General Solution for Laterally Loaded Monopile Foundation under Scour

Abstract

Scour is a common phenomenon for laterally loaded underwater piles. The underwater piles used in different projects have different pile head conditions and pile types (flexible and rigid). On the one hand, an increasing number of rigid piles have begun to be used for underwater piles; on the other hand, flexible piles may be transformed into rigid piles after scour. This study presents a general method for laterally loaded monopile foundations (flexible and rigid), with both a free head and a fixed head, based on the curved strain wedge model under scour. The rationality and accuracy of the presented analytical solution are verified by two different case studies. The influence of scour dimension on the response of piles with different pile head conditions and pile types is studied. The results reveal that pile head condition, pile type and scour dimension all have a great influence on monopile response and should all be carefully considered. Under the same conditions as other parameters, monopile with a fixed pile head is less susceptible to scour dimension than monopile with a free head, and flexible monopile is less susceptible to scour dimension than rigid monopile.

1. Introduction

Scour is a common phenomenon around monopile foundation in water and is caused by currents and waves. Scour includes local scour and general scour. Many reports show that scour has become one of the most critical factors leading to the failure of underwater structures [1,2]. The evaluation of the bearing performance of a monopile foundation after scour has become one of the most concerned problems of engineers. In the API design specification [3], scour dimension is considered to be known, and scour dimension is used to calculate the bearing capacity of a monopile foundation after scour. So, it is not possible to reflect all scour dimension cases, and the calculation results would be conservative or dangerous. Therefore, many scholars have begun to study the analytical or numerical method for the bearing capacity of a monopile foundation after scour. The analytical methods for evaluating the influence of scour on monopile foundation response have transitioned from simple to complex. As early as 1986, Diamamantidis and Arnesen [4] used the first-order reliability method to study the influence of both general scour and local scour on fixed structures, based on the p–y method from API RP 2A. Lin et al. [5] explored the influence of general scour on flexible monopile response with a free head by considering soil stress history based on the p–y method by Matlock [6] for soft clay. Lin et al. [7] performed research on the influence of general scour on flexible monopile response with a free head by considering soil stress history based on the p–y method by Reese et al. [8] for sand. Lin et al. [9,10] revealed the influence of local scour on flexible monopile response with a free head by considering scour dimension based on equivalent wedge failure mode for sand and soft clay in 2014 and 2016, respectively. And the p–y methods of sand and soft clay were also by Reese et al. [8] and Matlock [7], respectively. Zhang et al. [11] and Zhang et al. [12] studied the influence of local scour on flexible monopile response with a free head in soft clay based on the p–y method by Matlock [7] by considering the influence of soil stress history and scour dimension simultaneously. In 2018, Yang et al. [13] analyzed flexible monopile response with a free head under local scour based on the traditional strain wedge model. Then, flexible monopile with a free head under asymmetric scour in both sand and clay was analyzed by Zhang et al. [14], and a simplified calculation solution used in the preliminary design for small diameter flexible monopile foundation response under scour was first proposed.

However, most of the abovementioned analytical methods for pile response under scour are based on the available p–y methods. The available p–y methods are obtained from small diameter flexible piles, which may not be applicable to rigid piles [15]. The flexible monopile may be changed into rigid monopile under the condition of scour because of the shortening of the buried length of the monopile foundation. And more and more rigid monopiles are being used in the development of monopile dimensions [15]. The inaccurate calculation of the load bearing capacity of a monopile foundation after scour will also lead to the occurrence of disasters. Furthermore, many underwater monopile foundations have a fixed head; however, there are few studies focused on the influence of scour on fixed monopile response. Therefore, a general analytical solution for laterally loaded piles with different pile heads and different pile types (flexible or rigid) under the condition of scour is needed.

In this paper, a general method is proposed for both flexible monopile foundation and rigid monopile foundation, with both free and fixed heads, under scour, based on a curved strain wedge model. The measured results from field tests and the calculated results from this analytical method are consistent with each other. The influence of pile head condition and pile type (flexible or rigid) on monopile foundation response is discussed under different scour sizes. The results show that all the parameters (scour dimension, pile head type and pile type) should be carefully considered in the analysis of scour influence on monopile response. Rigid monopiles with a free head are more susceptible to scour dimensions than flexible monopiles with a free head, with all other parameters held equal. Flexible piles with a free head are more susceptible to scour dimensions than flexible piles with a fixed head, with all other parameters held equal.

2. General Solution for Piles under Scour

According to the criterion of rigidity and flexibility of monopile foundations [16], the dimensionless stiffness ratio is used to judge the rigidity and flexibility of pile foundation. When $E_{\mathrm{p}}{I}_{\mathrm{p}}/E{L}^4>0.208$, it is a rigid monopile foundation. And when $E_{\mathrm{p}}{I}_{\mathrm{p}}/E{L}^4<0.0025$, it is a flexible monopile foundation. E_pI_p is the bending rigidity of a monopile foundation. E is the elastic modulus of soil. L is the buried depth of a monopile foundation.

2.1. Review of a Curved Strain Wedge Method

A curved strain wedge method for flexible monopiles under lateral load was presented by Zhang et al. [17], according to deflection differential governing equation.

$$E_{\mathrm{p}}{I}_{\mathrm{p}}\frac{{\mathrm{d}}^{4}y}{\mathrm{d}{z}^4}+p\left(y,z\right)=0$$

(1)

where p(y, z) is the soil reaction when the displacement of the monopile is y at z depth (kN/m).

A more general curved strain wedge method is shown in Figure 1. Both flexible monopile and rigid monopile can be analyzed under scour in this model. This modified curved strain wedge is characterized by soil displacement S_i at the i^th sublayer, and the interaction between the soil and monopile is calculated by force equilibrium [17]. All parameters with subscript i/j represent parameters at the i^th/j^th sublayer in the following analysis.

$$S_i=\frac{\left|y_i\right|}{{\epsilon }_i}$$

(2)

$$p{\left(y,z\right)}_i=p_{pi}+p_{\tau i}$$

(3)

$$p_{pi}=2{\displaystyle {\int }_0^{2{\alpha }_i}{\left(D_{\mathrm{s}}/2\right)}_i{\left(\Delta {\sigma }_{\mathrm{h}}\right)}_i\cos \zeta }d\zeta$$

(4)

$$p_{\tau i}=2{\displaystyle {\int }_0^{\frac{\pi }{2}}\frac{D}{2}\left({\tau }_{\mathrm{h}1i}+{\tau }_{\mathrm{h}2i}\right)\sin \varphi }d\varphi$$

(5)

where $p_{pi}$ is the lateral soil resistance force caused by passive earth pressure; $p_{\tau i}$ is the lateral soil resistance force caused by lateral friction; y_i is pile displacement at the i^th sublayer (m); ${\epsilon }_i$ is soil strain at the i^th sublayer; D is pile diameter (m); $\frac{D_{\mathrm{s}}}{2}=\frac{1}{2}\left(S_i+\frac{D}{2}+\frac{D^2}{4S_i+2D}\right)$; ${\alpha }_i=\arctan \frac{D}{2S_i+D}$; $\Delta {\sigma }_{\mathrm{h}}$ is the stress increment in horizontal direction; ${\tau }_{\mathrm{h}}$ is shear stress between the monopile and soil; $\zeta$ and $\varphi$ are integration variables.

2.2. Solution of Soil Reaction Considering Scour

As a scour hole forms, the soil is unloaded, and the stress history of the remaining soil is changed. A scour hole is assumed to be a symmetrical inverted cone (scour depth S_d, scour top width S_wt, scour bottom width S_wb and scour slope angle β) [18,19]. The unloading pressure can be expressed by effective unit weight (${\gamma }^{\prime }$) by Equation (6), and the average stress reduction in the stress point from r = 0.5D to 3D can be expressed by Equation (8), as shown in Figure 1 [12,20,21].

$$p\left(r\right)=\left\{\begin{array}{l}{\gamma }^{\prime }{S}_d\begin{array}{c} \frac{D}{2}\le r\le S_{wb}+\frac{D}{2}\end{array}\\ {\gamma }^{\prime }{S}_d\left(\frac{S_{wt}+\frac{D}{2}-r}{S_{wt}-S_{wb}}\right)\begin{array}{c} S_{wb}+\frac{D}{2}\le r\le S_{wt}+\frac{D}{2}\end{array}\end{array}\right.$$

(6)

$$\begin{array}{c}\Delta {\sigma }_{r^{\prime }z}={\displaystyle {\int }_0^{2\pi }{\displaystyle {\int }_{D/2}^{S_{wb}+D/2}\frac{3{\gamma }^{\prime }{S}_d}{2\pi }\cdot \frac{{z^{\prime }}^3}{{\left({\left(r\sin \theta \right)}^2+{\left(r\cos \theta -r^{\prime }\right)}^2+{z^{\prime }}^2\right)}^{5/2}}}}r\mathrm{d}r\mathrm{d}\theta \\ +{\displaystyle {\int }_0^{2\pi }{\displaystyle {\int }_{S_{wb}+D/2}^{S_{wt}+D/2}\frac{3{\gamma }^{\prime }{S}_d}{2\pi }\cdot \frac{S_{wt}+D/2-r}{S_{wt}-S_{wb}}\cdot \frac{{z^{\prime }}^3}{{\left({\left(r\sin \theta \right)}^2+{\left(r\cos \theta -r^{\prime }\right)}^2+{z^{\prime }}^2\right)}^{5/2}}}}r\mathrm{d}r\mathrm{d}\theta \end{array}$$

(7)

$$\Delta {\sigma }_{3j}=\overline{\Delta {\sigma }_z}=\frac{1}{\pi \left({\left(3D\right)}^2-{\left(0.5D\right)}^2\right)}{\displaystyle {\int }_{-\pi }^{\pi }\Delta {\sigma }_{r^{\prime }z}}{r}^{\prime }\mathrm{d}{r}^{\prime }$$

(8)

The effective vertical stress (${\sigma }_{{}_{3j}}^{\mathrm{s}}$) within the embedded pile length after scour (below the mud line after scour) is expressed by Equation (9). Soil stress history also changes. To avoid an infinite OCR near the post-scour mud line, a limit value is defined by Equation (11) [22].

$${\sigma }_{3j}^{\mathrm{s}}={\sigma }_{3j}-\Delta {\sigma }_{3j}$$

(9)

$$\mathrm{OCR}=\frac{{\sigma }_{3j}}{{\sigma }_{3j}^{\mathrm{s}}}$$

(10)

$${\mathrm{OCR}}_{\mathrm{limit}}={\left[\frac{\left(1+\sin {\phi }^{\prime }\right)}{{\left(1-\sin {\phi }^{\prime }\right)}^2}\right]}^{\left(1/\sin {\phi }^{\prime }\right)}$$

(11)

${\phi }^{\prime }$ is the effective friction angle of soil. An equivalent effective friction angle is used for frictionless soil (clay) according to plasticity index $I_p^{\prime }$, according to the research of Sørensen and Okkels [23]. This equivalent effective friction angle is also used in the following section to calculate the equivalent modified effective friction angle for clay.

$${\phi }_i^{\prime }=43-10\log I_p^{\prime }\left(°\right)(\mathrm{For} \mathrm{normal} \mathrm{consolidated} \mathrm{clay})$$

(12)

$$\left\{\begin{array}{l}{\phi }_i^{\prime }=45-14\log I_p^{\prime }\left(°\right)\begin{array}{c} 4<I_p^{\prime }<50\end{array}\\ {\phi }_i^{\prime }=26-3\log I_p^{\prime }\left(°\right)\begin{array}{c} 50<I_p^{\prime }<150\end{array}\end{array}\right. (\mathrm{For} \mathrm{over-consolidated} \mathrm{clay})$$

(13)

The effective cohesion of the remaining soil after scour ($c^{{\mathrm{s}}^{\prime }}$) is changed according to the Cam-Clay model [24]. The change in effective friction angle is not significant [25]. $c^{{\mathrm{s}}^{\prime }}$ represents the undrained shear strength in clay.

$$c^{{\mathrm{s}}^{\prime }}=\frac{c^{\prime }{\sigma }_3^{{\mathrm{s}}^{\prime }}}{{\sigma }_3}{\mathrm{OCR}}^{\mathsf{\Lambda }}$$

(14)

where $\mathsf{\Lambda }=1-\frac{C_{\mathrm{ur}}}{C_{\mathrm{c}}}\approx \frac{\left(0.1~0.2\right)C_{\mathrm{c}}}{C_{\mathrm{c}}}$; $C_{\mathrm{c}}$ is the compression index; $C_{\mathrm{ur}}$ is the swelling index.

According to the research [7,11,12], the change in the effective cohesion has little influence on monopile response. As a result, for the sake of simplicity, the change in effective cohesion is ignored. The changed effective cohesion can be adopted in the derivation of the formulas in this paper for completeness.

The relationship between soil stress increment in the horizontal direction and soil strain after scour is determined based on the Duncan–Chang model [26].

$${\left(\Delta {\sigma }_{\mathrm{h}}\right)}_i={\left({\sigma }_1^{\mathrm{s}}-{\sigma }_3^{\mathrm{s}}\right)}_i=\frac{{\epsilon }_i}{\frac{1}{E_{\mathrm{s}i}}+\frac{{\epsilon }_i}{{\left({\sigma }_1^{\mathrm{s}}-{\sigma }_3^{\mathrm{s}}\right)}_{\mathrm{ult}i}}}$$

(15)

$${\left({\sigma }_1^{\mathrm{s}}-{\sigma }_3^{\mathrm{s}}\right)}_{\mathrm{ult}i}=\frac{{\left({\sigma }_1^{\mathrm{s}}-{\sigma }_3^{\mathrm{s}}\right)}_{\mathrm{f}i}}{R_{\mathrm{f}i}}$$

(16)

$${\left({\sigma }_1^{\mathrm{s}}-{\sigma }_3^{\mathrm{s}}\right)}_{\mathrm{f}i}=\frac{2c_i^{{\mathrm{s}}^{\prime }}\cos {\phi }_i^{\prime }+2{\sigma }_{3i}^{\mathrm{s}}\sin {\phi }_i^{\prime }}{1-\sin {\phi }_i^{\prime }}$$

(17)

According to the Material Models Manual of PLAXIS [27], the soil modulus is determined as follows:

$$\left\{\begin{array}{l}{E}_{\mathrm{s}i}=\frac{2E_{50i}}{2-R_{\mathrm{f}i}}\begin{array}{c} \mathrm{application} \mathrm{to} \mathrm{all} \mathrm{types} \mathrm{of} \mathrm{soil}\end{array}\\ E_{\mathrm{s}i}=E_{\mathrm{a}i}{p}^{\mathrm{ref}}{\left(\frac{{\sigma }_{3j}^{\mathrm{s}}}{p^{\mathrm{ref}}}\right)}_i^m\begin{array}{c} \mathrm{application} \mathrm{to} \mathrm{sand}\end{array}\end{array}\right.$$

(18)

$$\left\{\begin{array}{l}{E}_{50i}=E_{50i}^{\mathrm{ref}}{\left(\frac{c_i^{{\mathrm{s}}^{\prime }}\cos {\phi }_i^{\prime }+{\sigma }_{3i}^{\mathrm{s}}\sin {\phi }_i^{\prime }}{c_i^{{\mathrm{s}}^{\prime }}\cos {\phi }_i^{\prime }+p^{\mathrm{ref}}\sin {\phi }_i^{\prime }}\right)}^m\begin{array}{c} \mathrm{application} \mathrm{to} \mathrm{all} \mathrm{types} \mathrm{of} \mathrm{soil}\end{array}\\ E_{50i}=\frac{c_i^{{\mathrm{s}}^{\prime }}}{{\epsilon }_{50i}}\begin{array}{c} \mathrm{application} \mathrm{to} \mathrm{clay}\end{array}\end{array}\right.$$

(19)

where $R_{\mathrm{f}i}$ is the failure ratio; m and $E_{\mathrm{a}i}$ are modulus factors; $p^{\mathrm{ref}}$ is atmospheric pressure (kPa); $E_{50i}^{\mathrm{ref}}$ is the reference secant modulus (kPa); and ${\epsilon }_{50i}$ is the strain at half of the failure strength.

The relationship between soil stress increment in the horizontal direction and the modified effective cohesion ($c_{\mathrm{m}i}^{\prime }$) and the effective friction angle (${\phi }_{\mathrm{m}i}^{\prime }$) after scour is determined based on Equation (20).

$${\left(\Delta {\sigma }_{\mathrm{h}}\right)}_i={\left({\sigma }_1^{\mathrm{s}}-{\sigma }_3^{\mathrm{s}}\right)}_i=\frac{2c_{\mathrm{m}i}^{\prime }\cos {\phi }_{\mathrm{m}i}^{\prime }+2{\sigma }_{3i}^{\mathrm{s}}\sin {\phi }_{\mathrm{m}i}^{\prime }}{1-\sin {\phi }_{\mathrm{m}i}^{\prime }}$$

(20)

$$\left\{\begin{array}{l}{c}_{\mathrm{m}i}^{\prime }\cot {\phi }_{\mathrm{m}i}^{\prime }=c_i^{{\mathrm{s}}^{\prime }}\cot {\phi }_i^{\prime }\begin{array}{c} \mathrm{applicable} \mathrm{to} \mathrm{frictional} \mathrm{soil}\end{array}\\ c_i^{{\mathrm{s}}^{\prime }}{\phi }_{\mathrm{m}i}^{\prime }=c_{\mathrm{m}i}^{\prime }{\phi }_i^{\prime }\begin{array}{c}\begin{array}{c} \mathrm{applicable} \mathrm{to} \mathrm{frictionless} \mathrm{soil}\end{array}\end{array}\end{array}\right.$$

(21)

The relationship between pile displacement and soil strain can be determined according to the Mohr circle of strain [28].

$$\left|y_{i+1}-y_i\right|=h{\delta }_i=h\frac{{\gamma }_i}{2}=h\frac{\left(1+{\upsilon }_i\right){\epsilon }_i}{2}\cos {\phi }_{\mathrm{m}i}^{\prime }$$

(22)

where ${\upsilon }_i$ is the Poisson’s ratio of soil.

The shear stress between the monopile and soil is determined according to Ashour et al. [28].

$${\tau }_{\mathrm{h}1i}=\left({\sigma }_{3i}^{\mathrm{s}}+{\left(\Delta {\sigma }_{\mathrm{h}}\right)}_i\right)\tan {\phi }_{\mathrm{s}i}^{\prime }+a{c}_{\mathrm{s}i}^{\prime }$$

(23)

$${\tau }_{\mathrm{h}2i}={\sigma }_{3i}^{\mathrm{s}}\tan {\phi }_{\mathrm{s}i}^{\prime }+a{c}_{\mathrm{s}i}^{\prime }$$

(24)

$$\tan {\phi }_{\mathrm{s}i}^{\prime }=2\tan {\phi }_{\mathrm{m}i}^{\prime }\le \tan {\phi }_i^{\prime }$$

(25)

$$c_{\mathrm{s}i}^{\prime }=2c_{\mathrm{m}i}^{\prime }\le c_i^{{\mathrm{s}}^{\prime }}$$

(26)

where a is the adhesion factor, a = 1 (normal consolidated clay), and a = 0.5 (over-consolidated clay).

2.3. Pile Bottom Conditions for Flexible and Rigid Piles

For flexible piles, the shear force and bending moment at the pile bottom are all zero. To make the curved strain wedge method suitable for both flexible monopiles and rigid monopiles, the conditions at the pile bottom should be obtained by calculating the shear force, which is caused by displacement at the pile bottom, and the bending moment, which is caused by the deflection angle (${\delta }_{\mathrm{b}}$) at the pile bottom. As a result, the shear force and the bending moment are all zero for flexible monopiles. The shear force is determined by Equation (27), and the bending moment is determined by Equation (32). All parameters with subscript b represent parameters at the pile bottom in the following analysis. When scour occurs, the calculated parameters in the formula should be replaced by the parameters after scour.

$$F_{\mathrm{b}}=\frac{\pi \left(D^2-d^2\right){\tau }_{\mathrm{b}1}}{4}+\frac{\pi d^2{\tau }_{\mathrm{b}2}}{4}$$

(27)

$${\tau }_{\mathrm{b}1}={\sigma }_{3\mathrm{b}1}\tan {\phi }_{\mathrm{mb}}^{\prime }+a{c}_{\mathrm{mb}}^{\prime }$$

(28)

$${\tau }_{\mathrm{b}2}={\sigma }_{3\mathrm{b}2}\tan {\phi }_{\mathrm{mb}}^{\prime }+a{c}_{\mathrm{mb}}^{\prime }$$

(29)

$$\tan {\phi }_{\mathrm{mb}}^{\prime }\le \tan {\phi }_{\mathrm{b}}^{\prime }$$

(30)

$$c_{\mathrm{mb}}^{\prime }\le c_{\mathrm{b}}^{\prime }$$

(31)

where d is the inner diameter of the monopile (d = 0 for solid piles) (m); ${\sigma }_{3\mathrm{b}1}$ is the vertical effective stress of the monopile; and ${\sigma }_{3\mathrm{b}2}$ is the vertical effective stress of soil inside the monopile.

$$M_{\mathrm{b}}=k_{\mathrm{b}M}{\delta }_{\mathrm{b}}$$

(32)

where $k_{\mathrm{b}M}=\frac{E_{\mathrm{sb}}{D}^3}{6\left(1-{\upsilon }_{\mathrm{b}}^2\right)}$ [29].

2.4. Flow Chart for Flexible and Rigid Piles under Scour

An initial pile displacement (Y_l0) is specified, and the soil subgrade reaction modulus is calculated by Equation (33). The real pile displacement (Y_l) can be obtained by iterating Equations (1) and (33) with the finite difference method, according to the boundary conditions of the pile head and pile bottom. MATLAB is used for programming. The analysis process of the general solution for both flexible and rigid piles under scour is shown in Figure 2 and can be described as follows.

$$K{\left(z\right)}_i=\frac{p{\left(y,z\right)}_i}{y_i}$$

(33)

• Input the parameters of the layered soil and monopile foundation.
• Determine the position of the lateral load on the monopile foundation. Divide the monopile foundation and soil into n equal parts, according to the length of the monopile foundation and the thickness of soil layer, and determine the size of the scour hole.
• Calculate the vertical effective stress of soil without scour.
• Determine the lateral load and bending moment at the pile head.
• Assume initial lateral displacement of the monopile foundation on each node.
• Calculate the soil reaction and subgrade reaction modulus of each sublayer.
• Calculate the resistance force and bending moment at the pile bottom.
• Calculate the lateral displacement of the monopile foundation by the finite difference method.
• Compare the calculated lateral displacement of the monopile foundation on each node with the assumed initial lateral displacement of the monopile foundation on each node. If the absolute value of the difference is less than the specified precision, the final solution is obtained; otherwise, the initial lateral displacement of the monopile foundation is updated to the calculated lateral displacement of the monopile foundation in this step. Then, the iterative calculation is continued until the difference is less than the certain precision.
• At the end of the calculation, record the internal forces and displacement of the monopile foundation, such as the lateral displacement, the bending moment, the shear force of the monopile foundation and so on.

3. Verification

Verification has been performed by Zhang et al. [17] for flexible piles with both fixed and free heads. In this section, verification is performed on rigid piles and monopile foundations with scour.

The rigid monopile performed by Bhushan [30] was located in southern California. The diameter and pile length of the reinforced concrete monopile were 1.22 m and 4.96 m, respectively. The bending stiffness of the monopile was 2,350,000 kN·m². The soil was over-consolidated clay, and the effective unit weight of the clay was 10.4 kN/m³. The average undrained shear strength was 227.5 kPa, and the average ε₅₀ was 0.0072. Details of the parameters are shown in

Table 1. Parameters of monopile and soil.

Cases	Buried Length L (m)		Outer Diameter D (m)		Bending Stiffness Ep Ip (kN·m2)	Load Height above Mud Line e (m)
Bhushan et al.	4.73		1.22		2,350,000	0.23
Reese et al.	21		0.61		163,216	0.305
Cases	γ′ (kN/m3)	φ′ (°)	c′ (kPa)	υ	ε50/Dr (Ks/Es (kPa))	m	Rf
Bhushan et al.	10.4	0	227.5	0.5	0.0072 ε50	/	0.6
Reese et al.	10.4	39	0	0.3	1200 Es	0.7	0.6

. As shown in Figure 3a, the calculated results from the present solution correspond well with both the measured results from field tests and the calculated results from Ashour and Helal [31], which demonstrate that this method is also applicable to rigid piles.

Lin and Lin [32] performed research on the response of monopiles under scour according to the field tests by Reese et al. [8]. The soil was dense sand, and the pile was a flexible monopile. The parameters of the monopile and soil can be found in Table 1. The pile diameter was 0.61 m; the pile length was 21.305 mIthe pile bending stiffness was 163,216 kNI the effective unit weight of the soil was 10.4 Im³; the effective friction angle of the soil was 39°. Lin and Lin [32] established finite element models by PLAXIS based on the parameters and considering the condition of the scour. The scour depth was 1D, and the scour slope angle was 39°. The scour bottom widths were 0 and ∞, respectively. Figure 3b shows the comparison between the present solution and those from Lin and Lin [32]. The results are all consistent, and the present method is validated.

4. Results and Discussion

According to the research [9,10,12], under scour in clay or sand, the reaction variation rules of a monopile foundation are consistent. Only the parameters of the tests by Reese et al. [8] are adopted to analyze the influence of scour dimension on monopile response. The influence of the pile head condition and type (flexible or rigid) on monopile response is also discussed. The lateral capacity ratio is defined as the ratio between the lateral load after scour and the lateral load before scour at the same lateral deflection. In this section, the load capacity ratios are all shown at a lateral deflection of 0.05D.

4.1. Influence of Scour Depth on Response of Piles with Different Pile Heads and Lengths

Figure 4 shows the influence of scour depth on the response of piles with different pile heads and lengths. Local scour with a scour bottom width of 0 and a scour slope angle of 40° and general scour with a scour bottom width of ∞ and a scour slope angle of 40° are adopted to analyze the influence of scour width on the response of piles with different pile heads and lengths. According to Ettema [33], the maximum scour depth in real situations can be as large as 2.4D. The scour depths within 0–2.5D are adopted in the following analysis.

As shown in Figure 4, load capacity ratios of monopiles with different scour types, different pile heads and different pile types all decrease with increasing scour depth. When other parameters are held constant, the load capacity ratio of a monopile with a fixed head is greater than that of a monopile with a free head after scour, which demonstrates that scour depth has less influence on the bearing capacity of a monopile with a fixed head compared with that of a monopile with a free head. A rigid monopile is more susceptible to scour depth than a flexible monopile with the same pile head conditions. For local scour, the decreasing rate of the load capacity ratio increases with increasing scour depth. For general scour, the load capacity ratio decreases almost linearly with increasing scour depth. When the local scour depth is 2.5D, the lateral bearing capacity after scour is about 0.8, 0.64 and 0.71 times of that before scour for a flexible monopile with a free head, a flexible monopile with a fixed head and a rigid monopile with a free head, respectively, and the decreased bearing capacity varies from 20% to 29%. When the general scour depth is 2.5D, the lateral bearing capacity after scour is about 0.6, 0.5 and 0.37 times of that before scour for a flexible monopile with a free head, a flexible monopile with a fixed head and a rigid monopile with a free head, respectively, and the decreased bearing capacity varies from 40% to 63%. Therefore, when considering the impact of scour on pile bearing performance, scour dimension, pile head type and pile type (flexible or rigid) all need to be carefully considered.

4.2. Influence of Scour Width on Response of Piles with Different Pile Heads and Lengths

Figure 5 shows the influence of scour bottom width on the response of piles with different pile heads and lengths. Local scour with a scour depth of 0.5D and a scour slope angle of 40° and local scour with a scour depth of 2.5D and a scour slope angle of 40° are adopted to analyze the influence of scour bottom width on the response of piles with different pile heads and lengths. The scour bottom widths from 0 to 8D are adopted in the following analysis.

As shown in Figure 5, the load capacity ratios of monopiles with different scour depths, pile heads and pile types all decrease with increasing scour bottom width. The decreasing rates of load capacity ratios all decrease with increasing scour bottom width. The decreasing rates of load capacity ratios of a rigid monopile with a free head decrease with scour bottom width faster than those of a flexible monopile with a free head, and the decreasing rates of load capacity ratios of a flexible monopile with a free head decrease with scour bottom width faster than those of a flexible monopile with a fixed head. When the scour bottom width reaches 4D, the influence of local scour on pile bearing performance is almost the same as that of the general scour. With increasing scour depth, the scour bottom width decreases the load capacity ratio more. When the scour depth is 0.5D, the decreased bearing capacity varies from 1% to 15% for a rigid monopile with a free head with the variation of scour bottom width. When the scour depth is 2.5D, the decreased bearing capacity varies from 36% to 63% for a rigid monopile with a free head with the variation of scour bottom width.

4.3. Influence of Scour Slope Angle on Response of Piles with Different Pile Heads and Lengths

Figure 6 shows the influence of scour slope angle on the response of piles with different pile heads and lengths. Local scour with a scour depth of 0.5D and a scour bottom width of 0 and local scour with a scour depth of 2.5D and a scour bottom width of 0 are adopted to analyze the influence of scour slope angle on the response of piles with different pile heads and lengths. The scour slope angle varies from 0° to 90°.

As shown in Figure 6, the load capacity ratios of piles with different scour depths, pile heads and pile types all increase with increasing scour slope angle. Similar to the previous conclusion, under the same scour dimension, the load capacity ratio of a fixed flexible monopile is greater than that of a free flexible monopile, and the load capacity ratio of a free flexible monopile is greater than that of a free rigid monopile. When the scour depth is 0.5D, the load capacity ratio first increases with the scour slope angle and then basically remains unchanged. When the scour depth is 0.5D and the scour slope angle increases from 0° to 20°, the load capacity ratio increases from 0.85 to nearly 1. When the scour depth is 2.5D, the load capacity ratio increases with the scour slope angle, and the increasing rate of the load capacity ratio firstly decreases and then increases with the scour slope angle. When the scour depth is 2.5D and the scour slope angle increases from 0° to 80°, the load capacity ratio increases almost the same amount as when the scour slope angle increases from 80° to 90°.

5. Conclusions

This research presents a general analytical method for laterally loaded piles with different pile heads and pile types (flexible or rigid) under scour according to the curved strain wedge method. Scour dimension and stress history on the influence of soil parameters can all be considered. The following conclusion can be drawn according to the above analysis.

When other parameters are held constant, the increase in scour dimension has less influence on the bearing performance of a monopile with a fixed head compared with that of a monopile with a free head. A rigid monopile is more susceptible to scour dimensions than a flexible monopile under the same pile head conditions. The bearing capacities of monopiles with different pile heads and different pile types all decrease with increasing scour depth and scour bottom width but increase with increasing scour slope angle. Scour dimension, pile head type and pile type (flexible or rigid), which all influence the pile response, need to be carefully considered to perform the influence of scour on monopile bearing capacity.

When the local scour bottom width is 0, scour slope angle is 40° and scour depth is 2.5D, the lateral bearing capacity after scour is about 0.8, 0.64 and 0.71 times of that before scour for flexible monopiles with a free head, flexible monopiles with a fixed head and rigid monopiles with a free head, respectively. When the local scour bottom width is ∞, scour slope angle is 40° and scour depth is 2.5D, the lateral bearing capacity after scour is about 0.6, 0.5 and 0.37 times of that before scour for flexible monopiles with a free head, flexible monopiles with a fixed head and rigid monopiles with a free head, respectively. When the local scour depth is 0.5D and scour slope angle is 40°, the decreased bearing capacity varies from 1% to 15% for rigid monopiles with a free head with the variation of scour bottom width. When the local scour depth is 2.5D and scour slope angle is 40°, the decreased bearing capacity varies from 36% to 63% for rigid monopiles with a free head with the variation of scour bottom width. When the local scour depth is 0.5D, scour bottom width is 0 and scour slope angle increases from 0° to 20°, the load capacity ratio increases from 0.85 to nearly 1 for three different types of pile. When the local scour depth is 2.5D, scour bottom width is 0 and the scour slope angle increases from 0° to 80°, the load capacity ratio increases almost the same amount as when the scour slope angle increases from 80° to 90°.

6. Limitation and Future Research

In this paper, the influence of scour dimension, pile head type and pile type is investigated, which cannot reflect the design of pile dimension. In the future, the authors will conduct research on the pile dimension calculated by the analytical method and the conventional method when compared under the condition of scour. Future research can provide more theoretical support for practical engineering design.