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1. Introduction
Despite the current dominance of monopile
foundations for offshore wind turbines, there is
increasing interest in deploying suction caisson (or
suction bucket) foundations (e.g. for jacket
structures) for offshore wind farms located in deeper
waters due to economic advantages. Once installed,
the caisson foundations will experience vertical (V),
horizontal (Hx, Hy), overturning moment (Mx, My)
and torsional (T) loading during normal operations.
Although the ultimate capacity of the foundation is
important, the general operation of a wind turbine
means assessment of the dynamic and fatigue
performance of the foundation and structure is
particularly important. For such assessments, the soil
response can be approximated as linear elastic, as
the applied loads are within the lower ends of the
expected range during the lifetime. Care is needed,
however, in the selection of appropriate soil stiffness
parameters for use in these assessments.
For a linear elastic soil, it is known from previous
research (Doherty et al., 2005) that the resultant
forces (Hx, Hy, V, Mx, My, T) acting on a caisson
foundation are related to the displacements (Ux, Uy,
Uz, Θx, Θy, Θz) through global stiffness coefficients,
as shown in Eq. 1. KV, KH, KM, KT and KC are the
vertical, lateral, rotational, torsional and lateral-
rotational coupling stiffness respectively.
z
y
x
z
y
x
T
MC
MC
V
CH
CH
y
x
y
x
U
U
U
K
KK
KK
K
KK
KK
T
M
M
V
H
H
00000
0000
0000
00000
0000
0000
(1)
The main design challenge is that a large number of
analyses are required for fatigue assessments. Unlike
the small number of caisson foundations used for
bespoke offshore structures in oil and gas projects, a
typical new offshore wind farm may have hundreds
of such foundations (Byrne et al., 2015).
Optimisation of the caisson foundations for this new
application therefore requires design methods that
are both fast and reliable.
Unfortunately, existing design methods for assessing
stiffness of suction caisson foundations under low
operational loads are limited either by their
efficiency or the level of detail of soil profiles that
can be modelled. For example, some methods are
not efficient enough to handle the large number of
analyses required for fatigue design while others are
applicable only for relatively simple ground profiles.
There is clearly a need for new design methods that
are robust, fast and general enough to handle the
SIMPLIFIED MODEL FOR THE STIFFNESS OF SUCTION
CAISSON FOUNDATIONS UNDER 6 DOF LOADING
SK Suryasentana, BW Byrne and HJ Burd
University of Oxford, Oxford, UK
A Shonberg
Dong Energy Wind Power, London, UK
Abstract
Suction caisson foundations are increasingly used as foundations for offshore wind turbines. This paper pre-
sents a new, computationally efficient, model to determine the stiffness of caisson foundations embedded in
linearly elastic soil, when subjected to six degree-of-freedom loading; vertical (V), horizontal (Hx, Hy), over-
turning moment (Mx, My) and torsion (T). This approach is particularly useful for fatigue limit analyses,
where the constitutive behaviour of the soil can be modelled as linearly elastic. The paper describes the
framework on which the new model is based and the 3D finite element modelling required for calibration.
Analyses conducted using the proposed approach compare well with results obtained using 3D finite element
analysis. The possibility of low-cost analysis, coupled with a simple calibration process, makes the proposed
design method an attractive candidate for intensive applications such as foundation design optimisation.
widely varying heterogeneities in most real-world
ground profiles. This paper sets out a new design
method that addresses this need, and which can be
applied to large scale projects that require
optimisation, such as offshore wind farms.
2. Existing Design Methods
2.1 Macro Element model
The macro element model (e.g. Doherty et al., 2005)
represents the caisson foundation as a single
element, where the behaviour is described purely in
terms of the resultant forces acting on it and the
corresponding displacements. In other words, this
model directly provides the stiffness coefficients in
Eq. 1. This model has several key advantages such
as computational efficiency and easy integration
with most structural analysis programs.
However, there are some notable limitations. First,
the calibration process for this model is cumbersome
as a different set of stiffness coefficients is required
for every unique combination of soil and caisson
stiffness. Second, this model is accurate only for
soils where the stiffness increases continuously with
depth. As most ground conditions encountered in
practice involve layered soils, using simplified soil
profiles may introduce significant errors. This is
especially true when there is a stiff layer overlying
softer layers (Suryasentana et al., 2017).
2.2 3D Finite Element (3D FE) method
The 3D FE method is a rigorous design method and
is often the standard against which other design
methods are benchmarked. It can provide accurate
stiffness predictions for complex ground profiles,
soil constitutive behaviour and structural geometries.
However, it is limited by the high computational
cost and modelling complexities, relative to other
design methods. It is generally unsuitable for the
design and optimisation of foundations in large scale
projects such as for an offshore wind farm.
2.3 Winkler model
Winkler based models have been used successfully
for the design of deep foundations such as
monopiles (API, 2010; DNV, 2014). More recently,
this approach has been applied to shallow
foundations (e.g. Houlsby et al., 2005; Gerolymos &
Gazetas, 2006). In this modelling approach, the soil
continuum is represented by a series of independent
springs, each of which captures the local soil
reaction. This approach has several key advantages.
Similar to the macro element model, it is
computationally efficient and easily coupled with
structural analysis programs. However, a major
advantage over the macro element model is the
localised nature of the soil reactions, which allows
models that are based on the Winkler approach to be
used for any type of non-homogeneous elastic soil,
including layered soil.
Nevertheless, the Winkler approach is not without
limitations. The assumption that the springs are
independent ignores the continuum nature of soil.
This assumption may introduce significant errors in
stiffness predictions for highly heterogeneous
ground profiles. Furthermore, an issue with the
Winkler model, specific to caisson foundations, is
that it is incomplete. The available Winkler models
for caisson foundations are limited to lateral loading
only (Davidson et al., 1982; Gerolymos & Gazetas,
2006). The current Winkler approaches, unlike other
design methods, cannot be readily used to assess the
stiffness of caisson foundations under 6 degree-of-
freedom (dof) loading.
3. Proposed Design Method
This paper proposes a new Winkler-based model,
termed ‘1D caisson model’, to predict the stiffness
of caisson foundations in linear elastic soil. Unlike
existing Winkler models for caisson foundations,
this model is complete and can provide stiffness
predictions for 6 dof loading. Moreover, the
formulations of the Winkler spring forces, hereafter
referred to as 1D soil reactions, are calibrated
against rigorous 3D FE solutions. The model offers
the speed of the Winkler modelling approach and the
accuracy of the 3D FE method.
3.1 Theory
In this model, the global coordinate system is
defined at the centre of the suction caisson lid base
(i.e. the interface between the lid and the soil
medium). Furthermore, this origin is adopted as the
point of applied loading (LRP), as shown in Fig. 1.
Figure 1: Schematic representation of a caisson foundation
and the point of applied loading (D is the caisson diameter, L
is the skirt length and z is the depth below ground surface)
The caisson is assumed to be fully rigid and no slip
or gap is allowed between the foundation and soil.
For each cross section along the caisson skirt, the 3D
soil stresses acting on it can be resolved into 1D soil
reactions, which are essentially the resultant soil
forces acting on each cross section.
Figure 2: Sign conventions for the global applied loads and dof
of the foundation, with respect to the 1D soil reactions and
local dof of each cross section
There are six 1D soil reactions (hx, hy, v, mx, my, t)
and six local dof (ux, uy, uz, θx, θy, θz) associated
with each cross section. Figure 2 shows the sign
convention for the global and local dof with the
relation between the two defined by Eq. 2.
z
y
x
z
y
x
z
y
x
z
y
x
U
U
U
z
z
u
u
u
100000
010000
001000
000100
00010
00001
(2)
Since the caisson skirt can be divided into an infinite
number of infinitesimally thin cross sections, the 1D
soil reactions acting on it, henceforth known as the
skirt 1D soil reactions, would be distributed in
nature. Furthermore, there is an additional non-
distributed 1D soil reaction acting at the base of the
foundation, termed the base 1D soil reaction. This is
the resultant force acting across the base cross
section, which includes both the skirt tip annulus and
the soil plug base.
Fig. 3 shows a schematic diagram of the transfer of
the applied vertical load (V) into the respective 1D
soil reactions. As shown in Fig. 3a, V is balanced by
the soil reactions as follows:
lid
Lplugskirt
Lskirttip vdzvdzvvV 0
,
0
(3)
where vtip, vskirt, vskirt, plug, vlid are the soil reactions on
the skirt tip annulus, skirt exterior, skirt interior and
the lid base respectively. The skirt 1D soil reaction
is vskirt while the base 1D skirt reaction (vbase) is:
plugbasetipbase vvv ,
(4)
where vbase, plug is the soil reaction on the base of the
soil plug. From Fig. 3b, it is shown that:
lid
Lplugskirtplugbase vdzvv
0
,,
(5)
(a) Force equilibrium between caisson foundation, applied
load and local soil reactions
(b) Force equilibrium between internal soil plug and local
soil reactions
Figure 3: Transfer of vertical load into the 1D soil reactions
Substituting Eqs. 4 and 5 into Eq. 3 gives:
base
Lskirt vdzvV
0
(6)
Thus, the skirt and base 1D soil reactions complete
the set of soil reactions acting on the caisson.
3.2 Calibration of 1D soil reactions
To calibrate the 1D soil reactions, the caisson-soil
interaction problem is analysed using the 3D FE
method. Then, the 3D soil stresses from the adjacent
soil elements are resolved into 1D soil reactions.
It is assumed that the 1D soil reactions at each depth,
z, depend only on the soil properties at that depth.
This assumption implies that the model only needs
to be calibrated against the 3D FE results for a
homogeneous elastic soil, with the calibrated
reactions being applicable to non-homogeneous
elastic soil too. This model also assumes that the 1D
soil reactions are independent of the caisson stiffness
properties. Therefore, the model only needs to be
calibrated against a rigid caisson, with the calibrated
reactions also applying to caissons with flexible
skirts. An examination of these assumptions is not
provided here but will be addressed in future work.
To determine the 1D soil reactions, the nodal force
results from the 3D FE analyses are used.
Specifically, the 1D soil reactions are computed
from the contact nodal forces of the soil elements
adjacent to the foundation (including the soil plug).
For the skirt 1D soil reactions, contact nodal forces
refer to nodal forces from nodes shared by the skirt
exterior and surrounding soil elements. For each
‘ring’ of soil elements in contact with the skirt
exterior, the skirt 1D vertical and lateral reactions
are computed as the sum of the contact nodal forces
in the respective axes, divided by the soil element
thickness. The computed value corresponds to the
local soil reaction at the depth of the ‘ring’ of soil
elements. For the skirt 1D moment and torsional
reactions, the computation involves the sum of the
moment induced by each contact nodal force about
the centre of the cross section, divided by the soil
element thickness.
For the base 1D soil reactions, contact nodal forces
refer to nodal forces from nodes shared by the
interface between the bases of the internal soil plug
and skirt tip annulus and the soil elements directly
below them. The base 1D vertical and lateral soil
reactions are the sum of the contact nodal forces in
the respective axes while the base 1D moment and
torsional soil reactions are the sum of the moment
induced by each contact nodal force about the centre
of the cross section.
Finally, mathematical formulations are derived to
approximate these 1D soil reactions; these
formulations form the predictive basis of the 1D
caisson model.
3.3 Global stiffness equations
One advantage of the Winkler assumption is the
availability of analytical solutions to derive the
global stiffness of a rigid caisson directly from the
1D soil reactions, which are shown in Table 1.
Table 1: Analytical solutions to compute the global stiffness of
the foundation directly from the 1D soil reactions. L refers to
the caisson skirt length. KH, KM and KC can be similarly
defined in terms of hx and my, but with some minor
modifications
Equation
KV
z
base
L
z
skirt
u
v
dz
u
v
0
KH
y
base
y
L
y
skirt
yu
h
dz
u
h
0
KM
LL
u
hh
dzzz
u
hh
L
u
mm
dzz
u
mm
y
base
y
x
base
y
L
y
skirt
y
x
skirt
y
y
skirt
x
x
base
x
y
skirt
x
L
x
skirt
x
0
0
KT
z
base
L
z
skirt t
dz
t
0
KC
)()(
0
L
u
h
u
m
dzz
u
h
u
m
y
base
y
y
base
x
y
skirt
y
L
y
skirt
x
or
L
u
hh
dzz
u
hh
y
base
y
x
base
y
L
y
skirt
y
x
skirt
y
0
3.4 Relation to the approach of Byrne et al. (2015)
Whilst the 1D caisson model is similar to the PISA
design approach for short monopile foundations
(Byrne et al., 2015), there are also important
differences. First, it provides the 1D soil reactions
corresponding to the vertical and torsional dof. Thus,
it can handle fully three-dimensional loading.
Second, unlike the PISA approach, this model has
coupling between the lateral and rotational dof. A
local cross section rotation induces a local lateral
soil reaction and a local lateral displacement would
induce a local moment soil reaction. This coupling
has thus far been ignored by existing Winkler
models, such as the p-y method for pile foundations
(API, 2010; DNV, 2014).
4. Numerical Example
This section illustrates the process of calibrating the
1D soil reactions using the solutions of 3D FE
analyses. In this numerical example, a 3D FE model
of a caisson foundation embedded in incompressible
linear elastic soil was implemented in the finite
element program ABAQUS (version 6.13). The
global coordinate system adopted in the FE model is
the same as defined in Fig. 1.
The foundation has a unit diameter (D), a unit skirt
length (L = D) and a skirt thickness of 0.0025D.
Mesh convergence analyses were carried out to
determine the required mesh fineness. Moreover, a
mesh domain of 80D for both diameter and depth
was found to be sufficient to avoid boundary effects.
A typical mesh of the FE model is shown in Fig. 4.
Figure 4: Mesh of the complete 3D FE model, with an enlarged
partial view of caisson foundation
Displacements were fixed in all directions at the
bottom of the mesh domain and in the radial
directions on the periphery. Contact breaking
between the foundation and soil was not allowed and
this was implemented using tie constraints at the
foundation-soil interface.
The soil was weightless and homogeneous isotropic
linear elastic. A Young’s modulus of 100MPa and a
Poisson’s ratio of 0.49 was assigned to the soil
elements, for which eight-noded linear brick
elements C3D8RH (Dassault Systèmes, 2010) were
used. The foundation was assumed to be entirely
rigid and the rigid behaviour was simulated using
rigid body constraints. The reference point was set to
be the point of applied loading as defined in Fig. 1.
To fully calibrate the 1D soil reactions, four sets of
3D FE results are required. These four sets of results
are obtained from the 3D FE analyses of the caisson
foundation under four different prescribed
displacements. These prescribed displacements were
implemented by applying different boundary
conditions to the reference point of the caisson
foundation, as detailed in Table 2.
Table 2: Boundary conditions for the four types of prescribed
displacements applied in the 3D FE analyses
Ux/D
Uy/D
Uz/D
Θx
Θy
Θz
Vertical
0
0
0.1
0
0
0
Lateral
0
0.1
0
0
0
0
Rotational
0
0
0
0.1
0
0
Torsional
0
0
0
0
0
0.1
To verify that the 3D FE model has been set up
correctly, the normalized global stiffness
coefficients resulting from the prescribed
displacements are compared against known results
from previous work (in this case Doherty et al.,
2005), as shown in Table 3.
Table 3: Comparison of normalised stiffness coefficients from
the 3D FE results and values reported in previous work
Stiffness
Doherty et al. (2005)
3D FE
Difference
KV/GD
6.64
6.68
0.60 %
KH/GD
7.54
7.68
1.86 %
KM/GD3
7.40
7.11
-3.92 %
KT/GD3
4.04
4.07
0.74 %
KC/GD2
-4.69
-4.66
-0.64 %
The normalised stiffness coefficients computed by
the 3D FE model matched the values reported by
Doherty et al. (2005) well, with the maximum
deviation being only 3.92%. This provides
confidence that the FE model had been set up
correctly.
5. Results
Fig. 5 shows the 1D soil reactions profile that
resulted from the 3D FE analyses of the four sets of
prescribed displacements given in Table 2. Note that
the values depicted in Fig. 5 are with respect to the
global dof, and not the local dof.
(a)
(b)
(c)
(d)
(e)
(f)
Figure 5: Skirt and base 1D soil reactions from 3D FE results
As shown in the figure, most skirt 1D soil reactions
appear to be constant along the skirt length, apart
from the reactions nearest to the ground surface and
skirt tip. The only exception is the hy coupling
reaction, which changes with depth (see Fig. 5f).
This is expected as it is evident from Eq. 2 that a
pure global rotation Θx would result in local lateral
displacements that increase with depth.
Furthermore, Fig. 5d shows that a rigid lateral
displacement induces local moment reactions along
the skirt and at the base. Most existing Winkler
formulations for monopiles (API, 2010) or suction
caissons (Gerolymos & Gazetas, 2006) ignore these
coupling terms and doing so might introduce errors
in reproduction of the 3DFE results.
Next, simplifying approximations were made when
deriving the mathematical formulations for these 1D
soil reactions. Specifically, all the skirt 1D soil
reactions were assumed to be constant along the
skirt, apart from the hy coupling reaction, which
varies linearly with depth. This extra complexity is
necessary for accurate KM computations. The
constant or linearly varying profiles were found
using ordinary least square regression against the
true skirt 1D soil reactions and the best fit,
simplified skirt 1D soil reaction profiles are shown
in Fig. 5.
Table 4 shows the formulations that were derived
based on the simplified 1D soil reactions. A two-step
process was used to derive the formulations of these
reactions. First, these reactions were formulated with
respect to the global dof. Thereafter, these
formulations were transformed into the local dof
space using Eq. 2. The finalised formulations of the
1D soil reactions are as follows.
Table 4: Approximate formulations of the 1D soil reactions
following calibration against the 3D FE results. Formulations
for hx and my are similar to that of hy and mx
Formulations
Vertical
vskirt = 4.28 G uz
vbase = 2.4 GD uz
Torsional
tskirt = 3.66 GD2 θz
tbase = 0.41 GD3 θz
Rotational
mxskirt = GD2 (-0.12 uy/D + (1.17 - 0.12 z/D) θx)
mxbase = GD3 (-0.12 uy/D + 0.42 θx)
Lateral
hyskirt = GD (6.51 uy/D + (-19.83 * z/D + 10.28) θx)
hybase = GD2 (1.17 uy/D - 0.6 θx)
6. Discussion
To verify that the calibration was robust and that the
simplification in the formulations did not introduce
significant errors, the normalised global stiffness
coefficients computed using the formulated 1D soil
reactions were compared against the actual 3D FE
results. For this exercise, the global stiffness
coefficients were computed using the analytical
solutions in Table 1 and the 1D soil reaction
formulations in Table 4. Table 5 shows the
normalised global stiffness coefficients predicted by
the 1D caisson model and the original 3D FE results.
As can be observed, the formulated 1D soil
reactions, albeit simplified, can reproduce the 3D FE
results well.
Table 5: Comparison of normalised stiffness coefficients
computed by the formulated 1D soil reactions and the 3D FE
model. The first and second KC predictions by the 1D soil
reactions were computed using the mx and hy based equations
in Table 1 respectively
Stiffness
3D FE
1D Soil Reactions
Difference
KV/GD
6.68
6.68
0 %
KH/GD
7.68
7.68
0 %
KM/GD3
7.11
7.12
0.06 %
KT/GD3
4.07
4.07
0 %
KC/GD2 (1)
-4.66
-4.66
0 %
KC/GD2 (2)
-4.66
-4.66
0 %
An important result to note is the computational time
required for each set of predictions. While the 3D
FE analyses took an hour in total to compute the
stiffness coefficients shown in Table 3, the proposed
1D model takes only milliseconds. This shows the
potential of an efficient design process, which can be
broken down into an offline and online stage.
In the offline stage, the time intensive 3D FE
analyses are carried out to calibrate the proposed
model (which only needs to be done once). In the
online stage, the calibrated 1D caisson model is used
with minimal computational effort. This allows a
rapid turnover of design evaluations, which is a
crucial part of many time critical design activities
such as foundation design optimisation. This is a
very significant improvement over the current state
of practice.
Nevertheless, the proposed model does have some
limitations. The paper has shown that the modelling
approach is satisfactory and has been implemented
correctly; given the excellent agreement between the
model predictions and the 3D FE results as shown in
Table 5. However, these results do not provide any
evidence that the proposed model, in its current
form, has any predictive capabilities beyond the
single case of a caisson foundation of L/D = 1 in
incompressible elastic soil. Nevertheless, it is not
difficult to run a more extensive offline stage with
more 3D FE analyses to derive generalised 1D soil
reaction formulations for different caisson
dimensions and elastic soil properties. Although this
work has been completed it is not reported here as
the focus of this paper is on the underlying
modelling approach. The work on generalised 1D
soil reactions will be reported at a later stage.
7. Conclusion
Fatigue design of caisson foundations usually
requires a large number of analyses. Thus, a suitable
design method for fatigue design must be efficient,
in addition to being accurate. However, existing
design methods are limited by efficiency or the level
of detail of soil profiles that can be modelled.
This paper addresses this issue by proposing a
computationally efficient design method that can
provide accurate predictions of the stiffness of
caisson foundations in elastic soil. Compared to the
3D FE model, the proposed model can provide
stiffness predictions at similar levels of accuracy but
at a small fraction of the computational cost. Unlike
existing macro element models, the proposed model
is applicable for any non-homogeneous soil,
including layered soil.
It is evident that the proposed model offers
significant advantages over existing methods,
especially ease of calibration and computational
efficiency. Most of the limitations of the model are
related to the incomplete formulations of the 1D soil
reactions, which can be rectified with further
calibration against more 3D FE results, following
the methodology illustrated in this paper.
8. Acknowledgments
The first Author acknowledges the generous support
by DONG Energy Wind Power through a DPhil
Scholarship at the University of Oxford.
9. References
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methods for large diameter piles under lateral
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International Symposium on Frontiers in
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12.
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Stiffness of flexible caisson foundations
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Suryasentana, S.K., Byrne, B.W., Burd, H.J. and
Shonberg, A. 2017. Weighting functions for the
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