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Assessment of Numerical Procedures for Determining Shallow Foundation Failure Envelopes

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Abstract

The failure envelope approach is commonly used to assess the capacity of shallow foundations under combined loading, but there is limited published work that compares the performance of various numerical procedures for determining failure envelopes. This paper addresses this issue by carrying out a detailed numerical study to evaluate the accuracy, computational efficiency and resolution of these numerical procedures. The procedures evaluated are the displacement probe test, the load probe test, the swipe test (referred to in this paper as the single swipe test) and a less widely used procedure called the sequential swipe test. Each procedure is used to determine failure envelopes for a circular surface foundation and a circular suction caisson foundation under planar vertical, horizontal and moment (VHM) loading for a linear elastic, perfectly plastic von Mises soil. The calculations use conventional, incremental-iterative finite element analysis (FEA) except for the load probe tests, which are performed using finite element limit analysis (FELA). The results demonstrate that the procedures are similarly accurate, except for the single swipe test, which gives a load path that underpredicts the failure envelope in many of the examples considered. For determining a complete VHM failure envelope, the FEA-based sequential swipe test is shown to be more efficient and provide better resolution than the displacement probe test, while the FELA-based load probe test is found to offer a good balance of efficiency and accuracy.
Assessment of numerical procedures for determining
shallow foundation failure envelopes
STEPHEN K. SURYASENTANA, HELEN P. DUNNE, CHRISTOPHER M. MARTIN,
HARVEY J. BURD, BYRON W. BYRNEand AVI SHONBERG
The failure envelope approach is commonly used to assess the capacity of shallow foundations under
combined loading, but there is limited published work that compares the performance of various
numerical procedures for determining failure envelopes. This paper addresses this issue by carrying out
a detailed numerical study to evaluate the accuracy, computational efficiency and resolution of these
numerical procedures. The procedures evaluated are the displacement probe test, the load probe test, the
swipe test (referred to in this paper as the single swipe test) and a less widely used procedure called the
sequential swipe test. Each procedure is used to determine failure envelopes for a circular surface
foundation and a circular suction caisson foundation under planar vertical, horizontal and moment
(VHM) loading for a linear elastic, perfectly plastic von Mises soil. The calculations use conventional,
incremental-iterative finite-element analysis (FEA) except for the load probe tests, which are performed
using finite-element limit analysis (FELA). The results demonstrate that the procedures are similarly
accurate, except for the single swipe test, which gives a load path that under-predicts the failure
envelope in many of the examples considered. For determining a complete VHM failure envelope, the
FEA-based sequential swipe test is shown to be more efficient and to provide better resolution than the
displacement probe test, while the FELA-based load probe test is found to offer a good balance of
efficiency and accuracy.
KEYWORDS: bearing capacity; finite-element modelling; footings/foundations; limit state design/analysis;
numerical modelling; offshore engineering; soil/structure interaction
INTRODUCTION
In recent decades, there has been significant interest in the
failure envelope approach for assessing the ultimate capacity
of foundations under combined loading. The failure envelope
is a hypersurface that defines the n-dimensional combination
of loads (n1) that results in the ultimate limit state (or
plastic failure) of a foundation. The advantages of this
approach over classical bearing capacity methods (Terzaghi,
1943; Meyerhof, 1951; Vesic
´, 1973) are manifold and have
been widely discussed (Schotman, 1989; Tan, 1990; Nova
& Montrasio, 1991; Gottardi & Butterfield, 1993; Bransby &
Randolph, 1998; Martin & Houlsby, 2000; Houlsby &
Byrne, 2001; Gourvenec, 2007).
The failure envelope approach was first introduced
by Roscoe & Schofield (1957) to analyse the interaction
between a steel frame and its foundations using envelopes of
normalised forces. Since then, it has been widely adopted
to represent the results of numerical studies of foundation
bearing capacity, for a broad range of foundation types.
For example, failure envelopes have been determined for
surface foundations (Bell, 1991; Taiebat & Carter, 2000,
2010; Gourvenec, 2007; Vulpe et al., 2014; Shen et al., 2016,
2017), skirted or caisson foundations (Bransby & Randolph,
1998; Bransby & Yun, 2009; Gourvenec & Barnett, 2011;
Hung & Kim, 2014; Karapiperis & Gerolymos, 2014;
Gerolymos et al., 2015; Vulpe, 2015; Mehravar et al.,
2016), spudcan foundations (Zhang et al., 2011) and
mudmat foundations (Feng et al., 2014; Fu et al., 2014;
Nouri et al., 2014; Dunne & Martin, 2017). However,
there is limited published work that quantifies the per-
formance of the numerical procedures used to determine
these failure envelopes. Given the increasing need for
site- and foundation-specific failure envelopes, either for
macro-element modelling (e.g. Martin & Houlsby, 2001;
Cassidy et al., 2004) or for the assessment of ultimate
limit states using the failure envelope approach, the perfor-
mance of these numerical procedures is an important
consideration.
The contributions of this paper are two-fold. First, it
addresses the uncertainty around the performance of various
numerical procedures by carrying out a systematic compari-
son of the failure envelopes determined by each procedure
and by making an assessment of relative computational
efficiency, albeit for a limited range of foundation types and
loading conditions. The aim is to provide guidance for
researchers to identify which procedure they should adopt for
their studies, based on the criteria of accuracy, efficiency and
resolution. This paper does not make assumptions on which
particular parts of the failure envelope are more, or less,
significant for design and thus there is no attempt to quantify
or include the practical significance of errors (on the basis
of where they occur in load space) in the criteria of the
comparative study. Second, this study provides insights into
the implementation of one of the less widely used numerical
procedures called the sequential swipe test. As will be shown
Department of Engineering Science, University of Oxford,
Oxford, UK (Orcid:0000-0001-5460-5089).
Department of Engineering Science, University of Oxford,
Oxford, UK.
Ørsted Wind Power, London, UK.
Manuscript received 5 March 2018; revised manuscript accepted
28 January 2019.
Discussion on this paper is welcomed by the editor.
Published with permission by the ICE under the CC-BY 4.0 license.
(http://creativecommons.org/licenses/by/4.0/)
Suryasentana, S. K. et al.Géotechnique [https://doi.org/10.1680/jgeot.18.P.055]
1
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later, the number of discrete swipe stages used within a
sequential swipe test has a significant impact on the accuracy
of the failure envelope obtained.
NUMERICAL PROCEDURES FOR DETERMINING
FAILURE ENVELOPES
The numerical procedures investigated in this paper can
be categorised into two main groups: displacement-controlled
and load-controlled. The displacement-controlled analyses
(i.e. displacement probe test, single swipe test and sequential
swipe test) are performed using the three-dimensional (3D)
finite-element analysis (FEA) software, Abaqus version 6.13
(Dassault Systèmes, 2014). The load-controlled analyses (i.e.
load probe test) are performed using the 3D finite-element
limit analysis (FELA) software, OxLim (Makrodimopoulos &
Martin, 2006, 2007; Martin, 2011), which has been used to
analyse various bearing capacity problems in plane strain
(Martin & White, 2012; Mana et al., 2013; Dunne et al., 2015)
and more recently in three dimensions (Dunne & Martin,
2017).
All of the analyses reported in this paper are total stress
analyses carried out for undrained clay, modelled using the
von Mises yield criterion. The von Mises criterion was
chosen over the Tresca criterion primarily for convenience, as
it is more efficient to solve von Mises problems than Tresca
problems with the 3D FELA software, OxLim. However, it
has been shown by Gourvenec et al. (2006) that vertical
bearing capacity calculations using the von Mises criterion
(with the strength in simple shear set equal to the undrained
shear strength, s
u
) are reasonably close to those using the
Tresca criterion. Furthermore, once the failure envelope has
been normalised by the uniaxial capacities, the resulting
shape of the non-dimensional failure envelope is qualitatively
similar for both von Mises and Tresca soil for example,
compare the VHM failure envelopes for a circular surface
foundation in this paper (shown later as Fig. 11(a)) to
Fig. 10(a) in the paper by Gourvenec (2007). This paper is
concerned more with the numerical approaches, rather than a
particular soil model, and thus the adoption of a single soil
model for the comparative study is accepted as a limitation of
the scope of the paper.
In this paper, V,Hand Mrefer to the vertical, horizontal
and moment loads, respectively, and w,uand θrefer
to the corresponding vertical, horizontal and rotational
displacements. The loading reference points (LRP) are
located at the centres of the surface foundation base and
the suction caisson lid base (refer to Fig. 1 for the adopted
sign conventions). Note that this is different from some
previous research where the LRP is located at the level of
the caisson skirt base. Furthermore, failure envelopes are
presented in terms of normalised loads (V
˜
=V/V
0
,
˜
H=H/H
0
,
˜
M=M/M
0
), which refer to loads normalised by their
respective uniaxial capacities (V
0
,H
0
,M
0
) as determined
using the same numerical procedure.
Displacement probe test
In the displacement probe test, a displacement increment
in a prescribed direction is applied to the foundation from a
zero load state, with the final (steady) load state determining
a single point on the failure envelope. To find the full failure
envelope, a series of these probe tests with varying displace-
ment directions must be completed. The displacement probe
test has robust convergence properties, and provided that the
prescribed displacement magnitude is sufficiently large, a
well-defined failure load (or combination of loads) can be
obtained.
However, this approach is relatively inefficient as each
calculation only determines a single point on the failure
envelope. Furthermore, it does not allow a straightforward
investigation of the failure envelope as the load path followed
during a displacement probe test is typically non-linear and
difficult to predict. For example, the schematic diagram
in Fig. 2(a) shows a representative, non-linear load path
followed during a displacement probe. The initial load path is
determined by the elastic stiffness of the soilfoundation
system and the prescribed displacement direction. However,
as soil yielding occurs, the stiffness reduces by differing
amounts in each of the loading directions and the load path
changes direction before arriving at (and possibly tracking
along) the failure envelope, eventually maintaining a steady
load state as the displacements continue to increase.
Load probe test
In the load probe test, combined loading components in a
prescribed ratio are applied to the foundation until failure
occurs. It can be difficult to determine accurate failure
loads with load control in FEA, as convergence generally
cannot be obtained if the final prescribed load exceeds the
foundation capacity. A series of trial-and-error load cases, or
a careful approach to the failure envelope, is therefore
required to determine the maximum load that can converge.
However, the FELA technique does not suffer from such
issues and hence, FELA was adopted for the load probe tests
in this study. Furthermore, the use of both lower-bound and
upper-bound FELA provides a rigorous bracket on the
theoretical failure load. A key advantage of the load probe
test is that the predefined direction is followed throughout the
analysis, which enables a more straightforward approach
to determining the entire failure envelope. The schematic
diagram in Fig. 2(a) shows the process of determining a VH
failure envelope by probing in load space. Once a loading
ratio is defined, each of the load paths travels from the origin
M, θ
LRP H, u
V, w
M, θ
LRP H, u
V, w
(a) (b)
Fig. 1. Sign conventions for loads (V,H,M) and displacements (w,u,θ): (a) surface foundation; (b) caisson foundation. LRP denotes loading
reference point
SURYASENTANA, DUNNE, MARTIN, BURD, BYRNE AND SHONBERG2
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(or other initial load states) in the prescribed direction until
the failure envelope is reached.
Single swipe test
The original form of the single swipe test, also known as
the sideswipe test, was introduced by Tan (1990) to
investigate the VH failure envelope of a surface foundation
using centrifuge model tests. In a sideswipe test, the foun-
dation is first pushed vertically to a prescribed embedment,
after which the vertical displacement is held constant while
the foundation is swipedhorizontally. This test was gen-
eralised to VHM loading by Martin (1994), Gottardi
et al. (1999) and Byrne (2000), among other researchers.
Subsequent numerical studies (e.g. Bransby & Randolph,
1998; Gourvenec & Randolph, 2003) then applied this tech-
nique to a range of load spaces by following the same
principle of applying displacement in one degree of freedom
(DoF), followed by a displacement in another DoF while the
displacement in the first DoF is held constant. This process is
essentially two displacement probe tests applied in sequence.
A fundamental assumption underpinning this type of test is
that the swipe phase results in a load path that tracks closely
along the failure envelope, using analogies with hardening
plasticity theory as applied in critical state soil mechanics
(e.g. see the discussions in Tan (1990), Martin (1994) and
Martin & Houlsby (2000)). Unfortunately, this assumption
does not always hold when generalised single swipe tests are
applied to shallow foundations, as the load path may deviate
inside (or cut across) the failure envelope and thus under-
predict the capacity (Bransby & Randolph, 1998).
Sequential swipe test
Although the sequential swipe test is a less widely used
procedure for determining failure envelopes, it can resolve the
potential under-prediction behaviour of the single swipe
test referred to above. A sequential swipe test is a multi-swipe
test, which applies a more gradual change in direction
(in displacement space) by way of a series of discrete swipe
stages, compared with the abrupt directional change that
occurs in the single swipe test. This type of test first appeared
in physical experiments (Martin, 1994; Byrne, 2000; Martin
& Houlsby, 2000) under the term loop test, as a closed loop
path applied in displacement space. More recently, Taiebat &
Carter (2010) and Shen et al. (2017) used a similar approach,
called the modified swipe test, in which the displacement
increment in the first DoF is gradually reduced using a cosine
function while the displacement increment in the other DoF
is gradually increased using a sine function. Taiebat & Carter
(2010) suggested that this would maintain a greater plastic
displacement than the elastic displacement in the first DoF
while the plastic displacements were developing in the other
DoF, which would maintain normality over the whole load
path and, thus, the load path would stay on the failure
envelope.
Regardless of the different names adopted (loop test,
modified swipe test, sequential swipe test), the key principle
behind these tests is the same, which is that changes
in displacement direction should be applied gradually.
Fig. 2(b) shows the different load paths taken by representa-
tive displacement probe and sequential swipe tests. The
sequential swipe test can be considered as a discreteversion
of the loop or modified swipe test, in which the user can
control how gradually the displacement direction changes
through the number of discrete swipe stages (denoted below
as m). This will be made clearer in the following exposition.
Suppose that the directional change in the displacement
space is controlled by ψ, the angle between the current and
previous increments in displacement space. In this paper, the
sequential swipe test is implemented by keeping ψconstant
between all stages of the swipe sequence. For example, a
two-swipe sequential swipe test in the first quadrant of wu
displacement space (assuming the initial pre-swipe displace-
ment is in the wdirection) applies ψ=π/4 for all swipes,
resulting in δu/δw= tan(π/4) followed by δu/δw= tan(π/2),
where δuand δware the horizontal and vertical displacement
increments respectively. Correspondingly, an m-swipe
sequential swipe test in the same displacement space
applies ψ=π/2mfor all swipes, where the direction of the dis-
placement increment in the ith swipe is given in equation (1).
Here q
1
and q
2
denote generic normalised displacements
HH
V V
UB
LB
δH
δu
δw
δV
Displacement probe
Load probe
Average capacity value
Bound difference Displacement probes
Sequential swipe
Sequential swipe stage 3
Sequential swipe stage 2
Sequential swipe stage 1
(a) (b)
Fig. 2. (a) Schematic representation of load paths during displacement probe and load probe tests in VH space. For a displacement probe test, the
initial load path is determined by the elastic properties of the system that is δH/δV=(k
H
e
/k
V
e
)(δu/δw), where k
H
e
,k
V
e
,δuand δware the elastic
horizontal stiffness, elastic vertical stiffness, horizontal displacement and vertical displacement, respectively. As the soil starts yielding, the load
path changes non-linearly before arriving at the failure envelope and settling to a steady load state as the displacements continue to increase. Fora
load probe test, the load path is always co-directional with the load probe direction. (b) Difference in load paths taken by three displacement probe
tests and by a sequential swipe test using the same three probe directions
NUMERICAL PROCEDURES TO DETERMINE FOUNDATION FAILURE ENVELOPES 3
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corresponding to the first and second DoF, respectively, while
ψ
t
is the total directional change in the displacement space
during the swipe phase (e.g. q
1
=w/D,q
2
=u/Dand ψ
t
=π/2
for the above swipe).
δq2
δq1

i
¼tan iψt
m
for 1 imð1Þ
The larger mis, the more gradually the displacement
direction changes. A single swipe test can be obtained as a
special case of the sequential swipe test by letting m=1.Asa
preliminary investigation to illustrate the effect of m, different
m-valued sequential swipe tests were carried out for a surface
strip foundation on von Mises soil.
Figure 3 shows the two-dimensional (2D) FEA mesh for
the surface strip foundation, which consists of 7200 second-
order, fully integrated, hybrid quadrilateral elements (Abaqus
code CPE8H). The von Mises yield strength in pure shear,
k, was equated with the undrained shear strength of the clay,
s
u
, and was modelled as homogeneous throughout the
soil domain. The Poissons ratio of the soil, ν, was set as
0·49, while its Youngsmodulus,E, was set as 10003s
u
.The
soil was modelled as a weightless material, as soil weight does
not affect the capacity for this type of problem (i.e. horizontal
ground surface; pressure-insensitive von Mises yield criterion
for the soil; no contact breaking between foundation and soil).
The surface strip foundation was modelled indirectly by
applying a rigid body constraint to the soil nodes underneath
the foundation.
Figure 4 compares the VH failure envelopes obtained from
different m-valued sequential swipe tests with the analytical
solution (Green, 1954). Two types of swipe analysis were
carried out, with one reaching V
0
before swiping to H
0
and the
other taking the opposite route. For each analysis, three
sequential swipe tests were carried out, with mranging from 2
to 16. Key observations from Fig. 4 are listed below.
(a) All the tests swiping to H
0
end at point A, where the
analytical solution indicates no further change in failure
envelope gradient, as shown in Fig. 4(a).
B
5B
10B
Fig. 3. FEA mesh for sequential swipe testing of a surface strip foundation of width B(domain: 5Bin depth and 10Bin width)
Green (1954)
Single swipe
Sequential swipe (2 swipes)
Sequential swipe (8 swipes)
Sequential swipe (16 swipes)
Green (1954)
Single swipe
Sequential swipe (2 swipes)
Sequential swipe (8 swipes)
Sequential swipe (16 swipes)
1·2
1·0
0·8
H
~
V
~
0·6
0·4
0·2
0
1·2
1·0
0·8
H
~0·6
0·4
0·2
0
0 0·2 0·4 0·6 0·8 1·0 1·2
V
~
0 0·2 0·4 0·6 0·8 1·0 1·2
A
(a) (b)
Fig. 4. Comparison of various swipe tests in VH load space with the analytical solution of Green (1954): (a) swipe tests first reach maximum V
capacity before swiping to maximum Hcapacity; (b) swipe tests first reach maximum Hcapacity before swiping to maximum Vcapacity
SURYASENTANA, DUNNE, MARTIN, BURD, BYRNE AND SHONBERG4
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(b) The single swipe test marginally under-predicts the
failure envelope in Fig. 4(a) but significantly
under-predicts it in Fig. 4(b). In contrast, the sequential
swipe tests show accurate tracking of the failure
envelope, regardless of the starting point of the
swipe phase.
(c) It can be observed that the load paths of the sequential
swipe tests are essentially indistinguishable from the
analytical failure envelope when m8. This suggests
that if mis above some critical value, the load path
will track the failure envelope with negligible deviation.
When completing the analyses for Fig. 4, it was observed that
the rate of increase in the total computational time decreased
as the number of discrete swipe stages increased (this is
because the FEA requires fewer incrementation cutbacks and
equilibrium iterations for smaller ψthan for larger ψ). For
example, the total additional computational times (relative
to the single swipe test) taken by the two-swipe, eight-swipe
and 16-swipe tests were approximately 19%, 24% and 28%,
respectively. This indicates a marginal penalty in choosing a
higher number of stages for the sequential swipe test. Hence,
it is more practical to select a high number of stages at the
outset (e.g. m= 8) than to waste computational resources
attempting to find the optimal m, which in any case is likely
to vary with the problem type and the current load state.
For more systematic mapping of high-dimensional (n3)
failure envelopes, it is advisable that the sequential swipe
test is restricted to two dimensions, while constant load
conditions are applied for the other dimensions. In other
words, for a failure envelope with dimensionality n3, the
sequential swipe test should be used primarily to find 2D
contours of the failure envelope.
APPLICATION OF NUMERICAL PROCEDURES
To further evaluate the numerical procedures described
above, each procedure was used to find the failure envelopes
for planar VHM loading of two types of shallow foundation
(circular surface and suction caisson foundations) bearing on
undrained clay.
Foundation and soil properties
Both the surface and caisson foundations were modelled
as fully rigid, with a diameter D. The caisson foundation
was modelled as having an embedded length L=Dand
a skirt of thickness t
s
= 0·005D. The undrained clay was
modelled in FEA as a homogeneous, linear elastic (ν= 0·49;
E= 10003s
u
), perfectly plastic material and in FELA as a
homogeneous, rigid, perfectly plastic material. For both sets
of analyses, the von Mises yield criterion (with a yield
strength in pure shear of s
u
) and an associated flow rule were
adopted. The soil and foundations were modelled as weight-
less materials, as soil weight does not affect the capacity for
the problems considered here (for the same reasons as
described above).
The 3D FEA model
First-order, fully integrated, hybrid brick elements
(Abaqus code C3D8H) were used for the soil as these are
generally recommended for modelling near-incompressible
materials (Dassault Systèmes, 2014). Brick elements were
also used for the foundation, but the foundation was
made fully rigid by the application of a rigid body constraint.
Sliding and contact breaking between the foundation and soil
were not allowed.
Figure 5 shows the 3D FEA meshes for the surface and
caisson foundations, with symmetry exploited. Displacement
boundary conditions were set to prevent radial displacements
on the circumferential faces and out-of-plane displacements
on the plane of symmetry. In addition, the base of the mesh
was fixed in all directions. The meshes were sufficiently large
that boundary effects on the failure response of the foun-
dation were verified to be negligible. The meshes for the
surface and caisson foundations comprised approximately
40 000 and 44 000 elements, respectively.
The 3D FELA model
The FELA software OxLim first discretises the soil
domain into a mesh of tetrahedral elements using TetGen
(Si, 2015) and applies the boundary conditions. It then
sets up two constrained optimisation problems that together
bound the load multiplier (i.e. the factor by which the
specified live loads must be increased to cause failure). For
this study, the lower-bound (LB) analyses used a piecewise
linear stress field, and the upper-bound (UB) analyses used a
piecewise linear velocity field. The average of the bounds,
(LB +UB)/2, was taken as the best estimate solution for the
load multiplier. The use of the von Mises criterion meant that
both the LB and UB analyses could be cast as standard
second-order cone programming problems and solved with
high efficiency using specialised numerical optimisation
software (Mosek, 2014).
OxLim uses adaptive mesh refinement to improve the
bracketing of the exact collapse load multiplier, where the
RP
RP
(a) (b)
Fig. 5. FEA meshes for displacement probe and swipe tests: (a) surface foundation of diameter D(domain: 2·5Din depth and 3Din radius);
(b) caisson foundation of diameter Dand skirt length L=D(domain: 4·5Din depth and 3Din radius)
NUMERICAL PROCEDURES TO DETERMINE FOUNDATION FAILURE ENVELOPES 5
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adaptivity is based on the spatial variation of the deviatoric
strain rate in the UB velocity field. For the surface
foundation, the initial mesh was adaptively refined twice to
increase the number of elements from approximately 6500 to
25 000, as shown in Fig. 6(a). For the caisson foundation, the
initial mesh was adaptively refined once to increase the
number of elements from approximately 14 000 to 30 000, as
shown in Fig. 6(b). To keep the number of elements
comparable with the FEA mesh, a second refinement was
not undertaken for the caisson foundation. It should be
noted, however, that the average of the LB and UB solutions
(which is the main measure of comparison with the FEA
results) typically does not vary significantly as the bounds
converge. The mesh domain was sufficiently large to render
boundary effects negligible. Fixed boundary conditions were
applied to the base and sides of the domain (excluding the
symmetric plane).
Loading methodology
For this study, the failure envelopes were explored in
increasing dimensionality of load components. First, the
uniaxial capacities were identified for pure V,Hand M
loading. Thereafter, failure envelopes for combined VH,VM
and HM loading were found. Owing to the symmetry in the
VH and VM load spaces, only one quadrant of the failure
envelope needs to be determined. Similarly, symmetry in the
HM load space dictates that only two adjoining quadrants
are needed to define the full failure envelope.
For the displacement probe tests, nine equally spaced
displacement probe directions were used in each quadrant.
For comparison purposes, eight discrete swipe stages (using
the same set of probe directions) were adopted for the
sequential swipe tests. The displacement probe directions can
be identified from equation (1) by letting δq
1
and δq
2
be the
normalised displacement/rotation increments corresponding
to the load components (e.g. in the HM load space, q
1
=u/D
and q
2
=θ). Thereafter, let m= 8 and ψ
t
=π/2 (for VH and
VM)orm= 16 and ψ
t
=π(for HM). Single swipe tests were
also implemented for the study, with the probe directions
similarly identified from equation (1) by letting m= 1 and
ψ
t
=π/2 (for VH and VM)orm= 2 and ψ
t
=π(for HM).
With regard to the magnitude of the displacement increments
used in the displacement probe, sequential swipe or single
swipe tests, they were chosen to be sufficiently large for the
load to reach steady state by the end of each displacement
increment. For this study, the magnitude of each normalised
displacement increment (i.e. ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
δq2
1þδq2
2
q) was set at a
constant value of 0·1. For the load probe tests, nine equally
spaced loading directions were used in each quadrant.
Finally, the full VHM failure envelope was determined.
Mixed load and displacement controls were used for the
FEA-based tests. Load control was used for V, while
displacement control was used in the HM load space that
is, the VHM failure envelope was explored by determining
HM contours of the failure envelope at fixed levels of V.Five
vertical load levels were considered: V
˜
= 0·25, 0·5, 0·625, 0·75
and 0·875. A similar procedure was followed for the load
probe tests performed using FELA, with the HM contours
being determined by probing in HM load space under the
same set of fixed Vloads.
RESULTS
Pure V,Hand Mloading
To validate whether the FEA- and FELA-based pro-
cedures would provide similar answers for the same problems,
Table 1 compares the results obtained by the various
procedures for the uniaxial foundation capacities (V
0
,H
0
and M
0
), which shows that the results from the displacement
probes using FEA are within the bounds obtained using
the 3D FELA procedure, except for H
0
for the surface
foundation. Furthermore, the FELA load probe (average)
results generally agree very well with the FEA-based results.
Combined VH,VM and HM loading
Figures 79 show the VH,VM and HM failure envelopes
for both foundations. Because of symmetry, only one or two
quadrants are shown in these figures, as appropriate. The
small black markers in Figs 79 for the sequential swipe test
results represent intermediate equilibrium load states during
each discrete stage of the sequential swipe, which are
determined by Abaquss automatic step size incrementation
(a)
(b)
Fig. 6. FELA meshes for load probe tests. For surface foundation of
diameter D, mesh domain is 3·5Ddeep, 7Dwide and 3·5Dthick. For
caisson foundation of diameter Dand skirt length L=D, mesh domain
is 4·5Ddeep, 9Dwide and 4·5Dthick. (a) Surface foundation, refined
mesh under moment loading; (b) caisson foundation, refined mesh
under moment loading
Table 1. Uniaxial capacities of surface and caisson foundations
V0
Asu
H0
Asu
M0
ADsu
Surface
Displacement probe 5·63 1·02 0·714
Load probe (LB) 5·45 1·00 0·667
Load probe (UB) 5·77 1·00 0·715
Load probe (average) 5·61 1·00 0·691
Caisson
Displacement probe 13·12 5·86 3·64
Load probe (LB) 12·52 5·52 3·36
Load probe (UB) 13·68 6·28 3·96
Load probe (average) 13·10 5·90 3·66
A=πD
2
/4 refers to the foundation plan area. Note that the
procedure and results for the displacement probe, single swipe and
sequential swipe tests are identical for uniaxial loading.
SURYASENTANA, DUNNE, MARTIN, BURD, BYRNE AND SHONBERG6
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1·2
1·0
0·8
H
~
V
~
0·6
0·4
0·2
0
1·2
1·0
0·8
H
~0·6
0·4
0·2
0
0 0·2 0·4 0·6 0·8 1·0 1·2
(a)
V
~
0 0·2 0·4 0·6 0·8 1·0 1·2
(b)
Single swipe
Sequential swipe
Displacement probe
Load probe (average)
Load probe (bounds)
Single swipe
Sequential swipe
Displacement probe
Load probe (average)
Load probe (bounds)
Fig. 7. Dimensionless VH failure envelopes: (a) surface foundation; (b) caisson foundation
1·2
1·0
0·8
M
~
V
~
0·6
0·4
0·2
0
0 0·2 0·4 0·6 0·8 1·0 1·2
(a)
1·2
1·0
0·8
M
~
V
~
0·6
0·4
0·2
0
0 0·2 0·4 0·6 0·8 1·0 1·2
(b)
Single swipe
Sequential swipe
Displacement probe
Load probe (average)
Load probe (bounds)
Single swipe
Sequential swipe
Displacement probe
Load probe (average)
Load probe (bounds)
Fig. 8. Dimensionless VM failure envelopes: (a) surface foundation; (b) caisson foundation
1·4
1·2
1·0
0·8
0·6
0·4
–1·5 –1·0 –0·5 0
(a)
0·5 1·0 1·5 –2 –1 0 1 2
0·2
0
M
~
2·5
2·0
1·5
1·0
0·5
0
M
~
H
~
(b)
H
~
Single swipe
Sequential swipe
Displacement probe
Load probe (average)
Load probe (bounds)
Single swipe
Sequential swipe
Displacement probe
Load probe (average)
Load probe (bounds)
Fig. 9. Dimensionless HM failure envelopes: (a) surface foundation; (b) caisson foundation
NUMERICAL PROCEDURES TO DETERMINE FOUNDATION FAILURE ENVELOPES 7
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scheme. The density of the black markers (i.e. the resolution
of the failure envelope) can be controlled by changing
the step size incrementation scheme. The final load states
from the displacement probe tests, and the load paths
from the sequential swipe tests, were all within the bounds
obtained from the load probe tests. In fact, there are
no significant differences between these two sets of
FEA-generated results and the FELA load probe (average)
results. In contrast, there is noticeable under-prediction of
the failure envelopes by the single swipe test. The under-
prediction of the failure envelopes for the surface foundation
is minor for most cases, except for the HM failure envelope
when
˜
H0·7.However, under-prediction of the failure
envelopes for the caisson foundation is apparent for all the
load spaces explored.
Combined VHM loading
Figure 10 shows the HM failure envelopes obtained for
both foundations under three selected levels of normalised
vertical load V
˜
. Again, the results of the displacement probe
and sequential swipe tests are all within the LB and UB
envelopes obtained using FELA. Furthermore, Fig. 11
shows the HM failure envelopes obtained from both the
single swipe and sequential swipe tests, for all of the vertical
load levels considered. It is evident that the single swipe test
under-predicts the HM failure envelopes for all vertical load
levels.
DISCUSSION
To assess the performance of the various numerical pro-
cedures in determining the above failure envelopes, the
following performance criteria were adopted: accuracy, com-
putational efficiency and resolution.
To allow for a quantitative (albeit approximate) compari-
son of the accuracy of the various numerical procedures, an
accuracy measure η(relative to the displacement probe test) is
introduced as follows
η¼Ai
Aref
ð2Þ
where A
i
refers to the area enclosed within a failure envelope
that was determined by any numerical procedure, and A
ref
refers to the area enclosed within a reference failure envelope
that was determined by the displacement probe method
(which is the most widely used among the FEA-based
procedures). The area calculations were performed by
1·4
1·2
1·0
0·8
0·6
0·4
0·2
0
M
~
–1·5 –1·0 –0·5 0 0·5 1·0 1·5
(c)
H
~
2·5
2·0
1·5
1·0
0·5
0
M
~
–2 –1 0 1 2
(e)
H
~
2·5
2·0
1·5
1·0
0·5
0
M
~
–2 –1 0 1 2
(f)
H
~
1·4
1·2
1·0
0·8
0·6
0·4
0·2
0
M
~
–1·5 –1·0 –0·5 0 0·5 1·0 1·5
(b)
H
~
2·5
2·0
1·5
1·0
0·5
0
M
~
–2 –1 0 1 2
(d)
H
~
Single swipe
Sequential swipe
Displacement probe
Load probe (average)
Load probe (bounds)
1·4
1·2
1·0
0·8
0·6
0·4
0·2
0
M
~
–1·5 –1·0 –0·5 0 0·5 1·0 1·5
(a)
H
~
Single swipe
Sequential swipe
Displacement probe
Load probe (average)
Load probe (bounds)
Fig. 10. Dimensionless HM failure envelopes at selected V
˜
levels for surface foundations: (a) V
˜
= 0·25; (b) V
˜
= 0·5; (c) V
˜
= 0·75; and caisson
foundations: (d) V
˜
= 0·25; (e) V
˜
= 0·5; ( f) V
˜
= 0·75
SURYASENTANA, DUNNE, MARTIN, BURD, BYRNE AND SHONBERG8
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taking the set of failure points as the vertices of a polygon
(the failure points are taken to be the average of the bounds
for the FELA analyses). Fig. 12 shows an illustration of a
typical computation of the accuracy measure η. Note that
values of ηabove 1 do not necessarily imply inaccuracy, as
some of the procedures have either a higher number of failure
points (e.g. single swipe and sequential swipe tests) or more
evenly spaced failure points (e.g. load probe test) to better
approximate the computation of the failure envelope area.
Table 2 shows the comparison of ηfor each numerical
procedure, for each failure envelope shown in Figs 710. It
can be observed that the sequential swipe test and the load
probe test provide similar levels of accuracy to the displace-
ment probe test, with ηranging from 1·01 to 1·03 and 1·01 to
1·07, respectively. The single swipe test, however, generally
under-predicts the failure envelope areas, with ηranging from
0·78 to 1·01. On average (and relative to the displacement
probe test results), the single swipe test under-predicts the
reference failure envelopes by 11%, while the sequential
swipe test and load probe test over-predict the reference
failure envelopes by 2% and 3%, respectively.
Table 3 shows the total and average ( per probe)
computational time taken by each procedure to find the
VH,VM,HM and VHM failure envelopes presented in the
previous section. Using FEA, 120 displacement probe tests
were performed for each foundation. For the sequential swipe
tests, the same 120 probe directions were used for the discrete
swipe stages. In contrast, only 22 displacement increments
(corresponding to the first and last probe directions in each
quadrant of the displacement space) were performed for the
single swipe tests. Using FELA, 120 load probe tests were
performed to identify 120 failure loads. All the analyses were
set up using scripts and the difference in set-up time is thus
negligible. The computer used to run the analyses had an
Intel Xeon 3·60 GHz processor (eight central processing
units) with 16 GB RAM (random access memory).
Table 3 is revealing in several ways. The single swipe test
was found to be the most efficient procedure if the total time
1·4
1·2
1·0
0·8
0·6
0·4
0·2
0
M
~
–1·5 –1·0 –0·5 0
(a)
0·5 1·0 1·5
H
~
2·5
2·0
1·5
1·0
0·5
0
M
~
–2 –1 0 1 2
(b)
H
~
Single swipe
Sequential swipe
Increasing V
~Increasing V
~
Fig. 11. Dimensionless HM failure envelopes at selected V
˜
levels (V
˜
= 0, 0·25, 0·5, 0·625, 0·75, 0·85): (a) surface foundation; (b) caisson
foundation
Displacement probes
Single swipe test
Aref
Aref
Ai
Ai
η =
H
V
Fig. 12. Computation of the accuracy measure ηfor a typical single
swipe test in VH space, where A
i
is the area enclosed by the failure
envelope from the single swipe test (i.e. the shaded area) and A
ref
is the
area enclosed by the failure envelope from the displacement probe tests
Table 2. Comparison of the accuracy measure η(as per equation (2))
for the surface and caisson foundations, for each failure envelope
shown in Figs 710
Failure
envelope
η(single
swipe)
η(sequential
swipe)
η(load
probe)
Surface
Figure 7(a) 0·99 1·01 1·01
Figure 8(a) 1·01 1·03 1·04
Figure 9(a) 0·89 1·01 1·03
Figure 10(a) 0·87 1·01 1·04
Figure 10(b) 0·83 1·01 1·05
Figure 10(c) 0·81 1·01 1·07
Caisson
Figure 7(b) 0·92 1·01 1·01
Figure 8(b) 0·94 1·03 1·02
Figure 9(b) 0·88 1·04 1·00
Figure 10(d) 0·90 1·04 1·01
Figure 10(e) 0·87 1·04 1·02
Figure 10(f ) 0·78 1·04 1·04
Average 0·89 1·02 1·03
Table 3. Computational time taken by each numerical procedure to
find all failure envelopes (VH,VM,HM,VHM) of the surface and
caisson foundations
Number
of probes
Total
time: h
Average time
per probe: h
Surface
Displacement probe 120 68·6 0·571
Single swipe 22 21·4 0·971
Sequential swipe 120 25·5 0·212
Load probe 120 31·2 0·260
Caisson
Displacement probe 120 152·8 1·27
Single swipe 22 23·0 1·05
Sequential swipe 120 59·5 0·496
Load probe 120 22·3 0·186
NUMERICAL PROCEDURES TO DETERMINE FOUNDATION FAILURE ENVELOPES 9
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taken is adopted as the efficiency measure. However, different
procedures provide different numbers of reliable failure
points; only the final failure points which have reached
steady state at the end of each probe are dependably accurate
for all cases. Thus, an alternative efficiency measure, the
time taken per reliable failure point (defined as the average
time to analyse one probe), was compared. Based on this
efficiency measure, the load probe test was found to be the
most efficient procedure. For the analyses of the surface and
caisson foundations, the sequential swipe test was, respect-
ively, 2·7 and 2·6 times faster than the displacement probe
test. This is an interesting result as it shows the existence of a
numerical procedure capable of providing failure envelope
predictions that are as accurate as the displacement probe
test, but with greater efficiency. The single swipe test, on the
other hand, has lower efficiency than the sequential swipe
test when evaluated on a per probe basis.
In terms of resolution, the single swipe and sequential
swipe tests provide more failure points than the other pro-
cedures. However, Figs 79 have shown that the load path
followed during a single swipe test may be far from the FELA
load probe (average) results (and outside the bounds). Thus,
the intermediate points during the single swipe test may
not be accurate failure points. In contrast, the same figures
show that the intermediate points obtained during each stage
of a sequential swipe test are sufficiently close to the FELA
load probe (average) results (and within the bounds) to be
considered as reasonably accurate failure points. Thus, the
sequential swipe test provides higher failure envelope
resolution than the other procedures.
Overall, the sequential swipe test appears to provide the
best balance of accuracy, efficiency and resolution among the
FEA-based procedures, while the FELA-based load probe
test provides a good balance of accuracy and efficiency (if
resolution is not an important criterion). However, if the
accuracy of intermediate failure points is not an important
criterion, the load path from a single swipe test can provide a
quick and conservative estimation of the failure envelope;
although users should be aware that the shape of a failure
envelope determined from the intermediate points can
sometimes be significantly different from the reference
failure envelope (e.g. see Figs 4(b) and 11(a)).
There are some limitations of this comparative study. First,
the conclusions of this study have only been obtained for
von Mises soil. It is unknown whether the same conclusions
apply for other soil models such as the MohrCoulomb
model, especially if a non-associated flow rule is adopted.
Second, the influence of features such as non-homogeneous
soil strength profiles and the allowance for contact breaking
between foundation and soil have not been investigated.
Further studies are required to address these issues.
CONCLUSIONS
The primary goal of this paper was to evaluate the per-
formance of various numerical procedures for determining
undrained VHM failure envelopes of shallow foundations:
the displacement probe test, the single swipe test and the
sequential swipe test (all performed using FEA) as well as the
load probe test (performed using FELA). Two circular
foundation types with significantly different failure envelope
shapes were considered.
In general, there is little to differentiate between the
procedures in terms of accuracy, except for the single swipe
test, where the load path was sometimes found to under-
predict (i.e. deviate inside) the reference failure envelope. For
the examples considered in this paper, the sequential swipe
test appears to offer the best balance of accuracy, efficiency
and resolution. The FELA-based load probe test has higher
efficiency but lower resolution. The findings suggest that
the sequential swipe test offers an attractive alternative to
the widely used displacement probe test, since it is just as
accurate, but is faster and has the additional benefit of higher
failure envelope resolution.
Finally, this study investigated the influence of the number
of discrete swipe stages used in a sequential swipe test. It was
found that there is a critical number above which the load
path appears to track the failure envelope with negligible
deviation. Based on the findings of this paper, a minimum of
eight discrete swipe stages in each quadrant of the displace-
ment space is recommended to ensure that the load path stays
close to the failure envelope throughout the analysis. As the
number of discrete swipe stages decreases, the accuracy of the
sequential swipe test decreases and the load path becomes
more sensitive to the starting point of the swipe phase, as
shown by the single swipe test results in Fig. 4.
ACKNOWLEDGEMENTS
The first and second authors acknowledge the generous
support of Ørsted Wind Power and Subsea 7, respectively, for
funding their DPhil studentships at the University of Oxford.
NOTATION
A
i
area enclosed by failure envelope from any numerical
procedure
A
ref
area enclosed by reference failure envelope from displacement
probe tests
Bwidth of surface strip foundation
Hhorizontal load
˜
Hnormalised horizontal load
H
0
horizontal uniaxial capacity
Mmoment load
˜
Mnormalised moment load
M
0
moment uniaxial capacity
mnumber of discrete sequential swipe stages
nnumber of loading dimensions
q
1
normalised generalised first degree of freedom
q
2
normalised generalised second degree of freedom
uhorizontal displacement
Vvertical load
V
˜
normalised vertical load
V
0
vertical uniaxial capacity
wvertical displacement
ηrelative accuracy measure for failure envelopes
θrotational displacement
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NUMERICAL PROCEDURES TO DETERMINE FOUNDATION FAILURE ENVELOPES 11
Downloaded by [ UNIVERSITY OF OXFORD] on [05/03/19]. Published with permission by the ICE under the CC-BY license
... The uniaxial ultimate bearing capacity was obtained by applying a displacement in the uniaxial direction without restricting the displacement in the other two directions. The displacement probe method (Bransby and Houlsby, 1998;Feng et al., 2017;Liu et al., 2016Liu et al., , 2017aTong et al., 2020;Suryasentana et al., 2020b) was applied to obtain the failure envelope in this study. In each probe test, the foundation was given a fixed displacement ratio of δ u /δ w , δ u /δ θ , or δ w /δ θ to derive a failure point (δ u and δ w are respectively the displacement increments of the foundation in horizontal and vertical directions, δ θ is the rotational increment of the foundation). ...
... Applied Ocean Research 118 (2022) 103007 It should be noted that in practice, the combined action of wind and waves may result in non-planar horizontal-moment loading, and a fully 6 degrees of freedom (6DoF) failure envelope is more accurate for these non-planar loading situations Suryasentana et al., 2021). But due to expensive computing costs required for the 6DoF failure envelope, the 3DoF failure envelope (i.e., F V − F H − M failure envelope) as a simplified idealization was considered in majority of the literature (e.g., Bransby and Randolph, 1998;Vulpe, 2015;Zhao et al., 2020;Suryasentana et al., 2020aSuryasentana et al., , 2020b and appropriate for situations when horizontal-moment loading is approximately planar. Hence, the 3DoF failure envelope is researched in this study, and the effect of local scour on the 6DoF failure envelope is accounted for in further investigations. ...
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Local scour may induce bearing capacity loss of the foundation significantly by removing soil around the foundation and it has become one of the pivotal factors causing the failure of structures in offshore engineering. Therefore, it is indispensable to investigate the effect of local scour on bearing capacity behaviour of foundations. This paper focuses on the effects of dimensions of local scour hole on both ultimate bearing capacities and failure envelopes of the monopile and the caisson in undrained soil with different soil strength heterogeneity ratios using three-dimensional finite element method. The results show that the scour depth has a more significant effect on the ultimate bearing capacity than the scour hole angle. Meanwhile, the scour depth has a more significant effect on the lateral bearing capacity (i.e., horizontal bearing capacity and moment) than on the vertical bearing capacity. For given dimensions of local scour hole, the effect of local scour on the bearing capacity decreases as soil strength heterogeneity ratio increases. Moreover, local scour affects not only the size of the failure envelopes of the monopile and the caisson, but also their shape. Based on the above findings, a procedure of designing the monopile and the caisson to consider local scour effect is proposed, which could be helpful to the design of the foundations in practical engineering.
... However, foundations may be subjected not only to vertical loads (V), but also to horizontal loads (H) and bending moments (M) due to lateral loads that originate from earthquakes, wind and other constraints provided in the design and construction of structures. Following the pioneering seminal contributions of Meyerhof (1953), Hansen (1970) and Vesic (1975), several analytical, numerical, and experimental studies have been performed to simulate the drained and undrained response of shallow foundations under vertical-horizontal-moment (V-H-M) combined loading (Bransby and Randolph 1998;Georgiadis and Butterfield 1988;Ukritchon et al. 1998;Gottardi et al. 1999;Taiebat and Carter 2000;Zhu 2000;Houlsby and Cassidy 2002;Okamura et al. 2002;Gourvenec and Randolph 2003;Hjiaj et al. 2004;Cassidy 2007;Loukidis et al. 2008;Lyamin et al. 2009;Govoni et al. 2011;Krabbenhoft et al. 2012;Krabbenhoft et al. 2014;Cassidy et al. 2013;Rao et al. 2015;Tang et al. 2015;Yahia-Cherif et al. 2017;Zheng et al. 2019;Dastpak et al. 2020;Pham et al. 2020;Suryasentana et al. 2020;Zhao et al. 2020). Taiebat and Carter (2000) utilised the finite element method to present three-dimensional failure loci for a circular footing resting on a uniform clay deposit and subjected to various combinations of vertical-horizontal-moment loadings. ...
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Soil is a cross anisotropic particulate medium with different strengths in various directions; this is primarily due to its geological deposition process and the very fact that particles always settle in their most stable positions. This study examines the influence of inherent anisotropy on the ultimate bearing capacity of eccentrically and obliquely loaded strip footings that rest on cohesionless granular soils using a two-dimensional plane strain finite element simulation in conjunction with the lower bound limit analysis method and second-order cone programming (SOCP). The inherent anisotropy, manifested in the so-called parameter of anisotropy ratio, is simulated by considering variable internal friction angles along different directions. The nonlinear form of the universal Mohr-Coulomb failure criterion is also used to optimize the lower bound formulation. The failure envelopes of shallow foundations that correspond to inclined and eccentric loadings are depicted and discussed for various anisotropy ratios of the underlying soil deposit. It is observed that the failure locus generally decreases in size as the anisotropy ratio increases. Based on the results of numerical simulations, a general equation that describes the general bearing capacity of shallow foundations resting on inherently anisotropic cohesionless granular medium subjected to combined vertical-horizontal-moment loadings is presented and discussed.
... Although there has been much research in assessing soil-foundation response under multiple DoF loading, most of them are focused on the ultimate limit response (e.g. Gourvenec and Randolph 2003;Gourvenec 2007;Nouri et al. 2014;Vulpe et al. 2014;Shen et al. 2017;Dunne and Martin 2017;Suryasentana et al. 2020a, b;He and Newson 2020). The assessment of soil-foundation response at relatively small magnitudes of multiple DoF loading is, however, important for applications such as structural fatigue analysis and natural frequency analysis. ...
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Although there are many methods for assessing vertical stiffness of footings on the ground, simplified solutions to evaluate lateral, rotational, and torsional static stiffness are much more limited, particularly for nonhomogeneous profiles of shear modulus with depth. This paper addresses the topic by introducing a novel “work-equivalent” framework to develop new simplified design methods for estimating the stiffnesses of footings under multiple degrees-of-freedom loading for general nonhomogeneous soils. Furthermore, this framework provides a unified basis to analyze two existing design methods that have diverging results. 3D finite element analyses were carried out to investigate the soil–footing interaction for a range of continuously varying and multilayered nonhomogeneous soils, and to validate the new design approach.
... Moreover, the physical tests tend to be simplified, thereby only limited factors can be considered and it is thus hard to comprehensively capture the failure mechanism of caisson foundation. Compared to the experiment, the numerical modelling has been extensively used to obtain the failure envelopes [9][10][11][12] due to the fast development of computer techniques. However, the computational cost of numerical modelling is also expensive when conducting numerous simulations each time for a given caisson foundation. ...
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To reduce the computational cost and improve the accuracy in predicting failure envelopes of caisson foundations , this study proposes an intelligent method using random forest (RF) based on data extended from experiments and calibrated numerical simulations. Two databases are built from the numerical results by coupled Lagrangian finite element method and smoothed particle hydrodynamics with a critical state based simple sand model (CLSPH-SIMSAND). The first database involves the failure envelopes of caisson foundations with various specifications for a given sand, and the second database includes two additional failure envelopes of caisson foundations in other granular soils. The relationship between the characteristic measures of failure envelope and sand properties as well as the specification of caisson foundation is trained by RF using the prepared databases. The results indicate the RF based model is able to accurately learn the failure mechanism of caisson foundation from the raw data. Once a RF based model that can accurately reproduce the failure envelopes of caisson foundations in a given sand is developed, it can be easily modified to predict the failure envelopes of caisson foundations in a random granular soil as long as one numerical result in such soil is added to the database. Therefore, the RF based model is much more convenient than the calibration of parameters used in the conventional analytical solutions and the computational cost is much less than the conventional numerical modelling methods.
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