For every load station, the M_x and M_y moments calculated at any time step are reported into a x−y plane (cross section of the rotor blade at the load station). The x−y plane is divided into angular sectors (at least 12 according to DNV-GL (2015)). For example, when considering 30° sectors, the sector 0° is defined ±15° around the x-axis; sector 30° is defined between +15° and 45° and so on. All the load points in a given sector are projected to the centerline of the sector and the highest value of the projected load (or the load module) is taken. The methodology is visualized in Figure 2. This results, for each sector, in a moment distribution along the blade length axis (z). For all other load components (F_x, F_y, F_z, M_z) the envelopes are generated based on the min/max approach, if needed.

Figure 2.

Sector projection of bending loads with 30° sectors. The sector directions are depicted with dashed black lines, the load points as blue dots. The projection is illustrated by the red lines.

Fatigue (cyclic) loads

The most realistic results are obtained when fatigue calculations are performed directly with the load time series for all six load components (called direct or time series approach). Fatigue loads are composed by few hundred up to few thousand time series, each of which generally comprises around 10⁴ time points. The stress (or strain) for each time point needs to be calculated and the counting methods (e.g. Rainflow Counting) are applied on the obtained stress time series. This approach is computationally very expensive but is widely used in the industry because of its accuracy.

The complexity of the fatigue calculation can be reduced by applying cycle counting methods onto the load time series (instead of the stress time series). With the counting, the load time series for a given load component at each load station is clustered into cycles with given range and mean and the corresponding cycles are counted. The result is reported in form of a so called Range-Mean-Matrix or Markov-Matrix.

As counting can be performed only on a single load component, the loads components need to be considered individually, that is, one independent Markov matrix is obtained for each load component. This approach is highly non-conservative as the superpositioning of two load components, for example, Lead-Lag and Flapwise bending moments, is neglected.

A better representation can be obtained using the sector approach introduced in the previous chapter. The x−y-plane is divided into sectors and the counting is performed on the projection of the loads onto the centerline of the sector. Differently from the static loads, not only the loads in the sector but all loads are projected. Please note that only one half of the sectors needs to be considered (e.g. from 0° to 180° instead of from 0° to 360°) as loads can have positive to negative values. This approach allows to consider the superpositioning of the Lead-Lag and Flapwise loads but not of all different loads components (e.g. shear forces and torsional moments).

A consequence of the sector load representation is that for each sector, every point on the blade (for a FE analysis, each finite element) has only one corresponding principal stress vector whereas for time series the direction of the resultant stress vector will vary constantly with the varying combination of M_x and M_y.

Please note that, when considering sector loads, the fatigue analysis needs to be performed independently for all the sectors and the maximum stress exposure out of all the sectors has to be considered.

A further simplification is represented by the Damage Equivalent Loads (DEL). The time series for each load station and load component is rainflow-counted and condensed into one value which results in the same damage. When neglecting the mean value, the damage equivalent amplitude $M_{DEL}$ evaluated for $N_{ref}$ load cycles result:

M_{D E L a m p l} = {(\frac{1}{N_{r e f}} \sum^{​} n_{i} M_{a i} m)}^{\frac{1}{m}}

(1)

where $M_{a i}$ is the amplitude of a given bin, $n_{i}$ the associated number of cycles and m the slope parameter of the SN diagram.

Unfortunately, it is not possible to correctly account for the mean load value without considering the specific material behavior, for example, through a Goodman or constant-life diagram (Freebury and Musial, 2000).

One significant advantage of the DEL is the possibility of easily comparing different load sets, for example, for different wind conditions or turbine configurations. The DEL are also commonly used for the definition of loads for blade testing.

As the damage calculation depends on the negative inverse slope of the S-N curve m, the DEL evaluation needs to be performed for all the relevant m values. In order to consider the superpositioning of the Lead-Lag and Flapwise loads, DEL can also be calculated for sector loads.

Analysis of the sector projection

As illustrated in section 2.2, several approaches exist for representing cyclic loads. The sector approaches are faster compared to the time series approach but result in a loss of accuracy. The introduced approximation is assessed in the following. For the sake of simplicity, the assessment is limited to the bending moments M_x and M_y, which are considered to be dimensioning for the fatigue damage of the rotor blade.

The applied load (moment) for a specific load station (radial position) and time can be represented in polar coordinates $M_{0}$ and $φ$ :

{\begin{matrix} M_{x} = M_{0} \cdot \cos (φ) \\ M_{y} = M_{0} \cdot \sin (φ) \end{matrix}

(2)

A visual representation of the azimuth $φ$ is shown in Figure 1.

The strain (or stress) component in a particular point on the rotor blade and in a given direction (e.g. the UD fiber direction) is a function of the applied load:

ε = f (F_{x}, F_{y}, F_{z}, M_{x}, M_{y}, M_{z})

(3)

Using a Maclaurin series, the strain can be approximated as:

ε \approx f_{0} + \frac{\partial f}{\partial F_{x}} \cdot F_{x} + \frac{\partial f}{\partial F_{y}} \cdot F_{y} + \frac{\partial f}{\partial F_{z}} \cdot F_{z} + \frac{\partial f}{\partial M_{x}} \cdot M_{x} + \frac{\partial f}{\partial M_{y}} \cdot M_{y} + \frac{\partial f}{\partial M_{z}} \cdot M_{z}

(4)

where $f_{0}$ represents frozen stresses and can be neglected.³ Please note that the same approach can be used for describing other entities, for example, the peeling force under a shear web.

By considering only the effect of the bending moments, we obtain the transfer function (TF) to reconstruct the strain (or stress) from the applied load:

ε \approx \frac{\partial f}{\partial M_{x}} \cdot M_{x} + \frac{\partial f}{\partial M_{y}} \cdot M_{y} = T F_{Mx} \cdot M_{x} + T F_{My} \cdot M_{y}

(5)

The transfer function is a function of the position on the rotor blade and of the considered strain component; it is time invariant.

The transfer function for each point on the blade can be obtained by calculating (e.g. with FE or with analytical methods) the strain/stress under two linearly independent load cases,⁴ LC1 and LC2, and solving the following linear system:

[\begin{matrix} {{}^{LC 1}M}_{x} & {{}^{LC 1}M}_{y} \\ {{}^{LC 2}M}_{x} & {{}^{LC 2}M}_{y} \end{matrix}] \cdot {\begin{matrix} T F_{x} \\ T F_{y} \end{matrix}} = {\begin{matrix} {}^{LC 1}ε \\ {}^{LC 2}ε \end{matrix}}

(6)

Similarly to the loads, we represent the transfer function in polar coordinates $T F_{0}$ and $ϑ$ :

{\begin{matrix} T F_{Mx} = T F_{0} \cdot \cos (ϑ) \\ T F_{My} = T F_{0} \cdot \sin (ϑ) \end{matrix}

(7)

Please note that $ϑ$ is not a physical angle and is, like $T F_{0}$ , depending on the considered strain component. For the case of the lengthwise strain, $ϑ$ can be assumed to be related to the position on the blade. In particular, a point on the leading edge (LE) is mostly stressed by M_x loads ( $ϑ \approx 0$ ) whereas a point close to the spar caps is mostly stressed by M_y ( $ϑ \approx \pm 90 °$ ).⁵

During the transformation of the loads into sector loads, the moments are projected onto the i-th sector $α_{i}$ :

{{}^{α_{i}}M}_{0} = M_{0} \cdot \cos (α_{i} - φ)

(8)

The strain under the considered sector load results by replacing (2) in (5):

{}^{α_{i}}ε = {{}^{α_{i}}M}_{0} \cdot [\cos (α_{i}) \cdot T F_{x} + \sin (α_{i}) \cdot T F_{y}]

(9)

By replacing (7) and (8) into (9) we obtain:

{}^{α_{i}}ε = M_{0} \cdot T F_{0} \cdot \cos (α_{i} - φ) \cdot [\cos (α_{i}) \cdot \cos (ϑ) + \sin (α_{i}) \cdot \sin (ϑ)]

(10)

The direct strain (without the sector projection) is obtained by replacing (2) and (7) in (5):

ε = M_{0} \cdot T F_{0} \cdot [\cos (φ) \cdot \cos (ϑ) + \sin (φ) \cdot \sin (ϑ)]

(11)

The ratio between the projected and the direct (not projected) strain results:

f = \frac{{}^{α_{i}}ε}{ε} = \cos (α_{i} - φ) \cdot \frac{\cos (α_{i}) \cdot \cos (ϑ) + \sin (α_{i}) \cdot \sin (ϑ)}{\cos (φ) \cdot \cos (ϑ) + \sin (φ) \cdot \sin (ϑ)}

(12)

If the sector oriented in the load direction is selected ( $α_{i} = φ$ ), the factor f is equal to 1 and the sector projection introduces no approximation. However, as the load orientation $φ$ is variable in time, this condition will be satisfied for a given sector direction $α_{i}$ only for a few time instants.

If the sector direction $α_{i}$ is equal to the transfer function azimuth $ϑ$ , the factor results:

f = \frac{{}^{α_{i}}ε}{ε} = \cos (ϑ - φ) \cdot \frac{\cos (ϑ) \cdot \cos (ϑ) + \sin (ϑ) \cdot \sin (ϑ)}{\cos (φ) \cdot \cos (ϑ) + \sin (φ) \cdot \sin (ϑ)} = 1

(13)

As the factor is one independently of the load direction $φ$ , the condition holds for all time instants. Therefore, the direction $α_{i} = ϑ = atan (T F_{My} / T F_{Mx})$ is an invariant.

Validation

Effect of the sector projection

To offer a visual interpretation of the results above, three cases are investigated. The applied load is acting in the lead-lag direction ( $φ = 0 °$ , LE under tension) whereas the TF has orientation $ϑ = 0 °$ , $ϑ = 45 °$ and $ϑ = 90 °$ . According to the used definitions, the first case represents (for strains/stresses in the blade direction) an element close to the LE, the third case an element close to the SS spar cap.

The results are illustrated in Figure 3. The red line represents the load vector whereas the black line represents the transfer function vector (TF); please note that here only the direction is of interest and not the absolute value, which is assumed to be 1.

Figure 3.

Reconstructed strains along different sectors for different orientations of the transfer function: (a) $ϑ = 0^{°}$ , (b) $ϑ = 45^{°}$ , and (c) $ϑ = 90^{°}$ . The moments and the transfer function TF have unit modulus.

The black dashed line indicates the direct strain ( $ε_{real}$ ), which is the strain obtained under the really applied moments, equation (5) or (11); the value is independent from the considered sector and is therefore constant for all evaluated directions. The green marks indicate the strain reconstructed for the different (15°) sectors; the thin, green, dashed line represent the envelope potentially obtained for a finer discretization.

The results indicate that, when the load and the transfer function are aligned ( $φ \approx ϑ$ ), Figure 3(a), the maximum reconstructed strain matches the direct strain. When the load and the TF are perpendicular ( $(φ - ϑ) \approx \pm 90 °$ ), Figure 3(c), the direct strain is zero whereas the reconstructed strain is not. In an intermediate situation, Figure 3(b), we see that the maximum reconstructed strain is significantly greater than the direct one (phantom strains). The maximum reconstructed strain lies on the secant between the load and the TF azimuth.

Figure 3 shows that the strain is always correctly reconstructed along the TF azimuth ( $α_{i} = ϑ$ ) and along the moment direction ( $α_{i} = φ$ ), as predicted in sec. 3. Please note that while the loading direction $φ$ varies in time, the TF azimuth $ϑ$ for a given location and strain component remains constant.

Summarizing, the results show that the sector representation of the loads (Markov matrices or DELs) results in numerical artefacts (phantom strains) for certain sectors. Please note that, as the strain is correctly reconstructed for the sector corresponding to the moment direction ( $α_{i} = φ$ ), the phantom strains are not significant in the static case as the loads are projected only onto the adjacent sector direction ( $| α_{i} - φ |$ is smaller than half of the sector discretization).

Effect on the damage calculation

According to common practice, the following investigation is performed based on stresses. Methods and conclusions shown in the above sections are valid for stresses analogously.

As explained in the previous section, the use of a sector representation of the loads results in phantom strains/stresses. These strain/stresses can be very significant

The assessment of sec. 4.1 concentrated on the loading for a single point in time. The fatigue loading is however composed of a multitude of time points (load time series), for which the load vector has in general different directions and magnitudes. As the damage for a given point on the rotor blade is given by the sum of all the (Rainflow counted) time points, the impact of the phantom stresses on the final damage cannot be easily estimated.

To provide an estimation, two load time series are assumed and the damage is calculated for three points on the rotor blade. The first assumed bending moment time series (Figure 4) is flap-dominated (average azimuth $φ$ : 90.9°); the flap and lead-lag moments are not in phase.

Figure 4.

(First) arbitrary time series: (a) M_x/M_y representation and (b) module/azimuth time series.

Three exemplary transfer functions (element locations) are considered:

$T F_{1} = [0.1, 0.3]$ (element in the main girder area; $ϑ ~ 71.6 °$ )

$T F_{2} = [0.1, 0.1]$ (element between main girder and leading edge; $ϑ ~ 45 °$ )

$T F_{3} = [0.5, 0.1]$ (element close to the trailing edge; $ϑ ~ 11.3 °$ )

The damage is calculated both based on the time series (direct approach) and on the sector Markov matrices (sector approach).

In the direct approach, the bending moment time series is transformed to element stress time series for each considered point on the rotor blade using the transfer function. Each stress series is then rainflow-counted and, based on the obtained stress amplitudes and mean values, the damage d is calculated with the Palmgren-Miner linear accumulation (Miner, 1945) as:

d = \sum_{i} \frac{n_{i}}{N_{i}}

(14)

where $n_{i}$ is the number of load cycles in a given bin and $N_{i}$ the permissible number of cycles for the given load.

Several possible approaches are available for estimating $N_{i}$ . Although a multiple R-value constant-life diagram (Nijssen, 2006) would potentially offer the highest accuracy, it would require the definition of several material parameters. For the sake of simplicity, in this investigation a (shifted) Goodman diagram is therefore used. Under the formulated assumptions (Lloyd, 2010), $N_{i}$ is calculated as:

N_{i} = {(\frac{R_{k, A} - | S_{k, M} - R_{k, M} |}{S_{k, A}})}^{m}

(15)

where $R_{k, A}$ and $R_{k, M}$ are respectively the amplitude and mean resistances, $S_{k, A}$ and $S_{k, M}$ the amplitude and mean value of the stress in a given bin and m the slope parameter of the SN curve. For the current investigation, $R_{k, A} = 1000 MPa$ , $R_{k, M} = 0 MPa$ and $m = 10$ are considered.

Finally, the stress exposure e is obtained as:

e = \sqrt[m]{d}

(16)

In the sector approach, the same bending moment time series is projected to 12 × 15° sectors. Subsequently each sector time series is rainflow counted and sector range-mean (Markov) bending moment matrices are derived. These are transformed to stress range-mean matrices (one for each sector and for each considered point on the rotor blade) using the transfer function. Finally, the stress exposure is calculated as explained above.

Finally, the stress exposure is additionally calculated using damage equivalent loads (DEL).

The results of the different approaches are compared in Figure 5. The results indicate that the (maximum) calculated stress exposure for the sector approach is greater than the one for the direct approach for all three considered locations; thus, the sector approach is conservative. The calculated stress exposure for direct and sector approaches are in good agreement for the first two cases (point close to the girder and between girder and TE) whereas the sector approach results in a twice as high stress exposure as for the direct approach for the third case (Table 1). This last case, which represents a point close to the TE, has an almost 90° phase between the predominant load direction ( $φ = 90.9 °$ ) and the TF azimuth ( $ϑ ~ 11.3 °$ ) Particularly, the highest discrepancy for the third case is observed for the sector 45°, which represents the secant between the two directions. This result is also in good agreement with the prediction from sec. 3.

Figure 5.

Calculated damage based on stress time series (solid line) and sector range-mean matrices (dashed line) and for DEL loads (dotted line) for three individual transfer functions. The round markers indicate the TF azimuth $ϑ$ , the vertical line the load azimuth $φ$ . The lines connecting the markers represent the results obtained for an infinitely fine sector discretization.

Table 1.

Comparison of the stress exposure obtained with the direct and with the sector approach.

	Direct	Sector (Markov)	Error (%)	Sector (DEL)	Error (%)
TF₁	0.751	0.758	1	0.411	−45
TF₂	0.199	0.213	7	0.175	−12
TF₃	0.462	0.926	101	0.508	10

Noteworthy, the results are in good agreement for all cases when considering the sector oriented in the TF azimuth ( $α_{i} ~ ϑ$ ); this is marked with a circle in Figure 5. On the other hand, it should be noted that no good agreement is observed around the predominant load direction ( $90.9 °$ ).

The results for the DEL are lower than the ones for the Markov matrices. This is due to neglecting the mean component of the load.

To corroborate the produced results, a second time series is considered (Figure 6). This time series, which is more representative of the normal blade operation, considers also random noise to represent vibrations and disturbances (e.g. turbulence). Although the loads are still flap-dominated (average load azimuth $φ$ 88.4°), the lead-lag loads display a much higher amplitude.

Figure 6.

Second considered time series: (a) M_x/M_y representation and (b) module/azimuth time series.

The calculated stress exposure is reported in Figure 7. Due to the greater influence of lead-lag loads, the discrepancy for the main girder element (TF₁) has increased whereas the one for the TE element (TF₃) has reduced (Table 2). Also in this case, the results for the sector $α_{i} ~ ϑ$ are in good agreement with the direct method whereas no good agreement is observed around the predominant load direction ( $88.4 °$ ).

Figure 7.

Damage calculated with stress time series (solid line), with sector range-mean matrices (dashed line) and for DEL loads (dotted line) for three individual transfer functions. The round markers indicate the TF azimuth $ϑ$ , the vertical line the load azimuth $φ$ . The lines connecting the markers represent the results obtained for an infinitely fine sector discretization.

Table 2.

Comparison of the stress exposure obtained with the direct and with the sector approach.

	Direct	Sector (Markov)	Error (%)	Sector (DEL)	Error (%)
TF₁	0.307	0.394	29	0.272	−11
TF₂	0.161	0.176	9	0.1568	−3
TF₃	0.796	0.901	13	0.6551	−18

Also in this case, the results for the DEL are lower than the ones for the Markov matrices. Particularly, the results are non-conservative, that is, they underestimate the stress exposure.

When considering a particular part of the blade, it is common practice to concentrate on the sector producing the highest stress exposure. The curves of Figures 5 and 7 clearly indicate however that this sector is not the one representing the true damage to the considered part. Instead, the sector oriented in the invariant direction $α_{i} = ϑ$ (the TF azimuth of the considered part) shall be used. Please note that this does not mean that only the loads in the invariant direction damage a specific components, but that the projection of all the fatigue loads on the invariant direction provides the damage equivalent representation for a given part (independently of the SN slope).

Invariant-sector method

Based on the presented results, an improvement for the approach for calculating the fatigue damage under sector loads can be derived. Rather than considering the maximum of all sectors, only the sector corresponding to the invariant direction, that is, the TF azimuth $ϑ = atan (\frac{T F_{y}}{T F_{x}})$ , should be evaluated (invariant-sector method). In practical applications, since the direction $ϑ$ of the TF will not match perfectly a specific sector, the damage can be evaluated for the two nearest sectors (the higher value shall be considered); please note that the sector(s) to be evaluated depend(s) on the position on the rotor blade and on the considered stress/strain component.⁶

Alternatively, the stress exposure can be obtained per interpolation. Considering a linear interpolation between the two nearest sectors $α_{i}$ and $α_{i + 1}$ , the stress exposure e results:

e = e (α_{i}) + \frac{e (α_{i + 1}) - e (α_{i})}{α_{i + 1} - α_{i}} (ϑ - α_{i})

(17)

The results obtained with the sector-invariant method for the two time series of sec. 4.2 (Figures 4 and 6) are reported in the last columns of Table 3. The new approach allows for a decrease of the discrepancy from the direct method. Particularly, when considering the maximum stress exposure of the adjacent sectors (max), the obtained error is reduced but remains significant. When considering the interpolated result (interp.), the obtained error becomes negligible (<1%); however, the estimation might be slightly optimistic.

Table 3.

Comparison of the stress exposure obtained with the direct, the sector and the sector-invariant method.

Time series	Location	Direct	Sector (Markov)		Invariant-sector (max)		Invariant-sector (interp.)
Time series	Location	Stress exp.	Stress exp.	Error (%)	Stress exp.	Error (%)	Stress exp.	Error (%)
1	TF₁	0.751	0.758	1	0.758	1	0.749	−0.2
	TF₂	0.199	0.213	7	0.199	0	0.199	0.0
	TF₃	0.462	0.926	101	0.526	14	0.463	0.2
2	TF₁	0.307	0.394	29	0.383	25	0.305	−0.7
	TF₂	0.161	0.176	9	0.176	9	0.161	0.0
	TF₃	0.796	0.901	13	0.830	4	0.793	−0.3

Please note that, with the invariant-sector method, only two sectors have to be evaluated for each point (or FE element) independently of the sector discretization (e.g. 30° or 15°). As a finer sector discretization allows for more accurate results, the quality of the results can be further improved without increasing the computational cost (provided the availability of the loads). It is expected that a very fine sector discretization can reach a similar accuracy as the interpolation (while requiring the calculation of only one sector at the cost of the numerically more expensive rainflow counting when generating the Markov matrices).

The results clearly show that the invariant-sector method allows for a significant reduction of the error associated with the sector approach, thus allowing for a more efficient design. Furthermore, the reduction of the number of sectors to be evaluated per element translates into a decrease of the computational time,⁷ thus making the sector approach even more competitive compared with the direct approach.

Conclusion

The presented work investigates the approximation introduced when using a sector representation of the fatigue loads for the analysis of rotor blades. The investigation indicates that the sector representation results in higher reconstructed stresses/strains (phantom stresses). The stresses are however correctly reconstructed⁸ along the moment direction ( $α_{i} = φ$ ) and along the TF azimuth ( $α_{i} = ϑ$ ). The moment direction $φ$ varies in time whereas the TF azimuth $ϑ$ depends only on the structural properties of the blade in a specific point on the blade and is time independent. Thus, the $ϑ$ direction represents an invariant. The loads in this direction are the ones that best represent the damage of the considered point.

It has been verified with arbitrary transfer functions and arbitrary time series that the calculated maximum damage of all sectors provides a conservative estimation of the damage calculated with the direct method. However, under certain circumstances, the stress exposure is considerably overestimated (up to 101% in the considered examples): large discrepancies are to be expected for example, in the LE/TE areas under predominantly flapwise loading and in the main girder area for predominantly edgewise loading. The results also confirmed that the sector approach delivers the correct results when considering the sector in the TF azimuth ( $α_{i} = ϑ$ ).

Based on the above observations, the invariant sector method was developed. The method can be summarized in the following steps:

Calculate, for each point/element of the blade to be evaluated, the transfer function relating the strain/stress to the applied loads: $ε = T F_{Mx} \cdot M_{x} + T F_{My} \cdot M_{y}$

Calculate, for each point/element, the invariant direction $ϑ = atan (\frac{T F_{y}}{T F_{x}})$

Calculate the stress exposures e with the Markov matrices for the two sector directions $α_{i}, α_{i + 1}$ closest to the invariant direction $α_{i} \leq ϑ \leq α_{i + 1}$

Calculate the final stress exposure as $e = e (α_{i}) + \frac{e (α_{i + 1}) - e (α_{i})}{α_{i + 1} - α_{i}} (ϑ - α_{i})$ or, more conservatively, as $e = max (e (α_{i + 1}) - e (α_{i}))$

The method considers not the maximum sector stress exposure but only the sector(s) with the orientation closest to the invariant direction, that is, the TF azimuth. The sector to be considered depends, like the TF, on the location on the rotor blade and on the considered stress (or strain) component and is independent of the loading. The invariant-sector method allows for significantly more accurate results as well as for a reduction of the computational effort.

Footnotes

Acknowledgements

The authors would like to thank Stefan Steiner for his support to the writing of this paper and Senvion for authorizing the publication of this paper.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

F. Previtali

Notes

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