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An elastoplastic 1D Winkler model for suction caisson foundations under combined loading

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Most existing Winkler models use non-linear elastic soil reactions to capture the non-linear be-haviour of foundations. These models cannot easily capture phenomena such as permanent displacement, hys-teresis and the influence of combined loading on the failure states. To resolve these shortcomings, an elasto-plastic Winkler model for suction caisson foundations under combined loading is presented. The proposed model combines Winkler-type linear elastic soil reactions with local plastic yield surfaces to model the non-linear soil response using standard plasticity theory, albeit in a simplified one-dimensional (1D) framework. The results demonstrate that the model reproduces the appropriate foundation behaviour, comparing closely to three-dimensional finite element (3DFE) analyses but with the advantage of rapid computation time.
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1 INTRODUCTION
1.1 General Background
Winkler models are widely used to design deep foun-
dations such as piles. However, in recent work
(Gerolymos and Gazetas, 2006; Varun et al., 2009;
Suryasentana et al., 2017), Winkler models have also
been developed for shallow foundations such as cais-
son foundations. While these design methods may not
be as accurate as more rigorous approaches such as
the three-dimensional finite element (3DFE) method,
Winkler models have the advantages of being rela-
tively fast and easy to use.
Winkler models simplify the three-dimensional
(3D) foundation-soil interaction problem into a more
tractable one-dimensional (1D) problem, with the
foundation replaced by a beam and the soil continuum
by Winkler ‘springs’ (also termed as soil reactions in
this paper). To simulate the non-linear response of
soil, the Winkler models adopt non-linear elastic soil
reactions. Examples of such models include the p-y
and t-z methods (API, 2010; DNV, 2014) used to de-
sign laterally and axially loaded piles respectively.
Nevertheless, there are shortcomings with these
existing non-linear Winkler models. For example, the
non-linear elastic soil reactions used in the p-y or t-z
methods for piles cannot reproduce observed cyclic
loading phenomena such as permanent displacement
or hysteresis. Moreover, they cannot account for com-
bined loading effects on the failure state.
1.2 Proposed Model
Recently, Suryasentana et al. (2017) developed a
1D Winkler model, calibrated against 3DFE analyses,
to accurately predict suction caisson behaviour in lin-
ear elastic soil for six degrees of freedom (dof) load-
ing. However, this model can only be applied to load-
ing conditions where the soil response can be
approximated as linear elastic.
This paper extends the 1D Winkler model devel-
oped in Suryasentana et al. (2017) to allow predic-
tions of non-linear caisson behaviour in undrained
clay under combined planar vertical V, horizontal H
and moment M loading. The extension involves cou-
pling linear elastic soil reactions with local plastic
yield surfaces, which are calibrated against rigorous
3DFE failure state analyses.
The governing mechanics of the proposed 1D
model is based on the same elastoplasticity frame-
work used in 3DFE analyses. This allows straightfor-
ward reproduction of the 3DFE predictions, but with
higher efficiency due to dimensionality reduction.
Consequently, the proposed 1D model allows fast and
accurate solutions of caisson behaviour in elasto-
plastic soil for design assessment under fatigue, ser-
viceability and ultimate limit states (FLS, SLS, ULS).
This enables an efficient design process, with the 1D
model used to quickly shortlist potential designs from
a large candidate space, before further refinement is
conducted with 3DFE analyses.
An elastoplastic 1D Winkler model for suction caisson foundations under
combined loading
S.K. Suryasentana, B.W. Byrne & H.J. Burd
University of Oxford, UK
A. Shonberg
Ørsted Wind Power, London, UK
ABSTRACT: Most existing Winkler models use non-linear elastic soil reactions to capture the non-linear be-
haviour of foundations. These models cannot easily capture phenomena such as permanent displacement, hys-
teresis and the influence of combined loading on the failure states. To resolve these shortcomings, an elasto-
plastic Winkler model for suction caisson foundations under combined loading is presented. The proposed
model combines Winkler-type linear elastic soil reactions with local plastic yield surfaces to model the non-
linear soil response using standard plasticity theory, albeit in a simplified one-dimensional (1D) framework.
The results demonstrate that the model reproduces the appropriate foundation behaviour, comparing closely to
three-dimensional finite element (3DFE) analyses but with the advantage of rapid computation time.
2 METHODS
2.1 1D Model
The 1D model adopted in this paper is similar to that
detailed in Suryasentana et al. (2017) and it is briefly
described as follows. The 1D model is a simplified
representation of the original 3D caisson-soil interac-
tion problem, where the caisson structure and soil
continuum are replaced by a 1D rigid body and Win-
kler-type soil reactions respectively.
Figure 1 shows a schematic diagram of the original
3D caisson-soil problem and the 1D model represen-
tation. There are two types of soil reactions in this
model: distributed soil reactions that act along the
caisson skirt (referred to as the skirt soil reactions’
and indicated as hskirt, mskirt and vskirt in Figure 1) and
concentrated soil reactions that act on the caisson
base, including the soil plug (referred to as the ‘base
soil reactions’ and indicated as hbase, mbase, vbase in
Figure 1). hskirt, vskirt and mskirt represent the distrib-
uted horizontal force, vertical force and rotational
moment along the skirt length, while hbase, vbase and
mbase represent the concentrated horizontal force, ver-
tical force and rotational moment at the base.
Figure 1. Schematic diagram of an embedded suction caisson
foundation (left) and its corresponding simplified 1D represen-
tation (right), where RP is the loading reference point. v, h and
m are the vertical, horizontal and rotational soil reactions.
There are, however, a few notable differences be-
tween the 1D model adopted in this paper and that de-
scribed in Suryasentana et al. (2017). First, as this pa-
per is only concerned with planar VHM loading, there
are only 3 components (v, h, m) for each soil reaction,
which correspond to the vertical w, horizontal u and
rotational ϴ degrees of freedom (dof). Second, unlike
the linear elastic soil assumed in Suryasentana et al.
(2017), the current paper assumes a linear elastic-per-
fectly plastic soil. This gives an ultimate limit to the
soil response, which the previous 1D model was not
able to capture. To simulate this behaviour, the 1D
model in this paper couples the linear elastic soil re-
actions with local plastic yield surfaces. These local
yield surfaces are a direct analogy of the elemental
yield surfaces in the soil reactions space (consisting
of v, h, m components). Just as the canonical yield
surfaces determine the set of allowable elemental
stress states, the local yield surfaces determine the set
of allowable soil reaction states.
The mechanics of the coupled soil reactions-yield
surfaces model can be explained by standard plastic-
ity theory. For soil reaction states lying inside the lo-
cal yield surface, the soil response is linear elastic
with the incremental response given by:
 (1)
where p = soil reactions {v, h, m}, ke = elastic stiffness
matrix and u = local displacements {w, u, ϴ}. ke can
be obtained from Suryasentana et al. (2017) as the
caisson dimensions (L/D = 1) and elastic soil proper-
ties (ν = 0.49) adopted in this paper are identical.
However, for simplicity and faster numerical conver-
gence, the coupling terms between h and m in ke are
ignored (the exclusion of these coupling terms will
mainly impact the accuracy of the elastic horizontal
and rotational predictions). Thus, ke for the skirt and
base soil reactions are as follows:
   
 
  (2)
  
 
  (3)
where G = shear modulus of soil, D = caisson diame-
ter, z = depth below ground level (see Figure 1).
When the soil reaction states reach the local yield
surface, the soil response becomes elastoplastic, with
incremental behaviour given by:
 (4)
where kep = elastoplastic stiffness matrix. By conven-
tion, the local yield surface f(p) is defined as follows:
f < 0 for states inside the yield surface, f = 0 for states
on the yield surface, and f > 0 for inadmissible states
outside the yield surface.
When elastoplastic yielding occurs, permanent
plastic displacements accumulate with the total dis-
placement increment u composed of elastic and
plastic parts:
 (5)
The elastic displacement increment ue is determined
by the soil reaction increment through:
 (6)
The plastic displacement increment up is determined
by the flow rule:

 (7)
where g(p) is a plastic potential function and λ is a
non-negative, scalar plastic multiplier. When yielding
occurs, the incremental soil reaction p must remain
on the local yield surface. This is enforced by the con-
sistency condition:

 = 0 (8)
Following the conventional approach for linear elas-
tic-perfectly plastic models, kep is obtained from:
 




 (9)
For this paper, an associated flow rule is assumed i.e.
g(p) = f(p). The local yield surface f(p) is calibrated
using the limiting soil reactions extracted from the
3DFE analyses, which is described in Section 2.3.
The 1D model was implemented numerically using
the Galerkin finite element methodology, where two-
noded 1D soil elements (each with a linear shape
function and two Gauss points) representing the skirt
soil reactions are tied to two-noded 1D caisson rigid
bar elements. The base soil reaction is represented by
a lumped model tied to the bottom node of the deepest
caisson element. The explicit Runge-Kutta (4, 5) al-
gorithm (Dormand and Prince, 1980) was used for the
integration process during elastoplastic behaviour
and the full Newton-Raphson procedure was used to
obtain the system solution.
2.2 3DFE Model
The 3DFE analyses were carried out using the finite
element program ABAQUS v6.13 (Dassault Sys-
tèmes 2010). The 3DFE model consists of a suction
caisson foundation (of unit diameter D and unit skirt
length L = D) embedded in homogeneous soil, which
is similar to that used in Suryasentana et al. (2017).
The mesh domain is set as 8D for the diameter and
6D for the depth, which was verified to be large
enough to avoid boundary effects for load capacity
predictions. Mesh convergence analyses were also
carried out to determine the required mesh fineness.
Due to symmetry of the problem, only half of the cais-
son and soil domain was modelled. A typical mesh of
the 3DFE model is shown in Figure 2.
The soil was defined as weightless, homogeneous
and linear elastic-perfectly plastic. The soil is as-
sumed to obey the von Mises yield criterion with an
associated flow rule. The Young’s modulus E of the
soil is set as 10003su (where su is the undrained
shear strength) and the Poisson’s ratio is set as 0.49.
Figure 2. Mesh used for the 3DFE analyses. The diameter and
depth of the mesh domain is set as 8D and 6D respectively.
Fully-integrated, linear, brick elements C3D8H
were used to model the soil elements. The caisson
was modelled as being entirely rigid using rigid body
constraints. The caisson reference point was set at RP,
as shown in Figure 1. Contact breaking between the
caisson and soil was prevented using tie constraints at
the caisson-soil interface. Displacements were fixed
in all directions at the bottom of the mesh domain and
in the radial directions at the periphery.
2.3 Calibration of local yield surfaces
To calibrate the local yield surfaces in the proposed
1D model, a series of 3DFE analyses were carried out
to obtain the limiting soil reactions. These analyses
involved the determination of the global VHM failure
envelope of the caisson-soil interaction problem
(Bransby and Yun, 2009; Gourvenec and Barnett,
2011; Vulpe, 2015). This was done using mixed load
and displacement control, where load control was
used in the V load space while displacement control
was used in the HM load space. In total, four vertical
loads (V/V0 = 0, 0.25, 0.5, 0.75 where V is the vertical
load applied at RP and V0 is the uniaxial vertical load
capacity) were applied before displacement probes
were applied in the HM load space. This determines
the HM failure envelopes at fixed levels of V. It is hy-
pothesized that, just as there exists a failure envelope
that limits the global load space, there also exists a
limiting envelope in the soil reactions space (termed
as local yield surface in this paper), which can be
identified using the limiting soil reactions extracted
from the 3DFE analyses.
The limiting soil reactions were extracted from the
3DFE results at the end of each displacement probe,
corresponding to a global failure state of the caisson-
soil interaction problem. For simplicity, the limiting
skirt soil reactions are assumed to be constant along
the skirt and are computed as the average of the soil
reactions along the skirt.
To represent the local yield surface, an ellipsoid
function f(p) is adopted:

(10)
where v0, h0 and m0 are the limiting uniaxial vertical,
horizontal and moment soil reactions (i.e. the uniaxial
capacities in the soil reactions space) and α is a pa-
rameter that governs the rotation of the ellipsoid in
the hm space. This ellipsoid function was adopted as
it has favourable theoretical properties such as global
convexity.
The unknown parameters v0, h0, m0 and α were
identified by running least squares regression against
the limiting soil reactions extracted from the 3DFE
results. The best-fit parameters for the skirt and base
local yield surfaces are shown in Table 1.
Table 1. Best-fit parameters for the skirt and base lo-
cal yield surfaces
_________________________________________________
Parameter Skirt Base
_________________________________________________
v0/su Askirt 9.1Abase
h0/su 2.07Askirt 1.34Abase
m0/su 0.19AskirtD 0.72AbaseD
α -1.23 -0.47
_________________________________________________
where D is the caisson diameter, Askirt (skirt surface area per
metre length basis) = πD and Abase = πD2/4
The local yield surface contours generated by
Equation 17 and the best-fit parameters in Table 1 are
compared against the 3DFE limiting soil reactions in
Figure 3. Although the global vertical load V is fixed
while the HM failure envelope is probed, the distribu-
tion of the vertical load between the skirt and base soil
reactions is not constant. Thus, each of the limiting
soil reactions is associated with a different v/v0 value.
To simplify the process, the average of these v/v0 val-
ues (for each dataset corresponding to a fixed V) are
used in Equation 17 to predict the hm contours for
each V/V0; their values are shown in the contour labels
in Figure 3. For V/V0 = {0, 0.25, 0.5, 0.75}, v/v0 = {0,
0.1, 0.35, 0.7} for the skirt soil reactions and {0, 0.32,
0.57, 0.77} for the base soil reactions.
It was observed that the predicted local yield sur-
face contours are good approximations to the limiting
base soil reactions at low vertical loads. However, at
higher vertical loads (V/V0 ≥ 0.5), there is less agree-
ment as the ellipsoid function cannot capture the
change in yield surface geometry. The fit is less than
ideal for the limiting skirt soil reactions as they do not
conform closely to an ellipsoidal shape.
However, although these simplified local yield
surfaces do not match well on a local level, they pro-
duce reasonably accurate global predictions (as will
be shown in Section 3.2).
Figure 3. Comparison of the 3DFE limiting skirt and base soil
reactions (as depicted by the markers in the figure) against the
local yield surface contours (as depicted by the grey dotted lines)
predicted by Equation 17 and Table 1. The average v/v0 values
(for each dataset corresponding to a fixed V) are shown in the
contour labels and they are used in Equation 17 to produce the
contours.
2.4 Evaluation of Models
To compare the predictions between the 1D model
and the 3DFE model, three types of evaluations were
implemented. First, the uniaxial load capacities
(which are the load capacities under the application
of V, H and M individually) were evaluated to assess
the accuracy of the models for simple loading cases.
Next, the influence of combined loading on failure
states was assessed by using the 1D and 3DFE models
to find the failure envelopes of the caisson in the VHM
load space. Finally, a single cyclic load test was sim-
ulated using both models to assess the capability of
capturing permanent displacement and hysteresis.
3 RESULTS
3.1 Uniaxial Load Capacities
Table 2 shows the uniaxial vertical V0, horizonal H0
and moment load capacities M0 predicted by the 1D
and 3DFE models. It is evident that the 1D model pre-
dictions agree very well with the 3DFE model predic-
tions, with the largest difference being only 1.51% for
the horizontal load capacity H0.
Table 2. Comparison of the uniaxial global load ca-
pacities predicted by the 1D and 3DFE models
_________________________________________________
Capacity 1D 3DFE Diff (%)
_________________________________________________
V0/Abasesu 13.12 13.12 0.00
H0/Abasesu 6.01 5.93 1.51
M0/AbaseDsu 3.7 3.7 -0.07
_________________________________________________
where D is the caisson diameter and Abase = πD2/4
Figure 4 compares the global load-displacement
predictions under uniaxial loading, where wRP, uRP
and ϴRP are the vertical, horizontal and rotational dis-
placements of the loading reference point RP. As ob-
served, the 1D and 3DFE model predictions tend to
reach the load capacity at different displacements.
Under pure vertical loading, the 1D model reaches
load capacity at a smaller displacement than the
3DFE model. Moreover, it can be seen from the close-
up inset that the 1D model load-displacement predic-
tion is bilinear. The first linear response is the elastic
response while the second linear response occurs
when the base soil reaction has reached its local yield
surface but the skirt soil reaction remains elastic.
Under pure horizontal loading, the 1D model pre-
dicts uRP/D of 0.1 and ϴRP of 0.129, which compares
well with the 3DFE model predictions of uRP/D of 0.1
and ϴRP of 0.139. Similarly, under pure moment load-
ing, the 1D model predicts ϴRP of 0.1 and uRP/D of
0.05, which compares well with the 3DFE model pre-
dictions of ϴRP of 0.1 and uRP/D of 0.0556.
Under pure horizontal or moment loading, the 1D
model load-displacement predictions are not bilinear
as both horizontal h and rotational m soil reactions
occur during these loadings. The influence of com-
bined h and m loading forces the soil reaction path to
track on the local yield surface during elastoplastic
yielding, until the global load capacity is reached.
Figure 4. Comparison of global load-displacement predictions.
The close-up insets focus on the results at small displacements.
3.2 Failure Envelopes
Figure 5 compares the predictions of the VH and VM
failure envelopes of the caisson in normalised forms,
where the loads are normalised by their respective
uniaxial capacities. The 1D model predictions of the
VH and VM failure envelopes match the 3DFE results
very well, albeit with a slight overprediction for the
VH failure envelope for some load cases.
Figure 5. Comparison of global failure envelope predictions in
the VH and VM load space.
Next, Figure 6 compares the predictions of the
VHM failure envelopes in normalised forms. Despite
the poor match of the local yield surfaces at the local
level (see Figure 3), the 1D model predictions of the
global HM envelope under fixed V loads match the
3DFE predictions reasonably well.
Figure 6. Comparison of global failure envelope predictions in
the VHM load space.
Nevertheless, given the mismatch (especially that of
the skirt local yield surface) in Figure 3, it is encour-
aging to see that the global failure envelope predic-
tions are not too sensitive to the accuracy of these lo-
cal yield surfaces. Furthermore, most loading
scenarios are in the quadrants where H and M have
the same sign. Thus, the mismatch in the quadrants
where H and M have different signs are of less prac-
tical concern.
3.3 Cyclic Loading
To assess whether the 1D model can simulate hyster-
etic behavior, a single cycle of positive and negative
vertical displacements (wRP/D = ±0.05) was pre-
scribed onto the caisson. Figure 7 shows the compar-
ison of the global load-displacement behavior under
this cyclic loading. It is clear that the 1D model is able
to simulate hysteresis, although the 1D model predic-
tions is a rather crude piecewise linear approximation
of the 3DFE model predictions.
By comparing the displacements at zero V load, it
is evident that the permanent displacement predic-
tions of the 1D and 3DFE models are in good agree-
ment. The 1D model comes with a built-in capability
for simulating effects such as permanent displace-
ment and hysteresis. This is not surprising as both the
1D and 3DFE models are based on fundamentally the
same elastoplasticity concepts, but with different
measures of ‘stress’ and ‘strain’.
Figure 7. Comparison of global load-displacement behavior un-
der cyclic vertical loading
4 DISCUSSION
The 1D model ignores much of the detail of the orig-
inal 3D continuum-based problem, with the aim of
appropriate simplification to provide a fast proxy to
the original problem. Despite the simplifying abstrac-
tions, the loss in accuracy is minimal, relative to the
large gains in computational efficiency. For example,
the 3DFE model took about 28 hours in total to run
the analyses presented in Section 3. By contrast, the
1D model took about 0.8 hours in total, yielding a
time saving of 97%.
This computational efficiency is very important
for design optimization involving multiple founda-
tions, such as that for an offshore wind farm. Whilst
3DFE is perhaps practical for design projects involv-
ing only a few foundations, it is clearly impractical
when there are hundreds of foundations. A tool such
as the trained 1D model offers the 3DFE accuracy but
with much higher efficiency, and therefore allows
more of the design space to be explored.
Furthermore, the proposed 1D model offers ad-
vantages over existing macro-element models for
shallow foundations (e.g. Cassidy 2004, Salciarini et
al. 2011). Given the localised nature of the soil reac-
tions and the yield surfaces, the 1D model may be
simply adapted to non-homogeneous or multi-layered
grounds with arbitrary yield strength profiles. This
contrasts with macro-element models, which can only
be adapted to ground profiles similar to that in the
original calibration. In other words, the 1D model is
a more generalised model by comparison with the
macro-element model.
The focus of this paper is a presentation of the map-
ping process from a 3DFE elastoplastic continuum
model to a 1D elastoplastic Winkler model, and a
demonstration of the accuracy of the approach. As
such, generalised formulations of the yield surfaces,
although established for caissons of L/D ≤ 2, are not
presented. They will be described in future publica-
tions.
There are, of course, some observed limitations
with the 1D model. For example, the load predictions
under purely vertical loading is bilinear. This could
be resolved by adding multiple or nested local yield
surfaces but this increase in accuracy comes at the ex-
pense of increased computational effort. Also, there
is room for improvement for the global failure enve-
lopes predicted by the 1D model and this can be
achieved by adopting a more expressive function with
more parameters to represent the local yield surface.
However, while there are ready solutions to these lim-
itations, it is advisable to consider whether the addi-
tional complexity balances the aim of providing a
rapid but approximate solution to the caisson-soil in-
teraction problem for preliminary designs, which can
then be refined using more advanced 3DFE analyses.
5 CONCLUSION
The main concern with Winkler models that use non-
linear elastic soil reactions to approximate the soil
continuum response is that they do not easily repro-
duce observed phenomena such as permanent dis-
placement, hysteresis and influence of combined
loading on failure states. This paper resolves this
shortcoming by proposing a 1D Winkler model that
couple linear elastic soil reactions with local plastic
yield surfaces that limits the allowable soil reaction
states. The results indicate that the proposed 1D
model compares favourably with the 3DFE model
predictions in terms of accuracy across a range of
loading states. The principal advantage, however, is
efficiency, as it takes only 3% of the computational
time required by the 3DFE model. Furthermore, un-
like macro-element models which can only be used
for ground profiles that are similar to the original cal-
ibration, the 1D model can be used for non-homoge-
neous or multi-layered grounds with arbitrary yield
strength profiles, making it a more general, and argu-
ably, useful model. Thus, the proposed 1D model of-
fers an efficient method to predict realistic, non-linear
behavior of caissons in elastoplastic soil.
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... Compared to non-linear elastic Winkler models such as the p-y and t-z methods (API, 2010;DNV, 2014), this elastoplastic Winkler model offers significant advantages such as realistic modelling of phenomena such as hysteresis and the interaction of different local load and moment components at failure. The OxCaisson-LEPP model employed in this paper is similar to that described in Suryasentana et al. (2018), except that it has been calibrated for caissons of 0 ≤ L/D ≤ 2 using the approach described in Suryasentana et al. (2019b). The soil model used to calibrate OxCaisson-LEPP is von Mises soil (representing undrained clay) with s u profile assumed constant with depth. ...
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This paper describes an automated approach for determining the optimal dimensions (length and diameter) of a suction caisson foundation subject to lateral loads, to minimise the foundation weight, whilst satisfying installation requirements, serviceability and ultimate limit states. The design problem was cast as a constrained optimisation problem. Solutions were initially developed using a graphical approach; the solution process was then repeated with an automated approach using an optimisation solver. Both approaches were feasible because a computationally efficient elastoplastic Winkler model was used to model the suction caisson foundation behavior under applied loading. The automated approach was found to be fast and reasonably accurate (when compared to more computationally expensive design procedures using three-dimensional finite element analyses). The benefits of this approach, made possible by the efficiency of the models employed, include better design outcomes and reduced design time.
Article
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The response of skirted circular foundations with rough and smooth soil–skirt interface to combined loading is investigated. The shape and size of the failure envelopes of the skirted circular foundations under a practical range of embedment ratio, soil strength heterogeneity, soil–skirt interface and level of vertical mobilisation are compared to those of solid embedded circular foundations and skirted strip foundations. The results show that both the foundation geometry and soil plug inside the skirt compartment significantly influence the uniaxial capacity and the shape and size of the failure envelopes. Approximating expressions for solid embedded foundations do not capture the shifting eccentricity of the failure envelopes of the skirted circular foundations. A new approximating solution for describing the failure envelope as a function of embedment ratio, soil strength heterogeneity and soil–skirt interface is proposed. Scaling factors accounting for the decreasing uniaxial capacity of skirted foundations with smooth soil–skirt interface are also defined.
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The response of skirted foundations to combined vertical, horizontal and moment loading is important for the design of offshore installations. The vertical skirts beneath the footing interact with the soil and increase the foundation capacity, compared with a surface footing. Previous researchers have assumed that the soil within the skirts remains rigid during undrained loading, but this assumption has not been investigated rigorously. A series of plane-strain finite element analyses has been conducted to investigate directly how the skirt geometry affects the undrained strip foundation capacity under combined horizontal-moment loading and the mechanisms occurring at failure. Conditions of both uniform and non-uniform undrained strength soil have been considered. The results show that deformation of the soil between external skirts can lead to significantly less foundation capacity than that of an equivalent solid embedded foundation. Hence the specific geometry of the foundation must be considered in design. In addition, the failure envelopes for skirted foundations with different embedment ratios differed significantly. This makes general recommendations for failure envelope shapes problematical. Finally, significant increases in foundation bearing capacity may be achieved by adding an intermediate skirt to the foundation, which results in a foundation capacity that is almost equal to that of a solid embedded foundation.
Conference Paper
In this work, the hypoplastic macroelement model for shallow foundations proposed by Salciarini & Tamagnini (2009) is extended to six-dimensional loading conditions, including torsional moments and rotations. In addition, a new hardening law for the vertical bearing capacity Vf is adopted to include the effects of accumulated displacements and rotations. As pointed out in Bienen et al. (2006), the extension to six-dimensional loading conditions is of great importance in the analysis of foundations for offshore structures. The incrementally non- linear constitutive equations of the macroelement are formulated in terms of normalized generalized forces and displacements and are constructed based on the general approach proposed by Niemunis (2002). A suitable vectorial internal variable (internal displacement) is employed to provide the model sufficient memory of past displacement history to be able to reproduce the observed behavior under cyclic loading paths such as those associated with wind or wave loading. The model performance has been evaluated by comparing the model predictions with available experimental results from a series of small-scale model tests with complex loading paths reported by Bienen et al. (2006). The model parameters have been calibrated using independent experimental data for more conventional loading conditions. As compared to similar macroelements based on the theory of kinematic hardening elastoplasticity, the proposed approach has the advantage of a much simpler mathematical structure, which allows a straightforward implementation in existing structural analysis FE codes.
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Three-dimensional failure envelopes can be used to define the bearing capacity and proximity to failure of shallow foundations under general vertical, horizontal and moment loading (V, H, M/B). Different structures, and different load cases for the same structure, cover varying domains of (6V, 6H, 6M/B) load space; therefore, a fully encompassing failure envelope in (V, H, M/B) load space is a useful tool to define ultimate limit states for design. In this technical note, a closed-form expression is proposed that enables prediction of undrained bearing capacity of skirted foundations under general in-plane loading, valid for a range of embedment ratios and soil shear strength heterogeneities.
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The transient response of large embedded foundation elements of length-to-diameter aspect ratio D/B=2–6 is characterized by a complex stress distribution at the pier–soil interface that cannot be adequately represented by means of existing models for shallow foundations or flexible piles. On the other hand, while three-dimensional (3D) numerical solutions are feasible, they are infrequently employed in practice due to their associated cost and effort. Prompted by the scarcity of simplified models for design in current practice, we here develop an analytical model that accounts for the multitude of soil resistance mechanisms mobilized at their base and circumference, while retaining the advantages of simplified methodologies for the design of non-critical facilities. The characteristics of soil resistance mechanisms and corresponding complex spring functions are developed on the basis of finite element simulations, by equating the stiffness matrix terms and/or overall numerically computed response to the analytical expressions derived by means of the proposed Winkler model. Sensitivity analyses are performed for the optimization of the truncated numerical domain size, the optimal finite element size and the far-field dynamic boundary conditions to avoid spurious wave reflections. Numerical simulations of the transient system response to vertically propagating shear waves are next successfully compared to the analytically predicted response. Finally, the applicability of the method is assessed for soil profiles with depth-varying properties. The formulation of frequency-dependent complex spring functions including material damping is also described, while extension of the methodology to account for nonlinear soil behavior and soil–foundation interface separation is described in the conclusion and is being currently investigated.
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As an extension of the elastic multi-spring model developed by the authors in a companion paper [Gerolymos N, Gazetas G. Winkler model for lateral response of rigid caisson foundations in linear soil. Soil Dyn Earthq Eng; 2005 (submitted companion paper).], this paper develops a nonlinear Winkler-spring method for the static, cyclic, and dynamic response of caisson foundations. The nonlinear soil reactions along the circumference and on the base of the caisson are modeled realistically by using suitable couple translational and rotational nonlinear interaction springs and dashpots, which can realistically (even if approximately) model such effects as separation and slippage at the caisson–soil interface, uplift of the caisson base, radiation damping, stiffness and strength degradation with large number of cycles. The method is implemented in a new finite difference time-domain code, NL-CAISSON. An efficient numerical methodology is also developed for calibrating the model parameters using a variety of experimental and analytical data. The necessity for the proposed model arises from the difficulty to predict the large-amplitude dynamic response of caissons up to failure, statically or dynamically. In a subsequent companion paper [Gerolymos N, Gazetas G. Static and dynamic response of massive caisson foundations with soil and interface nonlinearities—validation and results. Soil Dyn Earthq Eng; 2005 (submitted companion paper).], the model is validated against in situ medium-scale static load tests and results of 3D finite element analysis. It is then used to analyse the dynamic response of a laterally loaded caisson considering soil and interface nonlinearities.
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As jack-ups have moved into deeper and harsher waters there has been an increased need to understand jack-up behaviour and develop analysis techniques. One of the areas of significant development has been the modelling of spudcan footing performance, where the load–displacement behaviour of the footings is required to be included in any overall numerical model. Because they can be incorporated into conventional structural analysis programs, force resultant models based on strain-hardening plasticity theory are appropriate replacements for the unrealistic assumptions of pinned or linear spring footings. The development of these models for the analysis of spudcans on both clay and sand is reviewed here. A formulation for a six-degrees-of-freedom model that describes the load–displacement behaviour in the vertical, moment, horizontal, and torsion directions is also detailed. Using this model any load or deformation path can be applied to the footing and the corresponding unknowns (deformations or loads) calculated. This formulation allows the model to be implemented into three-dimensional structural analysis programs, and examples of this are given. Some future challenges in this area are addressed, including the development of models that account for cyclic loading behaviour.
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A family of embedded Runge-Kutta formulae RK5 (4) are derived. From these are presented formulae which have (a) ‘small’ principal truncation terms in the fifth order and (b) extended regions of absolute stability.
OS-J101 -Design of Offshore Wind Turbine Structures
  • Dnv
DNV. 2014. OS-J101 -Design of Offshore Wind Turbine Structures. Oslo: Det Norske Veritas.