The Simplified Method of Head Stiffness Considering Semi-Rigid Behaviors of Deep Foundations in OWT Systems

Abstract

Simplified methods of static free head stiffness of the semi-rigid foundation under lateral loads were limited to flexible or rigid behavior by the critical length of piles. This would lead to errors when predicting the static or dynamic performance of their upper structures in OWT Systems. This paper presents a comprehensive analysis of the head static stiffness of the semi-rigid pile without considering the critical length. Firstly, case studies using the energy-based variational method encompassing nearly twenty thousand cases were conducted. These cases involved different types of foundations, including steel pipe piles and concrete caissons, in three types of soil: homogeneous soil, linearly inhomogeneous soil, and heterogeneous soil. Through the analysis of these cases, a series of polynomial equations of three kinds of head static stiffness, containing the relative stiffness of the pile and soil, the slenderness ratio, and Poisson’s ratio, were developed to capture the semi-rigid behavior of the foundations. Furthermore, the lateral deflection, the rotation for concrete caissons in the bridge projects, and several natural frequencies of three cases about the OWT system considering the SSI effect were carried out. the error of high-order frequency of the OWT system reached 13% after considering the semi-rigid effect of the foundation.

1. Introduction

Monopiles and well foundations (also known as caissons), with large diameters (2 m ≤ D ≤ 10 m) and smaller slender ratios (2 ≤ L_p/D ≤ 10, L_p is the length of the foundation), are extensively employed in offshore structures or bridges (Figure 1). The circular cross-sections of these foundations are designed as massive hollow bodies to comply with mechanics and reduce costs [1,2,3]. These foundations carry mechanisms of lateral loads to the surrounding soils caused by wind, waves, and earthquakes, and bring great challenges to predicting the supporting performance. It is important to note that the laterally loaded foundation belongs to a semi-rigid beam, which both covers rigid and flexible behavior (Figure 1). The soil-structure interaction (SSI) effect can lead to changes in the static or dynamic mechanical behavior of the upper structures, and the effect would be much more obvious when involving high-order derivative operations [4,5,6,7]. On the other hand, Structural monitoring or real-time analysis of dynamics leads to an urgent need for a more accurate expression of the SSI effect. Therefore, to maintain the stability and serviceability of a structure, it is imperative to accurately predict the foundation’s free-head stiffness under lateral loads. This precision is particularly critical for preserving the accuracy of upper structure results during high-order derivative mechanical calculations, especially when accounting for the SSI effect.

The free head stiffness of a foundation under lateral loads is commonly characterized by three parameters: K_L (lateral stiffness), K_R (rocking stiffness), and K_LR (cross-coupling stiffness). The critical length of piles, typically determined as a function of the relative stiffness between the pile and soil, plays a pivotal role in influencing foundation performance [8,9,10]. Depending on the specified threshold value for the critical length, various beam theories are employed to model the foundation’s bending behavior. Consequently, different equations for K_L, K_R, and K_LR are utilized. These stiffness parameters, particularly their initial values, serve two key purposes: firstly, they are instrumental in predicting the foundation’s head deflection and rotation under static loading conditions; secondly, they are essential for factoring in the Soil-Structure Interaction (SSI) effect during dynamic analysis, particularly within the framework of a Serviceability Limit State (SLS).

To characterize the head stiffness of laterally loaded piles, researchers have explored several approaches over the decades, including closed-form solutions based on Winkler-based theory, semi-analytical analyses grounded in continuum theory, and experimental testing.

Table 1. Different equations of stiffness formula by different researchers for Semi-infinite beams in various soil profile. m* is the equivalent ratio of shear modulus, m* = dGs/dz(1 + 0.75νs)

Source	KL, KR and KLR (If Have)	Constant Value				Beam and Soil Material
		KL	KLR	KR	Others
Randolph [13], Ko [24]	KL: $a_1{E}_{s0}Df\left({\nu }_s\right){\left(\frac{E_{eq}}{E_{s0}}\right)}^{b_1}$ KLR: $a_2{E}_{s0}{D}^{2}f\left({\nu }_s\right){\left(\frac{E_{eq}}{E_{s0}}\right)}^{b_2}$ KR: $a_3{E}_{s0}{D}^{3}f\left({\nu }_s\right){\left(\frac{E_{eq}}{E_{s0}}\right)}^{b_3}$	a1 = 3.147 b1 = 1/7	a2 = −0.5311 b2 = 3/7	a3 = 0.2458 b3 = 5/7	$E_{eq}=\frac{E_p{I}_p}{\frac{\pi D^4}{64}}$ $f\left({\nu }_s\right)=\frac{1+0.75{\nu }_s}{1+{\nu }_s}$ Es0 is the initial value of soil elastic modulus	The Semi-infinite beam in homogeneous and linear inhomogeneous soils (Triangular-linear strain element)
Pender [17]		a1 = 1.285 b1 = 0.188	a2 = −0.3075 b2 = 0.47	a3 = 0.18125 b3 = 0.738	f(vs) = 1 m* is the equivalent ratio of shear modulus, m* = dGs/dz(1 + 0.75νs)	The Semi-infinite beam in homogeneous soil
		a1 = 0.85 b1 = 0.29	a2 = −0.24 b2 = 0.53	a3 = 0.15 b3 = 0.77		The Semi-infinite beam in linear inhomogeneous homogeneous soil
		a1 = 0.735 b1 = 0.33	a2 = −0.27 b2 = 0.55	a3 = 0.1725 b3 = 0.776		The Semi-infinite beam in parabolic inhomogeneous soil
Gazetas [18] and Eurocode 8 Part 5 [25]		a1 = 1.08 b1 = 0.21	a2 = −0.22 b2 = 0.50	a3 = 0.16 b3 = 0.75		The Semi-infinite beam in homogeneous soil
		a1 = 0.60 b1 = 0.35	a2 = −0.17 b2 = 0.60	a3 = 0.14 b3 = 0.80		The Semi-infinite beam in linear inhomogeneous soil
		a1 = 0.79 b1 = 0.28	a2 = −0.24 b2 = 0.53	a3 = 0.15 b3 = 0.77		The Semi-infinite beam in parabolic inhomogeneous soil
		a1 = 0.1967 m* b1 = 18	a2 = −0.3472 m* b2 = 0.43	a3 = 0.2083 m* b3 = 0.72		-
		a1=0.1967 m* b1 = 33	a2=−0.3472 m* b2 = 0.54	a3=0.2083 m* b3 = 0.78		-
Syngros [26]; Anoyatis [22]		a1 = 1.24 b1 = 0.18	a2 = −0.21 b2 = 0.50	a3 = 0.15 b3 = 0.75		The Semi-infinite beam in parabolic inhomogeneous soil
Shadlou et al. [20]		a1 = 1.45 b1 = 0.186	a2 = −0.30 b2 = 0.50	a3 = 0.18 b3 = 0.73	$f\left({\nu }_s\right)=\frac{1}{1+{\nu }_s-0.25}$	The Semi-infinite beam in homogeneous soil
		a1 = 0.79 b1 = 0.34	a2 = −0.26 b2 = 0.567	a3 = 0.17 b3 = 0.78		The Semi-infinite beam in linear inhomogeneous homogeneous soil
		a1 = 1.02 b1 = 0.27	a2 = −0.29 b2 = 0.52	a3 = 0.17 b3 = 0.76		The Semi-infinite beam in parabolic inhomogeneous soil
Higgins et al. [19]	$K_L=\frac{100G_s{r}_p{\left(\frac{E_p}{E_s}\right)}^{0.18}}{3.4-{\left(\frac{E_p}{E_s}\right)}^{0.04}}$$K_{LR}=\frac{1}{0.3}\frac{G_s{r}_p^2{\left(\frac{E_p}{E_s}\right)}^{0.47}}{{\left(\frac{E_p}{E_s}\right)}^{0.04}-3.4}$$K_R=\frac{34}{9}\frac{G_s{r}_p^3{\left(\frac{E_p}{E_s}\right)}^{0.72}}{3.4-{\left(\frac{E_p}{E_s}\right)}^{0.04}}$				f(νs) = Gs(1 + 0.75νs)/Es m* = dGs/dz(1 + 0.75νs)	The Semi-infinite beam in homogeneous soil
	$K_L=\frac{1.23m^{\ast }{r}_p^2{\left(\frac{E_p}{m^{\ast }{r}_p}\right)}^{0.33}}{0.6765-0.2704{\left(\frac{E_p}{m^{\ast }{r}_p}\right)}^{0.03}}$$K_{LR}=\frac{0.52m^{\ast }{r}_p^3{\left(\frac{E_p}{m^{\ast }{r}_p}\right)}^{0.57}}{0.2704{\left(\frac{E_p}{m^{\ast }{r}_p}\right)}^{0.03}-0.6765}$$K_R=\frac{0.55m^{\ast }{r}_p^4{\left(\frac{E_p}{m^{\ast }{r}_p}\right)}^{0.78}}{0.6765-0.2704{\left(\frac{E_p}{m^{\ast }{r}_p}\right)}^{0.03}}$					The Semi-infinite beam in linear inhomogeneous soil
		a1 = 3\3.66\4.58 (Lp/rp = 40, Ep/Es = 100\300\1000)	a2 = 2.70\4.48\7.81 (Lp/rp = 6, Ep/Es = 100~1000	a3 = 6.02\13.20\31.31 (Lp/rp = 6, Ep/Es = 100~1000)		-

and

Table 2. Different equations of Stiffness formulae by different researchers for short beams in various soil profile.

Source	KL, KR and KLR (If Have)	Constant Value			Others	Beam and Soil Material
		KL	KLR	R
Randolph [13], Ko [24]	KL: $a_1{E}_{s0}Df\left({\nu }_s\right){\left(\frac{E_{eq}}{E_{s0}}\right)}^{b_1}$ KLR: $a_2{E}_{s0}{D}^{2}f\left({\nu }_s\right){\left(\frac{E_{eq}}{E_{s0}}\right)}^{b_2}$ KR: $a_3{E}_{s0}{D}^{3}f\left({\nu }_s\right){\left(\frac{E_{eq}}{E_{s0}}\right)}^{b_3}$	a1 = 3.150/(1 − 0.3345(Lp/D)0.25 b1 = 1/3	a2 = −2.045/(1 − 0.3345(Lp/D)0.25 b2 = 9/8	a3 = 3.969/(1 − 0.3345(Lp/D)0.25 b3 = 5/3	Finite element method	The rigid beam in homogeneous and linear inhomogeneous soils (Triangular-linear strain element)
Carter and Kulhawy [27]; Bouzid et al. [28]		a1 = 1.884 b1 = 0.627	a2 = −1.048 b2 = 1.483	a3 = 1.91 b3 = 2.049		The rigid beam in homogeneous soil
Higgns et al. [19]; Bouzid et al. [28]		a1 = 2.426 b1 = 0.71	a2 = −1.44 b2 = 1.67	a3 = 1.789 b3 = 2.459	The Fourier finite-element method	The rigid beam in homogeneous soil
		a1 = 0.929 b1 = 2.041	a2 = −0.633 b2 = 3.061	a3 = 0.672 b3 = 3.491		The rigid beam in linear inhomogeneous soil
Shadlou and Bhattacharya [20]		a1 = 3.2 b1 = 0.62	a2 = −1.7 b2 = 1.56	a3 = 1.65 b3 = 2.5	3D element analysis	The rigid beam in homogeneous soil
		a1 = 2.35 b1 = 1.53	a2 = −1.775 b2 = 2.5	a3 = 1.58 b3 = 3.45		The rigid beam in linear inhomogeneous soil
		a1 = 2.66 b1 = 1.07	a2 = −1.8 b2 = 2.0	a3 = 1.63 b3 = 3.0		The rigid beam in parabolic inhomogeneous soil
Abed et al. [29]; Bouzid et al. [28]		a1 = 1.708 b1 = 1.661	a2 = −1.233 b2 = 2.655	a3 = 0.672 b3 = 3.941	Fourier Series Aided Finite Element (FSAFE) approach	The rigid beam in linear inhomogeneous soil
		a1 = 2.841 b1 = 0.977	a2 = −2.933 b2 = 1.767	a3 = 3.894 b3 = 2.562		The rigid beam in soil whose stiffness increases with the square root of the depth
Aissa et al. [21]		a1 = 2.756 b1 = 0.668	a2 = −1.595 b2 = 1.636	a3 = 1.731 b3 = 2.495	The semi-analytical finite element analysis	The rigid beam in homogeneous soil
Carter et al. [27]	KL: $\frac{3.15G_s^{\ast }{D}^{\frac{2}{3}}{L_p}^{\frac{1}{3}}}{1-0.28{\left(\frac{2L_p}{D}\right)}^{\frac{1}{4}}}$$K_{\mathrm{LR}}:-\frac{2G_s^{\ast }{D}^{\frac{7}{8}}{L_p}^{\frac{5}{3}}}{1-0.28{\left(\frac{2L_p}{D}\right)}^{\frac{1}{4}}}$$K_{\mathrm{R}}:\frac{4G_s^{\ast }{D}^{\frac{4}{3}}{L_p}^{\frac{5}{3}}}{1-0.28{\left(\frac{2L_p}{D}\right)}^{\frac{1}{4}}}$				G* is the equivalent shear modulus, G* = Gs (1 + 0.75 νs)	The rigid beam in bedrock

provide a compilation of key findings from these studies. For example, Bergfelt [11] proposed a closed function of K_L based on the long flexible beam in a Winkler foundation. Building upon this work, Thomas [12] provided a solution considering the relative stiffness of the pile and soil. Randolph [13] developed the finite element method to modify the 3-D continuum behavior of the soil and put forward equations for K_L, K_R, and K_LR in flexible and rigid beams based on the relative stiffness of the pile and soil (E_eq/E_s). Additionally, Poulos and Davis [14], following Barber [15], presented a series of functions for K_L, K_R, and K_LR, considering flexible and rigid beams with the threshold value of β. Other researchers, such as Budhu and Davies [16], Pender [17], Gazetas [18], Higgins et al. [19], Shadlou and Bhattacharya [20], Aissa et al. [21]. and Anoyatis et al. [22], have proposed expressions for flexible piles and rigid piles. These studies collectively yield dimensionless expressions for K_L, K_LR, and K_R, which are denoted as K_L/(E_sD), K_LR/(E_sD²), and K_R/(E_sD³), respectively. Notably, these investigations reveal that the dimensionless head stiffness of long, flexible piles is primarily influenced by the parameter E_eq/E_s, while for short piles, the slenderness ratio (L_p/D) assumes a significant role. However, the foundations examined in this study primarily exhibit semi-rigid beam characteristics, demonstrating aspects of both rigid and flexible behavior. Specifically, when predicting the static or dynamic performance of upper structures—particularly those involving high-order derivative operations like high-order self-resonant frequencies—traditional methods may yield results with significant, non-negligible errors [23]. Therefore, there is a compelling need to develop a more precise equation for calculating the dimensionless head stiffness—those that consider both the rigid and flexible behavior of the foundation while remaining as practical to calculate as the methods employed in previous studies.

The energy-based variational method, characterized by its energy-based theory and virtual work principles, has emerged as a valuable approach for predicting the response of piles subjected to lateral loads. This method offers notable advantages, combining the efficiency of quick computations with the mathematical rigor comparable to that of three-dimensional finite-difference analyses [30,31,32]. Previously, Shadlou et al. [20] conducted an energy-based study that involved the modification of pile-head spring stiffness (K_L, K_R, and K_LR) using the Euler-Bernoulli beam theory in layered soils. As highlighted in previous research (Byron & Houlsby [33,34]; Gupta [35,36,37], Cao [38]), the Timoshenko beam model has gained recognition for its appropriateness in characterizing the response of laterally loaded piles with substantial diameters. More intricate models, such as shells or solid bodies, have found application in commercial software like ABAQUS, FLAC3D, or PLAXIS. A consensus has emerged within the research community, affirming that the Timoshenko beam strikes a commendable balance between accuracy and simplicity. Gupta [37] and Gupta and Basu [36] introduced the Timoshenko beam theory within the energy-based variational method, discussing pile-head spring stiffness under both dynamic and static lateral loads with no specific calculation equations provided in this work; Taking the method further, Li [39,40,41,42,43] extended its application by considering both vertical and horizontal soil displacement in the analysis of laterally loaded deep foundations.

The objective of this research is to investigate the head static stiffness of the semi-rigid pile without the critical length. To achieve this, a comprehensive study using the energy-based variational method with Timoshenko beam encompassing nearly twenty thousand cases was conducted. By using the efficiency of the variational method, different types of foundations (steel pipe piles and concrete caissons) with diameters and slenderness ratios (L_p/D) ranging from 2 m to 10 m, and three types of soil (homogeneous soil, linearly inhomogeneous soil, and heterogeneous soil) were studied, each characterized by varying elastic moduli and Poisson’s ratio. Through the analysis of these cases, the three kinds of static stiffness of the pile head, covering the relative stiffness of the pile and soil, the slenderness ratio as well and Poisson’s ratio without consideration of the critical length, were developed to accurately capture the semi-rigid behavior exhibited by these foundations. These new equations can ensure the accuracy of mechanical calculation involving high-order derivative operations of the upper structures.

2. Methodology

2.1. Energy-Based Variational Method

The energy-based variational method has the advantage of computational efficiency and open accessibility. To investigate the head stiffness of a semi-rigid pile, we examine a single circular beam, which may represent either a pile or a caisson. This beam has specific geometric parameters, including a radius (r_p), wall thickness (t), and length (L_p), and it is embedded within a three-dimensional continuous medium. The beam experiences lateral forces (F_a) and/or moments (M_a) at its head (Figure 1). In this analysis, we treat the pile as a vertical Timoshenko beam, characterized by a lateral deflection denoted as w(z), which is associated with the depth parameter z. and the shear rotation of the plane section is ϕ(z), the three-dimensional soil displacement distribution can be simplified as below in terms of r-θ-z:

$$\begin{array}{l}{u}_r=w\left(z\right){\varphi }_r\left(r\right)\cos \theta \\ u_{\theta }=-w\left(z\right){\varphi }_{\theta }\left(r\right)\sin \theta \\ u_r=\varphi \left(z\right)r_p{\varphi }_z\left(r\right)\cos \theta \end{array}$$

(1)

The boundary conditions of ϕ_r, ϕ_θ, and ϕ_z are given as:

$$\begin{array}{l}{\varphi }_r(r)=\left\{\begin{array}{cc}1\hfill & 0\le r\le r_p\\ 0\hfill & r\to \infty \end{array}\right.\\ {\varphi }_{\theta }(r)=\left\{\begin{array}{cc}1\hfill & 0\le r\le r_p\\ 0\hfill & r\to \infty \end{array}\right.\\ {\varphi }_z(r)=\left\{\begin{array}{cc}0\hfill & r_p=0\\ \frac{r}{r_p}\hfill & 0<r<r_p\\ 1\hfill & r=r_p\\ 0\hfill & r\to \infty \end{array}\right.\end{array}$$

(2)

where ϕ_z(r_p) is the value of ϕ_z(r) when r = r_p. In Equation (2), the functions ϕ_r and ϕ_θ both have a value of 1 when 0 ≤ r ≤r_p and 0 when r approaches infinity. As for ϕ_z, it equals 0 when r is 0 and r/r_p when 0 < r < r_p, ϕ_z becomes 0 again as r approaches infinity. These functions are related to the variables r, θ, and the pile radius r_p. In Equation (2), it is further assumed that ϕ_r, ϕ_θ, and ϕ_z are mutually independent in the r-direction. The boundary condition of ϕ_z at 0 ≤ r ≤ r_p is from the static Timoshenko beam theory (Timoshenko, 1932; https://ccrma.stanford.edu/~bilbao/master/node163.html, accessed on 16 February 2024.), where the displacements of the beam are u_r = w(z)cosθ, u_θ = −w(z)sinθ, and u_z = ϕ(z)cosθ.

For Timoshenko beam theory, depending on the total energy of the system and the principle of virtual work, the following equation is obtained:

$$\mathsf{\delta }\prod =E_{\mathrm{p}}{I}_{\mathrm{p}}{\displaystyle \underset{0}{\overset{L_{\mathrm{p}}}{\int }}(\frac{\mathrm{d}\varphi }{\mathrm{d}z}})\mathsf{\delta }(\frac{\mathrm{d}\varphi }{\mathrm{d}z})\mathrm{d}z+{\displaystyle \underset{0}{\overset{L_{\mathrm{p}}}{\int }}\kappa G_{\mathrm{p}}{A}_{\mathrm{p}}}(\frac{\mathrm{d}w}{\mathrm{d}z}-\varphi )\mathsf{\delta }(\frac{\mathrm{d}w}{\mathrm{d}z}-\varphi )\mathrm{d}z+{\displaystyle \underset{0}{\overset{L_{\mathrm{p}}}{\int }}{\displaystyle \underset{0}{\overset{2\pi }{\int }}{\displaystyle \underset{r_{\mathrm{p}}}{\overset{\infty }{\int }}{\mathsf{\sigma }}_{pq}\mathsf{\delta }{\mathsf{\epsilon }}_{pq}r\mathrm{d}r\mathrm{d}\theta \mathrm{d}z}}}+{\displaystyle \underset{L_{\mathrm{p}}}{\overset{\infty }{\int }}{\displaystyle \underset{0}{\overset{2\pi }{\int }}{\displaystyle \underset{0}{\overset{\infty }{\int }}{\mathsf{\sigma }}_{pq}\mathsf{\delta }{\mathsf{\epsilon }}_{pq}r\mathrm{d}r\mathrm{d}\theta \mathrm{d}z}}}-\delta W$$

(3)

where E_p is Young’s modulus of the beam; I_p is the second moment of inertia of the cross-section; κ is the shear correction factor; w is the lateral displacement of the beam central line; W is the work by outer force (lateral force F_a or moment M_a). The soil potential energy is in the region of r > r_p when L_p ≥ z ≥ 0 and r > 0 when z ≥ L_p; σ_pq is the stress in soil domain; ε_pq is the strain in soil domain.

2.2. Soil Conditions

To obtain the initial stiffness, the soil is in an elastic state under very small to small strain conditions, and the stress-strain relationship is elastic. In this section, four main categories are considered from the point of view of the current analysis:

(1) Homogenous soil conditions (E_s = E_s0(z/D)^α, α = 0): E_s0 is the initial value of soil elastic modulus, α is an index of the function. The elastic modulus is considered constant with a depth of the soil, often used in cohesive soils, weathered bedrock, and very dense sand for typical North Sea soils [7,44].

(2) Lineally inhomogeneous soil conditions (E_s = E_s0(z/D)^α, α = 1): G_s increases linearly with depth from zero value at the ground surface, also called Gibson’s soil, often used to describe normally consolidated cohesive soils [7], London clay reported by Skempton and Henkel [45], see also, Ward et al. [46], Burland and Lord [47], Butler [48], and Hobbs [49].

(3) Heterogeneous soil conditions (E_s = E_s0(z/D)^α, 0 < α < 1): G_s increases nonlinearly with depth from zero value at the ground surface; when α = 0.5, it is described as a parabolic soil profile, which has been used to describe soil profiles in several Europe offshore wind projects [7,50].

(4) Complex layered soil conditions: E_s is obviously not included in the above categories, different layers with obviously different E_s are often observed, and often used in real sites, the multilayered sedimentation in river or coast areas, i.e., interlaced layers with soft clay and sand. There are other soil conditions characterized by heterogeneity, such as G = G₁ + G₂ cos(z/D). These conditions often arise in geomaterials formed through periodic deposition, such as varved clays and sedimentary rocks [51].

This paper primarily centers its attention on deep foundation scenarios that pertain to bridges and offshore structures. These scenarios are categorized within groups (1) to (3). In elastic media, the constitutive model is σ_pq = λ_s(z)δ_pqε_pp +2G_s(z)ε_pq (p, q = 1, 2, 3), where λ_s(z) and G_s(z) are Lame’s constants; λ_s = E_sv_s/(1 + v_s)/(1 − 2v_s); G_s = E_s/2/(1 + v_s); E_s is the elastic modulus; v_s is Poisson’s ratio. Therefore, λ_s in categories (1) to (3) can be given by:

$${\lambda }_s=\frac{E_{s0}{\nu }_s{\left(z/D\right)}^{\alpha }}{\left(1+{\nu }_s\right)\left(1-2{\nu }_s\right)}$$

(4)

2.3. Governing Differential Equations of the Beam and Soils

The governing differential equations of the foundation and soils have been derived in the Appendix A and Appendix B, the governing differential equations of soils have been derived in the Appendix C. The stiffness of a beam under lateral loading is expressed in matrix form as follows:

$$\left[\begin{array}{cc}{K}_L& K_{LR}\\ K_{RL}& K_R\end{array}\right]\left[\begin{array}{c}{w}_{z=0}\\ {\left(\frac{\mathrm{d}w}{\mathrm{d}z}\right)}_{z=0}\end{array}\right]=\left[\begin{array}{c}F\\ M\end{array}\right]$$

(5)

where K_L is the swaying dynamic head stiffness, K_LR is the coupled swaying-rocking head stiffness, and K_R is the rocking dynamic head stiffness K_LR = K_RL.

To calculate K_L, K_LR, and K_R, three steps are needed:

(1) A response analysis with applied lateral force F_a and restricted pile head rotation θ to derive the lateral stiffness coefficient (K_L = F_a/w₀), the boundary conditions at the pile head are F_a = constant, θ₀ = 0, w₀ is unknown, and

$$\begin{array}{l}\kappa G_p{A}_p\left(\frac{dw}{dz}-\varphi \right)+k_2\varphi +2t\frac{dw}{dz}=-F_a(z=0\mathrm{m})\\ {\theta }_0=\frac{dw}{dz}=0\end{array}$$

(6)

where t and k₂ are shown in Appendix A.

(2) A response analysis for calculating to moment M_a that leads to zero rotation θ for a given pile head displacement w₀. This defines the off-diagonal term (K_LR = −M_a/w₀), the boundary conditions at the pile head are w₀ = constant, θ₀ = 0, M_a is unknown, and

$$\begin{array}{l}{w}_0=\mathrm{constant}(z=0\mathrm{m})\\ {\theta }_0=0\end{array}$$

(7)

(3) A response analysis with zero pile head deflection w₀ and an applied bending moment M_a. The rotational stiffness coefficient can be derived from the observed pile head rotation (K_R = −M_a/θ) the boundary conditions at the pile head are M_a = constant, w₀ = 0, θ₀ is unknown, and

$$\begin{array}{l}{w}_0=0\hspace{1em}\hspace{1em}(z=0\mathrm{m})\\ E_p{I}_p\frac{d\varphi }{dz}+\left(-k_1+k_2\right)w+2t_4\frac{d\varphi }{dz}=M_{\mathrm{a}}\end{array}$$

(8)

where k₁, k₂ and t₄ are shown in Appendix A.

From Equations (1)–(8), the values of K_L, K_LR, and K_R can be obtained, respectively. The derivation details are given in Appendix B.

2.4. Modeling Cases

The values of K_L, K_LR, and K_R are influenced by a multitude of factors, including the diameter of the pile (D), the slenderness ratio (L_p/D), the thickness of the pile wall (t), the relative stiffness of the pile and soil, the elastic modulus of the soil (E_s), and Poisson’s ratio (v_s). To comprehensively explore the impact of these parameters, a wide range of values has been considered in the analyses conducted for this paper, as summarized in

Table 3. Pile geometries and soil conditions in study cases.

Pile Diameter D (m)	Slenderness Ratio, Lp/D	Diameter-to-Wall Thickness Ratio	Elastic Modulus		vs
			Pile (Ep) GPa	Soil (Es)
2–10	2–10	t = (0.01 ± 0.005)D	210 (Steel pipe pile)	Es = Es0(z/D)α (α = 0, 0.25, 05, 0.75, 1)	0.20–0.45
2–10	2–10	t = (0.1 ± 0.05)D	30 (Concrete caissons)

. For instance, the wall thickness for monopiles typically falls within the range of D/80 to D/120, equivalent to 0.005D to 0.015D [14]. In the case of concrete caissons, the wall thickness is in the range of 0.05D to 0.15D. Additionally, the elastic modulus of the soil (E_s0) has been studied within the range of 2 MPa to 300 MPa, Poisson’s ratio (v_s) spans from 0.20 to 0.45. In fact, this study has examined over twenty thousand cases to comprehensively investigate the relationships and effects of these parameters on K_L, K_LR, and K_R. These cases encompass a wide array of deep foundation scenarios pertinent to bridges and offshore structures, ensuring a thorough exploration of the subject matter.

3. Results and Discussion

3.1. Validation of the Analysis Compared against Different Methods

Homogenous soil conditions (α = 0) To investigate the influence of ln(E_eq/E_s) and L_p/D on the head stiffness of laterally loaded foundations in homogeneous soil conditions, we present Figure 2, Figure 3 and Figure 4 as illustrative examples. These figures depict the effects of ln(E_eq/E_s) and L_p/D on K_L/(E_sD), K_LR/(E_sD²), and K_LR/(E_sD³) for steel monopiles, considering t = 0.01D and v_s = 0.30 in 3-D version. The results are compared with closed-form solutions from Randolph [13], Shadlou et al. [20], Higgins et al. [19], and Abed et al. [29] presented in Table 1 and Table 2.

As presented in previous studies, Randolph [13] and Carter et al. [27] respectively proposed the critical pile length of slender piles and rigid piles as:

$$\frac{L_p}{D}\left\{\begin{array}{cc}\ge {\left(\frac{E_{eq}}{E_s\left(1+0.75v_s\right)/\left(1+v_s\right)}\right)}^{2/7}& \mathrm{flexible}\mathrm{piles}\\ \le 0.05{\left(\frac{E_{eq}}{E_s\left(1+0.75v_s\right)/\left(1+v_s\right)}\right)}^{1/2}& \mathrm{rigid}\mathrm{piles}\end{array}\right.$$

(9)

Then Higgins et al. [19] proposed a method for determining flexible pile and rigid piles with the consideration of:

$$\frac{L_p}{D}\left\{\begin{array}{cc}\ge 40& \mathrm{flexible}\mathrm{piles}\\ \le {\left(\frac{1}{412.8}\frac{E_{eq}}{E_s\left(1+0.75v_s\right)/\left(1+v_s\right)}\right)}^{1/3.23}& \mathrm{rigid}\mathrm{piles}\end{array}(\mathsf{\alpha }=0)\right.$$

(10)

and

$$\frac{L_p}{D}\le \begin{array}{cc}{\left(\frac{1}{650}\frac{E_{eq}}{\frac{\mathrm{d}{E}_s}{\mathrm{dz}}D\left(1+0.5v_s\right)/\left(1+v_s\right)}\right)}^{1/3.45}& \mathrm{rigid}\mathrm{piles}\end{array}(\mathsf{\alpha }=1)$$

(11)

Figure 3a illustrates a similar relationship between K_LR/(E_sD²) and both (L_p/D) and ln(E_eq/E_s) in the 3-D context. Figure 3b highlights the trend of decreasing values of K_LR/(E_sD²) with increasing L_p/D for short piles (2 ≤ L_p/D ≤ 6, ln(E_eq/E_s) > 4.5). These values are consistent with those from Randolph [13] and Shadlou et al. [20] for rigid piles, falling within the range of K_LR/(E_sD²) values obtained in the present study. However, the rate of increase differs: the former exhibits a positive increasing rate concerning (L_p/D), while the results from this study indicate that K_LR/(E_sD²) becomes insensitive to L_p/D when L_p/D ≥ 6, suggesting a flexible pile behavior. The successful prediction of the threshold zone where K_LR/(E_sD²) becomes insensitive to L_p/D by Higgins et al. [19] further supports this observation. In Figure 3c, the equations provided by Shadlou et al. [20] and Randolph [13] for flexible piles in Table 1 also demonstrate good performance on K_LR/(E_sD²) when L_p/D ≥ 6.

A similar phenomenon is also evident in Figure 4, where the relationship between K_R/(E_sD³) and (L_p/D) and ln(E_eq/E_s) in the 3-D context has been analyzed. This phenomenon has also been observed in the case of concrete caissons.

Based on the findings presented in Figure 2, Figure 3 and Figure 4, it can be concluded that while traditional methods demonstrate good performance for K_L/(E_sD), K_LR/(E_sD²), and K_R/(E_sD³), it is essential for the functions of K_L/(E_sD), K_LR/(E_sD²), and K_R/(E_sD³) to incorporate the parameters E_eq/E_s and L_p/D. Therefore, the development of more accurate equations that consider both parameters is warranted. These observations highlight the need for more comprehensive and refined equations that account for a broader range of influencing factors, ultimately improving our ability to predict the head stiffness of laterally loaded foundations in various scenarios.

Linearly inhomogeneous soil conditions (α = 1). To comprehend the impact of ln(E_eq/E_s) and L_p/D on the head stiffness of laterally loaded foundations in linearly inhomogeneous soil conditions, Figure 5, Figure 6 and Figure 7 provide insights into the effects of these parameters on K_L/(E_sD), K_LR/(E_sD²), and K_LR/(E_sD³) for steel monopiles with t = 0.01D and v_s = 0.30. In Figure 5a, it is evident that the results presented in this study are influenced by ln(E_eq/E_s) and L_p/D. Figure 5b demonstrates that K_L/(E_sD) for short piles (2 ≤ L_p/D ≤ 4, ln(E_eq/E_s) ≥ 6) exhibits an increasing trend with rising L_p/D values, aligning well with the results from Shadlou et al. [20], Higgins et al. [19], Abed et al. [29] for rigid pile conditions in Table 2. Conversely, results from Randolph [13] tend to underpredict K_L/(E_sD) at smaller L_p/D values, while those from Higgins et al. [19], Shadlou et al. [20], and Abed et al. [29] tend to overpredict it at larger L_p/D values. In Figure 5c, it becomes evident that K_L/(E_sD) becomes less sensitive to L_p/D as ln(E_eq/E_s) decreases and L_p/D increases. This behavior indicates a flexible pile response. The threshold zone where K_L/(E_sD) becomes insensitive to L_p/D is successfully predicted by Higgins et al. [19], and equations from Pender [17] and Shadlou et al. [20] for flexible piles perform better in terms of prediction accuracy compared to the equation from Randolph [13].

Figure 6 demonstrates a similar relationship between K_LR/(E_sD²) and both (L_p/D) and ln(E_eq/E_s) in the 3-D context. In Figure 6a, it’s evident that the results presented in this study are influenced by ln(E_eq/E_s) and L_p/D. Figure 6b illustrates that for short piles (2 ≤ L_p/D ≤ 4, ln(E_eq/E_s) > 6), the values of K_LR/(E_sD²) exhibit a curvilinear decrease as L_p/D increases. The range of L_p/D values (2 ≤ L_p/D ≤ 4) covered by the results from the present study for K_LR/(E_sD²) includes the results from Randolph [13], Higgins et al. [19], Shadlou et al. [20], and Abed et al. [29] for rigid pile conditions. Similar to the previous case, as L_p/D increases beyond 6, K_LR/(E_sD²) becomes insensitive to L_p/D, with the threshold zone falling between the predictions of Randolph [13] and Higgins et al. [19]. In Figure 6c, the results from this paper at L_p/D = 3 align more closely with the predictions for flexible piles by Randolph [13] (as given in Table 1), while the results from the present study for L_p/D ≥ 6 better match the equations from Shadlou et al. [20] and Pender [17] for flexible piles (as given in Table 1). A similar trend is observed in Figure 6, where K_R/(E_sD³) versus (L_p/D, ln (E_eq/E_s)) in the 3-D version was analyzed, and the same phenomenon has also been observed in concrete well foundations (or caissons).

Based on the insights provided in Figure 5, Figure 6 and Figure 7, it becomes evident that the relationships governing K_L/(E_sD), K_LR/(E_sD²), and K_R/(E_sD³) should be expressed as functions of both E_eq/E_s and L_p/D when dealing with linearly inhomogeneous soils. This underscores the need for more precise equations that take into account the combined influence of these two parameters. It’s worth noting that the threshold zones for long flexible piles and short rigid piles in linearly inhomogeneous soils differ from those observed in homogeneous soils, indicating the importance of considering soil heterogeneity in foundation analysis.

Heterogeneous soil conditions (α = 0.5). Figure 8, Figure 9 and Figure 10 provide insights into the impact of ln (E_eq/E_s) and L_p/D on K_L/(E_sD), K_LR/(E_sD²), and K_LR/(E_sD³) for steel monopiles with t = 0.01D and v_s = 0.30. In Figure 8a, it is evident that the results presented in this study are influenced by ln(E_eq/E_s) and L_p/D. Figure 8b shows that the values of K_L/(E_sD) for short piles exhibit an increasing trend with rising L_p/D values, and the results at L_p/D = 2 and ln(E_eq/E_s) = 9.08 align well with those of Shadlou et al. [20] for rigid pile conditions, as presented in Table 1 and Table 2. In Figure 8c, K_L/(E_sD) becomes less sensitive to L_p/D as ln(E_eq/E_s) decreases and L_p/D increases, indicating a foundation behavior more akin to a flexible pile. The values from the present study at L_p/D = 2 are closer to those from Syngros [26], and the values at L_p/D > 5 fall within the range between the equations from Gazetas [18] and the equations from Shadlou et al. [20] used for flexible piles.

Figure 9 illustrates a similar relationship between K_LR/(E_sD²) and both (L_p/D) and ln(E_eq/E_s) in the 3-D context. In Figure 9b, it is evident that the values of K_LR/(E_sD²) decrease with increasing L_p/D, and the values from the present study at L_p/D = 2 align well with the results of Shadlou et al. [20] for rigid pile conditions, as presented in Table 1 and Table 2. Furthermore, K_LR/(E_sD²) from the present study becomes insensitive to L_p/D as L_p/D increases. In Figure 9c, the results from the present study, where L_p/D ≥ 6, better align with the equations from Shadlou et al. [20] for flexible piles, indicating that K_LR/(E_sD²) exhibits behavior characteristic of flexible piles in this regime. A similar phenomenon has been observed in Figure 10, where K_R/(E_sD³) versus (L_p/D, ln(E_eq/E_s)) in the 3-D context were analyzed, further emphasizing the importance of considering these parameters for accurate predictions in laterally loaded foundation analysis.

The observations made in Figure 8, Figure 9 and Figure 10 highlight the limitations of traditional methods in covering all calculating conditions for foundations with D ≥ 2 m and 2 ≤ L_p/D ≤ 10 in heterogeneous soil conditions. It becomes evident that the functions of K_L/(E_sD), K_LR/(E_sD²), and K_R/(E_sD³) should indeed be expressed as functions of E_eq/E_s and L_p/D in heterogeneous soil conditions, necessitating the development of more accurate equations that account for both of these parameters. Moreover, the threshold zones for long flexible piles and short rigid piles in heterogeneous soil conditions differ from those in inhomogeneous and linearly inhomogeneous soils, underlining the importance of considering specific soil characteristics when analyzing foundation behavior.

Heterogeneous soil conditions (α = 0.25 and 0.75). Additionally, a similar phenomenon observed in K_L/(E_sD), K_LR/(E_sD²), and K_R/(E_sD³) versus (L_p/D, ln(E_eq/E_s)) when α = 0.25 and 0.75 in the 3-D context further emphasizes the need for more accurate equations that can account for a wider range of scenarios without relying on the critical length parameter. This suggests the importance of developing equations that consider both the relative stiffness of the pile and soil and the slenderness ratio in future studies.

3.2. Parameter Analysis

The effect of materials and thickness. Figure 11 and Figure 12 provide valuable insights into the behavior of foundations in heterogeneous soil conditions with varying materials. In Figure 11, which considers different materials (steel piles with t = 0.01D and concrete piles with 0.1D) and α = 0.75, the results for K_L/(E_sD) show that the difference between these materials is within 3%. This suggests that using the function E_eq/E_s is an effective way to account for different materials of hollow piles when analyzing the head stiffness of foundations in heterogeneous soil conditions. Similarly, in Figure 12, which focuses on concrete piles with varying wall thicknesses (t = 0.1D and 0.15D) and α = 0.25, v_s = 0.30, the results for K_L/(E_sD) also indicate that the difference between these material variations is within 3%. This further supports the notion that the function E_eq/E_s is a robust approach for considering material variations in the analysis of foundation behavior in heterogeneous soil conditions. These findings suggest that, compared with Poulos and Davis [14] and Aissa et al. [21], the function E_eq/E_s can efficiently incorporate different materials in the analysis, simplifying the modeling process while maintaining accuracy in predicting the behavior of foundations with varying materials.

The effect of Poisson’s ratio. The consideration of Poisson’s ratio in the analysis of K_L/(E_sD), K_LR/(E_sD²), and K_R/(E_sD³) is relatively important. In previous research, the effect of Poisson’s ratio on K_L/(E_sD), K_LR/(E_sD²), and K_R/(E_sD³) is commonly neglected in some studies. Randolph [13] considered the effect by f(v_s) = (1 + 0.75 v_s)/(1+ v_s); Shadlou and Bhattacharya [20] gave the function $f({\nu }_s)=1+\left|{\nu }_s-0.25\right|$; Jabli et al. [52] presented solutions for stiffness of suction caissons having rigid skirted for 0.5 < L/D < 2 in three types of ground profiles with $f\left({\nu }_s\right)=1+0.6\left|{\nu }_s-0.25\right|$ and by f(v_s) = 1.1(0.096L_p/D + 0.6)v_s² − 0.7v_s + 1.06. In this paper, the effect of Poisson’s ratio is observed, and it’s noted that the effect is influenced by various factors, including ln(E_eq/E_s), L_p/D, and α. Comparing the values obtained for different Poisson’s ratios (v_s = 0.2 to v_s = 0.45) with those for v_s = 0.30 under the same pile and soil conditions, it’s observed that the error ranges from −2% to 14%. This suggests that accounting for varying Poisson’s ratios in soils is essential for accurate analysis. Compared with Randolph [13] and Jabli et al. [52], a more efficient way is proposed to take into account different Poisson’s ratios in soils, and the following equation can be used:

$$f^{\left(1\right)}\left({\nu }_s\right)=f^{\left(2\right)}\left({\nu }_s\right)=\left(-0.7146\alpha +2.837\right){\nu }_s^2-\left(-0.2666\alpha +1.4381\right){\nu }_s+1.17$$

(12)

$$f^{\left(3\right)}\left({\nu }_s\right)=1+0.4\left|{\nu }_s-0.3\right|$$

(13)

The proposed equation for considering different Poisson’s ratios (v_s) in soils has limited the errors in K_L/(E_sD), K_LR/(E_sD²), and K_R/(E_sD³) to a range of −1.5% to 5% across all cases in Table 3. Figure 13 gives an example of the modification effect of Equations (12) and (13) for K_L/(E_sD) in different conditions (α = 0 to 1, concrete and steel material), where the values of K_L at v_s = 0.2 to 0.45 were divided by the value of K_L when v_s = 0.3. results show that the values are in the range of 0.985 to 1.035, and the average values are nearly equal to 1.0. This suggests that the equation effectively accounts for variations in Poisson’s ratio and provides accurate predictions for the behavior of foundations under different soil conditions.

3.3. The Equations of Stiffness of Pile-Head Springs

From Section 3.1, the dimensionless function of the stiffness of pile-head springs, g⁽ⁿ⁾, can be described by a serial of polynomial equations containing ln(E_eq/E_s0) and L_p/D:

$$g^{\left(n\right)}\left(\ln \left(\frac{E_{eq}}{E_{s0}}\right),\frac{L_p}{D}\right)={\displaystyle \sum _{i=0}^{i=3}{\displaystyle \sum _{j=0}^{j=4}{P}_{ij}^{(n)}}}{\left(\ln \left(\frac{E_{eq}}{E_{s0}}\right)\right)}^i{\left(\frac{L_p}{D}\right)}^j$$

(14)

where g⁽ⁿ⁾ is the polynomial function related to K_L/E_s0D, K_LR/E_s0D² and K_R/E_s0D³ at v_s = 0.3, respectively; n = 1, 2, 3, g⁽¹⁾ is the constant of K_L, g⁽²⁾ is the constant of K_LR, g⁽³⁾ is the constant of K_R; P_ij is the dimensionless constant (i = 0, 1, 2, 3; j = 0, 1, 2, 3, 4); the value of P⁽¹⁾_ij is shown in

Table 4. The content of K_L.

Value	P00	P10	P01	P20	P11	P02	P30	P21	P12	P03	P31	P22	P13	P04
α = 0	−0.1946	1.585	0.5968	−0.1631	−0.4379	0.06025	0	0.07794	−0.01022	−0.001649	0	−0.005156	0.003405	−0.0006621
α = 0.25	0.3576	0.8363	−0.1893	−0.01239	−0.01615	0.06096	−0.01037	0.02333	−0.02733	0.003275	0.003736	−0.006168	0.004877	−0.001146
α = 0.5	1.387	0.1003	0.1399	0.1291	−0.1217	0.03236	−0.02091	0.04254	−0.03505	0.01073	0.004781	−0.008981	0.007158	−0.002031
α = 0.75	−0.6245	0.5882	0.7791	0.135	−0.4054	0.06082	−0.02838	0.07991	−0.04049	0.01065	0.006437	−0.01417	0.01121	−0.003105
α = 1.0	−0.828	0.7034	0.1463	0.1117	−0.2019	0.1179	−0.03042	0.07791	−0.08425	0.02335	0.008512	−0.01638	0.015	−0.004651

, the value of P⁽²⁾_ij is shown in

Table 5. The content of K_LR.

Value	P00	P10	P01	P20	P11	P02	P30	P21	P12	P03	P31	P22	P13	P04
α = 0	−12.96	7.616	0.1802	−1.802	−0.5758	0.1998	0.1437	0.2000	−0.06428	−0.00767	−0.03409	0.0266	−0.01054	0.002288
α = 0.25	−9.391	5.391	0.8606	−1.424	−0.5084	−0.04152	0.133	0.1489	−0.0149	0.003804	−0.03747	0.0351	−0.01815	0.003377
α = 0.5	−3.061	3.251	−0.6617	−1.265	−0.1393	0.1574	0.1443	0.1388	−0.05027	−0.01146	−0.04832	0.05271	−0.02807	0.006426
α = 0.75	−0.6676	4.592	−3.002	−1.937	0.2958	0.4112	0.2146	0.2178	−0.184	0.007226	−0.07174	0.08561	−0.04499	0.01009
α = 1.0	11.21	−0.8236	−1.624	−1.192	0.003626	0.1521	0.1971	0.216	−0.09182	−0.005945	−0.08304	0.1031	−0.06253	0.01531

, the value of P⁽³⁾_ij is shown in

Table 6. The content of K_R.

Value	P00	P10	P01	P20	P11	P02	P30	P21	P12	P03	P31	P22	P13	P04
α = 0	131.2	−82.5	−15.48	17.05	14.32	−2.878	−1.153	−3.607	0.9585	0.03846	0.3045	−0.09629	−0.008377	0.002295
α = 0.25	82.54	−56.65	−13.35	13.38	10.44	−0.9526	−1.048	−3.021	0.8748	−0.1454	0.319	−0.156	0.0314	0.001307
α = 0.5	76.44	−58.59	−10.91	14.67	11.83	−2.367	−1.209	−3.7	1.362	−0.1506	0.4116	−0.249	0.06592	−0.005961
α = 0.75	144.1	−102.1	−21.31	23.48	21.57	−5.23	−1.804	−6.166	2.572	−0.2827	0.6233	−0.4138	0.1193	−0.01369
α = 1.0	63.22	−67.89	−17.9	19.33	19.84	−4.724	−1.71	−6.207	2.814	−0.4226	0.6904	−0.5235	0.1853	−0.0248

.

From Section 3.2, the effect of Poisson’s ratio is given by Equations (12) and (13). Put Equations (12) and (13) into Equation (14), the semi-experimental equations of K_L, K_LR, and K_R considering the effects of Poisson’s ratio are given as:

$$\frac{K_L}{E_{s0}D}=g^{\left(1\right)}{f}^{\left(1\right)}\left({\nu }_s\right)$$

(15)

$$\frac{K_{LR}}{E_{s0}{D}^2}=g^{\left(2\right)}{f}^{\left(2\right)}\left({\nu }_s\right)$$

(16)

$$\frac{K_R}{E_{s0}{D}^3}=g^{\left(3\right)}{f}^{\left(3\right)}\left({\nu }_s\right)$$

(17)

The polynomial equations proposed in Equations (15)–(17) exhibit a strong fit with the outcomes obtained through the energy-based variational method in Table 3, with a coefficient of determination (R²) exceeding 0.997.

4. Application of the Methodology

4.1. The Lateral Deflection and Rotation of Caissons for Serviceability Limit State Calculations

In the design process, caissons cannot be decidedly classified as slender or rigid beams. By using Equations (15)–(17), the head stiffness of the caissons was given, and the values can be put into Equation (18) to predict the lateral deflection and rotation at the caisson’s head. The results are given in

Table 7. Detail of parameters in studies.

Fa (kN)	Ma/Fa/Lp (kNm)	D	Lp	t	Es	vs	Ip	ln(EP/ES0)	Lp/D
2000	0 to 1	5	12	0.4	60	0.25	15.40	5.5257	2.4
		2					0.68	5.5257	6

.

$$\begin{array}{l}w=\frac{K_R}{K_L{K}_R-K_{LR}^2}{F}_a-\frac{K_{LR}}{K_L{K}_R-K_{LR}^2}{M}_a\\ \theta =-\frac{K_{LR}}{K_L{K}_R-K_{LR}^2}{F}_a+\frac{K_L}{K_L{K}_R-K_{LR}^2}{M}_a\end{array}$$

(18)

Table 7 and Figure 14 present the head stiffness of the caissons obtained from Equations (15)–(17) and Table 4, Table 5 and Table 6. Results show that these values differ significantly from those provided by Shadlou et al. [20], who categorized caissons as either rigid or flexible beams. This suggests that the proposed method in this paper takes into account the caissons’ behavior more accurately by not categorizing them as strictly rigid or flexible.

Table 8. The values of K_L, K_LR and K_R.

Parameters	Present Study		Shadlou et al. [20] (Rigid)		Shadlou [20] (Flexible)
	D = 2 m	D = 5 m	D = 2 m	D = 5 m	D = 2 m
KL	1244	497	1652	1166	486
KLR	−7310	−1208	−9992	−6677	−1141
KR	78,716	6606	110,426	69,840	4879

outlines the parameters for comparing lateral deflection and rotation between the method in this paper, Shadlou [20], and FEM analysis performed by ABAQUS [53]. The relative parameters are presented in Table 8, where different M_a/F_a/L_p were covered. In the FEM, half models were built, both the method in this paper and the FEM analysis use consistent characteristics of the caisson and soil properties, grid size, boundary conditions, and method of force application, which is essential for a meaningful comparison. However, the method in this paper employs the Timoshenko beam to construct the caisson, whereas FEM analysis employs the solid element.

Figure 15a illustrates a comparison of lateral deflection errors between this study, FEM, and Shadlou et al. [20] for different values of L_p/D, specifically L_p/D = 2.4 and L_p/D = 6. The errors in lateral deflection range from 8% to 12% for L_p/D = 2.4 and from −2% to 6% for L_p/D = 6 when compared to FEA results. In contrast, the errors from Shadlou et al. [20] and an unspecified FEM show significantly higher deviations, with errors ranging from −26% to −21% for L_p/D = 2.4 and −54% to 36% for L_p/D = 6. Similarly, Figure 15b presents a comparison of the rotation errors obtained from the head stiffness equations (Equations (15)–(17)) in this study with FEM results and those from Shadlou et al. [20]. The rotation errors from this study are found to be more consistent with FEM results when compared to those from Shadlou et al. [20].

Therefore, in the case of a semi-rigid caisson, the prediction of head deflection and rotation under static loading conditions is significantly contingent on the accurate determination of its head stiffness. Hence, there is a clear imperative to develop a more precise method for forecasting head stiffness.

4.2. The Natural Frequency of OWT Considering the SSI Effect

Three cases from Belwind [54], Walney [36] and Kentish Flats Offshore Wind Farm [54] were adopted to verify the accuracy of Equations (15)–(17) in predicting the natural frequencies of Offshore Wind Turbine (OWT) models while considering the Soil-Structure Interaction (SSI) effect. The relative parameters of the OWT system are presented in Figure 16 and detailed in

Table 9. The relative parameters of OWT system.

Parameters		Belwind	Walney [36]	Kentish Flats Offshore Wind Farm
Rated power (MW)		3	3.6	-
Mass of rotor-nacelle assembly (mRNA) (kg)		130,800	234,500	132,000
Tower height		53	67.3	60.06
Tower bottom diameter (m)		4.3	5	4.45
Tower top diameter (m)		2.3	3	2.3
Tower wall bottom thickness (mm)		28	41	26
Tower wall top thickness (mm)		28	41	15
Tower Young’s modulus (GPa)		210	210	210
Tower density (kg/m3)		7860	7860	7960
Transfer piece diameter (m)		5	6	4.3
Platform height above mudline (m)		37	37.3	14.94
Monopile diameter (D) (m)		5	6	4.3
Monopile thickness (mm)		60	80	45
Monopile length (m)		35	23.5	29.5
Monopile young’s modulus (GPa)		210	210	210
Soil type		2	2	1
Es0 (Mpa)		15	30	52
vs		0.3	0.25	0.4
α		1	1	0
From previous studies (Laszlo, [54]; Gupat et al. [36])	lateral stiffness of foundation (GN/m3)	1.02	1.53	0.82
	cross stiffness of foundation, KLR (GN)	−7.59	−13.88	−5.42
	Rocking stiffness of foundation, KR (GN/rad)	91.93	205.72	58.77
From this paper	lateral stiffness of foundation, KL (GN/m3)	0.626	1.22	1.04
	cross stiffness of foundation, KLR (GN)	−5.74	−12.94	−5.72
	Rocking stiffness of foundation, KR (GN/rad)	89.24	205.26	66.69

. For the modification of the OWT system, FEM was conducted using ABAQUS [53], and a concentrated mass block was placed on top of a series of Timoshenko beams. To account for the monopile-soil interaction, three types of springs with initial stiffness values of K_L, K_LR, and K_R, which were calculated using Equations (15)–(17), were used to account for the SSI effect.

From Table 9, it is observed that in the second case, where the critical length of the pile aligns more closely with a slender pile (as indicated in Table 1), the results of K_L, K_LR, and K_R closely match those from the reference [54]. Conversely, in the first and the third cases, where the classification of the pile as slender or rigid is less clear based on Table 1 and Table 2, the results from the first case are lower than those expected for a typical rigid pile according to the reference [54], on the other hand, the results from the third case are higher than those expected for a typical flexible pile according to the reference [54].

The proposed method in this study has also been verified to enhance the accuracy of natural frequency predictions for OWT models considering the SSI effect. The natural frequencies of wind turbine structures, after accounting for the SSI effect, are provided in

Table 10. Influence of spring stiffness on the natural frequencies of the elastically supported immersed beam with eccentricity.

Offshore Wind Farm	Mode	Measured Frequency	Frequency (ABAQUS)		Error (%)
			From Previous Studies	From This Paper
Belwind	1	0.372	0.364	0.365	−0.27
	2	-	0.379	0.383	−1.04
	3	-	5.083	5.006	1.54
	4	-	6.279	6.483	−3.15
	5	-	15.827	15.720	0.68
	10	-	88.20	88.09	0.12
	18	-	303.26	300.36	0.97
Walney	1	0.350	0.337	0.337	0.00
	2	-	0.427	0.404	5.82
	4	-	10.876	10.195	6.68
	5	-	10.887	10.876	0.10
	10	-	50.187	49.823	0.73
	20	-	163.28	162.83	0.28
Kentish Flats Offshore Wind Farm	1	0.339	0.343	0.342	0.00
	3	-	7.3693	6.736	9.40
	5	-	13.395	11.954	12.05
	7	-	35.449	31.339	13.11
	10	-	79.674	73.455	8.47
	18	-	316.35	300.31	5.34

. Among the studied cases, the first natural frequency, as evaluated by both the references and this study, closely aligns with field testing results in the rotor-stop condition. However, discrepancies become more apparent in higher natural frequencies, with the largest error reaching 13%. This underscores the need for a more accurate equation for calculating dimensionless head stiffness, particularly when dealing with high-order self-resonant frequency calculations.

5. Conclusions

This study focuses on developing a precise method for calculating the head static stiffness of semi-rigid piles, without the need to consider the critical length, a parameter commonly used in traditional approaches. To achieve this objective, an extensive analysis comprising nearly twenty thousand cases was conducted using the energy-based variational method with Timoshenko beam theory. These cases encompassed a wide range of scenarios, including steel pipe piles and concrete caissons with diameters ranging from 2 m to 10 m, slenderness ratios between 2 and 10, and various soil types represented by the parameter α (ranging from 0 to 1) with a wide range of elastic moduli (E_s) from 2 Mpa to 300 Mpa and Poisson’s ratios (v_s) spanning 0.2 to 0.45.

The results demonstrate a limitation of traditional methods, as they fail to cover the full spectrum of foundation conditions within the aforementioned range. In contrast, the polynomial equations proposed in this paper exhibit a strong fit with the outcomes obtained through the energy-based variational method, with a coefficient of determination (R²) exceeding 0.997.

In practical applications, the method introduced in this study offers a distinct advantage. It enables engineers to achieve accurate results for head deflection and rotation of foundations under static loading conditions without the need for FDM or FEM software. This research also contributes to a more precise and accessible method for calculating foundation behavior, with potential applications in the analysis of high-order self-resonant frequencies in Offshore Wind Turbine (OWT) systems.

Notably, while these equations are applicable to a wide range of soil types and foundation scenarios, caution should be exercised when applying them to unique or unconventional cases without the aforementioned range, such as monopiles embedded in rocks.