Study on the Soil–Pile Interaction of Slender Piles in Multi-Layered Soil by the Variational Analysis Method

Abstract

To rapidly and precisely calculate the slender pile response and the soil resistance under lateral load in offshore wind power projects, the energy-based variational method was used to improve the calculation method of laterally loaded slender piles in layered elastic soil. Firstly, two cases were collected, the results were compared with that from the m-method, and numerical analysis or measured data were used to verify the accuracy of the improved method in this paper. Then, from the field test of a steel pipe pile in an offshore wind power project, the results from the improved method were compared with those from the numerical analysis and measured data, and the lateral soil resistance considering the layered shear effect and the distributed moment around the pile were discussed. The results show that the improved method in this paper can rapidly and precisely calculate the slender pile response under lateral load, and there is a 9~20% difference in the lateral soil resistance if the shear effect between each soil layer is considered. The ratio of the total distributed moment around the pile and the overturning moment at the pile head is less than 2%, and the resistance coefficient of m from this method provides the lower bound solution of the measured value.

1. Introduction

In the foundation design of offshore platforms and offshore wind power projects, the horizontal bearing capacity of the pile foundation is a key controlling factor which directly affects the foundation design scheme. It is of great engineering significance to clarify the pile–soil interaction under horizontal load [1,2,3]. At present, the commonly used calculation methods include the constant method, the m-method, the comprehensive stiffness principle, and the two-parameter method [4], among which the m-method has been compiled into relevant norms in China [5,6]. The development details of the m-method and other traditional methods are given in

Table 1. The development details of the m-method and other traditional methods.

Methods	Horizontal Soil Resistance Expression	Initial Stiffness Expression	Remarks
Constant method	$p=kw$	k is a constant.	Based on the distributed spring method, it cannot reflect the layered shear effect or the distributed moment around the pile.
m-method	$p=mzw$	k = mz. m is a constant; k increases linearly with soil depth.
C-method	$p=C{z}^{{\displaystyle \frac{1}{2}}}w$	k = Cz0.5, m is a constant; k increases linearly with soil depth.
K-method	--	k remains a constant below the first elastic zero point, while above this point, it decreases along a concave curve.
Comprehensive stiffness method	p = K(z)w	k = mzi, m is a constant, i > 0. The shape of the distribution curve of k along z is controlled by the different values of i.
Multi-parameter method	p = Kbwl	k = Kb, K = z0 + mz1/n or m(z0 + z)1/n. m and n are the deformation coefficient and exponent, b is the equivalent width of the foundation, and z0 is the equivalent depth. z0 = 0, n is a constant; when m is not a constant, it is referred to as the single-parameter method. z0 is a constant; when n is not a constant and m is also not a constant, it is referred to as the two-parameter method; when all three are not constants, it is referred to as the parameter method.
The method in this paper	$p(z,w)=-2t{\displaystyle \frac{{\mathrm{d}}^{2}w}{\mathrm{d}{z}^2}}+kw$	t and k refer to the following text.	Based on the continuum theory, it can comprehensively reflect the layered shear effect and the distributed moment around the pile.

. However, the above methods cannot comprehensively account for the effects of interlayer shear interactions and additional moments on the pile side. And these specifications are inadequate to determine the vertical bearing capacity of large-diameter piles (exceeding 2 m) [7,8]. On the other hand, the method of determining bearing capacity by a field test has the problem of difficult loading, as well as being time-consuming and laborious [9,10,11]. For this reason, Sun et al. [12] used the variational method based on the energy principle to solve the pile response under horizontal loads. Li et al. [13,14,15,16] presented an extensive usage of the energy-based variational method in laterally loaded deep foundations with very large diameters. Salgado et al. [17,18] and Basu et al. [19] improved the diffusion function expression of pile displacement in soil. Li et al. [4] proposed a simplified method for estimating the interface mechanical behavior of monopiles under initial lateral loads. Han et al. [20,21,22] further proposed a calculation method for laterally loaded piles in multi-layered nonlinear soil. Teja et al. [23] used laboratory model tests to understand the behavior of individual piles and pile groups under lateral loads in uniform sand and stratified soil. Compared with the traditional discrete element method, the greatest advantage of the variational method based on the energy principle is that the pile response and soil resistance distribution can be obtained without a field test, with only basic pile–soil parameters.

In this paper, the theoretical system proposed by Salgado et al. [17,18], Basu et al. [19], and Gupta et al. [24] is optimized, and the pile response of a slender pile in multi-layer soil under horizontal load is analyzed. By comparing the results with the numerical simulation and measured data, the rationality of this method is verified, and the influence of interlayer interaction on the horizontal soil resistance and the pile’s additional moment is discussed. The purpose of this study is to provide a more efficient approach to calculate the lateral response of large-diameter, long, slender piles in offshore wind farm projects.

2. Calculation Principle of Variational Method

2.1. Pile–Soil Interaction Model

The large-diameter, long, slender pile (the pile’s length-to-diameter ratio is greater than 10, and the diameter is larger than 1.5 m) appears in offshore wind power projects in the form of a high-pile-cap pile group. This paper reasonably simplifies the pile–soil interaction system of a single slender pile and derives the mechanical model in the r-θ-z coordinate system. The assumptions of this mechanical model are as follows:

• The pile is modeled as the Euler–Bernoulli Beam.
• The soil within each soil layer is a uniform, isotropic, ideal elastic material.
• The pile and the soil maintain contact throughout the deformation process.

For Hypothesis 1: According to classical mechanics theory, flexural deformation of the horizontal load-bearing, long, slender pile is dominant, while shear deformation is weak. Therefore, it is reasonable to choose the Euler–Bernoulli Beam.

According to the principle of conservation of energy [19], under the action of the horizontal force F_a and moment M_a at the pile head, the total energy of the pile–soil interaction system is:

$$П=E_{\mathrm{p}}{I}_{\mathrm{p}}{\displaystyle \underset{0}{\overset{L_{\mathrm{p}}}{\int }}(\frac{{\mathrm{d}}^{2}w\left(z\right)}{\mathrm{d}{z}^2}}{)}^2\mathrm{d}z+{\displaystyle \underset{\mathsf{\Omega }}{\iiint }{\sigma }_{pq}{\epsilon }_{pq}r\mathrm{d}r\mathrm{d}\theta \mathrm{d}z}-F_{\mathrm{a}}{w}_0+M_{\mathrm{a}}{\displaystyle \frac{\mathrm{d}{w}_0}{\mathrm{d}z}}=0$$

(1)

where E_p is the elastic modulus of the pile; I_p is the moment of inertia of the cross-section; L_p is the length of the pile; and w(z) is the horizontal displacement of the pile. When w₀ is the horizontal displacement of the pile at z = 0, d_w0/d_z is the rotation angle of the pile at the soil surface at z = 0; σ_pq and ε_pq are the stress and strain tensors of the soil, respectively, where p = r, θ, z, q = r, θ, z (σ_pq and ε_pq are tensors, representing 3 × 3 matrices); and Ω is the integration region of the soil.

Assuming that the soil is an ideal linear elasticity, the bulk modulus K_si of the i-th soil layer is given by K_si = E_siν_si/(1 + ν_si)(1 − 2ν_si), and the shear modulus G_si is given by G_si = E_si/2/(1 + ν_si), where E_si and ν_si are the elastic modulus and Poisson’s ratio of the i-th soil layer, respectively.

Based on the assumptions of the mechanical model, the relationship expression between the horizontal displacement of the pile and the displacement of the soil is obtained as:

$$\begin{array}{l}{s}_r=w(z){\varphi }_r(r)\cos \theta \\ s_{\theta }=-w(z){\varphi }_{\theta }(r)\sin \theta \\ s_z=0\end{array}$$

(2)

where s_r, s_θ, and s_z are the displacements of the soil in the r-, θ-, and z-directions, respectively. ϕ_r(r) and ϕ_θ(r) are the transfer functions of w(z) in the r and θ directions, respectively.

Based on the principles of virtual work and minimum potential energy, Equation (1) can be expressed as:

$$\mathsf{\delta }П=E_{\mathrm{p}}{I}_{\mathrm{p}}{\displaystyle \underset{0}{\overset{L_{\mathrm{p}}}{\int }}(\frac{{\mathrm{d}}^{2}w\left(z\right)}{\mathrm{d}{z}^2}})\mathsf{\delta }({\displaystyle \frac{{\mathrm{d}}^{2}w\left(z\right)}{\mathrm{d}{z}^2}})\mathrm{d}z+{\displaystyle \underset{\mathsf{\Omega }}{\iiint }{\sigma }_{pq}\mathsf{\delta }{\epsilon }_{pq}r\mathrm{d}r\mathrm{d}\theta \mathrm{d}z}-F_{\mathrm{a}}\mathsf{\delta }{w}_0+M_{\mathrm{a}}\mathsf{\delta }({\displaystyle \frac{\mathrm{d}{w}_0}{\mathrm{d}z}})=0$$

(3)

Based on Equations (1) and (2), σ_pq and ε_pq in Equation (3) can be transformed into functions of w(z), ϕ_r(r), and ϕ_θ(r). In Equation (3), if δП = 0 holds true under any infinitesimal horizontal displacement of the pile, the mechanical expression of pile–soil interaction is:

$$E_{\mathrm{p}}{I}_{\mathrm{p}}{\displaystyle \frac{{\mathrm{d}}^4{w}_i}{\mathrm{d}{z}^4}}-2t_i{\displaystyle \frac{{\mathrm{d}}^2{w}_i}{\mathrm{d}{z}^2}}+k_i{w}_i=0$$

(4)

In Equation (4),

$$t_i=\left\{\begin{array}{c}{\displaystyle \frac{\pi }{2}}{G}_{\mathrm{s}i}\left[{\displaystyle \underset{r_p}{\overset{\infty }{\int }}({\varphi }_r^2(r)+{\varphi }_{\theta }^2(r))r\mathrm{d}r}\right]\begin{array}{cc}& i=1,2,\dots ,n\end{array}\\ t_n+{\displaystyle \frac{\pi }{2}}{G}_{\mathrm{s}n}{r}_{\mathrm{p}}^2\begin{array}{cc}& i=n+1\end{array}\end{array}\right.$$

(5)

$$\begin{array}{l}{k}_i=\pi (K_{\mathrm{s}i}+2G_{\mathrm{s}i}){\displaystyle \underset{r_{\mathrm{p}}}{\overset{\infty }{\int }}r{\left(\frac{\mathrm{d}{\varphi }_r(r)}{\mathrm{d}r}\right)}^2\mathrm{d}r}+\pi G_{\mathrm{s}i}{\displaystyle \underset{r_{\mathrm{p}}}{\overset{\infty }{\int }}r{\left(\frac{\mathrm{d}{\varphi }_{\theta }(r)}{\mathrm{d}r}\right)}^2\mathrm{d}r}+2\pi G_{\mathrm{s}i}{\displaystyle \underset{r_{\mathrm{p}}}{\overset{\infty }{\int }}\left({\varphi }_r(r)-{\varphi }_{\theta }(r)\right)\frac{\mathrm{d}{\varphi }_r(r)}{\mathrm{d}r}\mathrm{d}r}\\ +2\pi G_{\mathrm{s}i}{\displaystyle \underset{r_{\mathrm{p}}}{\overset{\infty }{\int }}\left({\varphi }_r(r)-{\varphi }_{\theta }(r)\right)\frac{\mathrm{d}{\varphi }_{\theta }(r)}{\mathrm{d}r}\mathrm{d}r}+\pi (K_{\mathrm{s}i}+3G_{\mathrm{s}i}){\displaystyle \underset{r_{\mathrm{p}}}{\overset{\infty }{\int }}\frac{1}{r}{\left({\varphi }_r(r)-{\varphi }_{\theta }(r)\right)}^2\mathrm{d}r}\end{array}$$

(6)

where w_i is the horizontal displacement of the pile at z = h_i; h_i is the height from the ground surface to the i-th soil layer; t_i is the initial stiffness of the shear deformation resistance of the i-th soil layer; and k_i is the initial stiffness of the compressive deformation resistance of the i-th soil layer.

The expression for the displacement diffusion law of the soil around the pile is:

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

(7)

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

(8)

In the Equations (7) and (8),

$${\gamma }_1=\sqrt{{\displaystyle \frac{{\displaystyle \sum _{i=1}^n(K_{\mathrm{s}i}+3G_{\mathrm{s}i}){\displaystyle \underset{{\tilde{h}}_{i-1}}{\overset{{\tilde{h}}_i}{\int }}{\tilde{w}}_i^2\mathrm{d}\tilde{z}}+(K_{\mathrm{s}n}+3G_{\mathrm{s}n})\sqrt{\frac{{\tilde{t}}_{n+1}}{2{\tilde{k}}_n}}{\left.{\tilde{w}}_n^2\right|}_{\tilde{z}=1}}}{{\displaystyle \sum _{i=1}^n(K_{\mathrm{s}i}+2G_{\mathrm{s}i}){\displaystyle \underset{{\tilde{h}}_{i-1}}{\overset{{\tilde{h}}_i}{\int }}{\tilde{w}}_i^2\mathrm{d}\tilde{z}}+(K_{\mathrm{s}n}+2G_{\mathrm{s}n})\sqrt{\frac{{\tilde{t}}_{n+1}}{2{\tilde{k}}_n}}{\left.{\tilde{w}}_n^2\right|}_{\tilde{z}=1}}}}}$$

(9)

$${\gamma }_2=\sqrt{{\displaystyle \frac{1}{{\psi }^2}}{\displaystyle \frac{{\displaystyle \sum _{i=1}^n{G}_{\mathrm{s}i}{\displaystyle \underset{{\tilde{h}}_{i-1}}{\overset{{\tilde{h}}_i}{\int }}{\left(\frac{{\tilde{w}}_i}{\mathrm{d}\tilde{z}}\right)}^2\mathrm{d}\tilde{z}}+G_{\mathrm{s}n}\sqrt{\frac{{\tilde{k}}_n}{8{\tilde{t}}_{n+1}}}{\left.{\tilde{w}}_n^2\right|}_{\tilde{z}=1}}}{{\displaystyle \sum _{i=1}^n(K_{\mathrm{s}i}+2G_{\mathrm{s}i}){\displaystyle \underset{{\tilde{h}}_{i-1}}{\overset{{\tilde{h}}_i}{\int }}{\tilde{w}}_i^2\mathrm{d}\tilde{z}}+(K_{\mathrm{s}n}+2G_{\mathrm{s}n})\sqrt{\frac{{\tilde{t}}_{n+1}}{2{\tilde{k}}_n}}{\left.{\tilde{w}}_n^2\right|}_{\tilde{z}=1}}}}}$$

(10)

$${\gamma }_3=\sqrt{{\displaystyle \frac{{\displaystyle \sum _{i=1}^n(K_{\mathrm{s}i}+G_{\mathrm{s}i}){\displaystyle \underset{{\tilde{h}}_{i-1}}{\overset{{\tilde{h}}_i}{\int }}{\tilde{w}}_i^2\mathrm{d}\tilde{z}}+(K_{\mathrm{s}n}+G_{\mathrm{s}n})\sqrt{\frac{{\tilde{t}}_{n+1}}{2{\tilde{k}}_n}}{\left.{\tilde{w}}_n^2\right|}_{\tilde{z}=1}}}{{\displaystyle \sum _{i=1}^n(K_{\mathrm{s}i}+2G_{\mathrm{s}i}){\displaystyle \underset{{\tilde{h}}_{i-1}}{\overset{{\tilde{h}}_i}{\int }}{\tilde{w}}_i^2\mathrm{d}\tilde{z}}+(K_{\mathrm{s}n}+2G_{\mathrm{s}n})\sqrt{\frac{{\tilde{t}}_{n+1}}{2{\tilde{k}}_n}}{\left.{\tilde{w}}_n^2\right|}_{\tilde{z}=1}}}}}$$

(11)

$${\gamma }_4=\sqrt{{\displaystyle \frac{{\displaystyle \sum _{i=1}^n(K_{\mathrm{s}i}+3G_{\mathrm{s}i}){\displaystyle \underset{{\tilde{h}}_{i-1}}{\overset{{\tilde{h}}_i}{\int }}{\tilde{w}}_i^2\mathrm{d}\tilde{z}}+(K_{\mathrm{s}n}+3G_{\mathrm{s}n})\sqrt{\frac{{\tilde{t}}_{n+1}}{2{\tilde{k}}_n}}{\left.{\tilde{w}}_n^2\right|}_{\tilde{z}=1}}}{{\displaystyle \sum _{i=1}^n{G}_{\mathrm{s}i}{\displaystyle \underset{{\tilde{h}}_{i-1}}{\overset{{\tilde{h}}_i}{\int }}{\tilde{w}}_i^2\mathrm{d}\tilde{z}}+G_{\mathrm{s}n}\sqrt{\frac{{\tilde{t}}_{n+1}}{2{\tilde{k}}_n}}{\left.{\tilde{w}}_n^2\right|}_{\tilde{z}=1}}}}}$$

(12)

$${\gamma }_5=\sqrt{{\displaystyle \frac{1}{{\psi }^2}}{\displaystyle \frac{{\displaystyle \sum _{i=1}^n{G}_{\mathrm{s}i}{\displaystyle \underset{{\tilde{h}}_{i-1}}{\overset{{\tilde{h}}_i}{\int }}{\left(\frac{{\tilde{w}}_i}{\mathrm{d}\tilde{z}}\right)}^2\mathrm{d}\tilde{z}}+G_{\mathrm{s}n}\sqrt{\frac{{\tilde{k}}_n}{8{\tilde{t}}_{n+1}}{\left.{\tilde{w}}_n^2\right|}_{\tilde{z}=1}}}}{{\displaystyle \sum _{i=1}^n{G}_{\mathrm{s}i}{\displaystyle \underset{{\tilde{h}}_{i-1}}{\overset{{\tilde{h}}_i}{\int }}{\tilde{w}}_i^2\mathrm{d}\tilde{z}}+K_n\sqrt{\frac{{\tilde{t}}_{n+1}}{2{\tilde{k}}_n}{\left.{\tilde{w}}_n^2\right|}_{\tilde{z}=1}}}}}}$$

(13)

$${\gamma }_6=\sqrt{{\displaystyle \frac{{\displaystyle \sum _{i=1}^n(K_{\mathrm{s}i}+G_{\mathrm{s}i}){\displaystyle \underset{{\tilde{h}}_{i-1}}{\overset{{\tilde{h}}_i}{\int }}{\tilde{w}}_i^2\mathrm{d}\tilde{z}}+(K_{\mathrm{s}n}+G_{\mathrm{s}n})\sqrt{\frac{{\tilde{t}}_{n+1}}{2{\tilde{k}}_n}}{\left.{\tilde{w}}_n^2\right|}_{\tilde{z}=1}}}{{\displaystyle \sum _{i=1}^n{G}_{\mathrm{s}i}{\displaystyle \underset{{\tilde{h}}_{i-1}}{\overset{{\tilde{h}}_i}{\int }}{\tilde{w}}_i^2\mathrm{d}\tilde{z}}+G_{\mathrm{s}n}\sqrt{\frac{{\tilde{t}}_{n+1}}{2{\tilde{k}}_n}}{\left.{\tilde{w}}_n^2\right|}_{\tilde{z}=1}}}}}$$

(14)

where ${\tilde{w}}_i$ = w_i/L_p; $\tilde{z}$ = z/L_p; ${\tilde{t}}_i$ = t_iL_p²/(E_pI_p); ${\tilde{k}}_i$ = k_iL_p⁴/(E_pI_p); r_p is the radius of the pile’s cross-section.

2.2. Calculation of Horizontal Displacement of Pile

By solving Equation (4), the following results can be obtained:

$$w(z)={\mathrm{A}}_1\sinh a\tilde{z}\cos b\tilde{z}+{\mathrm{A}}_2\cosh a\tilde{z}\cos b\tilde{z}+{\mathrm{A}}_3\cosh a\tilde{z}\sin b\tilde{z}+{\mathrm{A}}_4\sinh a\tilde{z}\sin b\tilde{z}$$

(15)

where $a=\sqrt{{\displaystyle \frac{1}{2}}(\sqrt{\tilde{k}}+\tilde{t})}$; $b=\sqrt{{\displaystyle \frac{1}{2}}(\sqrt{\tilde{k}}-\tilde{t})}$. A₁, A₂, A₃, and A₄ are undetermined coefficients. These coefficients can be determined by solving a system of 4n equations, which is established based on the boundary conditions of a free pile head and a free pile tip, as well as the continuity conditions of displacement and internal forces at each segment of the pile [17,18,19,24].

2.3. Calculation of Transfer Function

By transforming the first-order and second-order derivatives in Equations (5) and (6) into difference expressions, two sets of n equations can be obtained: one set expresses ϕ_r(r) in terms of ϕ_θ(r), and the other set expresses ϕ_θ(r) in terms of ϕ_r(r). The decoupled equations are obtained:

$${\varphi }_r={\left(K_{{\varphi }_r}-\left(B{K}_{{\varphi }_{\theta }}^{-1}\right)D\right)}^{-1}\left(C+\left(B{K}_{{\varphi }_{\theta }}^{-1}\right)E\right)$$

(16)

$${\varphi }_{\theta }=K_{{\varphi }_{\theta }}^{-1}\left(E+D{\varphi }_r\right)$$

(17)

In Equation (16),

$$B={\left[\begin{array}{ccccccccc}0& & & & & & & & \\ & -{\left({\displaystyle \frac{{\gamma }_1}{r_2}}\right)}^2& {\displaystyle \frac{{\gamma }_3^2}{2r_2\mathsf{\Delta }r}}& & & & & & \\ & -{\displaystyle \frac{{\gamma }_3^2}{2r_3\mathsf{\Delta }r}}& -{\left({\displaystyle \frac{{\gamma }_1}{r_3}}\right)}^2& {\displaystyle \frac{{\gamma }_3^2}{2r_3\mathsf{\Delta }r}}& & & & & \\ & & .& .& .& & & & \\ & & & .& .& .& & & \\ & & & & .& .& .& & \\ & & & & & -{\displaystyle \frac{{\gamma }_3^2}{2r_{m-2}\mathsf{\Delta }r}}& -{\left({\displaystyle \frac{{\gamma }_1}{r_{m-2}}}\right)}^2& {\displaystyle \frac{{\gamma }_3^2}{2r_{m-2}\mathsf{\Delta }r}}& \\ & & & & & & -{\displaystyle \frac{{\gamma }_3^2}{2r_{m-1}\mathsf{\Delta }r}}& -{\left({\displaystyle \frac{{\gamma }_1}{r_{m-1}}}\right)}^2& \\ & & & & & & & & 0\end{array}\right]}_{m\times m}$$

(18)

$${C={\left[\begin{array}{ccccc}1& -{\displaystyle \frac{1}{\mathsf{\Delta }{r}^2}}+{\displaystyle \frac{1}{2r_2\mathsf{\Delta }r}}-{\displaystyle \frac{{\gamma }_3^2}{2r_2\mathsf{\Delta }r}}& 0& \dots & 0\end{array}\right]}^{\mathrm{T}}}_{m\times 1}$$

(19)

$$D={\left[\begin{array}{ccccccccc}0& & & & & & & & \\ & -{\left({\displaystyle \frac{{\gamma }_4}{r_2}}\right)}^2& -{\displaystyle \frac{{\gamma }_6^2}{2r_2\mathsf{\Delta }r}}& & & & & & \\ & {\displaystyle \frac{{\gamma }_6^2}{2r_3\mathsf{\Delta }r}}& -{\left({\displaystyle \frac{{\gamma }_4}{r_3}}\right)}^2& -{\displaystyle \frac{{\gamma }_6^2}{2r_3\mathsf{\Delta }r}}& & & & & \\ & & .& .& .& & & & \\ & & & .& .& .& & & \\ & & & & .& .& .& & \\ & & & & & {\displaystyle \frac{{\gamma }_6^2}{2r_{m-2}\mathsf{\Delta }r}}& -{\left({\displaystyle \frac{{\gamma }_4}{r_{m-2}}}\right)}^2& -{\displaystyle \frac{{\gamma }_4^2}{2r_{m-2}\mathsf{\Delta }r}}& \\ & & & & & & {\displaystyle \frac{{\gamma }_6^2}{2r_{m-1}\mathsf{\Delta }r}}& -{\left({\displaystyle \frac{{\gamma }_4}{r_{m-1}}}\right)}^2& \\ & & & & & & & & 0\end{array}\right]}_{m\times m}$$

(20)

$${E={\left[\begin{array}{ccccc}1& -{\displaystyle \frac{1}{\mathsf{\Delta }{r}^2}}+{\displaystyle \frac{1}{2r_2\mathsf{\Delta }r}}+{\displaystyle \frac{{\gamma }_6^2}{2r_2\mathsf{\Delta }r}}& 0& \dots & 0\end{array}\right]}^{\mathrm{T}}}_{m\times 1}$$

(21)

$$K_{\varphi r}={\left[\begin{array}{ccccccccc}1& & & & & & & & \\ & -{\displaystyle \frac{2}{\mathsf{\Delta }{r}^2}}-\left[{\left({\displaystyle \frac{{\gamma }_1}{r_2}}\right)}^2+{\left({\displaystyle \frac{{\gamma }_2}{r_{\mathrm{p}}}}\right)}^2\right]& {\displaystyle \frac{1}{\mathsf{\Delta }{r}^2}}+{\displaystyle \frac{1}{2r_2\mathsf{\Delta }r}}& & & & & & \\ & {\displaystyle \frac{1}{\mathsf{\Delta }{r}^2}}-{\displaystyle \frac{1}{2r_3\mathsf{\Delta }r}}& -{\displaystyle \frac{2}{\mathsf{\Delta }{r}^2}}-\left[{\left({\displaystyle \frac{{\gamma }_1}{r_3}}\right)}^2+{\left({\displaystyle \frac{{\gamma }_2}{r_{\mathrm{p}}}}\right)}^2\right]& {\displaystyle \frac{1}{\mathsf{\Delta }{r}^2}}+{\displaystyle \frac{1}{2r_3\mathsf{\Delta }r}}& & & & & \\ & & .& .& .& & & & \\ & & & .& .& .& & & \\ & & & & .& .& .& & \\ & & & & & {\displaystyle \frac{1}{\mathsf{\Delta }{r}^2}}-{\displaystyle \frac{1}{2r_{m-2}\mathsf{\Delta }r}}& -{\displaystyle \frac{2}{\mathsf{\Delta }{r}^2}}-\left[{\left({\displaystyle \frac{{\gamma }_1}{r_{m-2}}}\right)}^2+{\left({\displaystyle \frac{{\gamma }_2}{r_{\mathrm{p}}}}\right)}^2\right]& {\displaystyle \frac{1}{\mathsf{\Delta }{r}^2}}+{\displaystyle \frac{1}{2r_{m-2}\mathsf{\Delta }r}}& \\ & & & & & & {\displaystyle \frac{1}{\mathsf{\Delta }{r}^2}}-{\displaystyle \frac{1}{2r_{m-1}\mathsf{\Delta }r}}& -{\displaystyle \frac{2}{\mathsf{\Delta }{r}^2}}-\left[{\left({\displaystyle \frac{{\gamma }_1}{r_{m-1}}}\right)}^2+{\left({\displaystyle \frac{{\gamma }_2}{r_{\mathrm{p}}}}\right)}^2\right]& \\ & & & & & & & & 1\end{array}\right]}_{m\times m}$$

(22)

$$K_{\varphi \theta }={\left[\begin{array}{ccccccccc}1& & & & & & & & \\ & -{\displaystyle \frac{2}{\mathsf{\Delta }{r}^2}}-\left[{\left({\displaystyle \frac{{\gamma }_4}{r_2}}\right)}^2+{\left({\displaystyle \frac{{\gamma }_5}{r_{\mathrm{p}}}}\right)}^2\right]& {\displaystyle \frac{1}{\mathsf{\Delta }{r}^2}}+{\displaystyle \frac{1}{2r_2\mathsf{\Delta }r}}& & & & & & \\ & {\displaystyle \frac{1}{\mathsf{\Delta }{r}^2}}-{\displaystyle \frac{1}{2r_3\mathsf{\Delta }r}}& -{\displaystyle \frac{2}{\mathsf{\Delta }{r}^2}}-\left[{\left({\displaystyle \frac{{\gamma }_4}{r_3}}\right)}^2+{\left({\displaystyle \frac{{\gamma }_5}{r_{\mathrm{p}}}}\right)}^2\right]& {\displaystyle \frac{1}{\mathsf{\Delta }{r}^2}}+{\displaystyle \frac{1}{2r_3\mathsf{\Delta }r}}& & & & & \\ & & .& .& .& & & & \\ & & & .& .& .& & & \\ & & & & .& .& .& & \\ & & & & & {\displaystyle \frac{1}{\mathsf{\Delta }{r}^2}}-{\displaystyle \frac{1}{2r_{m-2}\mathsf{\Delta }r}}& -{\displaystyle \frac{2}{\mathsf{\Delta }{r}^2}}-\left[{\left({\displaystyle \frac{{\gamma }_4}{r_{n-2}}}\right)}^2+{\left({\displaystyle \frac{{\gamma }_5}{r_{\mathrm{p}}}}\right)}^2\right]& {\displaystyle \frac{1}{\mathsf{\Delta }{r}^2}}+{\displaystyle \frac{1}{2r_{m-2}\mathsf{\Delta }r}}& \\ & & & & & & {\displaystyle \frac{1}{\mathsf{\Delta }{r}^2}}-{\displaystyle \frac{1}{2r_{m-1}\mathsf{\Delta }r}}& -{\displaystyle \frac{2}{\mathsf{\Delta }{r}^2}}-\left[{\left({\displaystyle \frac{{\gamma }_4}{r_{n-1}}}\right)}^2+{\left({\displaystyle \frac{{\gamma }_5}{r_{\mathrm{p}}}}\right)}^2\right]& \\ & & & & & & & & 1\end{array}\right]}_{m\times m}$$

(23)

where r₁, r₂, ….r_m, ∆r are the distances from the pile axis to the 1st, 2nd, …, m-th nodes, respectively, and m is the number of nodes in the soil region divided into grids in the r- direction, where r₁ = r_p.

2.4. Iterative Solution Process

The pile–soil-related parameters are input, and initial values are assigned to γ₁~γ₆. By using MATLAB software (R2023b) for iterative computation, the results can be obtained efficiently. During the iteration process, when the error between the old and new values of γ₁~γ₆ is less than 1 × 10⁻⁴, the new values can be considered the true solution. The true solution can be substituted back into Equations (15)–(23) to obtain the soil displacement and pile displacement.

Compared to previous studies [17,18,19,20,21,22,24], the advantages of the solution method in this paper are as follows:

• The expression for the pile-tip boundary condition is improved, making it more consistent with the characteristics of long slender piles.
• The iterative calculation process is simplified, enhancing the computational efficiency.

2.5. Pile–Soil Interaction Analysis

The expression of soil resistance of the m-method in a discrete medium system is:

$$p\left(z,w\right)=mzw\left(z\right)$$

(24)

where m is the horizontal resistance coefficient of the foundation.

The horizontal soil resistance of the pile shaft in the i-th soil layer, obtained by the improved method in this paper, is:

$$p_i(z,w)=-2t_i{\displaystyle \frac{{\mathrm{d}}^2{w}_i}{\mathrm{d}{z}^2}}+k_i{w}_i$$

(25)

According to the basic definition of the shaft resistance f_i, the calculation formula is:

$$f_i={\displaystyle \frac{\mathrm{d}}{\mathrm{d}z}}{\displaystyle \underset{\mathrm{A}}{\iint }{\sigma }_{rz}\mathrm{dA}}=0$$

(26)

where A represents the unit length side surface area of the pile and σ_rz denotes the shear stress on the pile side. The calculated shaft resistance is 0.

Based on the basic definition of the additional moment on the pile side, the additional moment of the pile side in the i-th soil layer is:

$$M_i={\displaystyle \frac{\mathrm{d}}{\mathrm{d}z}}{\displaystyle \underset{A}{\iint }{\sigma }_{rz}{r_{\mathrm{p}}}^2\cos \theta \mathrm{d}\theta \mathrm{d}z}=-\pi {r_{\mathrm{p}}}^2{G}_{\mathrm{s}i}{\displaystyle \frac{\mathrm{d}{w}_i}{\mathrm{d}z}}$$

(27)

From this, it can be concluded that the additional moment on the pile side is related to the soil elastic modulus E_s, Poisson’s ratio V_s, pile radius r_p, and dw_i/dz, but is independent of the soil distribution function.

3. Example Verification

3.1. Example 1

To verify the accuracy of the method proposed in this paper, the horizontal displacement calculation results of the pile are compared with those obtained from the finite element method, the method from the reference [12], and the m-method. The value is described as follows: L_p = 40 m, D = 1.7 m, E_p = 25 GPa, F_a = 3000 kN, M_a = 0 kN·m. The relevant calculation parameters of soil mass are shown in

Table 2. Soil-related calculation parameters, adapted from [12].

Soil Layer	Layer Thickness/m	Elastic Modulus Es/MPa	Poisson’s Ratio vs
1	1.5	20	0.35
2	2.0	25	0.30
3	5.0	40	0.25
4	32	80	0.20

, and the comparison results are shown in Figure 1.

As shown in Figure 1, the calculation results of the method in this paper are in good agreement with the finite element simulation results, and the results of the m-method are closer to those obtained using the method in reference [12]. At z = 0 m, the pile head displacement calculated using the method described in this paper is 7.9 mm, while the numerical simulation gives the same result of 7.9 mm. However, Sun [12] and the m-method result in 5.9 mm. The method in reference [12] and the m-method only consider displacement along the horizontal loading direction, with 0 displacement in the direction perpendicular to F_a, which leads to an overestimation of the soil stiffness and an underestimation of the horizontal displacement of the pile body. Therefore, the results obtained by the method in this paper are more reasonable.

3.2. Example 2

According to the field test results of prefabricated pipe piles, McClelland and Focht [19] studied the pile response of horizontally loaded slender piles. The parameter values were as follows: L_p = 23 m, D = 0.71 m, E_p = 68.42 GPa, F_a = 300 kN, M_a = −265 kN. The relevant calculation parameters of soil mass are shown in

Table 3. Soil-related calculation parameters, adapted from [5].

Soil Layer	Layer Thickness/m	Elastic Modulus Es/MPa	Poisson’s Ratio vs
1	2.0	1.6	0.3
2	4.0	4.8	0.3
3	4.0	8.0	0.3
4	7.5	14.0	0.3

. The relevant soil calculation parameters are shown in Table 2. Based on the measured pile top displacement data combined with the “Code for Pile Foundation of Port Engineering 2012” [5] and the “Code for Pile Foundation Testing 2014” [6], the pile displacement curve obtained using the m-method was derived.

It can be seen from Figure 2 that the horizontal displacement curve of the pile obtained by this method is consistent with the measured data and the results of the m-method, which further proves the rationality of this method. At z = 0 m, the pile head displacement calculated using the method proposed in this paper is 18.7 mm, and result of the measured data is 19.8 mm, while the result from the m-method is 20.5 mm. The m-method only considers horizontal soil resistance and neglects the lateral soil resistance considering the layered shear effect and the distributed moment around the pile, leading to an overestimation of the horizontal displacement of the pile. This example further demonstrates the rationality and reliability of the improved method proposed in this paper for calculating the horizontal displacement of the pile.

4. Engineering Applications

4.1. Relevant Parameters

In the field test of an offshore wind power project [25], L_p = 69 m, D = 1.7 m, Ep = 208 GPa, and the pile wall thickness was 0.03 m. The soil parameters are shown in

Table 4. Related parameters of soil layer, adapted from [13,25].

Soil Description	Layer Thickness/m	Elastic Modulus Es/MPa	Poisson’s Ratio
Mucky silty clay	1.5	3.8	0.41
Mucky clay	6.8	4.5	0.41
clay	14.4	5.9	0.35
clay	3.2	9.1	0.35
Silty fine sand	7.1	9.1	0.32
Silty fine sand	19	45	0.32
Silty fine sand	17	50	0.30

, where the soil around the pile is in a state of small strain [13,25]. Horizontal forces F_a of 40 kN, 80 kN, and 120 kN are applied to the pile top, respectively, and the height of the loading point of F_a is 15.4 m.

4.2. Pile–Soil Interaction

In order to verify the accuracy of the calculation results of the improved method, FLAC^3D software modeling verification was also adopted in this case. The mesh size of the soil body around the pile was ∆r = 0.02 m, ∆θ = 5°, ∆z = 0.02 m; the soil depth at the end of the pile was 1.5 times the pile length; and the soil boundary at the side of the pile was 30 times the pile diameter. The time required for FLAC^3D calculation was 16 h, and the time required for the method in this paper was 23 s. This phenomenon proves the efficiency of the latter operation. The horizontal displacement and bending moment of the pile body obtained by this method were compared with the measured data and FLAC^3D results, and the results are shown in Figure 3. As can be seen from the figure, the results of the method in this paper are consistent with the FLAC^3D results, thereby proving the accuracy of the method. Although there is some discrepancy between the results of this method and the measured results, this error is within the acceptable range for engineering purposes. The reason for this is attributed to changes in the soil layer distribution caused by scour and differences in the soil parameter values.

Taking the FLAC^3D results as a reference, the horizontal soil resistance under different horizontal loading conditions, with and without considering the shear interaction between soil layers, is compared, as shown in Figure 4a. As can be seen from the figure, the horizontal soil resistance of the pile is smaller when considering the shear interaction between soil layers. Compared to the horizontal soil resistance without considering the shear interaction, the error ranges from 9% to 20%. Therefore, the impact of shear interaction between layers on the horizontal soil resistance should be considered in the design.

As shown in Figure 3, there is a certain difference between the results obtained in this paper and the measured data for the distribution of the pile-side bending moment. This difference is mainly caused by the following aspects:

(1)

There is a time difference between geotechnical testing and the start of pile testing. Due to the characteristics of the local marine environment, the scouring effect is significant, which may result in certain differences between the actual surface soil layer and the geological exploration soil layer.

(2)

There is a problem with determining the value of the elastic modulus, E_s. Since E_s is mainly obtained based on laboratory geotechnical tests and local experience, there may be some differences compared to the actual situation.

However, for the horizontal displacement and maximum bending moment value of the pile top, as well as the location where they occur, the results of this method are in good agreement with the actual measurements. Therefore, the use of the variational method of the pile–soil model in this section to calculate the pile body response and soil displacement of horizontally loaded flexible piles under small deformation is reliable and can be effectively applied in offshore wind power projects.

Figure 4b compares the bending moment values obtained using the method in this paper and FLAC^3D under different horizontal loading conditions. As can be seen from the figure, the results of the two methods are in good agreement, further proving the accuracy of the method in this paper. The additional moment on the pile side is in the opposite direction to the overturning moment at the pile top, indicating that M helps to improve the horizontal bearing capacity of the pile. The total bending moment values under different loads are −9, −19, and −26 kN·m, accounting for approximately 2% of the pile top overturning moment, indicating that this enhancement effect can be neglected in the design.

4.3. Estimation of m Value

According to the “Code for Pile Foundation of Port Engineering” [5] and the “Code for Pile Foundation Testing” [6], the variation in the m value with horizontal displacement at the mud surface of the pile is obtained (as shown in Figure 5), and the m values obtained by the method in this paper are compared with the measured data. The results show that, as the mud surface displacement increases, the m value obtained by the improved method in this paper increases linearly from 1070 kN/m⁴ to 1080 kN/m⁴, while the measured value decreases from 6480 kN/m⁴ to 1280 kN/m⁴. In the mechanical assumptions of the former, the soil is treated as an ideal linear elastic material, which leads to the m value obtained by the improved method varying linearly and being the lower bound of the measured results.

5. Conclusions

This paper discusses an improved method for the pile–soil interaction of slender piles in composite elastic foundations under horizontal loading. The expressions of the forces around the large-diameter slender steel pipe pile under horizontal load were obtained, and the accuracy of the method was verified through comparative analysis with three typical examples. The following conclusions are drawn:

• Horizontal soil resistance at the pile side is the dominant factor in the total soil resistance, the influence of shear interaction between soil layers on the horizontal soil resistance is about 9~20%, and the calculation of horizontal soil resistance should fully consider the impact of shear interaction between soil layers.
• The additional moment on the pile side is beneficial in terms of improving the horizontal bearing capacity of the pile body, but the enhancement effect is negligible.
• The obtained m-value from the method presented in this paper is the lower bound of the measured results.
• The method in this paper can accurately predict the responses of long slender piles under horizontal load and the distribution of soil resistance around the pile. Compared to the numerical analysis method, it has a faster calculation speed and accurate results. However, the constitutive model of the soil in this paper cannot reflect the influence of the loading path and plastic evolution on the pile–soil interaction, and it cannot simulate the detachment effect at the pile–soil interface. These aspects will represent research directions in the future.