Outline of presentation

1. Outline of presentation • Motivation for the work • Computational model • Non-dimensional static stiffness of the bucket foundation • Example results of the coupled BE/FE model • Non-dimensional dynamic stiffness of the bucket foundation • Conclusions Impedance of Bucket Foundations: Torsional, Horizontal and Rocking MotionLars Andersen†, Lars Bo Ibsen† & Morten Albjerg Liingaard‡† Department of Civil Engineering, Aalborg University, Aalborg, Denmark‡ DONG Energy A/S, Fredericia, Denmark

2. Motivation for the work Computational model Non-dimensional static stiffness of the bucket foundation Example results of the coupled BE/FE model Non-dimensional dynamic stiffness of the bucket foundation Conclusions Motivation for the work • A computationally efficient model of a foundation is required for the analysis of wind turbine response

3. Motivation for the work Computational model Non-dimensional static stiffness of the bucket foundation Example results of the coupled BE/FE model Non-dimensional dynamic stiffness of the bucket foundation Conclusions Motivation for the work Vacuum provides the necessary penetration force Bucket foundations are installed from a barge Can be used on shallow and medium water depth Prototype installed at Frederikshavn, Denmark A 3 MW wind turbine is installed on top

4. Motivation for the work Computational model Non-dimensional static stiffness of the bucket foundation Example results of the coupled BE/FE model Non-dimensional dynamic stiffness of the bucket foundation Conclusions Computational model Static and dynamic stiffness of a rigid footing – non-dimensional quantities

5. Motivation for the work Computational model Non-dimensional static stiffness of the bucket foundation Example results of the coupled BE/FE model Non-dimensional dynamic stiffness of the bucket foundation Conclusions Computational model Boundary elements for the subsoil Finite elements for the bucket foundation Boundary elements for the soil inside the bucket

6. Motivation for the work Computational model Non-dimensional static stiffness of the bucket foundation Example results of the coupled BE/FE model Non-dimensional dynamic stiffness of the bucket foundation Conclusions Computational model • Formulation of a Finite-Element domain • Formulation of a Boundary-Element domain • Coupling in terms of nodal forces: • Finite element assembly as usual • Boundary elements are convertedinto macro finite elements

7. Motivation for the work Computational model Non-dimensional static stiffness of the bucket foundation Example results of the coupled BE/FE model Non-dimensional dynamic stiffness of the bucket foundation Conclusions Non-dimensional static stiffness of the bucket foundation • Coupled Boundary-Element/Finite-Element model is based on time-harmonic response • True static solution cannot be established with the present code • The static response is approximated by the solution at a0 = 0.01 • Comparison with Finite-Element solution from ABAQUS

8. Motivation for the work Computational model Non-dimensional static stiffness of the bucket foundation Example results of the coupled BE/FE model Non-dimensional dynamic stiffness of the bucket foundation Conclusions Non-dimensional static stiffness of the bucket foundation

9. Motivation for the work Computational model Non-dimensional static stiffness of the bucket foundation Example results of the coupled BE/FE model Non-dimensional dynamic stiffness of the bucket foundation Conclusions Example results of the coupled Boundary Element/Finite Element model • Torsional motion at the frequency 2 Hz • Note: Poisson’s ratio = 0.4 ; H/D = 1 Torsion Horizontal Rocking

10. Motivation for the work Computational model Non-dimensional static stiffness of the bucket foundation Example results of the coupled BE/FE model Non-dimensional dynamic stiffness of the bucket foundation Conclusions Example results of the coupled Boundary Element/Finite Element model • Horizontal motion at the frequency 2 Hz • Note: Poisson’s ratio = 0.4 ; H/D = 1 Torsion Horizontal Rocking

11. Motivation for the work Computational model Non-dimensional static stiffness of the bucket foundation Example results of the coupled BE/FE model Non-dimensional dynamic stiffness of the bucket foundation Conclusions Example results of the coupled Boundary Element/Finite Element model • Rocking motion at the frequency 2 Hz • Note: Poisson’s ratio = 0.4 ; H/D = 1 Torsion Horizontal Rocking

12. Motivation for the work Computational model Non-dimensional static stiffness of the bucket foundation Example results of the coupled BE/FE model Non-dimensional dynamic stiffness of the bucket foundation Conclusions Non-dimensional dynamic stiffness of the bucket foundation • Torsional stiffness not influencedby Poisson’s ratio (only S-waves) • Like surface footing atlow frequencies • Smooth variation for H/D = 1/4 • Very spiky for H/D > 1 • Like infinite cylinder athigh frequencies

13. Motivation for the work Computational model Non-dimensional static stiffness of the bucket foundation Example results of the coupled BE/FE model Non-dimensional dynamic stiffness of the bucket foundation Conclusions Non-dimensional dynamic stiffness of the bucket foundation • Horizontal stiffness increaseswith increasing frequency • Increase is stronger than fora surface footing for allvalues of Poisson’s ratio • No obvious trend in changedue to changing Poisson’s ratio • BE/FE model of surface footingis very accurate compared tothe reference solution byVeletsos and Wei (1971) • Similar results for rocking

14. Motivation for the work Computational model Non-dimensional static stiffness of the bucket foundation Example results of the coupled BE/FE model Non-dimensional dynamic stiffness of the bucket foundation Conclusions Non-dimensional dynamic stiffness of the bucket foundation • Coupling stiffness looks similarto horizontal stiffness • BE/FE model fails to producereliable results • Alternative model based onhalf-space Green’s function • Very accurate compared tothe reference solution byVeletsos and Wei (1971)

15. Motivation for the work Computational model Non-dimensional static stiffness of the bucket foundation Example results of the coupled BE/FE model Non-dimensional dynamic stiffness of the bucket foundation Conclusions Non-dimensional dynamic stiffness of the bucket foundation • Rocking term increases withan increase in H/D from H/D = 1/4 to H/D = 1 • No additional increase fromH/D = 1 to H/D = 2 • Goes toward the solutionfor an infinite cylinder

16. Motivation for the work Computational model Non-dimensional static stiffness of the bucket foundation Example results of the coupled BE/FE model Non-dimensional dynamic stiffness of the bucket foundation Conclusions Non-dimensional dynamic stiffness of the bucket foundation • Rocking term increases withan increase in H/D from H/D = 1/4 to H/D = 1 • No additional increase fromH/D = 1 to H/D = 2 • Tips and dips coincide withthose of the sliding stiffness

17. Motivation for the work Computational model Non-dimensional static stiffness of the bucket foundation Example results of the coupled BE/FE model Non-dimensional dynamic stiffness of the bucket foundation Conclusions Non-dimensional dynamic stiffness of the bucket foundation • Coupling term varies with H/D • No clear pattern in locationof tips and dips • Very different from surfacefooting (alternative method)

18. Motivation for the work Computational model Non-dimensional static stiffness of the bucket foundation Example results of the coupled BE/FE model Non-dimensional dynamic stiffness of the bucket foundation Conclusions Conclusions • Torsional stiffness: • Independent of Poisson’s ratio and increasing almost linearly with increasing skirt length • Significant peaks observed at a0 ≈ 4, 7 and 10 as also observed for an infinite cylinder • Static stiffness of BE/FE model within approximately 5% of the referece solution • Horizontal sliding and rocking stiffnesses: • Horizontal sliding stiffness does not increase very much with increasing skirt length • Rocking stiffness increases more than linearly with increasing skirt length • Slight increase with increasing Poisson’s ratio due to P-wave generation • Tips and dips in the dynamic stiffness are not repeated with Δa0 = π • Significant coupling exists between horizontal sliding and rocking • Static stiffness of BE/FE model within approximately 7% of the referece solution • Coupling term is less accurate (the BE/FE model fails for a surface footing), andthe coupling terms KHM and KMH do not match exactly (11% difference) • High-frequency limit of non-dimensional dynamic stiffnesses presented in the paper • Next step is to calibrate and implement lumped-parameter models in aeroelastic codes

19. Thank you for your attention Lars Andersen: la@civil.aau.dk