Impact of shell structure stiffness on aero-structural coupling in wind turbine rotor blades

Wind turbine rotor blades are heavily loaded composite structures that experience a mixture of aerodynamic, inertial, gravitational, and gyroscopic forces during their operation life. Due to the high loads, the cross-sections of the blades are subjected to in-plane and out-of-plane deformations. The out-of-plane deformations are referred to as shear warping while the in-plane deformations are also called blade breathing. Blade breathing depends on the magnitude of the mechanical loads, which are expressed by means of internal forces and moments, and the stiffness of the blade shell. In this work, the relationships between in-plane cross-sectional deformations and internal loads are investigated. For the quantification of the deformation, a reference blade is studied via 3D finite shell element simulations for different loading scenarios. The cross-section of interest is located at the radial position of maximum chord. To compare the shape of the cross-sections in the undeformed and the deformed configurations, a procedure is proposed to relate the positions of nodes associated with the cross-section of interest in both configurations to a joint coordinate system. The shape of the deformed cross-section is then extracted and compared with the undeformed configuration. The comparison is executed for the individual internal forces and moments, namely flapwise and edgewise bending moments, normal force, shear forces, and torsion moment, respectively. The deformation patterns are discussed and it is addressed how these may influence the aerodynamic behavior of the cross-section under consideration.


Introduction
The performance of a wind turbine is governed by the aeroelastic characteristics of the rotor to a large extent.The aerodynamic characteristics of the rotor are defined by the characteristics of the airfoils used in the aerodynamic design.The lift-to-drag ratio as a measure of the airfoil efficiency is determined by the overall shape of the airfoil, which can be reduced to the relative thickness and the camber of the airfoil.A thin airfoil has a high lift-to-drag ratio, and the camber further increases the lift.Hence, a blade would consist only of thin airfoils, if it was designed just with respect to aerodynamic considerations.However, from a structural point of view, thick airfoils are beneficial for the higher loaded sections close to the blade root, as an increase of thickness increases the cross-section's moment of inertia and with it the structural features of the blade (i.e., stiffness, strength, natural frequencies, etc.).
Cross-sectional deformations are normally subdivided into out-of-plane deformations called shear warping, and in-plane deformations.In the general case, in-plane cross-sectional 1293 (2023) 012025 IOP Publishing doi:10.1088/1757-899X/1293/1/012025 2 deformations (IPCD) in a blade correspond to a variation of the airfoil shape and the aerodynamic characteristics on the one hand, and a variation of the cross-section's moment of inertia and the structural characteristics on the other.Consequently, cross-sectional deformations may compromise the aeroelastic performance of the blade, especially for very large and slender blade structures.
The effect of cross-sectional deformations on thin-walled shell-like structures was already demonstrated in [1].It was mentioned under the name Brazier effect as a geometric nonlinearity in wind turbine blades in [2].A solution for the calculation of cross-sectional deformations in symmetrical multi-cell airfoils was provided in [3].A simplified approach for the analysis of the Brazier effect in composite beams was proposed in [4].Cross-sectional deformations have been investigated in the context of damage analyses [5].They have also been considered to calculate peel stresses in a trailing edge adhesive joint [6].Buckling as another geometrically nonlinear effect was found to be a possible driver for trailing edge damages [7].However, to the best knowledge of the authors, there is no publication available on the effect of IPCD on the aerodynamic or aeroelastic behavior of the blade.
In this paper, the cross-sectional deformation is investigated for a reference wind turbine blade at the maximum chord position.A shell element model is used for 3D finite element (FE) simulations of the structural response.The blade is considered clamped at the root, and the force-like boundary conditions are applied in such a way that the internal force and moment distributions from loads analyses are approximated.The impact of the normal force, flapwise and edgewise shear forces, torsion moment, and flapwise and edgewise bending moments are analyzed independent of each other.Although this is unrealistic, as a blade in real life is always subjected to a combination of these loads, it allows to estimate what internal force or moment contributes in which manner to the IPCD.
To identify the shape of the deformed cross-section, a plane is identified considered the crosssectional plane in the deformed configuration.The nodal positions of the nodes associated with the cross-section under investigation in the deformed configuration are projected onto that plane.The method retains the assumptions according to Timoshenko beam theory, i.e., planarity of cross-sections and constant shear angles, but contains detailed information of the 3D model (e.g., blade compliance).The cross-sectional deformation contributions of each internal force and moment are described, both qualitatively and quantitatively, and conclusions are drawn related to their criticality in the context of the aeroelastic behavior of the blade.

Calculation of cross-sectional deformations
For the cross-sectional deformations, the following assumptions were made: • The cross-sections of interest undergo a translational and rotational deflection in space.
• The cross-section of interest in the undeformed and the deformed configuration is planar.
• The cross-section of interest is subjected to out-of-plane and in-plane warping, the latter of which called IPCD in this paper.
The aim was to extract the deformed cross-section from the solution of the 3D FE simulation and to compare it with the undeformed configuration.Both cross-sections thus had to be related to a joint coordinate system, which was the (global) blade root coordinate system.There, z was the basis vector pointing along the pitch axis of the blade, i.e., the director vector of the blade root cross-section, x was the in-plane basis vector perpendicular to the rotor plane (i.e., in wind direction), and y was the in-plane basis vector parallel to the rotor plane forming a right-hand Cartesian coordinate system, see also Figure 1.
The centroid of the undeformed and deformed cross-sections were calculated by the mean of all position vectors of the associated nodes in the undeformed and the deformed configuration, 1293 (2023) 012025 IOP Publishing doi:10.1088/1757-899X/1293/1/012025 Planform of the blade's 3D FE model, visualization of the cross-section at the marked maximum chord position, coordinate systems, and exemplary load boundary conditions.The blade model was created with MoCA [12], the in-house model creation and analysis tool (for visibility the mesh is coarser than that used for the analyses).The cross-section visualization was created with BECAS [13].The red dots show the load points.The external single forces F i approximate the internal forces and moments.The global blade coordinate system is plotted in red, the local cross-section coordinate system in blue.
respectively.The associated nodes were then translated by the centroid vectors.The centroid was thus positioned in the origin of the blade root coordinate system for both configurations.
For the undeformed cross-section, the (local) chord coordinate system was used1 to set up a rotation matrix.The rotation matrix was then utilized to transform the nodal coordinates so that the cross-section plane was aligned with the blade root coordinate system.Note that this step is only necessary if the blade exhibits prebend and/or presweep.Otherwise, the plane of the undeformed cross-section is parallel to the blade root coordinate system and the rotation matrix becomes the identity matrix.
The translational motion of the deformed cross-section was already accounted for by the translation of the cross-section centroid to the blade root center.For the rotational motion, the orientation of the cross-sectional plane in the deformed configuration had to be found.In the general case, the nodes in the deformed configuration are not located on a plane due to out-ofplane warping that is non-constant due to changes in shell stiffness around the circumference of the airfoil.However, we assumed that the deformed cross-section is planar (see the second bullet point above).It was thus necessary to identify a plane that is considered the crosssectional plane.In other words: The error of nodal positions not matching the cross-sectional plane had to be minimized.The mean value of the rotations of nodes associated with the crosssection of interest was calculated for setting up a rotation.This rotation matrix was then used for the coordinate transformation from the deformed to the undeformed cross-sectional plane.The nodal coordinates in the setting that was already translated to the blade root were then transformed via the rotation matrix and subsequently projected onto the plane parallel to the blade root.As a result, the shape of the deformed cross-section in the blade root coordinate system was obtained.

Rotor blade and load cases
The rotor blade under consideration was that of the reference wind turbine model developed in the IEA Wind TCP Task 37 [8].The turbine has a rated power of 15 MW, a rotor diameter of 240 m, and a blade length of 117 m.As the turbine exhibits a cone angle of -4 • and a prebend with a tip amplitude of 4 m, the cross-section plane in the undeformed configuration is not parallel to the blade root cross-section and the joint coordinate axes.Hence, the rotational transformation described in section 2 is necessary.
A loads assessment was executed for the design load case normal operation [9] at a mean wind velocity of 10 m/s, which is approximately the rated wind speed.OpenFast [10] was used with the ElastoDyn module [11] for the blades and the tower.The highest bending moment distribution was identified and the internal forces and moments were extracted for that time instance in the time series.
For creating the FE model, the in-house model creation and analysis tool called MoCA [12] was used.A sketch of the blade model in the planform view is shown in Figure 1.The coarse mesh depicted there is for visualization purposes only.The FE model used in the simulations consisted of 26,267 shell elements with linear shape functions and 25,839 nodes.Ansys R Mechanical APDL [14] was used for the solution of the FE problem.The cross-section of interest was chosen to be located at the maximum chord position, as the largest cross-sectional deformations were expected there.A sketch of the cross-section of interest is included in Figure 1, which was created with BECAS [13] for visualization via a MoCA2BECAS interface.The internal forces and moments extracted from the loads assessment at this position are given in Table 1.Note that a positive flapwise shear force or bending moment points in positive x direction, a positive edgewise shear force or bending moment in positive y direction, and a positive normal force or torsion moment in positive z direction on the positive cutting edge All 6 degrees of freedom were fixed in each node at the blade root, i.e., the blade was clamped.The force-like boundary conditions consisted of external forces and/or moments applied at a finite number of positions along the blade named load points.The load point positions and the magnitude of the forces and/or moments were chosen in such a way that the internal force and moment distributions from the loads analysis were approximated, with a focus on the crosssection of interest.In the load points, a single force or moment was applied to a master node that was linked to the nodes in the cross-section at the same radial position via multiple point constraints.The internal forces and the torsion moment were thus approximated as piecewise constant distributions with jumps in the load point positions, whereas the bending moments were approximated by continuous and piecewise linear distributions with kinks in the load point positions.The load points and the single force distribution for the maximum edgewise bending moment (i.e., compression in the trailing edge) is depicted in Figure 1.
Figure 2 exemplarily shows the minimum and maximum flapwise and edgewise bending moment distributions.As the blade root is clamped it cannot contribute to cross-sectional deformations.The load values at the blade root, i.e., at r/R = 0, are thus omitted in Figure 2. The convex graphs of the maximum bending moments as well as the concave graph of the minimum edgewise bending moments are well approximated.There is only a certain inaccuracy in the outboard region between load points 5 and 6 (counted from the root to the tip).This inaccuracy should be far enough away from the cross-section of interest to not influence the results.The minimum flapwise bending moment oscillates around zero, which is not well approximated between the load points 4 and 6.Here, an optimization of the load point positions would help to improve the solution.In any case, the bending moments in the cross-section of interest are all captured well.The convexity of the bending moment distribution determines whether the applied load distribution is an over-or an underestimation (i.e., bending moments in convex distributions are slightly overestimated, those in concave distributions are underestimated).
Recall that we intended to have only the desired internal force or moment in the cross-section of interest in order to be able to evaluate their impact on the IPCD separately.Hence, for the normal force, the shear force, and the torsion moment distributions the external force and moment were applied to the fourth load point, which is the load point next to the cross-section of interest towards the tip.For the normal force, the load point was moved inside the respective cross-section so that the bending moment induced by the prebend and the offset to the elastic center was compensated as much as possible, and a counter-moment was further applied to have a bending-free cross-section of interest.A similar strategy was followed for the shear forces, where, due to a small offset between the load point and the shear center, an additional counteracting torsion moment was applied so that the cross-section of interest was torsion-free.Note that shear forces and bending moments are not independent of each other, so that the distance between cross-section of interest and load point results in a bending moment in the cross-section of interest when a shear force is applied to the load point.A counter-moment was thus applied in the load point In order to obtain a bending-free cross-section of interest that is only loaded by shear forces.

Results
In this section, the results of the IPCD for the different load cases are presented and discussed.The order of load cases is normal force, flapwise shear force, edgewise shear force, torsion moment, flapwise bending moment, and edgewise bending moment.For each, deformations due to both minimum and maximum load is presented, respectively.
Figure 3 shows the IPCD for the applied internal forces.The nodes in the undeformed configuration are plotted by circles, and the arrows indicate the nodal displacements.IPCD for normal forces are plotted at the top, those for shear forces in the flapwise direction in the middle, those for shear forces in the edgewise direction at the bottom; maximum forces are on the left and minimum forces on the right.
For the normal force, there is no significant IPCD.Only the leading edge panel (between the leading edge and the left shear web) on the pressure side slightly deform inwards for the maximum normal force.However, the magnitude of deformation is negligible, given that the deformations are amplified by a scaling factor of s = 100.
For the maximum shear force in flapwise direction, the leading edge panel on the suction side is deformed inwards, that on the pressure side outwards.The trailing edge panel (between the right shear web and the trailing edge) on the suction side is mainly deformed inwards and that on the pressure side mainly outwards.The spar caps that are located in the shells between the shear webs are deformed upwards.Hence, the shear webs are translated upwards, but additionally deform partially towards the leading edge.The deformations in the leading edge panels are larger in magnitude than those in the trailing edge panels and the deformations in the left shear web are larger than those in the right shear web.The deformation pattern for the minimum shear force in flapwise direction looks qualitatively very similar, but is quantitatively smaller.This is because the minimum shear force has the same sign as the maximum shear force, but is two orders of magnitude smaller, see Table 1.
For the maximum shear force in edgewise direction, the leading edge panel on the suction side and the trailing edge panel on the pressure side deform outwards.All other parts of the crosssection hardly show any deformation.The deformation at the trailing edge is larger than that at the leading edge.For the minimum shear force in edgewise direction, the trailing edge panels (both at the suction and the pressure side) show deformations inwards.The spar caps around the right shear web deform upwards, so that the right shear web is translated upwards.The rest of the cross-section hardly deforms.The magnitude of the trailing edge deformations is orders of magnitude larger than those in the rest of the cross-section.Note that in this particular plot the scaling factor is s = 100, while the scaling factor in the other shear force-related plots is s = 10.The results thus must be treated with caution when it comes to quantitative comparisons.
Figure 4 shows the IPCD for the torsion and bending moments.Torsion moments are at the top, flapwise bending moments in the middle, and edgewise bending moments at the bottom.The maximum moments are on the left and the minimum moments on the right.The circles depict the nodes in the undeformed configuration, and the arrows are the nodal displacement vectors.
The deformation due to the maximum torsion moment is generally very small, given the amplification factor of s = 100.Qualitatively, the leading edge panel on the suction side deforms outwards, that on the pressure side outwards.In the trailing edge panels it is the other way round: That on the suction side deforms outwards and that on the pressure side inwards.The spar (spar caps and shear webs) hardly show any deformation.The deformation pattern for the minimum torsion moment is qualitatively the opposite, but has a larger amplitude.The reason is that the torsion moment has the opposite sign, but the magnitude is about 4 times larger.In the case of the maximum flapwise bending moment, the leading edge panel on the suction side and the trailing edge panel on the pressure side deform inwards, while the leading edge panel on the pressure side deforms outward.The trailing edge panel on the suction side does not show any significant deformation.The spar (spar caps and shear webs) stay also in place.The deformation pattern for the minimum flapwise bending moment looks qualitatively more or less like the negatively scaled version of that for the maximum flapwise bending moment, but with a much smaller amplitude, especially given the amplification factor that is one order of magnitude higher.The reason is that the minimum flapwise bending moment in the crosssection of interest is almost zero, while the maximum flapwise bending moment is quite high (two orders of magnitude higher than the minimum flapwise bending moment, see Table 1).
For the maximum edgewise bending moment, the deformation of the leading edge panel on the suction side is inwards, as well as that of the trailing edge panel on the pressure side.The leading edge panel on the pressure side and the trailing edge panel on the suction side deform outwards.Note that the shear webs are bended and rotated clockwise.It looks like the spar is undergoing a certain portion of torsion.This may be due to some bend-torsion coupling or an offset between the external forces and the shear center of the respective cross-sections resulting in an unintentional torsion moment.Generally, the deformation amplitudes are comparably IOP Publishing doi:10.1088/1757-899X/1293/1/0120259 high.For the minimum edgewise bending moment, the cross-section hardly deforms.Only the trailing edge is moved downwards, which results in a deformation inwards in the trailing edge panel on the suction side and a deformation outwards on the pressure side.

Discussion
The cross-sectional deformation of the investigated blade at the maximum chord position is generally small, given that the deformation had to be scaled by a factor of at least s = 10 to make it visible.However, a decrease in stiffness especially of the shell structure would result in more pronounced blade breathing.It is thus likely that for very large and slender blades, where the panels have a smaller local bending stiffness, suffer more from cross-sectional deformations.
The effect of normal forces on the cross-sectional deformations is in any case negligible.Even with a scaling factor of s = 100 the cross-sectional deformation is hardly visible.A decrease of the blade structure in such extent that the cross-sections show significant in-plane deformations would very likely result in loss of structural integrity.As the exterior shape of the airfoil is unaffected by normal forces, the aerodynamic performance is unaffected as well.
The cross-sectional deformations due to flapwise shear forces may become significant if the shell bending stiffness is reduced (or the length of the cross-section is increased).The deformation pattern shows a deflection of the leading edge towards the pressure side, which in turn affects the camber line (increase in camber).This may result in higher lift forces and thus in an increase of power output.However, an increase in lift force also leads to an increase of extreme loads and fatigue load amplitudes, and thus to a reduction of the fatigue life.The shape change at the leading edge and the trailing edge panel on the suction side may also be detrimental for the angle of attack at stall, which should be further investigated.
Edgewise shear forces mainly influence the shape of the trailing edge panels (deflection inwards or outwards, depending on the sign of the shear force).This can have an impact on the camber line and on the stall behavior and also deserves further attention in the future.
Naturally, a torsion moment triggers an elastic twist around the blade axis.This elastic twist will affect the angle of attack, a key parameter for the determination of aerodynamic forces.However, a torsion moment also affects the shape of the camber line significantly, as can clearly be seen in Figure 4.For the maximum torsion moment, the camber is increased (higher lift at the same angle of attack in the linear range of the lift coefficient).However, for the minimum torsion moment, the camber is reduced.Depending on the stiffness of the shell panels, the camber may be reverted in sign, which completely changes the aerodynamic behavior of the airfoil.
In case of bending, the ovalization appears only in the shell panels, as the shear webs are normally designed stiff enough to hold the spar caps in position.However, the deformation can affect the camber line and the airfoils shape significantly, in both the leading edge region and the trailing edge region.This holds independent of the direction of bending, i.e., for both flapwise and edgewise bending.
It is clear that the cross-sectional deformation depends on the magnitude of the respective internal forces and moments, as can clearly be seen when comparing Figure 3 and Figure 4 with Table 1.However, flapwise shear forces, torsion, and bending (both flapwise and edgewise) seem to be the most critical loads in the context of aerostructural coupling of cross-sectional deformations.
From the results obtained so far, we suspect an impact of the cross-sectional deformations on the aerodynamic airfoil characteristics, as discussed above.Due to a camber change, the lift coefficient in the attached flow regime may be shifted and the angle of attack at stall may be modified as well.Depending on the particular operation state of the turbine, this can have a beneficial or detrimental effect on the power production and the load distribution along the blade.However, a detailed evaluation requires cross-sectional airfoil polars for the undeformed IOP Publishing doi:10.1088/1757-899X/1293/1/01202510 and deformed configuration.The calculation of those is work in progress and a comparison of the aerodynamic performance will be published in the near future.
The aim of this paper was to analyze the impact of internal forces and moments individually in order to highlight the importance of each with respect to IPCD.However, a blade in real life is subjected to all loads simultaneously.Hence, the results of this paper cannot be used directly to draw conclusions for a blade in operation, as the cross-sectional deformations are geometrically non-linear in nature and cannot be superimposed.Furthermore, the internal loads not only appear simultaneously, but also vary with time at different frequencies.This induces multiaxial non-proportional fatigue in the blade's subcomponents, e.g., the adhesive joints, which is currently subject of extensive research [15,16].Hence, conclusions on the relations between cross-sectional deformations and fatigue damage evolution is very difficult and may be addressed in future work.

Conclusions
In this paper, in-plane cross-sectional deformations of a wind turbine blade were investigated.A methodology has been established for the extraction of the shape of a deformed cross-section from a 3D finite element simulation.This methodology has then been employed for the investigation of in-plane cross-sectional deformations of a 15 MW reference wind turbine blade at one particular position of interest.The position of interest was the maximum chord position, as the largest cross-sectional deformations are normally expected there due to large shell panel dimensions.The deformed cross-sections were compared with the undeformed cross-sections, and it was discussed how the deformation may influence the aerodynamic performance of the corresponding airfoil.
It was found that the deformations were principally quite small, as a magnification factor was required to make the deformations visible.However, the deformation amplitude depends on the bending stiffness of the shell panels.Hence, in very large rotor blades, where the local bending stiffness of the panels decreases due to very large panel dimensions, the crosssectional deformations may become significant.Generally, the airfoil shape and consequently the camber line is modified by the cross-sectional deformations.The relative thickness is hardly affected.The results give implications that the flapwise shear forces, torsion moments, and bending moments (both flapwise and edgewise) have the most significant impact on the crosssectional deformations.However, the impact of cross-sectional deformations on the aerodynamic performance have to be investigated in more detail, which is work in progress.

Figure 2 .
Figure 2. Bending moment distributions along the blade.Distributions from load assessments (exact) and the approximation from force-like boundary conditions (approx) are shown.Flapwise bending moments are plotted at the top, edgewise bending moments at the bottom.Maximum bending moments are plotted on the left, minimum bending moments on the right.Red dots are the load points where the external loads are applied.The maximum chord position is indicated by the vertical lines.

Figure 3 .
Figure 3. In-plane cross-sectional deformations (IPCD) for internal forces: Normal forces N (top), flapwise shear forces Q flap (middle), edgewise shear forces Q edge (bottom), maximum forces (left), minimum forces (right).The circles show the nodes in the undeformed configuration and the arrows indicate the nodal displacement vectors.The scaling factor s for the deformation vectors and the direction of the forces are given in the chart areas.View direction is in positive z direction, i.e., from towards the blade tip.

Figure 4 .
Figure 4. In-plane cross-sectional deformations (IPCD) for internal moments: Torsion moments M T (top), flapwise bending moments M flap (middle), edgewise bending moments M edge (bottom), maximum moments (left), minimum moments (right).The circles show the nodes in the undeformed configuration and the arrows indicate the nodal displacement vectors.The scaling factor s for the deformation vectors and the direction of the moments are given in the chart areas.View direction is in positive z direction, i.e., from towards the blade tip.

Table 1 .
Internal loads in the cross-section of interest (maximum chord position). 2.