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Computers and Geotechnics 169 (2024) 106241
Available online 19 March 2024
0266-352X/© 2024 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).
Research Paper
Investigation of local soil resistance on suction caissons at capacity in
undrained clay under combined loading
Stephen K. Suryasentana
a
,
*
, Harvey J. Burd
b
, Byron W. Byrne
b
, Avi Shonberg
c
a
Department of Civil and Environmental Engineering, University of Strathclyde, Glasgow, UK
b
Department of Engineering Science, University of Oxford, Oxford, UK
c
Ørsted Wind Power, London, UK
ARTICLE INFO
Keywords:
Bearing capacity
Failure
Foundations
Soil-structure interaction
Offshore engineering
Numerical modelling
ABSTRACT
Winkler modelling offers a exible and computationally efcient framework for estimating suction caisson ca-
pacity. However, there is a limited understanding of the local soil resistance acting on caissons at capacity under
combined six degrees-of-freedom (6DoF) loading, which is essential for accurately estimating caisson failure
envelopes. Furthermore, existing simplied design models for caissons cannot assess capacity under non-planar
lateral and moment loading, which is common in offshore wind applications. To address these limitations, this
paper presents a comprehensive three-dimensional (3D) nite element analysis (FEA) study, which investigates
the local soil resistance acting on the caisson at capacity in undrained clay under combined 6DoF loading. The
paper introduces the concept of ‘soil reaction failure envelopes’ to characterise the interactions between soil
reactions at capacity. Closed-form formulations are derived to approximate these soil reaction failure envelopes.
An elastoplastic Winkler model is then developed, incorporating linear elastic perfectly plastic soil reactions
based on these formulations. The results demonstrate that the Winkler model can provide efcient and
reasonably accurate estimations of caisson capacity under combined 6DoF loading, even for irregular soil proles
that pose much uncertainty and challenges to existing macro-element models.
1. Introduction
Offshore wind energy is expected to grow rapidly over the next few
decades, following ambitious renewable energy targets set by countries
worldwide. To support this growth in offshore wind energy generation,
offshore wind farms are moving into deeper waters, where traditional
foundations such as monopiles are gradually being replaced by more
cost-effective foundations such as jacket structures on suction caissons
or oating wind platforms anchored by suction caissons. For example,
jacket structures on suction caissons were recently deployed at several
offshore windfarms: Borkum Riffgrund 1 in 2014, Aberdeen Bay in 2018
and Seagreen in 2021. Suction caisson anchors were also recently
deployed for oating wind turbines at the Hywind offshore wind farm in
2017. Suction caissons (or suction buckets) are attractive as they can be
installed faster, quieter and cheaper than monopiles, which brings about
signicant cost advantages and environmental benets such as reduced
noise pollution.
While there are several simplied design models for estimating the
stiffnesses of suction caissons (e.g. He et al., 2017; Jalbi et al., 2018;
Efthymiou and Gazetas, 2019), including some that consider the full six
degrees of freedom (6DoF) load space (e.g. Doherty et al., 2005; Sur-
yasentana et al., 2017, 2022, 2023a,b), the available simplied design
models for estimating the ultimate capacity of suction caissons under
combined loading are more limited. In particular, existing models for
estimating caisson capacity are only applicable to planar HM loading
(where H and M represent the lateral and moment loads, respectively).
Planar HM loading refers to loading conditions where the lateral and
moment loads are in the ‘same plane’, or specically when the moment
vector is orthogonal to the lateral load vector. There is currently no
model for estimating the caisson capacity under non-planar HM loading,
which is common in offshore wind applications due to different di-
rections of wind and wave actions. It is worth noting that such models
exist for other foundation types, e.g. surface foundations (Shen et al.,
2017; Suryasentana et al., 2021) and mudmat foundations (Feng et al.,
2014a,b; Feng and Gourvenec, 2015; Feng et al., 2015).
The failure envelope approach is widely used to assess the ultimate
capacity of shallow foundations under combined loading, as recom-
* Corresponding author.
E-mail addresses: stephen.suryasentana@strath.ac.uk (S.K. Suryasentana), harvey.burd@eng.ox.ac.uk (H.J. Burd), byron.byrne@eng.ox.ac.uk (B.W. Byrne),
avish@orsted.com (A. Shonberg).
Contents lists available at ScienceDirect
Computers and Geotechnics
journal homepage: www.elsevier.com/locate/compgeo
https://doi.org/10.1016/j.compgeo.2024.106241
Received 27 October 2023; Received in revised form 28 February 2024; Accepted 13 March 2024
Computers and Geotechnics 169 (2024) 106241
2
mended by several design guidelines (e.g. Paikowsky, 2010; API, 2011;
ISO, 2016; Offshore Wind Accelerator, 2019). A failure envelope denes
the combination of loads that cause an ultimate limit state of a foun-
dation and is typically represented by a mathematical formulation that
provides a reasonably good t with failure load data generated by nu-
merical and/or experimental studies. With reference to the caisson
conguration in Fig. 1a, a failure envelope under 6DoF loading is typi-
cally represented by f(Hx,My,Hy,Mx,V,Q) = 0, where Hx,My,Hy,Mx,V,
Q refer to the applied lateral force (along x-axis), rotational moment
(about y-axis), lateral force (along y-axis), rotational moment (about
x-axis), vertical force and torsion respectively. In this paper,
Hx,
My,
Hy,
Mx,
V,
Q refer to normalised forces or moments where
Hx=Hx/H0,
My=My/M0,
Hy=Hy/H0,
Mx=Mx/M0,
V=V/V0,
Q=Q/Q0 and H0,
M0,V0,Q0 are the respective uniaxial capacities. The applied loads are
applied with reference to a loading reference point (LRP).
1.1. Existing failure envelopes for suction caissons
The initial research on evaluating the failure envelopes of suction
caissons can be traced back to studies on skirted strip foundations, which
can be considered as plane strain versions of suction caissons. Bransby
and Randolph (1998, 1999), Bransby and Yun (2009), and Gourvenec
and Barnett (2011) conducted plane-strain nite element analyses on
skirted strip foundations in undrained clay under planar vertical, lateral,
and moment (VHM) loading conditions. Based on their respective nd-
ings, they proposed failure envelope formulations. Subsequently, Vulpe
(2015) and Hung and Kim (2014) performed three-dimensional (3D)
nite element analysis (FEA) studies on failure envelopes for suction
caissons in undrained clay, considering planar VHM loading conditions.
Additionally, Liu et al. (2023) conducted 3D FEA studies on the failure
envelopes for suction caissons in normally consolidated undrained clay
under planar vertical, lateral, moment, and torsional (VHMQ) loading
conditions.
1.2. Winkler models
Recently, there has been growing interest in using a Winkler
modelling approach to analyse the behaviour of suction caissons, similar
to the approach used for piles (e.g., Byrne et al., 2020a,b; Suryasentana
and Lehane, 2014a,b,2016). This approach involves representing the
soil resistance through independent local soil reactions that act along the
length of the caisson skirt and at its base. Typically, there are two types
of soil reactions that are considered: (a) Skirt soil reactions: These are
distributed soil reactions that act along the length of the caisson skirt.
These reactions represent the net force or moment exerted by the soil on
the cross section of the caisson per metre of skirt length; (b) Base soil
reactions: These are concentrated soil reactions that act on the caisson
base. These reactions represent the net force or moment exerted by the
soil on the caisson base. Fig. 1c shows a schematic diagram illustrating
these soil reactions.
Suryasentana et al. (2022) proposed a Winkler model (called
OxCaisson) to estimate the caisson stiffness under 6DoF loading condi-
tions in both homogeneous and non-homogeneous linear elastic soil.
This model can represent a suction caisson as either fully rigid or with a
exible skirt, using one-dimensional (1D) nite element frame elements
Nomenclature
V vertical load
Hx horizontal load along x-axis
Hy horizontal load along y-axis
Mx moment about x-axis
My moment about y-axis
Q torque about z-axis
V
0
vertical uniaxial capacity
H
0
horizontal uniaxial capacity
M
0
moment uniaxial capacity
Q
0
torsion uniaxial capacity
V normalised vertical load
Hx normalised horizontal load along x-axis
Hy normalised horizontal load along y-axis
Mx normalised moment about x-axis
My normalised moment about y-axis
Q normalised torque about z-axis
Hi horizontal load along a general axis i in the x-y plane
Mj moment about a general axis j in x-y plane
Hi normalised horizontal load along a general axis i in x-y
plane
Mj normalised moment about a general axis j in x-y plane
Ux, Uy, Uzdisplacements in the x,y and z directions
Θx, Θy, Θzrotations about the x,y and z axes
v vertical soil reaction
hx horizontal soil reaction along x-axis
hy horizontal soil reaction along y-axis
mx moment soil reaction about x-axis
my moment soil reaction about y-axis
q torque soil reaction about z-axis
v
0
vertical soil reaction uniaxial capacity
h
0
horizontal soil reaction uniaxial capacity
m
0
moment soil reaction uniaxial capacity
q
0
torsion soil reaction uniaxial capacity
v normalised vertical soil reaction
hx normalised horizontal soil reaction along x-axis
hy normalised horizontal soil reaction along y-axis
mx normalised moment soil reaction about x-axis
my normalised moment soil reaction about y-axis
q normalised torque soil reaction about z-axis
hi horizontal soil reaction along a general axis i in the x-y
plane
mj moment soil reaction about a general axis j in x-y plane
hi normalised horizontal soil reaction along a general axis i in
x-y plane
mj normalised moment soil reaction about a general axis j in
x-y plane
su undrained shear strength
L suction caisson embedded length
D suction caisson diameter
fskirt local failure envelope formulation for skirt soil reactions
fbase local failure envelope formulation for base soil reactions
α
SΘ angle between the horizontal displacement direction and
the normal to the rotation axis
α
HM
′
angle between Hi axis and M
′
j axis (which is clockwise
orthogonal to Mj axis)
α
hm
′
angle between hi axis and m
′
j axis (which is clockwise
orthogonal to mj axis)
Askirt external surface area of the caisson skirt per metre skirt
length, equivalent to
π
D
Abase area of the caisson base, equivalent to
π
D2/4
patm atmospheric pressure
S.K. Suryasentana et al.
Computers and Geotechnics 169 (2024) 106241
3
(in which shear and bending in the skirts is represented by Timoshenko
beam theory). The formulations for the linear elastic soil reactions of
OxCaisson were derived from 3D FEA calibration results. Antoniou et al.
(2022) proposed a ‘Caisson-on-Winkler-Soil’ (CWS) model for
performance-based seismic design of suction caissons under planar VHM
loading. The soil reactions in this model are represented by ‘hysteretic
elements’ that are also calibrated using 3D FEA. Each hysteretic element
has an ultimate strength for each load direction, which contributes to the
ultimate capacity of the caisson. Similar to the soil reactions in pile
Winkler models, the ultimate strengths of the hysteric elements are
uncoupled (i.e. there is no effect of vertical loading on the ultimate
lateral strength, and vice versa).
Winkler models offer several advantages over macro-element or
force-resultant models. They provide a degree of separation between the
structural properties of the caisson and the soil properties, allowing for
more exible modelling. Additionally, Winkler models allow direct
modelling of caisson performance in ‘irregular’ soil proles that deviate
from idealised soil proles. These advantages have been demonstrated
for caisson stiffness estimations (Suryasentana et al., 2022,2023a), but
they have not been shown for caisson capacity estimations. One of the
main challenges in developing Winkler models for capacity estimation is
the need to account for interaction between local soil reactions at ca-
pacity. Neglecting this interaction would prevent accurate reproduction
of key failure envelope characteristics observed in caissons, such as the
reduction in lateral capacity with increasing vertical load. Moreover,
there is a lack of research on the local soil resistance acting on a caisson
at capacity under combined 6DoF loading. Such research would provide
valuable scientic insights into the interactions between soil reactions
when the caisson is at capacity, and the contributions of the skirt and
base soil reactions to the caisson capacity.
1.3. Objectives
The main aim of this paper is to examine the local soil resistance
acting on suction caissons when they are at capacity in undrained clay
under combined 6DoF loading. This is achieved by carrying out a
comprehensive 3D FEA study of the ultimate suction caisson behaviour
in undrained clay under 6DoF loading. Another goal is to develop a
simplied design model that can predict the caisson capacity under non-
planar HM loading. This model will facilitate routine capacity assess-
ments for offshore wind applications, where such loading conditions are
typical. Additionally, the caisson capacity for suction caisson anchor
applications can be inuenced by torsional loading. To this end, this
paper will develop an elastoplastic Winkler model called OxCaisson-
LEPP (Linear Elastic Perfectly Plastic) that can estimate the caisson ca-
pacity in undrained clay under combined 6DoF loading. OxCaisson-
LEPP uses elastoplastic soil reactions derived from calibration results
obtained through 3D FEA. These soil reactions are coupled to capture
their interactions at capacity. As a result, OxCaisson-LEPP can estimate
realistic failure envelope characteristics, including reduced lateral ca-
pacity under increased vertical loading (see Fig. 2). The advantage of the
Winkler modelling approach adopted in this study is demonstrated by
comparing OxCaisson-LEPP to a comparable macro-element model.
Using an irregular undrained shear strength prole from a real-world
case study, OxCaisson-LEPP provides more accurate estimations of the
caisson failure envelope. Notably, this accuracy is achieved despite the
signicant differences between the irregular undrained shear strength
prole used in the case study and the undrained shear strength prole
used in the calibration.
2. Methodology
This section describes the 3D FEA study that was performed using the
nite element software Abaqus v6.13 (Dassault Syst`
emes, 2014) to
determine the capacity of a suction caisson foundation in undrained clay
under combined 6DoF loading. The 3D FEA model consists of a rigid
suction caisson of diameter D and skirt thickness dskirt /D=0.005 on
homogeneous, elastoplastic soil. The foundation diameter D was held
constant at unit length, while ve skirt lengths (L/D=0.125,0.25,0.5,1,
2) were analysed. The diameter and depth of the mesh domain were set
to 6D and 2L+2.5D respectively. This domain size is sufciently large to
avoid signicant boundary effects on the computed failure loads, as the
maximum change in the uniaxial capacities of the foundation was about
0.2 % when the domain was doubled (i.e. diameter of 12D and depth of
Fig. 1. (a) 6DoF loading conguration for suction caisson foundation, consis-
tent with the conventions in Buttereld et al. (1997). The loading reference
point (LRP) is at the centre of the foundation base. (b) Plan view of a suction
caisson foundation embedded in the ground. (c) Schematic diagram of a one-
dimensional Winkler model for a suction caisson foundation, which is sub-
jected to distributed ‘skirt soil reactions’ along its skirt (with the gure illus-
trating a single example at a specic depth) and concentrated ‘base soil
reactions’ at the base of the caisson.
S.K. Suryasentana et al.
Computers and Geotechnics 169 (2024) 106241
4
4L+5D). A representative mesh is shown in Fig. 3.
The soil was dened as a homogeneous, isotropic linear elastic,
perfectly plastic material, with uniform undrained shear strength su,
adopting a fully-associated von Mises yield criterion. The Poisson’s ratio
ν
of the soil was set to 0.49, while its Young’s modulus E was set to
1000
3
√su. First-order, fully-integrated, linear, brick elements C3D8H
were adopted for the soil. The caisson was modelled as a weightless,
rigid body (using rigid body constraints), and the LRP was set at the
centre of its lid base, as shown in Fig. 1. Separation and slip at the soil-
foundation interface was prevented using tie constraints; this means that
the contact interaction between the soil and the caisson skirt/lid are
based on connected nodes.
2.1. Numerical procedures to determine failure load data
The applied loading conditions are identical to those described in
Suryasentana et al. (2021); these are briey described below. First, the
uniaxial load capacities (Q0,V0,H0,M0) of the foundation were
computed; note that the vertical uniaxial capacities computed apply to
both compressive and tensile capacity as the nite element nodes be-
tween soil and caisson structure are connected and the employed von
Mises soil model is not pressure dependent. Table 1 provides compari-
sons between some of the 3D FEA uniaxial load capacity results from this
study and the 3D nite element limit analysis (FELA) results from Sur-
yasentana et al., (2020b). The 3D FEA results generally agree well with
these previous solutions. Next, failure envelope data were determined
using combined displacement and load controls. Specically, Q and/or
V loads were rst applied on the foundation, before HM contours of the
failure envelope are explored by applying horizontal displacements and
rotations using the sequential swipe test described in Suryasentana
et al., (2020b). The values of the normalised torque and vertical load
applied in the current study are
Q=0,0.25,0.5,0.75 and
V=0,0.25,
0.5,0.75. Similar to Suryasentana et al. (2021), the sequential swipe
tests were applied at ve different angles (
α
SΘ=0,
π
/8,
π
/4,3
π
/8,
π
/2)
between the direction of the horizontal displacement and the normal to
the rotation axis (see Fig. 4a), which was sufcient to map the HM
contours of the failure envelope for both planar and non-planar HM
loading.
α
HM
′
is dened here as the angle between the Hi axis and the M
′
j
axis (which is clockwise orthogonal to the Mj axis), where Hi and Mj are
the resultant lateral and moment loads along some axes i,j (see Fig. 4b).
α
HM
′
=0 corresponds to planar HM loading.
2.2. Local soil reactions
The skirt and base soil reactions are calculated from the nodal forces
of the soil element nodes in contact with the caisson skirt and base
respectively, as detailed in Suryasentana et al. (2022). The current study
focuses on examining the values of these soil reactions and their in-
teractions with one another when the caisson is at capacity. For the
current paper, q0,v0,h0,m0 are the uniaxial capacities of the torsional,
vertical, lateral and moment soil reactions respectively. Consistent with
their global counterparts (H0,M0,V0,Q0), they represent the ultimate
value of each soil reaction, when the other soil reactions are zero. For
example, q0 is the ultimate value of q, when v=0,h=0 and m=0.
Fig. 2. Comparison of the true normalised VH failure envelope for a caisson of
L/D=1, with the corresponding failure envelope estimated by a Winkler
model with soil reactions that have no interaction at capacity.
Fig. 3. (a) Oblique view of the full 3D FEA model. The diameter and depth of
the mesh domain are 6D and 2L+2.5D respectively. (b) Partial view of the
suction caisson.
Table 1
Uniaxial capacities of a suction caisson of L/D=1, where A=
π
D2/4 refers to
the foundation base area.
L
D
Q0
ADsu
V0
Asu
H0
Asu
M0
ADsu
3DFE 1 2.42 13.12 5.92 3.71
3D FELA (Average) 1 – 13.10 5.90 3.66
3D FELA (LB) 1 – 12.52 5.52 3.36
3D FELA (UB) 1 – 13.68 6.28 3.96
Analytical 1 2.333 – – –
S.K. Suryasentana et al.
Computers and Geotechnics 169 (2024) 106241
5
Furthermore,
hi,
mj,
v,
q refer to normalised soil reactions where
hi=
hi/h0,
mj=mj/m0,
v=v/v0,
q=q/q0 and hi,mj are the resultant lateral
and moment soil reactions along some axes i,j.
α
hm
′
is dened as the angle
between the hi axis and the m
′
j axis (which is clockwise orthogonal to the
mj axis), as shown in Fig. 4c.
α
hm
′
can be calculated as,
α
hm
′
=cos−1hi⋅m
′
j
him
′
j.(1)
where hi is the resultant lateral soil reaction vector and m
′
j is a vector that
is clockwise orthogonal to the resultant moment soil reaction vector mj.
3. Results
The results of the 3D FEA study are mainly shown for a representa-
tive caisson of L/D=0.5, but the ndings are similar for caissons of
other L/D ratios. Due to symmetry, the failure envelope results were
obtained for only two quadrants.
3.1. Local soil resistance on caisson at capacity
Fig. 5 shows the normalised load–displacement curves for the skirt
soil reactions at various depths, as well as the base soil reaction, under
various uniaxial loading conditions applied at the LRP of a caisson of L/
D=0.5, where Uy,Uz,Θx,Θz are the caisson lateral displacement along
y-axis, vertical displacement, rotation about x-axis and torsional
displacement. It can be seen that the ultimate values of the skirt soil
reactions are approximately the same for all depths.
Fig. 6a illustrates the evolution of the lateral and moment skirt soil
reactions at different depths as a caisson of L/D=0.5 remains at ca-
pacity, while the global load state ‘travels’ along the caisson failure
envelope during a sequential swipe test in the HM load space for planar
HM loading (i.e.
α
HM
′
=0),
V=0,
Q=0. It can be observed that the
interactions between the lateral and moment skirt reactions are broadly
similar, albeit with different interaction paths at different depths. Fig. 6a
also shows the interaction between the average lateral and moment skirt
reactions to give an indication of the general interaction path. For better
context, Fig. 6b shows the interaction paths taken by the H and M loads
and the corresponding skirt (average) and base soil reactions during the
sequential swipe test. The distinct interaction paths of the global loads,
base soil reactions, and skirt soil reactions at caisson capacity are
evident. In this paper, the interaction paths of the skirt (average) and
base soil reactions are referred to as ’skirt failure envelopes’ and ’base
failure envelopes’ respectively, for brevity.
Fig. 7 shows the variation of the skirt and base failure envelopes with
increasing L/D during sequential swipe tests in the HM load space for
planar HM loading,
V=0 and
Q=0. It can be observed that the hm skirt
failure envelopes get more angular in shape as L/D increases, but the
shapes remain quite similar (all parallelogram-like). The hm base failure
envelopes for L/D>0 are largely the same, but are much more rounded
in shape than that for L/D=0.
3.2. Uniaxial capacities of soil reactions
Fig. 8 shows the uniaxial capacities of the skirt and base soil re-
actions for different L/D ratios. While the uniaxial capacities for the
vertical and torsional soil reactions can be easily determined from the
ultimate limiting values of the soil reactions under uniaxial loading (e.g.,
the limiting values from Fig. 5a and 5b), the calculations of the uniaxial
capacities for the lateral and moment soil reactions are much more
involved. This is because under uniaxial lateral loading (e.g., Fig. 5c),
the moment soil reactions are not zero at the ultimate limiting values of
the lateral soil reaction. Thus, the uniaxial capacities for the lateral and
moment soil reactions are approximated as the intersection of the hm
Fig. 4. (a) Conventions for prescribed displacements. (b) Conventions adopted
for general HM loading. LRP is the loading reference point of the caisson. (c)
Conventions adopted for general hm soil reactions pointing in different relative
directions. RC is the centre (and reference point) of a cross-section along the
caisson skirt or at the caisson base.
S.K. Suryasentana et al.
Computers and Geotechnics 169 (2024) 106241
6
failure envelope (e.g., Fig. 6a) with the h and m axes, respectively.
Fig. 8a shows a clear trend: as L/D increases, the lateral capacity of
the caisson skirt soil reactions increases, while the vertical, moment, and
torsional capacities remain relatively constant. In Fig. 8b, we observe
that the vertical capacity of the caisson base soil reactions increases with
L/D until it reaches a plateau after L/D>1. On the other hand, the
lateral, moment, and torsional capacities exhibit little change. These
ndings suggest that as L/D increases, the contribution of the skirt soil
reactions to the caisson capacity becomes more signicant compared to
the base soil reactions. The following functions were derived to
approximate the uniaxial capacities of the skirt and base soil reactions:
vskirt
0
Askirtsu=1(2)
hskirt
0
Askirtsu=1.73 +1.11(1−exp( − 0.75L/D)) (3)
mskirt
0
AskirtDsu=0.337 −0.171(1−exp( − 1.32L/D)) (4)
qskirt
0
AskirtDsu=0.5(5)
vbase
0
Abasesu=5.63 +3.8(1−exp( − 2.19L/D)) (6)
hbase
0
Abasesu=1+0.41(1−exp( − 2.56L/D)) (7)
Fig. 5. Normalised load–displacement curves of skirt soil reactions at various depths and the base soil reaction under different uniaxial loading applied at the LRP of
a caisson of L/D=0.5 (a) Vertical soil reactions under purely vertical loading (b) Torsional soil reactions under pure torsion (c) Lateral soil reactions under purely
lateral loading (d) Moment soil reactions under purely moment loading. Note that A
base =
π
4D2 and Askirt =
π
D.
S.K. Suryasentana et al.
Computers and Geotechnics 169 (2024) 106241
7
mbase
0
AbaseDsu=0.73 (8)
qbase
0
AbaseDsu=1
3(9)
where Abase =
π
D2/4 is the area of the caisson base and Askirt =
π
D is the
external surface area of the caisson skirt per metre skirt length. The
predictions of these approximating functions are also plotted in Fig. 8;
evidently, they agree well with the 3D FEA calculations.
3.3. qvhmskirt and base failure envelopes
Figs. 9a and 10a show the effect of vertical loading on the hm con-
tours of the skirt and base failure envelopes for a caisson of L/D=0.5,
for planar HM loading and
Q=0. The average
v of all the data points in
each hm contour (corresponding to a xed
V) were calculated and that
value is shown in the bottom left of each subgure in Figs. 9 and 10, in
the order of the outermost contour to the innermost. For example, in
Fig. 9a,
V=0,0.25,0.5,0.75 corresponds to average
v=0,0.06,0.24,
0.59 for the skirt failure envelopes. This means that when a vertical load
amounting to 75% of the caisson vertical capacity is applied to the
caisson during a sequential swipe test, approximately 59% of the skirt
soil reaction capacities were mobilised during the test. It is evident from
these gures that as V loading increases, the hm capacity for both the
skirt and base soil reactions decreases, as depicted by the smaller con-
tours. It can be observed that the drop in hm capacity is minor for
V≤
0.5 but increases rapidly for
V>0.5.
Figs. 9 and 10 also show the effect of non-planar HM loading on the
skirt and base failure envelopes for a caisson of L/D=0.5 and
Q=0.
Fig. 6. For a sequential swipe test in the HM load space for
α
HM
′
=0,
V=0,
Q=0, for a caisson of L/D=0.5: (a) Interactions between lateral and moment
skirt soil reactions at various depths (b) Comparison of the caisson failure en-
velope with the corresponding interactions for the skirt (average) and base soil
reactions during the swipe test. Note that Abase =
π
4D2 and Askirt =
π
D.
(a)
(b)
Fig. 7. (a) Skirt failure envelopes, and (b) base failure envelopes for caissons of
various L/D ratios for
α
HM
′
=0,
V=0,
Q=0. The black dashed lines are the
predictions of Eqs. (20) and (21).
S.K. Suryasentana et al.
Computers and Geotechnics 169 (2024) 106241
8
There are much less data points for
α
hm
′
>0, as the results from
α
SΘ=0,
π
/4,
π
/2 are scattered around 0 ≤
α
hm
′
≤
π
/2. The observable trend is
that the asymmetry in the hm space decreases as
α
hm
′
increases from 0.
Although there are few data points for
α
hm
′
=
π
/2, it can be shown from
theoretical reasoning (see Suryasentana et al., 2021 for details) that
there should not be any asymmetry at
α
hm
′
=
π
/2.
Fig. 11 shows the combined effects of torsional and vertical loading
on the skirt and base failure envelopes for a caisson of L/D=0.5 under
planar HM loading. It is evident that as torsional loading increases, the
available hm capacity for both the skirt and base soil reactions decreases.
However, the shapes of the skirt and base failure envelopes remain the
same as torsional loading increases. The average
q associated with all
the data points are shown at the bottom left of each subgure in Fig. 11.
3.4. Elastoplastic Winkler model
The above insights into the behaviour of the soil reactions are used to
develop a new simplied design model, termed ’OxCaisson-LEPP’, that
can predict the caisson capacity in undrained clay under 6DoF loading.
OxCaisson-LEPP is an elastoplastic Winkler model that extends upon the
OxCaisson model described in Suryasentana et al. (2022) to allow for
linear elastic perfectly plastic (LEPP) behaviour of the soil reactions.
The elastic behaviour of the Winkler model is dened using the linear
elastic soil reaction formulations described in previous works (Sur-
yasentana et al., 2022,2023a; Suryasentana and Mayne, 2022). The
proposed model combines the elastic soil reactions with plastic yield
surfaces. These yield surfaces are determined by new formulations
derived from the 3D FEA study, which approximates the skirt and base
failure envelopes. The motivation behind adopting the Winkler model-
ling approach is based on the hypothesis that these soil reaction failure
envelopes, calibrated at a local level, can also be applied to soil proles
different from the one used for calibration. If this hypothesis holds true,
it would signicantly enhance the applicability of the proposed model.
The validity of the hypothesis will be assessed in a case study later in the
paper.
The mechanics of the LEPP soil reactions can be explained by stan-
dard plasticity theory. For soil reaction states lying inside the yield
surface, the soil response is linear elastic with the incremental response
given by,
δb=keδu(10)
where b =soil reactions vector, ke =elastic stiffness matrix for the soil
reactions and u =local displacements vector corresponding to the soil
reactions. When the soil reaction states reach the yield surface, the soil
response becomes elastoplastic, with incremental behaviour given by,
δb=kepδu.(11)
where kep =elastoplastic stiffness matrix for the soil reactions. When
elastoplastic yielding occurs, permanent plastic displacements accu-
mulate with the total displacement increment δu composed of elastic
and plastic parts,
δu=δue+δup(12)
The elastic displacement increment δue is determined through the
soil reaction increment,
δue=ke−1δb(13)
The plastic displacement increment δup is determined using the ow
rule,
δup=λ
∂
g
∂
b(14)
where g(b)is a plastic potential function and λ is a non-negative, scalar
plastic multiplier. When yielding occurs, the incremental soil reaction δb
must remain on the yield surface. This is enforced by the consistency
condition,
∂
f
∂
bT
δb=0(15)
Finally, kep is obtained from,
Fig. 8. Variation of the normalised uniaxial capacities of the skirt and base soil
reactions with L/D (a) Uniaxial capacities of the skirt soil reactions (b) Uniaxial
capacities of the base soil reactions. The dotted lines are the uniaxial capacities
predicted by Eqs. (2)–(9).
S.K. Suryasentana et al.
Computers and Geotechnics 169 (2024) 106241
9
kep =ke−
ke
∂
g
∂
b
∂
f
∂
bT
ke
∂
f
∂
bT
ke
∂
g
∂
b(16)
For this paper, an associated ow rule is adopted i.e. g(b) = f(b).
OxCaisson-LEPP is implemented using a one-dimensional (1D) nite
element framework, as described in Suryasentana et al. (2022). For the
current study, the caisson skirt is discretised using twenty 1D skirt ele-
ments. Each skirt element is connected to a corresponding 1D soil re-
action element of identical length, which represents the distributed soil
reactions acting along the caisson skirt. The integration points of the soil
reaction element directly use the value of the soil properties (e.g. su) at
the depths of the integration points.
3.5. Soil reaction yield surface formulations
To derive the formulations that best match the skirt and base failure
envelopes determined from the 3D FEA study, the SOS-convex poly-
nomial framework (Suryasentana et al., 2020a,2021) is used. This
framework provides an automated approach for deriving globally
convex and numerically well-behaved formulations that t failure en-
velope data for different foundation types and for 6DoF loading. How-
ever, the framework detailed in Suryasentana et al. (2020a, 2021) is
concerned with deriving formulations for xed failure envelopes that do
not vary with foundation dimensions such as L/D ratio. The current
paper extends the previous work by describing an approach for deriving
6DoF failure envelope formulations that vary with L/D and are still
guaranteed to be globally convex and numerically well-behaved.
Following the procedures in Suryasentana et al. (2021), a 4th degree
homogeneous SOS-convex polynomial p
α
hm
′
,
hi,
mj,
v,
qis sought to
represent the soil reaction failure envelope f of the following form,
f
α
hm
′
,
hi,mj,v,q=p
α
hm
′
,
hi,mj,v,q−1=0(17)
The current paper seeks to determine a soil reaction failure envelope
formulation that varies with the caisson L/D ratio i.e.
fL
D,
α
hm
′
,
hi,
mj,
v,
q. This is not as straightforward as adding an addi-
tional input variable to the SOS-convex polynomial, i.e. replace p
α
hm
′
,
hi,
mj,
v,
qwith pL
D,
α
hm
′
,
hi,
mj,
v,
qin Eq. (17). This is because global
convexity is only required in the load space and not necessarily in the L/
D domain. Therefore, the current paper adopts a formulation based on a
convex combination of two SOS-convex polynomials (p1 and p2),
fLDL
D,
α
hm
′
,
hi,mj,v,q=wp1+(1−w)p2−1=0(18)
where 0 ≤w≤1 is some weight parameter that is a function of L/D, and
p1 and p2 are SOS-convex polynomials that are functions of
α
hm
′
,
hi,
mj,
v,
q. Since w and (1−w)are both non-negative, wp1+(1−w)p2 is convex
as it is known that a non-negative weighted sum of convex functions is
Fig. 9. hmContours of skirt failure envelopes for a caisson of L/D=0.5 and for
α
hm
′
=0,
π
8,
π
4,
π
2,
V=0,0.25,0.5,0.75,
Q=0. The black dashed lines are the skirt failure
envelopes predicted by Eq. (20), while the data points are the ultimate skirt soil reactions calculated from 3D FEA. The average
q and
v of the data points are shown in
the bottom left of each subgure.
q corresponds to all the data points, while
v corresponds to the data points in each contour.
S.K. Suryasentana et al.
Computers and Geotechnics 169 (2024) 106241
10
itself a convex function (Boyd and Vandenberghe, 2004). Thus, Eq. (18)
is guaranteed to be globally convex as long as 0 ≤w≤1. Further details
are provided in Suryasentana and Houlsby (2022). Through an analysis
of the variation of the skirt and base failure envelopes with respect to
L/D, wskirt =exp−aL
D2and wbase =exp−bL
Dwas adopted as the
weight parameter functions for the skirt and base failure envelopes
respectively, where a and b are unknown coefcients to be determined
later. Based on these weight parameters, p1 in Eq. (18) would represent
the skirt and base failure envelopes for L/D=0, while p2 would
represent the skirt and base failure envelopes as L/D approaches innity.
To determine the unknown coefcients for the weight parameters
and polynomials in Eq. (18), the skirt and base failure envelope data
from the 3D FEA calculations are rst transformed from the space of hx,
my,hy,mx,v,qto the standardised space of
α
hm
′
,
hi,
mj,
v,
q. The
standardised data is used to determine the unknown coefcients in Eq.
(18) by solving the following convex optimisation problem, which is
based on the conditions: (i) p1 and p2 are both SOS-convex, and (ii) the
convex combination of p1 and p2 provide a best t with the standardised
data in a ‘least-squares’ sense,
minimize
ndata
i=1wp1xdata
i+ (1−w)p2xdata
i−12(19)
subject to p1andp2are bothSOS-convex
where xdata
i=
α
hm
′
,
hi,
mj,
v,
qis a set of standardised data and ndata is
the total number of data points. The MATLAB toolbox ‘YALMIP’
(L¨
ofberg, 2004, 2009) was employed to solve Eq. (19) to determine the
unknown coefcients.
3.6. Skirt and base failure envelope formulations
The skirt and base failure envelope formulations are referred to as
fskirt and fbase , respectively. They have the following functional forms
based on Eq. (18),
fskirtL
D,
α
hm
′
,hi,mj,v,q=wskirtpskirt
1+1−wskirtpskirt
2−1=0(20)
Fig. 10. hmContours of base failure envelopes for a caisson of L/D=0.5 and for
α
hm
′
=0,
π
8,
π
4,
π
2,
V=0,0.25,0.5,0.75,
Q=0. The black dashed lines are the base
failure envelopes predicted by Eq. (21), while the data points are the ultimate base soil reactions calculated from 3D FEA. The average
q and
v of the data points are
shown in the bottom left of each subgure.
q corresponds to all the data points, while
v corresponds to the data points in each contour.
S.K. Suryasentana et al.
Computers and Geotechnics 169 (2024) 106241
11
fbaseL
D,
α
hm
′
,hi,mj,v,q=wbasepbase
1+1−wbasepbase
2−1=0(21)
pskirt
1,pskirt
2,pbase
1,pbase
2 all have the following functional form (but with
different polynomial coefcients),
p
α
hm
′
,
hi,mj,v,q=
hi
4+mj
4+v4+q4+Ihm +Ivh +Ivm +Iqh +Iqm +Iqv
+Ivhm +Iqhm
(22)
where
Ihm =a1
hi
2
himjcos
α
hm
′
+a2
himjcos
α
hm
′
2
+a3mj
2
himjcos
α
hm
′
Ivh =a4
hi
2v2
Ivhm =a5v2
himjcos
α
hm
′
Ivm =a6mj
2v2
Iqh =a7
hi
2q2
Iqhm =a8q2
himjcos
α
hm
′
Iqm =a9mj
2q2
Iqv =a10v2q2
pbase
1 in Eq. (22) is the SOS-convex polynomial component of the
failure envelope of a surface foundation, which is known from previous
work (Suryasentana et al., 2021) and the coefcients of pbase
1 are listed in
Table 3. By solving the convex optimisation problem dened in Eq. (19),
the unknown coefcients a0,⋯,a10 in the interaction terms for pskirt
1,pskirt
2,
pbase
2 are determined and their values are listed in Tables 2 and 3 for the
skirt and base failure envelopes, respectively. Furthermore, the opti-
mised weight parameters are wskirt =exp−2L
D2and wbase =
exp−10 L
D.
The predictions of Eqs. (20) and (21) (using the optimised coefcient
values) are included in Figs. 7, 9, 10 and 11 for comparison with the 3D
FEA calculations. In general, the predictions of Eqs. (20) and (21) agree
reasonably well with the 3D FEA calculations, especially the salient
features such as the size and shape of the local failure envelopes.
Following the procedures detailed in Suryasentana et al. (2021), Eqs.
(20) and (21) can be redened in terms of the 6DoF soil reactions
hx,my,hy,mx,v,q. This is achieved by making the following re-
placements in Eq. (22): (i)
hi replaced by
h2
x+
h2
y
, (ii)
mj replaced by
m2
x+
m2
y
, and (iii)
hi
mjcos
α
hm
′
replaced by
hy
mx−
hx
my.
Fig. 11. hmContours of (a) and (b) skirt failure envelopes, and (c) and (d) base failure envelopes for a caisson of L/D=0.5 and for
α
hm
′
=0,
V=0,0.25,0.5,0.75,
Q=0.5,0.75. The black dashed lines are the skirt and base failure envelopes predicted by Eqs. (20) and (21) respectively, while the data points are the ultimate soil
reactions calculated from 3D FEA.
S.K. Suryasentana et al.
Computers and Geotechnics 169 (2024) 106241
12
Table 2
Best-t coefcients of the SOS-convex polynomials that make up the skirt failure envelope fskirt in Eq. (22), as determined by the optimisation process in Eq. (6).
p a1 a2 a3 a4 a5 a6 a7 a8 a9 a10
pskirt
1 −0.93 0.65 −0.87 1.58 −2.42 5.66 0.3 −0.54 1.36 2.22
pskirt
2 −1.36 1.71 −1.95 1.03 −4.06 5.17 0.2 −0.94 1.5 2.85
Table 3
Best-t coefcients of the SOS-convex polynomials that make up the base failure envelope fbase in Eq. (23), as determined by the optimisation process in Eq. (6).
p a1 a2 a3 a4 a5 a6 a7 a8 a9 a10
pbase
1 −0.36 0.9 −1.43 0.4 0.84 1.64 2.61 −0.84 0.34 0
pbase
2 −0.79 2.73 −1.13 0.88 0.31 0.88 2.55 −0.11 0.59 0
Fig. 12. Comparison of a sequential swipe test in the HM load space for a suction caisson of L/D=0.5 and for
α
HM
′
=0,
V=0.25,
Q=0. (a)-(c) Load-displacement
behaviour (d) HM failure envelope.
S.K. Suryasentana et al.
Computers and Geotechnics 169 (2024) 106241
13
3.7. Validation of OxCaisson-LEPP
Eqs. (20) and (21) are used as the yield functions and plastic po-
tentials for the soil reactions in OxCaisson-LEPP. These formulations
require the uniaxial capacities of the skirt and base soil reactions, which
are dened in Eqs. (2)–(9). Fig. 12 compares the OxCaisson-LEPP pre-
dictions of the global load–displacement behaviour and HM failure en-
velopes for a caisson of L/D=0.5 with the 3D FEA results. In this gure,
a sequential swipe test in the HM space under planar HM loading was
carried out, with
V=0.25 and
Q=0. Evidently, there is excellent
agreement between the OxCaisson-LEPP predictions and the 3D FEA
results. For further insights into the prediction of the HM failure enve-
lope by OxCaisson-LEPP, Fig. 13 shows the history of the H and M loads
and the corresponding h and m soil reactions. As the HM failure envelope
is being explored, the skirt and base soil reactions travel along their
respective yield surfaces. The predictions of OxCaisson-LEPP in Fig. 13 is
very similar to that of the 3D FEA model in Fig. 6b. One of the main
advantages of OxCaisson-LEPP is its efciency. While the 3D FEA model
took about 2 h to generate the data points in Fig. 12, OxCaisson-LEPP
took about 2 min. This efciency enables computationally intensive
applications such as automated optimisation of suction caisson foun-
dations (Suryasentana et al., 2019; Suryasentana et al., 2018).
To validate the ability of OxCaisson-LEPP to estimate the caisson
capacity under non-planar HM loading and combined vertical and
torsional loading, Fig. 14 shows the normalised Hx-Mx and Hy-Mx failure
envelopes resulting from a sequential swipe test under non-planar HM
loading with
V=0.5 and
Q=0.25. It is evident that OxCaisson-LEPP is
able to reproduce the 3D FEA results well.
3.8. Cowden till case study
To evaluate the hypothesis that OxCaisson-LEPP can be applied to a
soil prole that differs from that used for calibration, OxCaisson-LEPP is
used to estimate the planar HM failure envelope of a suction caisson of D
=10 m and L=12 m, based on a real-world Cowden till undrained
shear strength su prole (Zdravkovi´
c et al., 2020) that varies irregularly
with depth (see Fig. 15a). The OxCaisson-LEPP estimations will be
compared with the calculations made by 3D FEA. Additionally, it would
be informative to compare the OxCaisson-LEPP estimations with those
by existing macro-element failure envelope models.
However, it is quite challenging to nd a suitable macro-element
failure envelope model that can be applied to this routine HM failure
envelope estimation task. The model proposed by Vulpe (2015) cannot
be used as it has only been calibrated for a nite number of L/D ratios
(L/D=0,0.1,0.25,0.5). The model proposed by Liu et al. (2023) is
calibrated only for undrained shear strength proles in the form of su=
kz, which cannot t well with the data presented in Fig. 15a. Given these
limitations, the most compatible among the existing models, proposed
by Hung and Kim (2014), is used to estimate the HM failure envelope. To
use this model, a representative design su prole has to be rst tted to
the irregular prole based on the parametric form su=sum +kz, which
was used to calibrate the model; this process is not straightforward for
the prole in Fig. 15a and can be very subjective, introducing additional
uncertainty to the design outcomes.
A design prole (su=85 +5.5z kPa) was adopted to capture the
major trend of the su prole, as shown in Fig. 15a. Based on this design
prole, the caisson failure envelope estimated by Hung and Kim (2014)
is,
f
H,
M,
V=
M2+
α
βλ
H
M+
H2+
V2−1=0(23)
Fig. 13. Sequential swipe load history of the OxCaisson-LEPP predictions in the
global HM and local hm spaces for a caisson of L/D =0.5, for
α
HM
′
=0,
V=
0.25,
Q=0. The grey dashed lines are the hm contours of the soil reaction
failure envelopes predicted by Eqs. (20) and (21).
Fig. 14. Normalised H
y-Mx and Hx-Mx failure envelopes computed for a
sequential swipe test on a caisson of L/D=0.25 under non-planar HM loading
for
α
SΘ=
π
4,
V=0.5,
Q=0.25.
S.K. Suryasentana et al.
Computers and Geotechnics 169 (2024) 106241
14
where
α
=0.323,β=5 and λ=0.94.
Fig. 15b compares the HM failure envelopes calculated by
OxCaisson-LEPP, Hung and Kim (2014) and 3D FEA. The OxCaisson-
LEPP calculations agrees reasonably well with the 3D FEA calcula-
tions, with some underestimation of the capacity, especially towards the
extremes of the H and M loads. In comparison to the model by Hung and
Kim (2014), the OxCaisson-LEPP calculations exhibit a closer match to
the 3D FEA calculations; this is despite the Hung and Kim (2014) model
being calibrated using a linearly increasing su prole that is more similar
to Fig. 15a than the uniform su prole that was used to calibrate
OxCaisson-LEPP.
4. Discussion
The current paper addresses two key gaps in understanding and
modelling the ultimate behaviour of suction caissons in undrained clay.
First, it examines the local soil response on caissons at capacity under
combined 6DoF loading; the interaction between the various soil reac-
tion components (e.g. interaction between horizontal and moment soil
reactions) at capacity is identied. It is shown that the uniaxial capacity
of the base soil reactions stops increasing after L/D>1; this indicates
that the relative importance of the base soil reactions diminishes as L/D
increases. Second, the paper proposes an elastoplastic Winkler model for
estimating the failure envelopes for caissons under combined 6DoF
loading. Previous research in this area is limited, particularly in
connection with the estimation of failure envelopes for non-planar HM
loading which is important for offshore wind applications due to the
different directions of wind and wave actions.
Despite the simplifying assumption of independent skirt soil re-
actions (i.e., the skirt soil reactions at one depth do not interact with the
skirt soil reactions at other depths), the Winkler model offers a more
versatile framework than traditional macro-element models. Macro-
element models are typically applicable only for the specic soil pro-
les that they were calibrated for; this limits their usefulness for general
design applications. This is evident in the Cowden till case study pre-
sented in the paper, where only one of the reviewed macro-element
models (Hung and Kim, 2014) appears to be applicable; in all other
cases the Cowden till soil prole differs signicantly from the soil pro-
les used to calibrate the macro-element models.
The Winkler model employed in the current work is capable of
conducting design calculations for actual design soil proles; this con-
trasts with macro-element models for which idealised soil proles – to
approximate actual site conditions - need to be adopted. Macro-element
models therefore require an additional, subjective, step of determining a
representative soil prole from the actual site data; this adds uncertainty
to the design outcomes. Therefore, a key advantage of Winkler-based
models such as OxCaisson-LEPP is that it can be used directly for sites
with irregular undrained shear strength proles.
The success of Winkler modelling for caissons has been noted in
other studies, such as those by Antoniou et al. (2022). The efcacy of
Winkler modelling for low L/D caisson-like structures is also demon-
strated by the PISA design model (Burd et al., 2020), which is applicable
to piles with L/D=2; this is close to typical aspect ratios for caissons.
This paper focuses on estimating the failure envelopes for caissons in
undrained clay; this is crucial for assessing the ultimate limit state
conditions for foundations, especially in the context of caisson-based
anchoring systems. However, there are other factors such as founda-
tion stiffness and cyclic behaviour that could signicantly inuence the
design of a caisson in actual design cases. The issue of foundation stiff-
ness has been explored in previous studies (e.g. Suryasentana et al.,
2022,2023a,2023b). The issue of cyclic behaviour under combined
loading, however, is complex and is planned for future research.
5. Conclusions
This paper carries out an extensive 3D FEA study to investigate the
ultimate response of the local soil reactions acting on a suction caisson in
undrained clay under combined 6DoF loading. The 3D FEA results show
the interaction paths of the skirt and base soil reactions at capacity,
which approximate soil reaction failure envelopes. SOS-convex poly-
nomial-based formulations are derived to approximate these soil reac-
tion failure envelopes. A new elastoplastic Winkler model called
OxCaisson-LEPP was developed, with the derived skirt and base fail-
ure envelope formulations acting as the yield functions and plastic
Fig. 15. (a) Undrained shear strength prole of Cowden Till site (b) Compar-
ison of the OxCaisson-LEPP and 3D FEA predictions for the normalised planar
HM failure envelope of a caisson of D=10 m and L/D=1.2, where p
atm is the
atmospheric pressure.
S.K. Suryasentana et al.
Computers and Geotechnics 169 (2024) 106241
15
potential for the soil reactions of the model. The results show that
OxCaisson-LEPP can accurately reproduce the 3D FEA results with high
efciency, even for soil proles that differ from that used to calibrate the
model.
CRediT authorship contribution statement
Stephen K. Suryasentana: Conceptualization, Data curation,
Formal analysis, Investigation, Methodology, Software, Writing – orig-
inal draft. Harvey J. Burd: . Byron W. Byrne: . Avi Shonberg: Meth-
odology, Supervision, Writing – review & editing.
Declaration of competing interest
The authors declare that they have no known competing nancial
interests or personal relationships that could have appeared to inuence
the work reported in this paper.
Data availability
Data will be made available on request.
Acknowledgments
Part of this work was completed during the DPhil studies of the rst
author and he would like to acknowledge the generous support of Ørsted
Wind Power for funding his DPhil studentship at the University of Ox-
ford. Byrne is supported by the Royal Academy of Engineering under the
Research Chairs and Senior Research Fellowships scheme.
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