The Method of the Natural Frequency of the Offshore Wind Turbine System Considering Pile–Soil Interaction

Abstract

Accurately and efficiently evaluating the influence of pile–soil interaction on the overall natural frequency of wind turbines is one of the difficulties in current offshore wind power design. To improve the structural safety and reliability of the offshore wind turbine (OWT) systems, a new closed-form solution method of the overall natural frequency of OWTs considering pile–soil interactions with highly effective calculations is established. In this method, Hamilton’s principle and the equivalent coupled spring model (ECS model) were firstly combined. In Hamilton’s theory, the Timoshenko beam assumption and continuum element theory considering the three-dimensional displacement field of soil were used to simulate the large-diameter monopile–soil interaction under lateral load in multilayer soil. Case studies were used to validate the proposed method’s correctness and efficiency. The results show that when compared with the data of 13 offshore wind projects reported in existing research papers, the difference between the overall natural frequency calculated by the proposed method and that reported in this study is within ±10%. This calculation method achieves the goal of convenient, fast and accurate prediction of the overall natural frequency of offshore wind systems.

1. Introduction

In recent years, offshore wind power development has continued to be a major focus for the expansion of renewable energy, representing a substantial opportunity to advance towards carbon neutrality within the foreseeable future. During the normal operating phase, it is crucial to avoid resonance between the overall natural frequency of the offshore wind turbine (OWT) system and the frequencies associated with environmental loads, such as wind, waves and blade rotation [1,2]. To ensure the structural stability of OWTs, the design process must maintain an eigenfrequency within the safe range of f_1P to f_3P. On the other hand, monopile foundations are commonly used in offshore wind power projects due to their simplicity in construction, ease of installation, cost-effectiveness and simple design process. In China’s coastal regions, monopiles are employed for over 80% of offshore wind turbines. However, the interaction between monopiles and the underlying soil plays a critical role in the overall modal analysis of the turbine system. Therefore, it is important to study how pile–soil interaction affects the natural frequency of the OWT system.

The simplification methods for the pile–soil interaction model of monopile foundations in offshore wind turbines can be categorized into three main types [3,4,5,6,7,8,9,10,11]: (1) The distributed spring (DS) model, which reduces the complexity of pile–soil interaction to a series of independent springs, aligning with the Winkler foundation model. This includes the m-method and p-y curve method, both prevalent in design codes. Byron and Houlsby [12,13] expanded on this model by proposing the “p-y method with pile end resistance” multi-spring model and corresponding calculation formulas, which have become a hot research topic in this area. (2) The equivalent fixed cantilever (EF) model, which simplifies a single pile to an equivalent cantilever beam with a fixed bottom boundary. It offers a straightforward mechanical interpretation and ease of computation, but its applicability is limited to long, flexible steel pipe piles. Finally, (3) the equivalent coupled spring (ECS) model represents the pile with three types of springs that account for translational (K_L), rotational (K_R) and coupled stiffness (K_LR) at the pile top on the seabed. The ECS model is the best choice among the above three models for its simplicity and clear mechanical interpretation. The model can be used to describe the pile–soil interaction as boundary conditions when calculating the overall natural frequency of offshore wind turbines [14,15,16,17]. For analytical solutions of the ECS model, Randolph [18] and Pender [19] provided formulas for the equivalent head stiffness of flexible piles in linearly elastic homogeneous soil; Gazetas et al. [20] provided solutions for flexible piles in linearly elastic heterogeneous soil; and Poulos et al. [21] and Shadlou et al. [22] derived formulas for flexible and rigid piles in layered soils. If lateral loads and overturning moments are not considered, the ECS model degrades into the uncoupled spring model [23]. Additionally, there are also combined forms of the three models mentioned above. For example, Wang et al. [24] used the “M_R-θ_R” model to simulate ideal rigid piles in sandy soil, with pile–soil interaction equivalent to a bending spring at a certain depth; Zha et al. [25] proposed the “p-y and M_R-θ_R” model, which combines features of the DS model and ECS model; Houlsby et al. [26] introduced the hyperplastic accelerated ratchet model (HARM) based on extensive laboratory test data, which better simulates the hysteretic effect of pile head stiffness; and Arany et al. [14] combined the strengths of the EF and ECS models by treating the wind turbine tower base as a fixed end. By incorporating various correction factors related to pile head displacement and rotation, analytical solutions for natural frequency calculation have been proposed [27,28]. However, there is a notable gap in the literature regarding a comprehensive reflection of the influence of pile–soil interaction on the dynamic response of offshore wind turbines. This gap stems from a lack of studies based on continuum theory that considers the three-dimensional displacement field of the soil.

The variational method based on Hamilton’s principle offers distinctive advantages in the calculation of pile response for large-diameter piles in offshore wind turbines [29,30,31,32]. It can successfully describe the intrinsic mechanisms of pile–soil interaction [33,34]. This is crucial for quickly and accurately adjusting monopile design. By using the variational method, the mechanical performance of the monopile was simplified as a Timoshenko beam. It can efficiently describe the displacement field of the surrounding soil using functions after introducing an appropriate soil constitutive model. If the displacement field of the surrounding soil is determined, the strain distribution in the soil can be obtained using continuous medium theory. Furthermore, by incorporating a suitable soil constitutive model, the stress distribution around the pile during loading can be determined, and thus we can achieve the goal of understanding the pile–soil interaction mechanism. Currently, research progress in incorporating soil constitutive models is relatively slow. Han et al. [29,30,31] tried to characterize the stress–strain relationships in sandy and cohesive soils by employing the modified D-P model and Mises model, respectively. Gupta et al. [32] used a nonlinear elastic stiffness degradation distribution model to describe the deformation of the surrounding soil under normal operating conditions. From principles of energy, the variational method has yielded a fitting expression for the dynamic stiffness of the pile head for wind turbine monopiles in layered soils. Depending on the previous studies, Li et al. [33,34,35,36,37,38,39] extended the soil displacement field into three dimensions, providing an accurate representation of the uplift or subsidence of soil near the seabed and the rotational motion of soil in the vicinity of the pile end. However, the above studies are not extensively applied to natural frequency calculations. To better reflect the influence of pile–soil interaction on the overall frequency of OWT systems, it is important to combine this method with the ECS model.

In this study, based on Hamilton’s principle and the variational method, a mechanical model for pile–soil interaction of offshore wind turbine monopile foundations in multilayered soils was constructed. By using the continuous medium assumption and Timoshenko beam theory, and combining the ECS model for simplified calculations, a rapid calculation method for the natural frequency of wind turbines considering pile–soil interaction was proposed. To verify the correctness and efficiency of the proposed method, the results were then compared with measured data from the literature. This method provides effective guidance for engineering practice, and it also establishes a theoretical foundation for analyzing the intrinsic mechanisms of pile–soil interaction and its impact on the overall dynamic response of wind turbines. Therefore, the novelty of this manuscript is as follows:

(1) To obtain a closed-form solution method of the overall natural frequency of OWTs with highly effective calculation, Hamilton’s principle and the equivalent coupled spring model (ECS model) were combined.

(2) To better describe pile–soil interactions with very large diameters, the pile is modeled as a Timoshenko beam, and soil is modeled as a continuous material considering the three-dimensional displacements of soil (existing studies consider only two-dimensional displacements).

2. Theoretical Analysis

2.1. Fundamental Assumptions in Mechanics

In Figure 1, a schematic diagram of typical frequencies for offshore wind turbines is presented, where the eigenfrequency of OWT systems should be in the range of f_1P to f_3P. Figure 2 gives several typical models for pile–soil interaction, including the ECS model. Figure 3 shows the mechanical model of the pile–soil interaction for a monopile subjected to lateral loads in an r-θ-z cylindrical coordinate system. In this model, the pile top is subjected to lateral displacement F_a(t) and moment M_a(t). The pile has a cross-sectional radius r_p and length L_p. The heights from the ground surface to the bottom of each soil layer are H_i (where i = 1, 2, …,n). The key assumptions for this mechanical model are as follows:

• The pile body is modeled as a Timoshenko beam, considering the shear deformation of the pile.
• The soil within each soil layer is treated as a laterally isotropic continuous medium.
• The pile–soil interface is not separated.

2.2. Governing Equations

2.2.1. Hamilton’s Theory

Depending on the Hamilton theory, the total system energy of the pile and soil is given as follows:

$$L=T-U_{\mathrm{T}}+W$$

(1)

where T is the kinetic energy, including T₁ and T₂; U_T is the total potential energy, including U_T1 and U_T2; and W is the work exerted by external force.

The kinetic energy of the pile (T₁) can be given as follows:

$$T_1={\displaystyle \frac{1}{2}}{\displaystyle \underset{0}{\overset{L_{\mathrm{p}}}{\int }}{\rho }_{\mathrm{p}}{\left(\frac{\partial w}{\partial t}\right)}^2\mathrm{d}z}+{\displaystyle \frac{1}{2}}{\displaystyle \underset{0}{\overset{L_{\mathrm{p}}}{\int }}{\rho }_{\mathrm{p}}{I}_{\mathrm{p}}{\left(\frac{\partial \varphi }{\partial t}\right)}^2\mathrm{d}z}$$

(2)

where ρ_p is the density of the pile (kg/m³); w is the deflection of the pile body; t is time; I_p is the bending moment inertia of a pile unit; and ϕ is the shear deformation of the pile shaft.

The kinetic energy of the soil (T₂) can be given as follows:

$$\begin{array}{c}{T}_2={\displaystyle \underset{0}{\overset{L_{\mathrm{p}}}{\int }}{\displaystyle \underset{0}{\overset{2\pi }{\int }}{\displaystyle \underset{0}{\overset{r_{\mathrm{p}}-t}{\int }}{\rho }_{\mathrm{s}}\left({\left(\frac{\partial u_{\mathrm{r}}}{\partial t}\right)}^2+{\left(\frac{\partial u_{\mathsf{\theta }}}{\partial t}\right)}^2+{\left(\frac{\partial u_{\mathrm{z}}}{\partial t}\right)}^2\right)}r\mathrm{d}r\mathrm{d}\theta \mathrm{d}z}}\hfill \\ +{\displaystyle \underset{0}{\overset{L_{\mathrm{p}}}{\int }}{\displaystyle \underset{0}{\overset{2\pi }{\int }}{\displaystyle \underset{r_{\mathrm{p}}}{\overset{\infty }{\int }}{\rho }_{\mathrm{s}}\left({\left(\frac{\partial u_{\mathrm{r}}}{\partial t}\right)}^2+{\left(\frac{\partial u_{\mathsf{\theta }}}{\partial t}\right)}^2+{\left(\frac{\partial u_{\mathrm{z}}}{\partial t}\right)}^2\right)}r\mathrm{d}r\mathrm{d}\theta \mathrm{d}z}}\hfill \\ +{\displaystyle \underset{L_{\mathrm{p}}}{\overset{\infty }{\int }}{\displaystyle \underset{0}{\overset{2\pi }{\int }}{\displaystyle \underset{0}{\overset{r_{\mathrm{p}}}{\int }}{\rho }_{\mathrm{s}}\left\{{\left(\frac{\partial u_{\mathrm{r}}}{\partial t}\right)}^2+{\left(\frac{\partial u_{\mathsf{\theta }}}{\partial t}\right)}^2+{\left(\frac{\partial u_{\mathrm{z}}}{\partial t}\right)}^2\right\}}r\mathrm{d}r\mathrm{d}\theta \mathrm{d}z}}\hfill \\ +{\displaystyle \frac{1}{2}}{\displaystyle \underset{L_{\mathrm{p}}}{\overset{\infty }{\int }}{\displaystyle \underset{0}{\overset{2\pi }{\int }}{\displaystyle \underset{r_{\mathrm{p}}}{\overset{\infty }{\int }}{\rho }_{\mathrm{s}}\left({\left(\frac{\partial u_{\mathrm{r}}}{\partial t}\right)}^2+{\left(\frac{\partial u_{\mathsf{\theta }}}{\partial t}\right)}^2+{\left(\frac{\partial u_{\mathrm{z}}}{\partial t}\right)}^2\right)}r\mathrm{d}r\mathrm{d}\theta \mathrm{d}z}}\hfill \end{array}$$

(3)

where ρ_s is the soil density; u_r is the displacement of soil in the r-direction; u_θ is the displacement of soil in the θ-direction; and u_z is the displacement of soil in the z-direction.

Depending on assumption (1) in Section 2.1, the total potential energy of pile, U_T1, is given as follows:

$$U_{T1}={{\displaystyle \underset{0}{\overset{L_{\mathrm{p}}}{\int }}\frac{1}{2}{E}_{\mathrm{p}}{I}_{\mathrm{p}}\left(\frac{\partial \varphi }{\partial z}\right)}}^2\mathrm{d}z+{{\displaystyle \underset{0}{\overset{L_{\mathrm{p}}}{\int }}\frac{1}{2}\kappa G_{\mathrm{p}}{A}_{\mathrm{p}}\left(\frac{\partial w}{\partial z}-\varphi \right)}}^2\mathrm{d}z$$

(4)

where E_pI_p is the pile bending stiffness; κ is the cross-section shear coefficient of the pile; and A_p is the cross-sectional area of the pile. If κG_p reaches to infinity, then the effect of shear deformations can be neglected; thus, Equation (4) reduces to the Euler–Bernoulli beam theory.

The total potential energy of soil can be given by

$$U_{T2}={\displaystyle \frac{1}{2}}{\displaystyle \underset{0}{\overset{\infty }{\int }}{\displaystyle \underset{0}{\overset{2\pi }{\int }}{\displaystyle \underset{rp}{\overset{\infty }{\int }}{\mathit{\epsilon }}_{ij}^T{\mathit{\sigma }}_{ij}}rdrd\theta dz}}+{\displaystyle \frac{1}{2}}{\displaystyle \underset{Lp}{\overset{\infty }{\int }}{\displaystyle \underset{0}{\overset{2\pi }{\int }}{\displaystyle \underset{0}{\overset{rp}{\int }}{\mathit{\epsilon }}_{ij}^T{\mathit{\sigma }}_{ij}}rdrd\theta dz}}$$

(5)

where σ_ij is the stress tensor of the soil; ε_ij is the strain tensor of the soil; and i,j = r, θ, z.

The work exerted by external forces is expressed as follows:

$$W=F_{\mathrm{a}}\left(t\right)w{|}_{z=0}-M_{\mathrm{a}}\left(t\right)\varphi {|}_{z=0}$$

(6)

where F_a(t) = F_a e^iωt; M_a(t) = M_a e^iωt.

If the external load is static, then the total energy system Π of the single pile and the soil is

$$\begin{array}{l}\prod ={\displaystyle \frac{1}{2}}{E}_{\mathrm{p}}{I}_{\mathrm{p}}{\displaystyle \underset{0}{\overset{L_{\mathrm{p}}}{\int }}(\frac{\mathrm{d}\varphi }{\mathrm{d}z}}{)}^{2}dz+{\displaystyle \frac{1}{2}}{\displaystyle \underset{0}{\overset{L_{\mathrm{p}}}{\int }}\kappa G_{\mathrm{p}}{A}_{\mathrm{p}}}{({\displaystyle \frac{\mathrm{d}w}{\mathrm{d}z}}-\varphi )}^2\mathrm{d}z+{\displaystyle \frac{1}{2}}{\displaystyle \underset{0}{\overset{L_{\mathrm{p}}}{\int }}{\displaystyle \underset{0}{\overset{2\pi }{\int }}{\displaystyle \underset{r_{\mathrm{p}}}{\overset{\infty }{\int }}{\mathit{\sigma }}_{ij}{\mathit{\epsilon }}_{ij}r\mathrm{d}r\mathrm{d}\theta \mathrm{d}z}}}+{\displaystyle \frac{1}{2}}{\displaystyle \underset{L_{\mathrm{p}}}{\overset{\infty }{\int }}{\displaystyle \underset{0}{\overset{2\pi }{\int }}{\displaystyle \underset{0}{\overset{\infty }{\int }}{\mathit{\sigma }}_{ij}{\mathit{\epsilon }}_{ij}r\mathrm{d}r\mathrm{d}\theta \mathrm{d}z}}}\\ -F_{\mathrm{a}}{\left.w\right|}_{z=0}+M_{\mathrm{a}}{\left.\varphi \right|}_{z=0}+W_{\mathrm{weight}}\end{array}$$

(7)

When the constitutive models of the pile and soil are elastic, the work exerted by the self-weight of the pile and soil is zero, i.e., δW_weight = 0 if the soil is in an elastic stage.

2.2.2. Three-Dimensional Displacement Expression of the Soil

In the above formula, the displacement of the soil around the laterally loaded pile is assumed to be

$$\left\{\begin{array}{l}{u}_r=w(z,t){\varphi }_{\mathrm{r}}(r)\cos \theta \\ u_{\mathsf{\theta }}=-w(z,t){\varphi }_{\mathsf{\theta }}(r)\sin \theta \\ u_{\mathrm{z}}=\varphi (z,t)r_{\mathrm{p}}{\varphi }_{\mathrm{z}}(r)\cos \theta \end{array}\right.$$

(8)

where ϕ_r, ϕ_θ and ϕ_z represent the displacement distribution functions in the r-, θ- and z-directions of the soil, respectively. All three functions are dimensionless and dependent on the radial distance r. The existence of soil displacement in the z-direction reduces the soil domain stiffness, resulting in relatively larger deformation of the foundation. If u_z is not considered, Equation (8) degenerates into the expression proposed by Gupta et al. [32,33,34,35,36,37,38,39,40].

Depending on assumption (3) in Section 2.1 and energy transfer laws, the boundary conditions of ϕ_r, ϕ_θ and ϕ_z can be defined as follows:

$${\varphi }_{\mathrm{r}}(r)=\left\{\begin{array}{cc}1& 0\le r\le r_{\mathrm{p}}\\ 0& r\to +\infty \end{array}\right.$$

(9)

$${\varphi }_{\mathsf{\theta }}(r)=\left\{\begin{array}{cc}1& 0\le r\le r_{\mathrm{p}}\\ 0& r\to +\infty \end{array}\right.$$

(10)

$${\varphi }_z(r)=\left\{\begin{array}{cc}0& r_{\mathrm{p}}=0\\ {\displaystyle \frac{r}{r_{\mathrm{p}}}}& 0<r<r_{\mathrm{p}}\\ 1& r=r_{\mathrm{p}}\\ 0& \infty \end{array}\right.$$

(11)

From Equation (11), within the radial domain inside the pile (0 ≤ r < r_p), the vertical displacement of the soil is consistent with the vertical component of the deformation caused by the Timoshenko beam. At r = r_p, the interface between the pile and the soil exhibits no interfacial separation, embodying an idealized complete rough contact. At the boundary of infinity, the soil displacement approaches zero.

According to Assumption 2 in Section 2.1, the displacement–strain relationship in the soil domain is as follows:

$$\left[\begin{array}{c}{\epsilon }_{rr}\\ {\epsilon }_{\theta \theta }\\ {\epsilon }_{zz}\\ {\epsilon }_{r\theta }\\ {\epsilon }_{rz}\\ {\epsilon }_{\theta z}\end{array}\right]=\left[\begin{array}{c}-{\displaystyle \frac{\partial u_{\mathrm{r}}}{\partial r}}\\ -{\displaystyle \frac{u_{\mathrm{r}}}{r}}-{\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial u_{\mathsf{\theta }}}{\partial \theta }}\\ -{\displaystyle \frac{\partial u_{\mathrm{z}}}{\partial z}}\\ {\displaystyle \frac{1}{2}}\left(-{\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial u_{\mathrm{r}}}{\partial \theta }}-{\displaystyle \frac{\partial u_{\mathsf{\theta }}}{\partial r}}+{\displaystyle \frac{u_{\mathsf{\theta }}}{r}}\right)\\ {\displaystyle \frac{1}{2}}\left(-{\displaystyle \frac{\partial u_{\mathrm{z}}}{\partial r}}-{\displaystyle \frac{\partial u_{\mathrm{r}}}{\partial z}}\right)\\ {\displaystyle \frac{1}{2}}\left(-{\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial u_{\mathrm{z}}}{\partial \theta }}-{\displaystyle \frac{\partial u_{\mathsf{\theta }}}{\partial z}}\right)\end{array}\right]=\left[\begin{array}{c}-w(z,t){\displaystyle \frac{\mathrm{d}{\varphi }_{\mathrm{r}}(r)}{\mathrm{d}r}}\cos \theta \\ {\displaystyle \frac{-w(z,t){\varphi }_{\mathrm{r}}(r)}{r}}\cos \theta +{\displaystyle \frac{1}{r}}w(z,t){\varphi }_{\mathsf{\theta }}(r)\cos \theta \\ -{\displaystyle \frac{\partial \varphi (z,t)}{dz}}{r}_{\mathrm{p}}{\varphi }_{\mathrm{z}}(r)\cos \theta \\ {\displaystyle \frac{1}{2}}\left({\displaystyle \frac{1}{r}}w(z,t){\varphi }_{\mathrm{r}}(r)\sin \theta +w(z,t){\displaystyle \frac{\mathrm{d}{\varphi }_{\mathsf{\theta }}(r)}{\mathrm{d}r}}\sin \theta -{\displaystyle \frac{w(z,t){\varphi }_{\mathsf{\theta }}(r)}{r}}\sin \theta \right)\\ {\displaystyle \frac{1}{2}}\left(-\varphi (z,t)r_{\mathrm{p}}{\displaystyle \frac{\mathrm{d}{\varphi }_{\mathrm{z}}(r)}{\mathrm{d}r}}\cos \theta -{\displaystyle \frac{\partial w(z,t)}{\partial z}}{\varphi }_{\mathrm{r}}(r)\cos \theta \right)\\ {\displaystyle \frac{1}{2}}\left({\displaystyle \frac{\varphi (z,t)}{r}}{r}_{\mathrm{p}}{\varphi }_{\mathrm{z}}(r)\sin \theta +{\displaystyle \frac{\partial w(z,t)}{\partial z}}{\varphi }_{\mathsf{\theta }}(r)\sin \theta \right)\end{array}\right]$$

(12)

2.2.3. Constitutive Model of Soil

From the study by Thomas [41], during the standard operation stage of offshore wind turbines, the soil surrounding a monopile foundation is subjected to small-amplitude cyclic loading conditions, and the small-amplitude cyclic loads caused by wind, waves and currents mobilize the soil with minimal to small strains (Figure 4). The soil is also in an elastic state under very small to small strain conditions if the overall natural frequency of wind turbines is mobilized. Therefore, it is reasonable to employ an elastic constitutive model for the soil within the scope of this study.

In this section, two kinds of foundation models were discussed:

(a)

The impact of embedment depth on the elastic modulus of the soil:

$$E_{\mathrm{s}}(z)=E_{\mathrm{s}0}{\left({\displaystyle \frac{z}{D_{\mathrm{p}}}}\right)}^{\alpha }$$

(13)

where E_s represents the elastic modulus of the soil (kPa), E_s0 is the initial value of the elastic modulus and α is an exponent, where 0 ≤ α ≤ 1. If α = 0, the soil is homogeneous; if α = 0.5, the distribution of the soil’s elastic modulus with depth follows a parabolic shape; and if α = 1, the soil conforms to the Gibson foundation model.

(b)

Considering the effects of multiple soil layers, the soil in each layer is divided into n segments along the z-direction. The elastic modulus of the soil is assigned within each segment Δz, resulting in a matrix representation:

$${\mathit{E}}_{\mathrm{s}}={{\left[\begin{array}{ccccccc}{E}_{\mathrm{s}1}& E_{\mathrm{s}2}& E_{\mathrm{s}3}& \dots & E_{\mathrm{s}i}& \dots & E_{\mathrm{s}n}\end{array}\right]}^T}_{1\times n}$$

(14)

where E_s represents the global soil matrix of the elastic modulus and E_si denotes the elastic modulus corresponding to the i th soil layer (i = 1,2,3, …,n).

Under dynamic loading, the Lame’s coefficients for elastic soil materials are given by

$$\begin{array}{l}{\stackrel{⌢}{\lambda }}_{\mathrm{s}}={\lambda }_{\mathrm{s}}\left(1+2i\varsigma \right)\\ {\stackrel{⌢}{G}}_{\mathrm{s}}{=\mathrm{G}}_{\mathrm{s}}\left(1+2i\varsigma \right)\end{array}$$

(15)

where λ_s and G_s represent the compression modulus and shear modulus of soil, respectively; i is the imaginary unit; and ζ is the damping ratio of soil material, which is independent of frequency. Under static loading conditions, λ_s = E_sv_s/(1 + v_s)/(1 − 2v_s) and G_s = E_s/2/(1 + v_s).

Therefore, the soil’s constitutive relationship can be expressed as follows:

(a)

Dynamic loading condition: ${\mathit{\sigma }}_{ij}={\stackrel{⌢}{\lambda }}_s\left(z\right){\mathit{\delta }}_{ij}{\mathit{\epsilon }}_{ij}+2{\stackrel{⌢}{G}}_s\left(z\right){\mathit{\epsilon }}_{ij}$;

(b)

Static loading condition: σ_ij = λ_s(z)δ_ijε_ij + 2G_s(z)ε_ij.

Here, δ_ij is the stress tensor, ε_ij is the strain tensor and z indicates the depth or layer within the soil profile.

2.2.4. The Application of the Variational Method

Substituting Equations (2)–(6) and Equations (8)–(11) into Equation (1) and applying the variational principle, Equation (2) yields the following expression:

$${\displaystyle \int \mathrm{M}\left(w\right)}\delta w\mathrm{d}z+{\displaystyle \int \mathrm{P}\left(\varphi \right)}\delta \varphi \mathrm{d}z+{\displaystyle \int \mathrm{Q}\left({\varphi }_{\mathrm{r}}\right)}\delta {\varphi }_{\mathrm{r}}\mathrm{d}z+{\displaystyle \int \mathrm{R}\left({\varphi }_{\mathsf{\theta }}\right)}\delta {\varphi }_{\mathsf{\theta }}\mathrm{d}z+{\displaystyle \int \mathrm{S}\left({\varphi }_{\mathrm{z}}\right)}\delta {\varphi }_{\mathrm{z}}\mathrm{d}z=0$$

(16)

where M(w), P(ϕ), Q(ϕ_r), R(ϕ_θ) and S(ϕ_z) are polynomials that contain δ(w), δ(ϕ), δ(ϕ_r), δ(ϕ_θ) and δ(ϕ_z), respectively. According to the variational method, the equality holds identically if and only if M(w) = 0, P(ϕ) = 0, Q(ϕ_r) = 0, R(ϕ_θ) = 0 and S(ϕ_z) = 0 when δ(ϕ), δ(w), δ(ϕ_r), δ(ϕ_θ) and δ(ϕ_z) are infinitesimal quantities. From this, the relational expressions for ϕ, w, ϕ_r, ϕ_θ and ϕ_z can be further derived.

3. The Governing Equation for Pile–Soil Interaction

3.1. The Governing Equation of Pile

For large-diameter single piles, the relationship between ϕ and w(z) obtained from M(w) and P(ϕ) in Equation (16) is expressed as (0 ≤ z ≤ L_p)

$$\begin{array}{l}\kappa G_{\mathrm{p}}{A}_{\mathrm{p}}({\displaystyle \frac{{\partial }^{2}w}{\partial z^2}}-{\displaystyle \frac{\partial \varphi }{\partial z}})+2t{\displaystyle \frac{{\partial }^{2}w}{\partial z^2}}+k_1{\displaystyle \frac{\partial \varphi }{\partial z}}-kw-\left({\rho }_{\mathrm{p}}{A}_{\mathrm{p}}+n_1\right){\displaystyle \frac{{\partial }^{2}w}{\partial t^2}}=0\\ E_{\mathrm{p}}{I}_{\mathrm{p}}{\displaystyle \frac{{\partial }^2\varphi }{\partial z^2}}+\kappa G_{\mathrm{p}}{A}_{\mathrm{p}}({\displaystyle \frac{\partial w}{\partial z}}-\varphi )-k_3\varphi -k_1{\displaystyle \frac{\partial w}{\partial z}}+2t_4{\displaystyle \frac{{\partial }^2\varphi }{\partial z^2}}-\left({\rho }_{\mathrm{p}}{I}_{\mathrm{p}}+n_2\right){\displaystyle \frac{{\partial }^2\varphi }{\partial t^2}}=0\end{array}$$

(17)

where $t={\displaystyle \frac{\pi }{2}}{\displaystyle \underset{r_{\mathrm{p}}}{\overset{\infty }{\int }}}{G}_{\mathrm{s}}({{\varphi }_{\mathrm{r}}}^2+{{\varphi }_{\mathsf{\theta }}}^2)r\mathrm{d}r\mathrm{d}r$; $k_1=\pi {\displaystyle \underset{r_{\mathrm{p}}}{\overset{\infty }{\int }}}\{G_{\mathrm{s}}{r}_{\mathrm{p}}{\displaystyle \frac{\mathrm{d}{\varphi }_{\mathrm{z}}}{\mathrm{d}r}}{\varphi }_{\mathrm{r}}+G_{\mathrm{s}}{\displaystyle \frac{r_{\mathrm{p}}}{r}}{\varphi }_{\mathrm{z}}{\varphi }_{\mathsf{\theta }}-{\lambda }_{\mathrm{s}}{r}_{\mathrm{p}}{\varphi }_{\mathrm{z}}{\displaystyle \frac{\mathrm{d}{\varphi }_{\mathrm{r}}}{\mathrm{d}r}}-{\lambda }_{\mathrm{s}}{r}_{\mathrm{p}}{\varphi }_{\mathrm{z}}{\displaystyle \frac{{\varphi }_{\mathrm{r}}-{\varphi }_{\mathsf{\theta }}}{r}}\}r\mathrm{d}r$; $k=\pi {\displaystyle \underset{r_p}{\overset{\infty }{\int }}}\{({\lambda }_{\mathrm{s}}+2G_{\mathrm{s}}){({\displaystyle \frac{\mathrm{d}{\varphi }_{\mathrm{r}}}{\mathrm{d}r}})}^2+2{\lambda }_{\mathrm{s}}{\displaystyle \frac{{\varphi }_{\mathrm{r}}-{\varphi }_{\mathsf{\theta }}}{r}}{\displaystyle \frac{\mathrm{d}{\varphi }_{\mathrm{r}}}{\mathrm{d}r}}+({\lambda }_{\mathrm{s}}+2G_{\mathrm{s}}){({\displaystyle \frac{{\varphi }_{\mathrm{r}}-{\varphi }_{\mathsf{\theta }}}{r}})}^2+G_{\mathrm{s}}{({\displaystyle \frac{{\varphi }_{\mathrm{r}}-{\varphi }_{\mathsf{\theta }}}{r}}+{\displaystyle \frac{\mathrm{d}{\varphi }_{\mathsf{\theta }}}{\mathrm{d}r}})}^2\}r\mathrm{d}r$; $k_3=\pi {\displaystyle \underset{r_{\mathrm{p}}}{\overset{\infty }{\int }}}\left\{\right[G_{\mathrm{s}}{\displaystyle \frac{\mathrm{d}{\varphi }_{\mathrm{z}}}{\mathrm{d}r}}{\displaystyle \frac{\mathrm{d}{\varphi }_{\mathrm{z}}}{\mathrm{d}r}}+G_{\mathrm{s}}{\displaystyle \frac{{\varphi }_{\mathrm{z}}}{r}}{\displaystyle \frac{{\varphi }_{\mathrm{z}}}{r}}]{r_{\mathrm{p}}}^{2}r\mathrm{d}r$; $t_4={\displaystyle \frac{\pi }{2}}{\displaystyle \underset{r_{\mathrm{p}}}{\overset{\infty }{\int }}}({\lambda }_{\mathrm{s}}+2G_{\mathrm{s}}){{\varphi }_{\mathrm{z}}}^2{r_{\mathrm{p}}}^{2}r\mathrm{d}r$; $n_1=\pi {\rho }_s{\displaystyle \underset{r_{\mathrm{p}}}{\overset{\infty }{\int }}\left\{({\lambda }_{\mathrm{s}}+2G_{\mathrm{s}}){{\varphi }_z}^{2}r+G_{\mathrm{s}}({{\varphi }_r}^2+{{\varphi }_{\theta }}^2)r\right\}}\mathrm{d}r$; $n_2=\pi {\displaystyle \underset{r_{\mathrm{p}}}{\overset{\infty }{\int }}}{\rho }_{\mathrm{s}}{r_{\mathrm{p}}}^2{{\varphi }_{\mathrm{z}}}^{2}r\mathrm{d}r$.

Equation (17) takes into account the dynamic response of the soil core within the pile. To solve Equation (17), it is necessary to introduce the following intermediate quantities:

$$\begin{array}{l}w\left(z,t\right)=W\left(z\right)e^{i\omega t}\\ \varphi \left(z,t\right)=\psi \left(z\right)e^{i\omega t}\end{array}$$

(18)

where ω represents the frequency of the load and W and ψ are the vibration modes of pile displacement and rotation angle, respectively.

For large-diameter monopiles, the boundary conditions are crucial for accurately modeling their behavior. The boundary conditions typically depend on the specific load and environmental conditions. Here are the common boundary conditions for a large-diameter monopile:

$$\begin{array}{l}{\left\{\kappa G_{\mathrm{p}}{A}_{\mathrm{p}}\left({\displaystyle \frac{\partial w}{\partial z}}-\varphi \right)+k_2\varphi +2t{\displaystyle \frac{\partial w}{\partial z}}\right\}}_{z=0}=F_{\mathrm{a}}\\ {\left\{E_{\mathrm{p}}{I}_{\mathrm{p}}{\displaystyle \frac{\partial \varphi }{\partial z}}+\left(-k_1+k_2\right)w+2t_4{\displaystyle \frac{\partial \varphi }{\partial z}}\right\}}_{z=0}=M_{\mathrm{a}}\end{array}\left(z=0\right)$$

(19)

where $k_2=\pi {\displaystyle \underset{r_{\mathrm{p}}}{\overset{\infty }{\int }}}\left\{G_{\mathrm{s}}\right\{{\displaystyle \frac{{\varphi }_{\mathrm{z}}}{r}}{\varphi }_{\mathsf{\theta }}+{\displaystyle \frac{\mathrm{d}{\varphi }_{\mathrm{z}}}{\mathrm{d}r}}{\varphi }_{\mathrm{r}}\}{r}_{\mathrm{p}}r\mathrm{d}r$.

Based on the conditions of deformation and force continuity, the calculation method for the intermediate pile segment in the i-th layer (i = 1, 2, 3, …, n) is

$$\begin{array}{l}{w}_{i,z=H_i}=w_{i+1,z=H_i}\\ {{\displaystyle \frac{\partial w}{\partial z}}}_{i,z=H_i}={{\displaystyle \frac{\partial w}{\partial z}}}_{i+1,z=H_i}\\ {\left\{\kappa G_{\mathrm{p}}{A}_{\mathrm{p}}\left({\displaystyle \frac{\partial w}{\partial z}}-\varphi \right)+\{k_2\varphi +2t{\displaystyle \frac{\partial w}{\partial z}}\}\right\}}_{i,z=H_i}={\left\{\kappa G_{\mathrm{p}}{A}_{\mathrm{p}}\left({\displaystyle \frac{\partial w}{\partial z}}-\varphi \right)+\{k_2\varphi +2t{\displaystyle \frac{\partial w}{\partial z}}\}\right\}}_{i+1,z=H_i}\\ {\left\{E_{\mathrm{p}}{I}_{\mathrm{p}}{\displaystyle \frac{\partial \varphi }{\partial z}}+\left(-k_1+k_2\right)w+2t_4{\displaystyle \frac{\partial \varphi }{\partial z}}\right\}}_{i,z=H_i}={\left\{E_{\mathrm{p}}{I}_{\mathrm{p}}{\displaystyle \frac{\partial \varphi }{\partial z}}+\left(-k_1+k_2\right)w+2t_4{\displaystyle \frac{\partial \varphi }{\partial z}}\right\}}_{i+1,z=H_i}\end{array}\left(0<z<L_{\mathrm{p}}\right)$$

(20)

At the pile end:

$$\begin{array}{l}{\left\{\kappa G_{\mathrm{p}}{A}_{\mathrm{p}}\left({\displaystyle \frac{\partial w}{\partial z}}-\varphi \right)+\{k_2\varphi +2t{\displaystyle \frac{\partial w}{\partial z}}\}\right\}}_{z=L_{\mathrm{p}}}={\left\{k_{2\mathrm{end}}\varphi +2t_{\mathrm{end}}{\displaystyle \frac{\partial w}{\partial z}}\right\}}_{z=L_{\mathrm{p}}}\\ {\left\{E_{\mathrm{p}}{I}_{\mathrm{p}}{\displaystyle \frac{\partial \varphi }{\partial z}}+\left(-k_1+k_2\right)w+2t_4{\displaystyle \frac{\partial \varphi }{\partial z}}\right\}}_{z=L_{\mathrm{p}}}={\left\{\left(-k_{1\mathrm{end}}+k_{2\mathrm{end}}\right)w+2t_{4\mathrm{end}}{\displaystyle \frac{\partial \varphi }{\partial z}}\right\}}_{z=L_{\mathrm{p}}}\end{array}\left(z=L_{\mathrm{p}}\right)$$

(21)

where $t_{\mathrm{end}}={\displaystyle \frac{\pi }{2}}{\displaystyle \underset{r_{\mathrm{p}}}{\overset{\infty }{\int }}}{G}_{\mathrm{s}}({{\varphi }_{\mathrm{r}}}^2+{{\varphi }_{\mathsf{\theta }}}^2)r\mathrm{d}r\mathrm{d}r+{\displaystyle \frac{\pi }{2}}{G}_{\mathrm{s}}{r}_{\mathrm{p}}^2$; $t_{4,\mathrm{end}}={\displaystyle \frac{\pi }{2}}{\displaystyle \underset{r_{\mathrm{p}}}{\overset{\infty }{\int }}}({\lambda }_{\mathrm{s}}+2G_{\mathrm{s}}){{\varphi }_{\mathrm{z}}}^2{r_{\mathrm{p}}}^{2}r\mathrm{d}r+{\displaystyle \frac{\pi }{2}}{\displaystyle \underset{0}{\overset{r_{\mathrm{p}}}{\int }}}({\lambda }_{\mathrm{s}}+2G_{\mathrm{s}}){{\varphi }_{\mathrm{z}}}^2{r_{\mathrm{p}}}^{2}r\mathrm{d}r$ $k_{1,\mathrm{end}}=\pi {\displaystyle \underset{r_{\mathrm{p}}}{\overset{\infty }{\int }}}\{G_{\mathrm{s}}{r}_{\mathrm{p}}{\displaystyle \frac{\mathrm{d}{\varphi }_{\mathrm{z}}}{\mathrm{d}r}}{\varphi }_{\mathrm{r}}+G_{\mathrm{s}}{\displaystyle \frac{r_{\mathrm{p}}}{r}}{\varphi }_{\mathrm{z}}{\varphi }_{\mathsf{\theta }}-{\lambda }_{\mathrm{s}}{r}_{\mathrm{p}}{\varphi }_{\mathrm{z}}{\displaystyle \frac{\mathrm{d}{\varphi }_{\mathrm{r}}}{\mathrm{d}r}}-{\lambda }_{\mathrm{s}}{r}_{\mathrm{p}}{\varphi }_{\mathrm{z}}{\displaystyle \frac{{\varphi }_{\mathrm{r}}-{\varphi }_{\mathsf{\theta }}}{r}}\}r\mathrm{d}r+\pi {\displaystyle \underset{0}{\overset{r_{\mathrm{p}}}{\int }}}\{-{\lambda }_{\mathrm{s}}{r}_{\mathrm{p}}{\varphi }_{\mathrm{z}}{\displaystyle \frac{\mathrm{d}{\varphi }_{\mathrm{r}}}{\mathrm{d}r}}+2G_{\mathrm{s}}{r}_{\mathrm{p}}{\displaystyle \frac{\mathrm{d}{\varphi }_{\mathrm{z}}}{\mathrm{d}r}}+G_{\mathrm{s}}{\displaystyle \frac{r_{\mathrm{p}}}{r}}{\varphi }_{\mathrm{z}}\}r\mathrm{d}r$; $k_{2,\mathrm{end}}=\pi {\displaystyle \underset{r_{\mathrm{p}}}{\overset{\infty }{\int }}}\left\{G_{\mathrm{s}}\right\{{\displaystyle \frac{{\varphi }_{\mathrm{z}}}{r}}{\varphi }_{\mathsf{\theta }}+{\displaystyle \frac{\mathrm{d}{\varphi }_{\mathrm{z}}}{\mathrm{d}r}}{\varphi }_{\mathrm{r}}\}{r}_{\mathrm{p}}r\mathrm{d}r+\pi {\displaystyle \underset{0}{\overset{r_{\mathrm{p}}}{\int }}}\{G_{\mathrm{s}}\{{\displaystyle \frac{{\varphi }_{\mathrm{z}}}{r}}+{\displaystyle \frac{\mathrm{d}{\varphi }_{\mathrm{z}}}{\mathrm{d}r}}\}{r}_{p}r\mathrm{d}r$.

If the pile end is considered to be fixed:

$$\begin{array}{l}{w}_{z=L_{\mathrm{p}}}=0\\ {\varphi }_{z=L_{\mathrm{p}}}=0\end{array}\left(z=L_{\mathrm{p}}\right)$$

(22)

The corresponding governing equation for the soil below the pile bottom (z > L_p) is

$$\begin{array}{l}2t_{\mathrm{end}}{\displaystyle \frac{{\partial }^{2}w}{\partial z^2}}+k_{1,\mathrm{end}}{\displaystyle \frac{\partial \varphi }{\partial z}}-kw-n_{1,\mathrm{end}}{\displaystyle \frac{{\partial }^{2}w}{\partial t^2}}=0\\ -k_{3,\mathrm{end}}\varphi -k_{1,\mathrm{end}}{\displaystyle \frac{\partial w}{\partial z}}+2t_{4,\mathrm{end}}{\displaystyle \frac{{\partial }^2\varphi }{\partial z^2}}-n_{2,\mathrm{end}}{\displaystyle \frac{{\partial }^2\varphi }{\partial t^2}}=0\end{array}$$

(23)

where $k_{3,\mathrm{end}}=\pi {\displaystyle \underset{r_{\mathrm{p}}}{\overset{\infty }{\int }}}\left\{\right[G_{\mathrm{s}}{\displaystyle \frac{\mathrm{d}{\varphi }_{\mathrm{z}}}{\mathrm{d}r}}{\displaystyle \frac{\mathrm{d}{\varphi }_{\mathrm{z}}}{\mathrm{d}r}}+G_{\mathrm{s}}{\displaystyle \frac{{\varphi }_{\mathrm{z}}}{r}}{\displaystyle \frac{{\varphi }_{\mathrm{z}}}{r}}]{r_{\mathrm{p}}}^{2}r\mathrm{d}r+\pi {\displaystyle \underset{0}{\overset{r_{\mathrm{p}}}{\int }}}\{[G_{\mathrm{s}}{\displaystyle \frac{\mathrm{d}{\varphi }_{\mathrm{z}}}{\mathrm{d}r}}{\displaystyle \frac{\mathrm{d}{\varphi }_{\mathrm{z}}}{\mathrm{d}r}}+G_{\mathrm{s}}{\displaystyle \frac{{\varphi }_{\mathrm{z}}}{r}}{\displaystyle \frac{{\varphi }_{\mathrm{z}}}{r}}]{r_{\mathrm{p}}}^{2}r\mathrm{d}r$; $t_{4,\mathrm{end}}={\displaystyle \frac{\pi }{2}}{\displaystyle \underset{r_{\mathrm{p}}}{\overset{\infty }{\int }}}({\lambda }_{\mathrm{s}}+2G_{\mathrm{s}}){{\varphi }_{\mathrm{z}}}^2{r_{\mathrm{p}}}^{2}r\mathrm{d}r+{\displaystyle \frac{\pi }{2}}{\displaystyle \underset{0}{\overset{r_{\mathrm{p}}}{\int }}}({\lambda }_{\mathrm{s}}+2G_{\mathrm{s}}){{\varphi }_{\mathrm{z}}}^2{r_{\mathrm{p}}}^{2}r\mathrm{d}r$; and $-k_{1,\mathrm{end}}+k_{2,\mathrm{end}}=\pi {\displaystyle \underset{r_{\mathrm{p}}}{\overset{\infty }{\int }}}[{\lambda }_{\mathrm{s}}{\displaystyle \frac{\mathrm{d}{\varphi }_{\mathrm{r}}}{\mathrm{d}r}}{\varphi }_{\mathrm{z}}+{\lambda }_{\mathrm{s}}{\displaystyle \frac{{\varphi }_{\mathrm{r}}-{\varphi }_{\mathsf{\theta }}}{r}}{\varphi }_{\mathrm{z}}]r_{\mathrm{p}}r\mathrm{d}r+\pi {\displaystyle \underset{0}{\overset{r_{\mathrm{p}}}{\int }}}[{\lambda }_{\mathrm{s}}{\displaystyle \frac{\mathrm{d}{\varphi }_{\mathrm{r}}}{\mathrm{d}r}}{\varphi }_{\mathrm{z}}+{\lambda }_{\mathrm{s}}{\displaystyle \frac{{\varphi }_{\mathrm{r}}-{\varphi }_{\mathsf{\theta }}}{r}}{\varphi }_{\mathrm{z}}]r_{\mathrm{p}}r\mathrm{d}r$.

Under static load conditions, the control equation for the pile body is the expression mentioned above with the ω term removed, and the calculation method can be found in the literature [33,34,35,36,37,38,39].

3.2. The Governing Equation of Soil

Based on Q(ϕ_r) from Equation (16), the governing equation of ϕ_r can be given as follows:

$${\displaystyle \frac{{\mathrm{d}}^2{\varphi }_{\mathrm{r}}}{\mathrm{d}{r}^2}}+{\displaystyle \frac{1}{r}}{\displaystyle \frac{\mathrm{d}{\varphi }_{\mathrm{r}}}{\mathrm{d}r}}-({\left({\displaystyle \frac{{\gamma }_1}{r}}\right)}^2+{\left({\displaystyle \frac{{\gamma }_2}{r_{\mathrm{p}}}}\right)}^2){\varphi }_{\mathrm{r}}={\displaystyle \frac{{\gamma }_3^2}{r}}{\displaystyle \frac{\mathrm{d}{\varphi }_{\mathsf{\theta }}}{\mathrm{d}r}}-{\left({\displaystyle \frac{{\gamma }_1}{r}}\right)}^2{\varphi }_{\mathsf{\theta }}+{\gamma }_0^2{\displaystyle \frac{\mathrm{d}{\varphi }_{\mathrm{z}}}{\mathrm{d}r}}$$

(24)

where ${\gamma }_0=\sqrt{{\displaystyle \frac{{\displaystyle \underset{0}{\overset{\infty }{\int }}}\left(G_{\mathrm{s}}\varphi r_{\mathrm{p}}\frac{\mathrm{d}W}{\mathrm{d}z}r-{\lambda }_{\mathrm{s}}\frac{\partial \varphi }{\partial z}{r}_{\mathrm{p}}Wr\right)\mathrm{d}z}{{\displaystyle \underset{0}{\overset{\infty }{\int }}}({\lambda }_{\mathrm{s}}+2G_{\mathrm{s}})WW\mathrm{d}z}}}$; ${\gamma }_1=\sqrt{{\displaystyle \frac{{\displaystyle \underset{0}{\overset{\infty }{\int }}}({\lambda }_s+3G_{\mathrm{s}})WW\mathrm{d}z}{{\displaystyle \underset{0}{\overset{\infty }{\int }}}({\lambda }_{\mathrm{s}}+2G_{\mathrm{s}})WW\mathrm{d}z}}}$; ${\gamma }_2=r_{\mathrm{p}}\sqrt{{\displaystyle \frac{{\displaystyle \underset{0}{\overset{\infty }{\int }}}{G}_{\mathrm{s}}\frac{\mathrm{d}W}{\mathrm{d}z}\frac{\mathrm{d}W}{\mathrm{d}z}\mathrm{d}z-{\omega }^2{\displaystyle \underset{0}{\overset{\infty }{\int }}}{\rho }_{\mathrm{soil}}WW\mathrm{d}z}{{\displaystyle \underset{0}{\overset{\infty }{\int }}}({\lambda }_{\mathrm{s}}+2G_{\mathrm{s}})WW\mathrm{d}z}}}$; ${\gamma }_3=\sqrt{{\displaystyle \frac{{\displaystyle \underset{0}{\overset{\infty }{\int }}}(G_{\mathrm{s}}+{\lambda }_{\mathrm{s}})WW\mathrm{d}z}{{\displaystyle \underset{0}{\overset{\infty }{\int }}}({\lambda }_{\mathrm{s}}+2G_{\mathrm{s}})WW\mathrm{d}z}}}$.

Based on R(ϕ_θ) from Equation (16), the governing equation of ϕ_θ can be given as follows:

$${\displaystyle \frac{{\mathrm{d}}^2{\varphi }_{\mathsf{\theta }}}{\mathrm{d}{r}^2}}+{\displaystyle \frac{1}{r}}{\displaystyle \frac{\mathrm{d}{\varphi }_{\mathsf{\theta }}}{\mathrm{d}r}}-{\left({\displaystyle \frac{{\gamma }_4}{r}}\right)}^2{\varphi }_{\mathsf{\theta }}-{\left({\displaystyle \frac{{\gamma }_5}{r_{\mathrm{p}}}}\right)}^2{\varphi }_{\mathsf{\theta }}=-{\displaystyle \frac{{\gamma }_6^2}{r}}{\displaystyle \frac{\mathrm{d}{\varphi }_{\mathrm{r}}}{\mathrm{d}r}}-{\left({\displaystyle \frac{{\gamma }_4}{r}}\right)}^2{\varphi }_{\mathrm{r}}+{\displaystyle \frac{{\gamma }_7^2}{r}}{\varphi }_{\mathrm{z}}$$

(25)

where ${\gamma }_4=\sqrt{{\displaystyle \frac{{\displaystyle \underset{0}{\overset{\infty }{\int }}}({\lambda }_{\mathrm{s}}+3G_{\mathrm{s}})WW\mathrm{d}z}{{\displaystyle \underset{0}{\overset{\infty }{\int }}}{G}_{\mathrm{s}}WW\mathrm{d}z}}}$; ${\gamma }_5=r_{\mathrm{p}}\sqrt{{\displaystyle \frac{{\displaystyle \underset{0}{\overset{\infty }{\int }}}{G}_{\mathrm{s}}\frac{\mathrm{d}W}{\mathrm{d}z}\frac{\mathrm{d}W}{\mathrm{d}z}\mathrm{d}z-{\omega }^2{\displaystyle \underset{0}{\overset{\infty }{\int }}}{\rho }_{\mathrm{soil}}WW\mathrm{d}z}{{\displaystyle \underset{0}{\overset{\infty }{\int }}}{G}_{\mathrm{s}}WW\mathrm{d}z}}}$; ${\gamma }_6=\sqrt{{\displaystyle \frac{{\displaystyle \underset{0}{\overset{\infty }{\int }}}(G_{\mathrm{s}}+{\lambda }_{\mathrm{s}})WW\mathrm{d}z}{{\displaystyle \underset{0}{\overset{\infty }{\int }}}{G}_{\mathrm{s}}WW\mathrm{d}z}}}$; ${\gamma }_7=\sqrt{{\displaystyle \frac{{\displaystyle \underset{0}{\overset{\infty }{\int }}}\left(G_{\mathrm{s}}\psi r_{\mathrm{p}}\frac{\mathrm{d}W}{\mathrm{d}z}-{\lambda }_{\mathrm{s}}\frac{\mathrm{d}\psi }{\mathrm{d}z}{r}_{\mathrm{p}}W\right)\mathrm{d}z}{{\displaystyle \underset{0}{\overset{\infty }{\int }}}{G}_{\mathrm{s}}WW\mathrm{d}z}}}$.

Based on S(ϕ_z) from Equation (16), the governing equation of ϕ_z can be given as follows:

$${\displaystyle \frac{{\mathrm{d}}^2{\varphi }_{\mathrm{z}}}{\mathrm{d}{r}^2}}+{\displaystyle \frac{1}{r}}{\displaystyle \frac{\mathrm{d}{\varphi }_{\mathrm{z}}}{\mathrm{d}r}}-({{\gamma }_9}^2+{\displaystyle \frac{1}{r^2}}){\varphi }_{\mathrm{z}}=-{{\gamma }_8}^2{\displaystyle \frac{\mathrm{d}{\varphi }_r}{\mathrm{d}r}}-{\displaystyle \frac{1}{r}}{{\gamma }_8}^2{\varphi }_{\mathrm{r}}+{\displaystyle \frac{1}{r}}{{\gamma }_8}^2{\varphi }_{\mathsf{\theta }}$$

(26)

where ${\gamma }_8=\sqrt{{\displaystyle \frac{{\displaystyle \underset{0}{\overset{\infty }{\int }}}\left({\lambda }_{\mathrm{s}}\frac{\mathrm{d}\psi }{\mathrm{d}r}{r}_{\mathrm{p}}W-G_{\mathrm{s}}\psi r_{\mathrm{p}}\frac{\mathrm{d}W}{\mathrm{d}z}\right)\mathrm{d}z}{{\displaystyle \underset{0}{\overset{\infty }{\int }}}{G}_{\mathrm{s}}\psi r_{\mathrm{p}}\psi r_{\mathrm{p}}\mathrm{d}z}}}$; ${\gamma }_9=\sqrt{{\displaystyle \frac{{\displaystyle \underset{0}{\overset{\infty }{\int }}}({\lambda }_{\mathrm{s}}+2G_{\mathrm{s}})\frac{\mathrm{d}\psi }{\mathrm{d}r}{r}_{\mathrm{p}}\frac{\mathrm{d}\psi }{\mathrm{d}r}{r}_{\mathrm{p}}\mathrm{d}z-{\omega }^2{\displaystyle \underset{0}{\overset{\infty }{\int }}}{\rho }_{\mathrm{soil}}\psi r_{\mathrm{p}}\psi r_{\mathrm{p}}\mathrm{d}z}{{\displaystyle \underset{0}{\overset{\infty }{\int }}}{G}_{\mathrm{s}}\psi r_{\mathrm{p}}\psi r_{\mathrm{p}}\mathrm{d}z}}}$.

The calculation methods for Equations (24)–(26) are given in Appendix A. Under static load conditions, the control equation for the soil domain is the expression obtained by removing the ω term from the above Equations (24)–(26).

3.3. Programming Method

To solve the pile–soil interaction under dynamic (or static) loading conditions, a series of linear equations need to be calculated. These equations can be translated to solving C_A = Λ⁻¹ F, where C_A is the global displacement vector, Λ is the global matrix which contains the stiffness matrix and the global mass matrix and F is the global load vector. The details about C_A, Λ⁻¹ and F are given in Appendix B. The structured approach to implementing the process in MATLAB (2021b) is presented in Figure 5. Therefore, the problem of pile–soil interaction under static and dynamic loads is essentially a matrix operation problem. Λ is mainly transformed through the following relationships:

(1) Mechanical assumptions of the pile and assumptions of soil displacement distribution (Equation (8));

(2) Constitutive models of the pile and soil (Section 2.2.3);

(3) Principles of force balance, deformation compatibility conditions and boundary conditions (Equations (17)–(23)).

4. Simplified Calculation Method of Natural Frequency

4.1. Pile Head Stiffness

After obtaining pile body displacement and rotation under the dynamic load state using the method introduced in the previous section, the pile head stiffness expression under each mode shape condition can be further obtained using Equation (27):

$$\left[\begin{array}{c}{W}_0\\ {\psi }_0\end{array}\right]={\left[\begin{array}{cc}{K}_{\mathrm{L}}& K_{\mathrm{RL}}\\ K_{\mathrm{LR}}& K_{\mathrm{R}}\end{array}\right]}^{-1}\left[\begin{array}{c}{F}_{\mathrm{a}}\\ M_{\mathrm{a}}\end{array}\right]$$

(27)

where K_L, K_R and K_LR represent the three spring stiffnesses for translation, rotation and coupling action at the mudline, respectively; W₀ and ψ₀ are the vibration modes of the pile displacement and rotation angle at z = 0, respectively.

According to the theoretical model of the cantilever beam [26], the methods for solving K_L, K_R and K_LR are as follows:

$$\left[\begin{array}{cc}{K}_{\mathrm{L}}& K_{\mathrm{LR}}\\ K_{\mathrm{RL}}& K_{\mathrm{R}}\end{array}\right]={\displaystyle \frac{1}{\left(\frac{w_1}{F_{\mathrm{a}1}}\times \frac{\mathrm{d}{w}_2/\mathrm{d}z}{M_{\mathrm{a}2}}\right)-\left(\frac{w_2}{M_{\mathrm{a}2}}\times \frac{\mathrm{d}{w}_1/\mathrm{d}z}{H_{\mathrm{a}1}}\right)}}\times \left[\begin{array}{cc}{\displaystyle \frac{\mathrm{d}{w}_2/\mathrm{d}z}{M_{\mathrm{a}2}}}& -{\displaystyle \frac{w_2}{M_{\mathrm{a}2}}}\\ -{\displaystyle \frac{\mathrm{d}{w}_1/\mathrm{d}z}{H_{\mathrm{a}1}}}& {\displaystyle \frac{w_1}{F_{\mathrm{a}1}}}\end{array}\right]$$

(28)

where w₁ represents the horizontal displacement of the pile head when only the horizontal load F_a1 is applied; dw₁/dz represents the rotation angle of the pile head when only the horizontal load F_a1 is applied; w₂ represents the horizontal displacement of the pile head when only the overturning moment M_a2 is applied; and dw²/dz² represents the rotation angle of the pile head when only the overturning moment M_a2 is applied. First, assume arbitrary values for the external force F_a1 and M_a1 = 0 and calculate the corresponding pile head displacement w₁ and rotation angle dw₁/dz according to the calculation process shown in Figure 5; then, reset and calculate the corresponding pile head displacement w₂ and rotation angle dw₂/dz when the external moment M_a2 is at an arbitrary value and F_a2 = 0, according to the calculation process shown in Figure 5. Finally, substituting the calculation results into Equation (28), the form of the pile head stiffness is obtained.

4.2. The Natural Frequency of Wind Turbines Considering Pile–Soil Interaction

In the analytical solution calculation of the natural frequency of wind turbines, the superstructure of the offshore wind turbine is equivalent to the structure shown in Figure 6, where the wind turbine and its blades are equivalent to a top mass block m_RNAJ, the body of the turbine is equivalent to a uniform cross-section rod with equivalent elastic modulus E_T, distributed mass m_T, outer diameter D_T and wall thickness t_T; the part from the connection to above the pile mudline is equivalent to a uniform cross-section rod with outer diameter D_P and wall thickness t_P; and the pile–soil interaction is equivalent to the ECS model. The calculation steps are as follows:

(1)

Calculate the natural vibration frequency f_FB when the wind turbine tower base is fixed [27,28]:

$$f_{\mathrm{FB}}={\displaystyle \frac{1}{2\pi }}{C}_{\mathrm{MP}}\sqrt{{\displaystyle \frac{3E_{\mathrm{T}}{I}_{\mathrm{T}}}{\left(m_{\mathrm{RNA}}+{\alpha }_{\mathrm{s}}{m}_{\mathrm{T}}\right)L_{\mathrm{T}}^3}}}$$

(29)

where C_MP = (1/(1 + (1 + ρ³)λ−λ))^0.5, λ = L_s/L_T, ρ = E_TI_T/(E_pI_p) and α_s varies depending on the tower section shape; here, α_s = 33/140 [27].

(2)

Calculate K_L, K_RL and K_R according to Equation (28). Use the procedure obtained from Figure 5 for the calculation, and then calculate η_L, η_RL and η_R according to the following equations:

$${\eta }_{\mathrm{L}}={\displaystyle \frac{K_{\mathrm{L}}{L_{\mathrm{p}}}^3}{E{I}_{\mathsf{\eta }}}}$$

(30)

$${\eta }_{\mathrm{R}}={\displaystyle \frac{K_{\mathrm{R}}{L}_{\mathrm{p}}}{E{I}_{\mathsf{\eta }}}}$$

(31)

$${\eta }_{\mathrm{LR}}={\displaystyle \frac{K_{\mathrm{LR}}{L_{\mathrm{p}}}^2}{E{I}_{\mathsf{\eta }}}}$$

(32)

where EI_η represents the equivalent bending stiffness of the superstructure [27,28].

(3)

Calculate the values of the coefficients C_R and C_L:

$$C_{\mathrm{R}}\left({\eta }_{\mathrm{L}},{\eta }_{\mathrm{LR}},{\eta }_{\mathrm{R}}\right)=1-{\displaystyle \frac{1}{1+0.6\left({\eta }_{\mathrm{R}}-\frac{{\eta }_{\mathrm{LR}}^2}{{\eta }_{\mathrm{L}}}\right)}}$$

(33)

$$C_{\mathrm{L}}\left({\eta }_{\mathrm{L}},{\eta }_{\mathrm{LR}},{\eta }_{\mathrm{R}}\right)=1-{\displaystyle \frac{1}{1+0.5\left({\eta }_{\mathrm{L}}-\frac{{\eta }_{\mathrm{LR}}^2}{{\eta }_{\mathrm{R}}}\right)}}$$

(34)

(4)

The final first-order natural frequency is given by

$$f_0=C_{\mathrm{L}}{C}_{\mathrm{R}}{f}_{\mathrm{FB}}$$

(35)

where f₀ represents the overall natural frequencies of the OWT system from the method proposed in this study.

Based on the previous procedures, detailed calculations and programming to calculate the pile–soil interaction under dynamic loading conditions are presented in Figure 5. If the geological survey report and wind turbine foundation parameters are known, the program can quickly adjust the design scheme for the single-pile foundation.

5. Results and Discussion

In this section, to verify the correctness of the calculation method presented in this study, 13 offshore wind projects in Europe and China from different references were collected. Based on existing cases from references [13,15,16,42,43,44,45,46,47], the values of the top mass block (m_RNAJ), the equivalent elastic modulus of the tower (E_T), the distributed mass of the tower (m_T), the outer diameter of the tower (D_T), the outer diameter of the tower at the end (D_s), the wall thickness of the tower (t_T), the wall thickness of the tower at the end (t_s), the outer diameter of the monopile (D_P), the wall thickness of the monopile (t_P) and the measured overall natural frequencies of the OWT system (f) from 13 projects are given in

Table 1. The related parameters of offshore wind turbines and the measured overall natural frequencies of the OWT system (f) from 13 projects.

No.	1	2	3	4	5	6	7	8	9	10	11	12	13
Literature	Blyth [44]	Belwind [44]	Walney [44]	NREL [44]	Kentish Flats [45]	North Hoyle [47]	Lely (A2) [42]	Irene Vorrink [42]	Barrow [42]	Thanet [42]	BurboBank [42]	Gunfleet [42]	Hua Neng [34]
Wind power (MW)	2	3	3.6	5	—	2	0.5	0.6	—	—	—	—	—
mRNAJ (t)	80	130.8	234.5	350	132	100	32	35.7	130.8	130.8	234.5	234.5	243
LT (m)	54.5	53	67.3	87.6	60.06	67.3	37.9	44.5	58	54.1	66	60	86
Ds (m)	4.25	4.3	5	6	4.45	4	3.2	3.5	4.45	4.3	5	5	5.5
DT (m)	2.75	2.3	3	3.87	1.9	2.3	1.9	1.7	2.3	2.3	3	3	3.1
ts (mm)	34	28	40	27	26	35	13	14	32	36	28	33	60
tT (mm)	34	28	40	19	15	35		8	32	36	28	33	60
Ep (GPa)	210	210	210	210	210	210	210	210	210	210	210	210	210
mT (t)	159	120	260	347.46	108	130	31.44	35.7	153	160	180	193	269.75
Ls (m)	16.5	37	37.3	20	16	7	12.1	5.2	33	41.1	22.8	28	44
Pile diameter Dp (m)	3.5	5	6	6	4.3	4	3.2	3.5	4.75	4.7	4.7	5	6.1
Wall thickness tp (mm)	50	60	80	60	45	50	35	28	45–80	65	45-75	35-50	70
Lp (m)	15	35	23.5	40	29.5	33	13.5	19	30.2–40.7	25-30	24	38	44.3
Es0 (MPa)	42.5	3.72	17.4	37.1	Multilayer soils 20~53.3	382.5	42.5	42.5	3.72	3.72	3.72	17.4	Multilayer soils 4~60
α	0	1	1	1		0	0	0	1	1	1	1
fFB (HZ)	0.514	0.401	0.38	0.592	0.38	0.364	0.713	0.583~0.586	0.387	0.402	0.322	0.352	0.388
f (HZ)	0.488	0.372	0.35	0.546/0.563	0.339	0.35	0.634~0.735	0.553~0.560	0.369	0.37	0.292	0.314	0.325

. The natural frequency of the OWT system with a fixed end (f_FB) can also be given by Equation (29) if they are not in the references.

In Table 1, the formulas presented in Section 2.2.3 were used to describe the distribution of soil elastic modulus from 13 projects. The soil type in Kentish Flats and Hua Neng can be described by Equation (14), where multilayered soil was modeled. The elastic modulus E_si from Kentish Flats ranges from 20 MPa to 53.3 MPa, the details of each soil layer can be seen in reference [45]; E_si from Hua Neng ranges from 4 MPa to 60 MPa, the details can be seen in reference [34]. The soil type in Blyth, North Hoyle, Lely (A2) and Irene Vorrink can be described by Equation (13) with α = 0 and an initial value of the elastic modulus E_s0, and the soil type in the rest of the projects can be described by Equation (13) with α = 1 and E_s0. In these projects, the contact condition of the pile and soil is regarded as completely rough [48,49], which depends on assumption (3) in Section 2.1 and the displacement distribution functions in the r-, θ- and z-directions of the soil (Equations (9)–(11)).

Based on the above parameters presented in Table 1, η_L, η_R and η_RL from 13 projects were given by Equations (30), (31) and (32), respectively, and the results are presented in

Table 2. The overall natural frequencies of OWT system from the method proposed in this study (f₀) and their errors compared with measured data.

No.	1	2	3	4	5	6	7	8	9	10	11	12	13
ηL	1983.79	2437.03	283.68	3635.19	4019.59	13739.26	324.08	644.32	2594.14	2037.40	2743.56	1757.82	442.38
ηR	199.53	25.92	148.6818	14.15	33.29	49.27	22.42	99.68	23.04	20.80	18.82	14.59	22.04
ηLR	−118.46	−149.64	−0.51183	−135.05	−217.81	−472.66	−0.77	−8.55	−145.56	−122.56	−135.28	−95.34	−75.60
CR	0.972	0.9094	0.988915	0.8457	0.9280	0.9519	0.9181	0.9974	0.8992	0.8896	0.8793	0.8496	0.8456
CL	0.965	0.9712	0.992999	0.9995	0.9995	0.9999	0.9949	0.9803	0.9992	0.9990	0.9993	0.9989	0.9955
f0 (HZ)	0.482	0.354	0.341	0.500	0.352	0.346	0.702	0.540	0.348	0.357	0.283	0.299	0.327
Error (%)	−1.2	−4.8	−2.6	−11.2~ −8.4	3.8	−1.1	−4.5~10.7	−3.6~ −2.4	−5.7	−3.5	−3.1	−4.8	0.6

. Then, the values of the coefficients C_R and C_L were also calculated by Equations (33) and (34), respectively, and results are presented in Table 2. Therefore, the overall natural frequencies of the OWT system from the method proposed in this study (f₀) were obtained by Equation (35), also presented in Table 2. To verify the correctness of the calculation method presented in this study, the error between the measured value (f) presented in Table 1 and the results from this study (f₀) is given in Table 2. The real values of the overall natural frequencies of the OWT system (f) in Kentish Flats, Irene Vorrink, Lely (A2) and Barrow were given as range values in references [42,45]. To better compare the errors in different projects, Figure 7 presents the values of the error from 13 offshore wind projects in Table 2 with the error bar, where the horizontal ordinate presents the serial number of the projects, and the longitudinal ordinate presents the measured overall natural frequencies (which can also be regarded as the real value) of the OWT system. Therefore, the results from this study fall into the range shown with statistics in Figure 7.

During calculations, we found that the computation time using the proposed method was less than 10 s (on a workstation with Intel(R) Core(TM) i7-7700HQ CPU @2.80GHz (California, United States)), demonstrating the efficiency of the algorithm in this study. As shown in Table 1 and Table 2 and Figure 7, compared with the measured overall natural frequencies of the OWT system from references, the results obtained from this study are generally smaller. This means the pile–soil interaction described by the method in this study is slightly softer than the real situation. Compared with the condition that the pile–soil interaction is regarded as an ideal fixed boundary condition, the method in this study is much safer in the design process. The statistical errors in Table 2 and Figure 7 indicate that the error in the computed results can be controlled between −5% and 5%. Specifically, for NREL and Lely (A2), the errors between the calculated results and measured values were −11.2% to −8.4% and −4.5% to 10.7%, respectively. This conclusion further confirms the accuracy and reliability of the method proposed in this study.

6. Conclusions

In the design of offshore wind turbines, accurately and efficiently quantifying the impact of pile−soil interaction on the overall natural frequency of wind turbines is important. Few studies have comprehensively reflected the influence of pile−soil interaction on the dynamic response of OWTs based on continuum theory considering the three-dimensional displacement field of soil. Therefore, by combining Hamilton’s principle and the ECS model, an efficient analytical approach for the overall natural frequencies of wind turbines was obtained in this study. Thirteen different offshore wind projects were used to verify the correctness and efficiency of the proposed method. Compared with measured data, the proposed method for analyzing the overall natural frequencies of wind turbines achieves errors generally within the range of 10%. Although the pile−soil interaction described by the method in this study is slightly softer than the real situation, it is still much safer in the situation if the pile−soil interaction is regarded as an ideal rigid foundation. The results prove that a Timoshenko beam, which considers shear deformation of the monopile, and a proper soil constitutive relationship are necessary in the method, and soil is modeled as continuous element considering that the three-dimensional displacement of soil is also important.

This computational method provides a convenient, rapid and accurate means for predicting the overall natural frequencies of offshore wind turbines. However, the current derivation does not account for the separation between the soil and the pile, and soil mechanical properties may change under long-term cyclic loading. Therefore, a factor reflecting the separation of the pile−soil interface under large lateral loads and a new soil constitutive model reflecting stiffness degradation under long-term cyclic loading is one of the future directions for this research.