# Simplified Design Method of Laterally Loaded Rigid Monopiles in Cohesionless Soil

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

_{em}to outer diameter D) are usually greater than 20. With the development of the energy industry, especially offshore wind turbines, large-diameter rigid monopiles are becoming widely used to resist the lateral load and moment transferred from offshore wind turbines [3,4,5]. The monopile foundation, consisting of an open-ended steel pipe with an outer diameter D generally ranging from 3.5 m to 8 m, is often driven into the seabed with an embedded length L

_{em}of (5~10)D. Compared with the long slender piles widely used in the offshore oil/gas sector, the large-diameter monopile behaves similar to a rigid pile and tends to move around a rotation point under lateral loading [6,7,8].

_{em}/D) greater than 20, the applicability and reliability issues of this method were reviewed by many researchers when applied to the design of larger diameter rigid monopiles with slenderness ratios generally smaller than 10 (e.g., Abdel-Rahman and Achmus [13], Hu et al. [14], Wang et al. [15]).

## 2. Proposed Design

#### 2.1. Depth of Rotation Point of Rigid Monopile

_{em}under a lateral load applied at a height of L

_{up}above the ground surface level. In general, if the relative stiffness between the subsurface soil and monopile is small enough, the monopile under lateral loading undergoes pure rotation as a rigid body around a point located at a depth of Zr below the ground surface [2]. In order to investigate the location of the rotation point, a series of loading tests on rigid monopiles have been collected. These test piles, having a wide range of pile dimensions, were installed in loose to dense sand and loaded monotonically. The details of these pile tests are summarized in Table 1.

_{b}are presented in Figure 2. The normalized load magnitude Q

_{b}is defined as Q

_{b}= F/F

_{u}, in which F is the applied lateral load and F

_{u}is the ultimate load capacity of the laterally loaded monopile. For test piles for which the ultimate load capacities were not specified in the literature, the ultimate load capacity is taken as the load corresponding to a pile-head displacement of 0.1D [27,28]. As shown in Figure 2, with some discrete in general, the normalized rotation depth Z

_{r}/L

_{em}is mainly located in the range of 0.7~0.81, and the depth of the rotation point is approximately constant, independent of the test pile’s dimensions, soil conditions, load eccentricity, and applied load magnitude. This agrees with the observations from the numerical modeling [28]. Based on the analysis above, the proposed design method in this paper assumes the depth of the rotation point Z

_{r}is constant and equal to 0.75L

_{em}, which is the same as the assumption proposed by Wang et al. [29]. From an engineering point of view, the error caused by the assumption of a fixed rotation depth is within the acceptable tolerances, with an inaccuracy less than 10%.

#### 2.2. Mobilization Coefficient of Soil Lateral Reaction

- (1)
- As the depth increases, the magnitude of the lateral soil reaction around a monopile generally increases to a maximum value and then decreases to zero at the depth of the rotation point, and following that, at the rear side of the monopile, it gradually increases from this rotation point to a maximum value at the pile tip (e.g., Prasad and Chari, [30]; Zhang et al. [17]; Li et al. [18]; Wang et al. [15]). The maximum soil lateral reaction in the front side is located at a depth of Z
_{m}. - (2)
- The maximum soil lateral pressure in the front side of the monopile is p
_{m}, which may be calculated using Rankine’s passive earth pressure theory (K_{p}γ′Z_{m}) and the mobilization coefficient of ultimate soil resistance η, where γ′ is the effective unit weight of soil, and K_{p}is the coefficient of Rankine’s passive earth pressure. The mobilization coefficient η is introduced to quantify the amount of soil pressure/reaction mobilized under a certain loading magnitude. At a given depth of the monopile, to account for the non-uniformity distribution of soil lateral pressure across the diameter of a circular monopile, a reduction factor of 0.8 is usually introduced (e.g., Zhang et al. [17]; Prasad and Chari [30]). - (3)
- According to the equilibrium of lateral force and moment on the monopile, the depth of the maximum soil pressure Z
_{m}in front of the monopile can be determined using Equation (1), while the correlation between the applied lateral load and the mobilization coefficient η is given by Equation (2). The derivation process of Equations (1) and (2) can be referred to Appendix A. Equation (1) demonstrates that the depth of the maximum soil pressure Z_{m}is only related to the pile embedded depth L_{em}and load eccentricity L_{up}, and it is independent of the magnitude of the applied lateral load F, which is in line with the findings by other researchers (e.g., Georgiadis et al. [21]; Zhu et al. [27]; Prasad and Chari [30]).

#### 2.3. Correlation between Pile Head Rotation and Mobilization Coefficient

_{i}, the mobilization coefficient η

_{i}is calculated using Equations (1) and (2). Secondly, the pile head rotation θ

_{i}or lateral displacement y

_{i}corresponding to this applied load F

_{i}can be read from the measured pile head response. If only y

_{i}is given, the pile head rotation θ

_{i}can be calculated using Equation (3), as shown in Figure 3. It should be noted that Equation (3) is based on assumptions that the test pile is 100% rigid and the depth of the rotation point Z

_{r}is a constant value of 0.75L

_{em}. Then, the mobilization coefficient η

_{i}and pile head rotation θ

_{i}under the applied lateral load F

_{i}are derived.

- (1)
- As expected, the value of mobilization coefficient η increases with pile head rotation θ in a nonlinear pattern.
- (2)
- The relationship between η and θ depends on the critical friction angle of soil ϕ
_{c}and the relative density D_{r}, i.e., piles in similar ground conditions generate nearly identical η–θ correlations. - (3)
- A power function, as shown in Equation (4), is capable of modeling the relationship between η and θ, where m and n are the model parameters.

_{c}and relative density D

_{r}, the power function for each pile case is presented, as shown in Figure 5. To establish a design formula or chart for the determination of model parameters, the normalized model parameters m′ and n are plotted with sand critical friction angle ϕ

_{c}and relative density D

_{r}, respectively, as shown in Figure 6. The normalized model parameter m′ is defined as m′ = m/D

_{r}, in which m is the model parameter illustrated in Equation (4) and D

_{r}is the sand relative density.

_{c}, which indicates that the higher the critical friction angle, the stiffer the load-deformation response will be. As the relative density D

_{r}is incorporated into the normalized model parameter m′, a higher relative density will induce a larger model parameter m, which agrees the observations in practical pile tests [27]. As shown in Figure 6b, the model parameter n is mainly located in the range of 0.4~0.5 and is irrelevant to the sand relative density D

_{r}. Based on the cases analyzed, an average value of n = 0.45 is employed in this study.

#### 2.4. General Design Procedures

- Set a specific value of pile head rotation θ
_{i}; - According to the ground conditions, including the sand critical friction angle ϕ
_{c}and relative density D_{r}, the value of mobilization coefficient η_{i}can be calculated using Equation (4), where m = (0.26 ϕ_{c}− 4.8)D_{r}; n = 0.45. - Calculate the corresponding pile head load F
_{i}using Equation (5), as well as the pile head displacement using Equation (6). - Repeating steps 1 to 3, the general pile head response of a monopile can be estimated.

## 3. Validation

#### 3.1. Database

#### 3.2. Pile Head Response

^{3}, corresponding to a relative density of 85%. Studies performed by Dickin and King [38], Laman et al. [39] and Dickin and Laman [40] on Erith sand show that:

- (4)
- Erith sand consists of pure quartz grains with subrounded shape with critical friction angle ϕ
_{c}of 35° [40]. - (5)
- The peak friction angle ϕ
_{p}of Erith sand can be determined according to Equation (7).

_{3}is the confining pressure in kPa.

_{p}of the sand in the centrifuge tests should be first determined. For simplicity, the peak friction angle is assumed to be constant within the depth of the pile, and the representative depth is 0.5L

_{em}. For sand at a depth of 0.5L

_{em}, the average stress σ

_{ave}= 9.8 kPa (the coefficient of lateral earth pressure is assumed as 0.4). Thus, based on Equation (7), the peak friction angle ϕ

_{p}can be determined as 51°, which is used to determine the coefficient of Rankine’s passive earth pressure (K

_{p}).

_{c}and relative density D

_{r}of Erith sand, the correlation between η and θ adopted in this study is η = 3.7θ

^{0.45}. As shown in Figure 8, the proposed design method agrees well with the measured pile head response, which demonstrates the validity of the proposed design method. In addition, the prediction given by Zhang [6] is also shown in Figure 8, and the proposed method in this paper agrees well with the lower boundary of the measured results and those obtained by Zhang’s method. The proposed method is slightly better than Zhang’s [6] when the pile rotation angle is larger than 2º. Comparing the computing efficiency, this study’s proposed design is more convenient and does not need computer-based modeling and analysis.

_{p}/F

_{m}, and the abscissa is the outer diameter of each test pile. F

_{p}is the predicted load calculated by the proposed method, and F

_{m}is the measured load in the collected case history. It can be seen from Figure 9 that the recommended design procedures generally produce relatively good predictions for these test piles. The load ratio between measured and predicted is mainly within 0.8~1.2. The η–θ relationship adopted in this study is based on the critical friction angle ϕ

_{c}and relative density D

_{r}of the soil conditions, and the specific η–θ correlation for each test pile is listed in Table 3. In addition, as shown in Figure 9, F

_{p}tends to be larger than F

_{m}in general, and the reason may be due to the fact that the bending of the pile is ignored in the proposed method. When pile bending is considered, it will result in greater pile head deformation with the same pile rotation angle. In other words, ignoring the bending of the monopile will overestimate the pile rotation angle under the same pile head deformation, which will lead to overestimation of the soil resistance around the pile, making the predicted value higher than the measured value.

#### 3.3. Soil Lateral Reaction Profile

## 4. Conclusions

_{em}, which has been validated by a series of field or laboratory tests results. In this method, mobilization coefficient η is introduced to quantify the magnitude of soil lateral reaction mobilized under a certain load. The correlation between coefficient η and pile head rotation θ is derived by back-analyzing measured results from 13 test piles reported in the published literature. Furthermore, it was found that the parameters in Equation (4) are related to the critical friction angle ϕ

_{c}and relative density D

_{r}of cohesionless soils. The normalized model parameter m′ generally increases linearly with the sand critical friction angle ϕ

_{c}, while the model parameter n is mainly located in the range of 0.4~0.5 and is irrelevant to the sand relative density D

_{r}, based on the cases analyzed.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Notation

D | outer diameter of pile |

D_{r} | relative density of sand |

E_{p} | elastic modulus of pile |

F | lateral force acted on pile head |

F_{u} | pile ultimate load capacity |

h | height of displacement measured |

K_{p} | coefficient of Rankine’s passive earth pressure |

L_{em} | embedded length of pile |

L_{up} | loading eccentricity of pile |

M_{0} | moment acted on pile head |

m, n | parameters of correlation between η and θ |

m′ | normalized model parameter |

n_{h} | constant of horizontal subgrade reaction |

P | lateral soil reaction |

p_{m} | maximum soil pressure in the front side of monopile |

y | lateral displacement of rigid pile |

Z_{m} | depth of maximum lateral soil reaction |

Z_{r} | depth of rotation point |

ϕ_{c} | critical friction angle of sand |

ϕ_{p} | peak friction angle of sand |

γ′ | effective density of sand |

θ | rotation of pile |

η | mobilization coefficient of ultimate soil resistance |

Q_{b} | normalized load magnitude |

## Appendix A. Derivation of Mobilization Coefficient η

**Figure A1.**Schematic force diagrams of rigid monopile: (

**a**) horizontal force equilibrium; (

**b**) moment equilibrium.

_{m}and the coefficient of earth pressure η can be obtained:

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**Figure 2.**Normalized rotation depth of laterally loaded monopiles: (

**a**) PR1~PR12; (

**b**) PR13~PR18 (modified from Zhu et al. [27]).

**Figure 5.**Correlation between pile rotation and soil resistance mobilization coefficient for each case.

**Figure 6.**Variation patterns of model parameters with ground conditions: (

**a**) normalized model parameter m′; (

**b**) model parameter n.

**Figure 8.**Comparison of the moment-rotation results [6].

Pile No. | Field/ Lab | Soil Condition | Prototype Pile Dimensions | Pile Type | Reference | ||
---|---|---|---|---|---|---|---|

D (m) | L_{em}/D | L_{up}/D | |||||

PR1 | Field | Dense sand | 0.34 | 6.5 | 1.18 | Steel Pipe Pile | Li et al. [3] |

PR2 | Field | Medium-dense | 0.245 | 6.1 | 1.63 | Steel Pipe Pile | Murphy et al. [20] |

PR3 | Field | Medium-dense | 0.245 | 6.1 | 1.63 | Steel Winged Pipe Pile | Murphy et al. [20] |

PR4 | Lab | Medium-dense to dense | 1.092 | 8.3 | 1.14 | Steel Pipe Pile | Georgiadis et al. [21] |

PR5 | Lab | Dense sand | 3 | 6.0 | 15 | Steel Pipe Pile | Klinkvort & Hededal [22] |

PR6 | Lab | Loose sand | 0.168 | 7.1 | 1.16 | Steel Pipe Pile | Naggar & Wei [23] |

PR7 | Lab | Dense sand | 0.032 | 15.6 | 3.59 | Steel Pipe Pile | Qin & Guo [24] |

PR8 | Lab | Dense sand | 0.032 | 12.5 | 3.59 | Steel Pipe Pile | Qin & Guo [24] |

PR9 | Lab | Dense sand | 6 | 5.2 | 5.5 | Steel Pipe Pile | Choo & Kim [25] |

PR10~ PR12 | Lab | Dense sand | 0.048 | 8.3 | 1.04 | Steel Pipe Pile | Mu et al. [26] |

PR13~ PR18 | Lab | Dense sand | 0.165 | 5.5 | 6 | Steel Pipe Pile | Zhu et al. [27] |

Pile No | Pile Dimensions in Prototype | Soil Properties | Test Description | Height of Displacement Measured (m) | Reference | |||||
---|---|---|---|---|---|---|---|---|---|---|

D(m) | L_{em}/D | L_{up}/D | γ′ (kN/m ^{3}) | ϕ_{p}(°) | ϕ_{c}(°) | D_{r}(%) | ||||

P1 | 0.075 | 13.3 | 1 | 15.25 | 46 | 32 ^{a} | 82 | Angular dry sand 1 g model test | 0 | Chari and Meyerhof [31] |

P2 | 1 | 6 | 15 | 16.7 | 43.2 | 30 | 93 | Dry/Saturated Fontainebleau sand, centrifuge test | 1.375 | Klinkvort and Hededal. [22] |

P3 | 3 | 8.25 | 10.3 | 42.5 | 88 | 4.125 | ||||

P4 | 3 | 10.5 | 10.4 | 43.4 | 93 | 4.125 | ||||

P5 | 3 | 12.75 | 10.2 | 42.3 | 87 | 4.125 | ||||

P6 | 0.102 | 6 | 1.5 | 18.3 | 43 | 33.3 ^{b} | 75 | Well-graded angular dry sand 1 g model test | 0 | Prasad and Chari [30] |

P7 | 0.165 | 5.55 | 6 | 9.1 | 41.5 | 35.5 | 88 | Saturated Hangzhou silt sand, 1 g model test | 0.99 | Zhu et al. [27] |

P8 | 0.495 | |||||||||

P9 | 0.165 | |||||||||

P10 | 8.8 | 37.4 | 70 | 0.99 | ||||||

P11 | 0.495 | |||||||||

P12 | 0.165 | |||||||||

P13 | 0.34 | 6.5 | 1.18 | 20 | 54 | 37 | 100 | Heavily over-consolidated Blessington sand | 0 | Li et al. [32] |

^{a}: defined by the critical friction angle of silica sand [33].

^{b}: defined by the loose condition of the test sand.

Pile No. | Prototype Pile Dimensions | Soil Properties | Test Description | Measured Curves ^{b} | η~θ | Reference | |||||
---|---|---|---|---|---|---|---|---|---|---|---|

D(m) | L_{em}/D | L_{up}/D | γ′ (kN/m ^{3}) | ϕ_{p}(°) | ϕ_{c}(°) | D_{r}(%) | |||||

1 | 0.073 | 10 | 2.33 | 15.1 | 41.2 | 32 ^{a} | 77 | 1 g model tests | F-y | η = 2.7θ^{0.45} | Joo [34] |

2 | 0.09 | 8.9 | 2.78 | ||||||||

3 | 0.102 | 8.8 | 2.75 | ||||||||

4 | 1.224 | 7.4 | 1 | 16.3 | 36 | 30 | 60 | centrifuge tests | F-y | η = 1.8θ^{0.45} | Georgiadis et al. [21] |

5 | 0.076 | 9 | 0.92 | 16.42 | 41.4 | 32.9 | medium dense | 1 g model tests | F-y | η = 1.8θ^{0.45} | Agaiby et al. [35] |

6 | 8.6 | 41.4 | |||||||||

7 | 6 | 41.7 | |||||||||

8 | 3 | 42.3 | |||||||||

9 | 0.152 | 3 | 0.53 | 41.7 | |||||||

10 | 6 | 41.2 | |||||||||

11 | 9 | 40.9 | |||||||||

12 | 0.076 | 3 | 3 | 42.3 | |||||||

13 | 9 | ||||||||||

14 | 15 | ||||||||||

15 * | 1 | 2 | 6 | 16.4 | 51 | 35 | 85 | centrifuge tests | M-θ | η = 3.7θ^{0.45} | Nazir [36] |

16 | 1 | 6 | 2.5 | 16.2 | 43 | 30 | 80 | centrifuge tests | F-y | η = 2.4θ^{0.45} | Leth [37] |

17 | 8 | 42.5 | |||||||||

18 | 10 | 42 | |||||||||

19 | 2 | 6 | 1.43 | 41.6 | |||||||

20 | 8 | 40.9 | |||||||||

21 | 10 | 40.4 | |||||||||

22 | 3 | 6 | 1 | 40.5 | |||||||

23 | 3 | 8 | 39.9 |

^{a}: values are determined according to silica sand [33].

^{b}: F-y: load-displacement curve of piles; M-θ: moment-rotation curve at ground line. * including 3 centrifuge tests.

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**MDPI and ACS Style**

Luo, R.; Wang, A.; Li, J.; Ding, W.; Zhu, B.
Simplified Design Method of Laterally Loaded Rigid Monopiles in Cohesionless Soil. *J. Mar. Sci. Eng.* **2024**, *12*, 208.
https://doi.org/10.3390/jmse12020208

**AMA Style**

Luo R, Wang A, Li J, Ding W, Zhu B.
Simplified Design Method of Laterally Loaded Rigid Monopiles in Cohesionless Soil. *Journal of Marine Science and Engineering*. 2024; 12(2):208.
https://doi.org/10.3390/jmse12020208

**Chicago/Turabian Style**

Luo, Ruping, Anhui Wang, Jie Li, Wenyun Ding, and Bitang Zhu.
2024. "Simplified Design Method of Laterally Loaded Rigid Monopiles in Cohesionless Soil" *Journal of Marine Science and Engineering* 12, no. 2: 208.
https://doi.org/10.3390/jmse12020208