# Dynamic Impedances of Offshore Rock-Socketed Monopiles

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## Abstract

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## 1. Introduction

- (1)
- To compare the dynamic impedances of the monopile under different soil depths, rock weathering conditions, and exciting frequencies;
- (2)
- To analyze the deformation of monopile under different loading conditions;
- (3)
- To find out the distributions of von-Mises stresses in rock-socketed monopile, and special attention is paid on the stresses in monopile near the interface of rock and soil.

## 2. The Finite Element Model Created by ABAQUS

#### 2.1. Introduction to the Model

_{s}, and Es mean Poisson’s ratio, density, and elastic modulus of the soil, respectively. Data related to the displacement (u(t)) and the rotation angle (φ(t)) of the monopile are extracted from the post-processing module, whose amplitude and phase lag can be obtained by data fitting. The horizontal dynamic impedance (Κ

_{H}), the coupled dynamic impedance (Κ

_{MH}), and the rotational dynamic impedance (Κ

_{M}) of monopile can then be calculated correspondingly. Taking Κ

_{H}as an example:

_{H}means the horizontal stiffness, while c

_{H}means the horizontal radiative damping. The horizontal dimensionless dynamic impedance (Κ

_{h}) can be obtained by non-dimensionalization:

_{mh}) and the dimensionless rotational dynamic impedance (Κ

_{m}) can be written as:

#### 2.2. Comparision with the Existing Solutions

_{p}), soil (E

_{s}), and rock (E

_{r}) are set to be 210 GPa, 30 MPa, and 40 GPa, respectively, with the embedded length of the monopile in rock (h

_{0}) ranging from 1D to 3D (D means the diameter of the pile). Besides, in order to study the influences of rocks with different weathering conditions (refer to case 2 in Table 1), the elastic modulus of the upper soil is set to be 10 MPa, with the elastic modulus of rock being 0.5 GPa, 5 GPa, and 60 GPa, i.e., the ratios of E

_{s}to E

_{r}are 1:50, 1:500, and 1:6000, respectively.

## 3. Numerical Results

#### 3.1. Dynamic Impedance

#### 3.1.1. Effects of Rock-Socketed Depth on Dynamic Impedance of Monopile

_{s}= 30 MPa, are used to calculate Κ

_{h}, Κ

_{mh}, and Κ

_{m}, as shown in Figure 5.

_{0}always remains unchanged (equal to 5D, refer to Figure 2), which means the thickness of the upper soil layer increases while the rock-socketed depth decreases.

_{h}), $\mathrm{Re}\left({K}_{\mathrm{m}h}\right)$, and $\mathrm{Re}\left({K}_{m}\right)$ increase by about 35.5%, 19.6%, and 5.6%, respectively; when the rock-socketed depth increases from 1D to 3D, the above physical quantities increase by about 148.6%, 97.3%, and 31.1%, respectively. It can be seen that the deeper the monopile is embedded in the rock, the more obvious the effect of the rock on increasing the stiffness will be.

#### 3.1.2. Influence of Elastic Modulus Ratio of Rock to Soil

_{h}), $\mathrm{Re}\left({K}_{\mathrm{m}h}\right)$, and $\mathrm{Re}\left({K}_{m}\right)$ decrease by about 11.9%, 7.7%, and 3.2%, respectively; when it decreases to 1/120, Re(K

_{h}), $\mathrm{Re}\left({K}_{\mathrm{m}h}\right)$, and $\mathrm{Re}\left({K}_{m}\right)$ decrease by about 31.3%, 19.6%, and 5.6%, respectively, among which the horizontal dynamic stiffness decreases the most.

#### 3.2. Analysis of Pile Deformation Under Simple Harmonic Horizontal Forces

#### 3.2.1. Effect of Dimensionless Frequency on Monopile Deformation

#### 3.2.2. Effect of Rock-Socketed Depth on Pile Deformation

_{0}increases from 1D to 2D, the deflection of pile top decreases about 9.6%, and 46.6% when it increases from 1D to 3D. The pile deflection gradually decreases to almost 0 near the interface of rock and soil. While the slight reverse deformation exists, when h

_{0}is 1D and 2D in the rock layer, it disappears when the pile is embedded deeper in the rock.

#### 3.2.3. Effect of Elastic Modulus Ratio Between Soil and Rock on Monopile Deformation

#### 3.3. Analysis of the Internal Force of Pile under Simple Harmonic Forces

#### 3.3.1. Effect of Dimensionless Frequency on von-Mises Stress of Pile

_{0}= 1D. The results are shown in Figure 13. Stress of the pile first increases and then decreases with the increase of dimensionless frequency, reaching the maximum, when a = 0.2, in this case. At the interface of rock and soil, the stress of monopile changes very sharply: the stress of the pile under the interface is far less than that in the upper soil layer. Stress of the socketed part decreases rapidly, close to zero at the bottom of the pile.

#### 3.3.2. Effect of Rock-Socketed Depth on Von-Mises Stress

#### 3.3.3. Effect of Elastic Modulus Ratio between Soil and Rock on Von-Mises Stress

_{0}= 1D. With the increase of rock weathering degree, the stress of monopile in the upper soil layer part decreases while it increases in the lower rock layer part. The drop of stress near the interface under strong weathering rock condition is gentler than the weak weathering rock condition.

## 4. Conclusions and Outlook

#### 4.1. Basic Conclusions

- (1)
- When rock-socketed depth increases:
- the dynamic stiffness of pile increases, while the sensitivity to dimensionless frequency decreases, indicating that the ability of pile to resist deformation increases under dynamic load, which is consistent with the results obtained from monopile deformation analysis;
- the radiative damping of pile decreases, and the horizontal radiative damping decreases the most. When the contact surface between the pile and the soil becomes smaller, less stress wave energies will be generated and radiated;
- the deformation of monopile reduces and the deformation of the rock-embedded part of the monopile is very small;
- von-Mises stress of the monopile in the soil layer increases, and there is a sudden drop at the soil–rock interface.

- (2)
- When the elastic modulus ratio of soil to rock increases, that is, the weathering degree of rock increases:
- the dynamic stiffness of the monopile reduces, and the closer the elastic modulus of rock is to that of soil, the faster its reduction rate is. When the elastic modulus of the rock is reduced, resulting in the weakened ability of the pile to resist deformation under external force;
- the radiative damping increases, with the rotational radiative damping increasing the most. Compared with rock, it seems that the capability of the soil to radiate stress waves is stronger;
- the deflection of the monopile increases and the point at which the displacement is 0 shifts downward, considering that the effect of rock on fastening the pile is reduced;
- von-Mises stress of monopile in the soil layer decreases while increasing in the rock layer. The phenomenon of stress drop at the soil–rock interface is no longer obvious.

#### 4.2. Outlook on Further Study

- (1)
- The deformation and stress of the soil/rock around monopile under dynamic loadings with different amplitudes can be analyzed, in order to know more about the soil-rock-monopile dynamic contact problem;
- (2)
- The dynamic impedances and responses of rock-socketed monopile under long-term alternating loads remain to be further studied in the future;
- (3)
- More complicated soil and rock models can be further used to study the dynamic responses of rock-socketed monopiles under extreme loading conditions.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 4.**Model Verification: (

**a**) comparison of horizontal dynamic impedance (Κ

_{h}); (

**b**) comparison of coupled dynamic impedance (Κ

_{mh}); (

**c**) comparison of rotational dynamic impedance (Κ

_{m}).

**Figure 5.**Effects of rock-socketed depth on dynamic impedance of monopile: (

**a**) variation of Re(Κ

_{h}) for various h

_{0}; (

**b**) variation of Im(Κ

_{h}) for various h

_{0}; (

**c**) variation of Re(Κ

_{mh}) for various h

_{0}; (

**d**) variation of Im(Κ

_{mh}) for various h

_{0}; (

**e**) variation of Re(Κ

_{m}) for various h

_{0}; (

**f**) variation of Im(Κ

_{m}) for various h

_{0}.

**Figure 6.**Effects of elastic modulus ratio of rock to soil on dynamic impedance of monopile: (

**a**) variation of Re(Κ

_{h}) for various E

_{s}to E

_{r}ratio; (

**b**) variation of Im(Κ

_{h}) for various E

_{s}to E

_{r}ratio; (

**c**) variation of Re(Κ

_{mh}) for various E

_{s}to E

_{r}ratio; (

**d**) variation of Im(Κ

_{mh}) for various E

_{s}to E

_{r}ratio; (

**e**) variation of Re(Κ

_{m}) for various E

_{s}to E

_{r}ratio; (

**f**) variation of Im(Κ

_{m}) for various E

_{s}to E

_{r}ratio.

**Figure 10.**Effect of rock-socketed depth on monopile deformation: (

**a**) deflection of monopile under pure horizontal load; (

**b**) deflection of monopile under pure bending moment.

**Figure 11.**Effect of elastic modulus ratio between soil and rock on pile deformation: (

**a**) deflection of monopile under pure horizontal load; (

**b**) deflection of monopile under pure bending moment.

**Figure 14.**Effect of rock-socketed depth on von-Mises stress: (

**a**) von-Mises stress of monopile under pure horizontal load; (

**b**) von-Mises stress of monopile under pure bending moment.

**Figure 15.**Effect of elastic modulus ratio between soil and rock on von-Mises stress: (

**a**) von-Mises stress of monopile under pure horizontal load; (

**b**) von-Mises stress of monopile under pure bending moment.

Variable | Case 1 h_{0} (m) | 7 | 14 | 21 |

Case 2 E_{r} (GPa) | 0.5 | 5 | 60 | |

Properties | Parts | Steel pile | Soil | Rock |

Density-ρ(kg/m^{3}) | 7900 | 1500 | 3000 | |

Poisson’s Ration-ν | 0.3 | 0.3 | 0.25 | |

Elastic Modulus-E(MPa) | 2.1 × 10^{5} |

_{s}= 30 MPa, E

_{r}= 40 GPa, E

_{p}= 210 GPa, variable: h

_{0}; In case 2, E

_{s}= 10 MPa, E

_{p}= 210 GPa, h

_{0}= 7 m, variable: E

_{r}.

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## Share and Cite

**MDPI and ACS Style**

He, R.; Ji, J.; Zhang, J.; Peng, W.; Sun, Z.; Guo, Z.
Dynamic Impedances of Offshore Rock-Socketed Monopiles. *J. Mar. Sci. Eng.* **2019**, *7*, 134.
https://doi.org/10.3390/jmse7050134

**AMA Style**

He R, Ji J, Zhang J, Peng W, Sun Z, Guo Z.
Dynamic Impedances of Offshore Rock-Socketed Monopiles. *Journal of Marine Science and Engineering*. 2019; 7(5):134.
https://doi.org/10.3390/jmse7050134

**Chicago/Turabian Style**

He, Rui, Ji Ji, Jisheng Zhang, Wei Peng, Zufeng Sun, and Zhen Guo.
2019. "Dynamic Impedances of Offshore Rock-Socketed Monopiles" *Journal of Marine Science and Engineering* 7, no. 5: 134.
https://doi.org/10.3390/jmse7050134