Mechanism-Aligned Simplified Soil–Pile Interaction Models for Offshore Wind Turbine Monopiles in Sand

Abstract

Monopiles are the predominant foundation type for offshore wind turbines (OWTs). Their diameters have increased substantially to accommodate larger structures, while current design approaches primarily rely on the API “p-y” model to simulate soil–pile interaction (SPI), which significantly underestimates the ultimate lateral pile capacity of large-diameter monopiles. Further, the API model accounts only for lateral soil resistance, neglecting mechanisms that substantially influence the lateral response of piles with low length-to-diameter (L/D) ratios, including pile toe shear, toe moment, and axial interfacial shaft friction. To address these problems, this study proposes a complete set of mechanism-aligned, spring-based SPI models capable of accurately simulating lateral pile response in sand across the full L/D spectrum typical of OWTs. The models include: a one-spring “p-y” model for flexible piles, capturing distributed lateral soil resistance; a two-spring “p-y + M_R-θ_R” model for semi-rigid piles, which additionally accounts for pile toe shear and bending moment resistance against rigid-body rotations; and a three-spring “p-y + M_R-θ_R + M_p-θ_p” model for rigid piles, which further includes rotational springs to account for distributed moment resistance due to rotation-induced shaft friction effects in sand. The derived spring parameter formulas have been calibrated using readily available engineering parameters, such as soil modulus, friction angle, and pile geometry. The three mechanism-aligned SPI models were validated against full-scale offshore monopile tests, centrifuge tests, and small-scale laboratory experiments, achieving less than 10% error in predicted pile capacities and less than 15% error in soil–pile coupled stiffness evolution.

1. Introduction

With the increasing demand for clean and sustainable energy, offshore wind power has emerged as a major contributor to global energy production [1]. Expanding energy generation via offshore wind turbine (OWT) farms along extensive coastlines has become a viable solution to achieve the goal of carbon neutrality. Consequently, OWT deployment is increasing worldwide, with an expected compound annual growth rate of 31% through 2027 [2]. For example, the installed offshore wind power capacity in China has been steadily increasing, reaching 40.55 GW in 2024.

Various foundation types are used in OWTs, including monopile, jacket, gravity-based, suction bucket, and floating structures. Due to their structural simplicity and ease of construction, monopile foundations account for over 60% of OWT foundations globally [3]. Early designs of OWT monopile foundations typically featured diameters ranging from 4 to 6 m [4], with length-to-diameter ratios (L/D) generally ranging from 4 to 8. Due to expansion into deeper waters, increasing environmental loads, and advances in blade and turbine design, the deployment of large-capacity turbines (8–12 MW) has become more widespread, resulting in monopile diameters increasing to 8–10 m [5,6]. Currently planned major wind farm developments point towards the deployment of ultra-large turbines (12–22 MW) founded on monopile foundations with diameters over 10 m and L/D ratios of 4 and below [7].

Slenderness, i.e., the L/D of monopiles, is the key parameter controlling their rigidity, and thus their load transfer mechanism onto the soil. In turn, the relative stiffness of the soil–pile system governs the soil–pile interaction (SPI) mechanism and the failure mode of the foundation system. Accordingly, monopiles can be classified into three categories showing distinct SPI mechanisms at failure: flexible piles, semi-rigid piles, and rigid piles. The authors’ previous study on the relation of L/D and OWT monopile mechanisms [8] found that flexible monopiles (large L/D) exhibit flexural responses, dominated by bending at the upper portion of the pile. As L/D decreases (i.e., semi-rigid to rigid piles), the pile response gradually transitions to a rigid-body type behavior, dominated by rotation. Fully rigid piles exhibit minimal flexure, pronounced rotation, and substantial pile toe displacements. Considering the distinct failure mechanisms of piles with different L/D, it becomes clear that mechanism-aligned SPI models are required to accurately capture the lateral response of monopiles.

During the operation of OWTs, SPI directly governs overall structural integrity and lifecycle safety [8,9]. Currently, the “p-y” method is widely used to model SPI in monopile-supported OWT design, where p represents soil resistance and y denotes the lateral displacement of the pile. In this approach, the surrounding soil is discretized into a series of laterally distributed nonlinear springs along the pile shaft. The “p-y” curves recommended by the American Petroleum Institute (API) [10], originally developed by Matlock [11] from static field tests on small-diameter (<2 m) flexible piles, remain the industry standard. The API formulation does not account for the scale effect of pile diameter, though OWT monopiles typically have diameters of 4–12 m. Moreover, OWT monopiles exhibit semi-rigid to rigid behavior and are subjected to combined lateral force and moment loading. Despite these substantial differences in geometry and behavior compared with the dataset used to derive the “p-y” curves, the API formulation has also been adopted in the DNV guidelines [12] for OWT design. Further, the API “p-y” curves are primarily calibrated to ultimate bearing capacity rather than pile–soil coupled stiffness, leading to initial stiffness estimates that are 30–50% more conservative than expected [13,14]. This limitation is further reflected in discrepancies of 11–20% in predicted dynamic responses (e.g., natural frequency shifts under long-term cyclic loading) when compared to the high-fidelity finite element model (FEM) or field measurements [15,16]. Thus, the “p-y” method has been shown to misrepresent the load transfer mechanisms of OWT monopiles, often overestimating soil–pile stiffness (i.e., spring stiffness) while underestimating ultimate lateral resistance [17,18,19,20]. A key reason for the observed errors when using the “p-y” method is that it exclusively considers lateral soil resistance under horizontal loading. For large-diameter monopiles, however, additional rotation-induced load transfer mechanisms, namely toe shear, toe moment, and shaft friction, play significant roles depending on pile slenderness (L/D), soil type, and soil stiffness [21,22].

To tackle these issues, alternative approaches to optimize “p-y” models for OWT monopiles have been proposed in the recent literature, including enhanced stiffness calibration [23,24] and model updating [9]. However, these methods rely heavily on experimental, field, or high-fidelity FEM data, undermining the primary advantages of simplified models: efficiency and ease of use. Similarly, correction factor-based approaches [24,25,26] are typically derived from limited datasets and fail to resolve the fundamental issue of inaccurately represented SPI mechanisms, thereby limiting their general applicability and physical rigor.

Moving towards mechanism-aligned spring-based SPI modeling, Zhang and Andersen [27] introduced scaling factors based on laboratory and FEM analyses of soil flow around cylindrical piles to incorporate shaft frictional resistance within the traditional “p-y” framework. The PISA project [6] further advanced SPI modeling by proposing a 1-dimensional SPI modeling framework for monopiles in sand [28] and clay [29], integrating distributed lateral resistance, shaft friction, toe shear, and toe moment. However, the PISA model employs Gauss-point-level coupling between a Timoshenko beam (pile) and soil elements, rather than using rheological spring elements with node-level coupling, commonly available in commercial FEM software. This necessitates custom finite element implementation and limits the model to one-dimensional in-plane loading scenarios, thereby restricting its practical adoption and generalizability.

To retain the simplicity of spring-based approaches while incorporating key physical mechanisms identified by PISA, several multi-spring models have been proposed in recent years. Zhang and Andersen [30] introduced the “p-y + s-u” dual-spring model for low L/D monopiles in clay, combining lateral resistance (p-y) with toe shear-displacement (s-u) springs. However, the use of translational springs to represent toe resistance contradicts the rotational flow-dominated behavior observed in such piles, and the model neglects both toe moment resistance and shaft friction. Wang et al. [31] proposed the “p-y + M_R-θ_R” model for cohesive soils, incorporating a moment-rotation spring at the pile’s center of rotation to capture rotation-induced soil resistance effects (toe moment and shear). While this improves the mechanistic representation of SPI for semi-rigid and rigid piles, it omits shaft friction, which is particularly important in sandy soils. Fu et al. [32] extended this approach for clays by introducing a three-spring “p-y + s-u + M_p-θ_p” model that includes shaft frictional resistance. However, similar to earlier work, toe resistance is still modeled using translational springs, failing to capture its fundamentally rotational nature. In summary, existing multi-spring SPI models for large-diameter monopiles are largely limited to clayey soils, while models for sand either do not provide a unified spring framework capable of capturing all SPI mechanisms across the full range of L/D categories (flexible, semi-rigid, and rigid) or require custom finite element implementations.

To address these limitations, this study proposes a complete set of SPI spring models for large-diameter monopiles in sand that explicitly account for all L/D-dependent soil resistance mechanisms under lateral loading. Three formulations are developed: “p-y”, “p-y + M_R-θ_R”, and “p-y + M_R-θ_R + M_p-θ_p”, corresponding to flexible, semi-rigid, and rigid pile behavior, respectively. The spring parameters are derived using readily available soil and pile properties, enabling rapid and straightforward calibration for a wide range of L/D ratios and sand conditions. Furthermore, the use of a conventional spring-based framework ensures their seamless use in any commercial FEM software. The proposed models are validated against both numerical simulations and experimental case studies, demonstrating improved accuracy in capturing SPI behavior.

2. Numerical Models and Mechanisms of Monopile Resistance

2.1. Review of Soil–Pile Interaction Mechanisms

Previous studies have shown that the L/D of monopiles has a significant influence on the soil flow mechanism around the pile and the associated SPI behavior [8,33,34]. To review the effect of L/D on SPI behavior and demonstrate the soil flow mechanisms around monopiles at failure, three representative piles analyzed using FEM are presented first: flexible (D = 2 m, L = 40 m, L/D = 20), semi-rigid (D = 8 m, L = 40 m, L/D = 5), and rigid (D = 3 m, L = 10 m, L/D = 3.3). A static pushover was simulated with a target displacement of 0.1D at the mudline. Resulting soil displacement vectors are plotted in Figure 1, alongside pile displacements as a function of depth. The results demonstrate that soil flow mechanisms and pile deformations for each representative pile are distinct. Note that the parameters of these models are discussed in the following section; the results are shown here first for demonstrative purposes.

Soil around the upper portion of the flexible pile (Figure 1a) exhibits a wedge-shaped zone followed by a horizontal full-flow zone, due to significant bending at the pile head. However, the pile exhibits negligible displacements below z/D = 14 (0.7L). This supports modeling a fixed-end condition with only distributed lateral springs to capture passive soil resistance near the pile head.

The semi-rigid pile (Figure 1b) displays the same wedge and horizontal flow zones but also develops rotational soil flow about a lower center of rotation (~0.78L), evidenced by nonlinear, parabolic pile displacement profiles and a ~0.05D displacement at the pile toe, commonly referred to as “toe kick”. This toe kick mobilizes pile toe moment and shear resistance. The observed coupled flexural and rotational response requires distributed lateral springs to capture the wedge and full-flow, along with a rotational spring to account for toe kick and rotational flow. This rotational flow mechanism has also been observed and analyzed by Hong et al. [33] through centrifuge model tests in sand. Furthermore, studies [35,36] have demonstrated that the axial interfacial shaft friction mobilized during rigid-body rotation contributes significantly to the horizontal load resistance of piles with low L/D ratios, particularly in sandy soils. Therefore, the behavior of piles with lower L/D ratios is examined further.

The rigid pile (Figure 1c) primarily exhibits a wedge-shaped zone combined with pronounced rotational flow, producing a nearly linear displacement profile with a toe kick roughly six times larger than the semi-rigid pile (~0.3D). Such pronounced rigid-body rotation generates sufficient vertical displacement to fully mobilize interfacial shaft friction, requiring explicit modeling of distributed moment resistance via rotational springs in addition to lateral springs and a central rotational spring.

Overall, the soil flow mechanism around monopiles is primarily governed by the L/D ratio, with the three distinct SPI behaviors exhibiting fundamentally different load transfer mechanisms that control ultimate resistance. Simplified SPI models must therefore explicitly account for L/D dependency. While classical distributed “p-y” spring approaches are sufficient to capture the purely lateral resistance mechanism near the pile head of flexible piles, accurate modeling of semi-rigid and rigid piles requires simultaneous consideration of lateral and rotational load transfer components. Accordingly, this study first evaluates the contributions of the four primary load transfer mechanisms, lateral soil pressure, toe shear, toe moment, and distributed moment resistance due to shaft friction, to the ultimate resistance of monopiles in sand across a range of L/D ratios spanning flexible to rigid pile behavior.

2.2. Numerical Models and Validation

To ensure the accuracy and applicability of the finite element and material models used in the parametric analyses, a prototype-scale validation model was constructed in ABAQUS (Version 2025). The model replicated measurements from a centrifuge test on the horizontal pushover response of a 2.5 m diameter and 50 m long (L/D = 20) steel monopile (EI = 56.66 GNm²) in saturated cohesionless Fujian standard sand (φ = 35°, γ′ = 0.936 g/cm³), conducted by Zhu et al. [33]. To facilitate large deformations in ABAQUS, the sand was modeled using C3D8R solid elements with the elasto-plastic Mohr–Coulomb constitutive relationship, while the monopile was modeled using thin C3D8R solid elements and linear elastic steel material. Contact between the pile and soil was modeled using a small-sliding surface-to-surface formulation, which incorporates Mohr–Coulomb frictional behavior in the tangential direction via the penalty method and a hard contact in the normal direction to capture interfacial load transfer. Within a radial distance of 1D from the pile center, a refined mesh with an element size of D/24 was adopted. To minimize boundary effects, the diameter of the cylindrical soil domain was set to 25D, following Kato et al. [8]. Horizontal displacements (x and y directions) were constrained at the outer soil boundary, and the bottom boundary was fixed. The horizontal loading was applied at the pile head, h = 10 m above the mudline. The generalized model setup is depicted in Figure 2. The resulting pile head load–displacement diagrams are compared with the centrifuge measurements in Figure 3. The FEM results show good agreement with the centrifuge measurements, yielding a mean absolute percentage error (MAPE) of 6.05%, verifying that the modeling approach is sufficiently accurate and reliable for subsequent parametric analyses.

After validation, variations in the validated finite element model were constructed to perform additional static pushover analyses of monopile foundations over a wide range of L/D ratios. The results of this parametric study form the basis of this paper, including analyses of soil flow and soil resistance mechanisms, as well as the dataset for deriving spring parameter formulas using readily available soil and pile properties for the proposed one-, two-, and three-spring SPI models. For the parametric analysis, typical soil parameters observed in the sand-dominated coastal region of the East China Sea were adopted [35] (see

Table 1. Mohr–Coulomb parameters for sand.

Soil Type	Effective Unit Weight (kN/m3)	Elastic Modulus (MPa)	Poisson’s Ratio	Internal Friction Angle (°)	Cohesion (kPa)	Dilatancy Angle (°)
Sand	9.76	0–82.5 (z: 0–40 m)	0.25	35°	0.1	5

), and ten monopiles with varying L/D ratios were analyzed, covering flexible, semi-rigid, and rigid piles (see

Table 2. Monopile properties.

Embedment Length, L (m)	Pile Diameter, D (m)	L/D	Pile Length, L0 (m)	Pile Wall Thickness, t (m)	Pile–Soil Relative Stiffness, EpIp/Estoe	Pile Mechanism
40	2	20	50	0.02635	0.000075	flexible
40	3	13.3	50	0.03635	0.00038	flexible
40	4	10	50	0.04635	0.0012	flexible
40	5	8	50	0.05635	0.0029	semi-rigid
40	6	6.7	50	0.06635	0.0061	semi-rigid
40	8	5	50	0.08635	0.019	semi-rigid
20	5	4	30	0.05635	0.071	semi-rigid
10	3	3.3	20	0.03635	0.22	rigid
10	4	2.5	20	0.04635	0.71	rigid
10	5	2	20	0.05635	1.73	rigid

). The critical pile length (L_cr) [37], which depends on the relative stiffness of the pile–soil system (see

Table 3. Pile classification criteria [37].

$L_{cr}=4.44{\left({\displaystyle \frac{E_p{I}_p}{E_s^{toe}}}\right)}^{{\displaystyle \frac{1}{4}}}$	$L>L_{cr}$, flexible
	${\displaystyle \frac{L_{cr}}{3}}<L<L_{cr}$, semi-rigid
	$L<{\displaystyle \frac{L_{cr}}{3}}$, rigid

), was used to classify SPI behavior because it has been shown to provide the most accurate categorization for OWT monopiles [8]. To ensure that the classification formula produces SPI categories consistent with observed soil-flow mechanisms around OWT monopiles [8,37], which form the scientific basis for accurate categorization, the pile’s second moment of inertia, I_p, is calculated assuming a solid cross-section, and the soil elastic modulus, E_s^toe, is evaluated at the pile toe. The sand elastic modulus followed a logarithmic function with depth, while the steel monopiles were modeled as linear elastic beams with a Young’s modulus of 210 GPa and a Poisson’s ratio of 0.3. Pile wall thickness was defined following the API design requirements for OWT monopiles.

$$t=0.00635+{\displaystyle \frac{D}{100}}$$

(1)

where t is the pile wall thickness and D is the pile diameter, both in meters.

The friction coefficient, μ, of the Mohr–Coulomb material assigned to the tangential component of the surface-to-surface interface element between the pile shaft and the sand is defined as μ = tanδ, where δ denotes the soil–pile interface friction angle. Following established practice for sand [38,39], δ was taken as two-thirds of the soil internal friction angle φ, resulting in μ = 0.44.

2.3. Mechanistic Analysis of the Contribution Ratios of Four Pile Resistance Components

As demonstrated in the previous sections, when piles of different L/D ratios (or SPI categories) are subjected to horizontal loads at their heads, the load transfer mechanism along the monopile varies significantly. This results in the ultimate resistance composed of different combinations of four soil-restraining effects: distributed lateral soil resistance, distributed interfacial shaft friction moment resistance, pile toe shear force resistance, and pile toe bending moment resistance. The relative influence of these components depends on the SPI mechanism and, consequently, on L/D. To quantify the contribution of each resistance component, the moment contribution ratio (MCR) defined by Murphy et al. [40] is employed. The contribution of each component is calculated as a percentage of the externally applied moment about the pile’s center of rotation (see Figure 1), which corresponds to the restoring moment exerted by the soil onto the monopile. The MCR analysis allows the quantitative identification of necessary spring combinations that can capture all aspects of the observed SPI mechanisms for piles with different L/D ratios.

For this study, three representative monopiles with a fixed diameter of D = 4 m and L/D ratios of 2.5, 5, and 10 were selected for MCR analysis. The evolution of MCR from each resistance component with increasing pile head displacement is shown in Figure 4. These curves clearly illustrate the mobilization of each component and provide insight into the SPI mechanisms of piles with various L/D ratios. Stars in Figure 4 indicate the ultimate limit state (ULS, y/D = 0.1) and the serviceability limit state (SLS, 0.25° pile head rotation [41] or equivalently: y/D = L/230/D), enabling evaluation of each component’s mobilization level at relevant design states.

The dependence of each resistance component on the pile L/D ratio is evident. For the flexible pile (L/D = 10), horizontal soil resistance, p, is fully mobilized by the SLS and dominates the ULS with approximately 95% contribution. In the semi-rigid pile, p shows a similar mobilization trend (91% at ULS), while the contributions from toe bending moment, M_t, and distributed shaft friction moment, m, increase substantially. Distributed shaft friction moment resistance becomes fully mobilized only at relatively large displacements (~0.05D), reflecting its dependence on significant relative vertical movement between the pile and soil. In contrast, the toe bending moment, M_t, is fully mobilized at small displacements. As deformations increase, M_t gradually decreases because the distributed moment, m, increasingly contributes to the ultimate lateral resistance through interfacial frictional resistance along the upper portions of the pile, where pile–soil relative axial displacements are largest. In the rigid pile, p contributes the least to ultimate resistance (79–82%) and, along with the pile toe shear force, Q_t, it fully mobilizes at small displacements but diminishes with increasing pile head displacements. Conversely, M_t and m gradually mobilize as displacement increases, ultimately overtaking a significant portion of the MCR from p and Q_t, respectively. The ultimate lateral resistance of rigid piles is particularly influenced by shaft friction (~7%) and the toe bending moment (~13%) due to pronounced rigid-body rotations.

The results demonstrate that the load transfer mechanisms of SPI differ significantly across piles with varying L/D ratios, indicating distinct contributions of the four resistance components to ultimate lateral pile resistance. Therefore, to capture these contributions, developing distinct mechanism-aligned, spring-based SPI models tailored to each SPI category is crucial for the simplified yet accurate modeling of laterally loaded monopiles.

3. Derivation of Spring Models for Monopiles with Different Length-to-Diameter Ratios

3.1. One-Spring “p-y” SPI Model for Flexible Piles

Based on the above analysis, lateral soil resistance constitutes the dominant component of resistance forces, with its contribution increasing with pile L/D ratio. For flexible piles (L/D > 10), lateral soil resistance governs the ultimate capacity (>95% contribution) as these piles undergo nearly pure bending deformations. The distributed “p-y” spring method, originally developed by Matlock [11] for such piles, has been widely validated and employed to model SPI for laterally loaded monopiles. Accordingly, this study adopts the distributed “p-y” spring approach, illustrated in Figure 5, with a reworked formulation to suit OWT monopiles in sand.

3.1.1. Defining the Proposed “p-y” Curve Formulation

The API “p-y” curve formulation was derived from pile pushover data on small-diameter concrete piles and shows poor performance when applied to large-diameter steel monopiles. Therefore, the hyperbolic “p-y” curve proposed by Kondner et al. [42] is adopted here. This curve offers a simple formulation, accommodates complex soil conditions, and more accurately captures nonlinear SPI, as demonstrated by Kim et al. [43]. The formulation given in Equation (2) is adopted, and data from the parametric analyses are used to derive best-fit expressions for its governing parameters: the ultimate horizontal soil resistance, p_u, and the initial spring stiffness, K.

$$p={\displaystyle \frac{y}{\frac{1}{K}+\frac{y}{p_u}}}$$

(2)

To enable wide applicability for typical OWT pile embedment depths and pile diameters, the formulations for p_u and K are corrected using the numerical data. Specifically, the ultimate horizontal soil resistance, p_u, is primarily influenced by soil properties, pile geometry, and depth effects (overburden). Values of p_u at depths from 1 m to 10 m were obtained from the pushover analysis of six monopiles with diameters listed in Table 2 (2–8 m). To ensure the compatibility of the proposed “p-y” springs with semi-rigid and rigid piles as well, pile lengths were selected such that L/D ratios range between 5 and 20. The depth, z, versus p_u results are plotted in Figure 6a. The Broms [44] formula, extended with correction factors, given in Equation (3), was used to fit the numerical data. This formula accounts for both depth and pile diameter effects on ultimate horizontal soil resistance:

$$p_u=a{K}_p{ \gamma }^{\prime }{ z}^b{ D}^c,$$

(3)

where K_p is the passive earth pressure coefficient, γ′ is the effective soil unit weight, z is soil depth, and D is the pile diameter. By fitting the p_u data obtained from six monopiles with different diameters and L/D ratios (see Figure 6b), the optimized correction factors for the ultimate soil resistance factor a, depth factor b, and diameter factor c yield the final fitted formula given in Equation (4):

$$p_u=2.98K_p {\gamma }^{\prime }{ z}^{1.22}{ D}^{1.07},$$

(4)

To assess its accuracy and applicability across a range of L/D ratios, the modified p_u formula was compared with FEM results and the API formulation in Figure 6b. For large L/D (flexible piles), the modified formula closely matches FEM results, with a MAPE below 5%. Although some discrepancy occurs at smaller L/D ratios, overall agreement remains good, with MAPE < 12%, confirming that the modified p_u formula is accurate and broadly applicable to rigid and semi-rigid piles as well.

The initial spring stiffness, K, which is equivalent to the initial subgrade modulus of the soil along the pile, k, is the other key parameter in the hyperbolic “p-y” formula. However, in sand, the initial subgrade modulus is highly dependent on overburden pressure (i.e., soil depth) and pile diameter, making its direct use impractical for simplified spring-based SPI modeling. Therefore, following Reese et al. [45], the initial subgrade modulus, k, is taken as a constant, and scaling factors with respect to depth and diameter are introduced to adjust the modulus along the pile. This approach enables a straightforward calibration procedure for the distributed “p-y” spring stiffness, K, along the pile. To develop these scaling factors, initial subgrade moduli for monopiles of varying diameters (with a fixed embedment length of 40 m) were extracted from soil reaction–displacement curves recorded at 1 m intervals along the pile embedment depth from FEM pushover analyses. Figure 7a plots the extracted values with depth for the six pile diameters listed in Table 2. The results show that the initial subgrade modulus varies nonlinearly with depth and increases with pile diameter at a given depth. The initial spring stiffness, K, is then calculated using the formula proposed by Kallehave et al. [13] with correction factors to ensure its applicability to OWT monopiles:

$$K=k{ z}_{ref}{\left({\displaystyle \frac{z}{z_{ref}}}\right)}^a{\left({\displaystyle \frac{D}{D_{ref}}}\right)}^b,$$

(5)

where z_ref is the reference depth, D_ref is the reference pile diameter, a is the depth correction factor, b is the pile diameter correction factor, and k is the initial modulus of subgrade reaction determined as a function of internal friction angle (φ) per [45].

The correction factors were optimized to fit the family of numerical curves in Figure 7a. First, an initial modulus, k₁, at z_ref = 1 m and D_ref = 2 m was used to construct a normalized model for the evolution of k/k₁ with embedment depths, from 1 m to 40 m. Dimensional analysis and parameter standardization yielded the relationship between k and z for a fixed D_ref, using the depth correction factor a = 0.65. Second, the variation in k/k₁ values at depths ranging from 1 to 40 m with respect to the normalized pile diameter, D/D_ref, was determined for D ranging between 2 and 8 m. The fitted pile diameter correction factor was determined as b = 0.43. The resulting formulation provides the initial spring stiffness along the pile:

$$K=k{z}_{ref}{\left({\displaystyle \frac{z}{z_{ref}}}\right)}^{0.65}{\left({\displaystyle \frac{D}{D_{ref}}}\right)}^{0.43},$$

(6)

Validation against FEM results of the calculated spring stiffnesses, K, along the embedment depth of the six monopiles (see Figure 7b) shows good agreement, with MAPE ranging between 4.5% and 12%.

3.1.2. Performance of the One-Spring “p-y” Model

The proposed one-spring “p-y” model was implemented in ABAQUS to simulate SPI for monopiles with varying L/D ratios. Monopiles were modeled as elastic beams with “p-y” springs distributed along the embedded shaft at 1 m intervals. Static pushover analyses were conducted up to a pile head displacement of 0.1D. The calibrated “p-y” curves capture the distributed lateral soil–pile stiffness along the shaft. Reaction forces and lateral displacements at the pile head were extracted and compared with results from traditional API “p-y” curves and high-fidelity FEM simulations in Figure 8. The error between the predicted lateral bearing capacity of the piles using the proposed model and FEM is shown in Figure 9.

Figure 8 shows that the traditional API “p-y” model significantly overestimates both the soil–pile coupled stiffness and ultimate lateral pile capacity. In contrast, the proposed one-spring model closely replicates both the stiffness evolution and ultimate capacity. For L/D ≥ 8, the model predicts lateral bearing capacity with errors below 2% relative to FEM. However, with decreasing L/D, the error increases sharply, exceeding 25% for L/D < 8, highlighting the necessity of accounting for rotation-induced SPI mechanisms observed in semi-rigid and rigid piles.

3.2. Two-Spring “p-y + M_R-θ_R” SPI Model for Semi-Rigid Piles

For both semi-rigid and rigid piles, in addition to horizontal soil resistance, p, the moment constraint provided by the soil against rigid-body rotation must be considered, as indicated by the preceding MCR analysis. For semi-rigid piles, where rotation is moderate, incorporating pile toe moment, M_t, and shear force, Q_t, arising from opposing lateral soil resistances above and below the rotation point may be sufficient. In these cases, the axial displacement component from the rigid-body rotation is not large enough to mobilize significant axial interfacial shaft friction, as evidenced by its MCR in Figure 4. Since pile toe resistance is primarily governed by rotational soil flow, this study proposes an “M_R-θ_R” spring at the center of rotation to simultaneously capture M_t and Q_t, where M_R is the moment about the pile’s center of rotation and θ_R is the rotation angle. By combining distributed “p-y” springs above the rotation point with a single “M_R-θ_R” spring at the rotation point, a two-spring “p-y + M_R-θ_R” SPI model is developed for monopiles in sand. A schematic of the proposed two-spring model is presented in Figure 10.

3.2.1. Defining the “M_R-θ_R” Rotational Spring Curves

Accurate identification of the monopile’s center of rotation is critical for the “M_R-θ_R” spring. Previous studies suggest that the rotation point of rigid monopiles is typically located at 0.75–0.80 L. In this study, a representative center of rotation for semi-rigid and rigid piles was determined through parametric FEM analyses. Six monopiles with L/D ratios ranging from 2 to 8 were selected. The center of rotation was defined as the point along the pile where displacement is zero at failure, corresponding to maximal rotational flow and full mobilization of M_t and Q_t (Figure 4). Figure 11 shows the locations of the centers of rotation for the analyzed piles, with the failure points (y = 0.1D) highlighted. The average depth of the center of rotation across all semi-rigid and rigid piles was found to be z/L = 0.78.

Next, the M_R-θ_R relationship was developed as a formula normalized by the ultimate pile toe moment, M_u, to enable straightforward calibration. FEM results of moment-rotation responses about the center of rotation for monopiles with L/D ratios between 2.5 and 10 are plotted in Figure 12. These responses exhibit a well-defined hyperbolic tangent trend. The ultimate moment, M_u, for each monopile was extracted at a mudline displacement of 0.1D. The M_R-θ_R curves were then normalized by M_u (see Figure 13) and fitted with a hyperbolic tangent function, yielding the normalized relationship:

$${\displaystyle \frac{M_R}{M_u}}=\mathit{tanh}\left(9.05{\left({\theta }_R\right)}^{0.71}\right)$$

(7)

To facilitate the calibration procedure for practical engineering applications, an empirical formula for estimating Mu in sand was derived based on pile L/D ratio, which controls the ultimate toe moment. Extracted M_u values were plotted against L/D in Figure 14 and fitted using the expression:

$$M_u=250.51{\left({\displaystyle \frac{L}{D}}\right)}^{-1.03}$$

(8)

Using the geometric parameters of a monopile alone, its ultimate moment resistance, M_u, in sand can be calculated from Equation (8) and substituted into Equation (7) to generate the appropriate “M_R-θ_R” curve. This provides a convenient means of calibrating the proposed nonlinear rotational spring. Placed at z/L = 0.78, this spring captures the soil resistance generated by rotation-induced mobilization of pile toe moment and shear force in semi-rigid and rigid OWT monopiles. Combined with the proposed “p-y” springs distributed along the pile shaft, this forms the mechanism-aligned two-spring “p-y + M_R-θ_R” SPI model for simulating lateral responses of semi-rigid monopiles in sand.

3.2.2. Implementation and Performance of the Two-Spring “p-y + M_R-θ_R” Model

The proposed two-spring “p-y + M_R-θ_R” model was implemented in ABAQUS to simulate the SPI of monopiles with various L/D ratios. Monopiles were modeled as elastic beams, with distributed “p-y” springs applied along the shaft at 1 m intervals up to z/L = 0.78, where the “M_R-θ_R” rotational spring was located. A zero-displacement constraint (in 2 DOF) was applied at the rotation point, while all nodal constraints on the beam below this point were removed. This setup is critical to avoid double-counting lateral soil resistance, because the “M_R-θ_R” spring already captures lateral soil resistance below the pile as a moment resistance arising from differential lateral soil reactions above and below the pile’s rotation point. In this configuration, the “p-y” springs above the rotation point represent the cantilever-type lateral SPI, while the “M_R-θ_R” spring models the soil resistance along the entire pile length against rigid-body rotations.

Using the proposed two-spring model, static pushover analyses were conducted up to 0.1D pile head displacement on eight models with L/D ratios ranging from 2 to 10. Reaction forces and lateral displacements at the pile head were extracted and compared in Figure 15 with results using the proposed one-spring model, the traditional API “p-y” curves, and high-fidelity FEM. Figure 16 shows the error in lateral pile capacity predicted by the two-spring “p-y + M_R-θ_R” model relative to high-fidelity FEM results.

For semi-rigid piles with L/D between 5 and 10, the two-spring model accurately reproduces the coupled soil–pile stiffness evolution as evident from the load–displacement curves in Figure 15. The model also predicts pile lateral bearing capacity very accurately, with errors increasing from 2 to 9% with decreasing L/D. For piles with L/D < 4, however, the error increases above 20%, indicating that the two-spring model is most suitable for semi-rigid piles with L/D ≥ 5. While the error is rather high for rigid piles, the two-spring model shows significant improvements over the one-spring “p-y” model. The primary reason for the observed high errors for rigid piles is the substantial MCR of axial interfacial shaft friction to their lateral capacity. Therefore, an additional spring component is required to capture this mechanism in rigid piles experiencing pronounced rotational flow.

3.3. Three-Spring “p-y + M_R-θ_R + M_p-θ_p” SPI Model for Rigid and Semi-Rigid Piles

As shown in the prior section, the two-spring model reproduces the nonlinear evolution of coupled soil–pile stiffness and lateral pile capacity for semi-rigid piles fairly accurately. However, as pile L/D decreases, the model fails to fully capture the SPI behavior. In rigid piles experiencing substantial rigid-body rotation, the pile cross-section undergoes significant rotation, generating relative axial displacements between the pile surface and surrounding soil. This relative motion mobilizes interfacial shaft friction, which resists bending moments through a distributed moment along the pile circumference. The contribution of interfacial shaft friction cannot be neglected in rigid piles.

To account for this mechanism, an “M_p-θ_p” spring is introduced to capture the nonlinear evolution of shaft friction moment resistance along rigid monopiles. By combining distributed “p-y” and “M_p-θ_p” springs above the pile center of rotation with a single “M_R-θ_R” spring at the rotation point, the proposed three-spring “p-y + M_R-θ_R + M_p-θ_p” SPI model is formed. This model captures all significant soil resistance mechanisms of laterally loaded monopiles in sand, as introduced in Figure 4. A schematic of the three-spring model is presented in Figure 17.

3.3.1. Defining the “M_p-θ_p” Rotational Spring Curves

The “M_p-θ_p” spring is designed to capture the length-wise distributed rotational response of interfacial shaft friction. Due to its role being similar to that of the “p-y” springs, the hyperbolic formula proposed by Kondner et al. [42] is adopted to capture the distributed moment-rotation relationship, rewritten as:

$$M_p={\displaystyle \frac{\theta }{\frac{1}{k_p}+\frac{\theta }{M_{pu}}}},$$

(9)

where the governing parameters are: k_p, the initial stiffness of the rotational spring and M_pu, the ultimate moment resistance due to interfacial friction generated by relative axial motion between the pile and surrounding sand.

To define the “M_p-θ_p” relationship, the ultimate shaft friction moment, M_pu, in Equation (9) must first be determined. Figure 18 illustrates the theoretical distribution of shaft friction shear and the resulting distributed moment induced by the rotation of the pile cross-section. When a semi-rigid or rigid pile is laterally loaded, rotation-driven vertical relative motion between the pile skin and sand mobilizes axial interfacial shaft friction that contributes to resisting the applied bending moments by generating a distributed reaction moment along the pile perimeter. Shaft friction resistance acts along the pile length and around its circumference. Following Figure 18b, the ultimate lateral shaft friction moment per unit pile length, M_pu, can be expressed as the circular surface integral:

$$M_{pu}=2{\int }_{-{\displaystyle \frac{\pi }{2}}}^{{\displaystyle \frac{\pi }{2}}}{\displaystyle \frac{D}{2}}\mathit{cos}\phi dm=2{\int }_{-{\displaystyle \frac{\pi }{2}}}^{{\displaystyle \frac{\pi }{2}}}\tau {\left({\displaystyle \frac{D}{2}}\right)}^2\mathit{cos}\phi d\phi$$

(10)

The shear stress generated by axial interfacial shaft friction, τ, is expressed as:

$$\tau ={\tau }_{max}\left(1-{\displaystyle \frac{{\mathit{sin}}^2\phi +{\mathit{sin}}^4\phi }{2}}\right)$$

(11)

where the peak axial interfacial shear friction, τ_max, is a function of the peak horizontal soil pressure, σ_max, and is calculated using Coulomb friction:

$${\tau }_{max}={\sigma }_{max}\mathit{tan}\delta$$

(12)

$${\sigma }_{max}=K_p\gamma \prime z$$

(13)

where δ is the soil–pile interfacial friction angle, γ′ is the soil effective unit weight, and K_p is the passive earth pressure coefficient. Solving Equations (10)–(13) yields a simple expression for the ultimate shaft friction moment as a function of embedment depth z and pile diameter D:

$$\begin{gathered} M_{pu}=2{\int }_{-{\displaystyle \frac{\pi }{2}}}^{{\displaystyle \frac{\pi }{2}}}{\sigma }_{max}\mathit{tan}\delta \left(1-{\displaystyle \frac{{\mathit{sin}}^2\phi +{\mathit{sin}}^4\phi }{2}}\right){\left({\displaystyle \frac{D}{2}}\right)}^2\mathit{cos}\phi d\phi = \\ {\displaystyle \frac{11}{5}}{D}^2\mathit{tan}\delta K_p\gamma \prime z \end{gathered}$$

(14)

This equation uses readily available engineering parameters and can determine M_pu at any depth for any monopile in sand.

The second influencing factor in Equation (9), the initial spring stiffness k_p, also requires a practical determination method. Due to the limited research and lack of experimental data on lateral shaft friction moments for OWT monopiles, a simple and practical formula for k_p was established based on parametric FEM analyses. The initial stiffness was nondimensionalized as k_p/E_sD², where E_s is the soil Young’s modulus and D is the pile diameter. Figure 19 presents the relationship between this nondimensional stiffness and pile embedment depth.

Figure 19a shows that for piles with identical L/D ratios, k_p/E_sD² exhibits the same depth distribution regardless of absolute pile length or diameter, indicating that k_p scales linearly with E_sD² and is independent of L and D for constant L/D. Figure 19b highlights the effect of L/D, indicating that k_p/E_sD² is highly dependent on the ratio. Lower L/D ratios correspond to “more rigid” piles, and thus, a higher shaft friction moment resistance stiffens the pile’s moment-rotation response. This effect is particularly pronounced at the transition between semi-rigid and rigid piles (L/D = 5 vs. L/D = 2.5). Overall, k_p is dependent on pile–soil relative stiffness, with L/D representing pile rigidity and E_sD² representing soil lateral stiffness around the pile.

Given the complexity of fitting the shape of k_p/E_sD², while accounting for the non-linear effect of L/D, an optimization method proposed by Wan et al. [46] was employed to construct a formulation for initial “M_p-θ_p” spring stiffness. First, a stiffness coefficient f(E′) was defined to capture the effect of pile–soil relative stiffness, E′, independent of embedment depth. The relative stiffness E′ follows Poulus and Davis [47] per Equation (16), the soil Young’s modulus defined at the pile toe (E_s^toe) to maintain consistency with the SPI classification formula in Table 3 [37]. The “M_p-θ_p” spring stiffness formula, accounting for depth-dependent soil stiffness (E_sD²) and L/D dependent pile–soil relative stiffness (f(E′)) then becomes:

$$k_p=f\left(E^{\prime }\right)E_s{D}^2$$

(15)

$$E^{\prime }={\displaystyle \frac{E_p{I}_p}{E_s^{toe}{L}^4}}$$

(16)

The optimization procedure to determine f(E′) is as follows:

• Construct a high-fidelity FEM soil-monopile model for a given L/D, conduct a static pile pushover, and extract the mudline displacement, y₀, corresponding to 0.1D.
• Assume f(E′) = E, calculate the “M_p-θ_p” initial spring stiffness k_p via Equation (15) and M_pu via Equation (14) at every 1 m embedment depth.
• Define the “M_p-θ_p” springs using Equation (9), the “p-y” springs using Equation (6), and the “M_R-θ_R” spring using Equation (7). Build a simplified beam-spring model for the same monopile using the combined “p-y + M_R-θ_R + M_p-θ_p” three-spring SPI model and run a pushover analysis (use M₀) to obtain mudline displacement y_m.
• Iteratively adjust f(E′) until the mudline displacement calculated via the simplified model satisfies the condition: |y_m − y₀|/y_m ≤ 10⁻³.
• Repeat the procedure for monopiles with different L/D ratios.
• Fit the optimized f(E′)-E′ data with a curve.

Through this procedure, the stiffness coefficient function f(E′) for piles with various E′ (approximately corresponding to different L/D) was obtained, as shown in Figure 20:

$$f\left(E^{\prime }\right)=2.0{\left(E^{\prime }\right)}^{0.23}$$

(17)

Reintroducing E_sD² per Equation (15), to account for depth effects (given that E_s is depth dependent) and substituting Equation (17) for f(E′) gives the practical formula for calibrating the initial “M_p-θ_p” spring stiffness, k_p:

$$k_p=2.003{\left(E^{\prime }\right)}^{0.2305}{E}_s{D}^2$$

(18)

The derived shaft friction moment resistance “M_p-θ_p” springs combined with the proposed “p-y” springs and a “M_R-θ_R” spring at the rotation center, constitute the mechanism-aligned “p-y + M_R-θ_R + M_p-θ_p” three-spring SPI model for simulating the lateral response of semi-rigid and rigid monopiles in sand.

3.3.2. Implementation and Performance of the Three-Spring “p-y + M_R-θ_R + M_p-θ_p” Model

The “p-y + M_R-θ_R + M_p-θ_p” three-spring SPI model was implemented in ABAQUS to simulate static pile pushover up to 0.1D pile head displacement. This model incorporates all significant soil resistance mechanisms identified for laterally loaded monopiles in sand: horizontal soil resistance along the pile via distributed “p-y” springs, toe shear and moment via the “M_R-θ_R” spring at the center of rotation, and shaft friction moment resistance via distributed “M_p-θ_p” springs along the pile shaft. The modeling approach follows the procedure outlined in Section 3.2.2, with the addition of the “M_p-θ_p” springs applied at 1 m intervals along the pile shaft up to the rotation point (z/L = 0.78). Pile head reaction forces and lateral displacements were recorded and compared with high-fidelity FEM results, the API standard “p-y” model, and the previously proposed one-spring and two-spring models, as shown in Figure 21. The errors in lateral pile capacity predicted by the three-spring SPI model relative to high-fidelity FEM results are plotted in Figure 22.

For semi-rigid and rigid piles with low L/D ratios (<6.7), the three-spring model closely reproduces the coupled soil–pile stiffness simulated via high-fidelity FEM, outperforming the two-spring model. Furthermore, the introduction of the “M_p-θ_p” springs significantly reduces errors in predicted lateral pile capacity (see Figure 22): for semi-rigid piles with L/D between 6.7 and 5, the error drops below 1%, representing a tenfold improvement over the two-spring model. For rigid piles (L/D < 5), the errors range between 5 and 10%, demonstrating a satisfactory level of accuracy while still showing substantial improvement relative to the two-spring model.

Overall, the three-spring model provides a highly accurate, simple, and computationally efficient method for simulating lateral SPI of monopiles with L/D < 6.7. The presented results highlight the importance of systematically incorporating observed soil resistance mechanisms into simplified spring-based SPI models as pile L/D decreases, rather than relying on a single generalized approach for OWT monopile design.

4. Validations and Discussions

To further validate the three proposed SPI models, comparative analyses were conducted using data from full-scale OWT field tests, centrifuge model tests, and small-scale physical model tests.

4.1. Validation of the One-Spring “p-y” Model

The one-spring “p-y” model was validated against full-scale static load tests on two flexible steel monopiles at the Xiangshui OWT farm, Jiangsu Province, China [48]. The soil profiles at both sites were heterogeneous but dominated by sand, with internal friction angles (φ) ranging from 11° to 30°. Both test piles had a diameter of D = 2 m, with L/D ratios of 17.1 (SY1) and 18.3 (SY2). The soil properties and site layouts are shown in

Table 4. The mechanical properties of soils.

Soil	Total Unit Weight (kN/m3)	Internal Friction Angle (°)	Compression Modulus (MPa)
Silty sand 1	19.1	30	6.8
Muddy-silty clay	17.4	10.9	3.1
Silty sand	19.8	33	13.9
Silty clay 2	18.1	14.1	4.4
Fine sand	20.2	33.4	17.4

and Figure 23, respectively. The piles were subjected to monotonic loading up to 1.48 MN, applied at 16.3 m and 16.7 m above the mudline, respectively. The loading was performed in 10 stages, with a load increment of 148 kN.

The one-spring “p-y” model accurately captures the measured load–displacement response, as illustrated in Figure 24. The MAPE values for SY1 and SY2 are 5.63% and 6.84%, respectively. The model accurately reproduces both the evolution of coupled soil–pile stiffness and the maximum pile head displacements, validating its accuracy and broad applicability across sands with a wide range of strengths and stiffnesses. In contrast, the traditional API “p-y” model significantly overestimates the initial coupled soil–pile stiffness and underestimates the maximum pile head displacement, consistent with prior studies [14,19]. These results confirm that the API “p-y” model is inadequate for simulating the SPI of OWT monopiles. Conversely, the proposed one-spring “p-y” model provides a reliable modeling approach for flexible monopiles, consistent with the classification criteria presented in Section 2.2 and Table 3.

4.2. Validation of the Two-Spring “p-y + M_R-θ_R” Model

The two-spring “p-y + M_R-θ_R” model was validated against a centrifuge test on a large-diameter OWT monopile conducted by Choo and Kim [49]. Their M1 test results were compared with the simulated responses using the proposed two-spring SPI model. The tested monopile was semi-rigid, with a diameter of D = 6 m, embedment depth of L = 31 m (L/D = 5.2), and a loading height of 33 m above the mudline. The soil was homogeneous sand with a relative density of D_r = 86% and a peak friction angle (i.e., φ + ψ) of 44.2°. For the numerical simulations, a common dilatancy angle ψ = 5° was adopted. A load of up to 20 MN was gradually applied at the pile head, and horizontal displacements were recorded at 5.5 m above the mudline.

Figure 25 compares the centrifuge results with the simulated results from the proposed two-spring and one-spring SPI models. The two-spring model accurately predicts the mudline displacements of the semi-rigid monopile, with a MAPE of 1.88% relative to the centrifuge results. It significantly outperforms the one-spring model, highlighting the importance of capturing pile toe shear and moment resistance even under moderate rigid-body rotations in semi-rigid piles. These results validate the safe application of the proposed two-spring model for SPI analysis and design of semi-rigid OWT monopiles in sand.

4.3. Validation of the Three-Spring “p-y + M_R-θ_R + M_p-θ_p” Model

The three-spring “p-y + M_R-θ_R + M_p-θ_p” model was validated against a series of small-scale monopile experiments conducted by Prasad and Chari [50] on rigid piles in sands of three relative densities: D_r = 25%, 50%, and 75%. The unit weights of the soils were 16.5, 17.3, and 18.3 kN/m³, and the internal friction angles were 33.3°, 39°, and 43°, respectively. The test monopile had a diameter D = 0.102 m, an embedment depth L = 0.306 m (L/D = 3), and an elastic modulus of 210 GPa. Loads of 1.0 kN, 1.25 kN, and 2.5 kN were applied to the pile head (0.15 m above the mudline), and horizontal displacements were recorded. The proposed three-spring model was used to simulate the pushover tests numerically, with results compared to the experimental data in Figure 26.

The numerical predictions closely align with the experimental responses across all three relative densities. The MAPE values corresponding to increasing D_r (Figure 26a–c) are 8.51%, 10.95%, and 16.75%, respectively. While the model maintains high accuracy for loose and medium-dense sands, errors increase slightly for dense sand due to the parametric basis of the spring constants being derived primarily from medium-dense sand. Nevertheless, the total error remains within acceptable bounds for a simplified model. Overall, the three-spring model substantially outperforms the two-spring model, emphasizing the importance of capturing axial interfacial shaft friction mobilized by the pronounced rigid-body rotation of rigid piles. Consequently, the three-spring model is particularly suitable for rigid monopiles in loose to medium-dense sands, with acceptable accuracy in dense sands.

5. Summary and Conclusions

This study introduced a complete mechanism-aligned set of SPI spring models for large-diameter monopiles in sand that explicitly account for all L/D-dependent soil resistance mechanisms under lateral loading. Based on an analysis of the contribution ratios of four SPI resistance components, distributed lateral soil resistance, interfacial shaft-friction moment resistance, pile toe shear, and pile toe bending moment, three model sets were developed: “p-y”, “p-y + M_R-θ_R”, and “p-y + M_R-θ_R + M_p-θ_p”, capturing flexible, semi-rigid, and rigid pile behavior, respectively. The derived spring formulas allow rapid and straightforward calibration using readily available soil and pile properties. Furthermore, the use of a conventional spring-based framework ensures their seamless use in any commercial FEM software. The proposed models were validated against both numerical and experimental case studies, demonstrating their improved accuracy in reproducing SPI behavior.

For flexible piles (L/D > 8), lateral soil resistance dominates SPI. A one-spring SPI model was proposed to capture this effect. The model was established by reformulating the traditional API “p-y” curves to align with the behavior of large-diameter steel pipe piles used for OWTs. Distributed every 1 m along the pile shaft, the one-spring model predicted lateral pile bearing capacities with <2% error relative to high-fidelity FEM and reproduced the pile head load–displacement response of two full-scale offshore monopile pushover tests with ~5% error.

For semi-rigid piles (8 ≥ L/D > 4), in addition to lateral soil resistance, the toe shear force and bending moment induced by rigid-body rotations were also shown to significantly contribute to ultimate lateral pile capacity. The proposed two-spring “p-y + M_R-θ_R” model, therefore, introduces a rotational spring (M_R-θ_R) at the pile center of rotation to capture pile toe shear, toe moment, and horizontal soil resistance below the rotation point. This model achieved 2–9% error in lateral pile capacity predictions relative to FEM, and ~2% error in predicted pile head load–displacement response compared to centrifuge tests. The two-spring “p-y + M_R-θ_R” model showed significantly improved accuracy over both the API and one-spring “p-y” models for semi-rigid piles.

For rigid pile and semi-rigid piles with low L/D, axial interfacial shaft friction was also shown to be significant due to relative axial displacement between the pile and surrounding soil during rigid-body rotation. The proposed three-spring “p-y + M_R-θ_R + M_p-θ_p” model, therefore, introduces distributed rotational springs to capture this shaft friction-induced moment resistance along the pile shaft. Combined with distributed “p-y” springs and a single “M_R-θ_R” spring at the pile center of rotation, the three-spring model achieved <1% error in predicted lateral capacity for semi-rigid piles with L/D ≤ 6.7 and <10% for rigid piles (L/D ≤ 4), compared to high-fidelity FEM. Furthermore, the three-spring model reproduced the load–displacement response of monopiles in sands of various densities, with 8–17% error compared to small-scale physical tests.

Based on the observed error trends across the full range of L/D ratios and across all three SPI models, the authors generally recommend using the one-spring SPI model for monopiles with L/D > 10, the two-spring model for 10 ≥ L/D ≥ 7, and the three-spring model for L/D < 7.

Overall, the three proposed mechanism-aligned SPI models were validated against full-scale field tests, centrifuge experiments, and small-scale laboratory tests. The validations showed excellent accuracy (<10%) in ultimate lateral pile capacity prediction, and faithful reproduction of soil–pile coupled stiffness evolution (<17% error in pile head load–displacement curves). Covering the full L/D range (2–20) used in OWT monopiles, these models offer high computational efficiency and spring constant formulas that can be calibrated using readily available engineering parameters, such as soil unit weight, Young’s modulus, internal friction angle, and pile geometry. Moreover, unlike other simplified SPI models of similar accuracy, the proposed approach relies solely on p-y and M-θ spring elements, available in most commercial FEM software, facilitating convenient and rapid adoption in industrial design. It should be noted that the models are limited to capturing static SPI responses, and the derived spring formulations are specific to drained sand behavior. While the formulas’ applicability to loose and dense sands has been validated, they were derived from simulations of a uniform layer of medium-dense sand. Extension to cyclic loading regimes and heterogeneous soil layering, including clay soils, requires consideration of undrained behavior and rate effects, which will be addressed in future work.