Evaluation of Curve Fitting Techniques in Deriving p–y Curves for Laterally Loaded Piles
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Abstract
The p–y method is one of the most popular methods for the analysis and design of laterally loaded piles. The mathematical relationship it provides between the bending moment, which can be easily measured at strain gauges along the pile, and the soil resistance and lateral pile displacement, facilitates the construction of p–y curves. Numerical techniques are required to fit smooth continuous curves to the discrete bending moment data in order to improve the accuracy of subsequent differentiation and integration operations. Due to the lack of guidance on the optimum positioning of strain gauges and the reliability and accuracy of curve fitting methods, a unifying study, inclusive of small (0.61 m) and large (3.8 and 7.5 m) diameter piles in clay, was carried out using 18 strain gauge layouts and cubic spline, cubic to quintic Bspline and 3rd to 10th degree global polynomial techniques. Bending moment data was obtained using 3D finite element analysis. Through a comprehensive evaluation, the cubic and cubic Bspline methods were found to be consistently accurate in deriving p–y curves for both the small and large diameter piles.
Keywords
Curve fitting Global polynomials Laterally loaded piles Monopiles p–y curves Splines1 Introduction
The p–y method, which is a subgrade reaction technique that describes the nonlinear relationship between the mobilised soil resistance, p, and the lateral deflection of the pile, y, is widely used for designing laterally loaded piles. A plot of the variables \(p\) and \(y\) at a discrete soil depth constitutes a p–y curve at that depth.
Due to the relative ease of determining \(M\), coupled with the difficulty of measuring \(p\) and \(y\) experimentally, this derivation procedure has continued to be adhered to in centrifuge modelling, 1g physical modelling and fullscale field testing.
Centrifuge modelling research on laterally loaded piles
Research  Soil type  Size of pile  Number of strain gauges  Curve fitting technique  

\(D\) (m)  \(L\) (m)  
Barton (1982)  Sand  0.61  9.6  8  Cubic spline interpolation 
Springman (1989)  Clay and Sand  1.27  17  8  4th to 9th degree global polynomial approximation 
Bouafia and Garnier (1991)  Sand  0.5, 0.9  5  12  4th to 7th degree global polynomial approximation, quintic spline approximation 
King (1994)  Sand  0.76  8.4  6  4th to 7th degree global polynomial approximation, cubic and quartic spline interpolation 
Kitazume and Miyajima (1994)  Clay  0.5  10  12  7th degree global polynomial approximation 
Ellis (1997)  Clay and Sand  1.2  19  12  Piecewise polynomial approximation 
Dyson and Randolph (1998)  Sand  2.08  41.6  12  3rd degree piecewise polynomial approximation 
Mezazigh and Levacher (1998)  Sand  0.72  12  20  Quintic spline approximation 
Remaud et al. (1998)  Sand  0.72  12  20  Quintic spline approximation 
Ilyas et al. (2004)  Clay  0.84  14.7  10  7th degree global polynomial approximation 
Bouafia and Bouguerra (2006)  Sand  0.5  3.1, 5  11–12  Quintic spline approximation 
Kong and Zhang (2007)  Sand  0.63  12  8  5th degree global polynomial approximation 
Jeanjean (2009)  Clay  0.91  20.2  11  Loess function interpolation, cubic spline interpolation and 6th degree global polynomial approximation 
Klinkvort (2012)  Sand  3  18  10  6th degree global polynomial approximation 
Choo and Kim (2015)  Sand  6  31  7  ‘Modified’ 5th degree global polynomial approximation 
Lau (2015)  Clay  3.8  20  6  Cubic spline interpolation 

The number of strain gauges used along the embedded length of the pile, \(L\), varied widely between 6 and 20. Strain gauges are usually fixed to the outside of the pile as their installation on the inside is relatively complicated. To avoid being damaged during pile installation, the strain gauges are attached to recesses within the pile wall and coated with an epoxy resin. Therefore, it is necessary to have an optimum number of strain gauges along the pile that would provide sufficient bending moment data yet cause minimal alteration to the sectional and surface properties of the pile.

Amongst the various mathematical techniques available for curve fitting, global polynomial approximation, piecewise polynomial approximation, cubic spline interpolation and quintic spline approximation, were the most commonly used. Although piecewise polynomials and splines are frequently referred to interchangeably, the continuity of their derivatives distinguishes the two. A piecewise polynomial need only be once continuously differentiable whereas a spline of degree \(k\) is (\(k  1\)) continuously differentiable. To mathematically define the polynomial correctly, it is vital to distinguish between the order and degree of a polynomial. The former refers to the total number of terms in a polynomial, including the constant, whereas the latter refers to the largest exponent in a polynomial. However, in the literature reviewed, the term order was frequently used to imply the degree of a polynomial. Hence, wherever possible, the polynomial structure was verified to ensure that the interpretation of the literature review was consistent.

Of the 16 publications reviewed, 13 were related to small diameter piles thus highlighting the absence of adequate curve fitting precedents for large diameter monopiles for offshore wind turbines, whose diameter, D, ranges between 3.8 and 7.5 m. Inconsistencies in the selection of the curve fitting method and strain gauge configuration were also obvious.
King (1994) and Yang and Liang (2006) compared the accuracy of curve fitting methods for laterally loaded piles. Through the analysis of centrifuge test results of a small diameter pile, King (1994) concluded that double integration of the bending moment curves was relatively accurate irrespective of whether global polynomials or splines were used. However, the double differentiation operation was found to be errorprone with the 7th degree polynomial producing the smoothest soil resistance profile. On the other hand, based on results of fullscale tests on small diameter piles and hypothetical numerical simulations, Yang and Liang (2006) deduced that approximation using the cubic piecewise polynomial was more accurate relative to the ‘modified’ 5th degree global polynomial (an exponent of 2.5, instead of 2, was used for the quadratic term), weighted residuals and smoothed weighted residuals. However, the omission of splines diluted the findings of this study.
This indicates the need for detailed guidance on the application of curve fitting techniques in the derivation of p–y curves for the general spectrum of laterally loaded piles, including small and large diameter piles, to obviate uncertainties with respect to reliability and accuracy.
2 Methodology
 Analysis, using Model 1, shown in Fig. 1a, of a 0.61 m diameter steel pile embedded 35 m in homogenous lightly overconsolidated stiff clay with \(s_{u}\) of 100 kPa. A lateral load, \(H\), of 0.06 MN and overturning moment, \(M_{\text{a}}\), of 2.32 MNm were applied to the pile in 100 equally spaced increments. The total stress approach was used with the clay assumed to be undrained and simulated by the Tresca constitutive model.
 Simulation, using Model 2, illustrated in Fig. 1b, of centrifuge test CT (Lau 2015), involving a 3.8 m diameter aluminium monopile embedded 20 m in heavily overconsolidated Speswhite Kaolin clay with \(s_{u}\) ranging from 3 kPa at mudline to 33.25 kPa at the pile tip. A lateral displacement, \(y_{\text{a}}\), of 6 m was applied to the pile head in 60 equally spaced increments. The poroelastoplastic approach was used with the clay assumed to be draining and represented by the Modified Drucker–Prager constitutive model. As shown in Fig. 2, good agreement was obtained between the centrifuge test and FEA results (Haiderali 2015).

Analysis, using Model 3, illustrated in Fig. 1c, of a 7.5 m diameter steel monopile embedded 30 m in soft normally consolidated clay with \(s_{u}\) ranging from 5 kPa at mudline to 67.6 kPa at the pile tip. \(H\) and \(M_{\text{a}}\) of 7.2 MN and 259.2 MNm respectively were applied to the pile in 100 equally spaced increments. The total stress approach was utilised with the clay assumed to be undrained and modelled by the Tresca yield criterion.
Although pile driving would lead to soil disturbance, it was assumed that this would be limited to a relatively thin region of soil around the pile in comparison to the much thicker zone of soil that would be subject to an increase in stress due to lateral loading. Therefore, pile installation was not modelled in these analyses.
Models with varying clay properties, element types, loading conditions and soil constitutive models were selected to ensure unbiased findings. Quadratic elements were not used as they brought about a negligible increase in solution accuracy that was offset by greater computational cost (Haiderali 2015). A fine mesh of between 27,072 and 255,936 elements, verified to be sufficiently accurate via a mesh sensitivity study (Haiderali 2015), was used. Finite sliding surfacetosurface contact pair formulation was used to accurately model the pilesoil interaction necessary for the computation of the soil resistance. In the normal direction, no contact stress was transmitted unless the pile nodes came into contact with the soil surface whilst in the tangential direction, the contact shear stress was related to the normal stress by the interface friction coefficient, \(\mu\) (Coulomb friction law).

To ensure that the sectional properties of the piles used in experimental research would not be adversely affected, the maximum number of idealised strain gauges was limited to 18, 11 and 16 for the 0.61, 3.8 and 7.5 m diameter piles respectively, in proportion to their corresponding embedded lengths.

Strain gauges were positioned to be roughly equidistant along the pile for the 3.8 and 7.5 m diameter piles, and for layouts 6, 9 and 12SG for the 0.61 m diameter pile (Fig. 3a, c, e respectively). Since the deformation of the 0.61 m diameter pile was found by Haiderali and Madabhushi (2012) to be confined to its upper portion, layouts 11, 15 and 18SG, shown in Fig. 3d, g, h, were designed to have a higher concentration of strain gauges within this segment of the pile.

Layout 6SG for the 3.8 m diameter monopile reproduced, at prototype scale, the strain gauge layout used for the centrifuge model monopile, shown in Fig. 4a, to enable the impact of curve fitting errors on p–y curves derived using centrifuge test results to be assessed.

Based on the conventional assumption that the bending moment at the pile tip, \(M_{\text{tip}}\), is zero, a dummy data point prescribed with this condition at the pile tip was included in all the strain gauge layouts for the 0.61 m diameter pile, the 6 and 10SG layouts for the 3.8 m diameter pile (Fig. 4b, d respectively), and the 6, 10 and 15SG layouts for the 7.5 m diameter pile (Fig. 5a, c, e respectively). However, it was found by Haiderali (2015) that large diameter piles have nonzero \(M_{\text{tip}}\) due to the restraint provided by the soil. The effect of the zero \(M_{\text{tip}}\) assumption on curvefitting accuracy for the 3.8 and 7.5 m diameter piles was therefore assessed by not employing a dummy point with this condition at the pile tip but by instead including an additional strain gauge at the pile tip in the 7 and 11SG layouts for the 3.8 m diameter pile (Fig. 4c, e respectively) and the 7, 11 and 16SG layouts for the 7.5 m diameter pile (Fig. 5b, d, f respectively). It is assumed that strain gauges at the pile tip can be calibrated experimentally.

Considering the bending moment in the lower half of the 0.61 m diameter pile tends to be negligible (Haiderali and Madabhushi 2012), layouts 7SG and 12BSG, illustrated in Fig. 3b, f respectively, were each designed to have 4 dummy points, between a depth of 20 and 35 m, at which the bending moment was specified to be zero.
Scripts were developed using MATLAB (2014) to automate curve fitting and numerical differentiation and integration operations at 100 load increments for the 0.61 and 0.75 m diameter piles, and 60 for the 3.8 m diameter pile. It is to be noted that the effect of strain gauge measurement errors on the accuracy of p–y curves is not covered in this study.
3 Curve Fitting Methods
In the context of this paper, curve fitting methods refer to both approximation and interpolation techniques. They were implemented using readily available software and are reproducible.
Global polynomials analysed
Pile diameter (m)  Strain gauge layout  Zero bending moment points  Number of data points  Polynomial degree 

0.61  6SG  1  7  3rd–5th, 6th^{a} 
7SG  4  11  3rd–9th, 10th^{a}  
9SG  1  10  3rd–9th^{a}  
11SG  1  12  3rd–10th  
12SG  1  13  3rd–10th  
12BSG  4  16  3rd–10th  
15SG  1  16  3rd–10th  
18SG  1  19  3rd–10th  
3.8  6SG  1  7  3rd–5th, 6th^{a} 
7SG  –  7  3rd–5th, 6th^{a}  
10SG  1  11  3rd–9th, 10th^{a}  
11SG  –  11  3rd–9th, 10th^{a}  
7.5  6SG  1  7  3rd–5th, 6th^{a} 
7SG  –  7  3rd–5th, 6th^{a}  
10SG  1  11  3rd–9th, 10th^{a}  
11SG  –  11  3rd–9th, 10th^{a}  
15SG  1  16  3rd–10th  
16SG  –  16  3rd–10th 
Cubic splines eliminate oscillations associated with Runge’s phenomenon and being of a lower degree lead to a reduction in roundoff errors as well.
Interpolation with cubic, quartic (4th degree) and quintic (5th degree) Bsplines was undertaken using the spapi function. A nondecreasing uniformlyspaced knot vector was specified using the aptknt (acceptable knot) algorithm. These Bsplines, together with the cubic spline, were analysed at all the strain gauge layouts considered in this study.

specified to be zero at the pile depth at which the soil resistance was interpolated to be zero, and

at mudline, was equated to the FEAderived value, since it could be measured during an experiment.
4 Curve Fitting Accuracy
Slight differences in the depth of the pivot point computed using FEA and derived via curve fitting exacerbated NRMSE at those locations. Although these errors could have been classified as ‘outliers’ and discarded from the sample, they were retained to ensure the error analysis was unbiased.
5 Digitisation of CurveFitted Bending Moment Curves
Parametric study undertaken to optimise the number of digital points
Number of digital points  NRMSE (%)  

\(M\)  \(p\)  \(y\)  
601  0.11  0.68  5.18 
1201  0.11  0.65  5.18 
4801  0.11  0.63  5.17 
9601  0.11  0.62  5.17 
14,401  0.11  0.62  5.17 
6 Results and Discussion
6.1 Small Diameter Pile

Across the methods, there was significant variance within the upper 6–7 m of the pile, especially between 0–2 and 5–7 m, at which sharp changes in the slope of the bending moment curve occurred.
 Global polynomials, at all the strain gauge layouts, exhibited considerable oscillation in comparison to the cubic spline and Bsplines. This is illustrated in Fig. 6a for 6SG 3rd degree and 11SG 6th degree polynomials. The only exceptions were the curvefitted profiles for 12SG 8th to 10th degree polynomials, in which the fluctuation was not as significant. However, as the degree of the polynomial increased, incidence of Runge’s phenomenon, as shown in Fig. 6b for the 18SG 10th degree polynomial, was manifested through large oscillations at the edge of the profiles.
 The accuracy of cubic spline and Bspline methods at 6SG, illustrated in Fig. 7a, was relatively low with NRMSE of 5–6 % in its fitted profiles in comparison to the other strain gauge layouts in which NRMSE did not exceed 1.9 %. It can therefore be inferred that the use of six strain gauges is not sufficient to accurately interpolate the bending moment profile of a small diameter pile of this embedded length.

The use of Bsplines did not lead to the expected improvement in accuracy relative to cubic splines. On the contrary, as the degree of the Bspline increased, the accuracy became slightly diminished. As shown in Fig. 7b, oscillatory behaviour albeit slight was detected in the lower portion of the profile curvefitted with the 12BSG quintic Bspline.
 The most accurate profiles, with NRMSE less than 0.04 %, were obtained using cubic splines, cubic Bsplines and quartic Bsplines with the 12BSG, 15SG and 18SG strain gauge layouts. Illustrated in Fig. 8 for the 12BSG cubic spline and 18SG quartic Bspline, the entire curve was accurately represented by these methods. Furthermore, the increase in accuracy between 12BSG and 18SG was trivial indicating the superiority of 12BSG in which fewer strain gauges, concentrated along the upper part of the pile, were used in conjunction with multiple dummy points along the lower portion of the pile at which the bending moment was specified to be zero.
Soil resistance profiles derived using splines were of considerably higher accuracy, with NRMSE of 4.717.6 %. Cubic splines and cubic Bsplines at 12B, 15 and 18SG were the most accurate whereas all the splines at 6SG were the least accurate. However, as shown in Fig. 11c, d, even the most accurate methods failed to capture the local maxima close to the soil surface. Finally, higher degree Bsplines, especially quintic Bsplines, were found liable to inaccuracies.
6.2 Large Diameter Piles
 Illustrated in Fig. 13a, b, the most accurate methods for the 3.8 m diameter pile were the 11SG cubic spline and cubic Bspline with NRMSE of 0.04 % whereas those for the 7.5 m diameter pile were the 16SG quintic Bspline and 10th degree polynomial with NRMSE of 0.02 %.
 For both piles, the 3rd and 4th degree polynomials were the least accurate with NRMSE of between 1.1 and 3.4 % (Fig. 14a).

As shown in Fig. 14b, the inaccuracy arising from the assumption of zero \({\text{M}}_{\text{tip}}\) in the 7.5 m diameter pile was not limited to the pile tip but extended up to a depth of 28 m.

The variance between profiles derived using these methods and FEA was mainly in the upper 2–3 m and the lower 35 m of the pile. With most of the methods, a distinction could not be made along the rest of the pile.
 The most accurate methods for the 3.8 m diameter pile were the 7SG cubic and cubic Bsplines with NRMSE of 3.8 and 4 % respectively. The accuracy of profiles derived using the 6SG cubic and cubic Bsplines was also similar indicating the effect of assuming zero bending moment at the pile tip to be insignificant (Fig. 16a). This was not entirely unexpected considering the magnitude of \(M_{\text{tip}}\) was quite small for this monopile.
Interestingly, there was no gain in accuracy by increasing the number of strain gauges to 10/11.
 For the 7.5 m diameter pile, illustrated in Fig. 17, the 16SG 10th degree polynomial and the 3rd degree polynomial at all three strain gauge layouts (7, 11 and 16SG) were the most and least accurate respectively with corresponding NRMSE of 3.1 and 8.1 %. However, the overall margin of error was quite low with the rest of the methods having NRMSE of between 3.2 and 4.7 %.

The sharp increase in soil resistance at the tip of the 7.5 m diameter pile was not accurately depicted by any of the methods used. To confirm that this was not due to an overestimation in FEA, the analysis was repeated using the augmented Lagrangian contact enforcement algorithm, which is considered to be more accurate than the penalty stiffness method used in the current analyses. However, as shown in Fig. 18b, the soil resistance curves derived from these analyses were identical. Therefore, it is concluded that the inability to predict this phenomenon is not due to deficiencies in the curve fitting methods but rather exposes the underlying limitations of the p–y method in modelling extremely large diameter monopiles.
Ranking of most accurate methods for the 7.5 m diameter monopile
Rank  Method  Strain gauge layout  Cumulative NRMSE (%) 

1  10th degree polynomial  16SG  3.4 
2  Quintic Bspline  16SG  3.5 
3  9th degree polynomial  16SG  3.6 
4  Quartic Bspline  16SG  3.9 
4  8th degree polynomial  16SG  3.9 
6  10th degree polynomial  11SG  4.0 
7  9th degree polynomial  11SG  4.1 
7  8th degree polynomial  11SG  4.1 
9  Cubic Bspline  16SG  4.3 
9  Cubic Bspline  16SG  4.3 
9  Quintic Bspline  11SG  4.3 
10  7th degree polynomial  16SG  4.4 
Although the optimum strain gauge layout would vary depending on pile characteristics, on the basis of this study, the cubic spline and cubic Bspline methods were found to be consistently accurate for the full range of laterally loaded piles. This is particularly important for piles that are difficult to categorise prior to an experiment, for instance, hybrid monopiles, piles in novel geomaterials, etc.
7 Conclusions

Fitting bending moment data for the small diameter pile was more challenging due to sharp changes in the slope of the bending moment profile. As a result, global polynomials generally performed poorly. In contrast, due to the bending moment curve of the large diameter piles being fairly smooth, all the methods considered, including the high degree polynomials, were relatively accurate.

For both pile categories, the numerical integration procedure was found to be insensitive to small errors in the fitted bending moment curve. On the other hand, the numerical differentiation procedure was errorprone resulting in the amplification of slight errors in the bending moment profile. Hence, soil resistance profiles for the small diameter pile had significantly larger errors that those for the large diameter piles.

Irrespective of the method or strain gauge layout used, the sharp increase in soil resistance at the tip of the 7.5 m diameter pile could not be derived. This was attributed to the failure of the p–y method in modelling the development of shear at the pile tip and the consequent increase in soil resistance.

For monopiles greater than 3.8 m in diameter, it is advisable to include a strain gauge at the pile tip as inaccuracies were noticeable when the bending moment there was assumed to be zero. However, for the 3.8 m diameter pile, this assumption had no accuracy implications.

For the small diameter pile, a strain gauge layout with a higher concentration of strain gauges within the upper third of the pile coupled with multiple dummy points with zero bending moment along the rest of the pile was found to be optimum. Such a strain gauge layout would also require fewer strain gauges and therefore minimise adverse effects on the sectional and surface properties of the pile.

An increase in the number of strain gauges for the large diameter piles did not lead to a significant increase in accuracy.

Across the full range of laterally loaded piles, cubic and cubic Bsplines were found to be most consistent.
Notes
Acknowledgments
The first author is grateful to University of Cambridge and the Engineering and Physical Sciences Research Council for the doctoral scholarship.
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