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Small-strain, non-linear elastic Winkler model for uniaxial

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loading of suction caisson foundations

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Stephen K. Suryasentana1

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Harvey J. Burd2

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Byron W. Byrne3

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Avi Shonberg4

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Affiliations

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1 Department of Civil and Environmental Engineering, University of Strathclyde,

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Glasgow, UK;

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stephen.suryasentana@strath.ac.uk (Orcid: 0000-0001-5460-5089)

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2 Department of Engineering Science, University of Oxford, Oxford, UK;

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harvey.burd@eng.ox.ac.uk (Orcid: 0000-0002-8328-0786)

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3 Department of Engineering Science, University of Oxford, Oxford, UK;

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byron.byrne@eng.ox.ac.uk (Orcid: 0000-0002-9704-0767)

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4 Ørsted Wind Power, London, UK

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avish@orsted.com

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Corresponding author information

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Stephen K. Suryasentana

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stephen.suryasentana@strath.ac.uk

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Number of words in the main text (excluding abstract and references)

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2519

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Number of figures

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10

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Number of tables

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4

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Date

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1 May 2023

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2

Abstract

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Soils exhibit non-linear stress-strain behaviour, even at relatively low strain levels.

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Existing Winkler models for suction caisson foundations cannot capture this small-strain,

39

non-linear soil behaviour. To address this issue, this paper describes a new non-linear

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elastic Winkler model for the uniaxial loading of suction caissons. The soil reaction curves

41

employed in the model are formulated as scaled versions of the soil response as

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observed in standard laboratory tests (e.g. triaxial or simple shear tests). The scaling

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relationships needed to map the observed soil element behaviour onto the soil reaction

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curves employed in the Winkler model are determined from an extensive numerical study

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employing 3D finite element analysis. Key features of the proposed Winkler model

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include: computational efficiency, wide applicability (it can be used for caisson design in

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clay, silt or sand) and design convenience (the required soil reaction curves can be

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determined straightforwardly from standard laboratory test results). The proposed model

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is suitable for small and intermediate caisson displacements (corresponding to fatigue

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and serviceability limit state conditions) but it is not applicable to ultimate limit state

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analyses.

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Keywords

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Footings/foundations, soil/structure interaction, offshore engineering, numerical modelling

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59

3

Notation

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depth below ground level

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suction caisson diameter

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suction caisson embedded skirt length

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suction caisson skirt thickness

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horizontal displacement of foundation along x-axis

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horizontal displacement of foundation along y-axis

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vertical displacement of foundation

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rotation of foundation about x-axis

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rotation of foundation about y-axis

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torsional rotation of foundation

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secant shear stiffness of soil

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small-strain shear modulus of soil

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parameter governing the shape of the small-strain shear modulus profile

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shear strain of soil

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reference shear strain of soil

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shape parameter governing the shape of the stiffness degradation curve for

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deviatoric stress of a soil element

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deviatoric strain of a soil element

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reference deviatoric strain of a soil element

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equivalent deviatoric strain due to normalised displacement of a soil reaction

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lateral soil reaction along x-axis

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lateral soil reaction along y-axis

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axial soil reaction

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moment soil reaction about x-axis

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moment soil reaction about y-axis

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torsion soil reaction

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lateral displacement of the soil reaction along x-axis

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lateral displacement of the soil reaction along y-axis

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axial displacement of the soil reaction

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rotational displacement of the soil reaction about x-axis

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rotational displacement of the soil reaction about y-axis

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torsional displacement of the soil reaction

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secant stiffness of soil reaction

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initial linear elastic stiffness of soil reaction

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4

axial secant stiffness of the soil reaction

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lateral secant stiffness of the soil reaction

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rotational secant stiffness of the soil reaction

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torsional secant stiffness of the soil reaction

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displacement scaling factor for the vertical soil reaction stiffness

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displacement scaling factor for the horizontal soil reaction stiffness

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displacement scaling factor for the moment soil reaction stiffness

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displacement scaling factor for the torsional soil reaction stiffness

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103

5

Introduction

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Suction caisson foundations are increasingly used to support offshore wind turbines in relatively

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deep waters, where they may be more economical than monopile foundations. To facilitate the

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efficient design of suction caissons for large-scale wind farm projects, design tools are needed

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that are both rapid and reliable. Winkler-based design models have recently been proposed for

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suction caissons (e.g. Suryasentana et al. 2022; Antoniou et al. 2022). Compared to detailed

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computational simulations such as three-dimensional (3D) finite element analysis (FEA),

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Winkler models offer advantages such as computational efficiency and ease of use.

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OxCaisson (Suryasentana et al. 2022) is a Winkler model that provides accurate predictions of

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the stiffness of suction caissons in linear elastic soil. However, its applicability is limited to low

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load levels for which the soil response can be approximated as being linear elastic. To achieve

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realistic predictions at higher load levels, the non-linear behaviour of soil must be considered.

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However, existing non-linear Winkler models for caissons (e.g. Suryasentana et al. 2018;

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Antoniou et al. 2022) do not consider the small-strain, non-linear behaviour of soil. The current

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paper describes a new small-strain, non-linear elastic Winkler model to predict the caisson

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response for uniaxial loading for small and intermediate soil strain levels, relevant to fatigue and

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serviceability limit state conditions. A key feature of the proposed Winkler model is that the soil

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reaction curves employed in the analysis are represented by scaled versions of the soil

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response as observed in standard laboratory soil element tests.

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Small-strain soil non-linearity

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Soils exhibit non-linear stress-strain behaviour, even at small strains (Atkinson 2000).

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Numerous studies have explored the non-linear variation of soil secant shear stiffness with

125

strain. For example, Hardin and Drnevich (1972) proposed a one-parameter () model to

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describe the reduction of with increasing shear strain ,

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(1)

where is the small-strain shear modulus and is a reference shear strain.

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Darendeli (2001) proposed a two-parameter (, ) variant of Eq. 1 to better match laboratory

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test data,

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(2)

6

Darendeli (2001) determined best-fit values for ranging from 0.8 to 0.9 for sand and 0.9 to 1.0

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for clay and silt. Vardanega and Bolton (2013) suggested values for ranging from 0.5 to 1.39

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for clays, and 0.73 to 0.94 for silts.

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Scaling relationships

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Previous FEA studies on laterally loaded pile slices (e.g. Bransby 1999; Zhang and Andersen

135

2017) suggest that the - soil reaction behaviour can be approximately represented as a

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scaled version of the stress-strain behaviour of a soil element as measured in laboratory tests.

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This concept allows for the generation of site-specific - curves by scaling measured data from

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laboratory tests. This scaling concept is employed in the current work.

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Objectives of the paper

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The paper has three principal objectives: (i) determine scaling relationships between soil

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element stress-strain behaviour and soil reaction curves; (ii) develop a small-strain, non-linear

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elastic Winkler model in which the soil reaction curves are formulated using these scaling

143

relationships; (iii) demonstrate that the proposed Winkler model provides an accurate

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representation of the performance of a caisson for uniaxial loading as determined by 3D FEA.

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Finite Element Calibration Analyses

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To determine the scaling relationships, 3D FEA were conducted for a rigid caisson embedded in

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a small-strain, non-linear elastic soil. Soil reaction curves extracted from the analysis are

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compared with the stress-strain behaviour of the soil (as implemented in the finite element

150

model) to determine the required scaling factors. These analyses, which were conducted using

151

Abaqus v6.13 (Dassault Systemes 2014), employed a UMAT implementation of a non-linear

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elastic soil constitutive model in which the secant shear stiffness is defined as,

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(3)

The deviatoric strain is defined in the Appendix and is a model parameter. Note that

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Eq. (3) is a generalised form of Eq. 2. The secant bulk modulus, , is determined via

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(4)

where is a (constant) Poisson’s ratio. The soil is weightless. The depth variation of adopted

156

in the analyses is,

157

7

(5)

where is a reference stiffness, is a parameter (values were considered), is

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the distance below ground level, and

is the caisson diameter. Four values of Poisson’s ratios

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(), three values (0.0001, 0.0005, 0.002) and three values (0.4, 0.7,

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1) were considered. The cases considered in this study are summarised in Table 2, which

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encompasses a parameter range extensive enough to account for the majority of practical

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applications. For example, the parameters in Table 2 include the approximate lower and

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upper bound of the values observed by Vardanega and Bolton (2013) based on an extensive

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database of soil laboratory tests.

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The 3D finite element model was based on the procedures described in Suryasentana et al.

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(2022). Five embedded skirt lengths () were analysed. The caisson was

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modelled as being rigid via the use of rigid body constraints. A cylindrical mesh was employed –

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as illustrated in Suryasentana et al. (2022) - with the diameter and depth both equal to .

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Displacements were fixed at the bottom of the mesh domain and in the radial directions on the

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vertical mesh boundaries. First-order, linear, brick elements C3D8 (C3D8H for ) were

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employed for the soil and caisson. Tie constraints were used at the soil-caisson interface to

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prevent contact breaking. Uniaxial displacements were applied at the caisson loading reference

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point (LRP, see Fig. 1a) as indicated in Table 1.

174

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The skirt soil reactions (i.e. the distributed force/moment applied by the soil to the caisson skirt)

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and the base soil reactions (i.e. the concentrated force/moment applied by the soil to the

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caisson base) are extracted from the FEA results following the procedures in Suryasentana et

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al. (2022). There are six types of skirt and base soil reactions: , representing

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lateral soil reaction along the x direction, lateral soil reaction along the y direction, axial soil

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reaction, moment soil reaction about the x axis, moment soil reaction about the y axis and

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torsional soil reaction, respectively.

The corresponding displacements are ,

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representing lateral displacement of the soil reaction along the x direction, lateral displacement

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of the soil reaction along the y direction, axial displacement of the soil reaction, rotational

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displacement of the soil reaction about the x axis, rotational displacement of the soil reaction

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about the y axis and torsional displacement of the soil reaction, respectively. Fig. 1b shows a

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schematic diagram of the skirt and base soil reactions acting on a one-dimensional Winkler

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model representation of the suction caisson structure. Based on the 3D FEA analysis in

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Suryasentana et al. (2022), it was observed that the caisson lid has insignificant effects on the

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distribution of the soil reactions on the skirt. Thus, the lid is not modelled in the Winkler model,

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consistent with other researchers (e.g., Antoniou et al. 2022).

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192

8

Results

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The computed load-displacement behaviour of the caisson for an example case (

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and ) for each of the four uniaxial displacement cases

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specified in Table 1 is shown in Fig. 2. Fig. 3 shows the corresponding soil reactions extracted

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from the FEA. As the skirt soil reactions are distributed soil reactions, the skirt soil reaction

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curve in Fig. 3 should be integrated along the caisson skirt length and then added to the base

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soil reaction curve in Fig. 3 in order to obtain the curve in Fig. 2. Fig. 4 presents the normalised

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secant stiffness of the soil reactions, where and represent the secant stiffness of

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the lateral, moment, axial and torsional soil reactions, respectively. It is evident that the

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normalised secant stiffnesses in Fig. 4 decrease with increasing displacement in a manner

202

similar to the reduction of with increasing strain.

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Scaling relationships

204

Fig. 4 suggests that the normalised displacements of the soil reaction curves can be scaled to

205

match the curves (Eq. 3). Suitable scaling factors () were determined using

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least squares regression, with and representing the scaling factors for , ,

207

(and ), (and ), respectively. Different scaling factors were identified for the skirt and base

208

soil reactions. The scaled normalised displacement is referred to as the ‘equivalent deviatoric

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strain’ , which is defined as the equivalent amount of deviatoric strain induced by the

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normalised displacement. The four definitions corresponding to each displacement mode

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are,

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(6)

(7)

(8)

(9)

Fig. 5 illustrates the excellent agreement between the normalised secant stiffness of the soil

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reactions and after applying the determined scaling factors.

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Figs. 6 to 9 show the best fit values of the scaling factors for ; similar trends were

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observed for and 1. Figs. 6 and 7 show the variations of the scaling factors for the skirt

216

soil reactions (

) and base soil reactions (

) with

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respect to and , for . For both the skirt and base scaling factors, decrease

218

9

with , while tends to increase with . Figs. 8 and 9 show the variations of the scaling

219

factors with respect to and , for . For both the skirt and base scaling factors,

220

increase with .

221

The scaling factors were found to depend only on and , with having negligible

222

influence. Approximating functions were derived to represent these scaling factors, based on

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the following functional forms:

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(10)

(11)

Eqs. 10 and 11 are used to estimate the scaling factors () to transform the

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normalised displacements into the ‘equivalent deviatoric strain’, as per Eqs. 6 to 9. Eq. 10 is

226

used for and Eq. 11 is used for

and

. Tables 3

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and 4 provide the best-fit parameters for Eqs. 10 and 11 obtained through least squares

228

regression. The scaling factors determined by Eqs. 10 and 11 using these parameters are

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shown as dashed lines in Figs. 6 to 9, for comparison with the FEA values. Good agreement

230

between the approximating functions and the FEA calculated scaling factors is clearly observed.

231

Soil reaction curves

232

On the basis of the scaling relationships described above, Eq. 3 can be used to represent the

233

soil reaction functions with a change of variables (

and ),

234

(12)

where is the secant stiffness of the soil reaction curves, and is the corresponding initial

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stiffness (Suryasentana et al. 2022; Suryasentana and Mayne 2022). Using Eqs. 6 to 9, the

236

following equations describe how the secant stiffness of the soil reaction curves for each loading

237

mode change with displacements,

238

(13)

10

(14)

(15)

(16)

where Eqs. 13 to 16 apply to both the skirt and base soil reactions. Values of

,

,

239

and

are listed in Table 3 in Suryasentana et al. (2022), and

,

,

and

240

are listed in Table 5 in Suryasentana et al. (2022).

241

Proposed small-strain, non-linear elastic Winkler model

242

The proposed small-strain, non-linear elastic Winkler model employs the soil reaction curves

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described in Equations 13 to 16, based on the soil-structure interaction framework described in

244

Suryasentana et al. (2022). It was observed that the -coupling soil reaction secant stiffness

245

do not follow the same ‘S-shaped’ stiffness degradation curve as per Fig. 4 and decreases

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rapidly at increasing strain. As no scaling relationship was identified for and the artificial

247

rotational stiffness

defined in Suryasentana et al. (2022), a constant displacement

248

scaling factor was defined for both and

for simplicity. These components

249

therefore contribute to the initial caisson stiffness but degrade rapidly and have negligible

250

contributions at increasing displacements. Note that there is still coupling between the global

251

lateral load and moment for the caisson, even if there are no contributions from and

.

252

The proposed Winkler model’s calculations are shown in Fig. 2, which shows good agreement

253

with the corresponding 3D FEA results.

254

Equations 13 to 16 are calibrated for uniaxial displacements, whereas practical design cases

255

are more likely to involve uniaxial loads. Uniaxial displacement refers to the application of

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displacement in a single direction while keeping displacements in all other directions at zero. On

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the other hand, uniaxial loading refers to applying a load solely in one direction while

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maintaining loads in all other directions at zero (with the caisson remaining free to displace in all

259

directions). To explore the performance of the Winkler model for uniaxial loads (rather than

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displacements), additional 3D FEA were carried out for uniaxial applications of vertical,

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horizontal, moment and torsional loading at the caisson LRP caisson. Fig. 10 compares the

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resultant 3D FEA calculations with the corresponding Winkler model’s predictions for the

263

11

example case where and (i.e., the same

264

conditions as Fig. 2); the two sets of results agree well.

265

Discussion

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The proposed Winkler model is suitable for predicting the performance of caisson foundations at

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intermediate displacements to support the assessment of fatigue and serviceability limit state

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design conditions. Site-specific calibrations are achieved by fitting the shear stiffness

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degradation model in Eq. 2 to the available laboratory test data to determine appropriate value

270

of and . The value of is then transformed into (see Appendix). Site specific data

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on small strain shear modulus and Poisson’s ratio are also required.

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Compared to 3D FEA, the proposed Winkler model requires significantly less computational

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time (e.g. 5 hours for 3D FEA versus 1 minute for the Winkler model in producing the

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calculations for Fig. 2). Adopting the Winkler modelling framework offers some potential

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advantages, such as the ability to use deformable elements (e.g. Timoshenko beam elements)

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to represent caissons with flexible skirt. Previous studies (Suryasentana et al. 2022) have

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shown that the soil reactions are largely decoupled from the caisson structural properties and

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the same soil reactions can be used to estimate the stiffness of rigid and flexible caissons. For

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users who prefer to use the caisson stiffness results directly, the caisson stiffness results from

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the FEA study in this paper are also available at

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https://github.com/autogeolab/caisson_smallstrain_fea. Caution is advised against extrapolating

282

beyond the parameter ranges in Table 2.

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The proposed Winkler model is developed for uniaxial loading, whereas typical design cases

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would likely involve the application of combined loads. The non-linear basis of the Winkler

285

model means that linear superposition does not strictly apply and so the model cannot be

286

guaranteed for combined loads. However, preliminary work has indicated that – when combined

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loading cases typical of offshore wind applications (e.g., lateral-moment loading at some

288

constant ratio) are applied to the proposed Winkler model – a good agreement with

289

independent 3D FEA is typically achieved. It is uncertain if the Winkler model can be directly

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applied to soil stiffness depth profiles that differ from those investigated in this paper, and if

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approaches used for estimating caisson stiffness in layered soils (Suryasentana et al. 2023)

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would be directly applicable to this model. Further work is needed to explore the applicability of

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the model to general combined load cases, different soil profiles and identify any limitations with

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the model. The proposed Winkler model is not suitable for the analysis of failure states. For

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ultimate limit state assessments, alternative approaches employing elastoplastic Winkler models

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(e.g. Suryasentana et al. 2018; Antoniou et al. 2022), macro-element models (e.g. Skau et al.

297

2018) or failure envelope models (e.g., Vulpe 2015; Suryasentana et al. 2020) would be

298

appropriate.

299

12

Previous studies (e.g., Bransby 1999; Zhang and Anderson 2017) have indicated that the lateral

300

pile soil reaction behaviour are scaled versions of soil element behaviour using their respective

301

elastoplastic soil constitutive models. This suggests the possibility of finding similar scaling

302

relationships for caisson soil reactions based on more advanced soil constitutive models.

303

However, advanced soil constitutive models often involve a large number of parameters, making

304

it more practical to develop non-scaling based formulations for the soil reactions based on a

305

smaller set of parameters, as demonstrated in the case of PISA soil reactions (Byrne et al.

306

2020).

307

Acknowledgments

308

Parts of the work described here were conducted during the DPhil studies of the first author at

309

the University of Oxford. The first author would like to acknowledge the generous support of

310

Ørsted Wind Power for funding his DPhil studentship at the University of Oxford. Byrne is

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supported by the Royal Academy of Engineering under the Research Chairs and Senior

312

Research Fellowships scheme.

313

314

315

13

Appendix: Calculation of from triaxial and simple shear test results

316

The deviatoric strain is defined,

317

A1

where

and are displacements and coordinate directions respectively.

318

For simple shear tests, Eq. A1 simplifies to:

319

A2

where is the shear strain. For triaxial tests, Eq. A1 simplifies to:

320

A3

where and are the axial and radial strain, respectively.

321

14

References

322

Antoniou, M., Kourkoulis, R., Gelagoti, F., & Anastasopoulos, I. (2022). Simplified method for

323

performance-based seismic design of suction caissons supporting jacket offshore wind

324

turbines. Soil Dynamics and Earthquake Engineering, 155, 107169.

325

Atkinson, J. H. (2000). Non-linear soil stiffness in routine design. Géotechnique 50(5), 487–508.

326

Bransby, M. F. (1999). Selection of p-y curves for the design of single laterally loaded piles.

327

International Journal for Numerical and Analytical Methods in Geomechanics 23(15), 1909–

328

1926.

329

Byrne, B.W., Houlsby, G.T., Burd, H.J., Gavin, K.G., Igoe, D.J., Jardine, R.J., Martin, C.M.,

330

McAdam, R.A., Potts, D.M., Taborda, D.M. and Zdravković, L., (2020). PISA design model for

331

monopiles for offshore wind turbines: application to a stiff glacial clay till. Géotechnique,

332

70(11), pp.1030-1047.

333

Darendeli, M. B. (2001). Development of a new family of normalized modulus reduction and

334

material damping curves. PhD thesis, University of Texas at Austin.

335

Dassault Systemes (2014). Abaqus user manual. Simula Corp., Providence, RI.

336

Hardin, B. O. and Drnevich, V. P. (1972). Shear Modulus and Damping in Soils: Design Equations

337

and Curves. Journal of Soil Mechanics and Foundation Division (SM7), 667–692.

338

Skau K. S., Grimstad G., Page A. M., Eiksund G. R. and Jostad H. P. (2018). A macro-element

339

for integrated time domain analyses representing bucket foundations for offshore wind

340

turbines. Marine Structures, 59, 158–78.

341

Suryasentana, S. K., Byrne, B. W., Burd, H. J., & Shonberg, A. (2018). An elastoplastic 1D Winkler

342

model for suction caisson foundations under combined loading. Numerical Methods in

343

Geotechnical Engineering IX, 973-980. CRC Press.

344

Suryasentana, S. K., Dunne, H. P., Martin, C. M., Burd, H. J., Byrne, B. W., & Shonberg, A. (2020).

345

Assessment of numerical procedures for determining shallow foundation failure envelopes.

346

Géotechnique, 70(1), 60-70.

347

Suryasentana, S. K., Burd, H. J., Byrne, B. W., & Shonberg, A. (2022). A Winkler model for suction

348

caisson foundations in homogeneous and non-homogeneous linear elastic soil.

349

Géotechnique, 72(5), 407-423.

350

Suryasentana, S. K., & Mayne, P. W. (2022). Simplified method for the lateral, rotational, and

351

torsional static stiffness of circular footings on a nonhomogeneous elastic half-space based

352

on a work-equivalent framework. Journal of Geotechnical and Geoenvironmental Engineering,

353

148(2), 04021182.

354

Suryasentana, S. K., Burd, H. J., Byrne, B. W., & Shonberg, A. (2023). Modulus weighting method

355

for stiffness estimations of suction caissons in layered soils. Géotechnique Letters, 13(2), 1-8.

356

Vardanega, P. J. and Bolton, M. D. (2013). Stiffness of Clays and Silts: Normalizing Shear

357

Modulus and Shear Strain. Journal of Geotechnical and Geoenvironmental Engineering

358

9(September), 1575–1589.

359

15

Vulpe, C. (2015). Design method for the undrained capacity of skirted circular foundations under

360

combined loading: effect of deformable soil plug. Géotechnique, 65(8), 669-683.

361

Zhang, Y. and Andersen, K. H. (2017). Scaling of lateral pile p-y response in clay from laboratory

362

stress-strain curves. Marine Structures 53, 124–135.

363

364

16

Table 1. Displacement boundary conditions applied at LRP (see Fig. 1a) to determine the soil

365

reaction responses. For the analyses in this paper, was set to 0.1, which was sufficient to

366

observe the small-strain, non-linear behaviour of the soil reactions.

367

368

(rad)

(rad)

(rad)

Rigid Axial Translation

0

0

0

0

0

Rigid Lateral Translation

0

0

0

0

0

Rigid Rotation

0

0

0

0

0

Rigid Torsion

0

0

0

0

0

369

370

Table 2. Combinations of 3D FEA input parameters analysed for the calculation of displacement

371

scaling factors for the skirt and base soil reactions

372

Values analysed

0, 0.25, 0.5, 1, 2

0, 0.2, 0.4, 0.49

0, 0.5, 1

0.4, 0.7, 1

0.0001, 0.0005, 0.002

373

374

Table 3. Best-fit parameters of Eq. 10 for the displacement scaling factors for the skirt and base

375

soil reactions

376

0.064

0.044

0.062

0

0.167

0.316

0.799

1.732

0.427

0

0.293

0

0.63

0

0.33

0.885

0.067

0

0.07

0

0.144

0.325

0.68

1.074

0.101

-0.043

0.105

0

0.055

-0.046

0.324

1.85

0.383

0

0.251

0

0.229

0

0.375

1.399

0.094

0

0.257

0

0.019

0.228

0.609

1.505

377

7.684

-2.536

-4.855

-0.652

9.424

0

-4.637

-0.179

5.709

1.603

-3.676

-0.394

7.436

8.477

-5.341

-2.105

14.453

0

-5.951

-0.629

8.412

1.854

-4.276

-1.054

378

379

Table 4. Best-fit parameters of Eq. 11 for the displacement scaling factors for the skirt and base

380

soil reactions

381

0.36

0.172

0.426

0.354

0.171

0

0

0.193

0.305

-0.232

0.233

0.219

0.231

0.437

0.115

0.111

382

17

383

(a)

384

385

(b)

386

Figure 1. (a) Schematic diagram of a suction caisson foundation, where LRP represents the

387

loading reference point. (b) Distributed skirt soil reactions and the concentrated base soil

388

reactions acting on a one-dimensional Winkler model representation of the suction caisson

389

structure.

390

391

18

392

393

(a)

(b)

(c)

(d)

394

Figure 2. The 3D FEA results for the ground-level load-displacement behaviour of suction

395

caissons in small-strain, non-linear elastic soil, for and

396

, under the prescribed displacements detailed in Table 1. The corresponding

397

calculations by the proposed Winkler model are also included in these figures for comparison.

398

399

400

401

402

403

404

19

(a)

(b)

(c)

(d)

405

Figure 3. Load-displacement behaviour of the skirt and base soil reactions for

406

and , under the prescribed displacements detailed in Table

407

1.

408

409

20

(a)

(b)

*

(d)

410

411

Figure 4. Secant stiffness of the skirt and base soil reactions from Fig. 3, normalised by their

412

linear elastic counterparts, for and . The soil

413

constitutive relationship (Eq. 3) is also plotted here as grey dotted lines for comparison.

414

415

416

417

418

21

(a)

(b)

(c)

(d)

419

420

Figure 5. Secant stiffness of the skirt and base soil reactions, normalised by their linear elastic

421

counterparts, after applying the displacement scaling factors (), for

422

and . The soil constitutive relationship (Eq. 3) is also plotted

423

here as grey dotted lines for comparison.

424

425

426

22

427

(a)

(b)

*

(c)

(d)

428

Figure 6. Variations of the best-fit displacement scaling factors (

) for the

429

skirt soil reactions with respect to and for and . The dotted lines represent

430

the predictions of the approximating functions (Eqs. 9 and 10).

431

432

433

23

(a)

(b)

*

(c)

(d)

434

435

Figure 7. Variations of the best-fit displacement scaling factors (

) for the

436

base soil reactions with respect to and for and . The dotted lines represent

437

the predictions of the approximating functions (Eqs. 9 and 10).

438

439

440

441

24

442

(a)

(b)

*

(c)

(d)

443

Figure 8. Variations of the best-fit displacement scaling factors (

) for the

444

skirt soil reactions with respect to and for and . The dotted lines represent

445

the predictions of the approximating functions (Eqs. 9 and 10).

446

447

448

25

449

(a)

(b)

*

(c)

(d)

450

Figure 9. Variations of the best-fit displacement scaling factors (

) for the

451

base soil reactions with respect to and for and . The dotted lines represent

452

the predictions of the approximating functions (Eqs. 9 and 10).

453

454

455

456

457

458

26

459

(a)

(b)

(c)

(d)

460

Figure 10. The 3D FEA results for the ground-level load-displacement behaviour of suction

461

caissons in small-strain, non-linear elastic soil, for and

462

, for prescribed uniaxial loading. The corresponding Winkler model predictions are

463

included in these figures for comparison.

464

465