Research on Offshore Foundations:
Papers at the International Symposium on Frontiers in
Offshore Geotechnics
Perth, Australia, 2005
by
G.T. Houlsby, C.M. Martin, B.W. Byrne, R.B. Kelly
E.J. Hazell, L. Nguyen-Sy, F.A. Villalobos and L-B. Ibsen
Report No. OUEL 2275/05
University of Oxford
Department of Engineering Science
Parks Road, Oxford, OX1 3PJ, U.K.
Tel. 01865 273162/283300
Fax. 01865 283301
Email [email protected]
http://www-civil.eng.ox.ac.uk/
Research on Offshore Foundations:
Papers at the International Symposium on Frontiers in Offshore Geotechnics
Perth, Australia, 2005
G.T. Houlsby, C.M. Martin, B.W. Byrne, R.B. Kelly
E.J. Hazell, L. Nguyen-Sy, F.A. Villalobos and L-B. Ibsen
This report consists of six papers that have been accepted for the International Symposium on
Frontiers in Offshore Geotechnics at Perth Australia in September 2005. The abstracts of the six
papers are:
a) Keynote Paper : “Suction caissons for wind turbines.”
Authors: Houlsby, G.T., Ibsen, L-B. and Byrne, B.W.
Abstract: Suction caissons may be used in the future as the foundations for offshore wind turbines.
We review recent research work on the development of design methods for suction caissons for
these applications. We give some attention to installation, but concentrate on design for in-service
performance. Whilst much can be learned from previous offshore experience, the wind turbine
problem poses the particularly challenging combination of a relatively light structure, with large
imposed horizontal forces and overturning moments. Monopod or tripod/tetrapod foundations result
in very different loading regimes on the foundations, and we consider both cases. The results of
laboratory studies and field trials are reported. We also outline briefly numerical and theoretical
work that is relevant. Extensive references are given to sources of further information.
b) “Bearing capacity of parallel strip footings on non-homogeneous clay.”
Authors: Martin, C.M. and Hazell, E.J.
Abstract: On soft seabed soils, subsea equipment installations are often supported by mudmat
foundation systems that can be idealised as parallel strip footings, grillages, or annular (ringshaped) footings. This paper presents some theoretical results for the bearing capacity of (a) two
parallel strip footings, otherwise isolated; (b) a long series of parallel strip footings at equal
spacings. The soil is idealised as an isotropic Tresca material possessing a linear increase of
undrained strength with depth. The bearing capacity analyses are performed using the method of
characteristics, and the trends of these (possibly exact) results are verified by a companion series of
upper bound calculations based on simple mechanisms. Parameters of interest are the footing
spacing, the relative rate of increase of strength with depth, and the footing roughness. An
application of the results to the design of perforated mudmats is discussed.
c) “Investigating six degree-of-freedom loading on shallow foundations.”
Authors: Byrne, B.W. and Houlsby, G.T.
Abstract: Previous laboratory studies of the response of shallow foundations have only considered
planar loading. This paper describes the development of a loading device capable of applying
general loading on model shallow foundations. Loading involving all six degrees of freedom
{vertical (V), horizontal (H2, H3), torsion (Q) and overturning moment (M2, M3)}, can be applied
experimentally to the model foundations. Aspects of the design, including the loading rig
configuration, development of a six degree-of-freedom load cell, numerical control algorithms and
an accurate displacement measuring system are described. Finally results from initial experiments
are presented that provide evidence for the generalisation of existing work-hardening plasticity
models from planar loading to the general loading condition.
d) “The tensile capacity of suction caissons in sand under rapid loading.”
Authors: Houlsby, G.T., Kelly, R.B. and Byrne, B.W.
Abstract: We develop here a simplified theory for predicting the capacity of a suction caisson in
sand, when it is subjected to rapid tensile loading. The capacity is found to be determined
principally by the rate of pullout (relative to the permeability of the sand), and by the ambient pore
pressure (which determines whether or not the water cavitates beneath the caisson). The calculation
procedure depends on first predicting the suction beneath the caisson lid, and then further
calculating the tensile load. The method is based on similar principles to a previously published
method for suction-assisted caisson installation (Houlsby and Byrne, 2005). In the analysis a
number of different cases are identified, and successful comparisons with experimental data are
achieved for cases in which the pore water either does or does not cavitate.
e) “Theoretical modelling of a suction caisson foundation using hyperplasticity theory.”
Authors: Nguyen-Sy, L. and Houlsby, G.T.
Abstract: A theoretical model for the analysis of suction caison foundations, based on a
thermodynamic framework (Houlsby and Puzrin, 2000) and the macro-element concept is
presented. The elastic-plastic response is first described in terms of a single-yield-surface model,
using a non-associated flow rule. To capture hysteresis phenomena, this model is then extended to a
multiple yield surface model. The installation of the caisson using suction is also analysed as part of
the theoretical model. Some preliminary numerical results are given as demonstrations of the
capabilities of the model..
f) “Moment loading of caissons installed in saturated sand.”
Authors: Villalobos, F.A., Byrne, B.W. and Houlsby, G.T.
Abstract: A series of moment capacity tests have been carried out at model scale, to investigate the
effects of different installation procedures on the response of suction caisson foundations in sand.
Two caissons of different diameters and wall thicknesses, but similar skirt length to diameter ratio,
have been tested in water-saturated dense sand. The caissons were installed either by pushing or by
using suction. It was found that the moment resistance depends on the method of installation.
Suction Caissons for Wind Turbines
Guy T. Houlsby1, Lars Bo Ibsen2 & Byron W. Byrne1
1: Department of Engineering Science, Oxford University, U.K.
2: Department of Civil Engineering, Aalborg University, Denmark
ABSTRACT: Suction caissons may be used in the future as the foundations for offshore wind turbines. We
review recent research on the development of design methods for suction caissons for these applications. We
give some attention to installation, but concentrate on design for in-service performance. Whilst much can be
learned from previous offshore experience, the wind turbine problem poses a particularly challenging
combination of a relatively light structure, with large imposed horizontal forces and overturning moments.
Monopod or tripod/tetrapod foundations result in very different loading regimes on the foundations, and we
consider both cases. The results of laboratory studies and field trials are reported. We also outline briefly
relevant numerical and theoretical work. Extensive references are given to sources of further information.
1. INTRODUCTION
The purpose of this paper is to review recent
research work on the design of suction caisson
foundations for offshore wind turbines. Most of the
relevant work has been conducted at, or in cooperation with, the universities of Oxford and
Aalborg, so we report here mainly the work of our
own research groups.
Suction caissons have been extensively used as
anchors, principally in clays, and have also been
used as foundations for a small number of offshore
platforms in the North Sea. They are currently being
considered as possible foundations for offshore wind
turbines. As discussed by Houlsby and Byrne (2000)
and by Byrne and Houlsby (2003), it is important to
realise that the loading regimes on offshore turbines
differ in several respects from those on structures
usually encountered in the offshore oil and gas
industry. Firstly the structures are likely to be
founded in much shallower water: 10m to 20m is
typical of the early developments, although deeper
water applications are already being planned.
Typically the structures are relatively light, with a
mass of say 600t (vertical deadload 6MN), but in
proportion to the vertical load the horizontal loads
and overturning moments are large. For instance the
horizontal load under extreme conditions may be
about 60% of the vertical load.
An important consideration is that, unlike the oil
and gas industry where large one-off structures
Figure 1: Offshore tests in Frederikshavn, Denmark. Front:
Vestas V90 3.0MW turbine. Back: Nordex 2.3MW turbine.
dominate, many relatively small and inexpensive
foundations are required for a wind farm
development, which might involve anything from 30
to 250 turbines.
The dominant device used for large scale wind
power generation is a horizontal axis, 3-bladed
turbine with the blades upwind of the tower, as
shown in Figure 1. The details of the generator,
rotational speed and blade pitch control vary
between designs. Most offshore turbines installed to
date generate 2MW rated power, and typically have
a rotor about 80m in diameter with a hub about 80m
above mean sea level. The size of turbines available
is increasing rapidly, and prototypes of 5MW
turbines already exist. These involve a rotor of about
128m diameter at a hub height of about 100m. The
loads on a “typical” 3.5MW turbine are shown in
Figure 2, which is intended to give no more than a
broad indication of the magnitude of the problem.
Figure 2: Typical loads on a 3.5MW offshore wind turbine
Note that in conditions as might be encountered
in the North Sea, the horizontal load from waves
(say 3MN) is significantly larger than that from the
wind (say 1MN). However, because the latter acts at
a much higher point (say 90m above the foundation)
it provides more of the overturning moment than the
wave loading, which may only act at say 10m above
the foundation. Using these figures the overturning
moment of 120MNm would divide as 90MNm due
to wind and 30MNm due to waves.
Realistic combinations of loads need to be
considered. For instance the maximum thrust on the
turbine occurs when it is generating at the maximum
allowable wind speed for generation (say 25m/s). At
higher wind speeds the blades will be feathered and
provide much less wind resistance. It is thus unlikely
that the maximum storm wave loading would occur
at the same time as maximum thrust. Turbine
designers must also consider important load cases
such as emergency braking. It is important to
recognise that the design of a turbine foundation is
not usually governed by considerations of ultimate
capacity, but is typically dominated by (a)
considerations of stiffness of the foundation and (b)
performance under fatigue loading.
An operational wind turbine is subjected to
harmonic excitation from the rotor. The rotor's
rotational frequency is the first excitation frequency
and is commonly referred to as 1P. The second
excitation frequency to consider is the blade passing
frequency, often called 3P (for a three-bladed wind
turbine) at three times the 1P frequency.
Figure 3 shows a representative frequency plot of
a selection of measured displacements for the Vestas
V90 3.0MW wind turbine in operational mode. The
foundation is a suction caisson. The measured data,
monitoring system and “Output-Only Modal
Analysis” used to establish the frequency plot are
described in Ibsen and Liingaard (2005). The first
mode of the structure is estimated, and corresponds
to the frequency observed from idling conditions.
The peak to the left of the first natural frequency is
the forced vibration from the rotor at 1P. To the right
of the first natural frequency is the 3P frequency. It
should be noted that the 1P and 3P frequencies in
general cover frequency bands and not just two
particular values, because the Vestas wind turbine is
a variable speed device.
To avoid resonances in the structure at the key
excitation frequencies (1P, 3P) the structural
designer needs to know the stiffness of the
foundation with some confidence, this means that
problems of deformation and stiffness are as
important as capacity. Furthermore, much of the
structural design is dictated by considerations of
high cycle fatigue (up to about 108 cycles), and the
foundation too must be designed for these
conditions.
2. CASES FOR STUDY
The two main problems that need to be studied in
design of a suction caisson as a foundation are:
• installation;
• in service performance.
In this review we shall discuss installation
methods briefly, but shall concentrate mainly on
design for in service performance. The relevant
studies involve techniques as diverse as laboratory
model testing, centrifuge model testing, field trials at
reduced scale, and a full-scale field installation.
Frequency Domain Decomposition - Peak Picking
Average of the Normalized Singular Values of
Spectral Density Matrices of all Data Sets.
dB | 1.0 / Hz
20
3P
1P
First mode
0
-20
-40
-60
-80
-100
1
Frequency
Figure 3. Frequency plot of measured displacements for a wind turbine in operational mode.
Complementing these experiments are numerical
studies using finite element techniques, and the
development of plasticity-based models to represent
the foundation behaviour.
Suction caissons may be installed in a variety of
soils, but we shall consider here two somewhat
idealised cases: a caisson installed either in clay,
which may be treated as undrained, or in sand. For
typical sands the combination of permeability value,
size of caisson and loading rates leads to partially
drained conditions, although much of the testing we
shall report is under fully drained conditions. In this
paper we report mainly work on sands.
We shall consider two significantly different
loading regimes, which depend on the nature of the
structure supporting the wind turbine. Most offshore
wind turbines to date have been supported on a
“monopile” – a single large diameter pile, which in
effect is a direct extension of the tubular steel tower
which supports the turbine. Some turbines have been
supported on circular gravity bases. An obvious
alternative is to use a single suction caisson to
support the turbine, and we shall call this a
“monopod” foundation, Figure 4(a). The monopod
resists the overturning moment (usually the most
important loading component) directly by its
rotational fixity in the seabed.
As turbines become larger, monopod designs may
become sufficiently large to be uneconomic, and an
alternative is a structure founded on three or four
smaller foundations: a “tripod” or “tetrapod”, Figure
4(b). In either of these configurations the
overturning moment on the structure is resisted
principally by “push-pull” action of opposing
vertical loads on the upwind and downwind
foundations. Alternatives using asymmetric designs
of tripod, and those employing “jacket” type
substructures are also under consideration.
(a)
(b)
Figure 4: caisson foundations for a wind turbine, (a) monopod,
(b) tripod/tetrapod
3. NORMALISATION PROCEDURES
A number of studies have been conducted at
different scales and it is necessary to compare the
results from these various studies. To do this it is
appropriate to normalise all the results so that they
can be represented in non-dimensional form. This
procedure also allows more confident extrapolation
to full scale.
The geometry of a caisson is shown in Figure 5.
The outside radius is R (diameter Do ), skirt length is
L and wall thickness t. In practice caissons may also
involve stiffeners on the inside of the caisson, these
being necessary to prevent buckling instability
during suction installation, but we ignore these in a
simplified analysis. Geometric similarity is achieved
by requiring similar values of L 2 R and t 2 R .
In sand it is straightforward to show that, for
similar values of dimensionless bearing capacity
factor, the loads at failure would be proportional to
γ ′ and to R 3 . We therefore normalise vertical and
horizontal loads as V 2πR 3 γ ′ and H 2πR 3 γ ′ ,
where we have included the factor 2π to give the
normalisation factor a simple physical meaning: it is
the effective weight of a cylinder of soil of the same
diameter of the caisson, and depth equal to the
diameter. In a similar way we normalise the
overturning moment as M 4πR 4 γ ′ .
Use of the above normalisation is appropriate for
comparing tests in sands with similar angles of
friction and dilation. We recognise that these angles
both decrease slightly with pressure and increase
rapidly with Relative Density (Bolton, 1986). This
means that comparable tests at smaller scales (and
therefore lower stress levels) will need to be at lower
Relative Densities to be comparable with field tests.
In clay the vertical capacity is proportional to a
representative undrained shear strength su and to
R 2 , so we normalise loads as V πR 2 su
Figure 5: Geometry of a caisson foundation
and
H πR 2 su , and the moment as M 2πR 3 su .
In order to be comparable, tests at different scales
will need the profile of undrained strength with
depth to be similar. If the strength profile is fitted by
a simple straight-line fit su = suo + ρz , then this
requires similar values of the factor 2 Rρ suo .
Scaling of results using the above methods should
give satisfactory results in terms of capacity. For
clays it should also lead to satisfactory comparisons
in terms of stiffness, provided that the clays being
compared have similar values of I r = G su . This
condition is usually satisfied if the clays are of
similar composition and overconsolidation ratio. For
sands, however, an extra consideration needs to be
taken into account. The shear modulus of a sand
does not increase in proportion to the stress level,
but instead can reasonably be expressed by:
Figure 6: Loading and displacement conventions for a caisson
foundation (displacements exaggerated).
The sign convention for applied loads and
displacements is shown in Figure 6.
The rotation of the caisson θ is already
dimensionless, and we normalise the displacements
simply by dividing by the caisson diameter, to give
w 2 R and u 2 R .
⎛ p′
G
= g ⎜⎜
pa
⎝ pa
⎞
⎟⎟
⎠
n
(1)
where g and n are dimensionless constants, and p a
is atmospheric pressure (used as a reference
pressure). The value of n is typically about 0.5, so
that the stiffness is proportional roughly to the
square root of pressure.
Comparing rotational stiffnesses on the basis of a
Table 1: Installations in shallow water
4
plot of M 4πR γ ′ against θ effectively makes the
assumption that the shear stiffness is proportional to
2 Rγ ′ , which may be regarded as a representative
stress level. Since in fact the stiffness increases at a
lower rate with stress level, this comparison will
result in larger scale tests giving lower apparent
normalised stiffness. This effect can be reduced by
multiplying the θ scale by the dimensionless factor
( p a 2 Rγ ′)1− n ,
which compensates for the stiffness
variation with stress level.
Thus we recommend that to compare both
stiffness and capacity data for sands one should plot
M 4πR 4 γ ′
against
θ( p a 2 Rγ ′)0.5
(assuming
Site
Soil
hw
D
L
Ref.
(m) (m) (m)
Installation
April 2005
Sand 1.0 12.0 6.0 30
2.0 2.0
Sand 0.2
4.0 4.0
Sand 0.5 4.0 2.5 23
Sand 2.0 2.0 2.0 23
Sand 0.5 2.0 2.0 3.0 1.5
Sand 0.2
27
1.5 1.0
3.0 1.5
Clay 0.2
26
1.5 1.0
Wilhelmshaven Sand 6.0 16.0 15.0
Frederikshavn
Frederikshavn
Sandy Haven
Tenby
Burry Port
Luce Bay
Bothkennar
n = 0.5 ) for moment tests, and V 2πR 3 γ ′ against
(w 2 R )( p a
2 Rγ ′)0.5 for vertical loading tests. A
fuller description of these normalisation procedures
is given by Kelly et al. (2005a).
4. INSTALLATION STUDIES
The principal difference between installation of a
suction caisson for an offshore wind turbine and for
previous applications is that the turbines are likely to
be installed in much shallower water. There is a
popular misconception that suction caissons can only
be installed in deep water, where a very substantial
head difference can be established across the lid of
the caisson. In shallow water the net suction that can
be achieved is indeed much smaller (being limited
by the efficiency of the pumps, as the absolute
pressure approaches zero), but the suctions that can
be achieved are nevertheless sufficient for
installation in most circumstances. Only in stiff clays
is it likely that some possible caisson designs, which
might otherwise be suitable as far as in-service
conditions are concerned, could not be installed by
suction in shallow water.
In Table 1 we list the main instances where
caissons have been installed in shallow water, as
appropriate to wind turbine installations. The water
depths hw are approximate only. In addition to the
field tests listed, a large number of small scale model
tests of installation have been carried out at Oxford
University (on caissons of 0.1m to 0.4m diameter),
the University of Western Australia (UWA),
Aalborg and elsewhere.
The largest completed installation in shallow
water is that of a prototype suction caisson, shown in
Figure 7, installed in the offshore research test
facility in Frederikshavn, Denmark. The prototype
(a)
(b)
Figure 7: Installation of the prototype foundation at the test site
in Frederikshavn: (a) during installation, (b) at the end of
installation.
has a diameter of 12m and a skirt length of 6m. The
operational water depth is 4m, and as the site is in a
basin, no wave or ice loads are applied. As seen in
Figure 7 the suction caisson was installed in only 1m
of water in the basin. The steel construction has a
mass of approximately 140t, and the caisson was
placed in late October 2002. The installation period
was about 12 hours, with the soil penetration time
being 6 hours. A computer system was used to
control the inclination, suction pressure and
penetration rate. Det Norske Veritas (DNV) has
certified the design of the prototype in
Frederikshavn to B level. The Vestas V90 3.0MW
turbine was erected on the foundation in December
2002. The development of the design procedure for
the bucket foundation is described in Ibsen and
Brincker (2004). An even larger installation is
currently in progress at Wilhelmshaven, Denmark.
There are two main ways of predicting firstly the
self-weight penetration of the caisson and secondly
the suction required to achieve full installation. The
first method (Houlsby and Byrne, 2005a,b) involves
use of adaptations of pile capacity analysis, in which
the resistance to penetration is calculated as the sum
of an end bearing term on the rim and friction on the
inside and outside. In sands the seepage pattern set
up by the suction processes alters the effective stress
regime in a way that aids installation.
The calculation has been implemented in a
spreadsheet program SCIP. Figure 8 shows for
example a comparison between variation of
measured suction in a model test installation with tip
penetration of the caisson (Sanham, 2003), and the
SCIP calculation.
0
500
1000
Suction, s (Pa)
1500
2000
2500
3000
0
Penetration, h (mm)
50
The penetration resistance is calculated from the
following expression, which is based on calibration
against measured data:
d
Rd (d ) = K t (d ) Atip qt (d ) + Aout ∫ K out ( z ) f s ( z )dz +
0
d
Ain ∫ K in ( z ) f s ( z )dz
0
where qt is the corrected cone resistance and f s the
sleeve friction at depth z. K t is a coefficient relating
qt to the unit tip resistance on the rim. This
resistance is adjusted for the reduction due to the
applied suction by the expression:
⎛
∆u ⎞
K t = kt ⎜1 − rt
⎟
∆ucrit ⎠
⎝
βt
(4)
where kt is an empirical coefficient relating qt to the
tip resistance during static penetration of the caisson,
rt is the maximum reduction in tip resistance. ∆ucrit
is the critical suction resulting in the critical
hydraulic gradient icrit = 1 along the skirt. β t is an
empirical factor.
K out and K in are coefficients relating f s to the
unit skin friction on the outside and inside of the
skirt. The water flow along the skirt changes the skin
friction. For the inside skin friction the coefficient
reduces the skin friction when suction is applied,
whereas on the outside the skin friction is increased.
The coefficients are established as:
K out
100
⎛
∆u ⎞
= α out ⎜1 + rout
⎟
∆ucrit ⎠
⎝
150
200
(3)
βout
(5a,b)
SCIP Results
Experimental Result
⎛
∆u ⎞
K in = α in ⎜1 − rin
⎟
∆ucrit ⎠
⎝
250
Figure 8: Comparison of SCIP with model test
The other approach involves use of CPT data to
infer directly the resistance Rd to penetration of the
caisson. The required suction u req to penetrate the
caisson to depth d is calculated as:
∆ureq (d ) =
Rd (d ) − G '(d )
Asuc
(2)
where G '(d ) is the self-weight of the caisson at
penetration depth d (reduced for buoyancy), and
Asuc is the area inside the caisson, where the suction
is applied.
βin
Where α out and α in are empirical coefficients relating
f s to the unit skin friction during static penetration
of the caisson. rout and rin are the maximum changes
in skirt friction. β out and β in are empirical factors.
The required suction ureq to penetrate the
prototype in Frederikshavn was predicted using
equation (2). The result of the analysis is shown in
Figure 9. The lower line represents ureq calculated
from the CPT tests. The curved line represents the
limiting suction upip which would cause piping to
occur. umax is the theoretical maximum net suction,
limited by the possibility of cavitation within the
caisson, as the absolute pressure approaches zero, so
-3
0
5
10
3
Volume, (10 m )
15
20
25
30
35
0
Cell 1
Cell 2
Penetration, h (mm)
50
100
150
200
250
300
350
Seepage
Volume
Total
Volume
Volume
Displaced
400
Figure 11: Volumes pumped from 2-cell caisson in sand.
Figure 9: Suction required for installation at Frederikshavn
Figure 10: The limiting suction upip has been achieved and soil
failure by piping has occurred.
that u max = 100kPa above water level and increases
linearly with the water depth, as shown by Figure 9.
umax is used to calculate the accessible net suction,
which is limited by the efficiency of the pumps,
upump. As is seen, the suction in shallow water can be
limited either by the suction causing piping or by the
accessible net suction available from the pumps.
The suction upip causing piping has been studied
at the test site in Frederikshavn by installation tests
on 2x2m and 4x4m caissons. Figure 10 shows a
4x4m caisson where the limiting suction upip has
been achieved, and soil failure by piping has
occurred. The soil outside of the skirt is sucked into
the caisson and the penetration of the caisson cannot
proceed.
If a tripod or tetrapod structure is to be installed,
then levelling of the structure can be achieved by
separately controlling the suction in each of the
caissons. For a monopod structure, however, an
alternative strategy has to be adopted. Experience
suggests that for installation in either clay or sand,
the level of the caisson is rather sensitive to the
application of eccentric loads (moments), especially
in the early stages of installation. This offers one
possibility for controlling the level of the caisson: by
use of an eccentric load that can be adjusted in
position to keep the caisson level.
An alternative strategy, which has proven to be
highly successful for installation in sand, is to divide
the rim into sections and to control the pressures at
the skirt tip in each section individually. By applying
pressure over one segment of the caisson rim the
upward hydraulic gradient within the caisson can be
enhanced locally, thus encouraging additional
downward movement for that sector. By controlling
the pressures at a number of points the caisson may
be maintained level.
This method would not be applicable in clays.
One possibility, as yet untried at large scale, for
controlling level in clays would be to use a
segmented caisson in which the suctions in the
different
segments
could
be
controlled
independently.
Some preliminary small scale tests suggest that
this approach might be successful in sand too
(Coldicott, 2005). Figure 11 shows the volumes of
water pumped from the two halves of a 400mm
diameter caisson split by a diametral vertical wall.
About 60% of the water pumped represents the
volume displaced by the descending caisson, whilst
about 40% represents seepage beneath the caisson
rim. Figure 12 shows that during the installation the
suctions developed in the two halves were (as would
be expected in a uniform material) almost equal.
5. CAISSON PERFORMANCE: MONOPOD
A large number of tests have been devoted to
studying the performance of a caisson under moment
loading at relatively small vertical loads, as is
relevant to the wind turbine design. Some details of
the test programmes are given in Table 2.
0
1000
2000
Suction, s (Pa)
3000
4000
Table 2: Moment loading tests
5000
6000
0
Penetration, h (mm)
50
Cell 1
Cell 2
100
150
200
250
Site
Soil
Frederikshavn
Frederikshavn
Sandy Haven
Burry Port
Luce Bay
Sand
Sand
Sand
Sand
Sand
Oxford laboratory
Sand
Aalborg laboratory
Sand
Bothkennar
Clay
Oxford laboratory
Clay
UWA centrifuge
(100g)
Clay
300
350
400
Figure 12: Suctions required for installation of 2-cell caisson in
sand.
5.1 Sand: field tests
The largest test involves the instrumented Vestas
V90 3.0MW prototype turbine at Frederikshavn,
Denmark. The caisson is installed in a shallow 4m
depth lagoon next to the sea, and the turbine is fully
operational. The only significant difference between
this installation and an offshore one is that the
structure is not subjected to wave loading.
The test program involving the prototype (turbine
and caisson) is focusing on long-term deformations,
soil structure interaction, stiffness and fatigue. The
prototype has been equipped with:
• an online monitoring system that measures the
dynamic deformation modes of the foundation
and the wind turbine,
• a monitoring system that measures the long-time
deflection and rotation of the caisson
• a monitoring system that measures the pore
pressure along the inside of the skirt.
The online monitoring system that measures the
modes of deformation of the foundation and wind
turbine involves 15 accelerometers and a real-time
data-acquisition system. The accelerometers are
placed at three different levels in the turbine tower
and at one level in compartments inside the caisson
foundation. The positions are shown in Figure 13,
and the locations and measuring directions are
defined in Figure14.
Output-only Modal Analysis has been used to
analyze the structural behaviour of the wind turbine
during various operational conditions. The modal
analysis has shown highly damped mode shapes of
the foundation/wind turbine system, which the
present aero-elastic codes for wind turbine design
cannot model. Further studies are to be carried out
with respect to soil-structure interaction. A detailed
description of the measuring system and the OutputOnly Modal Analysis is given by Ibsen and
Liingaard (2005).
D
(m)
12.0
2.0
4.0
2.0
3.0
0.1
0.15
0.15
0.2
0.2
0.3
0.2
0.3
0.4
3.0
0.2
0.3
0.06
L
(m)
6.0
2.0
2.5
2.0
1.5
0.0-0.066
0.05
0.1
0.1
0.2
0.15
0.0 – 0.2
0.0 – 0.3
0.0 – 0.4
1.5
0.1
0.15
0.02
0.03
0.06
Ref.
21
27
2,4
2,7
42,43
34,42,43
11,43
11,42,43
26
34
12
Level IV: 89 m
Level III: 46 m
Level II: 13 m
Level I: 6 m
Figure 13: Sensor positions in tower and foundation.
The static moment tests referred to in Table 2 at
Sandy Haven and at Burry Port were relatively
straightforward, with very simple instrumentation,
but those at Frederikshavn test site and at Luce Bay
were detailed investigations.
The large scale tests at Frederikshavn is part of a
research and development program concerning
caisson foundation for offshore wind turbines. The
research program is a co-operation between Aalborg
Table 3. Loading heights in the Aalborg test program
Prototype
D p = 12m
Field
Model
Laboratory Model
D m = 0 .2 m
h p [m]
104.4
69.6
38.0
20.0
10.0
0.3m
0.4m
h m [m]
1.74
1.16
0.63
0.33
0.17
2.61
1.74
0.95
0.50
0.25
2.0m
h m [m]
3.48
2.32
1.27
0.67
0.33
17.40
11.60
6.33
3.33
1.67
Figure 14: Sensor mountings in the tower and foundation at
Frederikshavn.
University and MBD offshore power (Ibsen et al.
2003). The large scale tests are complemented by
laboratory studies. The laboratory and large scale
tests are intended to model the prototype in
Frederikshavn directly. In order to design a caisson
foundation for offshore wind turbines several load
combinations have to be investigated. Each load
combination is represented by a height of load h
above the foundation and a horizontal force H. The
moment at the seabed is calculated as M = hH. Table
3 shows that the resulting loading height varies from
10m (for a wave force in shallow water) to 104.4m
(force from normal production of a 3MW turbine in
20m of water). Scaling of the tests is achieved by:
hm = h p
Dm
Dp
(6)
where D is the diameter of the caisson and index m
and p are for model and prototype. The values of the
loading height in the test program are shown in
Table 3.
The large scale tests at Frederikshavn employ
loading by applying a horizontal load at a fixed
height, under constant vertical load. A steel caisson
with an outer diameter of 2m and a skirt length of
2m has been used. The skirt is made of 12mm thick
steel plate. Figure 15 shows the caisson prior to
installation, and Figure 16 the overall test setup.
Currently 10 experiments have been conducted, but
the testing program is ongoing. Each test has three
phases:
Figure 15: Caisson for large scale test at Frederikshavn
3 MW Vestas
windmill on bucket
foundation
loading tower
tower located on
bucket foundation
loading wire
Figure 16: Setup for combined loading of 2x2m caisson at
Frederikshavn. (Back: prototype 3MW Vestas wind turbine on
the 12 x 6m caisson)
1. Installation phase: The caisson is installed by
means of suction. CPT tests are performed
before and after installation of the caisson.
2. Loading phase: An old tower from a wind
turbine is mounted on top of the caisson. The
caisson is loaded by pulling the tower
frequency, cycles were applied using a hydraulic
jack. A diagram of the loading rig, which allowed
both moment and vertical loading tests, is shown in
Figure 18.
The SEMV test involve cycles of moment
loading at increasing amplitude as the frequency
increases. Figure 19 shows the hysteresis loops
obtained from a series of these cycles at different
amplitudes. As the cycles become larger the stiffness
reduces but hysteresis increases. The tests were
interpreted (Houlsby et al., 2005b) using the theory
of Wolf (1994), which takes account of the dynamic
effects in the soil, and the equivalent secant shear
modulus for each amplitude of cycling determined.
Figure 20 shows the moment rotation curves for
much larger amplitude cycling applied by the
hydraulic jack. Again hysteresis increases and secant
stiffness decreases as the amplitude increases. The
unusual “waisted” shape of the hysteresis loops at
very large amplitude is due to gapping occurring at
the sides of the caisson.
The secant stiffnesses deduced from both the
SEMV tests and the hydraulic jacking tests are
combined in Figure 21, where they are plotted
against the amplitude of cyclic rotation. It is clear
that the two groups of tests give a consistent pattern
of reduction of shear modulus with strain amplitude,
similar to that obtained for instance from laboratory
tests.
horizontally with a wire. The combined loading
(H,M) is controlled by changing the height of
loading.
3. Dismantling phase. The caisson is removed by
applying overpressure inside the bucket.
Figure 17 shows the moment rotation curve for a
test on the 2x2m caisson at Frederikshavn. The test
is performed with hm = 17.4m and a vertical load on
the caisson of 37.3kN. The fluctuations in the curve
are caused by wind on the tower.
Figure 17: Moment-rotation test on 2x2m caisson.
Tests at Luce Bay were designed by Oxford
University and conducted by Fugro Ltd.. The
moment loading tests were of two types. Firstly
small amplitude (but relatively high frequency)
loading was applied by a “Structural Eccentric Mass
Vibrator” (SEMV) in which rotating masses are used
to apply inertial loads at frequencies up to about
12Hz. Secondly larger amplitude, but lower
5.2 Sand: laboratory tests
Turning now to model testing, a large number of
tests have been carried out both at Aalborg and at
Oxford. Almost all the model tests have involved “in
plane” loading (in which the moment is about an
L
H
V
V
A
A
W
H
L
R
L
1500
4000
B
B
H
L
C
R
4000
L
3000
L
L
C
C
6000
(a)
(b)
Figure 18: Field testing equipment, dimensions in mm. Water level and displacement reference frames not shown. (a) arrangement
for jacking tests on 1.5m and 3.0m caissons, (b) alternative arrangement during SEMV tests. Labels indicate (A) A-frame, (B)
concrete block, (C) caissons, (H) hydraulic jacks, (L) load cells, (R) foundations of reaction frame, (V) SEMV, (W) weight
providing offset load for SEMV tests.
30
6Hz
20
7Hz
Moment (kNm)
8Hz
10
9Hz
10H
-0.00005
0
-0.000025
0
0.000025
0.00005
-10
-20
-30
Rotation (radians)
Figure 19: Hysteresis loops from SEMV tests on 3m caisson.
500
400
300
Moment (kNm)
200
100
-0.08
-0.06
-0.04
0
-0.02 -100 0
0.02
0.04
0.06
0.08
-200
-300
-400
-500
-600
Rotation of caisson centre (2R θ) (m)
Figure 20: Hysteresis loops from hydraulic jacking tests on 3m
caisson.
100
Jacking
SEMV
Hyperbolic curve fit
90
80
G (MPa)
70
60
50
40
30
20
10
0
0.000001
0.00001
0.0001
0.001
0.01
0.1
∆θ (radians)
Figure 21: Shear modulus against rotation amplitude.
axis perpendicular to the horizontal load). However,
a test rig capable of applying full 6 degree-offreedom loading has recently been developed by
Byrne and Houlsby (2005).
The model tests at Aalborg are performed by the
test rig shown in Figure 22. The rig consists of a test
box and loading frame. The test box consists of a
steel frame with an inner width of 1.6m x 1.6m and
an inner total depth of 0.65m. The test box is filled
with Aalborg University Sand No 0. After each
experiment the sand in the box is prepared in a
systematic way to ensure homogeneity within the
box, and between the different test boxes. The sand
Figure 22: The caisson test rig at Aalborg University
is saturated by the water reservoir shown in Figure
22. Before each experiment CPT-tests are performed
to verify the density and strength of the sand. The
caisson is then installed and loaded with a constant
vertical load. The vertical load is kept constant
through the experiment, while the horizontal force is
applied to the tower by the loading device mounted
on the loading frame, see Figure 21. The tower and
the loading device are connected by a wire. The
combined loading (H, M) is controlled by the height
of loading h. The loading frame allows the
possibility of changing h from 0.1m to 4.0m above
the sand surface (Table 3). The horizontal force H is
measured by a transducer connected to the wire. The
deformation of the foundation and the moment are
measured with the measuring cell mounted on the
top of the caisson, as shown by Figure 23.
Laboratory tests at Oxford University have used a
versatile 3 degree-of-freedom loading rig designed
by Martin (1994) and adapted by Byrne (2000) (see
also Martin and Houlsby (2000) and Gottardi et al.
(1999)). The rig is shown in Figure 24, and is
capable of applying a wide range of combinations of
vertical, horizontal and moment loading under either
displacement or load control.
Typical moment loading tests involve applying a
fixed vertical load, and then cycling the rotation at
100
Moment Load, M/2R (N)
80
60
40
20
0
-20
-40
-60
-80
-100
-2.0
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
Rotational Displacement, 2Rθ (mm)
Figure 25: Moment-rotation test on sand
Figure 23: The measuring cell connecting the caisson and the
tower.
Fitted Yield Surface
Soil Plug Weight
Moment Load, M/2R (N)
100
Experiment, M/2RH = 1
80
60
40
20
-160
-120
-80
0
-40
0
Vertical Load, V (N)
40
80
120
Figure 26: Experimentally determined yield surface in V-M
plane
Figure 24: Three degree-of-freedom testing rig at Oxford
University
increasing amplitude. An example is given in Figure
25.
The first interpretation of such tests is to
determine the yield surface for a single surface
plasticity model (see section 7.2 below, and also
Martin and Houlsby (2001), Houlsby and Cassidy
(2002), Houlsby (2003), Cassidy et al. (2004)). An
example of the yield points obtained, plotted in the
vertical load-moment plane, is given in Figure 26.
Of particular importance is the fact that at very low
vertical loads there is a significant moment capacity,
and that this extends even into the tensile load range.
In these drained tests the ultimate load in tension is a
significant fraction of the weight of the soil plug
inside the caisson.
Sections of the yield surface can also be plotted in
H-M space as shown in Figure 27, where the data
here have been assembled from many tests at
different stress levels. The flow vectors are also
plotted in this figure, and show that in this plane
(unlike the V-M plane) associated flow is a
reasonable approximation to the behaviour. Feld
(2001) has observed similar shapes of a yield surface
for a caisson in sand.
We now consider the possibility of scaling the
results of laboratory tests to the field. The test at
Frederikshavn shown in Figure 17 was on a caisson
with a ratio L 2 R = 1 , at an M 2 RH value of
approximately 8.7, and with a value of V 2πR 3 γ ′ of
about 0.62. Using the data from the Oxford
laboratory on 0.2x0.2m caissons this requires a
vertical load of about 60N. In fact a test had been
carried out with L 2 R = 1 and V = 50 N . According
to the scaling relationships discussed in section 3,
the moment should be scaled according to R 4 γ ′ (a
factor of 6250) and the rotational displacement 2 Rθ
according to R 3 γ ′ (a factor of 25). Figures 26 and
27 suggest that for a vertical load of 60N rather than
0.3
V = -50 N
80
0.2
V= 0N
0.1
V = 50 N
3
40
M/[su(2R) ]
Moment Load, M/2R(N)
Incremental Rotation, 2Rdtheta (mm)
120
0
-0.1
-0.2
-40
-0.3
-80
-120
-180
-0.4
-0.015
-140
-100
-60
-20
20
60
100
140
Horizontal Load, H(N)
Incremental Horizontal Displacement, du (mm)
-0.01
-0.005
0
0.005
0.01
0
0.005
0.01
θ
180
(a) field test
0.3
Figure 27: Yield surfaces and flow vectors in H-M space.
0.2
0.1
3
M/[su(2R) ]
150
100
Moment, M (kNm)
0
50
0
-0.1
-0.2
-0.3
0
-0.4
-0.015
-50
-0.01
-0.005
θ
-100
-150
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
Rotational Displacement, 2Rtheta (m)
Figure 28: Laboratory moment test scaled to field conditions
for comparison with Figure 17
50N a moment capacity say 5% higher might be
expected, and that for the higher value of M 2 RH a
further increase of say 15% is appropriate. We
therefore apply a factor of 7500 to the moments and
25 to the rotational displacements. The result is
shown in Figure 28. It can be seen that after scaling
the moment at a 2 Rθ value of 0.04 m is about
120kNm, compared to about 280kNm measured in
the field. Although there is a factor of about 2
between these values, it must be borne in mind that
there are a number of possible causes of difference
between the tests (e.g. the sand in the field test may
be much denser), and also that a factor of 7500 has
already been applied: a factor of 2 is relatively small
by comparison.
5.3 Clay: field and laboratory tests
Less work has been carried out on clay than on sand.
The large scale trials at Bothkennar (Houlsby et al.
2005b) are complemented by laboratory studies
intended to model these trials directly, and therefore
add confidence to the scaling of the results to
prototype size caissons (Kelly et al., 2005a).
(b) model test
Figure 29: Moment-rotation results presented in nondimensional form for laboratory and field tests.
At Bothkennar, moment loads were applied to a
3m x 1.5m caisson by two means. Small amplitude,
but relatively high frequency (10Hz) loading was
applied by means of the SEMV device described
above, and larger amplitude cycles, but at much
lower frequency, were applied using a hydraulic
jack. In both cases the loading was 4m above the
caisson, so that hload D = 1.33 . The most important
observation from these tests was the gradual
reduction of secant stiffness (and increase in
hysteresis) as the amplitude of the load cycles
increases.
The laboratory tests, specifically modelling the
field tests, involved just relatively low frequency
loading. After the scaling relationships described in
section 3 were applied, there was a satisfactory
agreement between laboratory and field data,
especially at relatively small amplitudes of
movement. As an example, Figure 29(a) shows the
results (in dimensionless form) for rotation of the
3.0m diameter caisson in the field, and Figure 29(b)
the equivalent results, also in dimensionless form,
from the small scale model test. The pattern of
behaviour is remarkably similar in the two tests.
This sort of comparison is vital to establish
In the following, in which we consider multiple
footing designs to support the wind turbine, we shall
refer principally to a tetrapod (four footings) rather
than a tripod. As a tripod is perhaps the most
obvious multiple footing design to use, and has the
obvious advantage of simplicity, our preference for
the tetrapod deserves some explanation.
As is discussed below, prudent design of a
multiple footing structure will avoid tension being
applied to any of the foundations (except under the
most extreme of circumstances). This in effect
dictates the separation of the foundations for a given
overturning moment and weight of structure.
Approximate calculations indicate that the tetrapod
structure is usually a more favourable configuration
to avoid tension, as it requires somewhat less
material. The differences are not large, and a tripod
may be preferred in some circumstances, but we
shall refer to a tetrapod, as this will probably be
more efficient. The important mechanism is the
same in both cases: the overturning moment is
resisted by opposing “push-pull” action on the
foundations.
In Table 4 we list the tests that have been carried
out on vertical loading of caissons relevant to the
wind turbine problem. In addition to these studies
there are a number of other relevant studies which
have been directed towards vertical loading of
caissons for structures in the oil and gas industry or
for use as anchors.
6.1 Sand: field and laboratory tests
The simplest tests on vertical loading of caissons in
sand, which are relevant both to installation and to
subsequent performance, simply involve pushing
caissons vertically into sand to determine the vertical
load-displacement response. Figure 30 shows the
results of a set of such tests on caissons of different
L/D ratios, Byrne et al. (2003). It is clear from the
figure that there is a well-established pattern. While
the caisson skirt is penetrating the sand there is
relatively low vertical capacity, but as soon as the
top plate makes contact with the sand there is a
sudden increase in capacity. The envelope of the
ultimate capacities of footings of different initial L/D
ratios also forms a single consistent line.
Of most importance, however, is the performance
of the caissons under cyclic vertical loading. Figure
Clay
D
(m)
1.5
0.05
0.1
0.15
0.15
0.2
0.28
1.5
Clay
0.06
Site
Soil
Luce Bay
Sand
Oxford
laboratory
Sand
Bothkennar
UWA
centrifuge
(100g)
L
(m)
1.0
0.0 - 0.1
0.0 - 0.066
0.05
0.1
0.133
0.18
1.0
0.02
0.03
0.06
Ref.
27
11
2,5
2,5
34
34
25,32,33,35
26
3
Normalised Vertical Displacement, w/D
0
0.5
1
1.5
2
2.5
3
3.5
400
350
Vertical Load, V (N)
6. CAISSON PERFORMANCE: TETRAPOD OR
TRIPOD
Table 4: Vertical loading tests
300
250
200
150
100
50
0
0
50
100
150
Vertical Displacement, w (mm)
Figure 30: Vertical load-penetration curves for caissons of
different L/D ratios
1600
1400
Vertical Stress (kPa)
confidence in the use of model testing to develop
design guidelines.
1200
1000
800
600
400
200
0
-200
-400
200
210
220
230
240
250
260
270
Vertical Displacement (mm)
Figure 31: Cyclic vertical loading of model caisson.
31 shows the results of tests on a 300mm diameter
caisson subjected to rapid cyclic loading. Smallamplitude cycles show a stiff response, with larger
cycles showing both more hysteresis and more
accumulated displacement per cycle. The most
important observation is that as soon as the cycles go
into tension, a much softer response is observed, and
the hysteresis loops acquire a characteristic “banana”
shape. Clearly the soft response on achieving tension
should be avoided in design. Closer examination of
the curves reveals that the softening in fact occurs
150
0
160
Vertical Displacement (mm)
170
180
190
200
210
Vertical Stress (kPa)
-50
-100
-150
-200
-250
Direction of
movement
-300
-350
5mm/s, 0kPa
100mm/s, 0kPa
100mm/s, 200kPa
-400
-450
Figure 32: Tensile capacity of model caisson pulled at different
rates and at different ambient pressures.
5
1.5m Field
4
0.15m Suction
0.2m Pushed
0.15m Pushed
3
V/[γ'(2R) ]
3
2
1
0
-1
0
0.01
0.02
0.03
0.04
0.05
1/2
[w/(2R)][pa/(2Rγ')]
Figure 33: Hysteresis loops from tests at different scales and
rates.
once the drained frictional capacity of the skirts has
been exceeded, rather than simply the transition into
tension.
Paradoxically, although additional accumulated
displacement is observed once tension is reached,
this accumulated displacement is downwards (not
upwards as one might expect because of the tensile
loading).
The above observations mean that tension must
be avoided in a prudent design of a tripod or tetrapod
foundation for a wind turbine. However, in all but
the shallowest of water, avoiding this tension means
that either the foundation must have a large spacing
between the footings, or that ballasting must be used.
The latter may in fact be a cost effective measure in
deep water.
Some designers may wish to reduce conservatism
by allowing for the possibility of tension under
extreme circumstances. It is therefore useful to
examine the ultimate tensile capacity under rapid
loading. Figure 32 shows the result of three such
tests. The slowest test (at 5mm/s) is almost drained,
and a very low capacity in tension is indicated. The
capacity in this case is simply the friction on the
skirts. The test at 100mm/s (but zero ambient water
pressure) shows a larger capacity, and it is
straightforward to show that this is controlled by
cavitation beneath the foundation. This means that at
elevated water pressures (as in the third test) the
capacity rises approximately in step with to the
ambient water pressure, as correspondingly larger
pressure changes are required to cause cavitation.
This problem is studied in more detail by Houlsby et
al. (2005a).
It is important to note, however, that although
ambient water pressure increases the ultimate
capacity, it has negligible influence on the tensile
load at which a flexible response begins to occur.
Comparison of cyclic loading tests at different
scales and at different speeds shows that it is
difficult to scale reliably the accumulated
displacements, which reduce with larger tests and
higher loading rates. However, when the scaling
rules described earlier are applied, the shapes of
individual hysteresis loops at different scales and at
different rates become remarkably similar. Figure 33
shows a comparison, for instance, of loops at three
different load amplitudes from four different tests.
At each particular load amplitude the loops from the
different tests are very similar.
The accumulation of displacement after very
large numbers of cycles is difficult to predict, and so
far few data are available. Rushton (2005) has
carried out vertical loading tests to about 100000
cycles on a model caisson in sand, using a simple
loading rig which employs a rotating mass and a
series of pulleys to apply a cyclic load. A typical
result is shown in Figure 34, on a caisson 200mm
diameter and 100mm deep, with cycling between
210 ± 260 N . The caisson is therefore subjected (at
the minimum vertical load) to a small tension, but
less than the frictional capacity of the skirts. The
dimensionless accumulated vertical displacement is
seen in Figure 34 to increase approximately with the
logarithm of the number of cycles of loading (after
about 1000 cycles). Note that even in this case where
there is a tensile loading in part of the cycle, the net
movement is downwards. The displacement is of
course very sensitive also to the amplitude of the
cycling.
6.2 Clay: field and laboratory tests
Very few vertical loading tests relevant to the wind
turbine problem have been completed on caissons in
clay, although there have been a number of studies
directed towards suction caissons used as tension
0.05
[w/(2R)][pa/(2Rγ’)]
1/2
0.00
1
10
100
1000
10000
100000
-0.05
-0.10
-0.15
-0.20
Min
Max
-0.25
Number of Cycles
Figure 34: Accumulated displacement during long term cyclic
vertical loading on sand
anchors, e.g. El-Gharbawy (1998), Watson (1999),
House (2002).
At Bothkennar tests were carried out in which
inclined (but near vertical) loading was applied to a
1.5m diameter caisson (Houlsby et al., 2005b).
Difficulties were encountered with the control of the
loads using a hydraulic system, and the resulting
load paths are therefore rather complex, leading to
difficulties in interpretation. Further work on vertical
loading in clay is required before definitive
conclusions can be drawn, and in particular the issue
of tensile loading in clay needs attention. Some
preliminary results (Byrne and Cassidy, 2002),
shown in Figure 35, show that the tensile response
may be sensitive to prior compressive loading.
Footings loaded in tension immediately after
installation showed a stiff tensile response, whilst
those loaded after first applying a compressive load
to failure showed a more flexible tensile response.
Vertical Stress, V/A (kPa)
60
40
20
0
-20
0
0.2
0.4
0.6
0.8
1
-40
Test 1: Post Bearing Capacity
-60
Test 2: Pre Bearing Capacity
-80
Normalised Displacement, (w + L)/D
Figure 35: Tension tests on caisson foundations in clay
7. NUMERICAL STUDIES
7.1. Finite element studies
A number of analyses of suction caissons for
offshore wind farms have been carried out as part of
commercial investigations for possible projects. A
more detailed research project was carried out by
Feld (2001).
Finite element analysis is particularly appropriate
for establishing the effects of design parameters on
the elastic behaviour of caissons, and has been used
by Doherty et al. (2004a,b) to determine elastic
stiffness coefficients for caisson design which take
into account the flexibility of the caisson wall as
well as coupling effects between horizontal and
moment loading.
7.2 Plasticity models
An important tool for the analysis of soil-structure
interaction problems, particularly those involving
dynamically sensitive structures are “force resultant”
models. In these the behaviour of the foundation is
represented purely through the force resultants
acting upon it, and the resulting displacements (see
Figure 4). Details of stresses and deformations
within the soil are ignored. The models are usually
framed within the context of work-hardening
plasticity theory. Examples include models for
foundations on clay (Martin and Houlsby, 2001) and
on sand (Houlsby and Cassidy, 2002). Overviews of
the development of these models are given by
Houlsby (2003) and Cassidy et al. (2004)
These models have been further developed
specifically for the offshore wind turbine
application. The developments include:
• Generalisation to full three-dimensional loading
conditions,
• Inclusion of special features to represent the
caisson geometry,
• Expression of the models within the “continuous
hyperplasticity” framework to allow realistic
description of hysteretic response during cyclic
loading.
A model with all these features is described by
Lam and Houlsby (2005). The fitting of cyclic data
to a continuous hyperplastic model is discussed by
Byrne et al (2002a).
8. OTHER CONSIDERATIONS
We have concentrated here on the design of caisson
foundations as far as capacity and stiffness are
concerned for in-service conditions. However, there
a number of other issues which need to be addressed
in a caisson design, and we mention them here
briefly.
8.1 Scour
Scour is more important for caissons, since they are
relatively shallow, than for piles. The size of
caissons, and the fact that part of the caisson
inevitably protrudes above mudline level, creates
rather aggressive conditions for scour. The fact that
the caissons may be installed in mobile shallowwater environments means that proper consideration
of this problem is essential, especially in sands.
If the scour depth can be determined with
sufficient confidence (e.g. from comprehensive
model testing) then it may be possible to permit the
scour to occur, and simply allow for this in the
design by ensuring that the caisson is deep enough.
It is more likely, however, that scour protection
measures such as rock-dumping will need to be
employed. Practical experience suggests that such
protection must be placed very soon after caisson
installation, as scour can occur very rapidly. In
highly mobile environments, significant scour can,
for instance, occur due to the currents in a single
tide. Model testing indicates, however, that scour
protection measures can be effective in preventing
further
erosion
(R.
Whitehouse:
private
communication). For in-service conditions regular
monitoring for the possibility of scour would be
prudent.
8.2 Liquefaction
The transient pore pressures induced in the seabed
can induce liquefaction, especially if the seabed is
partially saturated due to the presence of gas (as can
occur in shallow seabeds, largely due to decay of
organic matter).
The problem is a complex one, but typically, at
one stage in the wave cycle, the pore pressure in the
seabed can become equal to the overburden stress,
and the effective stress falls to zero. This problem is
further complicated by the presence of a structure,
which clearly modifies the pore pressure pattern that
would occur in the far field. Although some progress
has been made, the interactions are complex, and
theoretical modelling of the problem is not
straightforward.
8.3 Wave-induced forces
A quite different problem from liquefaction is also
related to the fact that the principal forces on the
structure are wave induced. As a wave passes the
column of the structure it exerts large horizontal
forces (of the order of a few meganewtons for a
large wave), which also cause overturning moments.
However, at the same time the wave causes a
transient pressure on the seabed, and on the lid of the
caisson. Because the caissons are in shallow water
these pressures are quite large. The pore water
pressure within the caisson is unlikely to change as
rapidly as the pressure on the lid, so there will be
pressure differentials across the lid of the caisson
which result in net vertical forces, and overturning
moments on the caisson.
The relative phase of the different sources of
loading is important. As the crest of the wave just
reaches the structure, the wave kinematics are such
that the horizontal forces are likely to be largest. At
this stage the pressure on the upwave side of the
caisson is likely to be larger than on the downwave
side. The net result is that the moment caused by the
pressures on the caisson lid opposes that caused by
the horizontal loading, so this effect is likely to be
beneficial to the performance of the caisson. Little
work has, however, yet been completed on the
magnitudes of these effects. The problem is
complicated by the fact that the kinematics of large
(highly non-linear) shallow water waves is still a
matter of research, as is their interaction with
structures.
8. CONCLUSIONS
In this paper we have provided an overview of the
extensive amount of work that has been carried out
on the design of suction caisson foundations for
offshore wind turbines. Further verification of the
results presented here is still required, and in due
course it is hoped that this will come from
instrumented caisson foundations offshore. Our
broad conclusions at present are:
• Suction caissons could be used as foundations
for offshore wind turbines, either in monopod or
tripod/tetrapod layout.
• The combination of low vertical load and high
horizontal load and moment is a particular
feature of the wind turbine problem.
• Stiffness and fatigue are as important for turbine
design as ultimate capacity.
• Monopod foundation design is dominated by
moment loading.
• Tripod/tetrapod foundation design is dominated
by considerations of tensile loading.
• The moment-rotation response of caissons in
sand has been extensively investigated by model
tests and field trials, and modelled theoretically
by finite element analyses and force resultant
(yield surface) models.
• As amplitude of moment loading increases,
stiffness reduces and hysteresis increases.
• Moment loading in clay has been less
extensively investigated in the laboratory and
field.
• Vertical loading in sand has been extensively
investigated in the laboratory and field.
•
The as the amplitude of vertical loading
increases, stiffness reduces and hysteresis
increases. Once tension is reached there is a
sudden reduction of stiffness.
• Whilst high ultimate tensile capacities are
possible (especially in deep water) this is at the
expense of large movements.
• Application of scaling procedures for tests in
both sand and clay allows model and field tests
to be compared successfully as far as stiffness
and the shapes of hysteresis loops is concerned.
• Cumulative displacements after very many
cycles are harder to model.
• The design of caisson foundations also needs to
take into consideration issues such as scour and
liquefaction.
It is hoped that the conclusions above lead in due
course to application of suction caissons as
foundations for offshore wind turbines, thereby
making an important renewable energy source more
economically viable.
ACKNOWLEDGEMENTS
The work at Oxford University has been supported
by the Department of Trade and Industry, the
Engineering and Physical Sciences Research
Council and a consortium of companies: SLP
Engineering Ltd, Aerolaminates (now Vestas),
Fugro Ltd, Garrad Hassan, GE Wind and Shell
Renewables. An outline of the project is given by
Byrne et al. (2002b). The work of Richard Kelly,
Nguyen-Sy Lam and Felipe Villalobos on this
project is gratefully acknowledged.
REFERENCES
1.
2.
3.
4.
5.
6.
7.
Bolton, M.D. (1986) “The strength and Dilatancy of
Sand”, Geotechnique, Vol. 36, No. 1, pp 65-78
Byrne, B.W. (2000) "Investigations of Suction Caissons in
Dense Sand", D.Phil. Thesis, Oxford University
Byrne, B.W. and Cassidy, M.J. (2002) “Investigating the
response of offshore foundations in soft clay soils”, Proc.
OMAE, Oslo, Paper OMAE2002-28057
Byrne, B.W. and Houlsby, G.T. (1999) "Drained
Behaviour of Suction Caisson Foundations on Very Dense
Sand", Offshore Technology Conference, 3-6 May,
Houston, Paper 10994
Byrne, B.W. and Houlsby, G.T. (2002) “Experimental
Investigations of the Response of Suction Caissons to
Transient Vertical Loading”, Proc. ASCE, J. of Geot. Eng.,
Vol. 128, No. 11, Nov., pp 926-939
Byrne, B.W. and Houlsby, G.T. (2003) "Foundations for
Offshore Wind Turbines", Phil. Trans. of the Royal
Society of London, Series A, Vol. 361, Dec., 2909-2930
Byrne, B.W. and Houlsby, G.T. (2004) “Experimental
Investigations of the Response of Suction Caissons to
Transient Combined Loading”, Proc. ASCE, J. of Geotech.
and Geoenvironmental Eng., Vol. 130, No. 3, pp 240-253
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
Byrne, B.W. and Houlsby, G.T. (2005) "Investigating 6
degree-of-freedom loading on shallow foundations", Proc.
International Symposium on Frontiers in Offshore
Geotechnics, Perth, Australia, 19-21 September, in press
Byrne, B.W., Houlsby, G.T. and Martin, C.M. (2002a)
"Cyclic Loading of Shallow Offshore Foundations on
Sand", Proc. Int. Conf on Physical Modelling in Geotech.,
July 10-12, St John's, Newfoundland, 277-282
Byrne, B.W., Houlsby, G.T., Martin, C.M. and Fish, P.
(2002b) "Suction Caisson Foundations for Offshore Wind
Turbines", Wind Engineering, Vol. 26, No. 3, pp 145-155
Byrne, B.W., Villalobos,, F. Houlsby, G.T. and Martin,
C.M. (2003) "Laboratory Testing of Shallow Skirted
Foundations in Sand", Proc. Int. Conf. on Foundations,
Dundee, 2-5 September, Thomas Telford, pp 161-173
Cassidy, M.J., Byrne, B.W. and Randolph, M.F. (2004) “A
comparison of the combined load behaviour of spudcan
and caisson foundations on soft normally consolidated
clay”, Géotechnique, Vol. 54, No. 2, pp 91-106
Cassidy, M.J., Martin, C.M. and Houlsby, G.T. (2004)
"Development and Application of Force Resultant Models
Describing Jack-up Foundation Behaviour", Marine
Structures, (special issue on Jack-up Platforms: Papers
from 9th Int. Conf. on Jack-Up Platform Design,
Construction and Operation, Sept. 23-24, 2003, City Univ.,
London), Vol. 17, No. 3-4, May-Aug., 165-193
Coldicott, L. (2005) “Suction installation of cellular
skirted foundations”, 4th year project report, Dept. of
Engineering Science, Oxford University
Doherty, J.P., Deeks, A.J. and Houlsby, G.T. (2004a)
"Evaluation of Foundation Stiffness Using the Scaled
Boundary Method", Proc. 6th World Congress on
Computational Mechanics, Beijing, 5-10 Sept., in press
Doherty, J.P., Houlsby, G.T. and Deeks, A.J. (2004b)
"Stiffness of Flexible Caisson Foundations Embedded in
Non-Homogeneous Elastic Soil", Submitted to Proc.
ASCE, Jour. Structural Engineering Division
El-Gharbawy, S.L. (1998) “The Pullout Capacity of
Suction Caisson Foundations”, PhD Thesis, University of
Texas at Austin
Feld T. (2001) “Suction Buckets, a New Innovative
Foundation Concept, applied to offshore Wind Turbines”
Ph.D. Thesis, Aalborg University Geotechnical
Engineering Group, Feb..
Gottardi, G., Houlsby, G.T. and Butterfield, R. (1999)
"The Plastic Response of Circular Footings on Sand under
General Planar Loading", Géotechnique, Vol. 49, No. 4,
pp 453-470
Houlsby, G.T. (2003) "Modelling of Shallow Foundations
for Offshore Structures", Proc. Int. Conf. on Foundations,
Dundee, 2-5 Sept., Thomas Telford, pp 11-26
Houlsby, G.T. and Byrne, B.W. (2000) “Suction Caisson
Foundations for Offshore Wind Turbines and Anemometer
Masts”, Wind Engineering, Vol. 24, No. 4, pp 249-255
Houlsby, G.T. and Byrne, B.W. (2005a) “Design
Procedures for Installation of Suction Caissons in Clay and
Other Materials”, Proc. ICE, Geotechnical Eng., Vol. 158
No. GE2, pp 75-82
Houlsby, G.T. and Byrne, B.W. (2005b) “Design
Procedures for Installation of Suction Caissons in Sand”,
Proceedings ICE, Geotechnical Eng., in press
Houlsby, G.T. and Cassidy, M.J. (2002) "A Plasticity
Model for the Behaviour of Footings on Sand under
Combined Loading", Géotechnique, Vol. 52, No. 2, Mar.,
117-129
Houlsby, G.T., Kelly, R.B. and Byrne, B.W. (2005a) "The
Tensile Capacity of Suction Caissons in Sand under Rapid
26.
27.
28.
29.
30.
31.
32.
33.
34.
35.
36.
37.
38.
39.
40.
41.
42.
43.
Loading", Proc. Int. Symp. on Frontiers in Offshore
Geotechnics, Perth, Australia, September, in press
Houlsby, G.T., Kelly, R.B., Huxtable, J. and Byrne, B.W.
(2005b) “Field Trials of Suction Caissons in Clay for
Offshore Wind Turbine Foundations”, Géotechnique, in
press
Houlsby, G.T., Kelly, R.B., Huxtable, J. and Byrne, B.W.
(2005c) “Field Trials of Suction Caissons in Sand for
Offshore Wind Turbine Foundations”, submitted to
Géotechnique
House, A. (2002) “Suction Caisson Foundations for
Buoyant Offshore Facilities”, PhD Thesis, the University
of Western Australia
Ibsen, L.B., Schakenda, B., Nielsen, S.A. (2003)
“Development of bucket foundation for offshore wind
turbines, a novel principle”. Proc. USA Wind 2003
Boston.
Ibsen, L.B. and Brincker, R. (2004) “Design of New
Foundation for Offshore Wind Turbines”, Proceedings of
The 22nd International Modal Analysis Conference
(IMAC), Detroit, Michigan, 2004.
Ibsen L.B., Liingaard M. (2005) “Output-Only Modal
Analysis Used on New Foundation Concept for Offshore
Wind Turbine”, in preparation
Kelly, R.B., Byrne, B.W., Houlsby, G.T. and Martin, C.M.
(2003) "Pressure Chamber Testing of Model Caisson
Foundations in Sand", Proc. Int. Conf. on Foundations,
Dundee, 2-5 Sept., Thomas Telford, pp 421-431
Kelly, R.B., Byrne, B.W., Houlsby, G.T. and Martin,
C.M., 2004. Tensile loading of model caisson foundations
for structures on sand, Proc. ISOPE, Toulon, Vol. 2, 638641
Kelly, R.B., Houlsby, G.T. and Byrne, B.W. (2005a) "A
Comparison of Field and Laboratory Tests of Caisson
Foundations in Sand and Clay" submitted to Géotechnique
Kelly, R.B., Houlsby, G.T. and Byrne, B.W. (2005b)
"Transient Vertical Loading of Model Suction Caissons in
a Pressure Chamber", submitted to Géotechnique
Lam, N.-S. and Houlsby, G.T. (2005) "The Theoretical
Modelling of a Suction Caisson Foundation using
Hyperplasticity Theory", Proc. Int. Symp. on Frontiers in
Offshore Geotechnics, Perth, Australia, Sept., in press
Martin, C.M. (1994) "Physical and Numerical Modelling
of Offshore Foundations Under Combined Loads", D.Phil.
Thesis, Oxford University
Martin, C.M. and Houlsby, G.T. (2000) "Combined
Loading of Spudcan Foundations on Clay: Laboratory
Tests", Géotechnique, Vol. 50, No. 4, pp 325-338
Martin, C.M. and Houlsby, G.T. (2001) “Combined
Loading of Spudcan Foundations on Clay: Numerical
Modelling”, Géotechnique, Vol. 51, No. 8, Oct., 687-700
Rushton, C. (2005) “Cyclic testing of model foundations
for an offshore wind turbine”, 4th year project report, Dept.
of Engineering Science, Oxford University
Sanham, S.C. (2003) “Investigations into the installation of
suction assisted caisson foundations”, 4th year project
report, Dept. of Engineering Science, Oxford University
Villalobos, F.A., Byrne, B.W. and Houlsby, G.T. (2005)
"Moment loading of caissons installed in saturated sand",
Proc. Int. Symp. on Frontiers in Offshore Geotechnics,
Perth, Australia, Sept., in press
Villalobos, F., Houlsby, G.T. and Byrne, B.W. (2004)
"Suction Caisson Foundations for Offshore Wind
Turbines", Proc. 5th Chilean Conference of Geotechnics
(Congreso Chileno de Geotecnia), Santiago, 24-26
November
44. Watson, P.G. (1999) “Performance of Skirted Foundations
for Offshore Structures”, PhD Thesis, the University of
Western Australia
45. Wolf, J.P. (1994) “Foundation Vibration Analysis Using
Simple Physical Models”, Prentice Hall, New Jersey
Bearing capacity of parallel strip footings on non-homogeneous clay
C.M. Martin & E.C.J. Hazell
Department of Engineering Science, University of Oxford, UK
ABSTRACT: On soft seabed soils, subsea equipment installations are often supported by mudmat foundation
systems that can be idealised as parallel strip footings, grillages, or annular (ring-shaped) footings. This paper
presents some theoretical results for the bearing capacity of (a) two parallel strip footings, otherwise isolated;
(b) a long series of parallel strip footings at equal spacings. The soil is idealised as an isotropic Tresca material possessing a linear increase of undrained strength with depth. The bearing capacity analyses are performed
using the method of characteristics, and the trends of these (possibly exact) results are verified by a companion series of upper bound calculations based on simple mechanisms. Parameters of interest are the footing
spacing, the relative rate of increase of strength with depth, and the footing roughness. An application of the
results to the design of perforated mudmats is discussed.
1 INTRODUCTION
Shallow foundations are usually designed on the assumption that they act in isolation. When two footings (or a group of footings) are closely spaced,
however, there is a beneficial interaction that can be
quantified in terms of the ‘efficiency’, i.e. the ratio
of the overall (group) bearing capacity to the sum of
the individual (isolated) bearing capacities. The literature on this topic has been surveyed by Hazell
(2004). For footings on sand, numerous theoretical
and experimental studies have shown that the effect
of interaction becomes highly significant for friction
angles greater than about 30° and spacings less than
about one footing width B. In contrast, the undrained
bearing capacity of closely spaced footings on clay
has received very little attention, perhaps because the
early theoretical work of Mandel (1963) showed that
the beneficial effect of interaction was insignificant,
even for fully rough footings. This was confirmed
experimentally by Hazell (2004), though only a few
of his tests were conducted on clay.
When considering the relevance of these findings
to the design of grillages or closely spaced footings
on soft offshore soils, it is important to note that the
theoretical studies by Mandel (1963) were confined
to homogeneous soil, and although in the experiments of Hazell (2004) there was a marked increase
of undrained strength with depth, the dimensionless
ratio kB/su0 was no more than 0.2 for the small
model footings tested (su0 = mudline strength intercept, k = rate of increase of strength with depth). In
water depths greater than a few hundred metres, the
undrained strength at seabed level can be as low as 2
to 10 kPa, increasing with depth at 1 to 2 kPa/m
(Randolph 2004). A typical offshore mudmat might
be 5 m wide, and if supported by a single strip footing (without perforations) this would imply typical
values of the ratio kB/su0 in the range 0.5 to 5. When
calculating the bearing capacity of an isolated footing at the upper end of this range, the influence of
non-homogeneity on the bearing capacity would certainly be accounted for, either by adopting appropriate plasticity solutions (Davis & Booker 1973,
Houlsby & Wroth 1983), or by selecting a representative strength su > su0. It is therefore of interest to
investigate the undrained bearing capacity of closely
spaced footings for a similar range of kB/su0, and to
assess the effect of using a mudmat with perforations
in place of a continuous foundation (this is sometimes done to save weight, and to make the structure
easier to remove). For a small degree of perforation
it might be envisaged that there will be arching over
the gap(s), such that there is no loss of bearing capacity, and even if some soil is squeezed through,
there may still be a beneficial interaction effect. Here
we investigate these issues using plasticity analyses.
2 BEARING CAPACITY ANALYSES
2.1 Isolated footings
The bearing capacity of an isolated strip footing on
non-homogeneous clay was first studied by Davis &
Booker (1973). They showed that for any value of
kB/su0 (zero to infinity) the stress and velocity fields
computed using the method of characteristics furnished lower and upper bounds that were coincident.
Davis & Booker’s analyses have since been verified
by several authors, and they can also be replicated
using the free computer program ABC (Martin
2004). Figure 1a shows the variation of the bearing
capacity factor Nc as a function of kB/su0. Note that
Nc is defined with respect to the mudline strength,
i.e. as Qu/Bsu0 where Qu is the ultimate bearing capacity (per unit run). Figure 1b shows, for the same
range of kB/su0, the extent of the zone of plastic deformation adjacent to each side of the footing. This
distance is important because it also corresponds to
the critical spacing at which parallel strip footings
first begin to interact and give an overall bearing capacity that is greater than the sum of the individual
capacities. As kB/su0 increases, the zone of plastic
deformation becomes smaller, so the footings need
to be closer for any interaction to occur.
Also shown in Figure 1 are the results of upper
bound calculations performed using the generalised
Hill- and Prandtl-type mechanisms of Kusakabe et
al. (1986), which were originally developed for circular footings on non-homogeneous clay. There is
close agreement with the exact curves when kB/su0 is
small, but this deteriorates somewhat with increasing
non-homogeneity, especially for the rough footing.
2.2 Interacting footings – methodology
In principle, bearing capacity analyses can be performed for an arbitrary number of parallel strip footings at arbitrary spacings, but the calculations can
become tedious, particularly when using the method
of characteristics. Figure 2 shows the two problems
considered here, both of which allow a favourable
exploitation of symmetry: a pair of parallel strips,
and an infinite number of parallel strips at equal
spacings. Although impossible to realise in practice,
the latter case is relevant to the interior members of a
grillage containing many bearing elements. Note that
in this paper S refers to the edge-to-edge spacing, not
the centre-to-centre spacing as preferred by some authors. Note also that the footings are assumed to be
rigidly connected, such that they move down together without any horizontal displacement or rotation (for a pair of closely spaced footings there is a
tendency for separation and tilting to occur).
The main bearing capacity analyses for interacting footings were performed using the method of
characteristics. A modified version of the the Matlab
program InterBC, developed by Hazell (2004) for interacting footings on a homogeneous c-φ-γ soil, was
used. For a pair of footings, the program considers
the right-hand footing and builds two meshes of
characteristics, one commencing from the exterior
soil surface, and one from the gap between the foot-
ing and the axis of symmetry (see Fig. 3). An iterative adjustment process is used to ensure that the two
meshes are fully compatible at their common point C
(same coordinates, same stresses). Having calculated
the stress field, the program works back through the
mesh and constructs the associated velocity field.
These calculations are more complicated than those
for an isolated footing, since the inward-flowing soil
crosses a velocity discontinuity AD. Note that for
non-homogeneous soil, the velocities outside AOD
are not always parallel to the characteristics.
Two separate calculations of the bearing capacity
are performed: one by integrating the stresses along
ACB (deducting the self-weight of the ‘false head’ if
applicable), and the other by equating the internal
and external work rates of the collapse mechanism.
In all of the analyses for this study it was found that,
as the mesh of characteristics was refined, the two
calculations of the bearing capacity converged to
identical values. While this indicates that there are
no regions of negative plastic work, it does not necessarily mean that the calculated bearing capacity is
the exact collapse load, since it has not been demonstrated that the stress field can be extended throughout the soil. Construction of such an extension is
straightforward for isolated footings (see e.g. Davis
& Booker 1973, Martin 2005), but for interacting
footings it would be necessary to incorporate a nonplastic zone to allow the major principal compression to flip from horizontal to vertical at some point
on the z axis. This is not easy, and suggests that the
method of characteristics cannot readily be used to
obtain strict lower bounds for interacting footings
(finite element limit analysis could be used).
For a pair of smooth footings, the ‘squeezing’
type failure of Figure 3a is always critical (except on
homogeneous soil, where there is no interaction effect, and an infinite number of one- and two-sided
mechanisms giving Nc = 2 + π can be devised). For a
pair of rough footings that are very closely spaced,
overall failure as a single footing of width 2B + S
may be more critical than the squeezing failure of
Figure 3b. When determining the variation of bearing capacity with spacing, it is always necessary to
check this alternative mechanism, for which the
bearing capacity can be determined using ABC.
Figure 4 shows some typical solutions for the
case of infinitely many, equally spaced footings on
non-homogeneous clay. Here the double symmetry
means that it is only necessary to analyse half of one
footing, so analysis using the method of characteristics is relatively straightforward. A single mesh is
constructed, starting from the soil surface and
‘bouncing’ characteristics off the centerline of the
gap as before. The mesh is adjusted until point C lies
directly beneath the centre of the footing, with the
major principal compression aligned vertically. Calculation of the associated velocity field – again incorporating a discontinuity along AD – can then be
performed. As in the two-footing case, it was always
found that the stress- and velocity-based calculations
of the bearing capacity converged to the same value
as the mesh was refined. This converged value
represents a rigorous upper bound, but not a rigorous
lower bound since the stress field is incomplete. If
the number of footings is truly infinite then there is
no alternative overall failure mechanism – squeezing
failure is the only option, and the bearing capacity
must approach infinity as the spacing tends to zero.
As well as the analyses performed using InterBC,
some simple Hill- and Prandtl-type upper bound
mechanisms were devised for the problems shown in
Figure 2. These were based on the mechanisms of
Kusakabe et al. (1986), but modified to allow a rigid
wedge of soil to be extruded vertically between the
interacting footings. In Figures 3 and 4 the outlines
of the optimal upper bound mechanisms are superimposed on the method of characteristics solutions,
and there is a fairly close correspondence between
the two. Note that for rough footings, the Prandtltype mechanism shown in Figures 3b and 4b is only
critical when kB/su0 is small; otherwise a Hill-type
mechanism (similar to Figs 3a, 4a) governs.
2.3 Interacting footings – results
Concentrating first on the pair of interacting footings, Figure 5 shows the variation of efficiency (as
defined at the start of the paper) with the normalised
spacing S/B. For a spacing of zero, the two footings
behave as a single footing of width 2B. Although
this has no net effect when the soil is homogeneous,
there is a beneficial interaction when the strength increases with depth since the influence of nonhomogeneity is enhanced (2kB/su0 is greater than
kB/su0, so the operative Nc is greater in Fig. 1a). For
smooth footings the efficiency immediately begins to
drop as soon as a gap is introduced, while for rough
footings there is a brief increase in efficiency prior to
the transition between overall and squeezing failure
(see Section 2.2). In all cases there is a gradual decline towards unit efficiency as the critical spacing
plotted in Figure 1b is approached.
The results for homogeneous soil (Fig. 5a) agree
with those of Mandel (1963): there is no gain in efficiency for a pair of smooth footings, and a peak of
just 1.07 (at S/B = 0.15) for a pair of rough footings.
When the strength increases with depth, the potential
gains in efficiency are rather greater, but the spacing
needs to be small (< 0.1B), and the benefit from interaction is almost all attributable to the effective
augmentation of kB/su0 rather than a genuine arching
effect. In fact, the rough footing curves in Figure 5
clearly show that the influence of arching diminishes
rapidly as non-homogeneity becomes more significant; when kB/su0 ≥ 2 there is an almost immediate
transition from overall failure to squeezing failure as
the spacing is increased from zero.
The predictions from the method of characteristics are consistent with those from the simple upper
bound analyses, shown as dotted lines in Figure 5.
The efficiency curve for a pair of rough footings on
homogeneous clay (Fig. 5a) also agrees remarkably
well with that obtained by Galloway (2004) using
the finite element program ABAQUS. This suggests
that the results obtained here from the method of
characteristics may well be exact, though it is not
immediately clear why the squeezing stress field
should suddenly cease to become extensible at the
same moment that overall failure becomes critical. It
is noteworthy – and surely no coincidence – that
Galloway’s analyses also predict a peak efficiency of
1.07 at a spacing of S/B = 0.15, coinciding with an
abrupt transition from overall to squeezing failure.
Corresponding results for an infinite number of
equally spaced footings are shown in Figure 6. The
higher the value of kB/su0, the closer the footings
need to be before there is any significant gain in efficiency (> say 10%). For a given spacing, the increase
in efficiency is greatest for the homogeneous soil,
and considerably higher (by a factor of up to 2) for
rough footings than for smooth footings. The critical
spacings at which the curves in Figure 6 reach unit
efficiency are the same as those in Figure 5.
The dotted lines in Figure 6 show that the upper
bound calculations give the same general trend, but
they overpredict the efficiency factor quite seriously
as the spacing becomes small. This is because the
simple collapse mechanisms (consisting only of fan
zones and rigid blocks) are unsuitable for modelling
the increasingly complex velocity field as S/B → 0.
3 APPLICATION: PERFORATED MUDMATS
A common situation involving closely spaced footings is the design of a mudmat with perforations. In
practice such structures would usually have square
or circular geometry, with the bearing elements taking the form of a bidirectional grillage, or perhaps
created by making a series of regularly spaced holes
in an initially solid base. Although the actual failure
mechanisms in these cases are complex (requiring
3D analysis), it is nevertheless instructive to perform
some simplified calculations for plane strain conditions, to examine the general effect of introducing
perforations. For brevity, only the two extreme cases
shown in Figure 7 are considered. The notation
adopted is the same as that used in the paper by
White et al. elsewhere in these proceedings: W is the
overall width of the mudmat, B* is the width of the
individual bearing elements, and R is the perforation
ratio (i.e. the fraction of W that has been removed).
In Figure 7a, the ratios S/B* and kB*/su0 both change
constantly as R is increased from zero (the former
increases and the latter decreases). In Figure 7b,
S/B* again increases as R is increased, but the ratio
REFERENCES
Davis, E.H. & Booker, J.R. 1973. The effect of increasing
strength with depth on the bearing capacity of clays.
Géotechnique 23(4): 551-563.
Galloway, M. 2004. Interaction between adjacent footings in
offshore foundation systems. Final year project report, Department of Engineering Science, University of Oxford.
Hazell, E.C.J. 2004. Interaction of closely spaced strip footings. Final year project report, Department of Engineering
Science, University of Oxford.
Houlsby, G.T. & Wroth, C.P. 1983. Calculation of stresses on
shallow penetrometers and footings. OUEL Report No.
1503/83, Department of Engineering Science, University of
Oxford.
Kusakabe, O., Suzuki, H. & Nakase, A. 1986. An upper-bound
calculation on bearing capacity of a circular footing on a
non-homogeneous clay. Soils and Found. 26(3): 143-148.
Mandel, J. 1963. Interférence plastique de fondations superficielles. Proc. Int. Conf. on Soil Mech., Budapest: 267-280.
Martin, C.M. 2004. ABC – Analysis of Bearing Capacity.
Software and documentation available for download from
www-civil.eng.ox.ac.uk/people/cmm/software/abc.
Martin, C.M. 2005. Exact bearing capacity calculations using
the method of characteristics. Issues lecture, Proc. 11th Int.
Conf. of IACMAG, Turin, to appear.
Randolph, M.F. 2004. Characterisation of soft sediments for
offshore applications. Keynote lecture, Proc. 2nd Int. Conf.
on Site Investigation, Porto.
(a)
10
Smooth
Rough
9
Nc = Qu/Bsu0
kB*/su0 is effectively zero from the outset (since the
number of perforations is very large, the width B* of
each individual bearing element is very small). In
both cases, as R is increased there eventually comes
a point where there is no longer any interaction between adjacent bearing elements of the mudmat.
Results for the single perforation scenario of Figure 7a are shown in Figure 8. If the overall width W
is taken as given, it is appropriate to characterise the
non-homogeneity by kW/su0, and to define a ‘gross’
bearing capacity factor with respect to W, i.e.
Nc = Qu/Wsu0. Using this convention, Nc for a
smooth, singly-perforated mudmat on homogeneous
soil decreases linearly from 2 + π to 0 as R increases
from 0 to 1. The other curves in Figure 8a show that
breaking W into two smaller widths causes most of
the benefit derived from the increase of strength with
depth to be lost quite quickly, followed by a more
gradual decline once R passes about 0.1. The corresponding curves for rough-based mudmats with a
single central perforation (Fig. 8b) have an initial
plateau where overall failure is more critical than
squeezing failure. However the beneficial effect of
this arching across the perforation is only significant
when kW/su0 is small.
Figure 9 presents results for the other scenario of
a mudmat with numerous perforations (Fig. 7b). In
this case both the smooth and rough curves exhibit
plateaus corresponding to overall failure, but once
squeezing failure becomes critical the bearing capacity starts to decline (somewhat more rapidly than in
Fig. 8). Regardless of the value of kW/su0, the appropriate squeezing curve is always that for homogenous soil, for the reason mentioned above: the ratio
kB*/su0 is effectively zero because B* << W. (The
curved envelopes in Figs 9a, b are simply alternative
presentations of the data for kB/su0 = 0 in Figs 6a, b.)
Perhaps the most interesting prediction in Figure 9 is
that for a rough mudmat on homogeneous soil, a perforation ratio in excess of 15% can be tolerated
without any reduction in capacity. Unfortunately this
is clearly not the case when kW/su0 > 0.
8
7
6
5
0
1
2
3
4
5
kB/su0
(b)
1
4 CONCLUSIONS
Smooth
Rough
This paper has presented theoretical solutions for the
vertical bearing capacity of rigidly connected, parallel strip footings on clay exhibiting a linear increase
of undrained strength with depth. Both a pair of footings and a large group of equally spaced footings
have been considered. Results have been obtained
using the method of characteristics, and confirmed
by independent upper bound calculations based on
simple mechanisms. The former results are believed
to represent exact solutions, though they have only
been established as upper bounds at this stage. A
practical application to the design of perforated
mudmats on soft offshore soil has been explored.
Critical S/B
0.75
0.5
0.25
0
0
1
2
3
4
5
kB/su0
Figure 1. Isolated strip footing on non-homogeneous clay: (a)
bearing capacity (b) critical edge-to-edge spacing for interaction between parallel footings. Results from simple UB calculations (after Kusakabe et al. 1986) shown dotted.
(a) kB/su0 = 0
(a)
B
S
1.3
B
x
Efficiency
su = su0 + kz
φu = 0
z
(b)
S
B
S
B
S
B
S
B
1.1
S
etc.
etc.
x
0
0.1
0.2
φu = 0
Figure 2. Parallel strip footings on non-homogeneous clay: (a)
pair of footings (b) many footings, equally spaced.
(a) Smooth
B
0.3
0.4
0.5
S/B
(b) kB/su0 = 1
1.3
Efficiency
z
C
Smooth effic. = 1 for all S/B
1
su = su0 + kz
A
Smooth
Rough
1.2
Smooth
Rough
1.2
1.1
O
D
1
0
0.1
0.2
0.3
0.4
0.5
S/B
(b) Rough
(c) kB/su0 = 2
1.3
A
B
D
Efficiency
O
C
Smooth
Rough
1.2
1.1
1
Figure 3. Pair of parallel strip footings on non-homogeneous
clay (kB/su0 = 1, S/B = 0.15): characteristics and velocity vectors, with mechanism outlines from simple UB calculations.
0
0.1
0.2
0.3
0.4
0.5
S/B
(d) kB/su0 = 5
1.3
A
C
Note: half of one
footing shown
O
D
(b) Rough
Efficiency
(a) Smooth
Smooth
Rough
1.2
1.1
1
0
A
O
C
Note: half of one
footing shown
D
Figure 4. Many parallel strip footings on non-homogeneous
clay (kB/su0 = 1, S/B = 0.15): characteristics and velocity vectors, with mechanism outlines from simple UB calculations.
0.1
0.2
0.3
0.4
0.5
S/B
Figure 5. Pair of parallel strip footings: variation of efficiency
with edge-to-edge spacing. Efficiency = ratio of overall (group)
capacity to sum of individual (isolated) capacities. Results from
simple UB calculations shown dotted.
(a) Smooth
(a) Smooth
1.5
10
9
8
7
6
5
4
3
2
1
0
Nc = Qu/Wsu0
Efficiency
1.4
kB/su0 =
0, 1, 2, 5
1.3
1.2
1.1
1
0
0.1
0.2
0.3
0.4
0.5
kW/su0 =
0, 1, 2, 5
0
0.1
0.2
S/B
(b) Rough
kB/su0 =
0, 1, 2, 5
Nc = Qu/Wsu0
1.4
Efficiency
0.4
0.5
0.3
0.4
0.5
(b) Rough
1.5
1.3
1.2
1.1
1
0
0.3
R = S/W
0.1
0.2
0.3
0.4
10
9
8
7
6
5
4
3
2
1
0
0.5
kW/su0 =
0, 1, 2, 5
0
0.1
0.2
S/B
R = S/W
Figure 6. Many parallel strip footings: variation of efficiency
with edge-to-edge spacing. Efficiency = ratio of overall (group)
capacity to sum of individual (isolated) capacities. Results from
simple UB calculations shown dotted.
Figure 8. Mudmat with single central perforation: variation of
bearing capacity with perforation ratio. Initial plateaus in (b)
correspond to overall failure of mudmat.
Nc = Qu/Wsu0
(a) Smooth
(a)
W
B*
S
B*
Perforation ratio R =
S
W
10
9
8
7
6
5
4
3
2
1
0
kW/s
=
kB/su0
u0 =
0, 1, 2, 5
0
0.1
envelope continues
linearly to (1, 0)
0.2
0.3
0.4
0.5
R = S/(B*+S)
(b)
(b) Rough
W
B*
S
Perforation ratio R =
S
B * +S
Figure 7. Mudmat with (a) single central perforation (b) many
identical perforations. Perforation ratio R = fraction of overall
width W that has been removed.
Nc = Qu/Wsu0
.....
10
9
8
7
6
5
4
3
2
1
0
kW/s
kB/su0
u0 =
0, 1, 2, 5
envelope continues
linearly to (1, 0)
0
0.1
0.2
0.3
0.4
0.5
R = S/(B*+S)
Figure 9. Mudmat with many identical perforations: variation
of bearing capacity with perforation ratio. Initial plateaus correspond to overall failure of mudmat.
Investigating 6 degree-of-freedom loading on shallow foundations
B.W. Byrne & G.T. Houlsby
Department of Engineering Science, University of Oxford, United Kingdom
ABSTRACT: Previous laboratory studies of the response of shallow foundations have only considered planar
loading. This paper describes the development of a loading device capable of applying general loading on
model shallow foundations. Loading involving all six degrees of freedom {vertical (V), horizontal (H2, H3),
torsion (Q) and overturning moment (M2, M3)}, can be applied experimentally to the model foundations.
Aspects of the design, including the loading rig configuration, development of a six degree-of-freedom load
cell, numerical control algorithms and an accurate displacement measuring system are described. Finally
results from initial experiments are presented that provide evidence for the generalisation of existing workhardening plasticity models from planar loading to the general loading condition.
1 INTRODUCTION
1.1 Motivation
The response of shallow foundations subjected to
general loading is an important area of civil
engineering, particularly in the offshore industry,
where foundations must be designed for loadings
due to harsh environmental conditions. These
conditions may lead to large vertical (V), horizontal
(H) and moment (M) loads on the foundations.
Whilst earlier studies considered overall stability,
more recent studies have attempted to model the
displacements, using model tests to calibrate work
hardening plasticity theories (Houlsby et al., 1999;
Martin and Houlsby, 2000, 2001; Byrne and
Houlsby, 2001; Cassidy et al. 2002; Houlsby and
Cassidy, 2002).
Recently, this work has focussed on suction
caisson foundations (Byrne et al., 2002; Byrne and
Houlsby, 2003). With geometry rather like an
upturned bucket, the caisson is simply installed by
sucking the water out, and thus forcing the skirts into
the seabed. This type of foundation has potential
applications in the developing offshore wind energy
industry. In this application the loading consists of
very high moment and horizontal loads, but low
vertical loads. This is a very different pattern of
loading from that experienced by heavier structures
in the oil and gas sector. In addition, the wind and
wave directions may not coincide, so the base shear
and moment are not in the same direction.
Considerable uncertainty surrounds how these
foundations may perform under these loading
conditions (Byrne and Houlsby, 2003).
1.2 Background Theory
Figure 1 shows a shallow foundation under three
degree-of-freedom loading as defined by Butterfield
et al. (1997). This problem has received much
attention over the past twenty years, and the load
displacement behaviour of the foundation can be
captured well by work-hardening plasticity theories
(as shown by the papers cited above). A key
component of the plasticity theories is the definition
of a suitable yield surface. Figure 2 shows the shape
of a yield surface that has been defined
experimentally, for shallow foundations under three
degree-of-freedom loading. This shape can be
expressed mathematically as equation 1.
 h
f = 
 ho
2
2
  m
h m
 − 2a
 + 
ho mo
  mo 
− β 12 v 2 β1 (1 − v )
(1)
2β2
=0
V
M
H
, m=
, h=
, ho is the
Vo
2 RVo
Vo
normalised horizontal load capacity, mo is the
normalised moment capacity, a is the eccentricity of
the
ellipse
in
the
h:m
plane,
where v =
 (β 1 + β 2 )( β1 + β 2 ) 
 and β1 and β2 are shaping
= 
β1
β2

β
β
1
2


2
β 12
2
2R
Reference position
H2
2r
w
Current position
M
M3
H
H3 3
θ
u
Q
V
Figure 1: Sign conventions for 3 degree-of-freedom loading
(Butterfield et al., 1997).
parameters for the section in the vertical load plane.
Numerous studies have identified the parameter
values for the yield surface for a variety of footing
types and for different soils - for example see
Houlsby et al. (1999) for shallow circular
foundations on sand, or Martin and Houlsby (2000)
for spudcans on clay. A natural extension of the
these theories is to six degrees-of-freedom and
Martin (1994) proposed an expression for this case:
2
2
2
H2
,
Vo
h3 =
2
2
 q 
m 
m 
h 
h 
f =  2  +  3  +  3  +  2  +  
 q o  (2)
 mo 
 mo 
 ho 
 ho 
 h m − h2 m3 
 − β12 v 2β1 (1 − v )2β 2 = 0
− 2a 3 2
ho mo


h2 =
where
M2
H3
,
Vo
m2 =
M2
,
2 RVo
M3
Q
and q =
. Figure 3 shows the
2 RVo
2 RVo
definitions of the loads from Butterfield et al.
(1997). The displacements work-conjugate to the
(V , H 2 , H 3 , Q, M 2 , M 3 )
are
loads
(w, u 2 , u 3 , ω, θ 2 , θ3 ) . There has been no systematic
study of footing response to full six degree-offreedom loading to verify the extension of the planar
loading theories to the general case. In the following
the development of a loading device capable of
applying the general loading is discussed, and some
m3 =
H
M/2R
V
Yield surface
Figure 2: Yield surface for shallow foundations.
V
1
Figure 3: 6dof loading on a circular foundation.
initial experimental results are presented that can be
used to verify equation 2.
2 DESIGN OF A 6 D-O-F LOADING RIG
2.1 The loading system
Previous experimental work at Oxford has used a
three degree-of-freedom (3dof) loading device
designed by Martin (1994). This planar loading
device achieves vertical and horizontal motion by
using a system of sliding plates, and rotational
movement by rotating the loading arm relative to
these plates. All motions are independent of each
other, and are each driven by a stepper motor – these
features are useful for implementing load and
displacement control systems. However, this type of
system would become too cumbersome for six
degree-of-freedom (6dof) motions, and so an
alternative approach is required. Typically, in
robotics applications, the Stewart Platform (Stewart,
1965) is considered to be the most elegant approach
to achieving 6dof movement of a platform. There are
numerous applications of this system in robotics, but
the authors do not believe the system has been used
for the testing of civil engineering structures, and in
particular testing of foundations The arrangement
described in this paper is a variant of the Stewart
platform, and similar arrangements are used, for
instance, in the automobile industry for dynamic
testing of vehicles.
The system uses six actuators which, at one end,
are connected to the loading platform, and at the
other are connected to a stiff reaction frame.
Provided that six properly arranged actuators are
used, and are pinned at both ends, then it is possible
to achieve 6dof motion of the loading platform by
changing the lengths of the actuators in a coordinated fashion. By careful selection of the
actuator geometry, it is possible to ensure that the
control problem is well-conditioned, so that
calculations proceed in a straightforward fashion.
The disadvantage with the Stewart Platform is
that the simple motions are not linearly or
independently related to the motion of any individual
actuator, unlike the 3dof system of Martin (1994).
Actuator
Lengths
A1
A2
A3
A4
A5
A6
Forward
Kinematics
Inverse
Kinematics
Platform
Pose
w
u2
u3
ω
θ2
θ3
Figure 5: Calculation procedures used in computer program.
Figure 4: Photos of the 6dof loading rig including a close-up of
the small LVDT measurement system.
Therefore, quite complex control routines are
required to ensure that all actuators move in concert
to achieve the desired motion. Figure 4 shows the
loading rig as constructed, showing three actuators
approximately vertical and three actuators
approximately horizontal. This arrangement ensures
that the problem is well conditioned, as the main
motions can be directly related to the motions of a
sub-set of the actuators. For example, to achieve
vertical movement the three vertical actuators must
move the same distance, while only a slight
adjustment of the horizontal actuators is required.
The actuators, supplied by Ultra Motion, are
linear actuators each powered by an Animatics
SmartMotor. This brushless DC servo-motor
incorporates an integrated control system featuring a
motion controller, encoder and amplifier. The
actuators have a maximum extension of 200mm and
can move at rates of up to 5mm/s. Commands to the
actuators can specify relative motions, position,
velocity or acceleration. The actuators are daisychained together and commands can be sent to
individual
actuators
and
then
executed
simultaneously with a global command. More
importantly, a number of moves can be downloaded
to on-board memory on the motors, and then
executed according to a synchronised clock system
common to all actuators. This makes it possible to
execute complicated platform motions provided one
can determine, in advance, a time history of the
individual actuator motions required.
2.2 The control program
A program has been written in Visual Basic to
control the loading system. The program allows
input of a sequence of moves in terms of the motions
(w, u 2 , u3 , ω, θ 2 , θ3 ) of the platform, known as the
pose. These motions can be described in terms of a
rotation and translation matrix (i.e. a transformation
matrix). This matrix can be applied to the coordinates of the pinned connections of the actuators
with the loading platform to produce a new set of coordinates for the platform in its new position. If the
co-ordinates of the other (fixed) ends of the actuators
are known, then it is possible to determine the
required lengths of each actuator. To move the
platform to the new position simply requires
extending/retracting each actuator to its required
length. This calculation procedure is known as the
inverse kinematics problem and is a simple
analytical calculation.
The opposite calculation, called the forward
kinematics problem, is not so straightforward, and
requires a numerical solution. If the lengths of each
actuator are known, then it is possible to calculate
the new pose of the platform. Within the actuators
are linear potentiometers that allow the user to
determine the current length of the actuator, and
therefore the pose of the platform. Both inverse and
forward kinematics procedures are used within the
software as shown in Figure 5.
A typical test proceeds by determining the initial
platform pose using the forward procedure. The user
then specifies a sequence of moves in terms of
platform pose. These moves are broken into a series
of small moves so that the non-linearity of motion of
each actuator can be captured. The inverse procedure
is used to calculate for each of the moves the
required length of each actuator. A file of actuator
lengths with time (position-time data) is recorded.
The relevant data from this file are sent to each
actuator, and each movement is executed
simultaneously. An on-board buffering system
allows moves to be downloaded to each actuator.
The actuators themselves use sophisticated control
processes to determine the velocity and accelerations
required, so that the actuator reaches each position at
the time required, thereby ensuring a smooth motion.
While the moves are being performed the control
program logs the data. In particular the actuator
lengths are recorded and the platform pose is
calculated and displayed.
2.3 The load cell
The load cell was constructed using a thin walled
cylinder of radius r = 27.5mm, wall thickness t =
0.475mm and length 70mm. It was fabricated from
Aluminium alloy with a Young’s Modulus of 72
GPa and a shear modulus of 27.1 GPa. The thin
2.4 Small LVDT system
Figure 6: The 6 degree-of-freedom load cell.
walled section was machined from a larger block,
leaving heavy end flanges. The transition from thinwalled section to flange was smoothed at an
appropriate radius to minimise stress concentrations.
A total of 32 strain gauges are fixed to the outer
surface of the cylinder to measure the appropriate
strains. Figure 6 shows the completed cell. The
strain gauges were arranged in six Wheatstone
bridge circuits, each corresponding to the
measurement of a particular load component. Each
circuit was fully compensated for temperature. Eight
gauges were used for the vertical and torque circuits,
and four gauges for the moment and horizontal load
circuits. The cell was calibrated by applying known
loads and measuring the output from all six circuits.
By varying the loads one at a time, it is possible to
determine components of the matrix X relating loads
to voltages in the equation C = XF where C is the
circuit output vector and F is the load vector. Figure
7 shows the results from the six circuits for changes
in the vertical load. The slopes for these six curves
represent the components of the part of the matrix
relating to vertical load (i.e. the first column of the
matrix X). Inverting X produces a six by six
calibration matrix that can be incorporated into the
control program, so that loads are calculated during
the experiment. Note that the design of the circuits is
such that the off-diagonal terms are small. This is
indicated in Figure 7 where only one circuit is
responsive to the change in applied load.
One determination of the platform pose is by using
the linear potentiometers within the actuators. This,
however, provides only a coarse measurement of the
platform pose. In particular there are issues of
electrical noise and rig stiffness which have a
significant impact on both the resolution and
accuracy of this measurement. To achieve a more
accurate determination of the foundation movement
a system of small LVDTs (20mm range) are used.
These are placed in a similar configuration to the
actuators, but supported on a separate frame as
shown in Figure 4. The program carries out the
forward kinematics calculation to determine the pose
of the platform, given the measured lengths of the
LVDTs. This allows very fine resolution of the
foundation movement to the order of a few microns
(Williams, 2005).
3 EXPERIMENTAL RESULTS
Some preliminary experimental results on a 150mm
diameter flat circular footing using only
displacement control are presented here. At the time
of writing load control routines were being
developed and are anticipated to be implemented in
the near future. The experiments were carried out on
Leighton Buzzard 14/25 silica sand. This is a
uniform sand with particle sizes ranging from
0.6mm to 1.18m. The maximum and minimum void
ratios are 0.79 and 0.49 respectively. The sand was
prepared in a loose state with a relative density
estimated as 20%. Fuller details of the experimental
work are reported by ap Gwilym (2004), Stiles
(2004) and Williams (2005). The experiments were
designed to determine the shape of the yield surface
in the six dimensions. A number of ‘swipe tests’
were performed with various combinations of
translations and rotations at a constant vertical
displacement. The swipe test has been used
extensively to determine the shape of yield surfaces,
see Martin (1994), Gottardi et al. (1999), Martin and
Houlsby (2000), Byrne and Houlsby (2001).
600
1
C1
C2
C3
C4
C5
C6
0.6
500
Vertical Load (N)
Circuit Output (V)
0.8
0.4
0.2
400
300
200
100
0
0
-0.2
0
50
100
150
200
250
300
350
400
450
500
Vertical Load (N)
Figure 7: Calibration curves for the loadcell under vertical
loading.
0
5
10
15
20
Vertical Displacement (mm)
Figure 8: Typical vertical loading results.
25
30
3.1 Vertical loading
0.14
0.12
Horizontal Swipes
H /Vo , M /2RVo , Q /Vo
Prior to carrying out the swipe tests it was necessary
to perform vertical loading tests, as these give
information for the hardening law. Five experiments
are shown in Figure 8, showing good repeatability of
the results. Note that the measurement of the
displacement is coarse, as in these experiments the
small LVDT system was not used.
0.1
0.08
Rotational Swipes
0.06
0.04
0.02
Twisting Swipes
3.2 Swipe tests
0
0
A number of swipe tests were performed to
investigate the suitability of equation 2. A typical
experimental result for a swipe test is shown in
Figure 9. In this test the footing was displaced
vertically to a pre-specified distance at which point
the vertical load reached approximately 530N. At
this load the footing was translated horizontally. The
figure shows that as the footing translates
horizontally the relevant horizontal load traces a path
around a yield surface. In this particular test the
translation was u2 so the only horizontal load
developed was H2. It is instructive to observe that
the other load components are all relatively
unaffected by the translation, as was expected. It is
also possible to carry out tests involving translations
u2, u3, -u2 and -u3. The results of these translations
are shown in Figure 10 where the load paths for H2
and H3 are plotted. Note that each of the tests starts
at a different vertical load. However, it is clear that
the magnitudes and the shapes of the load paths are
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
V /Vo
Figure 11: Results normalised by Vo.
similar for the different translations. This confirms
the expectation that similar load paths will be traced
out regardless of the translation direction. Similar
experiments were carried out for rotations and twists
with the same results (i.e. the results were
independent of direction).
The data, such as shown in Figure 10, can be
easily compared by normalising all the loads by Vo
as suggested in equation 2. Results are plotted in
Figure 11 for all possible pure horizontal, rotational
and twisting swipes with the negative swipes
reflected about the vertical load axis. It is clear that
the results depend on the mode (i.e. translation/
twisting/rotation) of the swipe test but not on the
direction. Equation 2 can be fitted to the above
results to give the parameter values in Table 1,
which are compared to data for footings on sand
under planar loading.
80
H2
H3
M2/2R
M3/2R
Q/2R
Loads, H , M /2R , Q /2R (N)
70
60
Table 1: Parameter values for work-hardening model
Parameter
50
40
ho
mo
qo
β1
β2
a
30
20
10
0
-10
-20
0
100
200
300
400
500
Figure 9: A horizontal swipe result.
80
Horizontal Loads, H 2, H 3 (N)
60
40
20
0
-20
-40
-60
-80
100
200
300
Vertical Load, V (N)
Figure 10: Horizontal swipe results.
400
500
Gottardi et
al., 1999
0.122
0.090
N/A
1.0
1.0
-0.223
Byrne and
Houlsby, 2001
0.154
0.094
N/A
0.82
0.82
-0.25
600
Vertical Load, V (N)
0
This
study
0.122
0.077
0.033
0.688
0.709
-0.212
600
In determining these parameters it was also
necessary to use results for combined swipes, that is
swipes involving simultaneous rotation and
translation and other combinations of movements.
For instance Figure 12 shows the results from a test
where a translation of u3 and rotation of -θ3 were
applied simultaneously to the foundation. A number
of these tests (twenty included in the above analysis)
were performed as they are necessary in determining
the fit, and in particular determining the parameter a
which gives the rotation of the ellipse in the h:m
plane. The test shown in Figure 12 could not have
been performed using the previous 3dof loading rig
as it involves non co-planar loads. Equally Figure 13
shows a test unique to the 6dof device in that during
the swipe test the footing was first rotated by θ2 and
in constructing the 6dof load cell and Chris Waddup
who made the actuator frame and LVDT support
frame. We also acknowledge the work carried out by
final year undergraduate project students: Llywelyn
ap Gwilym, Ed Stiles and Rachel Williams. The
experimental work described here was conducted by
these students under the direction of BWB.
0.1
Normalised Loads, H/Vo, M/2RVo
0.08
H3
M3/2R
0.06
0.04
0.02
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
-0.02
-0.04
6 REFERENCES
-0.06
-0.08
V/Vo
Figure 12: Non co-planar loading applied to the foundation.
0.08
M2/2R
M3/2R
0.07
0.06
0.05
M /2RVo
0.04
0.03
0.02
0.01
0
-0.01
-0.02
-0.03
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
V /V o
Figure 13: A swipe test where consecutive rotations are
performed.
then rotated by θ3. (i.e. orthogonal and consecutive
rotations). Initially under the rotation θ2 the load
path for M2 tracks around a yield surface. When θ2
stops and θ3 starts the response for M2 drops off and
the response for M3 picks up and eventually tracks
around the same yield surface that M2 tracked.
4 CONCLUSIONS
In this paper the description of a unique loading
device capable of applying six degree-of-freedom
motion to a model foundation is presented. The
resulting loads on the foundation are measured using
a six degree-of-freedom load cell. A number of
experiments, mainly displacement controlled swipe
tests, are presented and interpreted to provide
verification of the extension of a three degree-offreedom plasticity model to six degrees-of-freedom.
Further experimental work is required to verify the
model fully.
5 ACKNOWLEDGEMENTS
The authors acknowledge the funding from the
Lubbock Trustees (pilot project grant), the Royal
Society (equipment grant), the Department of
Engineering Science at Oxford University and
EPSRC. We acknowledge the work of Clive Baker
ap Gwilym, T.Ll. ab E. (2004). Control of a six degree of
freedom loading rig. Fourth year project report, Department
of Engineering Science, University of Oxford.
Butterfield, R., Houlsby, G.T. and Gottardi, G. (1997).
Standardised sign conventions and notation for generally
loaded foundations. Géotechnique 47, No 5, pp 1051-1054;
corrigendum Géotechnique 48, No 1, p 157.
Byrne, B.W. and Houlsby, G.T. (2001). Observations of
footing behaviour on loose carbonate sands. Géotechnique
51, No 5, pp 463-466.
Byrne, B.W. and Houlsby, G.T. (2003). Foundations for
offshore wind turbines. Phil. Trans. Roy. Soc. A 361, Dec.,
pp 2909-2930.
Byrne, B.W., Houlsby, G.T., Martin, C.M. and Fish, P.M.
(2002). Suction caisson foundations for offshore wind
turbines. Wind Engineering 26, No 3.
Cassidy, M.J., Byrne, B.W. and Houlsby, G.T. (2002).
Modelling the behaviour of a circular footing under
combined loading on loose carbonate sand. Géotechnique
52, No 10, pp 705-712.
Gottardi, G., Houlsby, G.T. and Butterfield, R. (1999). The
plastic response of circular footings on sand under general
planar loading. Géotechnique 49, No 4, pp 453-470.
Houlsby, G.T. and Cassidy, M.J. (2002). A plasticity model for
the behaviour of footings on sand under combined loading.
Géotechnique 52, No 2, pp 117-129.
Martin, C.M. (1994). Physical and numerical modelling of
offshore foundations under combined loads. DPhil Thesis,
University of Oxford.
Martin, C.M. and Houlsby, G.T. (2000). Combined loading of
spudcan foundations on clay: laboratory tests. Géotechnique
50, No 4, pp 325-338.
Martin, C.M. and Houlsby, G.T. (2001). Combined loading of
spudcan foundations on clay: numerical modelling.
Géotechnique 51, No 8, pp 687-700.
Stewart, D. (1965) A platform with six degrees of freedom. The
Institution of Mechanical Engineers 180, No 15, pp371384.
Stiles, E. (2004). Experiments using a six degree of freedom
loading rig. Fourth year project report, Department of
Engineering Science, University of Oxford.
Williams, R. (2005). Six degree of freedom loading tests on
sand and clay. Fourth year project report, Department of
Engineering Science, University of Oxford, in preparation.
The tensile capacity of suction caissons in sand under rapid loading
Guy T. Houlsby, Richard B. Kelly & Byron W. Byrne
Department of Engineering Science, Oxford University
ABSTRACT: We develop here a simplified theory for predicting the capacity of a suction caisson in sand,
when it is subjected to rapid tensile loading. The capacity is found to be determined principally by the rate of
pullout (relative to the permeability of the sand), and by the ambient pore pressure (which determines whether
or not the water cavitates beneath the caisson). The calculation procedure depends on first predicting the
suction beneath the caisson lid, and then further calculating the tensile load. The method is based on similar
principles to a previously published method for suction-assisted caisson installation (Houlsby and Byrne,
2005). In the analysis a number of different cases are identified, and successful comparisons with
experimental data are achieved for cases in which the pore water either does or does not cavitate.
1. INTRODUCTION
2. TENSILE CAPACITY CALCULATIONS
Suction caissons are an option for the foundations
for offshore structures. Under large environmental
loads the upwind foundations of a multiple-caisson
foundation might be subjected to tensile loads.
Recent research indicates that serviceability
requirements will often dictate that, under working
and frequently encountered storm loads, tensile
loads on caissons should be avoided, as they are
accompanied by large displacements. However, it
may be appropriate to design structures so that under
certain extreme conditions the caissons are allowed
to undergo tension. It is therefore necessary to have
a means of estimating the tensile capacity of a
caisson foundation, whilst recognising that large
displacements may be necessary to mobilise this
capacity. The calculations are also relevant to the
holding capacity of caisson anchors subjected to
pure vertical load, and to calculation of forces
necessary to extract a caisson rapidly (for whatever
reason).
Under rapid tensile loading, a suction caisson in
sand will exhibit a limiting load which will typically
consist of a suction developed within the caisson,
and friction on the outer wall. However, a number of
different possible modes of failure exist. The
purpose of this paper note is to set out simple
calculations for capacities under various failure
modes, and to compare these with experimental
results.
2.1 Drained capacity
If the tensile load is applied very slowly, then pore
pressures will be small, and a fully drained
calculation is applicable for calculating the capacity.
For the purposes of calculation an idealised case of a
foundation on a homogeneous deposit of sand is
considered here.
The resistance on the caisson is calculated as the
sum of friction on the outside and the inside of the
skirt. The effective stresses on the annular rim are
likely to be sufficiently small that they can be
neglected, and it is assumed that the soil breaks
contact with the lid of the caisson. The frictional
terms are calculated in the same way as for the
installation calculation (Houlsby and Byrne, 2005),
by calculating the vertical effective stress adjacent to
the caisson, then assuming that horizontal effective
stress is a factor K times the vertical effective stress.
Assuming that the mobilised angle of friction
between the caisson wall and the soil is δ then we
obtain the result that the shear stress acting on the
caisson is σ′v K tan δ . Note that in the subsequent
analysis the values of K and δ never appear
separately, but only in the combination K tan δ , so it
is not possible to separate out the effects of these
two variables. Allowance is made, however, for the
possibility of different values of K tan δ acting on
the outside and inside of the caisson. A difference
between this analysis and conventional pile design is
that the contribution of friction in reducing the
vertical stress further down the caisson is taken into
account.
If, as a preliminary, no account is taken of the
reduction of vertical stress close to the caisson due
to the frictional forces further up the caisson, then
the tensile vertical load on the caisson for
penetration to depth h is given by:
V′ = −
′ 2
γ ′h 2
(K tan δ )o (πDo ) − γ h (K tan δ )i (πDi )
2
2
friction on outside
(1)
friction on inside
where the dimensions are as in Figure 1, and γ ′ is
the effective weight of the soil. V ′ is the buoyant
weight of the caisson and structure.
A check should always be made that the friction
calculated inside the caisson does not exceed the
weight of the trapped soil plug γ ′hπDi2 4 .
Ignoring the reduction of the stress in this case
proves unconservative (i.e. it overestimates the force
that can be developed), so we develop here a theory
which takes this effect into account. Consider first
the soil within the caisson. Assuming that the
vertical effective stress is constant across the section
of the caisson, the vertical equilibrium equation for a
disc of soil within the caisson (Figure 2) leads to:
σ′ (K tan δ )i (πDi )
4σ′ (K tan δ )i
dσ′v
= γ′ − v
= γ′ − v
2
dz
Di
πDi 4
(2)
Writing Di (4(K tan δ )i ) = Z i , Eq. (2) becomes
dσ′v σ′v
+
= γ′ ,
which
has
the
solution
dz
Zi
σ′v = γ ′Z i (1 − exp(− z Z i )) for σ′v = 0 at z = 0 . The
total frictional terms depend on the integral of the
vertical effective stress with depth, and we can also
h
obtain
∫ σ′v dz = γ ′Z i (exp(− h Z i ) − 1 + (h Z i )) .
2
For
0
V'
h
above ∫ σ′v dz = γ ′Z i2 y (h Z i ) .
0
A similar analysis follows for the stress on the
outside of the caisson. We assume that (a) there is a
zone between diameters Do and Dm = mDo in
which the vertical stress is reduced through the
action of the upward friction from the caisson, (b)
within this zone the vertical stress does not vary with
radial coordinate and (c) there is no shear stress on
vertical planes at diameter Dm . We then obtain the
same results as for the inside of the caisson, but with
(
)
Z i replaced by Z o = Do m 2 − 1 (4(K tan δ )o ) .
Alternative assumptions could be made for the
variation of Dm with depth, but at present there is
little evidence to justify any more sophisticated
approach. If Dm is taken as a variable, then the
differential equation for vertical stress will usually
need to be integrated numerically.
Accounting for the effects of stress enhancement,
Eq. (1) becomes modified to:
h
V ′ = ∫ σ′vo dz (K tan δ )o (πDo ) +
0
h
(3)
∫ σ′vi dz (K tan δ)i (πDi )
0
In the special case where m is taken as a constant
and uniform stress is assumed within the caisson this
becomes:
 h 
 (K tan δ )o (πDo )
V ′ = − γ ′Z o2 y 
 Zo 
 h 
− γ ′Z i2 y   (K tan δ )i (πDi )
 Zi 
(4)
σ'vπDi2/4
σ'v(Ktanδ)iπDidz
h
γ'(πDi2/4)dz
z
Di
Do
Figure 1: Caisson geometry
Mudline
small h Z i the integral simplifies to γ ′h 2 2 as in
Eq. (1). For brevity in the following we shall write
the function y ( x ) = (exp(− x ) − 1 + x ) , so that in the
(σ'v + dσ'v)πDi2/4
Figure 2: Vertical equilibrium of a slice of soil within the
caisson
The calculation accounting for stress reduction
obviates the need to check that the internal friction
does not exceed the soil plug weight, as the capacity
asymptotically approaches that value at large h.
2.2 Tensile capacity in the presence of suction
If the caisson is extracted more rapidly, then
transient excess pore pressures will occur, and the
suction within the caisson will need to be taken into
account. We return later to the calculation of the
relationship between the rate of movement and the
suction, but first address the calculation of load in
terms of the suction. If the pressure in the caisson is
s with respect to the ambient seabed water pressure,
i.e. the absolute pressure in the caisson is
p a + γ w hw − s (where p a is atmospheric pressure,
γ w is the unit weight of water and hw the water
depth), then we at first assume that the excess pore
pressure at the tip of the caisson is as , i.e. the
absolute pressure is pa + γ w (hw + h ) − as . There is
therefore an average downward hydraulic gradient of
as γ w h on the outside of the caisson and upward
hydraulic gradient of (1 − a )s γ w h on the inside.
We assume that the distribution of pore pressure
on the inside and outside of the caisson is linear with
depth. A detailed flow net analysis shows that this
approximation is reasonable. The solutions for the
vertical stresses inside and outside the caisson are
exactly as before, except that γ ′ is replaced by
γ ′ + as h outside the caisson and by γ ′ − (1 − a ) s h
inside the caisson. The capacity, accounting for the
pressure differential across the top of the caisson and
pore pressure on the rim (only relevant for a thick
caisson), is again calculated as the sum of the
external and internal frictional terms:
 πD 2
V ′ + s i
 4

(
2
2


 + as π Do − Di


4


h
h
0
0
) =


(5)
In the special case of m constant and a uniform
stress assumed within the caisson, this gives:
(
)
2
2 


 + as π Do − Di  =



4



 h 
as 

 (K tan δ )o (πDo )
−  γ ′ +  Z o2 y
h

 Zo 
(1 − a )s Z 2 y h

−  γ′ −
 i 
h 

 Zi

 (K tan δ )i (πDi )

(
)
 πD 2 
 π Do2 − Di2 
=
V ′ + s i  + as
 4 


4




 h 
s
 (K tan δ )o (πDo )
−   Z o2 y
h
 Zo 
(7)
In the case either that the thickness of the caisson
is small, or that a ≈ 1 this simplifies to the following
(writing the outer diameter as D, and the caisson
area πD 2 4 = A ):
s
h
V ′ = −  Z 2 y   (K tan δ )(πD ) − sA
h
Z

  4Z 2   h 
 y  (K tan δ )
= − sA1 + 

  Dh   Z 


 
(
(8)
)
where Z = D m 2 − 1 (4 K tan δ ) . Neglecting the
effects of stress reduction would give:
  2h 

V ′ = − sA1 +   (K tan δ )
 D

(9)
which means that the capacity is simply calculated
by applying a linearly varying factor to the suction
force beneath the lid.
2.3 Undrained failure
∫ σ′vo dz (K tan δ)o (πDo ) + ∫ σ′vi dz (K tan δ)i (πDi )
 πD 2
′
V + s i
 4

We can often make a further simplifying
assumption, that the suction is sufficiently large that
the soil within the caisson liquefies and therefore
(1 − a )s = 0 . For a large suction this means that
γ′ −
h
a ≈ 1 and almost all of the suction appears at the
caisson tip. The above rearranges to give
as s
γ′ +
= , and equation (6) can be simplified to:
h h
(6)
A further condition should be considered: that of
“undrained failure” of the sand. In any dilative sand,
however, the pore pressures developed under
undrained conditions are potentially so large that
invariable (except in very deep water) the cavitation
mechanism would intervene first. Since the
undrained strength of sand is in any case very
difficult to determine, we do not pursue this case
here.
3. RELATIONSHIP BETWEEN SUCTION
AND DISPLACEMENT RATE
At low displacement rates, the rate of influx of water
q to the caisson can be calculated by Darcy’s law,
and equated to the rate of displacement times the
area of the caisson. Flow calculations were presented
by Houlsby and Byrne (2005), and yield:
πDi2
k Ds
dh
q= o F =−
4 dt
γw
(10)
where F is a dimensionless factor as determined by
the procedures in Houlsby and Byrne (2005), which
may be fitted approximately by the equation
F = 3.6 (1 + 5h D ) for 0.1 ≤ h D ≤ 0.8 .
If the displacement rate is increased, the above
condition is interrupted by one of two conditions (a)
the suction becomes large enough for liquefaction of
the sand within the caisson to occur or (b) cavitation
occurs within the caisson.
When liquefaction occurs, the permeability of the
liquefied sand increases to a large value, with the
result that the a factor in the calculation of the load
changes (as noted above) to near unity. The
displacement rate may still be estimated from a flow
calculation, but the appropriate boundary condition
now becomes one of the suction applied at the base
rather than top of the caisson. Modified values of F
(termed FL for this case) are given in Figure 3, and
may be fitted approximately by the equation
F = 1.75 + 1.9 exp(− 5h D ) for 0.1 ≤ h D ≤ 1.0 .
When cavitation occurs, either before or after
liquefaction, the displacement rate becomes
unlimited and (assuming that cavitation occurs at an
absolute pressure fp a where f is a constant), the
suction will be constant and determined by
pa + γ w hw − s = fp a , or s = p a (1 − f ) + γ w hw . In
practice it appears that the factor f is near zero.
4. SUMMARY OF ANALYSIS CASES
The following summary presents equations for the
above cases, for a thin-walled caisson. To simplify
the equations we neglecting here the stress reduction
effect, although this should be included in more
Dimensionless flow factor FL
3.5
3.0
2.5
2.0
1.5
1.0
0.5
accurate calculations:
(a) Small − dh dt
π Dγ w dh
s=−
(from Eq. (10)) and:
4 F k o dt
V ′ = − sA −

as 
(1 − a )s  (K tan δ)  πDh 2

  γ ′ +  (K tan δ )o +  γ ′ −

i
h
h 


 2
(for − dh dt = 0 , s = 0 and these reduce to the
equations for the fully drained case).
(b) Liquefaction without cavitation
Onset of liquefaction occurs at s =
s=−
π Dγ w dh
and:
4 FL k o dt
γ ′h
, after that
(1 − a )
  2h 

V ′ = − sA1 +   (K tan δ )o 
 D

Note that this will imply a sudden jump in s and V ′
at the onset of liquefaction.
(c) Cavitation without liquefaction
Onset of cavitation occurs at s = p a (1 − f ) + γ w hw .
After that − dh dt is unbounded, s is constant and:
V ′ = − sA −

(1 − a )s  (K tan δ)  πDh 2
as 

  γ ′ +  (K tan δ )o +  γ ′ −

i
h 
h


 2
as in case(a).
(d) Cavitation with liquefaction
Since s is constant once cavitation occurs, this
condition can only occur when liquefaction occurs
before cavitation. Onset of cavitation is at
s = p a (1 − f ) + γ w hw , after which − dh dt is
unbounded, s is constant and:
  2h 

V ′ = − sA1 +   (K tan δ )o  as in case (b).
 D

Note that the above cases only occur in order (a),
(b), (d) or (a), (c). When several possibilities exist
for calculating load capacity it is often true that the
correct case is simply found by calculating all cases
and then taking the lowest value. Note in this
analysis that this simple approach cannot be adopted
as the onset of some states can preclude other cases
occurring, and the calculated load is not necessarily
the lowest of the cases.
0.0
0.0
0.2
0.4
0.6
0.8
1.0
Aspect ratio h/D
Figure 3: Dimensionless flow factor for liquefaction case
5. COMPARISONS WITH DATA
We present here a number of pullout tests conducted
two sands and at different pullout rates. The tests
pressure (kPa)
0.0
3042.8
-20.0
3043.3
-40.0
3043.8
3044.3
3044.8
3045.3
3044.8
3045.3
Experiment
Theory
-60.0
-80.0
-100.0
-120.0
t (s)
Figure 4: Pressure v. time for Test 9
2.0
0.0
3042.8
-2.0
3043.3
3043.8
3044.3
Experiment
Theory
V (kN)
-4.0
-6.0
-8.0
-10.0
-12.0
t (s)
Figure 5: Vertical load v. time for Test 9
were conducted in a pressure chamber: some tests at
an ambient (mudline) water pressure equal to
atmospheric, and some at atmospheric plus 200kPa.
The model caisson was 280mm diameter, 180mm
skirt length. In the following the loads presented
include the caisson weight.
The first test reported here (Test 9) was
conducted on Redhill Sand, at a pullout rate of
100mm/s and atmospheric pressure. Figure 4 shows
the record of suction developed beneath the lid of
the caisson against time, and Figure 5 shows the
corresponding vertical load. It can be seen that (with
a minor initial fluctuation) the suction rapidly
approaches 100kPa, at which stage cavitation occurs.
At around 3044.5s there is a sudden loss of both
suction and vertical load, but this is of little practical
interest since by then the displacements are
enormous and about three-quarters of the caisson
had been pulled out of the soil.
Figure 6 shows the ratio V / sA , showing that this
ratio remains approximately constant during most of
the pullout.
It can readily be shown that the suction in this
case rapidly increased to sufficient value to cause
liquefaction (which would occur at a suction of only
about 3kPa), and that the relevant case for analysis
here is case (d). The predicted values from the
theory described above (including stress reduction)
are also shown on each of Figures 4 to 6, and it is
clear that the theory (whilst not capturing some of
the detail at the beginning of the pullout) predicts the
broad trends of the test correctly.
Figures 7 and 8 show corresponding results for
Test 10 (at the same pullout rate) but at an ambient
pressure of atmospheric plus 200kPa. The suctions
developed at this rate of loading are insufficient to
cause cavitation, which would occur at -300kPa
relative to ambient. It can be seen that again the
theory predicts the overall pattern of behaviour well.
This time it is case (b) that applies. The fluctuations
in predicted suction (and hence load) are due to
minor variations in the calculated velocity of
extraction.
Figures 9 and 10 show the results from Test 11,
which is directly comparable to Test 9, but this time
0
4257
4258
4258.5
-100
4259.5
4259
4259.5
Experiment
Theory
-150
-200
-300
t (s)
Figure 7: Pressure v. time for Test 10
5.0
0.0
4257
1.5
V (kN)
-5.0
1.0
Experiment
Theory
0.5
-10.0
4257.5
4258
4258.5
Experiment
Theory
-15.0
-20.0
-25.0
0.0
3042.8
4259
-250
2.0
V / sA
4257.5
-50
pressure (kPa)
20.0
3043.3
3043.8
3044.3
t (s)
Figure 6: V/sA v. time for Test 9
3044.8
3045.3
-30.0
t (s)
Figure 8: Vertical load v. time for Test 10
at a pullout rate of only 5mm/s. Although the
suctions are sufficient to cause liquefaction, the
pullout rate is such that the suction is sufficiently
small so that cavitation does not occur, and the
vertical loads are correspondingly lower too. The
predicted suction and load are also shown on the
Figures. The match to the data could be improved by
adjusting the permeability, but the value used in the
predictions were deliberately kept the same for all
three tests discussed. The permeability value used
was k = 0.5 × 10 −3 m/s, which is somewhat higher
than estimated previously for this sand (Kelly et al.
2004). The other parameters used are K tan δ = 0.7
and m = 1.5 .
Finally, Figures 11 and 12 present equivalent data
for a test on HP5 sand, which is much finer that
Redhill Sand, and has an estimated permeability of
k = 0.2 × 10 −4 m/s. The extraction rate was 25mm/s,
and in this case, although the extraction rate is
lower, the pore pressures are sufficient to cause
cavitation even with the ambient pressure of
atmospheric plus 200kPa.
The predicted and measured values of maximum
tensile load for the three tests on Redhill sand and
one on HP5 are shown in Table 1. The order of
magnitude of the tensile load is correctly predicted
in all cases, even though the actual capacity of the
caisson varies greatly in the different tests.
Table 1: Predicted and measured tensile loads
Max. tensile load (kN)
Predicted Measured
Test 11 (5mm/s, 0kPa)
1.1
2.4
Test 9 (100mm/s, 0kPa)
10.1
11.1
Test 10 (100mm/s, 200kPa)
25.6
24.2
HP5 sand:
30.6
33.2
Test 23 (25mm/s, 200kPa)
Test
6. CONCLUSIONS
In this paper we develop a simplified theory for
predicting the maximum tensile capacity of a caisson
foundation in sand. The calculated capacity depends
critically on the rate of pullout (in relation to the
permeability) and the ambient water pressure (which
determines whether cavitation occurs). The theory is
used successfully to explain widely differing
experimental results for caissons pulled out under
different conditions.
REFERENCES
Houlsby, G.T. and Byrne, B.W., 2005. Design procedures for
installation of suction caissons in sand, Proc. ICE,
Geotechnical Engineering, in press.
Kelly, R.B., Byrne, B.W., Houlsby, G.T. and Martin, C.M.,
2004. Tensile loading of model caisson foundations for
structures on sand, Proc. ISOPE, Toulon, Vol. 2, 638-641
50
5
2990
3000
3010
3020
0
8890
-50
3030
pressure (kPa)
pressure (kPa)
0
2980
-5
-10
8894
8896
8898
8900
8902
-100
Experiment
Theory
-150
-200
-250
Experiment
Theory
-15
8892
-300
-350
-20
t (s)
t (s)
Figure 9: Pressure v. time for Test 11
Figure 11: Pressure v. time for Test 23
10
1.0
0.5
2990
3000
3010
3020
0
8890
3030
8892
8894
8896
8898
8900
-10
V (kN)
V (kN)
0.0
2980
-0.5
-1.0
-20
-1.5
Experiment
Theory
-2.0
Experiment
Theory
-30
-2.5
-3.0
t (s)
Figure 10: Vertical load v. time for Test 11
-40
t (s)
Figure 12: Vertical load v. time for Test 23
8902
The theoretical modelling of a suction caisson foundation using
hyperplasticity theory
Lam Nguyen-Sy & Guy T. Houlsby
Department of Engineering Science, Oxford University
ABSTRACT: A theoretical model for the analysis of suction caison foundations, based on a thermodynamic
framework (Houlsby and Puzrin, 2000) and the macro-element concept is presented. The elastic-plastic
response is first described in terms of a single-yield-surface model, using a non-associated flow rule. To
capture hysteresis phenomena, this model is then extended to a multiple yield surface model. The installation
of the caisson using suction is also analysed as part of the theoretical model. Some preliminary numerical
results are given as demonstrations of the capabilities of the model.
1. INTRODUCTION
In developments of offshore wind turbines, the
foundations account for a significant fraction of the
overall installed cost, approximately 15% to 40% of
the total cost (Houlsby and Byrne, 2000). To satisfy
the increasing need for renewable energy, there are a
number of offshore wind farms to be constructed off
the coast of the UK within the next few years. The
possible use of caisson foundations for these
turbines is therefore an important economical issue.
From previous research, there are elastic-plastic
theoretical models available for analysis of shallow
offshore foundations, such as “Model B” for a jackup footing on clay (Martin, 1994) and “Model C” for
footings on sand (Cassidy, 1999). These models are
based on the idea of a “macro-element”, representing
the foundation behaviour. The loading on the footing
is represented by force resultants at a chosen
reference point on the footing, and the movement by
the corresponding displacements of this point. In this
paper, a macro-element model for a caisson is
presented in outline. The main goal of this work is to
establish a theoretical framework to model correctly
the cyclic behaviour of a caisson foundation, and
this necessitates extension of previous modelling
concepts to use of multiple yield surfaces.
2. CAISSON FOUNDATIONS
A caisson foundation consists essentially of two
parts: a circular top plate and a perimeter skirt, see
Figure 1. The whole foundation is installed by the
combination of gravity and suction within the
caisson. In Figure 1, d is the distance between the
Load Reference Point (LRP) and an idealised soil
surface position just as installation begins. The
position of the LRP is arbitrary, but is conveniently
taken at the joint between the caisson and the
support structure. The conventions for forces are
shown in Figure 2. The forces VR, H2R, H3R, QR, M2R,
and M3R are applied at the LRP. In the analysis,
however, it is often convenient to use the force
system, σ i = (V, H 2 , H 3 , Q, M 2 , M 3 ) at the
idealised soil surface level. The relationships
between these two systems are: V = VR; H2 = H2R; H3
= H3R; Q = QR; M2 = M2R + dH3R; M3 = M3R – dH2R
The displacement vector at soil surface level is
ε i = (w, u 2 , u 3 , ω, θ 2 , θ 3 ) . The corresponding
displacements at the LRP are given w = wR; u2 = u2R
+ dθ3R; u3 = u3R - dθ2R; ω = ωR; θ2 = θ2R; θ3 = θ3R.
3. RATE-INDEPENDENT SINGLE YIELD
SURFACE HYPERPLASTICITY MODEL
Based on the hyperplasticity framework (Houlsby
and Puzrin, 2000), a mechanical model can be
derived from two scalar functions: the Gibbs free
energy g, and either the dissipation function d or
yield function y. The yield function is used here,
since it can be identified directly from test results.
The macro-element model is expressed in terms of
the force vector σ i and displacement vector ε i . It is
also necessary to introduce the generalized force
vector χ i = χV , χ H 2 , χ H 3 , χ Q , χ M 2 , χ M 3 and
(
)
finally the plastic displacement (or internal variable)
(
)
vector α i = αV , α H 2 , α H 3 , α Q , α M 2 , α M 3 . In
general, for a model without elastic-plastic coupling,
the free energy g and yield function y can be
expressed as:
g = g1 (σ i ) − σ i α i + g 2 (α i )
(1)
y = y (σ i , χ i , α i ) = 0
(2)
VR
HR
g2 =
LRP
MR
+
mudline
2
*
H1*αV 2 H 3*α H 2 2 H 3*α H 3 2 H 5 α Q
+
+
+
2
2
2
2
H 2*α M 2 2
2
+
H 2*α M 3 2
2
(4)
+ H 4* (α M 2 α H 3 − α M 3 α H 2 )
where H1* … H5* are hardening parameters which can
conveniently be expressed in terms of the elastic
stiffness factors K1 … K5. Details of these functions
are discussed later. For the time being, however, we
simply take g 2 = 0 for the single surface model
since the yield surface in this case does not undergo
kinematic hardening.
d
V
H
foundation, in which the elastic stiffness of the
caisson itself is taken into account.
The g2 term, which is the work of the plastic
displacements, specifies the kinematic hardening of
the model. A simple linear hardening relationship is
achieved if g2 is a quadratic function of the plastic
strains, and this form will later be used here as the
basis for the multiple-surface model:
M
Soil surface level
2R
Figure 1. Geometry of caisson footing
3.2 The yield function y
H2R
2R
M3R
M2R
H 3R
2
d
QR
H2
VR
M2
M3
H3
3
Q
1
V
Figure 2. Conventions for forces
3.1 The Gibbs free energy function g
In Eq. (1), g1 represents the elastic response of the
foundation and is independent of plastic
displacements. For linear elasticity it takes the
following form:
g1 = −
K H 2 K H 2 Q2
V2
− 2 2 − 2 3 −
2K1
2D
2D
2K 5
K 3 M 22 K 3 M 32 K 4 H 3 M 2 K 4 H 2 M 3
−
−
+
−
2D
2D
D
D
(3)
in which: K1 = 2GRk1; K2 = GR3k2 – 8GR2k4d +
2GRd2k3; K3 = 2GRk3; K4 = 4GR2k4 – 2GRk3d; K5 =
8GR3k5 and D = K2K3 – K42. G is the shear modulus
of the soil, and the factors k1 … k5 are dimensionless
stiffness coefficients as proposed by Doherty et al.
(2004), who give elastic solutions for a caisson
In the hyperplasticity framework (Houlsby and
Puzrin, 2000), the yield function has been
regconised as the singular Legendre transform of the
dissipation function d in case of rate-independent
materials. The yield surface is therefore expressed as
a function of the generalised forces χi. Furthermore,
the appearance of force components, (analogous to
the true stresses σij in continuum plasticity models),
in the form of the yield function leads automatically
to non-associated flow rules (Collins and Houlsby,
1997), which are known as the appropriate to
describe soil behaviour. Consequently, “association
factors” which play an interpolation role between
true and generalised forces are proposed in the yield
function. Martin (1994) and Cassidy (1999) have
establised the yield functions in elastic-plastic
models (Model B and Model C for jack-up and
circular footings of offshore structures on clay and
sand). Based on these results, a yield function for a
caisson footing is proposed with certain
modifications to include aspects such as the nonassociated flow rule:
y = t − Sβ12 v1 + t 0
β1
1 − v2
β2
=0
(5)
in which:
S = sgn[(v1 + t 0 )(1 − v 2 )] , β12 =
further definitions follow below:
 β1 + β 2

 1 + t0
β1



β1 +β2
(β1 ) (β 2 )
β2
and
t = h22 + h32 + m 22 + m32 + q 2 + 2a(h2 m3 − h3 m 2 ) (6)
a χ + (1 − aV 1 )(V − ρV
v1 = V 1 V
V0
)
a (χ + ρV ) + (1 − aV 2 )V − ρV
v2 = V 2 V
V0
a (χ
+ ρ H 2 ) + (1 − a H )H 2 − ρ H 2
h2 = H H 2
h0V0
a (χ
+ ρ H 3 ) + (1 − a H )H 3 − ρ H 3
h3 = H H 3
h0V0
a Q χ Q + ρ Q + 1 − aQ Q − ρ Q
q=
2 Rq 0V0
(
) (
)

 a M (χ M 2 + ρ M 2 + d (χ H 3 + ρ H 3 ))


(
)(
)
(
)
1
a
M
dH
d
+
−
+
−
ρ
+
ρ
H
2
3
M2
H3 


 a M (χ M 3 + ρ M 3 − d (χ H 2 + ρ H 2 ))
1


m3 =
2 Rm0V0  + (1 − a H )(M 3 − dH 2 ) − (ρ M 3 − dρ H 2 )
m2 =
1
2 Rm0V0
(7)
(8)
developments for a continuous hyperplasticity model
can be made. The Gibbs free energy function now
becomes a functional as follows:
1
gˆ = g1 − ∫ σ i αˆ i dη +
0
1 ˆ* 2
H 1 αˆ V
(9)
(10)
+∫
0
2
2
1 Hˆ * α
5ˆQ
+∫
0
1
2
2
1 Hˆ * α
3 ˆ H2
dη + ∫
0
2
2
1 Hˆ * α
2 ˆ M2
dη + ∫
0
2
2
1 Hˆ *α
3 ˆ H3
dη + ∫
2
0
2
1 Hˆ * α
2 ˆ M3
dη + ∫
2
0
(14)
dη
(11)
+ ∫ Hˆ 4* (αˆ M 2 αˆ H 3 − αˆ M 3 αˆ H 2 )dη
(12)
where η is a dimensionless parameter which varies
from 0 to 1 and expresses the relative sizes of the
yield surfaces. When η = 0, no plastic behaviour
occurs. Once η = 1, fully plastic behaviour occurs.
The hat notation is used to denote any function of η .
The hardening parameters in Eq. (4) now become
functions of η. These functions determine the shapes
of the force-displacement curves. Hyperbolic curves
may conveniently be used and for this case the
hardening functions have the form:
Hˆ * (η) = A K (b − η)ni
(15)
(13)
It is convenient to note that the vertical load at which
the maximum dimension of the yield surface is achieved
β1 t 0 + α
β − β 2t0
V
=
, leading to
.
is
=α= 1
β2 1−α
V0
β1 + β 2
V0 is the vertical bearing capacity of the
foundation (the intercept of the yield surface on the
positive V-axis). Appropriate values of the
parameters specifying the yield surface shape for a
typical fully embedded caisson are β1 = β2 = 0.99; t0
= 0.1088. The parameters m0, h0 and q0 are factors
which determine the sizes of the yield surface in the
moment, horizontal and torsion directions: typical
values are 0.15, 0.337 and 0.1 respectively.
The parameters aV1, aV2, aM, aH, aQ are the
“association factors”; ρV, ρH2, ρH3, ρQ, ρM2 and ρM3
are the “back stresses” which are the difference
between true force and generalised force, and are in
turn expressed as functions of the internal variable
αi .
4. RATE-INDEPENDENT CONTINUOUS
HYPERPLASTICITY MODEL
The main reason for the introduction of continuous
hyperplasticity, which is in effect models an infinite
number of yield surfaces, is to simulate a smooth
transition between elastic and plastic behaviour, and
capture with reasonable precision the hysteretic
response of a foundation under cyclic loading. Such
behaviour can not be described by a conventional
single yield surface model.
4.1 The Gibbs free energy function g
Starting from the form of Gibbs free energy function
for a single yield surface model as in Eq. (1), further
0
i
i
i
i
where Ai, bi and ni are parameters defining the shape
of the curves.
4.2 The yield function y
For a certain value of η, the yield function can be
expressed as follows:
β
β
yˆ = tˆ − ηSˆβ 12 vˆ1 + t 0 1 1 − vˆ 2 2 = 0
(16)
where
(
tˆ = hˆ22 + hˆ32 + mˆ 22 + mˆ 32 + qˆ 2 + 2a hˆ2 mˆ 3 − hˆ3 mˆ 2
) (17)
All the definitions of variables in Eqs. (16) and (17)
are as for single-yield model, but these variables are
determined for the yield surface corresponding to η.
5. MULTIPLE-YIELD SURFACE
HYPERPLASTICITY MODEL
The concept of an infinite number of yield surfaces
can model very well the response of a foundation
under cyclic loading. However, to implement the
model in a numerical analysis, it is necessary to
discretise the continuous plasticity model to a
multiple-yield-surface model. Firstly, the integrals in
the Gibbs free energy become summations.
Secondly, the continuously varying functions of η
are replaced by discrete variables.
5.1 The Gibbs free energy function g
The hat notation as in Eq. (14) is now abandoned to
express the fact that the variables are no longer
functions, but a series of discrete values. N is the
number of yield surfaces chosen to simulate the
continuous yield surface. We replace η by the factor
j/N where j is the number of the yield surface which
is being considered. In the summation, the increment
dη in the integral becomes:
dη =
j
j −1 1
−
=
N
N
N
N σα
i ij
j =1 N
N H * α2
1 j Vj
+∑
j =1 2 N
j =1
2N
N H*
4j
+∑
j =1 N
N H * α2
3j H2j
+∑
j =1
N H * α2
5 j Qj
+∑
+
2N
N H * α2
2j M2j
+∑
2N
j =1
j =1
2N
+∑
j =1
AK
H i* ( j N ) = i i (bi − j N )ni
2
(20)
5.2 The yield function y
Using the same style of yield function as in Eq. (16),
the jth yield surface can be expressed as:
β
β
j
Sβ12 v1 j + t 0 1 1 − v 2 j 2 = 0
N
(21)
Where equations exactly similar to (6) to (13) apply,
but with each definition applying for the jth surface,
thus Eq. (7) becomes for example:
v1 j =
(
)
aV 1 χVj + ρVj + (1 − aV 1 )V − ρVj
( j N )V0
(22)
The factors, S, β12, β1, β2 and t0, have the same
values as in the single yield surface model. The
definitions of the generalised forces can be
expressed as follows:
χVj = − N
n
A K 
∂g
j V
= V − V 1  bV −  αVj
∂αVj
2 
N
(23)
αH 2 j
nM 2
nH 3
αH 3 j
(25)
αM 2 j
n
AQ K 5 
∂g
j Q
=Q−
 bQ −  α Qj
∂α Qj
2 
N
χM 2 j = − N
A K 
j
∂g
= M 2 − M 2 2  bM 2 − 
2 
N
∂α M 2 j
nH 3
(24)
αM 3 j
A K 
j
∂g
= H 3 − H 3 3  bH 3 − 
N
2 
∂α H 3 j
nM 2
αM 2 j
(26)
(27)
αH 3 j
A K 
j
∂g
= M 3 − M 3 2  bM 3 − 
N
2
∂α M 3 j

A K 
j
+ H 2 4  bH 2 − 
2
N

(α M 2 j α H 3 j − α M 3 j α H 2 j )
nH 2
χ Qj = − N
χM 3 j = − N
2N
nM 3
A K 
j
− M 2 4  bM 2 − 
2
N

(19)
N H * α2
2 j M3j
The above is appropriate provided that the N value
chosen is large enough to result in a small dηi to
achieve a reasonable approximation to the integral
by use of a summation. Eq. (15) now becomes:
yj =tj −
χH 3 j = − N
A K 
j
− H 3 4  bH 3 − 
2 
N
N H * α2
3 j H3j
+∑
∂g
A K 
j
= H 2 − H 2 3  bH 2 − 
2 
N
∂α H 2 j
A K 
j
+ M 3 4  bM 3 − 
2 
N
(18)
The free energy function is therefore:
g = g1 − ∑
χH 2 j = − N
nH 2
nM 3
αM3 j
(28)
αH2 j
The coordinates of center of jth yield surface in stress
space can be defined as:
ρVj = V − χVj
(29)
and likewise for the other variables.
Figure 4 shows the form of yield surfaces after a
purely vertical loading. The size of the smallest yield
surface in the vertical load direction is set as a
certain fraction of the size of the outer yield surface.
Between the inner and outer surfaces a uniform
distribution of sizes of yield surfaces is used. The
purpose of using a non-zero size of the first yield
surface on the V-axis is to control the development
of vertical plastic displacement on vertical
unloading.
5.3 Incremental response
In the multiple-yield-surface model, using rateindependent behaviour, the loading point must
always be within or on each yield surface. This
condition requires that the y-values for all active
yield surfaces must be identically zero. The
imposition of these “consistency conditions” is not
straightforward in numerical analyses. The use of
rate-dependent behaviour has been proposed as a
means to simplify the numerical difficulties by
Houlsby and Puzrin (2002) and Puzrin and Houlsby
(2003). The dissipation function d is in this case
separated into two functions; force potential function
z and flow potential w. Houlsby and Puzrin (2002)
note that w can take alternative forms, depending on
the form of the viscosity assumed. In this paper,
linear viscosity is used and the flow potential
functions can be defined as:
(
)2
y j σ i , α ij , χ ij
w j σ i , α ij , χ ij =
(30)
2µ
Where µ is the viscosity; y j is the jth yield function
(
)
146.5mm; the length of the perimeter skirt H =
146.5mm. The caisson is installed to the full
penetration position and then the horizontal and
moment loads are applied. The vertical load V
increases to the value of V = 945N during the
penetration process and decreases to the value of V =
50N before the lateral loads are applied. During the
which no longer needs to be identically zero. Note
that w j is only zero when the rates of change of
t
300
Vertical penetration w ithout
suction
250
Vertical loads (kN)
plastic displacements are all zero. The incremental
changes of plastic displacements caused by the jth
yield surface can be defined as:
Vertical penetration w ith
suction assistance
200
150
100
50
0
1.0
0
v = V/V0
t0
0.5
1
1.5
2
Vertical penetrations (m)
Figure 5. Installation processes with and without suction
Initial fraction = 0.8
Figure 4. Multiple yield surfaces
20
15
y j ∂y j
dα ij =
dt =
dt
∂χ ij
µ ∂χ ij
The total displacement
calculated as:
(31)
increments
are
N
5
-0.004
0
-5 0
-0.002
0.002
-10
y j ∂y j j
∂ g
∂ g
dε i = −
dt
σk − ∑
N
∂σ i ∂σ k
∂
σ
∂
α
µ
∂
χ
i
ij
ij
j =1
2
now
10
M3 (Nm)
∂w j
2
test results
-15
(32)
theoretical results
-20
theta3 (rad)
Figure 6. Rotation under cyclic loading
6. NUMERICAL ILLUSTRATIONS
60
test results
40
Theoretical results
20
H2 (N)
Firstly, a result modelling the suction assisted
penetration process using the concepts of Houlsby
and Byrne (2005) is shown in Figure 5.
Secondly, a numerical example is given to
illustrate test results which are obtained from
laboratory testing of model caissons.
In this
example, AV = 1.0; AH2 = AH3 = AQ = AM2 = AM3 =
0.5; bV = bH2 = bH3 = bQ = bM2 = bM3 = 1.0; nV = nH2
= nH3 = nQ = nM2 = nM3 = 3.0. Twenty yield surfaces
are used. The values of yield function parameters
are: am = ah = 0.7; aV1 = 0.297; aV2 = 1.0; t0 =
0.1088; m0 = 0.15; h0 = 0.337; the shear modulus of
the soil is G = 0.7MPa, self-weight γ = 15.74kN/m2,
Poisson ratio ν = 0.2; initial fraction for the first
yield function = 0.8. The radius of caisson R =
0.004
0
-0.0004 -0.0003 -0.0002 -0.0001
-20
0
0.0001 0.0002 0.0003
-40
-60
u2 (m)
Figure 7. Horizontal displacement under cyclic loading
w (m)
0.1445
0.1444
0.1443
0.1442
0.1441
test results
theoretical
0.144
0.1439
0.1438
0.1437
-0.004
-0.002
0
0.002
0.004
theta3 (rad)
Figure 8. Vertical movements under cyclic loading
application of the cyclic loads, the vertical load is
kept constant at 50N. The resulting moment-rotation
behaviour is shown in Figure 6, and horizontal loaddisplacement in Figure 7. Finally the vertical
movements during the cycling are presented in
Figure 8. In each case the analyses are compared to
test results, and it can be seen that a satisfactory
agreement is achieved.
7. DISCUSSION
There are four main points that must be addressed in
this model: the choice of the hardening functions,
the values of the association factors, the effects of
suction pressures and the use of the rate dependent
solution.
Firstly, the hardening functions, Hi*, determine
the distributions of plastic displacements which are
caused by each yield surface. Therefore, the
solutions can become stiffer or softer by increasing
or decreasing the factors Ai, bi or ni. It is very
important to determine the appropriate value of the
shear modulus G. Since the hardening functions
depend on the elastic stiffness factors, which include
the shear modulus, the value of G strongly affects
the solutions.
Secondly, the association factors play the role of
determining the direction of the flow vectors of the
plastic strains. To choose suitable values for these
factors, it is necessary to consider some special
aspects of the yield yield functions, such as the
positions of the “parallel points” where the vertical
plastic displacement incerments are zero. The
directions of the flow vectors in the (V, M) plane,
(M, H) plane or (V, H) plane can be obtained from
tests. Furthermore, during the application of lateral
loads, the upward or downward movements of the
footing are also depend on the position of the
parallel point and the value of the vertical load.
However, the details of these expressions are beyond
the scope of this paper.
Thirdly, as shown in Figure 5, by using suction,
the vertical load which must be applied for
installation is rather small compared with that for
installation using purely vertical load. This feature
is very useful because it is impossible to apply a
large value of vertical load to install the caisson in
practice. Consequently, by using suction assisted
penetration, this obstacle can be overcome.
Lastly, in order to avoid numerical difficulties,
the rate-dependent solution has been proposed. The
most important aspect of using the rate-dependent
solution is the relationship among the viscosity µ,
the time step dt and the load step. Suitable values
must be chosen to maintain accuracy and stability
for the numerical solution. There are as yet no
precise procedures for selecting for these
parameters. However, by using some trials, one can
determine suitable values for µ, dt and load step.
8. CONCLUSIONS
This paper presents a multiple yield surface
hyperplasticity model for caisson foundations.
Preliminary choices for the parameters are made.
The model captures reasonably well the behaviour of
a caisson foundation under cyclic loading, and could
be incorporated in numerical analyses of
caisson/structure systems.
REFERENCES
Cassidy, M.J., 1999. Non-linear analysis of Jack-Up structures
subjected to random waves. DPhil thesis, University of
Oxford.
Collins, I.F. and Houlsby, G.T., 1997. Application of
Thermomechanical Principles to the Modelling of
Geotechnical Materials. Proc. Royal Society of London,
Series A, Vol. 453, pp 1975-2001
Doherty, J.P., Deeks, A.J. and Houlsby, G.T., 2004. Evaluation
of Foundation Stiffness Using the Scaled Boundary
Method. Proc. 6th World Cong. on Comp. Mech., Beijing, 510 Sept.
Houlsby, G.T. and Byrne, B.W., 2000. Suction caisson
foundations for offshore wind turbines and anemometer
masts. Wind Engineering, Vol. 24, No. 4, pp. 249-255.
Houlsby, G.T. and Byrne, B.W., 2005. Design Procedures for
Installation of Suction Caissons in Sand, Proc. ICE,
Geotechnical Engineering, in press.
Houlsby, G.T and Cassidy, M.J., 2002. A plasticity model for
the behaviour of footings on sand under combined loading.
Géotechnique, 52, No 2, pp. 117-129
Houlsby, G.T. and Puzrin, A.M., 2000. A Thermomechanical
Framework for Constitutive Models for Rate-Independent
Dissipative Materials. Int. J. of Plasticity, Vol. 16, No. 9,
pp 1017-1047.
Houlsby, G.T. and Puzrin, A.M., 2002. Rate-Dependent
Plasticity Models Derived from Potential Functions. J. of
Rheology, Vol. 46, No. 1, Jan./Feb., pp 113-126.
Martin, C.M., 1994. Physical and numerical modelling of
offshore foundations under combined loads. DPhil thesis,
University of Oxford.
Puzrin, A.M. and Houlsby, G.T., 2003. Rate Dependent
Hyperplasticity with Internal Functions, Proc. ASCE, J.
Eng. Mech. Div., Vol. 129, No. 3, March, pp 252-263
Moment loading of caissons installed in saturated sand
Felipe A. Villalobos, Byron W. Byrne & Guy T. Houlsby
Department of Engineering Science, Oxford University
ABSTRACT: A series of moment capacity tests have been carried out at model scale, to investigate the
effects of different installation procedures on the response of suction caisson foundations in sand. Two
caissons of different diameters and wall thicknesses, but similar skirt length to diameter ratio, have been
tested in water-saturated dense sand. The caissons were installed either by pushing or by using suction. It was
found that the moment resistance depends on the method of installation.
1. INTRODUCTION
Suction caisson foundations are increasingly being
used in offshore applications. They have been used
for fixed structure applications, as described by Bye
et al. (1995), and also for floating facilities (House,
2002). More recently they are being considered as
foundations for offshore wind turbines (Byrne and
Houlsby, 2003). The wind turbine structures may be
founded on single or multiple caissons. The multiple
caisson problem is addressed by Kelly et al. (2004),
so in this paper we concentrate on the single caisson
problem. Typical dimensions and loads for this
problem are shown in Figure 1. Byrne and Houlsby
(2003) describe this problem in detail, but the main
differences in loads on the foundations for offshore
wind turbines as compared to typical oil and gas
structures are that: (a) the vertical load is much
smaller, (b) the horizontal and moment loads are
proportionately larger. New design methods must be
developed to allow safe designs to be engineered for
this regime of loading. As a result Byrne et al.
(2002) describe a research project aimed at
developing such design guidelines. This paper
outlines the results from a part of that project.
Initial studies of the moment capacity of caisson
foundations in the laboratory were carried out in
drained sand. Preliminary results from these
experiments are described by Byrne et al. (2003). As
the sand used during the tests was dry, the caissons
were installed into the prepared sand bed by
applying a vertical load. The advantage of using dry
sand is that the test bed can be prepared quickly, and
a large number of tests can be carried out at
specified densities. To mitigate the effects of scale,
Figure 1: Dimensions and magnitude of loads for a 3.5MW
turbine structure founded on a monopod suction (adapted from
Byrne and Houlsby, 2003)
the tests beds were chosen to be relatively loose.
Clearly using installation by applying vertical loads
is different from the procedure that has to be used in
the field i.e. the suction installation process. The
different installation techniques may impose
different stress paths on elements of soil around the
caisson, which in turn may affect the response of the
caisson to the applied loads. Therefore it is
necessary to carry out experiments similar to those
in the dry sand, but on caissons installed by suction,
to observe if there are any fundamental differences
in behaviour.
Combined vertical, moment and horizontal
loading tests have been conducted on caissons
installed by suction and by vertical load in a watersaturated, dense sand. Load-displacement data are
presented and interpreted for installation and for
moment loading tests.
2. EQUIPMENT AND MATERIALS
2.1 Sand samples
The sand used during the experiments was a
commercially produced sand called Redhill 110. The
properties of this sand are given in Table 1.
Figure 2: 3DOF-loading rig
Table 1: Redhill 110 properties (Kelly et al., 2004)
D10, D30, D50, D60 (mm)
Coefficients of uniformity, Cu and
curvature Cc
Specific gravity, Gs
Minimum dry density, γmin (kN/m3)
Maximum dry density, γmax (kN/m3)
Critical state friction angle, φcs
0.08, 0.10,
0.12, 0.13
1.63, 0.96
2.65
12.76
16.80
36º
The sand samples were saturated with water
inside a tank of diameter 1100mm and depth
400mm. Preparation of the test bed involved an
initial phase of fluidisation by an upward hydraulic
gradient induced in the sand bed. The sample was
then densified by vibration under a small confining
stress. The density was determined by measuring the
weight and the volume of the sample. The
preparation process was halted once a target density
was reached. The peak triaxial angle of friction was
estimated as 44.1o to 45.2o from the correlation of
Bolton (1986), for the range of relative densities
tested (see Table 3).
2.2 Testing procedure
Tests were performed using a three degree-offreedom loading rig (3DOF) designed by Martin
(1994). This rig, shown in Figure 2, can apply any
combination of vertical, rotational and horizontal
displacement (w, 2Rθ, u) to a footing by means of
computer-controlled stepper motors (R is the radius
of the footing). Byrne (2000) has installed a software
control program, so that any combination of vertical,
moment or horizontal load (V, M/2R, H) can also be
applied to the footing. All displacements and loads
are monitored and recorded using appropriate dataacquisition routines as well as being used within
feedback control routines. It is possible to apply
Figure 3: Suction device
loads and displacements to the footing which
represent the offshore environment loads of gravity,
wind, waves and currents. The geometry of the
model suction caissons used in the experiments is
given in Table 2. The model caissons were
fabricated from aluminium alloy, with a relatively
smooth (but not polished) surface.
Table 2: Geometry of the model caissons tested
Diameter, 2R (mm)
Length of skirt, L (mm)
Thickness of the skirt wall, t (mm)
Aspect ratio, L/2R
Thickness ratio, 2R/t
293
146.5
3.4
0.5
86
200
100
1.0
0.5
200
The loading apparatus was modified to allow the
footings to be suction installed. Previous
experiments had only used caissons forced into the
ground by vertical load. To enable the suction
installation phase to be carried out, the equipment
was modified as shown in Figure 3. The suction
caisson, attached to the 3DOF loading rig, was
pushed into the ground about 30mm with the air
valve open. This allowed the pressure inside the
Theory (Houlsby and Byrne, 2005)
Experiments, D = 293mm
Experiments, D = 200mm
2000
1800
1600
Vertical Load, V (N)
caisson to equilibrate to the outside pressure. On
reaching a penetration of 30mm the air valve was
closed, and the fluid valve opened. The fluid from
inside of the caisson was connected to a reservoir,
which was slowly lowered to increase the head
difference, hf, between the inside and outside of the
caisson. The head difference was increased to a
maximum of 300mm (3kPa), whilst the vertical load
applied to the footing was kept constant using
feedback control. The reservoir was connected to the
suction caisson by a pipe of 6mm internal diameter,
chosen to allow sufficient water flow with minimal
head loss.
This procedure allowed the caisson to be installed
by suction whilst connected to the loading rig. Once
the suction phase was complete, it was possible to
carry out experiments similar to those carried out on
the dry sand as described by Byrne et al. (2003).
1400
1200
1000
800
600
400
200
0
0
20
40
60
80
100
120 140
Vertical Displacement (skirt penetration), h (mm)
Figure 4: Pushing installation tests and theoretical calculations
for both caissons
caissons. This expression is used below for
comparison with the experimental data.
4. RESULTS OF THE INSTALLATION TESTS
2.3 Comments on Installation Methods
The two different installation methods have been
described by Houlsby and Byrne (2005). Installation
by vertical load involves pushing the caisson into the
ground. The resistance to penetration is given by the
friction on the inside and outside of the wall, and the
bearing resistance on the skirt tip. Due to ‘silo
effects’ the stresses around the skirt, and at the tip,
are enhanced, leading to larger resistances than may
be given by a more conventional pile calculation.
Houlsby and Byrne (2005) developed expressions
for predicting the resistance to penetration for
caissons, taking account of these ‘silo’ effects. The
equations they developed are used below to provide
a comparison with the experimental results. In these
calculations it will be assumed that (Ktanδ)i,o takes a
value of 0.9, and the stress enhancement factors m
and n are taken as 1.
Installation by suction requires an initial
penetration to create a seal at the skirt tip. Typically
10% to 20% of the caisson skirt penetrates into the
ground under its own weight. The seal allows the
suction process to begin and should prevent the
occurrence of an unconfined flow failure (i.e. a
piping failure). Once sealed, the caisson will
penetrate into the ground under the application of
suction. Typically a pump will remove fluid from
inside the caisson, creating a pressure differential on
the caisson lid, as well as inducing hydraulic
gradients in the soil. The hydraulic gradients lead to
changes in the effective stresses around the caisson
skirt that are beneficial to installation. Houlsby and
Byrne (2005) have developed expressions to
calculate the required suction for installation of
Figure 4 shows the load-displacement results for
caissons pushed into the ground at a constant rate.
The results are shown as vertical load V against
vertical penetration h. The maximum values of V
obtained during these tests were approximately
400N for the footing of diameter 200 mm and
1700N for the 293 mm diameter footing. These
maximum values of V (denoted by Vo in Table 3)
represent “preconsolidation” vertical loads which
might be used for interpreting the results within the
context of a yield surface model (Gottardi et al.,
1999; Houlsby and Cassidy, 2002). Rd in Table 3 is
the Relative Density. Also shown on Figure 4 is a
theoretical prediction calculated using the methods
of Houlsby and Byrne (2005).
Table 3: Installation tests (Suction and Pushing)
Test
FV6_5_1S
FV7_5_1P
FV6_2_1S
FV6_3_1P
FV6_8_1S
FV7_1_1P
FV8_1_1S
FV8_2_1P
FV7_3_1S
FV7_4_1P
FV7_1_3S
FV7_2_1P
2R
mm
200
200
200
200
200
200
293
293
293
293
293
293
V
N
5
40
60
10
60
120
-
Vo
N
398
425
410
428
400
469
1700
1772
1700
1740
1500
1741
Rd
%
75
74
75
75
75
74
81
81
74
74
74
74
h&
mm/s
0.04
0.50
0.04
0.50
0.04
0.50
0.04
0.20
0.06
0.20
0.07
0.40
30
1800
1600
Moment Load, M/2R (N)
Vertical Load, V and V + S (N)
2000
V
1400
by pushing FV7_4_1
1200
by suction FV7_3_1 V = 60N
1000
800
600
V +S
400
V
200
20
40
60
80
20
Suction installation
10
0
0
0
Pushed installation
100
120
0
140
0.25
0.5
0.75
1
1.25
1.5
1.75
2
Rotational Displacement, 2R( theta) (mm)
Vertical Displacement (skirt penetration), h (mm)
Figure 5: Comparison between pushed installation and suction
installation for 293mm diameter caisson
Figure 7: Moment capacity tests, load-rotation response showing
yield points (M/2R)y
Test FV7_3_1: V
Calculated V
Test FV7_3_1: S
Calculated S
Vertical Load, V and S (N)
200
100
0
0
20
40
60
80
100
120
140
-100
-200
Horizontal Load,H (N)
30
300
Pushed installation
20
Suction installation
10
0
0
-300
0.25
0.5
0.75
1
Horizontal Displacement, u (mm)
-400
Vertical Displacement (skirt penetration), h (mm)
Figure 6: V and S comparison between experimental result and
calculation for a suction installed test of the 293mm diameter
caisson
Figure 5 shows the load penetration curves for a
pushed test and a suction-installed test. In the latter
the vertical load was kept constant at 60 N during
the suction phase. For this test the curve labelled
V+S shows the net vertical load due to applied load
plus the pressure differential on the caisson lid. It is
clear that there is a significant difference between
this net vertical load and the vertical load for the
caisson installed by pushing. The difference between
these curves represents the beneficial effects of the
hydraulic gradients set up within the soil due to the
suction.
Figure 6 shows one of these tests compared to the
theoretical predictions of Houlsby and Byrne (2005).
In all of the experimental tests the suction was
applied after approximately 30mm of penetration.
The suction force shown in Figure 6 is slightly
underestimated by the calculations.
5. MOMENT CAPACITY
Once the caissons were installed, moment capacity
tests were carried out. These tests are similar to
Figure 8: Two moment capacity tests, load-displacement
response showing yield points Hy
those reported by Byrne et al. (2003), and consist of
rotation and translation of the footing at a specified
ratio of M/2RH under a constant vertical load. The
tests were carried out slowly, so that the conditions
were fully drained. They are thus relevant to only
one of a series of possible conditions in the field,
where, depending on caisson size, sand type and
loading rate, conditions may vary from fully drained
to almost undrained. Summary data for the moment
tests are presented in Table 4, and further data about
initial conditions can be found in Table 3
(installation method, Vo, initial Rd, etc.)
Figures 7 and 8 compare the moment and
horizontal load capacities for different installation
methods for the 200mm diameter caisson. It is clear
from these figures that the installation method has a
strong effect on the load-displacement behaviour.
The load-displacement curves have been interpreted
using the method described by Byrne et al. (2003),
with fitting linear expressions to the elastic and
plastic components of the curve. The intersection of
the lines represents a yield point. These points are
shown on the figure and given in Table 4 for all the
tests.
by suction FV6_2_2 :V = 40N; D = 200mm
by pushing FV6_3_2: V = 40N; D = 200mm
60
Moment Load, M/2R (N),
2Rd (theta)
Horizontal Displacement,u (mm)
1.25
1
0.75
0.5
0.25
40
30
20
10
0
0
0.25
0.5
0.75
1
1.25
1.5
1.75
2
0
-40
-30
-20
Rotational Displacement, 2R( theta) (mm)
Vertical Displacement, w (mm)
-10
10
20
30
40
50
60
Figure 11: Pushing and suction installation calculations for a
200mm diameter caisson and V = 5N, 40N and 60N
2
2
⎛ H ⎞
⎛ M ⎞
⎛ H ⎞⎛ M ⎞
⎟⎟ + ⎜⎜
⎟⎟ − 2a⎜⎜
⎟⎟⎜⎜
⎟⎟ −
y = ⎜⎜
⎝ hoVo ⎠
⎝ 2RmoVo ⎠
⎝ hoVo ⎠⎝ 2RmoVo ⎠ (1)
by suction FV6_2_2 :V = 40N; D = 200mm
by pushing FV6_3_2: V = 40N; D = 200mm
-0.75
0
Vertical Load, V (N) - dw
Figure 9: Plastic displacements increments during the tests:
horizontal displacement with respect to rotational displacement
-1
suction installed
pushing installed
50
⎛
⎞
β12
⎜
⎟
⎜ (t + 1)(β1+β2 ) ⎟
⎝ o
⎠
-0.5
2
⎛V
⎞
⎜⎜
+ t o ⎟⎟
⎝ Vo
⎠
2β1
⎛
V ⎞
⎜⎜1 − ⎟⎟
⎝ Vo ⎠
2β2
=0
-0.25
in which ho , mo , t o , a, β1 and β 2 define the shape of
0
the surface an β12 = (β1 + β 2 )
β
β1 +β2
(β1 ) 1 (β 2 ) β2
0.25
0
0.25
0.5
0.75
1
1.25
1.5
1.75
.
2
Rotational Displacement, 2R( theta) (mm)
Table 4: Moment capacity tests
Figure 10: Plastic displacements increments during the tests:
vertical displacement with respect to rotational displacement
The displacements paths from the tests are shown
in Figures 9 and 10. In general the rotational
displacement causes an initial elastic response that
gradually changes to an almost perfectly plastic
response, which can be fitted with a straight line.
The values of the slopes of these plastic
displacement increments are presented in Table 4 as
a ratio between horizontal and rotational
displacement increments u& / 2 Rθ& and between
vertical and rotational displacement increments
w& / 2 Rθ& .
Figure 10 illustrates the change of vertical
displacement during the rotation of the caisson. The
suction installed caisson experiences a lower
magnitude of uplift compared with the caisson
installed by pushing.
5.1 Yield Surface and velocity vectors
Using the yield points ((M/2R)y, V) in Table 4, a plot
of the yield surface for low vertical loads is
illustrated in Figure 11. It is possible to fit through
these data points a surface such as expressed by the
formula:
Test
M
2 RH
FV6_5_2S
FV7_5_2P
FV6_2_2S
FV6_3_2P
FV6_8_2S
FV7_1_2P
FV8_1_2S
FV8_2_2P
FV7_3_2S
FV7_4_2P
FV7_1_4S
FV7_2_2P
1.03
1.02
1.06
1.05
1.05
1.03
1.04
1.03
1.04
1.03
1.04
1.03
⎛M ⎞
⎟
V ⎜
2R ⎠ y
⎝
N
N
5.5 6.7
6
14.3
40 11.8
40 24.1
60 18.3
60 29.2
10 14.8
10 33.4
60 30.9
60 42.0
120 39.7
120 56.3
Hy
N
u&
2 Rθ&
w&
2 Rθ&
4.8
12.8
10.4
21.7
16.4
26.9
15.1
32.9
27.7
40.4
40.4
53.3
0.391
0.490
0.463
0.465
0.501
0.505
0.310
0.569
0.404
0.446
0.362
0.477
-0.397
-0.445
-0.122
-0.284
-0.051
-0.253
-0.409
-0.551
-0.299
-0.483
-0.125
-0.289
The surface is fitted through the data points using
parameter values given in Table 5. These values
were found for a series of rotational tests performed
with the same 293mm diameter caisson in dry sand.
Also shown on Figure 11 are the directions of the
displacement increment vectors.
The data for both footing diameters can be
presented on the same figure by normalising with
Moment Load, (M/2RVo ),
2Rd(theta)
0.15
suction installed ; D = 293 mm
pushing installed ; D = 293 mm
suction installed ; D = 200 mm
pushing installed ; D = 200 mm
0.1
0.05
0
-0.1
-0.05
0
0.05
Vertical Load, V/Vo , dw
0.1
0.15
Figure 12: Summary of experimental yield points (normalized
by Vo) and incremental plastic displacement vectors
respect to Vo, the maximum applied vertical load.
These results are shown on Figure 12. Equation (1)
has been included in this plot with a value of to =
0.064 for the smaller footing and 0.040 for the larger
footing. It is necessary to use different values of to in
this plot because the tensile capacity scales with
2RL2 whilst the Vo value scales principally with
2 RtL . Since the two footings have the same L / 2 R
value but different t / 2 R values their tensile
capacities differ on the normalised plot. However,
the normalisation by Vo merges the two curves
shown in Figure 11 for any one caisson, thus suction
or pushing installation has only a minor effect on the
normalised curve.
In more detail, however, the yield surfaces
presented in Figure 12 serve as lower bounds for the
moment capacity in the case of a caisson installed by
pushing. On the other hand, it represents an upper
bound for a suction installed caisson. The
differences are thought to be due to disturbance in
the installation process due to suction.
The incremental plastic displacement vectors
were also compared. The vectors for suction
installation tests have a smaller component in the wdirection compared with the vectors for pushed
installation tests (see last column in Table 4 for
values). Therefore, there was less uplift during the
rotation of a caisson when a suction installation
procedure was used.
Table 5: Yield surface parameters for L/2R = 0.5
Eccentricity of yield surface, a
-0.75
Horizontal dimension of yield surface, ho 0.337
Moment dimension of yield surface, mo
0.122
0.99
Curvature factor at low V, β1
0.99
Curvature factor at high V, β2
6. CONCLUSIONS
A series of experiments comparing the moment
response for suction installed caissons and those
installed by pushing have been carried out. The main
results are:
(a) The use of suction beneficially reduces the
resistance to penetration of the caisson.
(b) The moment resistance of a suction caisson
depends on the method of installation.
(c) The ratio of horizontal and rotational plastic
displacement increments, u& p / 2 Rθ& p , was
independent of the installation method.
(d) Under rotations more vertical uplift was
observed for the pushed installed caisson than
the suction installed caisson although this was
also dependent on the applied vertical load.
(e) The yield surface (equation (1)) was applied
successfully to two different size suction
caissons after normalisation by Vo, but requires
differing values of to, to account for different
ratios of 2R/t.
REFERENCES
Bolton, M.D. (1986) The strength and dilancy of sands.
Géotechnique, Vol 36, No. 1, pp65-78
Bye, A., Erbrich, C. T., Rognlien, B., and Tjelta, T. I. 1995.
Geotechnical design of bucket foundations. Offshore
Technology Conference, Houston, paper 7793
Byrne, B.W. 2000. Investigations of suction caissons in dense
sand, DPhil thesis, University of Oxford
Byrne, B.W. and Houlsby, G.T. 2003. Foundation for offshore
wind turbines, Phil. Trans. of the Royal Society of London,
Series A 361, 2909-2300
Byrne, B.W., Houlsby, G.T., Martin, C.M. and Fish, P.M.
2002. Suction caisson foundations for offshore wind
turbines. Wind Engineering, Vol. 26, No 3.
Byrne, B.W., Villalobos, F.A., Houlsby, G.T. and Martin, C.M.
2003. Laboratory Testing of Shallow Skirted Foundations
in Sand, Proc. Int. Conf. on Foundations, Dundee, 2-5
September, Thomas Telford, 161-173.
Gottardi, G., Houlsby, G.T. and Butterfield, R. 1999. The
Plastic Response of Circular Footings on Sand under
General Planar Loading, Géotechnique, 49, 4, pp 453-470.
Houlsby, G.T. and Cassidy, M.J. 2002. A plasticity model for
the behaviour of footings on sand under combined loading,
Geotéchnique, Vol. 52, No. 2, pp 117-129
Houlsby, G.T. and Byrne, B.W. 2005. Calculation procedures
for installation of suction caissons in sand. Proc ICE,
Geotechnical Engineering, in press
House, A.R. 2002. Suction Caisson Foundations for Bouyant
Offshore Facilities, PhD thesis, the University of Western
Australia
Kelly, R.B., Byrne, B.W., Houlsby, G.T. and Martin, C.M.
2004. Tensile loading of Model Caisson Foundations for
Structures on Sand, Proc. ISOPE. Conf., Toulon
Martin, C.M. 1994. Physical and Numerical Modelling of
Offshore Foundations under Combined Loads, DPhil
thesis, University of Oxford