the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 26 Oct 2023
Comparison of different cross-sectional approaches for the structural design and optimization of composite wind turbine blades based on beam models
Abstract. During the preliminary multidisciplinary design phase of wind turbine blades the evaluation of many design candidates plays an important role. Computationally efficient methods for the structural analysis are needed to cover the required effects, e.g., correct prediction of stiffness matrix entries including the (bend-twist) coupling terms. The present paper provides an extended overview of available approaches and shows their ability to fulfill the requirements for the composite design of rotor blades. Three cross-sectional theories are selected and implemented to compare the cross-sectional coupling stiffness terms and the stress distribution based on different multi-cell test cross-sections. The cross-sectional results are compared with the 2D finite element code BECAS and discussed in the context of accuracy and computational efficiency. The most promising approach achieved a better resolution of the stress distribution compared to BECAS and an order of a magnitude less computation time when the same discretization is used. The deviations of the stress distributions are below 10 percent for the most test cases. The results can serve as a basis for the beam-based design of wind turbine rotor blades.
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CC1: 'Comment on wes-2023-147', Vengalattore Nagaraj, 18 Nov 2023
Comments on “Comparison of different cross-sectional approaches for the structural design and optimization of composite wind turbine blades based on beam models”
Authors: Edgar Werthen, Daniel Hardt, Claudio Balzani, and Christian Hühne
During the preliminary design phase of composite rotor blades of wind turbines and helicopters it is necessary to evaluate of a large number of design candidates. These blades are idealized as beams for the aeromechanics analysis for the prediction of the loads and aeroelastic stability of the rotors. It is necessary to include the effects of laminate stacking and structural coupling effects between bend-twist and extension-twist deformations. A number of design candidates result from these different structural topologies (e.g., number and/or of spars) and concepts for materials used and how they are combined in laminate lay-ups. Consequently, the basic requirement of the approach used in the analysis is a significant reduction of the computation time for model creation and the calculation of internal stresses compared to a high-fidelity FE model. The computation time for the stress calculation scales with the number of iterations of the optimization process. Available computational models include 3-D FE models, 2-D cross-sectional analysis models, and 2-D analytical models.
The use of 3-D FE model in the preliminary design phase can be ruled out due to the high modelling effort and the long computation times. 2-D FE cross-section models also have similar problems due to the remeshing required for each iteration and also due to expensive solving effort compared with a 2-D analytical approach. The requirements of an acceptable high fidelity 2_D analytical model are that it must be capable of modeling beams with closed, single- or multi-cell cross-section geometries that can vary along the beam axis. The cross-sections of the rotor blades are usually thin-walled,andmay have material lay-ups that exhibit bend-twist and extension-twist couplings. In addition, it is necessary to accurately model the influence of cross-section warping and the shear deformation on the stiffness and stress distribution in the cross-section.
The paper under review provides a comprehensive review of available cross-sectional approaches.. The authors have selected three cross-sectional theories to compare their performance in the prediction of the stiffness properties including the influence of cross-sectional coupling stiffness and the stress distribution in the cross-section.. The cross-sectional results are validated with the 2D-finite element code BECAS. (BECAS itself has been validated against VABS, another 2-D FE that is extensively used in the helicopter field). The comparison is carried out using two different cross-sections with different material distributions. The first cross-section as a thin-walled and the second cross-section is a NACA 2412 airfoil with two shear webs , which is representative for a cross-section of a wind turbine rotor blade.
The advantage of the analytical approaches is that they do not need a discretization in contour-thickness-direction, while the 2-D FE approaches like BECAS requires a discretization for each layer of the laminate in the thickness-direction. The authors show that the analytical approach of Jung et al has the advantages of including the influence of transverse shear stress and torsional warping (Vlasov) and gives results that are close to the FE approach while it is an order of magnitude (sometimes even more) faster than BECAS. Also, the computing time for a single load case is around an order of magnitude faster than BECAS.
As a co-author of the Jung et al. papers, I commend the authors’ summary the analytical details of our approach. In their comparisons they have included the details of the cross-section stiffnesses, stress distributions, and the time required for the date preparation and solution of each case.
In summary, I find the paper under review to be a systematic and timely contribution to the literature and is worth publication.
Vengalattore T Nagaraj
Research Scientist. Department of Aerospace Engineering
University of Maryland, College Park, MD.
Disclaimer: this community comment is written by an individual and does not necessarily reflect the opinion of their employer.Citation: https://doi.org/10.5194/wes-2023-147-CC1 -
AC1: 'Reply on CC1', Edgar Werthen, 22 Nov 2023
Dear Dr. Nagaraj,
thank you very much for your comment. We appreciate your feedback.
Kind Regards
Edgar Werthen, Daniel Hardt, Claudio Balzani and Christian Hühne
Citation: https://doi.org/10.5194/wes-2023-147-AC1
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AC1: 'Reply on CC1', Edgar Werthen, 22 Nov 2023
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RC1: 'Comment on wes-2023-147', Anonymous Referee #1, 03 Dec 2023
In this paper, the comparison of the estimated cross-sectional stiffness matrix of a rectangular and a blade cross-section by the three semi-analytical approaches (Jung, Song and Wiedemann) and the 2D FE code BECAS (which is considered as the reference method). The paper is well written with a clear structure. Some of the minor comments are provided below for the authors to consider when preparing their revision.
- When obtaining the shear stiffness terms, a calculation model is considered with the blade tip loaded. Do you mean that a blade is assumed with the same cross-sections from the root to the tip? Moreover, is the tip load realistic to consider? In fact, a distributed line load is often used when designing the blade.
- Secondly, do you assume that the blade will experience relatively small deformations and behave as a linear beam globally? Will the geometrical nonlinearity as a blade beam influence the cross-sectional coupling stiffness terms?
- Only one blade cross-section is considered. It might be interesting to consider at least two cross-sections with different aerodynamic profiles.
Citation: https://doi.org/10.5194/wes-2023-147-RC1 -
AC2: 'Reply on RC1', Edgar Werthen, 08 Dec 2023
Comments of Reviewer in bold, Answers of Authors in italic
Dear reviewer,
Thank you very much for your valuable and helpful comments on our manuscript. It seems that we have not described the methods well enough. Hence, your feedback will help to improve the manuscript and to better present its content. Please find below the response to your questions:
When obtaining the shear stiffness terms, a calculation model is considered with the blade tip loaded. Do you mean that a blade is assumed with the same cross-sections from the root to the tip? Moreover, is the tip load realistic to consider? In fact, a distributed line load is often used when designing the blade.
Thank you very much for this comment. You are right, a blade certainly consists of several different cross-sections along the blade, and a blade is certainly not loaded by just a single force at the blade tip. The focus of this paper is the calculation of the cross-sectional properties (stiffness and mass matrices) of a beam model, not about a beam model itself. Like in most analytical cross-sectional theories (see e.g. Jung and Nagaraj (2002) equation 23), the approach to integrate the shear stiffness terms in the displacement based respectively mixed formulation of the cross-sectional stiffness matrix, is to assume a prismatic beam with a unit load at the free end, following the first-order shear deformation theory.
Once the cross-sectional properties of all cross-sections are calculated, a beam model consisting of several cross-sections certainly needs to be constructed and can subsequently be used to do loads simulations, obtaining the real load distribution along the blade. We will modify the text in order to make the approach more clear.
Secondly, do you assume that the blade will experience relatively small deformations and behave as a linear beam globally? Will the geometrical nonlinearity as a blade beam influence the cross-sectional coupling stiffness terms?
For the calculation of the cross-sectional properties, we indeed apply a linear theory. This is reasonable to our point of view, unless in-plane cross-sectional deformations are to be considered. In fact, this is work in progress, but is out of scope of this paper. When the linear cross-sectional properties are used to set up a beam model, the beam model itself must include geometrical nonlinearity in the sense of large deflections, as blades undergo very large deflections in operation. Large deflections in turn result in additional coupling effects. E.g., when considering equilibrium in the deformed state (which is the definition of geometrical nonlinearity), large flapwise deflections trigger edgewise bending / torsion coupling, which is not accounted for in linear beam theory. As mentioned above, this paper is about the calculation of cross-sectional properties. The inclusion of large deflections has to be included in a beam model, but beam model formulations are out of scope of this paper. We add this point as a discussion in the outlook of the paper.
Only one blade cross-section is considered. It might be interesting to consider at least two cross-sections with different aerodynamic profiles.
We have 2 different profiles. One rectangle, allowing a visual verification of expecting stress distributions for simple load cases and a NACA 2412 with 2 shear webs to represent the rotor blade. Adding the material combinations, 6 different variants were created in total, which - in the opinion of the authors - is enough to conclude that the method works. This said, we would like to emphasize that we are principally open to add another application example, but we are unsure what type of cross-section really adds value and insight instead of just extending the length of the manuscript. May we ask the reviewer for advice? Which kind of profile would you suggest?
Thank you very much!
Kind Regards
Edgar Werthen, Daniel Hardt, Claudio Balzani and Christian Hühne
Citation: https://doi.org/10.5194/wes-2023-147-AC2 -
AC4: 'Reply on RC1', Edgar Werthen, 18 Feb 2024
Dear Reviewer,
Thank you very much for your valuable and helpful comments on our manuscript. Please find our detailed answers in the attachment of our additional comment “Final response”.
Kind Regards
Edgar Werthen, Daniel Hardt, Claudio Balzani and Christian Hühne
Citation: https://doi.org/10.5194/wes-2023-147-AC4
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RC2: 'Comment on wes-2023-147', Anonymous Referee #2, 16 Jan 2024
In this manuscript, a number of beam cross-sectional models are compared for use in the context of early-stage wind turbine blade optimisation. This is a timely investigation into an interesting area, which is certainly worth researching. The paper is complete and generally well written. I am happy to recommend publication following minor revision for optimal clarity, but I would invite the authors to consider the points below to enhance their manuscript.
- The paper is dotted with typos, grammatical and formatting imprecisions. I would strongly recommend another round of careful proofreading.
- It would be good to add some comments into the manuscript on whether or not the predictions of the stiffness coefficients are mesh insensitive or, in other words, to have more details in the paper about the meshes adopted. It would also be good to have figures showing the meshes, especially around geometric details. This is because there is evidence in the literature that BECAS and VABS predictions are mesh sensitive, with fine meshes and accurate geometric representation of the cross-section being required for accurate results. See, e.g., https://wes.copernicus.org/preprints/wes-2023-85/. So, are the BECAS reference solutions converged?
- A similar comment applies to the accuracy of the stresses. The authors do comment on the link between mesh and stress predictions, but it would be useful if that discussion could be expanded. Similarly, it would be good to see how the different models perform with the stress recovery of all stress components, not just a few. That’s particularly important for the composite models, where through-the-thickness ply-by-ply stresses are notoriously difficult to resolve.
- I would strongly recommend considering an additional beam model that seems to have been omitted from the paper. Models from the book 'Mechanics of Composite Structures' by Kollár and Springer have proven useful in various projects, providing accuracy and efficiency.
- I am generally diffident of code-to-code performance comparison. Computer scientists have methods to do it accurately, but, in an engineering context, so many caveats need to be added that the results very quickly lose meaning. For instance, for a fair comparison, an accurate baseline needs to be established. I’d expect the comparison to be done between models that all deliver the same accuracy, otherwise one may compare, e.g., models that are quick and inaccurate with models that are slow and accurate. Also, can the authors discern if the speed of each model is related to the mathematic formulation thereof or to the specific software implementation of that model? More basically, a computer’s OS manages the machine’s resources continuously. Comparing run time not knowing what else the computer was doing during the analysis can be very misleading.
Citation: https://doi.org/10.5194/wes-2023-147-RC2 -
AC5: 'Reply on RC2', Edgar Werthen, 18 Feb 2024
Dear Reviewer,
Thank you very much for your valuable and helpful comments on our manuscript. Please find our detailed answers in the attachment of our additional comment “Final response”.
Kind Regards
Edgar Werthen, Daniel Hardt, Claudio Balzani and Christian Hühne
Citation: https://doi.org/10.5194/wes-2023-147-AC5
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RC3: 'Comment on wes-2023-147', Alexander Krimmer, 17 Jan 2024
The submitted manuscript reviews different methods to determine the cross-sectional stiffness properties of a wind turbine rotor blade on an analytical basis in comparison with BECAS. Since BECAS is a well known tool for this task it perfectly serves as a reference. The advantage of the analytical basis is adequately identified in terms of the calculation speed. This not only serves quick design space iinvestigation but as well high iteration speed in preliminary design. Therefore, the manuscript is of high relevance.
Anyhow, the results show differing deviations from BECAS for bending and torsional stiffness properties depending on complexity of the chosen cross-section. Here the deviation is reduced with increasing complexity. This seems counterintuitive. Thus, more insight into the actual calculation procedure would be helpful. Two of three chosen approaches are only mentioned and not elaborated on, which is why it will be difficult to repeat parts of the work.
All in all, the manuscript is worth of being published with minor revisions. Additional comments can be found in the pdf document attached.
Best regards.
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AC6: 'Reply on RC3', Edgar Werthen, 18 Feb 2024
Dear Reviewer,
Thank you very much for your valuable and helpful comments on our manuscript. Please find our detailed answers in the attachment of our additional comment “Final response”.
Kind Regards
Edgar Werthen, Daniel Hardt, Claudio Balzani and Christian Hühne
Citation: https://doi.org/10.5194/wes-2023-147-AC6
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AC6: 'Reply on RC3', Edgar Werthen, 18 Feb 2024
- AC3: 'Comment on wes-2023-147, Final response', Edgar Werthen, 18 Feb 2024
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Daniel Hardt
Claudio Balzani
Christian Hühne
A comprehensive overview is provided showing available cross-sectional approaches and their properties in relation to derived requirements for the design of composite rotor blades. The analytical approach of Jung shows the best results in terms of accuracy of stiffness terms (coupling and transverse shear) and stress distributions. An improved performance compared to 2D FE codes could be achieved making the approach applicable for optimization problems with a high number of design variables.
A comprehensive overview is provided showing available cross-sectional approaches and their...