To follow industrial standards, certain safety factors must be considered in the design of fatigue tests, which are omitted in this work for simplification.

All procedures described in this work follow certain simplifying assumptions, which are listed below. If any of these assumptions are considered non-applicable, the methods described here would need to be adjusted accordingly:

The validity of beam theory with small deformations is assumed, i.e., negligible in-plane warping of blade sections and negligible Brazier effect . Otherwise, sectional stiffness components would become dependent on these deformations (e.g., Brazier effect reduces outer dimensions, which in turn reduces the bending stiffness).

First, the DLCs which are to be considered are chosen (e.g., from ). Corresponding aero-elastic turbine simulations are performed, resulting in sets of time series f(t) for load distributions along the blade length, i.e., sectional bending moments Mx(t) and My(t) and longitudinal force Fz(t). These loads are derived for a local reference coordinate system in which the x–y plane of the section is perpendicular to the blade's beam axis, which includes following the orientation of any curvature of the blade reference line (e.g., pre-bend). The coordinate system's position and orientation in which the loads are reported must follow the blade deformation during simulation. Otherwise, the longitudinal z axis would not be perpendicular to the cross-section plane, and the beam theory formulas used subsequently would not be valid. Here, it is assumed that, in the undeformed state, the projection of this local coordinate system's x axis onto the blade's root section is parallel to the global lead-lag direction of the blade for any section and does not follow the blade's twist angle. Following the twist angle or other orientations is also possible. For the same sections along the blade, for which the loads are derived, the following properties are computed (see Fig. ):

coordinates of the elastic center (EC), i.e., the point where a force applied normal to the cross-section produces no bending curvatures, xEC and yEC;

The load time series are transformed into the EC and into the principal axis orientation of the corresponding section of the blade according to Eq. (): 1MxeMyeFze=cos⁡θpasin⁡θpa0-sin⁡θpacos⁡θpa0001⋅10-yEC01xEC001⋅MxMyFz.

This load transformation is necessary as the following equations for strain are only valid for the EC in the principal orientation (see Appendix ).

From this, the longitudinal strain at any given point of interest P within the corresponding blade section (see Fig. ) can be computed: 2εz(P)=yPeEIxe⋅Mxe-xPeEIye⋅Mye+FzeEA.

Assuming the longitudinal force contribution is negligible compared to the bending moment contribution, i.e., FzeEA≈0, and utilizing the distance rP from EC to P and its angle αP, the strain can be written as 3εz,M(P)=rP⋅sin⁡αPEIxe⋅Mxe-rP⋅cos⁡αPEIye⋅Mye.

In this work, the strain time series εz(P,t) are used as reference because they are assumed to be the most realistic representation of material fatigue behavior.

The bending moment Mβ perpendicular to the direction of rP, which is usually assumed to contribute the most to the strain εz,M, can be calculated through coordinate transformation: 4Mβ=sin⁡αP⋅Mxe-cos⁡αP⋅Mye.

In the assumed conventional procedure, only the bending moment Mβ is considered, particularly for the global blade main directions, i.e., flapwise and lead-lag, which, under the assumed coordinate system orientation, correspond to αP,f=-θpa and αP,l=-θpa+90°, respectively.

Any given load (e.g., bending moment, strain) must be further processed. In the following, L is used as a placeholder for any available load measure. To accumulate the load time series L(t) from simulations into corresponding DELs, the time series are converted via the rainflow counting algorithm into a list of occurring load amplitudes LA,i with corresponding mean loads LM,i and cycle numbers ni. This list can be compressed further into so-called Markov matrices by sorting the loads into discrete intervals (binning). Here, no binning was applied.

Only after rainflow counting and before the next processing step can mean load correction (MLC) be applied. This step accounts for the effect of the mean load on material fatigue. It entails changing a load amplitude LA,i, which corresponds to a specific mean load LM,i, to a corrected load amplitude LA,MLC,i, which in turn corresponds to the mean load Mi,MLC=0. This corrected amplitude is computed such that it contributes the same material fatigue damage as the original LA,i-LM,i pair. This correction requires the use of constant-life diagrams (CLDs), which are material-specific.

The simplest form of this is a linear symmetric CLD (also known as the Goodman or Goodman–Haigh diagram), which only requires one ultimate load LU and assumes symmetric behavior in tension and compression. For this, the MLC can be performed according to Eq. (): 5aLA,MLC,i=LA,i⋅LULU-|LM,i|.

As most composite materials have different properties in tension and compression, a shifted Goodman diagram is proposed in . This uses different ultimate loads for tension LUt and compression LUc, which results in Eq. () for the MLC: 5bLA,MLC,i=LA,i⋅LU,avg-|LU,mid|LU,avg-|LM,i-LU,mid| with LU,avg=|LUt-LUc|2,LU,mid=LUt+LUc2.

More complex CLDs, as proposed by , can also be employed. In that case, the implementation of Eq. (5) in the load evaluation would need to be replaced by corresponding methods.

Since the required material properties for MLC are only available for stress or strain data, this correction is not possible for bending moments. Therefore, when employing the conventional bending-moment-based approach, the impact of the mean load cannot be taken into account and must be neglected, resulting in Eq. (): 5cLA,MLC,i=LA,i.

After MLC, the corrected load amplitudes LA,MLC,i for each simulation are accumulated into a single DEL amplitude LDEL with an arbitrary cycle number NDEL, using linear damage accumulation according to and , assuming a linear stress–life relationship according to : 6LDEL=∑ini⋅(LA,MLC,i)mnDEL1m, where m denotes the negative inverse Basquin curve exponent of the material under investigation.

There are several approaches to defining this arbitrary number of cycles nDEL. Some research has suggested using the dominant frequency of the blade if contained in the load spectrum or otherwise the zero or mean crossing frequency . Another approach is to pick a frequency of 1 Hz, which is representative of a turbine size of the time . The latter approach was widely adopted because simulation time t in seconds is equal to the number of cycles nDEL, and nowadays a 1 Hz equivalent load is the commonly accepted practice, resulting in nDEL=t.

After the loads for each separate simulation j are accumulated into one damage-equivalent load LDEL,j, each with nDEL,j=1 according to Eq. (), the loads from different simulations are accumulated into one total damage-equivalent load amplitude LDEL,total with a cycle number of NDEL,total using probabilities of occurrence as weighting factors: 7LDEL,total=∑j(LTtj⋅nDEL,j⋅pws,j⋅pyaw,j⋅pDLC,j⋅(LDEL,j)m)NDEL,total⋅∑j(nts,j⋅pws,j⋅pyaw,j⋅pDLC,j)1m, where LT denotes the total expected turbine design lifetime; tj is the duration of the time series; nts,j is the number of turbulence seeds (i.e., the number of simulations with the same conditions); and pws,j, pyaw,j, and pDLC,j are the probabilities of the simulation's wind speed, yaw angles, and design load case (DLC), respectively. If further variables are differentiated with more simulations, the probabilities need to be adapted accordingly. As each simulation contains three blades, the loads from each blade can be regarded as separate simulation runs. This effectively triples the number of turbulence seeds nts,j if loads for all three blades are evaluated and accumulated.

Note that this damage accumulation is only valid for a linear stress–life relationship (Assumption 7 in Sect. ). To consider more complex fatigue behavior (e.g., ), the damage accumulation (Eqs.  and ) must be adjusted accordingly.

The resulting load DELs can then be used as target loads for blade fatigue testing. Depending on the scope of the fatigue test, the number of investigated angles αP and blade sections must be chosen correspondingly. The fatigue tests then have to be designed to match or exceed these target loads.

From Eqs. () and (), it can be seen that the strain and the swept bending moment are generally not proportional, εz,M∝Mβ. There are only two cases when they are proportional: (i) when the two principal stiffnesses of the section are equal (EIxe=EIye), which is usually only the case at the cylindrical root of the rotor blade, or (ii) when the position of interest P is on the principal axes, i.e., αP = 0°, ± 90°, and 180°. As conventional target loads are usually based on these bending moments and material fatigue damage is based on stresses or strains, this non-proportionality leads to discrepancies between the fatigue loads in blade fatigue testing and material fatigue damage arising from the design loads.

Further discrepancies can arise if the moments are converted into DELs while omitting the load transformation into the EC (see path 0 in Fig. ). The impact of this is outside the scope of this work as it is highly dependent on the arbitrary position of the coordinate systems used.

To mitigate the non-proportionality of bending moments and strain, proposed the modified bending moment Mβ′ to be used as the basis for target loads instead of the regular bending moments Mβ (see path 2.1 in Fig. ). In this work, Mβ′ has been slightly modified compared to in to more closely represent strain values: 8Mβ′=sin⁡αP⋅Mxe-cos⁡αP⋅EIxeEIye⋅Mye.

Translating the loads into Mβ′ for the test design instead of transforming data directly into strains has the benefit that the data required for the translation do not contain sensitive blade design data, which helps with data transfer between the OEM and test center, as highlighted by . With information on geometry and stiffness properties, Mβ′ can be transformed into the following corresponding strain: 9εz,M=rPEIxe⋅Mβ′.

This transformation is only valid if the assumption holds that the longitudinal force contribution to strain is negligible. Therefore, the impact of this assumption is investigated in Sect. .

For the MLC of Mβ′, proposed an approach based on the symmetric Goodman–Haigh diagram but without the use of material data; instead, unspecified ultimate loads derived from the test institution's experience were used. Moreover, the symmetry which only requires angles αP = 0°…180° does not hold anymore once the MLC is applied, and angles αP = -180°…180° are required. As Eq. () allows for simple conversion between strain εz,M and Mβ′, material-based MLC can be used in the Mβ′ domain by converting CLD data into the Mβ′ domain (see Fig. , path 2.2). For example, to enable Eq. (), the ultimate tension and compression loads must be evaluated: 10Mβ,Ut′=EIxerP⋅εz,Ut,Mβ,Uc′=EIxerP⋅εz,Uc.

However, this requires the derivation of individual CLD data for every position of interest along the blade. Moreover, Mβ′ after MLC is no longer independent of rP and is only valid for the position for which the corresponding CLD data are derived. If multiple positions along the αP direction with different rP are of interest, multiple CLDs must be considered for the same Mβ′. The confidentiality benefit still holds because no direct material data need to be disclosed for MLC.

The first study investigates the assumption that the longitudinal force is negligible. Therefore, the resulting strain DELs including MLC without the longitudinal force contribution, εz,M,DEL,MLC, and with it, εz,DEL,MLC, are compared. These measures are evaluated exactly as shown in Fig.  (path 3.2) with the only difference of utilizing Eqs. () or (), respectively. The relative differences between them are shown in Fig. , where they are projected on the blade geometry (left) and plotted over blade span and angle. The trailing edge (TE), the leading edge (LE), and the boundaries of the spar caps on the suction side (SS-SC) and pressure side (PS-SC) are marked for reference. The results show that considering the influence of the longitudinal force, compared to neglecting it, raises the accumulated DELs on the suction side by a maximum of 1.8 % and lowers them by a minimum of -1.8 % on the pressure side, especially close to the root. This deviation is deemed small enough to be neglected and confirms the assumption that the longitudinal force does not need to be considered in the fatigue test target loads. For cases when the longitudinal force should be considered anyway, target loads can be evaluated based on strains first (Fig. , path 3.2 using Eq. ) and then be translated into Mβ,DEL,MLC′ using Eq. ().

The next study investigates the impact of the MLC on the accumulated DELs. Therefore, the DELs are evaluated with and without MLC, utilizing Eqs. () and () (Fig.  paths 2.2 and 2.1), respectively. The relative differences between the modified bending moment DELs are shown in Fig. . Due to their proportionality, the same differences are found for the strain; i.e., εz,M,DEL,MLC (path 3.2)εz,M,DEL (path 3.1)=Mβ,DEL,MLC′ (path 2.2)Mβ,DEL′ (path 2.1). The results show that the DELs along the LE and TE are not affected by the MLC. However, on the SS panels, the MLC raises the DELs by up to 7.5 %, and on the whole SS-SC between 20 and 100 m, they range from about 8 % to up to 10.4 % around the 80 m span. On the PS, up to the 110 m span, the DELs are lowered by at least -3 %, with the lowest of -6.1 % on the spar cap around the 25 m span. These deviations are considered significant, and the increased load in particular confirms the necessity of MLC. Using the conventional methods without MLC to define the target loads would therefore lead to an insufficiently tested SS-SC: a 9.4 % decrease (opposite to a 10.4 % increase) of DELs leads to a 75 % decrease in applied fatigue damage (with m=14), which is missing compared to the predicted fatigue damage from the time series with MLC. This would lead to a fatigue test confirming only 25 % of the intended design fatigue life. The 10.4 % discrepancy shown in this study may also be exceeded when using different material properties or different CLD formulations.

To investigate the impact of using the modified bending moments Mβ′ instead of the regular bending moments Mβ for defining target loads for fatigue testing, these measures cannot be simply compared by values because their formulations are inherently different. Hence, for comparison, multiple sets of simplified uniaxial fatigue test loads for flapwise and lead-lag are computed here. These are designed to satisfy the different target loads, respectively, i.e., to match or exceed the corresponding field loads. As these are uniaxial tests, only the loads in the global main directions (i.e., φ = 180, 0, 90, -90°) are considered targets. In conventional fatigue test designs, the blade is fixed at the root with the blade axis horizontal and with the suction side facing downwards. Therefore, the blade's self-weight generates tension in the pressure side and compression in the suction side, which is assumed to be the mean load for the simplified test loads (neglecting any additional masses used in real fatigue tests, e.g., load frames). The cycle number for both tests is set to ntest = 2 × 10⁶. As test amplitude, the scaled load vectors resulting from the first two natural bending mode shapes of the blade are considered (assuming Fz=0), as fatigue tests are usually excited in resonance: 12LM=MxMy=0Fz=0self-weightLA,f=Sf⋅MxMyFz=0first mode shapeLA,l=Sl⋅MxMyFz=0second mode shape, where LM is the mean load vector, and LA,f and LA,l are the amplitude load vectors for the flapwise and lead-lag tests, respectively, with the corresponding scaling factors Sf and Sl. The amplitude load vectors are scaled for both tests such that the accumulated loads from both tests satisfy the target loads. Therefore, the test loads are evaluated in the same way as the field loads starting from Eq. (). This is done for each section along the blade individually and independently. To find the scaling factors for each section, an optimization problem is solved: 13aminimizeSf,Sl∑φ∈Φ(LDEL,φ,test(LM,LA,f,LA,l)-LDEL,φ,field load)13bsubject to LDEL,φ,test(LM,LA,f,LA,l)≥LDEL,φ,field load,φ∈Φ{180°,0°,90°,-90°}.

The optimization problem is solved using the Nelder–Mead algorithm . This results in load amplitude distributions for each test. Note that these load distributions do not represent actual fatigue tests that could be performed in reality but rather the best-case scenario, where the target loads are matched as closely as possible along the whole blade span. This optimization is executed to generate test loads designed for three different cases (cases 1, 2.1, and 2.2, corresponding to the paths in Fig. ) to match the corresponding field load data for LDEL=Mβ,DEL (case 1), Mβ,DEL′ (case 2.1), or Mβ,DEL,MLC′ (case 2.2).

From this, the impact of using these different approaches can be determined by evaluating the test loads in terms of damage-equivalent strain amplitude εz,DEL,MLC. Figure  shows the difference in the test loads for case 2.1 and case 2.2, each relative to case 1. Designing the test loads for Mβ,DEL′ (case 2.1) compared to Mβ,DEL (case 1) requires higher loads to achieve the target. In the lead-lag direction, the load needs to be raised by up to 14 % around 25 m blade length, which corresponds to the maximum chord length. Toward the tip outboard of 107 m, the load needs to be raised even more, though this area is usually not within the area of interest for fatigue testing. Only in the area between 77 and 96 m does the load need to be lowered. In the flapwise direction, the load only needs to be raised by up to 3 % close to the root around 15 m. Between 60 and 90 m, the flapwise loads for case 1 and case 2.1 are almost identical. Designing the loads for Mβ,DEL,MLC′ (case 2.2), i.e., considering MLC, requires even higher loads. In the lead-lag direction, the loads are similar to those in case 2.1 and need to be raised by up to 16 % around 25 m compared to case 1. In the flapwise direction, the load is raised by 5 %–8 % almost along the whole blade (8–115 m) compared to case 1.

This shows that the assumed conventional method used to design fatigue test loads (case 1) can lead to severe under-testing of the blade, as the more detailed methods (cases 2.1 and 2.2) require up to 16 % higher loads.

To elaborate further on these differences, Fig.  shows the fatigue damage from case 1 relative to the damage from case 2.1 and case 2.2. This damage ratio is derived from the load ratio and m for the corresponding material: D(1)D(2.1 or 2.2)=(εz,DEL,MLC(1)εz,DEL,MLC(2.1 or 2.2))m. This reveals that the 16 % required load raise in the lead-lag direction corresponds to a fatigue damage deficit of just under 80 %. In the flapwise direction between 15 and 105 m, case 1 produces only 33 %–43 % of the fatigue damage of case 2.2.

However, this method only considered the main flapwise and lead-lag blade directions; i.e., φ = 180, 0, 90, -90°. If other directions are also considered, the test loads compared to field loads for case 1 and case 2.2 are shown in Figs.  and , respectively. For case 1, the loads along the main directions are not matched as suggested above, but for case 2.2, the loads along the main directions are tested sufficiently. But both test scenarios show large areas that are loaded less from the test than from the field loads (hatched areas in Figs.  and ). These areas may include features which should be tested, such as critical structural details or significant load transitions between design elements. This suggests that uniaxial fatigue testing is insufficient to test the whole blade, and more sophisticated testing methods (e.g., biaxial testing) will be required to test the whole blade sufficiently.