A Winkler model for suction caisson foundations in homogeneous and
non-homogeneous linear elastic soil
Stephen K. Suryasentana1
Harvey J. Burd2
Byron W. Byrne3
1 Department of Civil and Environmental Engineering, University of Strathclyde, Glasgow, UK;
formerly Department of Engineering Science, University of Oxford, Oxford, UK
2 Department of Engineering Science, University of Oxford, Oxford, UK
3 Department of Engineering Science, University of Oxford, Oxford, UK
4 Ørsted Wind Power, London, UK
Corresponding author information
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
Suction caisson foundations, soil-structure interaction, offshore engineering, Winkler model
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.
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.
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
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
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,
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
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
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
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.
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,
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.
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
The corresponding local cross section load and moment soil reactions are (Fig 2b),
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,
The cross section displacements at the base of the caisson are related to the global
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, ).
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.
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,
where the angular position is defined in Fig. 3. Similarly, the base soil reaction vector
is determined from the base tractions by,
where is an element of area.
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
The virtual work done on the soil by the soil tractions acting on the caisson skirt is (see Fig 3),
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,
Similarly, the virtual work done on the soil by the soil tractions acting on the base cross section
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,
Substituting Equations 5 and 7 into Equation 15 gives,
where . Since is arbitrary, Equation 16 implies the equilibrium equation,
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,
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
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
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,
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
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.
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,
Similarly, the local stiffness matrix for the base is determined from,
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.
). 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
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,
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,
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
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,
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), ;
. 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,
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,
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
. The value of the local rotational stiffness
employed in the
OxCaisson model is therefore,
is the rotational stiffness coefficient determined from the parametric model (as
specified in Table 3). Least squares regression was conducted to determine the following
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
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
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
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
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
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).
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
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.
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.
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 ().
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.
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,
Incorporating the stiffness relationships and into Equation A1 gives,
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
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.
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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
Rigid Axial Translation
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).
Doherty et al.
3D finite element
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
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).
3D finite element results (this study)
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).
Base soil stiffness function parameters
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.
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 .
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.
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
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
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.
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.
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
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).
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).
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
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.
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.