# Model Uncertainties for Soil-Structure Interaction in Offshore Wind Turbine Monopile Foundations

^{1}

^{2}

^{3}

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## Abstract

**:**

## 1. Introduction

## 2. Wave Loads

#### 2.1. Load Model for Vertical Surface Piercing Circular Cylinders

^{3}, and the pile radius employed in the present study is ${R}_{\mathrm{p}}=3$ m. This radius is significantly smaller than the wavelengths resulting from the models considered below, which justifies use of the Morison equation.

#### 2.2. Linear and Nonlinear Wave Kinematics

#### 2.3. Load Scenarios

- Load Case 1: Linear irregular waves with significant wave height ${\mathrm{H}}_{\mathrm{m}0}=10$ m, peak wave period ${\mathrm{T}}_{\mathrm{p}}=13$ s. The duration of the simulated time series is 3 h corresponding to around 1000 waves. Loads are calculated by Morison’s equation without slamming contribution.
- Load Case 2: Same as Load Case 1 but where one wave in the irregular wave train is substituted by a stream function wave with $\mathrm{H}=18.7$ m, $\mathrm{T}=11.45$ s corresponding approximately to the maximum wave in the irregular series (Rayleigh distributed wave heights). This leads to a highly nonlinear wave that is merged into and out of the irregular wave by a cosine taper window. The regular wave is fully active for one wave period and merged in and out in half a period to either side.
- Load Case 3: Same as Load Case 2 but including slamming loads for the stream-function wave.

## 3. Lumped-Parameter Models for the Foundation

#### 3.1. Semi-Analytical Model of Monopile Response in the Frequency Domain

- The mid-surface of the monopile wall is discretized into a total of $n$ points, forming a ring with 18 points at the level of each FE node as illustrated in Figure 1. Each point has three translational degrees of freedom.
- Three modes of rigid-body motion (translations, ${\overline{u}}_{x}$, ${\overline{u}}_{z}$, and rotation, ${\overline{\theta}}_{y}$) are prescribed for each of the $m$ rigid rings, defining the rigid-body motion matrix $\overline{\mathbf{U}}$ of dimensions, $3n\times 3m$.
- The displacements ${G}_{ijkl}\left(\omega \right)$ at receiver point $i$ in direction $j$ due to a unit-magnitude load applied at source point $k$ in direction $l$ are determined for all combinations of source and receiver points and assembled into the Green’s function matrix $\mathbf{G}\left(\omega \right)$. It is noted that this matrix is not symmetrical, but that reciprocity of the Green’s function implies that ${G}_{ijkl}\left(\omega \right)={G}_{jilk}\left(\omega \right)$.
- The load magnitudes of the contact forces associated with the rigid-body modes of the rings are determined as $\overline{\mathbf{P}}\left(\omega \right)={\mathbf{G}}^{-1}\left(\omega \right)\overline{\mathbf{U}}\left(\omega \right)$. Hence, the dynamic stiffness matrix for the soil interacting with the pile becomes ${\mathbf{D}}_{\mathrm{s}}\left(\omega \right)={\left[\overline{\mathbf{U}}\left(\omega \right)\right]}^{T}\overline{\mathbf{P}}\left(\omega \right)={\left[\overline{\mathbf{U}}\left(\omega \right)\right]}^{T}{\mathbf{G}}^{-1}\left(\omega \right)\overline{\mathbf{U}}\left(\omega \right)$.
- The dynamic stiffness matrix of the pile is found as ${\mathbf{D}}_{\mathrm{p}}\left(\omega \right)={\mathbf{K}}_{\mathrm{p}}+\mathrm{i}\omega {\mathbf{C}}_{\mathrm{p}}-{\omega}^{2}{\mathbf{M}}_{\mathrm{p}}$, where ${\mathbf{K}}_{\mathrm{p}}$, ${\mathbf{C}}_{\mathrm{p}}$ and ${\mathbf{M}}_{\mathrm{p}}$ are the static stiffness, viscous damping and consistent mass matrices obtained from the FE model of the pile using two-node beam elements with Hermitian interpolation. Then, ${\mathbf{D}}_{\mathrm{ps}}\left(\omega \right)={\mathbf{D}}_{\mathrm{p}}\left(\omega \right)+{\mathbf{D}}_{\mathrm{s}}\left(\omega \right)$ provides the dynamic stiffness of the pile–soil system.
- A matrix, $\overline{\mathbf{F}}$, is constructed with dimensions, $3n\times 3$. A $3\times 3$ identity matrix is put into the submatrix associated with the three degrees of freedom (d.o.f.) at the pile cap, and the remaining entries are set to zero. Afterwards, the matrix ${\mathbf{Z}}_{\mathrm{ps}}\left(\omega \right)$ is taken as the $3\times 3n$ submatrix of ${\mathbf{D}}_{\mathrm{ps}}^{-1}\left(\omega \right)$ associated with the three d.o.f. of the pile cap. Hence, ${\mathbf{D}}_{\mathrm{cap}}\left(\omega \right)={\left[{\mathbf{Z}}_{\mathrm{ps}}\left(\omega \right)\overline{\mathbf{F}}\right]}^{-1}$ provides a $3\times 3$ dynamic stiffness matrix for the pile cap at the angular frequency $\omega $.

^{3}. The shear factor ${k}_{\mathrm{p}}=0.5$ has been used for the thin-walled circular tubular pile, having a wall thickness of ${t}_{\mathrm{p}}=50$ mm. The soil has the properties: ${E}_{\mathrm{s}}=100$ MPa, ${\nu}_{\mathrm{s}}=0.3$, and ${\rho}_{\mathrm{s}}=2000$ kg/m

^{3}. Further, hysteretic damping with a loss factor of 0.05 has been assumed for the soil.

#### 3.2. Consistent Lumped-Parameter Models of the Monopile Cap Response

#### 3.3. Properties of the Model for the Structure, Foundation and Subsoil

^{3}). The shear factor 0.5 is applied to account for the shear deformation in all parts of the system. The remaining data for the FE model of the structure are given in Table 1, where ${M}_{0}$ and ${J}_{0}$ are the mass and mass moment of inertia related with the distributed masses applied to model the nacelle, the boat landing, the transition piece and the platforms inside the tower. The total height of the structure is 135.0 m above mudline. It is noted that the parameters are largely based on a real, existing OWT to ensure realistic results. However, the data have been masked due to confidentiality.

#### 3.4. Calibration and Test of Lumped-Parameter Model

## 4. Influence of Model Simplifications

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Undamped tower eigenmodes and corresponding eigenfrequencies (blue/yellow shades indicate compression/tension).

**Figure 3.**Fitting of lumped-parameter models for the coupled horizontal translation and rocking rotation of the pile cap (dots indicate target solution; dashed lines indicate high-frequency limit).

**Figure 4.**Time series and Fourier amplitude spectra of external moments and mudline moments for the lumped-parameter model (LPM) of the pile–subsoil system, considering all load cases.

**Figure 5.**Time series and Fourier amplitude spectra of external moments and mudline moments for the model with static pile-cap stiffness, considering all load cases.

Segment | Length (m) | Radius (mm) | Wall Thickness (mm) | ${\mathit{M}}_{0}\text{}\mathbf{at}\text{}\mathbf{Top}\text{}\left(\mathbf{kg}\right)$ | ${\mathit{J}}_{0}\text{}\mathbf{at}\text{}\mathbf{Top}\text{}(\mathbf{kg}\cdot {\mathbf{m}}^{2})$ | Mesh Size (m) |
---|---|---|---|---|---|---|

Tower Seg. 1 | 15.0 | 2000 | 24 | 350,000 | 75,000,000 | 5.0 |

Tower Seg. 2 | 15.0 | 2200 | 24 | $-$ | $-$ | 5.0 |

Tower Seg. 3 | 15.0 | 2400 | 24 | 5000 | 10,000 | 5.0 |

Tower Seg. 4 | 15.0 | 2600 | 24 | $-$ | $-$ | 5.0 |

Tower Seg. 5 | 15.0 | 2800 | 24 | 5000 | 10,000 | 5.0 |

Tower Seg. 6 | 15.0 | 3000 | 24 | $-$ | $-$ | 5.0 |

Supp. Struct. Seg. 1 | 15.0 | 3000 | 80 | 5000 | 10,000 | 1.0 |

Supp. Struct. Seg. 2 | 30.0 | 3000 | 80 | 5000 | 10,000 | 2.0 |

Emb. Pile Seg. 1 | 40.0 | 3000 | 80 | $-$ | $-$ | 1.0 |

Layer | Depth (m) | Shear Modulus (MPa) | Poisson’s Ratio (−) | Mass Density (kg/m^{3}) | Loss Factor (−) | Shear Wave Speed (m/s) |
---|---|---|---|---|---|---|

Layer 1: Soft soil | 20.00 | 20.00 | 0.450 | 2000 | 0.050 | 100.0 |

Layer 2: Hard soil | ∞ | 80.00 | 0.450 | 2000 | 0.030 | 200.0 |

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**MDPI and ACS Style**

Andersen, L.V.; Andersen, T.L.; Manuel, L.
Model Uncertainties for Soil-Structure Interaction in Offshore Wind Turbine Monopile Foundations. *J. Mar. Sci. Eng.* **2018**, *6*, 87.
https://doi.org/10.3390/jmse6030087

**AMA Style**

Andersen LV, Andersen TL, Manuel L.
Model Uncertainties for Soil-Structure Interaction in Offshore Wind Turbine Monopile Foundations. *Journal of Marine Science and Engineering*. 2018; 6(3):87.
https://doi.org/10.3390/jmse6030087

**Chicago/Turabian Style**

Andersen, Lars Vabbersgaard, Thomas Lykke Andersen, and Lance Manuel.
2018. "Model Uncertainties for Soil-Structure Interaction in Offshore Wind Turbine Monopile Foundations" *Journal of Marine Science and Engineering* 6, no. 3: 87.
https://doi.org/10.3390/jmse6030087