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A Winkler model for suction caisson foundations in homogeneous and

non-homogeneous linear elastic soil

Stephen K. Suryasentana1

Harvey J. Burd2

Byron W. Byrne3

Avi Shonberg4

Affiliations

1 Department of Civil and Environmental Engineering, University of Strathclyde, Glasgow, UK;

formerly Department of Engineering Science, University of Oxford, Oxford, UK

(Orcid: 0000-0001-5460-5089)

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

(Orcid: 0000-0002-8328-0786)

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

(Orcid: 0000-0002-9704-0767)

4 Ørsted Wind Power, London, UK

Corresponding author information

Stephen Suryasentana

stephen.suryasentana@strath.ac.uk

2

Abstract

Suction caisson foundations provide options for new foundation systems for offshore structures,

particularly for wind turbine applications. During the foundation design process, it is necessary

to make reliable predictions of the stiffness of the foundation, since this has an important

influence on the dynamic performance of the overall support structure. The dynamic

characteristics of the structure, in turn, influence its fatigue life. This paper describes a

thermodynamically-consistent Winkler model, called OxCaisson, that delivers computationally

efficient estimates of foundation stiffness for caissons installed in homogeneous and non-

homogeneous linear elastic soil, for general six degrees-of-freedom loading. OxCaisson is

capable of delivering stiffness predictions that are comparable to those computed with three-

dimensional finite element analysis, but at a much lower computational cost. Therefore, the

proposed model is suited to design applications where both speed and accuracy are essential,

such as large-scale fatigue assessments of offshore wind farm structures. The paper

demonstrates that the OxCaisson model can also be applied to short rigid monopile

foundations.

KEYWORDS

Suction caisson foundations, soil-structure interaction, offshore engineering, Winkler model

3

INTRODUCTION

There is increasing interest in employing suction caisson foundations (also referred to as

suction bucket foundations, see Fig. 1a) for offshore wind turbine support structure applications,

because of potential advantages such as quiet installation (important for projects with noise

restrictions), faster installation and simpler decommissioning (Byrne & Houlsby, 2003)

compared with other foundation options. The design of suction caissons in this application relies

on the availability of efficient procedures to predict foundation stiffness (relevant to dynamic

analyses of the overall structure) and strength (to facilitate the analysis of storm loading events).

However, the range of available design procedures for suction caissons in this application is

rather more limited than is the case for monopile foundations, which are currently the dominant

foundation system for offshore wind turbines in shallow waters.

This paper describes a new design model – ‘OxCaisson’ - to predict the stiffness of suction

caisson foundations, especially for wind turbine support structure applications. The design

model is developed for full six degrees-of-freedom loading and is calibrated for homogeneous

and non-homogeneous elastic soils for which the stiffness varies with depth. OxCaisson is fast

to compute; it is therefore well-suited to the design of offshore wind farms, which typically

require a large number of computations to assess different foundation geometries and wind

farm layouts to determine optimal configurations.

The OxCaisson model is based on an underlying concept, known as the Winkler assumption

(Winkler 1867), in which the soil reactions acting on the foundation are considered to be

functions of the local caisson displacement and rotation only. The spatial coupling that occurs

within the soil is therefore excluded from the model. This approach has the advantage of

simplicity, although the shortcomings of the Winkler assumption has certain consequences for

the calibration of the model. The Winkler approach also means that the design model is only

reliable for the particular parameter space that was employed to calibrate it.

The Winkler modelling concept forms the basis of the - method’ (e.g. API 2010, DNV 2016)

that is widely used for the design of piles (including monopiles for offshore wind applications)

that are subjected to lateral loading. Standard forms of the p-y method employ a single soil

reaction component, namely a lateral distributed load, that is related to the local lateral pile

displacement by a nonlinear function (p-y curve) that is calibrated to the local soil conditions.

Although the accuracy of current forms of the p-y method has been questioned (e.g. Doherty &

Gavin 2011), its widespread use is indicative of the usefulness of the Winkler concept as the

basis for simplified design models. Piles subjected to axial loading are routinely designed using

another Winkler-type modelling approach known as the ‘- method’ (API 2010, DNV 2016).

The established nature of the - and - methods supports the use of the Winkler approach in

the current work.

4

Existing Winkler models

Extended forms of the Winkler model have been previously developed for solid caisson

foundations embedded in linear elastic soil (Gerolymos & Gazetas 2006a, Tsigginos et al. 2008

and Varun et al. 2009) in which four separate soil reactions (distributed lateral load and moment

reactions on the caisson shaft and horizontal force and moment reactions at the base) are

employed. These models are calibrated by inferring the local soil reaction components based

on the overall stiffness of the foundation computed at a convenient reference point (e.g. at

ground level); this general calibration approach is referred to in the current paper as ‘inferred

Winkler model’. Suction caisson foundations (the focus of the current paper) comprise a steel lid

and skirt structure that is embedded in the ground by suction; the performance of the caisson is

governed by interactions between the base of the soil plug and the surrounding ground, as well

as interactions between the exterior surface of the skirt and the surrounding ground.

Conversely, solid caissons consist of a homogeneous mass of embedded material, typically

concrete. Solid and suction caissons with the same diameter and embedded length are likely to

exhibit certain differences in behaviour (with solid caissons likely to be stiffer). Nevertheless,

modelling procedures developed for solid caissons are considered likely to be broadly

applicable to the suction caisson configuration considered in the current paper.

An enhanced form of the p-y method– known as the ‘PISA design model’ (Byrne et al, 2019)

has recently been developed for the design of monopiles with embedment ratios, in the

range (where is embedded length and is foundation diameter). Consistent with

the Gerolymos & Gazetas (2006a,b), Tsigginos et al. (2008) and Varun et al. (2009) models, the

PISA design model employs four separate soil reaction components, although the model is

further enhanced to incorporate nonlinear soil behaviour. The PISA design model employs 3D

finite element analyses to develop site-specific calibrations of the local soil reaction models.

Importantly, the functions employed to represent the soil reactions in the PISA model are

calibrated at a local level, using soil-pile tractions computed in the calibration analyses.

Modelling approaches such as this, that calibrate the model at a local level, are referred to here

as ‘calibrated Winkler models’.

Inferred Winkler models are calibrated to ensure that the performance of the model in predicting

the overall stiffness of the foundation is satisfactory. There is no guarantee or expectation,

however, that the model provides a realistic representation of the soil-foundation interaction

behaviour that actually occurs at a local level. Conversely, calibrated Winkler models consider

the actual local behaviour at the soil-foundation interface, as computed using the calibration

analyses. Provided that a suitable local model can be formulated and calibrated, it is expected

that the overall performance of the foundation will be predicted to a reasonable level of accuracy

by the model. Although Winkler approaches can only provide an imperfect representation of the

physics of the soil-foundation interaction, calibrated models seem likely to provide a closer

representation of reality than inferred models.

5

OxCaisson is a linear Winkler model for the analysis of suction caisson foundations embedded

in homogeneous and non-homogeneous elastic ground. Local soil stiffness coefficients are

calibrated using data from 3D finite element calibration analyses. Data interpreted from the 3D

analyses demonstrate that the lateral and rotational response of the soil is coupled at a local

level. As it is desirable for OxCaisson to provide a realistic representation of the caisson

behaviour at the soil-foundation interface, this local coupling is included in the OxCaisson

model. This local coupling means that the stiffness coefficients employed in the current model

cannot be compared directly with previous models (e.g. Gerolymos & Gazetas 2006a; Tsigginos

et al. 2008; Varun et al. 2009) in which this local coupling is absent. OxCaisson also

incorporates the influence of soil Poisson’s ratio on the value of the local stiffness coefficients,

which is important as some of the local stiffness coefficients (e.g. vertical stiffness) may vary as

much as 40% for . This is an improvement over previous Winkler caisson models

(e.g. Varun et al. 2009) in which the influence of soil Poisson’s ratio is not considered.

Several Winkler models for solid caisson foundations embedded in elastic soils have been

previously proposed in which the ‘inferred’ Winkler modelling approach is employed. Although

models of this form can be made to approximate the overall foundation stiffness obtained from

3D finite element analysis, they do not consider the local soil-foundation interactions.

Consequently, they do not provide insights into the local physics of the soil response. The

OxCaisson model adopts the alternative ‘calibrated’ approach in which closed-form formulations

are derived for the Winkler soil reactions (for the full six degrees-of-freedom load space) that

closely approximate the local physics of the soil-foundation interaction as determined from 3D

calibration analyses. It is considered that this procedure provides a more fundamental basis for

the development of more complex design models (e.g. for non-linear dynamics analyses,

elastoplastic soil behaviour or multi-directional cyclic loading design models) than the inferred

modelling approach. Moreover, the calibrated approach results in more generalised soil

reactions that are not tightly coupled with the structural properties of the foundation used for

calibration.

Alternative non-local Winkler models have been previously developed for pile foundations (e.g.

Versteijlen et al. 2018). These models have certain advantages for elastic analysis, but they

cannot readily be developed for non-linear soil behaviour. Non-local models are therefore not

further considered in the current work.

Existing macro-element models

A macro-element model to estimate the global stiffness coefficients for suction caissons

embedded in homogeneous and non-homogeneous elastic soil have previously been proposed

by Doherty et al. (2005) using the scaled boundary finite-element method. The results presented

in Doherty et al. (2005) relate to caissons with uniform skirt thickness. The Winkler approach

developed in the current paper offers certain advantages over this approach, such as enabling

6

caisson design optimisation by varying the wall thickness along the caisson skirt length;

foundation optimisation of this sort has been previously described for monopile design by

Kallehave et al. (2015).

The current paper describes an OxCaisson calibration for soils with uniform values of Poisson’s

ratio in the range and for caissons with embedment ratios . This

embedment ratio range is considered to be representative of caissons for offshore wind

applications. The calibration presented in the current paper is restricted to soils with a shear

modulus that varies with depth according to,

1

where is a reference stiffness, is a parameter , is distance below the ground

surface and is the caisson diameter (see Fig. 1). The OxCaisson model considers the

complete set of six degrees-of-freedom of loading applied to the foundation at the reference

point (RP) via the superstructure. It is initially calibrated for rigid caissons foundations, but the

application of the model to flexible caissons (i.e. incorporating skirt flexibility) is also

demonstrated.

Applications of OxCaisson

Although the current paper is concerned principally with caissons, it is of interest to consider

whether the modelling framework is also applicable to monopiles. (In the context of simplified

models, a monopile is considered to be equivalent to a caisson foundation in which the lid is

absent.) Exploratory, 3D finite element calculations provide a means of identifying the

foundation and soil configurations for which the rigid lid has a relatively small effect on the

overall stiffness; in these cases, the design model developed in the current paper (for caissons)

is considered also to be applicable to monopiles. This aspect of the paper draws on previous

work by Efthymiou & Gazetas (2018) that indicates, on the basis of 3D finite element studies,

that the influence of the lid on the overall stiffness of a caisson foundation becomes increasingly

less significant as the embedment ratio is increased.

A key application of OxCaisson is the formulation of foundation models to incorporate within an

overall dynamic analysis of a wind turbine support structure for the purpose of estimating its

fundamental natural frequency. Wind turbine structures during their lifetime are subjected to

large numbers of relatively small amplitude cyclic loads, e.g. due to ambient wind and wave

conditions. These loading cycles have the potential to cause high-cycle fatigue failures in the

support structure. To minimise the risk of fatigue, it is important to ensure that the fundamental

natural frequency of the system, for small amplitude loading, avoids the principal loading

frequencies (to minimise dynamic amplification effects). Two key frequencies to avoid are the

7

rotor frequency (1P) and the blade passing frequency (3P). A commonly employed design

philosophy (e.g. Bhattacharya 2019), known as ‘soft-stiff’ design, is to ensure that the

fundamental natural frequency of the support structure falls between the frequency ranges

implied by the 1P and 3P loading. For current offshore systems, this typically implies a target

natural frequency in the range 0.25-0.3 Hz (Bhattacharya 2019). Since, for this high-cycle

fatigue design condition, the amplitude of the loading is relatively small low, linear elastic soil

models are regarded as appropriate for design purposes (e.g. Arany et al. 2016, Jalbi et al.

2018). Additionally, since the frequencies of the applied loads are also relatively low, dynamic

soil stiffness effects associated with radiation damping, as routinely incorporated in models

intended for seismic analysis (e.g. Tsigginos et al. 2008) are typically considered negligible (e.g.

Arany et al. 2016).

Consistent with the considerations outlined above, the current model employs linear frequency-

independent elasticity to represent the soil. It is acknowledged that the foundation may

contribute a certain amount of damping to the overall support structure due to soil hysteresis

(not included in the current model). In practical cases, however, the damping ratio contributed

by the soil is small (typically less than 1%, Arany et al. 2016); damping at this level has a

minimal influence on the structural natural frequency.

Consistent with much previous work in this area, the coupling between the soil and foundation is

represented in this study with tie constraints. The possibility of gapping and sliding is not

incorporated in the model on the basis that these aspects are not considered to be significant

for the low levels of loading that occur under ambient conditions.

FORMULATION OF OXCAISSON

Problem definition

The OxCaisson model considers a suction caisson, circular in plan, with skirt length and

diameter , embedded in an isotropic non-homogeneous elastic soil as shown in Fig. 1a. In

developing the model, the foundation is constrained to deform as a rigid body, although the

model is also applicable to cases when skirt flexibility is incorporated.

External loads are applied to the caisson at the reference point

(RP), Fig. 1c. The corresponding energetically-conjugate global displacements at the RP are

, Fig 1b. The purpose of the OxCaisson model is to estimate

the global stiffness matrix, , where , for practical design applications.

8

For a linear elastic system (such as the one considered here) is guaranteed to be symmetric.

It is also positive definite (for positive values of soil shear and bulk modulus). Moreover,

geometric symmetry of the problem implies that the global stiffness matrix has the general form,

2

where , , , and are independent stiffness coefficients that depend on the stiffness

characteristics of the soil, the dimensions of the foundation and (for flexible caissons) the

stiffness characteristics of the caisson structure.

Cross sections

The model is based on the kinematics of horizontal planes, known as ‘cross sections’. Two

types of cross sections are considered; ‘skirt cross section’ and ‘base cross section’. Skirt cross

sections are horizontal sections through the caisson skirt for ; the base cross section

is the horizontal cross section at the base of the caisson (including the soil plug) at Fig.

1a). Each cross section has associated with it a set of local cross section displacements (Fig

2a),

3

The corresponding local cross section load and moment soil reactions are (Fig 2b),

4

These local cross section displacements and soil reactions are defined with respect to a

reference point (RC) at the centre of each cross section. The local soil model employed in

OxCaisson is specified by where is the local stiffness matrix.

For a rigid caisson, the local skirt cross section displacements, , are related to the global

caisson displacements, , by,

5

where,

9

6

The cross section displacements at the base of the caisson are related to the global

displacements by,

7

This rigid caisson form of the model is employed to develop and calibrate the local stiffness

coefficients (i.e. the coefficients in the local stiffness matrix, ).

Soil reactions

Displacements of the caisson (due to external loads) will lead to the development of soil

reactions at the soil-foundation interface. To calibrate the local stiffness model, , the

cross section soil reactions are determined directly from the local tractions

acting at the soil foundation interface, as determined from the 3D calibration analyses.

The soil reactions acting on the skirt, termed ‘skirt soil reactions’, are,

The first three terms in relate to distributed loads in the x, y and z directions respectively.

The

and

terms represent distributed moments and is distributed torque.

The skirt soil reaction vector is computed at specific depths from the local tractions, ,

determined from the 3D calibration analyses by,

8

where the angular position is defined in Fig. 3. Similarly, the base soil reaction vector

is determined from the base tractions by,

9

10

where is an element of area.

Equilibrium

An equilibrium equation for the caisson is established by virtual work as follows. Rigid body

virtual displacements are imposed on the caisson via virtual global displacements applied at

the RP. The corresponding local virtual displacements at an arbitrary point on the

caisson skirt and base are . The local and global virtual displacements are

related by,

10

The virtual work done on the soil by the soil tractions acting on the caisson skirt is (see Fig 3),

11

On the basis of Equation 5 (applied to the virtual displacements in Equation 10) and Equation 8,

this equation can be expressed in terms of the local cross section virtual displacements and

skirt soil reactions as,

12

Similarly, the virtual work done on the soil by the soil tractions acting on the base cross section

is,

13

The total internal virtual work is . The total external virtual work

is the work done by at RP. Equating the internal and external virtual work gives,

14

15

Substituting Equations 5 and 7 into Equation 15 gives,

11

16

where . Since is arbitrary, Equation 16 implies the equilibrium equation,

17

Equation 17 demonstrates that the current formulation reduces the original problem (specified in

three spatial dimensions) is to a single dimension (z).

Local stiffness matrix

By considering the symmetries in the problem, it can be demonstrated that the local stiffness

matrix, , has the general form,

18

The general form of the stiffness matrix therefore has six independent coefficients. Analysis of

the 3D finite element results has shown that if is determined directly from the results of a 3D

finite element calibration analysis, then symmetry of the local stiffness matrix is not guaranteed

(i.e. the off-diagonal coefficients and are not necessarily equal). There is also no

guarantee that the matrix is positive definite. This general lack of symmetry and positive

definiteness is a consequence of the Winkler concept, in which spatial coupling within the soil is

excluded.

It is a feature of OxCaisson that the global stiffness matrices (i.e. Equation 2) determined by the

model are guaranteed to be both symmetric and positive definite; this ensures consistency with

the laws of thermodynamics (e.g. Ottosen & Ristinmaa 2005); matrices with these

characteristics are referred to in this paper as ‘thermodynamically consistent’. If the global

stiffness matrices are not thermodynamically consistent, unrealistic results may be obtained,

such as non-unique solutions in a static linear elastic analysis, or negative and/or imaginary

natural frequencies in an eigenvalue analysis. The approach adopted to ensure this outcome

(sufficient but not necessary) is to constrain the local stiffness matrices to be symmetric and

positive definite at all locations on the skirt and also at the base (see Appendix). This procedure

12

requires certain adjustments to one of the local coupling stiffness coefficients determined from

the 3D calibration analyses.

Computing the global stiffness matrix

For a rigid caisson, the global stiffness matrix is determined from the local stiffness matrices (on

the basis of the equilibrium equation in Equation 17) by,

19

The integration is conducted numerically.

An alternative form of the model, incorporating a flexible skirt, has also been formulated. This

employs a one dimensional (1D) finite element model in which the skirt is represented as a line

mesh of 2-noded frame elements. The frame elements represent torsional and axial strains in

the caisson skirt using linear Lagrangian shape functions. Lateral displacements and rotations

due to bending and shear are modelled using Timoshenko beam theory (employing the five

degrees-of-freedom Timoshenko beam element formulation in Astley 1992).

The current model is concerned with suction caissons that have relatively small values of L/D. In

this case, it is open to question whether Timoshenko beam theory is sufficient to provide a

robust model for the deformations that occur in the skirt. The use of Timoshenko theory (rather

than higher order beam theories) is preferred in the current context, however, to maintain the

overall simplicity of the model. It is demonstrated in a subsequent section of the paper that the

Timoshenko beam approximation on OxCaisson provides values of the global stiffness

coefficient that compare well with 3D finite element analysis, therefore supporting the proposed

formulation.

FINITE ELEMENT CALIBRATION CALCULATIONS

Specification of the calculations

3D finite element calibration analyses have been conducted for suction caissons with the

configuration in Fig 1. Six shear modulus profiles (employing the stiffness profile in Equation 1),

corresponding to , are considered, where represents the

homogeneous soil case. For each case, six values of Poisson's ratios (

) were analysed. Caissons with embedment ratios

and skirt thickness were considered. Other values of

skirt thickness were analysed () but the influence of skirt thickness was

found to have a negligible effect on the results; the results presented below therefore all relate

to . The case represents a rigid, circular surface foundation.

13

The finite element calibration analyses were conducted using the commercial finite element

software Abaqus v6.13 (Dassault Systèmes 2014). First-order, fully-integrated, linear, brick

elements C3D8 (C3D8H for ) were employed for the soil. These elements were also

used to represent the caisson. These elements were found to be adequate for the current work

as comparisons with initial analyses using higher-order alternatives (C3D20 or C3D20H)

showed insignificant differences. The mesh domain was set to for both width and depth;

this was considered sufficiently large for boundary effects to be negligible. The maximum

change in the global stiffness matrix components was 0.2% when the domain was increased

from 100 to 200. Displacements were fixed in all directions at the bottom of the mesh and in

the radial directions on its periphery. Mesh convergence analyses were carried out to determine

an appropriate level of mesh refinement. The maximum change in the global stiffness matrix

components was 0.3% when the number of elements was increased from 80,000 to 160,000.

An example finite element mesh is shown in Fig. 4.

Displacements of the caisson were imposed in the model using rigid body constraints. Contact

breaking between the soil and caisson was prevented using tie constraints at the soil-caisson

interface. Due to symmetry, only four independent prescribed displacement modes are required

to compute the behaviour of the foundation for six degrees-of-freedom loading; the prescribed

displacement modes employed in the analyses are listed in Table 1.

To validate the finite element model, computed values of the global stiffness coefficients in

Equation 2 were compared with previous results in Doherty et al. (2005). Examples of the

comparison of normalised forms of the global stiffness coefficients are shown in Table 2, for

and . A close agreement is apparent between the two data sets, with the

maximum deviation being 3.2%; this supports the suitability of the 3D finite element calibration

procedures employed in the current work.

Determining the local stiffness matrices

To determine the local stiffness coefficients from the finite element results, the contact nodal

forces of the soil elements adjacent to the caisson are resolved into tractions in each of the

directions. For the skirt, the relevant nodal forces are determined from rings of soil

elements in contact with the skirt exterior (Fig. 5). For the base soil reactions, the relevant nodal

forces are determined from the soil elements directly below the caisson base (Fig. 5). Local soil

reaction vectors are computed from the tractions using Equations 8 and 9.

Stiffness relationships between the local soil reactions and the global displacements are initially

determined directly from the calibration analyses in matrix form as

for the skirt and

for the base. The local stiffness matrices for the skirt are then obtained from,

14

20

Similarly, the local stiffness matrix for the base is determined from,

21

The local stiffness coefficients are presented below, employing the dimensionless forms that are

defined in the left column of Table 3 (for the skirt) and the left column of Table 5 (for the base).

For the skirt stiffness coefficients, the dimensionless forms employ the local shear modulus .

The base coefficients, however, are normalised using , which is the shear modulus at a depth

of below the caisson base; the local shear modulus is not suitable for normalising the base

soil reactions, as it is zero-valued for and .

Fig. 6 shows normalised local skirt and base stiffness coefficients, for the example case of

and three embedment ratios (). Fig. 6 e, f indicates that the local stiffness

matrices determined from the 3D calibration analysis are, in general, non-symmetric (i.e.

and

). Similar lack of symmetry in the local stiffness matrices was

observed in the other cases considered. As described earlier, the OxCaisson model requires

that the local stiffness matrix employed in the model is symmetric and positive definite. Special

procedures, described in a subsequent section, are therefore employed to develop symmetric

and positive definite local stiffness matrices from the non-symmetrical stiffness data determined

from the calibration analyses.

LOCAL SOIL STIFFNESS FUNCTIONS

To represent the local soil stiffness behaviour in the OxCaisson model, appropriate

functional forms (referred to as ‘local soil stiffness functions’) are developed in terms of the

normalised forms. The diagonal and off-diagonal components of the local stiffness matrices are

considered separately, as described below.

Diagonal components of the local stiffness matrix

The data in Fig. 6 indicate a tendency for the normalised local skirt stiffness coefficients on the

diagonal of the local stiffness matrix

to increase slightly with depth, with a

higher rate of stiffness increase close to the base. Similar patterns were observed in the other

calibration data. For simplicity, it was decided to adopt local soil stiffness functions in which

these normalised local stiffness coefficients are constant with depth; values of the normalised

stiffness coefficients were computed for each calibration case by least squares regression. The

15

values of the stiffness coefficients determined in this way depend on

, and . After some

experimentation, it was found that the normalised stiffness coefficients could be accurately

represented by a parametric equation of the form,

22

where are parameters determined separately for each normalised stiffness

coefficient. Values of the parameters determined by least squares regression over the whole

data set are listed in Table 3. Values of the normalised stiffness parameters determined by

Equation 22 are shown in Fig. 7 for the example case of A close fit between the

calibration data and the parametric model is evident, with the average coefficient of

determination R2 for the four fits in Fig. 7 being 0.999. A similarly close match was obtained for

all other cases considered.

The normalised base stiffness coefficients determined from the calibration analyses were found

to be closely represented by the parametric equation,

23

The term in Equation 23 is the normalised stiffness coefficient for a rough, rigid, circular

surface foundation and are parameters. Polynomial functions of Poisson’s ratio

and are employed for such that the function reduces to standard cases for homogeneous

soil (Table 4) when . Values of and the parameters required to calibrate were

determined by least squares regression over the whole data set; the resulting parameters are

listed in Table 5.

Comparisons between the data extracted from the 3D finite element calibration analyses and

the parametric model are shown in Fig. 8 for the example case A close fit between the

data and the parametric model is apparent, with the average R2 for the four fits in Fig. 8 being

0.934.

Off-diagonal components of the local stiffness matrix

The local stiffness matrices incorporated in the OxCaisson model are required to be positive

definite with the symmetric form,

16

24

To develop a local stiffness matrix from the (non-symmetrical) form determined directly from the

3D calibration analyses, three possible straightforward choices exist to define ; (i), is set

equal to the value of determined from the 3D calibration analyses, (; (ii), ;

or (iii)

. On the basis of Sylvester’s criterion for positive definite matrices (i.e.

the determinants associated with all upper-left submatrices are positive), positive definiteness of

the matrix requires that,

25

26

The conditions in Equation 25 are satisfied by the current model, at least in the parameter space

explored in this paper. After evaluating the three possible options outlined above, only option (i)

was found to satisfy Equation 26 in all cases. Option (i) was therefore adopted to

develop the local stiffness matrices, for both the skirt and the base, in the OxCaisson model. To

ensure that the local stiffness matrices are symmetric and positive definite, the acceptable

ranges for the Poisson’s ratio and embedment ratio are and .

Incorporating the artificially-symmetric local stiffness matrix into the model means that a

correction needs to be applied to ensure that the adjustments made to the local stiffness matrix

(to make it symmetric) do not propagate errors to the calculation of the global stiffness matrix.

For the rigid caisson form of the model (Equation 19), the local modifications cause variations in

the global stiffness terms and . The variation in was found to be small (less than 2%

deviation) and this is neglected in the current model. However, the difference, , in the

computation of was found to be significant. For example, if left uncorrected, the model

prediction of for the typical case of would be 30% lower than the

value of determined from the corresponding 3D finite element model. On the basis of the

rigid caisson model in Equation 19,

27

To correct the model, the stiffness difference , is applied uniformly along the skirt length by

means of an additional artificial local rotational stiffness coefficient,

, determined from

17

. The value of the local rotational stiffness

employed in the

OxCaisson model is therefore,

28

where

is the rotational stiffness coefficient determined from the parametric model (as

specified in Table 3). Least squares regression was conducted to determine the following

approximation for

,

29

Fig. 9 shows an example comparison for of the values of

determined directly from

the 3D finite element results and the values determined by Equation 29. This shows a close

match between the numerical data and the assumed model, with the R2 of the fit being 0.999; a

similarly close match was obtained for the other cases considered. It should be emphasised that

is an artificial stiffness; it should not be interpreted beyond being a simple correction for

.

EVALUATION OF THE OXCAISSON MODEL

Fig. 10 compares the global stiffness coefficient predictions of OxCaisson (using Equation 19)

with the corresponding 3D finite element results for = 0.125, 0.25, 0.5, 1, 2, = 0.2 and

. Note that is a new test value, not included in the calibration set. To quantify the

prediction errors, Table 6 shows the average percentage differences (APD) and root-mean-

square percentage errors (RMSPE) of the global stiffness coefficient predictions in Fig. 10. It is

evident from Table 6 that the OxCaisson predictions agree well with the 3D finite element

computations for the homogeneous case (), especially for . The maximum

RMSPE is 6.41% for . For the non-homogeneous cases (, the overall agreement is

still reasonable, although the maximum RMSPE increases to 12% for . It can be observed

that there is some under-prediction for and (i.e. negative values of APD). This is likely

due to the OxCaisson computations for and being significantly influenced by

. The

uniform-with-depth idealisation employed in the model excludes a significant moment

contribution from the relatively large lateral soil reaction, apparent in the finite element results,

that develop near the base of the caisson (e.g. Fig. 6c).

Application to suction caissons with a flexible skirt

Separate analyses have been conducted to investigate the extent to which the OxCaisson

model, calibrated using the rigid caisson approach described above, is applicable when skirt

flexibility is incorporated. The 1D finite element version of the OxCaisson model – which

18

incorporates the effects of skirt flexibility – represents the caisson skirt with a line mesh of frame

elements. Separate finite element analyses were conducted in which the rigid constraints

imposed on the skirt were removed (although the rigid body constraints applied to the lid are

retained). For these analyses, a single layer of second-order, fully-integrated, brick elements

C3D20 was employed for the caisson skirt. The Young’s modulus and Poisson’s ratio for the

skirt was set at 206 GPa and 0.3 respectively, Separate OxCaisson analyses were conducted

employing the 1D finite element version of the model (in which shear and bending in the skirts is

represented by Timoshenko beam theory). The thin walled approximation was employed for the

area and second moment of area of the caisson skirt. A shear factor of

(Hutchison

2001) was adopted.

Table 7 lists the ratios of the global stiffness coefficients for flexible and rigid caissons computed

by the 3D finite element model for = 0.125, 0.25, 0.5, 1, 2, = 0.2 and . It is

evident that there are significant differences between the flexible and rigid cases, especially for

high values of and non-homogeneous soil (). Thus, it is important that the skirt

flexibility is considered when designing suction caissons. Fig. 10 compares the OxCaisson

global stiffness coefficient predictions with those computed from the 3D finite element analyses;

this shows generally good agreement. Table 6 also shows the APD and RMSPE calculations for

the OxCaisson predictions, relative to the corresponding 3D finite element results, for the

flexible skirt case. The OxCaisson predictions agree well with the 3D finite element

computations for the cases listed in the table, with a maximum RMSPE of 6.77% for .

Application to short rigid monopiles

Further exploratory analyses have been conducted using the 3D finite element model in which

the lid is absent from the analysis. The model in this case represents a ‘short’ rigid monopile.

Calculations were conducted by applying rigid body constraints to the monopile. Fig. 11 shows

the computed global stiffness coefficients and the local stiffness coefficients for the skirt and

base soil reactions of the monopile, relative to those of the caisson, for (homogeneous

soil).

Fig. 11 indicates that the presence of the rigid caisson lid has negligible influence on the global

torsional and lateral stiffness coefficients for embedment ratios , and for the

global vertical, moment and coupling stiffness coefficients for . As

decreases from 0.25, however, the global vertical, moment and coupling stiffness coefficients

for the monopile decreases slightly relative to those of the corresponding caisson foundation.

This tendency is consistent with previous finite element studies described by Efthymiou &

Gazetas (2018). It is evident from Fig. 11a, d, e that as decreases from 0.5, the local

stiffness coefficients for the skirt and base soil reactions for the monopile foundation increase

19

and decrease respectively, relative to the caisson case. The caisson lid therefore appears to

increase the load on the base at the expense of the skirt for .

The apparent equivalence in caisson and short monopile performance for certain cases, as

computed with the 3D finite element model, suggests that the current OxCaisson model could

be employed for the analyses of monopiles with embedment ratios in the range

(or for torsional and lateral stiffness).

DISCUSSION

The Winkler suction caisson model described in the paper – OxCaisson - provides a rapid

analysis tool to assess the stiffness of suction caissons embedded in homogeneous and non-

homogeneous elastic soil. For example, using a computer with an Intel Xeon 3.60 GHz

processor (eight central processing units) with 16 GB RAM (random access memory), a single

OxCaisson analysis usually takes less than a second, while its corresponding 3D finite element

analysis usually takes approximately 5 minutes. The total run time for all of the 3D finite element

analyses covered in this study is in the order of days, while it took only seconds to do the same

with OxCaisson. The model is therefore highly suited to time-critical design tasks such as

optimisation of large-scale offshore windfarm design.

Two forms of the model are proposed; (i) a rigid caisson model in which kinematic conditions

are used to relate the local caisson displacements to the displacements at a specified reference

point, and (ii) an alternative form, to incorporate the influence of skirt flexibility, which employs

embedded frame finite elements. A principal intended application of the model is to formulate

foundation models for incorporating in integrated analyses to assess the fundamental natural

frequency of wind turbine support structures under ambient loading conditions. These natural

frequency analyses are required to assess the high-cycle fatigue damage induced in the support

structure by the many millions of relatively small-amplitude load cycles applied to it during its

lifetime. For these small-amplitude load cycles, the linear elastic modelling approach

considered in the current work provides a means of minimising the complexity of the model

while also capturing the relevant aspects of soil behaviour. It is noted that the current model

employs a fully-tied condition at the soil-foundation interface. This condition is considered

appropriate for the small displacement loading envisaged in the fatigue assessment calculations

referred to above. For higher levels of loading, the possibility of gapping between the caisson

and the soil may need to be considered.

The elastic model employed in the current work does not incorporate the influence of soil

drainage conditions on the performance of the caisson. For cases where the soil can be

considered to be undrained or fully drained, appropriate values of the elastic soil parameters

() for incorporation in the model can be determined readily. In partial drainage cases

(determined on the basis of the loading frequency, the soil permeability and the dimensions of

20

the foundation), appropriate values of the elastic soil parameters for incorporation in the model

may be estimated based on previous research (e.g. Damgaard et al. 2014).

The OxCaisson model represents the local soil-foundation interaction via a local coupled

stiffness matrix; the local stiffness coefficients are determined via 3D finite element calibration

calculations. To ensure thermodynamic consistency of the overall model, certain adjustments

are required at the local level. Approximate analytical expressions are presented for the local

stiffness coefficients in the calibration range and , (where is

the stiffness exponent employed in the depth varying shear modulus model). In spite of the

approximations inherent in the OxCaisson approach, the rigid and flexible forms of the model

have been found to agree well with the global stiffness predictions by the existing macro-

element model for suction caisson foundations (Doherty et al. 2005). Differences between the

predictions of the current model and Doherty et al. (2005) are most significant for the global

rotational moment stiffness . For the example case of

, differences between the current work and the data in Doherty et al. (2005) for amount

to 4.8% and 7.4% for rigid and flexible caissons respectively. The OxCaisson model also

conforms closely to the 3D finite element results presented in the current paper. The OxCaisson

model is therefore able to approximate the calculations of relatively complex numerical

modelling procedures, but at a fraction of the computational cost.

Consistent with previous work (e.g. Doherty et al. 2005), the current study highlights the

significant influence that the flexibility of the skirts can have on the overall stiffness of the

caisson. This differentiates the performance of suction caissons from solid caissons for which

the foundation is typically regarded as entirely rigid. Although not included in the current study,

previous work by Skau et al. (2019) has highlighted the potential influence of caisson lid

flexibility on the vertical component of the foundation stiffness. A consideration of lid flexibility

effects would require an analysis of plate bending effects in the lid. It is considered that these

effects could be combined with the current model in an approximate way, although this has not

been attempted in the current work.

The current model is calibrated for power law depth variations in the shear modulus; this is

consistent with previous work (e.g. Doherty et al 2005). Although stiffness variations of this sort

are regarded as a useful approximate representation of the characteristics of natural deposits,

this approach may not be appropriate for cases where a suction caisson is embedded in soils

with distinct layers with significant differences in stiffness. Additionally, it is noted that suction

caisson foundations are commonly employed as foundations for jacket-type structures. In this

application, three or four foundations are likely to be employed to support the jacket and

interactions may occur between each foundation. The current modelling approach is based on

the assumption that each foundation is independent; the possibility of interactions between

nearby foundations is therefore excluded.

21

A key feature of the flexible skirt form of the OxCaisson model is that it can be employed,

straightforwardly, to analyse caissons with skirt thickness that varies with depth. More generally,

the calibrated Winkler modelling approach described in the paper is capable of being extended

to incorporate more complex aspects of behaviour such as soil plasticity and slip/gapping at the

soil-foundation interface. Developments of the model to incorporate these effects will form the

basis of future publications.

CONCLUSION

The objective of this study is to develop a computationally efficient Winkler model that can

provide accurate estimates of the stiffness of a rigid or flexible suction caisson foundation in

homogeneous and non-homogeneous linear elastic soil under the full six degrees-of-freedom

loading. The proposed model, OxCaisson, comprises thermodynamically consistent soil

reactions that have been calibrated against the local soil response data from 3D finite element

analyses. Generalised formulations of the calibrated soil reactions have been derived for six

degrees-of-freedom loading; this facilitates OxCaisson predictions of the stiffness of suction

caissons of any dimensions and soil properties within the calibration space. The results indicate

that OxCaisson can approximate the 3D finite element stiffness calculations well, but at a small

fraction of the computational time. Although OxCaisson was calibrated using a rigid caisson in

the 3D finite element analyses, it was found that the calibrated soil reactions were sufficiently

decoupled from the caisson structural properties that they may also be applied to flexible

caissons and ‘short’, rigid monopiles ().

ACKNOWLEDGEMENTS

The first author would like to acknowledge the generous support of Ørsted Wind Power for

funding his DPhil studentship at the University of Oxford. Byrne is supported by the Royal

Academy of Engineering under the Research Chairs and Senior Research Fellowships scheme.

22

APPENDIX

A positive definite symmetric local stiffness matrix guarantees a positive definite

symmetric global stiffness matrix

For linear elastic soil and a rigid caisson, the work done on the Winkler model is equal to the

strain energy stored in the elastic soil reactions,

A1

Incorporating the stiffness relationships and into Equation A1 gives,

A2

If is positive definite at all depths then (on the basis of the definition of a positive definite

matrix), for all non-zero . This implies from Equation A2 that for all non-

zero , and is therefore guaranteed to be positive definite.

For both a rigid or flexible caisson, a matrix can be found such that . Equation A2

gives:

A3

If is symmetric (i.e. ), the integrand must also be symmetric, since

. Given that

and the sum of symmetric matrices is also

symmetric, it follows that is guaranteed to be symmetric.

23

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behaviour of saturated soil subjected to cyclic loading from offshore monopile wind turbine

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Doherty, J. P., Houlsby, G. T. & Deeks, A. J. (2005), ‘Stiffness of Flexible Caisson Foundations

Embedded in Nonhomogeneous Elastic Soil’, Journal of Geotechnical and Geoenvironmental

Engineering 131, 1498–1508.

Doherty, P. & Gavin, K. (2011) Laterally loaded monopile design for offshore wind farms.

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Efthymiou, G. & Gazetas, G. (2018), Elastic Stiffnesses of a Rigid Suction Caisson and Its

Cylindrical Sidewall Shell, Journal of Geotechnical and Geoenvironmental Engineering 145

(2), 06018014.

Gerolymos, N. & Gazetas, G. (2006a), ‘Winkler model for lateral response of rigid caisson

foundations in linear soil’, Soil Dynamics and Earthquake Engineering 26(5), 347–361.

Gerolymos, N., & Gazetas, G. (2006b), Static and dynamic response of massive caisson

foundations with soil and interface nonlinearities—validation and results. Soil Dynamics and

Earthquake Engineering, 26(5), 377-394.

Hutchinson, J. R. (2001), ‘Shear Coefficients for Timoshenko Beam Theory’, Journal of Applied

Mechanics 68(1), 87–92.

Jalbi, S., Shadlou, M., & Bhattacharya, S. (2018). Impedance functions for rigid skirted caissons

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25

Table 1. Displacement boundary conditions applied to the caisson at the RP (see Fig. 1a) in the

3D finite element analysis to determine the global stiffness coefficients. The value of is

arbitrary

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

Table 2. Normalised global stiffness coefficients for compressible () and nearly

incompressible () homogeneous linear elastic soil for a caisson with .

Comparison of the results of 3D finite element analysis with data from Doherty et al. (2005).

Normalised

stiffness

coefficient

Doherty et al.

(2005)

3D finite element

analysis

Deviation (%)

0.2

3.88

3.94

1.81

0.2

2.43

2.45

0.82

0.2

4.55

4.66

2.42

0.2

2.1

2.14

1.9

0.2

-1.58

-1.63

3.16

0.49

5.39

5.43

0.74

0.49

2.43

2.45

0.82

0.49

5.45

5.56

2.02

0.49

2.5

2.42

3.2

0.49

-1.73

-1.73

0

Table 3. Dimensionless forms adopted for the normalised local skirt soil stiffness coefficients

(left column) where is the local value of shear modulus. Parameters for the skirt local stiffness

coefficients for incorporation in Equation 22 (right column). Note that a correction

(defined in Equation 29) needs to be added to

to obtain the value of

incorporated in OxCaisson, as per Equation 28.

Normalised local stiffness coefficient

Soil stiffness function parameters

2

Table 4. Normalised global stiffness coefficients for a rough, rigid, circular surface foundation

(except for which is for the smooth case) on a homogeneous elastic half-space with shear

modulus . All the stiffness solutions are analytical solutions, except for , which is

approximated from the current 3D finite element model. Most of the analytical solutions are

conveniently summarised in Poulos & Davis (1974).

Normalised global

stiffness

coefficient

Formula

Source

Spence (1968)

Reissner (1944)

Mindlin (1949)

Borowicka (1943)

3D finite element results (this study)

3

Table 5. Dimensionless forms adopted for the normalised base soil stiffness coefficients (left

column) where is the shear modulus at a distance

below the base. Parameters for the

base local stiffness coefficients for incorporation in Equation 23 (right column).

Normalised local

stiffness coefficient

Base soil stiffness function parameters

4

Table 6. Average percentage difference (APD) and root mean square percentage error

(RMSPE) of the global stiffness coefficients computed by OxCaisson relative to the 3D finite

element results for rigid and flexible forms of the model (for = 0.125, 0.25, 0.5, 1, 2, = 0.2

and .). The APD is computed as

and the RMSPE is computed as

, where is number of data points,

is the relative percentage difference,

and and are the global stiffness coefficients computed by OxCaisson and the

3D finite element model respectively.

Stiffness

APD

(%)

RMSPE

(%)

APD

(%)

RMSPE

(%)

APD

(%)

RMSPE

(%)

0.95

1.51

0.26

0.94

-4.42

5.95

0.42

0.50

-4.00

4.04

-0.75

5.87

1.18

2.53

-5.24

5.96

-5.08

6.40

-4.08

4.86

-9.21

9.58

-11.8

11.9

-6.14

6.41

-11.7

12.0

-10.4

11.3

0.95

1.48

-0.24

0.80

-4.62

6.08

0.75

0.97

-2.97

3.32

0.56

4.77

3.61

3.89

-1.00

2.43

0.34

2.78

5.33

6.77

-0.65

4.60

-3.42

6.54

4.22

4.63

-1.81

3.76

-0.372

3.10

5

Table 7. Ratios of the global stiffness coefficients for a flexible caisson relative to those of a rigid

caisson, as computed by the 3D finite element model for and .

Stiffness

L/D

0.125

0.25

0.5

1

2

0

1.00

1.00

0.99

0.97

0.93

0.99

0.98

0.94

0.83

0.62

0.96

0.95

0.91

0.81

0.60

0.98

0.94

0.87

0.68

0.31

0.90

0.87

0.81

0.64

0.33

0.5

1.00

1.00

0.98

0.94

0.83

1.00

0.98

0.93

0.77

0.46

0.95

0.93

0.88

0.70

0.41

0.99

0.95

0.86

0.59

0.19

0.91

0.88

0.80

0.56

0.20

1

1.00

0.99

0.98

0.90

0.69

1.00

0.99

0.93

0.71

0.31

0.95

0.93

0.85

0.59

0.26

0.99

0.96

0.85

0.50

0.10

0.93

0.89

0.78

0.46

0.12

6

(a)

(b)

(c)

Figure 1; (a) Vertical cross section view of a caisson foundation with diameter , skirt length

and skirt thickness . RP indicates the reference point and origin of the (x, y, z) coordinate

system (b) global displacements and rotations at the RP (c) loads

and moments/torque applied at the RP. Rotations and moments are

defined with right-hand rule conventions with respect to the (x, y, z) coordinate system.

7

(a)

(b)

Figure 2; Schematic diagram of a caisson cross section. RC is the centre (and reference point)

of the cross section; (a) displacements and rotations defined at RC (b)

distributed loads and moments/torque applied at RC. Rotations and

moments are defined with right-hand rule conventions with respect to the (x, y, z) coordinate

system

8

Figure 3; Horizontal slice through the caisson to illustrate the polar coordinate employed to

determine the skirt cross section soil reactions from the computed tractions at the soil-caisson

boundary

f

x

dz

y

tractions (tx,ty,tz)

Diameter, D

9

(a)

(b)

(c)

Figure 4; (a) Example 3D finite element mesh employed for the calibration calculations. The

depth and width of the mesh domain are both , where is the caisson diameter. (b) Plan

view of the 3D finite element mesh, with enlarged partial views of the suction caisson

foundation, (c) Enlarged partial view of the soil mesh near the suction caisson.

10

Figure 5; Vertical cross section view of the soil elements (not to scale), from which the skirt and

base and skirt soil reactions are determined. The base soil reactions include both the soil

response on the skirt tip and on the soil plug. The grey markers in the figure refer to the contact

nodes from which the nodal forces are extracted to determine the tractions. The ring of skirt soil

elements shown in the figure is just one of several rings from which the skirt soil reactions are

determined. In the current 3D finite element analyses, 10 rings are employed along the skirt.

11

(a)

(b)

(c)

(d)

(e)

(f)

Figure 6; Normalised local stiffness coefficients for the skirt and base soil reactions for

; (a) Vertical stiffness (b) Torsional stiffness (c) Lateral stiffness (d) Rotational

stiffness (e) Lateral coupling stiffness (f) Rotational coupling stiffness

12

(a)

(b)

(c)

(d)

Figure 7; Normalised local stiffness coefficients for the skirt, diagonal terms only, with respect

to and for and . The dotted lines indicate the OxCaisson local stiffness

coefficients (Equation 22 and Table 3).

13

(a)

(b)

(c)

(d)

Figure 8; Normalised local stiffness coefficients for the base, diagonal terms only, with respect

to and for . The dotted lines represent the OxCaisson model for the local

stiffness coefficients (Equation 23 and Table 5).

14

Figure 9; Artificial local rotational stiffness coefficient

computed from the 3D finite

element results, with respect to and for . The dotted lines are the estimations of

Equation 29.

15

(a)

(b)

(c)

(d)

(e)

Figure 10; Comparison of the global stiffness coefficients computed by OxCaisson (normalised

by the 3D finite element results), for , and .

Values of 1 represent perfect agreement between the OxCaisson computations and the 3D

analyses. White symbols correspond to a fully rigid suction caisson; grey symbols correspond to

a suction caisson with flexible skirt.

16

(a)

(b)

(c)

(d)

(e)

Figure 11; Comparison of the global stiffness coefficients and the local stiffness coefficients for

the skirt and base soil reactions for a rigid monopile foundation, relative to those of the rigid

suction caisson foundation, for , and

. All the stiffness coefficients shown in this figure are computed using

3D finite element analyses.