A Review of Static and Dynamic p-y Curve Models for Pile Foundations

Abstract

In addition to supporting vertical loads from superstructures, piles are frequently subjected to horizontal soil pressures, long-term wind, wave, and current forces, as well as seismic loads. Presently, the p-y curve method is widely employed for calculating the horizontal forces acting on piles due to its ability to replicate the nonlinear interaction between piles and soil. This paper provides a thorough review and analysis of the current research on p-y curve models for piles, examining literature across various conditions such as horizontal static loads, cyclic loads, and seismic loads. Special emphasis is placed on the development, classification, and analysis of the key factors influencing major p-y curve models. It also discusses future research directions and prospects, considering emerging trends and prevailing challenges in the field. For instance, future studies should investigate p-y curves for piles under various combined loads, considering the influence of construction methods and the installation effect. Additionally, the development of a comprehensive p-y curve database and the application of existing research to new foundation systems are essential for advancing pile technology and fostering innovative designs.

1. Introduction

In early construction, wooden piles were used for living and production purposes. It was not until the 1820s that cast iron sheet piles were used to build dams and docks. With the advancement of materials such as steel, cement, and concrete, steel piles, concrete piles, and reinforced concrete piles have become widely used in high-rise buildings, landslide control systems, cross-river and cross-sea bridges, port terminals, offshore wind power projects, and drilling platforms, as shown in Figure 1a–f. Despite the evolution of construction techniques, pile foundations remain a cornerstone in modern engineering practices.

Pile foundations are critical in high-rise buildings where they provide the necessary support to handle heavy loads. For landslide control, anti-slide piles stabilize slopes and prevent landslides. In aquatic environments, such as those encountered in cross-river and cross-sea bridges, pile foundations ensure stability against strong currents and varying water levels. At port terminals, they support heavy loads from cranes and storage facilities. To reduce the reliance on fossil fuels, wind power installed capacity has been steadily increasing, as shown in Figure 1g,h. According to the Global Wind Energy Council (GWEC), the total wind power installed capacity for 2023 reached 1021 GW, a 13% year-over-year increase. Monopiles are critical foundations for offshore wind turbines, bearing self-weight, wind loads, and ocean wave loads. These foundations typically have large diameters and significant lengths, requiring the reassessment of pile foundation application conditions to propose more accurate calculation methods.

Pile foundations are subjected to lateral soil pressures, prolonged wind–wave-flow actions, and seismic loads, making their load-bearing characteristics inherently complex. Consequently, the design and calculation methods for pile foundations under horizontal static and dynamic loading have become a focal point and research hotspot for numerous scholars. The p-y curve method is an improvement on the elastic foundation reaction method, which is based on the Winkler foundation model. It is a widely used approach for analyzing horizontal forces in pile foundations because it enhances the discrete and linear soil springs in the elastic foundation reaction method into discrete and nonlinear soil springs, allowing for the effective capture of nonlinear soil–structure interaction (SSI). The basic concept of this method is to establish the relationship between the soil reaction p and the horizontal displacement y of the pile at different depths below the ground surface, as shown in Figure 2.

Currently, the commonly used m-method and k-method [1] consider the soil resistance to displacement as a linear relationship applicable only to small deformations. In contrast, the p-y curve method not only suits elastic deformations but also captures the pile–soil interaction after the soil undergoes elastic–plastic deformation [2]. It is recommended for use by the Geotechnical Consideration and Foundation Design for Offshore Structures of USA [3] and Specifications for Design of Foundation of Highway Bridges and Culverts of China [4]. The concept of the p-y curve was initially introduced by Matlock [5] and Reese et al. [6], among others [7], validated this method and thus laid the foundation for the p-y curve, establishing its credibility. Since then, numerous scholars worldwide have continuously followed up, promoting the rapid development of the p-y curve method. Substantial research has improved existing models [8,9] or proposed new ones [10,11,12], significantly enhancing its applicability. These research achievements have propelled the continuous refinement of the p-y curve method.

This paper, based on an extensive review of the relevant research literature, provides a comprehensive discussion on the current research status of p-y curve models (Figure 2) under horizontal static loads, cyclic loads, and seismic loads. By examining the development, classification, and influencing factors of these curve models, this review seeks to highlight the progress made and identify areas for further research. Through an extensive literature review, this paper will discuss the foundational concepts, advancements, and future directions in the study of p-y curve models. The continuous evolution of pile foundation applications, driven by the need for more accurate and reliable calculation methods, underscores the importance of p-y curve models in modern engineering. As the demand for sustainable and resilient infrastructure grows, the insights provided by this review will be invaluable for advancing the design and implementation of pile foundations in various engineering contexts.

2. Static p-y Curve Model

The study of p-y curves under static loading began in the 1970s and has matured over the past half-century. Many studies have proposed models for static p-y curves. Due to variations in soil properties, the bearing characteristics of pile foundations vary significantly. This paper summarizes the current research status of static load p-y curves, classified by clay and sandy soils.

2.1. Static p-y Curves for Clay

The clay p-y curve model was first proposed by Matlock and subsequently incorporated into the specifications of the American Petroleum Institute (API) [3,5]. Since then, it has been widely utilized as a basis for improving and modifying p-y curves. Reese et al. [7] introduced a p-y curve for stiff clay below the water table, which, similar to Matlock’s soft clay curve, is segmented and eventually converges to a limiting constant. Together with the achievement by Lee and Gilbert [13], Sullivan et al. [14], Dunnavant and O’Neill [15], it laid the foundation for clay p-y curves research. In China, research on p-y curves started relatively later. Earlier studies on static clay p-y curves were based on field tests in different-sized pile diameters between 1984 and 1986. These tests formed the basis for establishing a unified calculation model for clay p-y curves, commonly referred to as the Hohai University method in China [16,17]. Additionally, Zhang and Chen [18], based on the aforementioned field test data and analysis using Matlock’s soft clay p-y curve method, identified discrepancies when applying it to soft clay in the middle and lower reaches of the Yangtze River in China. Therefore, they proposed a new p-y curve model, also known as the Tongji University method. Unlike Matlock’s approach, their study incorporated the effects of finite length calculation depth and introduced a reduction factor F to reduce the ultimate soil resistance.

It can be observed that the various p-y curves for cohesive soils mentioned above are all segmented. The difference lies in how the segmenting of the functions and the calculation of ultimate soil resistance are determined. This discrepancy may stem from variations in test results observed by different researchers or differences in test piles, soils, and applied loading conditions. Figure 3 compares Matlock’s soft clay p-y curve with other cohesive soil p-y curves [19]. It is evident that different researchers have conducted detailed measurements and derivations for soft clay p-y curves. Although the proposed models for soft clay p-y curves are not identical, they are all based on actual measurements. Therefore, the selection of models when applying p-y curves to design the horizontal bearing capacity of piles in clay is particularly important.

Pile foundations face increasingly complex working conditions in soils, prompting many studies to propose more suitable p-y curve models for specific pile types, soil conditions, and loading scenarios to adapt to the evolving interactions between piles and soil. These proposed p-y curve models often require obtaining numerous parameters, posing challenges in parameter acquisition. To better apply them in engineering practice, some studies have explored methods for the easier acquisition of geotechnical parameters. Specifically, the applicability of p-y curves has been a focal point for researchers, with a wealth of literature documenting contributions to p-y curves in cohesive soils, as showed in Table 1. For example, Zhou et al. [20] proposed a p-y curve model that facilitates the acquisition of geotechnical parameters based on the unified method of Hohai University to facilitate practical engineering applications. Zeng et al. [21] through comparisons of numerical calculations, centrifuge tests, and API specification calculations, explored p-y curve methods for cohesive soils. Additionally, Wang et al. [22] proposed corresponding p-y curve calculation methods for the widespread application of rigid composite piles.

Wang et al. [19] established an ideal elastoplastic p-y curve model applicable to different pile diameters in cohesive soil. Similarly, attention is paid to the pile diameter. Given that large-diameter pile foundations are predominant in offshore structures, Zhang et al. [23] adjusted the ultimate soil resistance and pile deformation parameters. Based on the established isotropic hardening model, the effects of soil strength non-uniformity, pile–soil friction, and pile diameter are considered in the proposed p-y curve model for large-diameter rigid piles. Winged piles contribute to increasing horizontal bearing capacity and are commonly used in marine port engineering. Hu et al. [24], based on the port engineering specifications, modified two important parameters of the single-pile p-y curve to obtain the calculation formula for this type of pile. Similarly, Chen et al. [25] also proposed a p-y curve calculation model applicable to such piles.

The future development of marine engineering urgently requires stronger foundation piles capable of withstanding wind, wave, and seismic loads. Large-diameter piles are the preferred choice to meet these requirements. Therefore, the bearing capacity of large-diameter pile foundations requires more advanced theoretical support. In recent years, multi-spring coupled analysis based on the extension of a single p-y spring has become a trend. The four-spring model (including the $s_{\mathrm{b}}-y_{\mathrm{b}}$ spring for base shear, the $M_{\mathrm{b}}-{\theta }_{\mathrm{b}}$ spring for base moment, the $M_{\mathrm{p}}-{\theta }_{\mathrm{p}}$ spring for pile moment, and the p-y spring), the three-spring model (including the $s_{\mathrm{b}}-y_{\mathrm{b}}$ spring for base shear, the $M_{\mathrm{p}}-{\theta }_{\mathrm{p}}$ spring for pile moment, and the p-y spring), and the two-spring model (either the $M_{\mathrm{R}}-{\theta }_{\mathrm{R}}$ spring for representing moment-rotation at a specific rotation center and the p-y spring, or the $s_{\mathrm{b}}-y_{\mathrm{b}}$ spring for base shear and the p-y spring combination) have been proposed. In the above, $s_{\mathrm{b}}$ represents the shear force at the base, $y_{\mathrm{b}}$ is the lateral displacement at the base, $M_{\mathrm{p}}$ is the pile moment, ${\theta }_{\mathrm{p}}$ is the pile rotation, $M_{\mathrm{b}}$ is the base moment, ${\theta }_{\mathrm{b}}$ is the base rotation, $M_{\mathrm{R}}$ is the moment at a specific rotation center, and ${\theta }_{\mathrm{R}}$ is the rotation at a specific rotation center. For instance, Jiao et al. [26], based on summarizing different types of pile–soil interaction spring models, have established a theoretical model for calculating the horizontal bearing capacity of pile foundations in clay, considering the lateral soil resistance, pile shaft friction, base shear force, and base moment. This model incorporates three types of springs: lateral resistance (p-y) spring, pile shaft rotation spring ($M_{\mathrm{p}}-{\theta }_{\mathrm{p}}$), and base moment-rotation spring ($M_{\mathrm{R}}-{\theta }_{\mathrm{R}}$), coupling the p-y spring with other force springs. This p-y curve method for large-diameter piles in clay has a higher computational accuracy and has been validated by finite-element numerical simulations, as shown in Figure 4.

Figure 4. Conceptual models of pile–soil interaction with different springs: (a) lateral load spring, pile shaft distributed moment spring, pile base moment spring, and base shear force spring; (b) lateral load spring and base shear force spring; (c) lateral load spring, pile shaft distributed moment spring, and base shear force spring; (d) lateral load spring and base moment-rotation spring; (e) lateral load spring, pile shaft distributed moment spring, and base moment-rotation spring [26].

Figure 4. Conceptual models of pile–soil interaction with different springs: (a) lateral load spring, pile shaft distributed moment spring, pile base moment spring, and base shear force spring; (b) lateral load spring and base shear force spring; (c) lateral load spring, pile shaft distributed moment spring, and base shear force spring; (d) lateral load spring and base moment-rotation spring; (e) lateral load spring, pile shaft distributed moment spring, and base moment-rotation spring [26].

However, this computational method warrants further validation through more practical engineering tests, especially for application in pile foundations buried in weak seabeds subjected to long-term cyclic loading. Further breakthroughs are needed to overcome the existing limitations. In summary, coupling the p-y spring with other soil interaction springs provides a valuable reference for further improving the prediction of the pile foundation horizontal bearing capacity using the p-y curve method.

Table 1. Clay static load p-y curve.

References	p-y Curve Model	Model Description
Matlock (1970) [5]	$p=\left\{\begin{array}{l}0.5p_{\mathrm{u}}\left(\frac{y}{y_{50}}\right){ }^{\frac{1}{3}}\hspace{1em}\hspace{1em}y<8y_{50}\\ p_{\mathrm{u}}\hspace{1em}\hspace{1em}y>8y_{50}\end{array}\right.$	Divided into two segments with 8$y_{50}$ as the boundary, applicable to soft clay
Reese (1975) [7]	$p_{\mathrm{u}}=\min \left\{\begin{array}{l}11C_{\mathrm{u}}D\\ \left(2+\frac{\gamma z}{C_{\mathrm{avg}}}+\frac{2.83z}{B}\right)C_{\mathrm{avg}}D\end{array}\right.$	Applicable to underwater stiff clay
Dunnavant and O’Neill (1989) [15]	$\frac{p}{p_{\mathrm{u}}}=1.02\tanh \left(0.537{\left(\frac{y}{y_{50}}\right)}^{0.7}\right)$	Based on full-scale pile load tests
Wang (1991) [16]	$p=\left\{\begin{array}{l}\frac{y/y_{50}}{a+by/y_{50}}{p}_{\mathrm{u}}\hspace{1em}\hspace{1em}\mathrm{y}<\beta y_{50}\\ p_{\mathrm{u}}\hspace{1em}\hspace{1em}\mathrm{y}>\beta y_{50}\end{array}\right.$	$a, b, \beta$ are all parameters
Zhang (1992) [18]	$p=\left\{\begin{array}{l}0.5p_{\mathrm{u}}{\left(\frac{y}{y_{50}}\right)}^{\frac{1}{3}}\\ p_{\mathrm{u}}\hspace{1em}\hspace{1em}x>x_{\mathrm{rs}}\\ p_{\mathrm{u}}\left[F_{\mathrm{s}}+\frac{\left(1-F_{\mathrm{s}}\right)x}{x_{\mathrm{rs}}}\right]\hspace{1em}\hspace{1em}x\le x_{\mathrm{rs}}\end{array}\right.$	Piecewise function with the critical depth $x_{\mathrm{rs}}$ as the boundary
Zhou (2013) [20]	$p=0.8p_{\mathrm{u}}{\left(\frac{y}{y_{50}}\right)}^{\frac{1}{8}}\hspace{1em}\hspace{1em}p\le p_{\mathrm{u}}$	Applicable to soft clay
Chen (2018) [25]	$p=\left\{\begin{array}{l}0.5p_{\mathrm{u}}{\left(\frac{Y}{Y_{50}}\right)}^{\frac{1}{1.84}}\hspace{1em}\hspace{1em}Y<3.58Y_{50}\\ p_{\mathrm{u}}\hspace{1em}\hspace{1em}Y\ge 3.58Y_{50}\end{array}\right.$	Piecewise function for large-diameter winged piles in soft clay
Zhang (2020) [11]	$p=\Delta p_{\mathrm{r}}^{\prime }+\Delta p_{\mathrm{r}}$	Based on the stress increment theory, errors exist under small displacements in sand
Zhang (2020) [23]	$p=0.5p_{\mathrm{ult}}{\left(\frac{y}{4.5e_{50}{\left(d/d_{\mathrm{ref}}\right)}^{0.5}{d}_{\mathrm{ref}}}\right)}^{\frac{1}{3}}$	Applicable to large-diameter single piles

Table 1. Clay static load p-y curve.

References	p-y Curve Model	Model Description
Matlock (1970) [5]	$p=\left\{\begin{array}{l}0.5p_{\mathrm{u}}\left(\frac{y}{y_{50}}\right){ }^{\frac{1}{3}}\hspace{1em}\hspace{1em}y<8y_{50}\\ p_{\mathrm{u}}\hspace{1em}\hspace{1em}y>8y_{50}\end{array}\right.$	Divided into two segments with 8$y_{50}$ as the boundary, applicable to soft clay
Reese (1975) [7]	$p_{\mathrm{u}}=\min \left\{\begin{array}{l}11C_{\mathrm{u}}D\\ \left(2+\frac{\gamma z}{C_{\mathrm{avg}}}+\frac{2.83z}{B}\right)C_{\mathrm{avg}}D\end{array}\right.$	Applicable to underwater stiff clay
Dunnavant and O’Neill (1989) [15]	$\frac{p}{p_{\mathrm{u}}}=1.02\tanh \left(0.537{\left(\frac{y}{y_{50}}\right)}^{0.7}\right)$	Based on full-scale pile load tests
Wang (1991) [16]	$p=\left\{\begin{array}{l}\frac{y/y_{50}}{a+by/y_{50}}{p}_{\mathrm{u}}\hspace{1em}\hspace{1em}\mathrm{y}<\beta y_{50}\\ p_{\mathrm{u}}\hspace{1em}\hspace{1em}\mathrm{y}>\beta y_{50}\end{array}\right.$	$a, b, \beta$ are all parameters
Zhang (1992) [18]	$p=\left\{\begin{array}{l}0.5p_{\mathrm{u}}{\left(\frac{y}{y_{50}}\right)}^{\frac{1}{3}}\\ p_{\mathrm{u}}\hspace{1em}\hspace{1em}x>x_{\mathrm{rs}}\\ p_{\mathrm{u}}\left[F_{\mathrm{s}}+\frac{\left(1-F_{\mathrm{s}}\right)x}{x_{\mathrm{rs}}}\right]\hspace{1em}\hspace{1em}x\le x_{\mathrm{rs}}\end{array}\right.$	Piecewise function with the critical depth $x_{\mathrm{rs}}$ as the boundary
Zhou (2013) [20]	$p=0.8p_{\mathrm{u}}{\left(\frac{y}{y_{50}}\right)}^{\frac{1}{8}}\hspace{1em}\hspace{1em}p\le p_{\mathrm{u}}$	Applicable to soft clay
Chen (2018) [25]	$p=\left\{\begin{array}{l}0.5p_{\mathrm{u}}{\left(\frac{Y}{Y_{50}}\right)}^{\frac{1}{1.84}}\hspace{1em}\hspace{1em}Y<3.58Y_{50}\\ p_{\mathrm{u}}\hspace{1em}\hspace{1em}Y\ge 3.58Y_{50}\end{array}\right.$	Piecewise function for large-diameter winged piles in soft clay
Zhang (2020) [11]	$p=\Delta p_{\mathrm{r}}^{\prime }+\Delta p_{\mathrm{r}}$	Based on the stress increment theory, errors exist under small displacements in sand
Zhang (2020) [23]	$p=0.5p_{\mathrm{ult}}{\left(\frac{y}{4.5e_{50}{\left(d/d_{\mathrm{ref}}\right)}^{0.5}{d}_{\mathrm{ref}}}\right)}^{\frac{1}{3}}$	Applicable to large-diameter single piles

2.2. Static p-y Curves for Sand

The sand p-y curve can be primarily classified into two types based on their functional forms: the API specification type and the hyperbolic type. Both types of curves are controlled by parameters such as the internal friction angle, effective unit weight, pile diameter, and depth.

2.2.1. API Specification Type p-y Curve

The sand p-y curve model originated from the segmented sand p-y curve model proposed by Reese after full-scale tests conducted on Mustang Island [6]. However, its applicability has significant limitations. Together with Scott and Murchison and O’Neil, Reese laid the foundation for the sand p-y curve [27,28]. The American Petroleum Institute incorporated this calculation method into its specifications to guide the design and construction of pile foundations. The accuracy of the calculations has gradually improved with successive versions [29]. The expression for the API specification type p-y curve is given as Equation (1):

$$p=A{p}_{\mathrm{u}}\tanh \left[\frac{kz}{A{p}_{\mathrm{u}}}y\right]$$

(1)

where z is the soil depth, k represents the stiffness of the soil, A is a load coefficient, A ≥ 0.9 for static loading and calculated as shown in Equation (2), and $p_{\mathrm{u}}$ is the ultimate soil resistance, calculated using Equation (3) for shallow soil and Equation (4) for deep soil.

$$A=(3.0-0.8H/D)$$

(2)

$$p_{\mathrm{us}}=\left(C_{1}z+C_{1}D\right)\gamma z$$

(3)

$$p_{\mathrm{ud}}=C_{3}D\gamma z$$

(4)

where $\gamma$ is effective unit weight, D is pile diameter, and $C_1,C_2,C_3$ are coefficients related to the internal friction angle of the soil.

It is worth noting that, although the API specifications introduce a load factor A to plot the p-y curve for piles subjected to cyclic loading, this solution seems unable to perfectly explain the degradation of stiffness and strength of the soil around the pile foundation due to cyclic loading, known as the soil cyclic weakening effect. This will be further discussed in the next section.

2.2.2. Hyperbolic-Shaped p-y Curve

The hyperbolic-shaped p-y curve was originally derived based on the stress–strain relationship obtained from triaxial compression tests on soil [30]. Due to its ability to better represent the nonlinear relationship between the pile and the soil, it has since been widely used for the modification of p-y curves, as shown in Equation (5), with the ultimate soil resistance and variables representing soil stiffness often being the objects of modification.

$$p=\frac{y}{\frac{1}{k_{\mathrm{ini}}}+\frac{y}{p_{\mathrm{u}}}}$$

(5)

where $k_{\mathrm{ini}}$ is the initial stiffness of the soil and $p_{\mathrm{u}}$ is the ultimate soil resistance.

Kim et al. [31] proposed a hyperbolic p-y curve model for sand considering the pile installation method and pile head constraints, represented by the proportion factors F₁ and F₂, which affect the initial soil reaction modulus $k_{\mathrm{ini}}$ and the ultimate soil resistance $p_{\mathrm{u}}$. The calculation reliability of the ultimate soil resistance $p_{\mathrm{u}}$ in this study is relatively high and has therefore been widely referenced [32,33]. Figure 5 shows a comparison of different sand p-y curves [31,34].

2.2.3. Discussion on Sand p-y Curves

The p-y curves of pile foundations vary due to differences in soil mechanics characteristics. Studies have reported the establishment of p-y curves applicable to both sand and soft clay. Different researchers have proposed targeted modifications to the sand p-y curves from various perspectives, with improvements reflected in different soil parameters, pile diameter, pile type, embedment depth, and other factors. For example, Wang and Yang [35] considered the influence of different internal friction angles of sand, pile embedment depth, and pile diameter on p-y curves. Similarly, Hu et al. [36] only considered the effects of pile diameter and embedment depth, while Sun et al. [37] not only considered the effects of pile diameter and soil depth but also the influence of the initial soil reaction modulus. In terms of pile types, Cao et al. [38] and Ling et al. [39] focused on battered piles in sand, while Fu et al. [40] explored the p-y curves of small-scale piles (d = 100–300 mm, and L/d > 30) in sand. Additionally, with the increasing demand for the horizontal bearing capacity of pile foundations, large-diameter piles have become a good choice, attracting attention from researchers like Sun and Huang [41] focusing on large-diameter single piles. Zhang et al. [42], based on finite-element numerical simulations, point out that both the API standard p-y curve and the hyperbolic p-y curve underestimate the ultimate soil reaction for large-diameter single piles in sandy soils. The actual initial stiffness of the soil is intermediate between these two methods. Therefore, taking into account the effects of pile diameter and soil layer depth, they propose modifications to the calculations of ultimate soil reaction and initial soil stiffness, establishing a p-y curve method for single piles with diameters greater than 2 m in sandy soils.

Some studies have specifically focused on the horizontal bearing capacity of pile foundations in silty soil. Wang et al. [43], based on horizontal static load tests on steel pipe model piles in the Yellow River silt, derived a formula for calculating the p-y curve applicable to Yellow River silt foundations. After comparing the experimental silty soil p-y curve with other p-y curves, Liu et al. [44] suggested using the API specification sand soil p-y curve for silty soil foundations. Zhang et al. [45], based on indoor model tests, established a silty soil p-y curve model related to the pile diameter.

The discussed research explores the influence of pile foundation p-y curves in sand or proposes optimized p-y curve calculation models; these improved models of sand p-y curves are listed in

Table 2. Sand static load p-y curve.

References	p-y Curve Model	Model Description
Reese (1974) [6]	$p_{\mathrm{a}}=ky\frac{z}{d}\hspace{1em}\hspace{1em}{p}_{\mathrm{b}}=p_{\mathrm{u}}\frac{B}{A}$	Based on full-scale experiments
Scott (1980) [27]	$p_{\mathrm{k}}=\frac{n{k}_0\gamma zd}{\sqrt{\frac{1}{\sin \phi }+\frac{1}{3-4v}}}$	Developed based on centrifuge experiments
Murchison and O’Neill (1984) [28]	$p=\eta A{p}_{\mathrm{u}}\tanh \left[\frac{k_{\mathrm{h}}y}{A\eta p_{\mathrm{u}}}\right]$	$\eta$ is the pile type factor, and $A$ is the load empirical factor
Gao (1988) [46]	$p=\eta p_{\mathrm{u}}y{\left[y+p_{\mathrm{u}}/(k_1{d}_{\mathrm{p}})\right]}^{-1}$	$\eta$ is the correction factor, based on sand model experiments
Kim (2004) [31]	$p=\frac{y}{\frac{1}{F_1{k}_{\mathrm{ini}}}+\frac{y}{F_2{p}_{\mathrm{u}}}}$	$F_1$ and $F_2$ consider the influence of installation methods and pile head constraint conditions
Wang (2009) [43]	$p=1.26p_{\mathrm{u}}\tanh \left(\frac{kz}{p_{\mathrm{u}}}y\right)$	Modified according to API specifications for application in silty soil
Wang (2011) [35]	$p=\frac{y}{\frac{1}{k_{\mathrm{ini}}}+\frac{y}{p_{\mathrm{u}}}}$	$k_{\mathrm{ini}}$ considers the influence of the pile diameter and $p_{\mathrm{u}}$ is calculated at the soil parameter selection location
Sun (2021) [37]	$p=A{p}_{\mathrm{ult}}\tanh \left[\frac{K_{\mathrm{s}}{z}_0{\left(\frac{D}{D_0}\right)}^m{\left(\frac{z}{z_0}\right)}^n}{A{p}_{\mathrm{ult}}}\right]$	Stiffness correction based on API, considering pile diameter and depth
Zhang (2022) [45]	$p=A{p}_{\mathrm{u}}\tanh \left(\frac{nz}{A{p}_{\mathrm{u}}}y\right)$	Modified according to API specifications for application in silty soil
Zhang (2023) [42]	$p=A{p}_{\mathrm{u}}\tanh \left(\frac{ky}{A{p}_{\mathrm{u}}}\right)$	Modified according to API specifications, applicable to large-diameter single pile in sand

. The improved sand p-y curves are based on either the standard or hyperbolic types, considering various factors such as different types of pile foundations, depths, diameters, embedment depths, and soil conditions. Future requirements for the safe and stable operation of pile foundations under more complex conditions demand the comprehensive consideration of various influencing factors. Large-diameter piles in offshore applications remain noteworthy, whether in cohesive or sandy soils.

3. Cyclic p-y Curves

3.1. Cyclic p-y Curve Models

Static load p-y curves cannot reflect the cyclic shear action between the pile and the soil. Therefore, it is necessary to modify the static load p-y curves or propose new models to accurately represent the cyclic action between the pile and the soil. After collecting and organizing a large amount of literature, it was found that some studies only analyze the factors influencing cyclic behavior based on experimental results, while others propose improved calculation models on this basis, as shown in

Table 3. Cyclic load p-y curve model.

References	p-y Curve Model	Model Description
Gerolymos et al. (2009) [48]	$p=\alpha ky+\left(1-\alpha \right)p_{\mathrm{y}}\zeta$	Correction for the influence of soil strength and stiffness parameters
Bienen et al. (2012) [49]	$\frac{y}{D}=f_N{A}_1{\left(100\frac{H_0}{D^{2}L{q}_{\mathrm{c}}^{\prime }}\right)}^{\alpha }$	$f_N$ is a correction factor and $\frac{H_0}{D^{2}L{q}_{\mathrm{c}}^{\prime }}$ represents the load conditions
Zhu et al. (2013) [50]	$p=\frac{y}{\frac{1}{k_{\mathrm{ini}}}+\frac{y}{p_{\mathrm{u}}}}$	Considering cyclic impact factors
Liu et al. (2015) [51]	$p=0.9p_{\mathrm{u}}\tanh \left(\frac{922.67z}{p_{\mathrm{u}}}y\right)$	$\mathrm{Calculate} p_{\mathrm{u}}$ in segments based on critical depth (for silty soil)
Zhong et al. (2015) [52]	$p=0.5\delta p_{\mathrm{u}}{\left(\frac{y}{y_{\mathrm{c}}}\right)}^{\frac{1}{3}}$	$\delta$ is the reduction coefficient
Baek et al. (2017) [53]	$p=\frac{y}{\frac{1}{c_{\mathrm{i}}{k}_{\mathrm{ini}}}+\frac{y}{c_{\mathrm{u}}{p}_{\mathrm{u}}}}$	Considering the influence of relative density and depth
Lee et al. (2019) [54]	$p=\frac{y}{\frac{1}{k_{\mathrm{ini}}}+\frac{y}{p_{\mathrm{u}}}}$	$p_{\mathrm{u}}$ calculated based on Broms [55]
Hu et al. (2020) [56]	$p=\frac{y}{\frac{1}{\lambda k_{\mathrm{ini}}^{\mathrm{d}}}+\frac{y}{p_{\mathrm{u}}}}$	$\lambda$ is the stiffness reduction factor, $k_{\mathrm{ini}}^{\mathrm{d}}$ considers the influence of pile diameter
Fuentes et al. (2021) [57]	$p=p_{\mathrm{u}}\tanh \left[c_{\mathrm{p}}{\left|\overline{y}\right|}^{n_{\mathrm{y}}}{D}^{n_{\mathrm{D}}}{L}^{n_{\mathrm{L}}}\right]f_{A}sign\left(\overline{y}\right)$	Based on API

. Among them, Wang et al. [47] summarized and discussed the models of horizontal cyclic p-y curves for the first time.

In summary, previous research results have contributed to the application of the p-y curve method. The p-y curve models are categorized into four types: the reduction coefficient method (reduction in soil resistance and parameters), the empirical fitting method calibrated using experimental data, the normalization method integrating multiple calculation models, and the unified method. Factors related to loads and pile–soil interactions are considered in these categories.

3.1.1. Reduction Coefficient Method

The reduction coefficient method considers the influence of cyclic loads on the pile–soil system. It is based on the original static p-y curve, where a reduction coefficient considering cyclic effects is multiplied by the p-value of the p-y curve or a cyclic influence factor is introduced to reduce the p-y curve parameters. The ultimate soil resistance and soil stiffness are often the objects of correction in the latter reduction methods.

The API specification method reduces the static p-y curve to obtain the cyclic p-y curve by introducing a load empirical coefficient A. Existing studies indicate errors in estimating the ultimate soil resistance and soil initial stiffness using the API specification method when compared to experimental results [58]. Zhang et al. [59], based on single-pile horizontal cyclic load tests, suggest taking the A value as 0.52. Zhong et al. [52] propose a reduction coefficient that considers the influence of cycle number, load amplitude, and different soil strengths to modify the original static p-y curve. Wu et al. [60] introduce the soil resistance reduction coefficient $r_{\mathrm{c}}$ affected by the cycle number and load amplitude in their p-y curve model. Zhu et al. [50] propose a reduction coefficient r considering the weakening factor influenced by cycle number N and cycle stress level to correct the soil resistance for the first cycle, obtaining a calculation model for N cycles. Bienen et al. [49] propose a winged pile p-y curve model that reduces the load horizontally through a modification factor  $f_N$ considering the cycle number. These studies provide a comprehensive analysis of load factors, but the influence of the pile and surrounding soil must also be considered.

In parameter reduction models, Hu et al. [56] establish a hyperbolic p-y curve model applicable to long-term cyclic loads, considering the stiffness decay of sandy soil. The model accounts for initial foundation reaction modulus correction with reductions based on the pile diameter and stiffness. Baek et al. [53], contrasting cyclic and static p-y curves in different relative density sandy soils, propose a hyperbolic p-y curve model with dual-parameter reduction, considering the influence of the relative density and depth in cyclic loading corrections.

3.1.2. Empirical Fitting Method

The empirical fitting method is based on the existing p-y curve and involves continuously fitting and correcting the curve using data from cyclic load tests.

Liu et al. [51] combined indoor model test data of silty soil in the Yellow River Delta with the API specification curve to propose a p-y curve model applicable to silty soil foundations. Sun et al. [61], relying on results from indoor model tests of rigid piles, fitted a function f(N) for the number of cyclic loads and modified the relationship between soil resistance and the number of cycles into a calculation formula for soil resistance and displacement. Gerolymos et al. [48] build upon the BWGG model, calibrating model parameters through centrifuge model tests and introducing an empirically fitted p-y curve model. It is noteworthy that these types of p-y curve models have relatively simple calculations, high reference value for specific pile and soil conditions, and significant advantages in practical applications. However, their application often relies on data from specific tests, and further validation through more case studies is required to assess their applicability.

3.1.3. Unified Method

The unified method primarily combines experimental data to propose cyclic p-y curves from a comprehensive perspective. In China, notable approaches include Tongji University’s method and Hohai University’s unified method. Tian and Wang [13], based on the results of field tests on horizontally loaded piles, corrected the static load p-y curve from the perspectives of the degradation of ultimate soil resistance and soil strain softening. They established a unified method for segmented cyclic p-y curves in clay. Zhang and Chen [62], summarizing previous p-y curve models and combining them with indoor model tests, proposed the effective depth $x_{\mathrm{rc}}$ and effective length $L_{\mathrm{c}}$ based on improved methods, incorporating a reduction factor for ultimate soil resistance. The displacement $y_{50}$ corresponding to half of the ultimate soil resistance was obtained through summarizing relevant test data, and a segmented p-y curve model was established.

In conclusion, the characteristic parameters in the aforementioned p-y curve models are related to the undrained shear strength $S_{\mathrm{u}}$ from triaxial tests and the strain ${\epsilon }_{50}$ corresponding to half of the maximum principal stress in unconsolidated undrained conditions. These parameters can represent the characteristics of cohesive soil and are thus referred to as the unified method. Furthermore, both unified methods are constructed as segmented functions and can reflect soil cyclic softening.

3.1.4. Normalization Method

Due to the complexity of analyzing the pile–soil system under cyclic loads, it is often challenging to consider the influence of a single factor. Therefore, it is necessary to integrate various modified models. Wang et al. [63] observed significant differences between the experimental p-y curve obtained from monotonic load tests and the code-specified curve, and they proposed a hyperbolic p-y curve model based on centrifuge tests and calculations by Zhang et al. [64]. Fuentes et al. [57], considering the relationship between ultimate soil resistance, initial foundation modulus, and soil depth, along with cyclic influence factors such as soil density, load amplitude, and cycle count, proposed a comprehensive p-y curve model referencing Barton’s ultimate soil resistance calculation and improving upon the API specifications [65]. Although the form of the model is relatively complex, it comprehensively considers various influencing factors, ensuring a more comprehensive analysis. Kim et al. [66,67] introduced a cyclic correction factor and presented a hyperbolic p-y curve calculation framework based on the cone penetration test (CPT). Subsequently, they proposed a CPT-based cyclic p-y curve model.

3.2. Analysis of Influencing Factors

The number of cyclic loads, load amplitude, load frequency, loading type, long-term load action, pile type, pile diameter, and embedment depth are all factors influencing the p-y curve. Previous studies have explored the impact of these factors on the p-y curve through experiments [54,68].

In the process of static loading, the influence of loading rate is often considered. However, the application of horizontal cyclic loading typically involves maintaining a relatively low loading rate. As a result, the impact of loading rate on the p-y curve is generally small, and it has not been a major focus. Recently, some studies have highlighted the correlation between the loading rate of cyclic loading and the initial stiffness of the p-y curve, particularly in relation to excess pore pressure (EPP) [69]. However, the load frequency is a crucial influencing factor in cyclic loading.

Sandy soil particles tend to become arranged more densely with an increase in load frequency, leading to an increase in soil resistance, as shown in Figure 6a [68]. Excessively high load frequencies can induce dynamic effects in the soil. In practice, the load frequencies for horizontal cyclic loading are generally within a small range (0.01–1.0 Hz). Consensus has been reached in current research on the p-y curve method, particularly in studies involving low-frequency cyclic loading [70].

Typically, attention is focused on the unidirectional and bidirectional cyclic loading of pile foundations. Zhang and Chen [62] and Yang and Zhang [71] indicate that bidirectional loading has the greatest impact on the p-y curve in cohesive soils. In contrast, in sandy soils, bidirectional loading results in less cumulative pile displacement at the same load value compared to unidirectional loading [72]. It is noteworthy that when considering the type of load application, it is necessary to study different soil types. Additionally, Hong et al. [73] indicated that in four-direction (i.e., starting from 0° and increasing by 90° each time) cyclic loading, the cumulative pile displacement includes lateral cyclic displacement relative to the current direction and the previous adjacent direction, further softening the surrounding soil, which is detrimental to the bearing performance of the pile foundation.

There has been considerable research on the influence of the number of cyclic load cycles on the p-y curve. However, the results consistently show that significant changes in the p-y curve occur within the first few dozen or hundred cycles. With an increase in the number of cycles, there is a consistent trend of decreasing ultimate soil resistance, decreasing soil stiffness, and increasing displacement, as shown in Figure 6b [56]. The horizontal bearing capacity of the pile foundation decreases with the increasing number of cycles. Leblanc et al. [74] pointed out that previous studies were based on a limited number of cycles of testing. In recent years, research has increased the number of cycles to tens of thousands or hundreds of thousands, considering the long-term effects of cyclic loading. After a certain number of cycles, the soil structure stabilizes, and subsequent displacement development becomes very slow.

Yang et al. [75], through numerical simulation, found that the pile head displacement increases with an increase in the load amplitude. Figure 6c shows the backbone curves of the p-y under different load amplitudes for the same load frequency in a group pile [68]. Fu studied the impact of five different levels of load amplitude on the p-y curve and found that the soil resistance increased with the gradual increase in load amplitude [76]. Zhu et al. [77] found that for two sets of tests with symmetric bidirectional loading, the p-y curves of the set with a larger load amplitude showed a more pronounced decrease in horizontal bearing capacity compared to the set with a smaller load amplitude. Therefore, when studying the factors influencing the horizontal cyclic p-y curve, it is necessary to consider the load amplitude.

In recent years, piles with diameters of 5 m or even larger have been widely used in offshore wind power projects. Existing studies have shown that the estimation of the lateral bearing capacity of large-diameter single piles by the code method is not very accurate, making the pile diameter an important influencing factor [9]. Figure 6d shows different p-y curves for different pile diameters [56], illustrating that increasing the pile diameter has a positive effect on the horizontal bearing capacity of the pile foundation. Embedment depth also affects the cyclic p-y curve, as shown in Figure 6e [54]. In deeper soil layers, the pile foundation experiences a greater soil resistance. Increasing the pile diameter and embedment depth involves more soil being mobilized laterally and vertically, expanding the scope of influence and thereby improving the bearing capacity of the pile foundation.

Additionally, under sufficiently large horizontal cyclic loading amplitudes, gaps may form between the pile and the soil, which often occur in clays. It can be inferred that the formation of these gaps is related to the properties of the soil. When the pile recontacts the surrounding soil under the guidance of cyclic loading, some energy dissipates in this process, significantly affecting the cyclic response of the pile. This is manifested in the p-y curve under cyclic loading, exhibiting a narrower shape [78,79]. Zhu [80] reduced the ultimate soil resistance under cyclic loading to 0.56–0.64 times that under static loading, indicating the influence of the pile–soil gaps on the p-y curve.

4. Seismic p-y Curves

4.1. Seismic p-y Curve Models

Based on previous research and a survey of the existing literature, p-y curve models for seismic loading can be classified into the p-multiplier method, empirical fitting method, and other p-y curve models according to the proposed methods.

4.1.1. p-Multiplier Method

The p-multiplier method multiplies the API standard p-y curve by a coefficient to represent the post-earthquake reduction in soil strength. The strength reduction factor closely related to the relative density of the soil is considered a type of p-multiplier method.

Liu and Dobry [81], based on centrifuge model tests, found a linear correlation between soil resistance and the pore pressure ratio. They fitted the relationship between the strength reduction factor and the pore pressure ratio using experimental data, indicated by the black dots in Figure 7. When designing pile foundations for liquefied soil, consider reducing the soil reaction to 0.1 times that before liquefaction. Boulanger et al. [82] quantified the relationship between the strength reduction factor and the pore pressure ratio as shown by the red line in Figure 7.

Wang and Feng [83] similarly utilized a decay factor s, correlated with the pore pressure ratio, to attenuate the soil resistance of the API specification p-y curve, addressing post-earthquake soil strength degradation concerns. Observations from shake table tests indicate that the strength of liquefied sand with a relative density $D_{\mathrm{r}}$ between 20% and 40% decreases to 0.1 of its pre-liquefaction strength. This finding aligns closely with the conclusions of Liu and Dobry regarding liquefied sand with $D_{\mathrm{r}}$ ≈ 62%.

4.1.2. Empirical Fitting Method

Gerber [84] triggered liquefaction in the soil by blasting on Treasure Island, conducting on-site tests on both single piles and pile groups to obtain experimental p-y curves. A new liquefied soil p-y curve equation was established by mathematically fitting the experimental curves. Rollins et al. [85] quantified this expansive effect using an empirical p-y curve model. Chang and Hutchinson [86], based on vibration table test results, demonstrated the alignment between the model and the p-y curve when the pore water pressure $r_u$ = 100%.

4.1.3. Other Methods

For pile–soil systems subjected to seismic loads, the presence of excess pore water pressure weakens the soil strength. Zhang et al. [87] theoretically derived a three-stage weakened sandy soil p-y curve, using the ultimate soil reaction force as the starting point. Li et al. [88] based on the API specification p-y curve, proposed two soil liquefaction impact correction parameters $\alpha$ and $\beta$, which are related to the pile depth-to-length ratio and the relative density of saturated sandy soil. These parameters are used to modify the ultimate soil reaction force and the initial modulus of the soil in the p-y curve.

Yoo et al. [89] obtained experimental p-y backbone curves through centrifuge model tests with different pile diameters, load frequencies, and acceleration amplitudes. In contrast, Lim and Jeong [32], in addition to considering the influence of load frequency and acceleration amplitude, the natural frequency of the system and the influence of soil confining pressure were also considered. Yang et al. [33] focused on the bending stiffness of the pile and the pile head mass. These studies all improved upon the same ultimate soil reaction force and soil stiffness calculations. Yoo et al. [90], drawing inspiration from Matlock’s ultimate soil reaction force calculation, proposed a hyperbolic dynamic p-y curve model for soft clay.

Franke and Rollins [91] proposed a composite p-y curve model by integrating Wang and Reese’s p-y curve model with Rollins’s residual shear strength model. They used ${\epsilon }_{50}$ as a function of ${(N_1)}_{60-\mathrm{cs}}$ and some studies considered dynamic loading as equivalent static loading, establishing dynamic p-y curve models for single piles based on the cone penetration test (CPT) to consider damping effects [10].

Sandy soils that have fully liquefied exhibit characteristics similar to those of soft clays, leading to the use of a p-y curve similar to that of soft clays to describe the p-y curve for sandy soils after liquefaction due to seismic or other vibratory loads. This approach replaces the soil pressure in the soft clay p-y curve with the undrained residual shear strength of liquefied sandy soils. However, obtaining accurate values for undrained residual shear strength poses challenges. Although there is a weak correlation between this strength and the ${(N_1)}_{60}$ obtained from the standard penetration test, the relationship is not strong enough, making the application of this method in engineering projects quite limited [92].

4.2. Discussion on Seismic p-y Curve Models

The research on seismic p-y curves has gradually evolved from initially focusing solely on whether the soil undergoes excessive liquefaction during earthquakes to investigating the influencing factors of seismic p-y curves. Numerous studies, employing indoor experiments, shake table tests, and centrifuge tests, have explored the impact of seismic loads on p-y curves. Their research consistently indicates significant discrepancies when using pseudo-static methods or API code for seismic design. The initial exploration of influencing factors on seismic p-y curves was not comprehensive. In recent years, studies have gradually considered factors such as pore water pressure, soil relative density, pile diameter, depth, vibration intensity, and frequency on seismic p-y curves.

4.2.1. Shape of Seismically Liquefied Soil p-y Curves

In accurately representing the relationship between the soil reaction force and pile displacement under seismic loading, guiding the seismic design of pile foundations has attracted a significant number of researchers to address this research problem. The outcomes of their studies have made substantial contributions to the seismic design of pile foundations.

The conventional liquefied soil p-y curves suggest that the initial interaction between piles and soil exhibits significant stiffness, followed by a softening effect in the soil. Some researchers propose that the liquefied soil p-y curve takes on an inverted “S” shape, indicating an upward concave trend, exhibiting a hardening effect when the surrounding soil is completely liquefied [93]. This is due to the expansion of liquefied soil during pile shearing, leading to the dissipation of excess pore water pressure and an increase in soil stiffness. Figure 8 illustrates the suggested shapes of p-y curves before and after liquefaction [94], revealing distinct differences in the p-y curve shapes before and after liquefaction.

4.2.2. Discussion on the p-Multiplier Method

Many researchers have focused on the determination of the p-multiplier in the p-y curves for seismic loads, particularly in liquefiable soils. Despite variations in their findings, indicating different values for the p-multiplier (e.g., 0.1, 0.1–0.2, and 0.25–0.35, between 0.26 and 0.30, and between 0.14 and 0.38), these values are influenced by differences in the mechanical properties of the tested soils, liquefaction severity, groundwater conditions, seismic loading, and other factors. This does not hinder their common approach of using a relatively small value to assess the post-liquefaction reduction in bearing capacity of soil subjected to seismic loading [81,82,95,96].

Despite variations in the p-multiplier values proposed by different studies, these differences fall within a relatively small range. The discrepancies arise from significant uncertainties in inversely calculating the p-multiplier through experimental comparisons. The influence of factors such as the relative density of liquefied soil, drainage conditions, and loading scenarios should be comprehensively considered. While the p-multiplier method aims to realistically capture the changes in soil bearing capacity after seismic loading, some research suggests that it may overestimate soil stiffness and fail to account for the expansion effects of liquefied soil, potentially leading to an overestimation of a small displacement soil resistance [97].

In the p-multiplier method, only one strength reduction factor is used to adjust the strength of liquefied soil after liquefaction. In the original p-y curve method, the initial stiffness of the soil is also an important parameter. In large-scale shake table tests conducted by Cubrinovski et al. [98], it was observed that the stiffness of the soil differed by 30–80 times before and after liquefaction. Considering the soil dilation effect aims to assess the dissipation of excess pore water pressure in the soil as it expands during shear. In this zone, the soil transitions from a liquefied state to a non-liquefied state, where strength and stiffness partially recover, exhibiting strain hardening characteristics. However, expressing this influence using the p-multiplier method is actually challenging.

Until the complete liquefaction of the soil, when designing pile foundations, the soil resistance p is considered to be reduced by a factor of 0.1 in the calculation. In fact, utilizing the apparent strength reduction factor to adjust the p-y curve is a simplified approximation, and further research is needed to evaluate its reliability.

Figure 9 compares the liquefied soil p-y curves proposed with the p-multiplier method and the non-liquefied soil API code-based p-y curves [97]. A more notable aspect is that the proposed p-y curves can overcome some limitations of the p-multiplier method by dynamically reflecting the interaction between piles and soil.

4.2.3. Factors Influencing Seismic p-y Curves

Summarizing the errors in liquefied soil p-y curves caused by API specification, it is noted that the standards significantly underestimate the ultimate soil resistance of shallow soils and overestimate that of deep soils. Additionally, they exhibit excessive linearity under small strains [99]. Correcting the errors in the API standards for dynamic load p-y curves requires the consideration of more factors.

The dynamic p-y curves are influenced by the pore water pressure. The fundamental reason is that external loads disrupt the original structure between soil particles, preventing the timely drainage of water from the soil. This leads to an increase in pore water pressure, affecting the soil’s bearing capacity. Figure 10 illustrates the dynamic p-y backbone curves for a 3.5 m-deep single pile under different excess pore pressure ratios (EPPRs) [100]. It can be observed that the ultimate bearing capacity of the pile foundation decreases with an increase in the EPPR. Similarly, Zhang et al. [101] identified a strong correlation between the number of cyclic vibrations and pore water pressure, suggesting that the pore pressure in sandy soils increases with continued vibrations, leading to a steady reduction in soil resistance around the pile as the sand undergoes liquefaction. This results in increased relative displacement between the pile and soil. Building on this, they proposed a theoretical framework that correlates the number of vibrations with pore water pressure. This approach was incorporated into the dynamic p-y curve model for saturated liquefied sandy soils, as developed by Dash et al. [94] to assess the bearing capacity of pile foundations.

Furthermore, research indicates that the resistance of coarse sand under loose conditions is significantly greater than that of fine sand. Even in densely packed sandy soil, the resistance of coarse sand remains higher than that of fine sand under small displacements [102]. Therefore, under the same soil relative density conditions, choosing coarse sand can enhance the seismic performance of pile foundations.

5. Discussion

(1) In the design of building pile foundations, the m-method is commonly employed, which considers the relationship between soil reaction force p and displacement y as linear, as shown in Equations (6) and (7) [1].

$$p=k_{\mathrm{s}}(z)\cdot b_0\cdot y$$

(6)

$$k_{\mathrm{s}}(z)=m\cdot (z-h)$$

(7)

where h is the design depth, $b_0$ is the pile calculated width, z is the calculation depth, and m is a constant related to soil properties and design requirements.

Similarly, the c-method and k-method, frequently used in engineering practice, also consider the foundation reaction coefficient as a linearly changing parameter to different extents. These methods fall under the category of linear elastic foundation reaction methods, which view the pile–soil relationship as discrete and linear springs, making them more suitable for addressing small displacement problems.

In contrast, the p-y curve method treats the pile–soil relationship as discrete and nonlinear. In this method, the foundation reaction coefficient changes nonlinearly with displacement. Here, the soil is considered as an elastic–plastic body. When dealing with large displacement problems (y > 10 mm), the p-y curve method demonstrates a significant advantage over the linear elastic foundation beam method in accurately representing the nonlinear behavior of the soil. Thus, it is recommended to prioritize the p-y curve method in the design of horizontal forces on pile foundations.

(2) The API specification p-y curves are derived from tests conducted on small-diameter piles. However, larger diameter single piles (D = 2–8 m) are widely employed in offshore wind power and large-span bridge foundations. Research on p-y curves focuses on the modification of ultimate soil resistance ($p_{\mathrm{u}}$) and initial foundation modulus ($k_{\mathrm{ini}}$). Specific formulas are proposed for calculating the ultimate soil resistance and initial foundation modulus tailored to different piles and soil conditions. For instance, in Reference [42], formulas for calculating the ultimate soil resistance considering the effects of pile diameter and depth are proposed based on the API specification p-y curves, as shown in Equations (8) and (9).

$$p_{\mathrm{u}}=0.53K_{\mathrm{p}}^2{\gamma }^{\prime }{z}^{a_n}{D}^{b_n}$$

(8)

$$k_{\mathrm{ini}}=n_{\mathrm{h}}{z}^{0.7}{\left(\frac{D}{D_0}\right)}^{0.55}$$

(9)

where $K_{\mathrm{p}}$ is the passive earth pressure coefficient, ${\gamma }^{\prime }$ is the effective unit weight of soil, D is the pile diameter, $D_0$ = 1.0 m, z is the depth, and ${\eta }_{\mathrm{h}}$ is the foundation reaction coefficient.

Reference [31] introduces factors $F_1$ and $F_2$ to adjust the API specification p-y curves considering the effects of pile installation methods and pile head restraint conditions, as shown in Equations (10) and (11). However, it is regrettable that these influencing factors are not quantified, which imposes certain limitations on their use.

$$p_{\mathrm{u}}=F_1{K}_{\mathrm{p}}{\gamma }^{\prime }{z}^{n}D$$

(10)

$$k_{\mathrm{ini}}=F_2{n}_{\mathrm{h}}z$$

(11)

The differences in these calculations lie in the depth-related factors (linear and nonlinear correlations), which essentially arise from researchers’ efforts to predict the pile horizontal bearing capacity under specific conditions. Developing a unified calculation strategy to reduce these differences poses significant challenges. Accurately predicting ultimate soil resistance and initial stiffness is necessary to improve the accuracy of p-y curve predictions. Furthermore, in the absence of a widely accepted method to improve calculation accuracy, standards such as the API provide an effective means of evaluating the pile horizontal bearing capacity, albeit being considered conservative.

(3) The p-y curve method, based on beam bending theory, uses the method of small elements to derive the control equation for piles based on the moment equilibrium, as shown in Equation (12) (where EI is the pile flexural stiffness). In the process of deriving the horizontal load p-y curve for piles, the equilibrium condition of the axial load on the elemental body is ignored. When the pile foundation bears the load, it not only experiences horizontal loads (H) but also transmits the loads from the superstructure and structural loads to the soil through the pile [103]. At the pile head, there are bending moments (M), vertical loads (V), and pile shaft frictional resistance (f). For example, in offshore wind turbines, due to their asymmetric structure, bending moments always exist at the pile head. Additionally, when horizontal loads or seismic actions induce horizontal displacement in the pile, it triggers additional bending moments at the pile head due to vertical loads, exacerbating the development of horizontal displacement, known as the p-Δ effect. This effect leads to asymmetric frictional resistance on both sides of the pile. The control equation is rewritten as Equation (13). Combining the multi-spring coupled model for horizontal load analysis of pile foundations mentioned in Section 2.1 of this paper, considering the combined load effects into the pile moment–angle spring ($M_{\mathrm{p}}-{\theta }_{\mathrm{p}}$), provides a possible reference for improving the accuracy of the p-y curve.

$$EI\frac{{\mathrm{d}}^{4}y}{\mathrm{d}{z}^4}+p\left(z\right)=0$$

(12)

$$EI\frac{{\mathrm{d}}^{4}y}{\mathrm{d}{z}^4}+p\left(z\right)-V_{\mathrm{v}}\frac{\mathrm{d}y}{\mathrm{d}z}+p_{\mathrm{z}}\frac{{\mathrm{d}}^{2}y}{\mathrm{d}{z}^2}+\frac{\mathrm{d}{M}_{\mathrm{R}}}{\mathrm{d}z}=0$$

(13)

Furthermore, in the direction of the pile foundation section, it may be subject to horizontal loads from different directions, such as those encountered in offshore wind turbines where loads from wind, waves, and currents originate from various horizontal directions. Under the action of multi-directional horizontal loads over time, the cyclic softening zone around the pile is expanded, further weakening the horizontal bearing capacity of the pile foundation. Therefore, the author suggests that when establishing the p-y curve method, practical conditions should be fully considered, including the combined effects of H-V-M loads and the historical loading of multi-directional horizontal loads.

(4) The p-y curve method effectively captures the nonlinear relationship in the pile–soil interaction. However, most p-y curve models are developed under idealized conditions, overlooking the soil disturbance caused by pile installation. Consequently, it is essential to account for the effects of pile foundation construction methods and the resulting soil disturbance on p-y curves.

For combined loads, the focus is primarily on vertical loads and multidirectional horizontal loads. Permanent vertical loads usually exist above the pile foundation, and when horizontal displacement occurs, these loads create additional moments at the pile head, known as the p-Δ effect. This effect further increases the difference in vertical friction on both sides of the pile, resulting in equivalent moments in the pile shaft. Both factors exacerbate horizontal displacement while also increasing the soil resistance along the pile shaft.

Regarding the p-y curve’s consideration of combined loads and construction impacts, some preliminary studies have already been conducted, providing us with methodological guidance and references. For instance, Mu et al. [104] incorporated vertical loads into the factors influencing p-y curves, thereby improving the calculation of the ultimate soil resistance of shallow soils in the hyperbolic p-y curve method. Additionally, when multidirectional horizontal loads are applied, the cyclic weakening zone of the soil around the pile foundation expands, causing greater soil displacement compared to unidirectional horizontal loads.

To enhance the practicality of p-y curves for pile foundations, it is essential to consider real-world influencing factors such as the installation effect. Kim et al. [31] conducted model tests on pile foundations under monotonic horizontal loading in the Nak-Dong River sand in Korea, primarily investigating the effects of different pile head constraints and installation methods. Their findings indicated that the initial modulus of subgrade reaction at the same depth varied significantly between fixed and free pile heads, with differences of nearly fivefold in medium-dense sand and nearly threefold in dense sand. For piles driven with different energy levels (0.5 J, 1.0 J, 1.5 J), the initial modulus of subgrade reaction was found to be 1.5, 2.0, and 2.5 times greater than that of pre-installed piles, respectively, while the ultimate soil resistance was 2.0, 3.0, and 4.0 times greater. Based on these observations, a hyperbolic model was chosen to correct the initial modulus of subgrade reaction and ultimate soil resistance, considering the installation method. The correction factors F1 and F2 were introduced, resulting in a sand p-y curve model that accounts for the installation method. Although Kim et al. [31] addressed the effects of pile installation and pile head constraints, they did not specifically quantify the reduction factors F1 and F2 in the p-y curve model for broader applications, thus limiting its use.

(5) The p-multiplier method employs a reduction factor directly related to the degree of soil liquefaction to reduce the strength of post-earthquake soil, offering a convenient approach. It provides a reference for predicting the p-y curves of piles in liquefied soil after earthquakes. However, its limitation lies in not considering the change in soil stiffness. As discussed in Section 4.2.2, the difference in soil stiffness before and after liquefaction can range from 30 to 80 times [98], which is a crucial aspect for adjusting p-y curves. Additionally, expressing the shear dilatancy effect and excess pore water pressure dissipation through the p-multiplier method is challenging. The soil shear stress–strain relationship (as shown in Figure 8a) indicates a significant increase in stiffness with displacement after an initial phase of low stiffness. Therefore, the suggested p-y curves for post-earthquake liquefied soil, as depicted by the red line in Figure 8b, account for this behavior.

In fact, providing an accurate prediction of pile behavior under seismic loads with p-y curves is challenging. Pile foundations are widely used, as discussed in the Introduction. It is necessary to consider more influencing factors such as seismic load accelerations, strength, and post-earthquake pore water pressure to develop an accurate and reliable p-y curve prediction model

(6) Various p-y curve models have been proposed to date, encompassing static loads, horizontal cyclic loads, and seismic loads, and considering numerous influencing factors such as loads, piles, and soil properties. Despite these advancements, the practical engineering design of pile foundations still tends to rely on conservative specifications. While these specifications provide convenient solutions, the development and application of the p-y curve method still have a long way to go. To further enhance the practicality and accuracy of p-y curve applications, it is suggested that a shared p-y curve database be established in the future. This database would allow engineers to easily access necessary models and perform calculations based on existing parameters using selected p-y models during pile foundation design. Such a database would significantly improve the efficiency of p-y curve computations and address current challenges associated with p-y curve applications. Additionally, it would integrate existing p-y curve models and their influencing factors, utilizing advanced methods such as convolutional neural networks to predict future p-y curve models based on temporal and spatial attributes.

For instance, Zhang et al. [105] introduced a convolutional long short-term memory network (conv-LSTW) with spatial and temporal attributes, as shown in Figure 11. By integrating existing p-y curve models, this approach can overcome the limitations of current p-y curves, which are only applicable to specific soil conditions, pile parameters, and load conditions.

The conv-LSTW method would incorporate ReLU, logistic sigmoid, and tanh functions into the neural network to enhance the fitting ability for the two main types of p-y curve functions (Hyperbolic and Tangent). Moreover, a vast amount of historical data extracted from existing large-scale field tests and model experiments (numerical simulation data are not recommended) would be fed into this neural network system, including load factors, pile and soil factors, and existing models, to establish a massive p-y curve database for neural network deep learning. Additionally, an interface would be provided for researchers to train and engineers to access the model for the calibration and application of p-y curve prediction models.

Overall, this approach would greatly advance the development and application of p-y curves, but it would also face challenges such as data sharing, researcher capability, data quality and quantity, which would require collaborative efforts to overcome.

6. Conclusions

(1) Although the p-y curve method explores the lateral load-carrying capacity, these loads often impact the horizontal load-carrying capacity, for example, the $p-\Delta$ effect caused by vertical loads. Therefore, it is necessary to consider the p-y curve of building foundations under combined loads and to analyze the coupling of p-y springs with other spring elements.

(2) The p-y curves for horizontal cyclic loads are mainly derived from static p-y curves, where the key to accurately predicting pile–soil interaction lies in the ultimate soil resistance and soil stiffness. Under cyclic loads, the soil undergoes degradation in strength and stiffness, making it crucial to accurately capture the soil degradation effects when solving the p-y curves for pile foundations under horizontal cyclic loads.

(3) Horizontal cyclic p-y curves are typically related to the soil conditions, pile embedment depth, pile diameter, and load conditions. The normalized p-y curve model comprehensively considers these influencing factors, but its practical implementation may pose significant challenges. Empirical fitting models have undeniable advantages in addressing pile–soil interaction problems under specific conditions but come with inevitable limitations. Overall, for practical engineering problems, the reduction coefficient method is widely used.

(4) Extensive research has been conducted on seismic p-y curve models, especially on the conditions of pre- and post-liquefaction. The pre-liquefaction p-y curve shows a gradually decreasing tangent slope, indicating soil softening. Therefore, the p-multiplier method is proposed to compensate for the deficiencies in calculating pre-liquefaction p-y curves according to the API standards. Additionally, the undrained shear strength method and some novel p-y models are proposed to improve the p-multiplier method. The post-liquefaction p-y curve gradually increases from a small initial stiffness, exhibiting hardening characteristics, leading to the development of various post-liquefaction p-y curve models. These studies collectively advance the development of the p-y curve method for seismic loads on pile foundations.

(5) The p-y curve method can better reflect the nonlinear relationship in the pile–soil interaction. However, the establishment of the most p-y curve method is based on idealized conditions, neglecting the disturbance of the soil caused by pile installation. Therefore, it is necessary to consider the impact of pile foundation construction methods and soil disturbance on p-y curves. Furthermore, existing research experiences and achievements need to be extended to new types of foundation systems to promote further development in pile foundations and other novel foundation technologies.

(6) Future research on static and dynamic p-y curves for pile foundations should be more comprehensive, integrating advanced technologies such as artificial intelligence, deep learning, and big data to enhance reliability and practicality. These technologies can effectively improve research efficiency and accuracy, aiding in the establishment of more precise numerical models and analysis of data. Simultaneously, integrating various data and building a comprehensive p-y curve database will facilitate data sharing and sustainable utilization.