The Importance of Cross-Sectional Analysis in Wind Turbine Blade Design
During the preliminary design phase of wind turbine blades, the evaluation of many design candidates in a short period of time plays a critical role. Computationally efficient methods for structural analysis that accurately predict stiffness matrix entries for beam models, including the crucial bend-twist coupling terms, are essential.
The present article provides a comprehensive review of available cross-sectional approaches and their capabilities in fulfilling the requirements for the composite design of rotor blades. Three cross-sectional theories are selected and implemented to compare the prediction quality of the cross-sectional coupling stiffness terms and the stress distribution across different multi-cell test cross-sections.
The cross-sectional results are compared with the industry-standard 2D finite element code BECAS, which serves as a reference solution. The analytical solution performing best shows very small deviations in the stiffness matrix entries compared to BECAS (below 1% in the majority of test cases). It also achieved a better resolution of the stress distribution and a computation time that is more than an order of magnitude smaller using the same spatial discretization.
These findings demonstrate the analytical solution as a feasible approach for the beam-based pre-design of wind turbine rotor blades, enabling efficient evaluation of a large number of design candidates within a limited time frame.
Beam-Based Approaches in Wind Turbine Blade Design
Beam-based approaches are commonly used in the conceptual and preliminary structural design of wind turbine blades. They are often embedded in a multi-disciplinary optimization (MDO) process due to their superior computational performance compared to high-fidelity finite element (FE) models using shell and/or solid elements.
A typical application of MDO is the design of rotor blades with tailored bend-twist coupling. The blade flexibility affects the angle of attack along the blade, thereby changing the lift and drag force distribution and reducing the flapwise bending moments.
For structural optimization in general, a common objective function is the reduction of mass or costs. For larger blades, mass and costs increase disproportionately with the blade radius, whereas the annual energy production (AEP) increases proportionally to the square of the blade radius. Hence, it is crucial to investigate new technologies, materials, or designs to withstand the increased mass-related loads and limit the blade costs, which are a significant part of the overall turbine costs.
The usage of beam models becomes necessary within the structural optimization in the preliminary design phase due to the evaluation of many design candidates. The number of design candidates results from the investigation of different designs for the structural topology (e.g., number and/or positions of spars) and concepts for materials used and how they are combined in laminate layups, which in turn have to be linked to a manufacturing concept.
Consequently, a basic requirement is a significant reduction of the computation time for model creation and the calculation of 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.
Requirements for Cross-Sectional Calculation Approaches
For the shell or solid FE model case, variations of the internal structure of the blade, e.g., the spar position, often require a 3D CAD (computer-aided design) model update and the subsequent translation into a new FE mesh. The higher modeling effort and the longer computation times with 3D models are not acceptable in the preliminary design phase.
FE beam models require the input of accurate cross-sectional properties, i.e., stiffness and mass matrices. In many design processes, the cross-sectional properties are determined using 2D FE models that serve to calculate the mass and stiffness properties and the stress distribution within the cross-section. These 2D FE approaches suffer from the need of expensive model updating, with re-meshing if the internal structure or layup changes during the optimization process, and from higher computation costs for solving the governing equations compared to analytical approaches.
The requirements for an analytical cross-sectional calculation module are as follows:
- Composite blades are modeled as beams with closed, different single- or multi-cell cross-sections that vary along the beam axis.
- The parts of the blade, e.g., shell panels and spars, consist of different materials, and different materials within one part can occur.
- The structure of the blade is mostly thin-walled, except near the blade root, and undergoes in-plane and out-of-plane cross-sectional deformations.
- Beside the classical loading of thin-walled beams such as bending or extension, shear forces play an important role and can be design drivers.
- The couplings of the beam’s degrees of freedom that result from the structural topology or the material layup have to be considered for an accurate representation of the blade.
- The computational efficiency, i.e., fast output with high accuracy, is of high importance to allow the assessment of a large number of design candidates in the preliminary design phase.
Comparison of Cross-Sectional Approaches
The present article provides a comprehensive review of available cross-sectional approaches based on the aforementioned requirements for the design of composite wind turbine rotor blades. Three cross-sectional theories are selected and implemented:
- Wiedemann Approach: A simple and fast approach that can be used for determining stress distributions, showing good agreement with the results from the reference 2D FE code BECAS.
- Song Approach: A displacement-based formulation that fulfills the requirements with respect to elastic coupling and shear stiffness terms, including transverse shear and restrained warping.
- Jung Approach: A mixed (displacement- and force-based) formulation that is expected to lead to better shear stress distributions compared to the Song approach.
The cross-sectional results of these three approaches are compared with the reference solution from BECAS, which is a well-established industry standard in rotor blade design.
Evaluation of Stiffness Matrix Entries
The comparison of the estimated cross-sectional stiffness matrix shows that the Song approach has high deviations in the shear stiffness terms compared to BECAS, due to the use of the first-order shear deformation theory (FSDT). In contrast, the Jung approach exhibits deviations below 5%, indicating a significant improvement.
The deviations of the main stiffness terms for extension, bending, and torsion are below 1% for the Jung approach, as well as for the Song and Wiedemann approaches, except for the CUS layup test case, where deviations up to 10% occur.
The coupling stiffness terms of the Jung and Song approaches show a good accordance with the BECAS results. The stiffness term for extension-torsion coupling and bend-twist coupling are calculated almost exactly by the Jung approach.
Comparison of Stress Distributions
A qualitative comparison of the shear stress distributions caused by a transverse force reveals that the Song approach wrongly calculates the shear stress distribution due to the FSDT assumption of constant shear strain over the entire cross-section. The shear stress magnitude using Song’s approach is only a third of that using the other approaches.
In contrast, the qualitative stress distributions of the Jung and Wiedemann approaches show a good agreement with the results from BECAS. Some deviations in the absolute values can be observed for the NACA 2412 airfoil cross-section, which require further investigation.
A quantitative comparison of the normal and shear stress distributions along the cross-section contour shows that the median of the deviations for the Jung and Wiedemann approaches is below 1%, with some outliers up to 25% for the NACA 2412 airfoil under transverse load in the y-direction.
Computational Efficiency
The analytical approaches achieve the same accuracy already with a coarse mesh of only four elements in the contour direction, while BECAS requires a fine FE mesh to obtain a converged solution. This results in a significant benefit in terms of computation time.
The analytical approaches are an order of magnitude (partly even more) faster than BECAS for the cross-sectional calculation. For a single load case, the difference is around 2 orders of magnitude. This underlines the usability of analytical cross-section approaches within a design process where many design candidates need to be evaluated and a high number of design iterations occur.
Conclusion and Outlook
The present article provides a comprehensive evaluation of different analytical cross-sectional approaches based on requirements derived for the preliminary design of wind turbine blades. The analytical solution of the Jung approach shows the best results in terms of accuracy of stiffness terms (including coupling and transverse shear) and stress distributions, while achieving a significantly higher computational efficiency compared to the reference 2D FE code BECAS.
These findings demonstrate the Jung approach as a feasible solution for the cross-sectional calculation module of a beam-based design tool, enabling the efficient evaluation of a large number of design candidates within the preliminary design phase.
Future work will focus on the comparison of the cross-sectional properties on the level of a complete beam model, including the effect of geometric nonlinearity on the blade dynamics. Additionally, the influence of warping on the cross-sectional stiffness terms requires further investigation.
The code implementing the analytical cross-sectional approaches is open-source and available under the MIT License, allowing the community to reproduce the results of this article and potentially contribute to its further development.
