p-y curves from in-situ ROBOCONE tests: a similarity approach for laterally loaded piles in clay
Authors: Bateman Abigail H. , Mylonakis George , Creasey James , Hajjar Ahmad El , White David , Cerfontaine Benjamin , Gourvenec Susan , Diambra Andrea 
Published: October 2023

The design of offshore piles requires assessment of the lateral load-displacement response, achieved using lateral nonlinear springs (p-y curves) distributed along the pile [1]. Specifically, a p-y curve describes the horizontal displacement, y, resulting from a distributed lateral load, p­, at the pile-soil interface, at a specific depth. Numerous methods are available to obtain p-y curves, either relating them to soil properties [2, 3, 4], or using empirical relationships with cone penetration test (CPT) data [5]. An advantage of the latter method is that the p-y curves are obtained from in-situ testing, which offers undisturbed, continuous measurements and does not require additional laboratory test data. However, the CPT approach relies heavily on empirical correlations and the loading applied to the soil by the CPT is notably different to that of a laterally loaded pile. An alternative solution is being pursued within the EPSRC-SFI funded ROBOCONE project, which involves the development of a horizontal loading module that can be included in a standard cone penetrometer [6]. The aim of this proposed module is to directly obtain lateral load-displacement curves, referred to herein as a ROBOCONE PR-yR curve, by moving part of the module horizontally, perpendicular to the long axis of the cone penetrometer (Figure 1(a)), closely replicating the soil loading in the pile problem under consideration.
The soil resistance from the horizontal loading of the ROBOCONE module, , is achieved by applying a constant displacement, , over a short length of pile, , at a specific depth; while for a pile, a force-displacement relationship at the pile head arises due to the effect of individual p-y relationships applied over a much greater portion (“active length”) of the pile, which is closer to a plane-strain “horizontal soil slice” problem. Therefore, while the horizontal loading applied to the soil by the ROBOCONE module is similar to that applied by a pile under lateral load, the ROBOCONE - curve may not be directly substituted for a p-y curve for any soil-pile configuration. Instead, a correction factor in some form must be employed to transform between the two curves, without the intermediate stage of recovering soil properties from the ROBOCONE data and then evaluating the resulting p-y curve.
A simple analytical method to obtain a correction factor for piles in clay is the similarity approach, which takes the form of a linear-transformation of axes. This method was first introduced by Skempton [7] who suggested that a stress-strain curve of
a representative clay soil sample can be considered similar in shape to a load-settlement curve of a vertically loaded foundation. It has since been employed to relate a stress-strain curve from an element test with a p-y curve [1, 2] or a t-z curve (describing the vertical pile displacement as a result of a distributed axial force) [8], the former of which is still used in current design codes [3]. Additionally, the similarity approach has not only been used to obtain curves transformed from element test results, but also to obtain an m-θ curve (describing the pile rotation as a result of a distributed moment) directly from a t-z curve [9, 10].
To employ the similarity approach to this scenario, it must be assumed that a ROBOCONE PR-yR curve can be considered similar in shape to a p-y curve, at a specific depth. Following this assumption, two steps are required to obtain a p-y curve, illustrated in Figure 1. Firstly, the y-axis of each curve is normalised by a bearing capacity factor, , the soil undrained shear strength, , and their respective dimensions, which naturally bound the ordinates of the two curves between 0 and 1. Secondly, the x-axis of each curve is normalised by a characteristic length, the pile diameter , and a factored ROBOCONE diameter, , respectively; where is a linear-transformation factor intended to collapse the two curves into a single curve.
To employ this method, a value of is needed. In the original work, Skempton [7] determined this factor by matching the dimensionless initial stiffnesses of the two curves he considered; this approach is followed here with the ROBOCONE PR-yR and p-y curves (see Figure 1). Firstly, from dimensional arguments, the initial stiffness of the ROBOCONE PR-yR curve, , is a function of the soil shear modulus and Poisson’s ratio, and , and the dimensions and , multiplied by a constant . can be estimated from elastic solutions for rectangular footings or embedded or buried cylinders (e.g. [11]). Secondly, the initial stiffness of the p-y curve is the Winkler spring stiffness, , for which many solutions are available (see a recent summary [12]). The resulting expression for is shown in Figure 1(c), which can be used to directly relate a ROBOCONE PR-yR curve with a pile p-y curve in practice. By taking as double the stiffness of a rigid rectangular footing on an elastic half-space [11] and for a fixed-head slender pile [12], to was obtained. This suggests that (1) the ROBOCONE PR-yR curve should be “stretched” to obtain a p-y curve, (2) the multiplier is independent of and , (3) tends to increase with increasing (e.g. free-head over fixed-head pile [12]) and ROBOCONE slenderness . This initial result indicates the magnitude of the scaling factor for estimating a p-y curve at a specific depth. However, this method requires numerical verification and/or field or physical modelling observations to be undertaken prior to application in design practice. This will be undertaken as part of the ROBOCONE project.