Statistical Evaluation of API P-Y Curve Model for Offshore Piles in Cohesionless Soils

Abstract

Pile foundations are widely used to support offshore wind turbines. While the p-y curve method is adopted for analysis of pile–soil interactions in popular design specifications, including the American Petroleum Institute (API), its accuracy remains unassessed systematically and quantitatively. This study established a database by collecting 491 sets of pile p-y curves from multiple offshore wind turbine projects. The database was used to statistically evaluate the accuracy of the API p-y curve method for cohesionless soils. The model accuracy is represented by a model factor defined as the ratio of measured to predicted values of soil resistance around the pile. The results showed that accuracy assessment using the field data is significantly different from that using the laboratory model test data. On average, the API p-y curve method overestimates the true soil resistance in the field by about 30%, but underestimates that in the laboratory by about 8%. The dispersions in prediction accuracy of both cases are high. Correction terms are introduced to calibrate the current API p-y curves. The calibrated API methods were shown to be accurate in general and medium dispersive in prediction accuracy. Last, the model factors for the current and calibrated API methods were demonstrated to be lognormal random variables.

1. Introduction

Thermal power is the most traditional form of energy supporting the daily life of the entirety of human society for the past centuries. In recent decades, new forms of energy that are more clean, renewable, and sustainable have been advocated for and developed, aiming at reducing carbon emission and easing extreme climate changes. Among these, wind energy has garnered widespread attention and rapid development [1,2]. In particular, offshore wind power has become a focus because of its higher energy capture efficiency, smaller land use, and advantages such as abundant resources, wide distribution, and ease of large-scale development. For example, over 50 countries and regions have initiated offshore wind power projects [3,4].

There are various types of foundations installed in the ocean to support wind turbines generating power, including monopile [5,6], suction bucket [7,8], jacket [9], gravity-based foundation [10,11], and floating foundation [12,13]. At the current stage, most wind farms are built nearshore in shallow water areas (e.g., less than 100 m depth) and thus monopiles are the most commonly used foundation type. According to investigative reports [14], about 75% offshore wind turbine foundations are monopiles. A pile foundation bears not only the gravitational loads from the weights of the upper structures it supports, but also dynamic loads from wind, ocean currents, waves, sea ice, and even earthquakes. These loads could cause foundation failures [15,16], such as the exceedance of ultimate lateral capacity, plastic hinge of the pile, and exceedance of allowable tilt angle [17]. Moreover, due to the large dimensions and complicated ocean environment, the construction of pile foundations accounts for approximately 20–30% of the total project cost [18]. If failed, the cost for pile repair or replacement would also be a huge number. Therefore, designing a safe while cost effective pile foundation is still challenging but desirable to engineers.

An accurate analysis of pile–soil interaction subjected to loads is a key step to achieve this goal. In the literature, the pile–soil interaction was extensively investigated experimentally [19,20,21], numerically [22,23,24,25], and theoretically [26,27,28,29]. A series of large-scale field tests within the framework of the Pile–Soil Analysis (PISA) project have been completed and a four-spring model was proposed [30,31]. In this four-spring model, the distributed load, distributed moment, base moment, and base shear are simultaneously applied to the pile shaft, which provides a comprehensive and reliable method to represent the stress modes along monopiles. While in practice, the p-y curve method has been widely adopted in design manuals, guidelines, and specifications (e.g., API specifications [32], DNV [33], China manual [34]), given its simplicity and practicality. The term “p” denotes the surrounding soil horizontal resistance along the pile while “y” denotes the pile shaft horizontal displacement. The curve is usually nonlinear, depending on soil type, depth, pile stiffness, loads, etc.

It is reported that McClelland and Focht [35] firstly proposed the p-y curve method based on results from consolidated undrained triaxial tests in 1956. Reese, Cox and Koop [36] further expanded this approach by fully considering the nonlinear characteristics of soil, and their expanded approach was then recommended by the American Petroleum Institute (API) specifications. Zhang, Zhao and Xu [37], Franke and Rollins [38] broadened the application of the p-y curve to saturated and liquefied scenarios. Wang, Wang, Hong, He and Zhu [39] introduced a new approach considering the failure mechanism of single piles in dense sands, further improving accuracy and applicability. Jiang, Fu, Wang, Chen and Ou [40] proposed an analytical solution of the p-y curve for sandy soils, claiming that their calculation method is more accurate and reliable. For silty soil, Zhang et al. [41] conducted two lateral loaded pile model tests with different diameters in silt, which was taken from the construction site. The measurement results showed that the p-y curve model for the pile in silt is similar to the model for sands. However, it was also verified that the cohesion of silty soil could affect the critical soil resistance around piles [42].

Despite the significant advancements made for the p-y curve method, its predictive capability still remains questionable under certain conditions. For instances, Achmus et al. [43] found that the p-y curve tends to overestimate the initial stiffness of small-diameter piles in sandy soils but underestimates the soil resistance of large-diameter piles. Wang, Wang, Hong, He and Zhu [39] and Lu and Zhang [44] further pointed out that the stiffness of piles would decrease with increasing horizontal loads, which is inconsistent with the p-y predictions. In addition, Hearn and Edgers [45] emphasized that the p-y curve is primarily applicable to small-diameter piles. For monopile foundations with bigger diameters, the p-y curve approach tends to either under- or over-estimate the lateral bearing capacity of piles to varying degrees. The above studies indicate that the accuracy of existing p-y curve models in predicting the horizontal bearing characteristics of individual piles is yet to be systematically examined and quantified. The model uncertainty of the p-y curve in the API specification [32] is still unknown, making it very challenging to strike the balance between safety and cost-effectiveness for offshore pile foundation designs. Without model uncertainty at hand, the development of rational reliability-based design methods such as load and resistance factor design (LRFD) [46] or direct reliability design [47,48] for pile foundations is far from possible. All these reasons highlight the need to carry out accuracy evaluation of the current p-y curve method widely used in API specifications.

To fill this gap, the present study collects a large number of test data from pile foundations constructed in cohesionless soil layers for offshore wind turbines, establishing a p-y curve measurement database. Based on this database, the accuracy of the API p-y curve model is statistically evaluated. The model accuracy is quantified through a model factor defined as the ratio between the measured and predicted soil resistance around the pile. After evaluation, correction terms are introduced to the current API p-y curve model for accuracy enhancement. Last, the probability density functions for the model factors are characterized.

2. The API P-Y Curve Method

The horizontal displacement at the pile head is typically the largest compared to other locations along the pile. Hence, it actually controls the horizontal bearing capacity of the monopile. The p-y curve depicts the displacement (y) of the pile shaft at a certain soil depth versus the horizontal resistance (p) of the soil around the pile at that depth. Generally, the mathematical expression of a p-y curve is determined from the regression of data from field or laboratory tests.

For piles in sandy soils, the API specifications [32] recommends a hyperbolic tangent-shaped p-y model, written as follows:

$$p=A_s{p}_u\tanh \left({\displaystyle \frac{kH}{{A_{s}p}_u}}y\right)$$

(1)

where $A_s$ represents the correction factor of soil ultimate resistance, defined as $A_s=\max \left(3-{\displaystyle \frac{0.8H}{D}},0.9\right)$ with $H$ being the depth below the soil surface (m) and $D$ being the pile diameter (m); $p_u$ is the ultimate resistance of the soil (kN·m⁻¹) given by $p_u={\gamma }^{\prime }H·\min \left(C_{1}H+C_{2}D,C_{3}D\right)$; the coefficients $C_1$, $C_2$, and $C_3$ depend on the soil internal friction angle; $k$ represents the initial modulus of the subgrade reaction (kN·m⁻³). The values of the parameters can be found in the API specification [32].

3. Database of Measured P-Y Curves for Cohesionless Soil

This paper establishes a database of measured p-y curves reported for offshore wind turbine monopile foundations from the literature, i.e., [49,50,51,52,53,54,55,56,57,58,59,60,61]. The data are collected from 15 monopile foundations installed and monitored in fields and in laboratory model tests.

Table 1. Summary of soil type, properties, and design parameters for the piles reported in the source documents.

Test Type		Pile	Soil Type	Soil Strength Properties		Monopile Geometry				Measure Point Depth	Installation Method	Reference
				$\mathit{\phi }$ (°)	$\mathit{\gamma }$ (kN·m−3)	$\mathit{L}$ (m)	$\mathit{D}$ (m)	t (mm)	Lem (m)	H (m)
Field		P1	fine sand	33.8	21.0	78.5	2.0	30.0	47.5	14.27	driven	[57]
		P2	fine sand	33.8	21.0	82.0	2.0	30.0	50.5	15.57	driven	[57]
		P3	medium-coarse sand	33.0	18.7	55.0	1.8	30.0	36.6	0.1–1.0	driven	[55]
		P4	medium-coarse sand	33.0–35.0	18.5–18.7	53.0	1.9	30.0	40.0	0.1–4.0	driven	[55]
		P5	silty sand	32.0–34.0	18.3–18.6	51.0	1.8	25.0	29.0	1.0–5.0	driven	[58]
		P6	silty sand	34.0	19.0	60.0	1.2	16.0	43.0	3.5–5.0	driven	[54]
Model	75× g level centrifuge test	P7	sand	42.0	19.5	0.85	0.080	1.2	0.41	0.02–0.16	driven	[49]
	1× g level test	P8	saturated silt	25.0	17.8	3.00	0.089	4.0	2.30	0.09–0.71	embedded	[59]
	1× g level test	P9	saturated silt	27.0	18.1	1.10	0.032	7.0	0.85	0.10–0.20	driven	[56]
	1× g level test	P10	sand	38.0	16.5	1.52	0.102	6.4	1.40	0.17–0.32	driven	[51]
	1× g level test	P11	silty sand	28.5	17.5	7.00	0.114	2.5	4.40	0.11–0.80	driven	[60]
	1× g level test	P12	saturated silt	35.5	17.5	2.00	0.165	3.0	0.85	0.10–0.54	driven	[61]
	1× g level test	P13	saturated silt	30.0	19.3	4.50	0.159	4.5	3.30	0.16–0.80	driven	[53]
	1× g level test	P14	dense sand	37.0	20.0	3.00	0.340	14.0	2.20	0.34–2.04	driven	[50]
	100× g level centrifuge test	P15	sand	34.0	17.1	0.23	0.018	1.0	0.09	0.01–0.07	driven	[62]

Note: $\phi$ = soil internal friction angle; $\gamma$ = soil unit weight; $L$ = monopile length; t = monopile thickness; L_em = monopile embedded depth; H = horizontal displacement and resistance monitoring depth.

summarizes the geotechnical information (including soil type and soil strength properties) and the design specifications of the monopiles. The corresponding source documents are also provided for the reader’s convenience to follow the work.

The soil strength parameters mainly include the soil internal friction angle ($\phi$) and soil unit weight ($\gamma$). All piles were made of steel pipes. The design parameters of the monopiles mainly include the pile length ($L$), pile outer diameter ($D$), pile wall thickness ($t$), and pile embedded depth under soil surface ($L_{em}$). Each pile was equipped with multiple monitoring sensors to record the pile shaft horizontal displacement and corresponding surrounding soil lateral resistance for the objective position. The measurement point depths of each monopile are also present in the database table. In total, there are 491 sets of p-y curve data: 170 sets were from field tests and the remaining 321 sets were from laboratory model tests. For the installation of the collected monopiles, they almost all used the driven method, which is also a widely recommended construction method for offshore wind turbine monopile foundation. Only one pile (P8) was embedded into soil mass in order to ensure the smooth progress of the model test. However, it was verified that the pile–soil interaction around the pile (P8) was also well simulated [59].

A preliminary analysis of the database finds that the monopiles in field tests were constructed in various cohesionless soil layers along the pile lengths, typically including fine sand, medium-coarse sand, and silty sand. In contrast, the model tests used sand, silty sand, and saturated silt that were artificially prepared. It can be concluded that the soils used in field and model monopile lateral bearing capacity tests all are classified as cohesionless soil. According to Table 1, the soil internal friction angle ($\phi$) surrounding the field piles varies less, ranging from 32.0° to 35.0°, with an average of 33.9°. The soil for the model pile cases exhibit significant variation in internal friction angle ($\phi$), ranging from 25° to 42°, with an average of 32.9°. The soil internal friction angle in the two types of tests is highly relative at the average level. For field tests, the sandy soil unit weight ($\gamma$) around the field test monopiles averages 19.5 kN·m⁻³, with values ranging from 18.3 to 21.0 kN·m⁻³. The soil unit weight of the model test is 17.1–20.0 kN·m⁻³, with an average of 18.1 kN·m⁻³.

Regarding the monopile design parameters, the lengths ($L$) of all field test piles exceed 50 m, with a maximum of 78.5 m (P1) and an average of 56.2 m, while for the model test piles in laboratory, the lengths are much shorter, ranging from 0.23 to 7.00 m, with an average of 2.98 m. Among them, the model tests of the P7 and P15 piles are centrifuge tests with 75× g and 100× g acceleration levels, respectively. Therefore, the prototype pile lengths are 63.75 m and 23 m. In terms of diameter ($D$), field piles ranged from 1.2 m to 2 m, while model piles ranged from 0.032 m to 0.34 m. The thicknesses ($t$) of the field piles mostly ranged from 16 to 30 mm, whereas the model pile thicknesses ranged from 1.0 to 14 mm. Clearly, the pile dimension varies extremely widely between field and model cases, even reaching one to two orders of magnitude. This could cause a great difference in model accuracy evaluation, and this can be referred to as the size effect. The size effect will be discussed later in the paper.

In addition, previous research has shown that the monopile embedded depth has a significant effect on pile–soil interaction, which controls pile failure condition and mode. The embedded depths (L_em) of collected piles are also present in the table. The embedded depths of test piles account for 56.9–75.5% of the pile total length in field tests and 39.1–77.3% in the main model tests. Among model tests, the embedded length of the P10 pile has a most distinctive value, 92.1% of total length, and is driven into quite a deep position. Multiple sensors were arranged along the pile depth direction to monitor pile shaft horizontal displacement and surrounding soil lateral resistance. The measurement point depth below the soil surface for field tests ranges from 0.1 to 15.57 m, with 80% being less than 5 m, while the monitoring depth for model tests ranges from 0.01 to 2.04 m, with 80% being less than 0.80 m.

In order to have a clearer understanding of the target information in the established database, the histograms and cumulative distributions of pile shaft horizontal displacement and surrounding soil lateral resistance are drawn in Figure 1 and Figure 2. The horizontal displacements observed in the field and model tests are less than 122.44 mm (80% less than 35 mm) and 310.104 mm (80% less than 25 mm), respectively, as shown in Figure 1. The soil lateral resistance ranges from 2.53 to 814.05 kN·m⁻¹ for the field cases and from 0.04 to 4793.89 kN·m⁻¹ for the model cases (see Figure 2). The majority of values are less than 85 kN·m⁻¹ in model tests and 408 kN·m⁻¹ in field tests.

To meet the installation requirements of steel pipe piles and to prevent local buckling when the pile reaches yield strength, the thickness should satisfy the following conditions [17,32]:

$$t\ge 6.35+{\displaystyle \frac{D}{100}}$$

(2)

Figure 3 shows the plot of thickness against diameter in the database, with Equation (2) and data from other references also presented for comparison. The linear line in the figure represents the minimum wall thickness value suggested by the API specification [32]. Most field test piles in the database met this requirement, whereas none of the model test piles did. However, this is possible as the model testing piles were designed to serve prescribed research targets rather than real projects. The relaxation of this $D/t$ criterion was not expected to cause severe damage or consequences. Note that the API also specifies that the requirement for $D/t$ may be relaxed when the pile is not damaged during installation [32].

It should be noted that offshore wind turbine monopile foundation horizontal bearing capacity tests are typically conducted in nearshore areas, where factors such as currents, waves, and tides can influence the test results. Additionally, the accuracy and stability of the testing equipment are crucial to the reliability of the test outcomes. During the tests, soil deformation and stress release can cause variations in the applied loads, further affecting the accuracy of the results. Unfortunately, these influencing factors are either missing or incomplete in the source documents, making it impossible to study their impact on model evaluation and calibration. On the other hand, large-diameter monopile tests for offshore wind turbines are usually highly expensive, time-consuming, difficult, and risky; as a result, empirical data, even only a few, on the horizontal bearing capacity of these foundations should be truly appreciated.

4. Evaluation and Calibration of the API P-Y Curve Method

This section conducts a quantitative uncertainty analysis of the API p-y curve model recommended for piles in sand based on the established database. The analysis primarily involves a statistical examination of 170 field test data points and 321 model test data points to identify the main factors contributing to model error. The model accuracy is represented by a model factor ($\lambda$), defined as the ratio of the measured soil resistance to the soil resistance predicted by the model. The measured soil resistance values are sourced directly from the database, while the predicted values are calculated using the API p-y curve model, i.e., Equation (1).

4.1. Calculation of Model Factor

The uncertainty of the API p-y curve model is evaluated using the model factor ($\lambda$) defined as

$$\lambda =p_m/p_p$$

(3)

where $p_m$ = measured surrounding soil horizontal resistance; $p_p$ = predicted surrounding soil horizontal resistance using Equation (1).

The model factor is a random variable, with its mean (${\mu }_{\lambda }$) representing the average accuracy of the model and its coefficient of variation ($CO{V}_{\lambda }$) indicating the dispersion between the predicted and actual values. Figure 4 plots the measured soil resistance around the piles ($p_m$) versus the API-predicted values ($p_p$) for both the field and model tests. Lines defining ratios of $p_m/p_p$ equal to 0.1, 0.2, 1.0, and 5.0 are also shown to help interpretation. For the field case, most data points lie below the $p_m/p_p=1.0$ line, indicating that in most cases, the model predictions are greater than the measured values, suggesting a risky bias. In other words, the soil resistance is not as large as thought. The data scatter basically between $p_m/p_p=0.1$ and $p_m/p_p=5$, implying a large divergence in prediction accuracy.

On the other hand, the majority of the model test data points distribute around the $p_m/p_p=1.0$ line when $p_p$ is less than 20 kN/m. For a larger $p_p$, such as over 100 kN/m, the data points tend to scatter below the $p_m/p_p=1.0$ line. Additionally, it can be observed from the figure that there is a significant deviation between the measured soil resistance and the API predicted values, with model factor values ranging from less than 0.2 to 5. The prediction accuracy in this case is also expected to be highly dispersive.

The calculation results show that the mean and COV of the model factor for the field tests are ${\mu }_{\lambda }$ = 0.703 and $CO{V}_{\lambda }$ = 0.788, respectively. This means that on average the current API p-y curve overestimates the soil resistance by about 30%. Indeed, this is an undesirable outcome as the true soil resistance is not as high as expected, which would then lead to an under-design of the pile. Nevertheless, this risk will be compensated by the factor of safety in practice. Even then, the true factor of safety would still be lower than the nominal value. The ${\mu }_{\lambda }$ value is consistent with the observations made from Figure 4. The $CO{V}_{\lambda }$ falls within 0.6 to 0.9, suggesting a high dispersion in prediction accuracy if ranked by the four-tier scheme proposed by Phoon and Tang [63]. As a matter of fact, a COV value within this range is not uncommon for geotechnical models, see [64].

While for the model tests, the mean and COV of the model factor are ${\mu }_{\lambda }$ = 1.077 and $CO{V}_{\lambda }$ = 0.624, respectively. There are only about 8% errors on average for this case, and the accuracy dispersion is around the border between medium and high. From these statistics of model factors, the current API p-y curve method appears to be more accurate in predicting soil resistance in physical model tests than in field tests. This is understandable as pile–soil interaction conditions are more controllable and simpler in laboratory settings. Also, the API p-y curve method was basically developed based on data from pile dimensions comparable to those for physical model tests shown in Table 1. A more accurate prediction in laboratory tests is undesirable as it would mislead researchers in understanding the performance of the p-y curve method for application in field or real projects.

The differences in model factors between field and model tests are too large for them to merge into one dataset. This is confirmed by applying a Mann–Whitney test to the two datasets. The p-value is 0.000, which is less than 0.05, indicating that the two model factor datasets differ significantly at a significance level of 0.05; they cannot be considered to be from the same population. Thus, the analyses to follow have to be on each individual dataset.

Specifically, for the database compiled by the present paper, the differences in model factors can be attributed to two main aspects: firstly, the soil condition in field tests was complex and varied within a wide range of internal friction angles, whereas the soils in model tests were homogeneous and showed less variation in internal friction angles. Secondly, the size of the pile differed. The piles in field were much larger in length and diameter with wall thickness designed to meet API specifications; the model test piles, on the other hand, were relatively shorter, smaller, and thinner. As mentioned earlier, this is the size effect.

4.2. Statistical Correlations of Model Factors

The data are separated into two subsets: (1) one subset for field tests, n = 170, (2) the other for model tests, n = 321. Statistical correlation analyses are then conducted on the subset level.

Figure 5 illustrates the plots of the model factors $\lambda$ against the predicted values ($p_p$) for both the field and model test cases. In general, as the predicted soil resistance increases, the model factor visually exhibits an overall decreasing trend. Spearman rank correlation tests were applied to the datasets, showing correlation coefficients of $\rho =$ −0.24 for the field tests with a p-value of 0.002 and $\rho =$ −0.55 for the model tests with a p-value of 0.001. For both cases, the correlation $\rho$ values are negative and the p-values are less than 0.05. These results confirm a statistically significant negative correlation at the 0.05 significance level. This is an undesirable outcome as it means that the prediction accuracy would be dependent on the magnitude of the predicted value. The randomness criterion of the model factor is then not satisfied.

To examine the source of the above model dependency, the model factors were tested against the input parameters by Spearman rank correlation tests. These parameters include the internal friction angle ($\phi$), ratio of depth to pile length ($H/L$), ratio of horizontal displacement to pile length ($y/L$), ratio of horizontal displacement to pile diameter ($y/D$), pile length ($L$), pile diameter ($D$), and ratio of pile diameter to pile length ($D/L$). The corresponding plots are shown in Figure 6.

Table 2. Summary of Spearman’s rank correlation test results between model factors and input parameters.

Test Type	(Model Factor, Input Parameter)	Spearman Test
		p	$\mathit{\rho }$
Field	($\lambda ,\phi$)	0.48 > 0.05	−0.05
	($\lambda ,H/L$)	0.18 > 0.05	−0.10
	($\lambda ,y/D$)	0.03	0.17
	($\lambda ,y/L$)	0.01	0.20
	($\lambda ,L$)	0.00	−0.40
	($\lambda ,D$)	0.00	−0.27
	($\lambda ,D/L$)	0.00	0.22
Model	($\lambda ,\phi$)	0.00	−0.61
	($\lambda ,H/L$)	0.00	−0.32
	($\lambda ,y/D$)	0.00	0.17
	($\lambda ,y/L$)	0.04	0.11
	($\lambda ,L$)	0.00	0.42
	($\lambda ,D$)	0.39 > 0.05	0.05
	($\lambda ,D/L$)	0.00	−0.53

summarizes the results of the Spearman rank correlation tests between the model factor and each input parameter. The results indicate that at the 0.05 significance level, the API p-y curve model factor for field tests is statistically positively correlated to $y/L$, $y/D$, and $D/L$, but negatively correlated to $L$ and $D$. The correlation coefficients $\rho$ were about 0.20 for the positive cases and up to −0.40 for the negative cases. Nevertheless, the data points in Figure 6 scatter widely. For the model tests, the API model factor is statistically correlated negatively to $\phi$, $H/L$, and $D/L$, but positively to $y/D$, $y/L$, and $L$. The two negative correlations related to $\phi$ and $D/L$ are relatively strong, i.e., both $\rho$ values exceeded −0.50 whereas the positive correlations were all less than 0.20. These findings reveal the subtle differences between field and model tests, and the main reasons could still be the size effects and soil conditions.

4.3. Calibration of the API P-Y Model for Piles in Cohesionless Soils

The above analyses show that the API p-y curve model for predicting soil resistance around piles is not satisfactory due to the overall bias, high dispersion, and model accuracy dependency with model inputs and predictions. Therefore, there is a great need to use the database to correct the model for accuracy improvement. This is performed by introducing an empirical correction factor to the model, as

$$p_p^{\prime }=p_p\cdot \alpha$$

(4)

where $\alpha$ is the empirical corrective factor. Based on Table 2, it is straightforward to take $\alpha$ as a function of $\alpha =f(y,D,L)$ for the field test case and $\alpha =f(\varphi ,H,y,D,L)$ for the model test case. These two $\alpha$ mathematical expressions can be further simplified based on two facts: first, there are high correlations between the input parameters; second, the Spearman’s $\rho$ values are low. With these ideas in mind and the philosophy to develop simple yet practical models, five simple functions are employed to regress $\lambda$ against the input parameters so that the $\alpha$ expressions can be feasibly determined. These five simple functions include exponential $y=a{e}^{bx}$, linear $y=ax+b$, logarithmic $y=a\ln x+b$, quadratic $y=a{x}^2+b$, and power $y=a{x}^b$. Here, $y$ is $\alpha$ and $x$ represents the input parameters. To examine the regression performance of the five simple functions above, the coefficients of determination $R^2$ for the regressions were computed and are summarized in

Table 3. Summary of coefficient of determination $R^2$ using the five regression functions for $\alpha$ against several input parameters of $p_p$.

Test Type	$\mathit{\alpha }$ Versus	Coefficient of Determination, ${\mathit{R}}^2$
		Exponential $\mathit{y}=\mathit{a}{\mathit{e}}^{\mathit{b}\mathit{x}}$	Linear $\mathit{y}=\mathit{a}\mathit{x}+\mathit{b}$	Logarithmic $\mathit{y}=\mathit{a}\mathbf{ln}\mathit{x}+\mathit{b}$	Quadratic $\mathit{y}=\mathit{a}{\mathit{x}}^2+\mathit{b}\mathit{x}+\mathit{c}$	Power $\mathit{y}=\mathit{a}{\mathit{x}}^{\mathit{b}}$
Field	y/D	0.035	0.034	0.037	0.036	0.000
	y/L	0.036	0.037	0.002	0.037	0.004
	L	0.100	0.107	0.109	0.108	0.096
	D	N/A	0.000	0.004	0.100	0.000
	D/L	0.048	0.046	0.043	0.061	0.040
Model	φ	0.314	0.310	0.313	0.311	0.318
	H/L	0.123	0.119	0.095	0.122	N/A
	y/D	0.047	0.046	0.017	0.048	0.010
	y/L	0.111	0.080	0.107	0.127	0.005
	L	N/A	0.153	0.226	0.158	0.184
	D/L	0.192	0.192	0.194	0.192	0.186

Note: N/A = not applicable.

. The input parameters only include those with a statistically significant correlation with the model factor at the 0.05 level as shown in Table 2.

Table 3 shows that for field tests, quadratic, linear, and logarithmic equations capture the data trend between $\lambda$ and $L$ better than other expressions, with the highest $R^2$ value of 0.109. Nevertheless, the $R^2$ values are overall very low given Spearman’s $\rho$ reaching −0.40, indicating regression using one of the simple equations above may not yield satisfactory outcomes. For model tests, all the expressions mentioned yield higher $R^2$ values in the regression between $\lambda$ and $\phi$. The table also presents the $R^2$ values between $\lambda$ and the other input parameters, all of which are relatively small, suggesting that using these input parameters for data fitting is inefficient and insufficient.

Based on the above analyses, for the field test, the final expression for $\alpha$ can be expressed as $\alpha =f\left(L\right)$; and for model testing, the final expression can be determined as $\alpha =f(\phi ,L)$. With Equation (4), the model factor after calibration can be written as

$${\lambda }^{\prime }=p_m/{p_p}^{\prime }$$

(5)

Clearly, the calibrated model factor ${\lambda }^{\prime }$ is the ratio of the measured value $p_m$ (taken directly from the database) to the calibrated predicted value ${p_p}^{\prime }$. The optimal values of the empirical constants a, b, and c in Table 3 can be easily determined by satisfying the following criteria: (1) the mean of ${\lambda }^{\prime }$ is equal to 1, (2) the coefficient of variation (COV) of ${\lambda }^{\prime }$ is minimized. Multiple combinations of the five simple equations were tried so as to find the best regression expressions, excluding those marked as “N/A” in Table 3. The results of the model calibration are shown in

Table 4. Results of model calibration for different combinations of simple regression.

Test Type	$\mathit{\alpha }$ Versus	Expression of $\mathit{\alpha }$	Empirical Constant	Calibrated Model Factor ${\mathit{\lambda }}^{\mathbf{\prime }}$		Model Factor $\mathit{\lambda }$
				Mean	COV	Mean	COV
Field	$f\left(L\right)$	$\alpha =a_1{L}^2+b_{1}L+c_1$	$a_1$ = 1.537 $b_1$ = 6.908 $c_1$ = 40.202	1.000	0.778	0.703	0.788
		$\alpha =a_{1}L+b_1$	$a_1$ = 0.023 $b_1$ = 2.004	1.000	0.690
		$\alpha =a_1\ln x+b_1$	$a_1$ = 1.530 $b_1$ = 6.876	1.000	0.611
Laboratory	$f(\phi$)	$\alpha =a_2{\phi }^{b_2}$	$a_2$ = 368.905 $b_2$ = 1.699	1.000	0.564	1.077	0.624
		$\alpha =a_2\ln \phi +b_2$	$a_2$ = 2.067 $b_2$ = 8.238	1.000	0.557
		$\alpha =a_2{\phi }^2+b_2\phi +c_2$	$a_2$ = 0.004 $b_2$ = 0.187 $c_1$ = 0.956	1.000	0.521
		$\alpha =a_2\phi +b_2$	$a_2$ = 0.067 $b_2$ = 3.268	1.000	0.550
	$f\left(L\right)$	$\alpha =a_3\ln \left(L\right)+b_3$	$a_3$ = 0.426 $b_3$ = 0.738	1.000	0.553
	$\phi \times L$	$\alpha =a_4{\left(\phi L\right)}^{b_4}$	$a_4$ = 0.365 $b_4$ = 0.247	1.000	0.593
		$\alpha =a_4\ln \left(\phi L\right)+b_4$	$a_4$ = 0.409 $b_4$ = −0.650	1.000	0.582
		$\alpha =a_4\left(\phi L\right)+b_4$	$a_4$ = 0.002 $b_4$ = 0.857	1.000	0.603
	$f(\phi$)+$f\left(L\right)$	$\alpha =a_5{\left(\phi \right)}^{b_5}+a_6{\left(L\right)}^{b_6}$	$a_5$ = 1.326 × 103 $b_5$ = −2.275 $a_6$ = 0.348 $b_6$ = 0.450	1.000	0.516
		$\alpha =a_5{e}^{b_5\phi }+a_6{e}^{b_6\left(L\right)}$	$a_5$ = 0.679 $b_5$ = −2.083 × 107 $a_6$ = 0.850 $b_6$ = 0.068	1.000	0.594
		$\alpha =a_5\ln \phi +a_6\ln \left(L\right){+b}_5$	$a_5$ = −1.163 $a_6$ = 0.257 $b_5$ = 4.925	1.000	0.496
	$f(\phi$) interacts with $f\left(L\right)$	$\alpha =a_7{\left(\phi \right)}^{b_7}{\left(L\right)}^{b_8}$	$a_7$ = 89.404 $b_7$ = −1.344 $a_8$ = 0.220	1.000	0.516
		$\alpha =a_7{e}^{b_7\phi +b_8\left(L\right)}$	$a_7$ = 4.866 $b_7$ = −0.051 $a_8$ = 0.032	1.000	0.529

.

These results indicate that for field tests, choosing the logarithmic function as the expression for the correction factor $\alpha$ allows the corrected model factor to achieve the smallest COV of ${\lambda }^{\prime }$. The specific expression is

$${\alpha }_1=a_1\ln \left(L\right)+b_1$$

(6)

where ${\alpha }_1$ is the correction factor of the field test, $L$ is the pile length, and $a_1,b_1$ are empirical constants with values given in Table 4. For model tests, a logarithmic function was also selected as the expression for $\alpha$, which provided the smallest COV of ${\lambda }^{\prime }$. The specific expression is

$${\alpha }_2=a_5\ln \left(\phi \right)+a_6\ln \left(L\right)+b_5$$

(7)

where ${\alpha }_2$ is the correction factor of the model test, $\phi$ is the soil friction angle, $L$ is the pile length, and $a_5,b_5,a_6$ are empirical constants.

After calibration, the model factors have mean values of 1.000 for both the field test and laboratory test cases. On average, both models are unbiased now. The COVs of $\lambda$ are reduced from 0.788 to 0.611 for the field test case, and from 0.624 to 0.496 for the laboratory test case. Approximately, a great reduction of up to 13% to 20% in COV is reached. The calibrated models are obviously superior to the original p-y model. These improvements provide more accurate practical value for predicting pile–soil resistance according to the API specifications. Figure 7 shows the plots of measured values versus predicted values with correction. Compared to Figure 4, the data points in Figure 7 are much more concentrated towards the 1:1 line, as expected. The computed statistics of the calibrated model—mean = 1.00 and COV = 0.611 for the field test case, and mean = 1.00 and COV = 0.496 for the laboratory test case—serve as essential inputs for calibrating resistance factors in Reliability-Based Design frameworks, particularly load and resistance factor design (LRFD) codes. These parameters enable direct reliability targeting during code calibration, bridging the gap between empirical data and modern design practice.

5. Characterization of the Probabilistic Distribution of Model Factor

In addition to the mean and COV, the probability density distribution of the model factor is also required for the reliability-based design of geotechnical structures. The histograms and cumulative distributions of model factors for the p-y curve models for sands are presented in Figure 8.

Kolmogorov–Smirnov normality tests (K–S) were applied to the logarithm of the bias sets (i.e., ln $\lambda$). All the p-values exceeded 0.05, suggesting that the ln $\lambda$ sets can be treated as normally distributed at the significance level of 0.05. In other words, the model factors of the p-y curve methods for sands can be taken as a lognormal random variable, which is typical for geotechnical structures, e.g., Phoon and Tang [63]^, Phoon and Tang [65], Phoon [66]. In addition, using a lognormal distribution to characterize the probability distribution of model factors may not be applicable for some special scenarios and it is suggested to use high-order Gaussian distribution [67].

6. Conclusions

This paper constructed a database of p-y curves for offshore wind turbine monopile foundations installed in cohesionless soils, including 170 sets of field test data and 321 sets of model test data. The API-recommended p-y curve models were evaluated and calibrated, with the main conclusions as follows.

(1)

Based on field test data, the API-recommended p-y curve method is found to overpredict the soil resistance around monopile foundations for offshore wind turbines in cohesionless soils by about 30% on average, with a high dispersion in prediction accuracy reaching 80%. This is risky as the real soil resistance is not as large as the predicted one. While based on laboratory model tests, on the contrary the API p-y curve method underpredicts the soil resistance by about 8% overall, the dispersion in prediction accuracy is still high, i.e., over 60%. Researchers should be cautious when using laboratory-scale tests to validate or develop models intended for full-scale field applications. The prediction accuracies for both field and laboratory test cases are statistically correlated to the predicted values, mainly due to the inherent statistical correlations to the input parameters of pile length, soil friction angle, and depth.

(2)

Correction factors expressed as simple equations with two to three empirical constants are introduced to calibrate the current API p-y curve method for the enhancement of prediction accuracy. The calibrated API p-y curve method has mean values of 1.00, suggesting it is unbiased on average. The dispersions in prediction accuracy are about 20% less than that of the original method.

(3)

The model factors for both the current and calibrated API p-y curve methods can be treated as lognormal random variables.

The model uncertainties of the current p-y curve method based on field tests and laboratory model tests are shown to be significantly different. This inconsistency may be due to the size effects and controllable conditions in the laboratory tests. While the simple model calibration has effectively reduced the dispersion in prediction accuracy, it is still high. More advanced regression techniques such as machine learning can be employed for model calibration. All these aspects should be explored in future studies.