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A systematic framework for formulating convex failure
envelopes in multiple loading dimensions
STEPHEN K. SURYASENTANA,HARVEYJ.BURD†, BYRON W. BYRNE†and AVI SHONBERG‡
The failure envelope approach is widely used to assess the ultimate capacity of shallow foundations for
combined loading, and to develop foundation macro-element models. Failure envelopes are typically
determined by fitting appropriate functions to a set of discrete failure load data, determined either
experimentallyor numerically. However, current procedures to formulate failure envelopes tend to be ad
hoc, and the resulting failure envelopes may not have the desirable features of being convex and well-
behaved for the entire domain of interest. This paper describes a new systematic framework to
determine failure envelopes –based on the use of sum of squares convex polynomials –that are
guaranteed to be convex and well-behaved. The framework is demonstrated by applying it to three
data sets for failure load combinations (vertical load, horizontal load and moment) for shallow
foundations on clay. An example foundation macro-element model based on the proposed framework
is also described.
KEYWORDS: bearing capacity; failure; footings/foundations; numerical modelling; offshore engineering;
soil/structure interaction
INTRODUCTION
Existing failure envelope formulations
The failure envelope approach is widely used to assess the
ultimate capacity of shallow foundations for combined load-
ing. There are numerous advantages of this approach com-
pared to classical bearing capacity methods (Terzaghi, 1943;
Meyerhof, 1951; Vesic
´, 1973), as has been discussed exten-
sively in previous work (e.g. Tan, 1990; Gottardi & Butterfield,
1993; Bransby & Randolph, 1998; Martin & Houlsby, 2000;
Houlsby & Byrne, 2001). Failure envelopes can also be used to
develop plasticity-based macro-element models for shallow
foundations (e.g. Nova & Montrasio, 1991; Cassidy, 1999;
Martin & Houlsby, 2001; Bienen et al., 2006).
A failure envelope for a specific shallow foundation
concept can be determined by measuring, or computing,
a set of discrete failure load combinations for different
loading conditions (e.g. the experimental data reported in
Martin & Houlsby (2000) and the numerical data in Taiebat
& Carter (2000)). These failure envelope data are then
approximated using an appropriate mathematical formu-
lation (referred to in this paper as a ‘failure envelope
formulation’) together with an optimisation process to
achieve a best fit.
In the current paper, V,Hand Mrefer to the applied
vertical and horizontal loads, and moment, respectively.
Furthermore,
˜
H¼H=H0,
˜
M¼M=M0and
˜
V¼V=V0refer
to the normalised loads, where H
0
,M
0
,V
0
are the respective
uniaxial capacities. (Uniaxial capacity refers to the failure
load for one load component when all other load
components are maintained at zero throughout the
loading.) In the notation employed in this paper, vectors
are denoted in bold face, italic, lower case (e.g. x) and
matrices are denoted in bold face, upper case, upright font
(e.g. A).
Current failure envelope formulations can generally be
categorised into one of two groups (although the wide-
ranging literature in this area also includes a number of
formulations that do not fit into either group, e.g. Taiebat &
Carter (2000)). One group represents the failure envelope
using a sum of power functions of the applied loads. This
approach has been used to represent the ‘cigar-shaped’failure
envelopes determined from experimental data for shallow
foundations on sand (Nova & Montrasio, 1991; Gottardi &
Butterfield, 1993; Cassidy, 1999; Gottardi et al., 1999; Byrne
& Houlsby, 2001; Bienen et al., 2006) and clay (Martin &
Houlsby, 2000). Published formulations of this type are
mainly defined for planar VHM loading, although exten-
sions to six degrees-of-freedom loading have been described
by Martin (1994), Byrne & Houlsby (2005) and Bienen
et al. (2006). An example of a formulation in this group is
‘Model A’(Martin, 1994), which was developed to represent
the failure envelope for a spudcan foundation in clay. Model
A is defined as
fð
˜
H;
˜
M;
˜
VÞ¼
˜
H2þ
˜
M216
˜
V2ð1
˜
VÞ2¼0ð1Þ
although for numerical implementation, Martin & Houlsby
(2001) recommend the alternative form
fð
˜
H;
˜
M;
˜
VÞ¼ð
˜
H2þ
˜
M2Þ1=24
˜
Vð1
˜
VÞ¼0ð2Þ
The second group of failure envelope formulations, termed
‘HM-based’, has primarily been used in connection with
numerically determined failure envelope data (Feng et al.,
2014; Vulpe et al., 2014; Vulpe, 2015; Shen et al., 2016,
2017). Formulations in this group are defined in terms
of composite horizontal and moment loads, where the
composition provides a means of incorporating the influence
of additional load components within a basic HM frame-
work. An example of a formulation in this group (from
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 30 October 2018; revised manuscript accepted
28 March 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.251]
1
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Vulpe et al. (2014)) is
fð
˜
H;
˜
MÞ¼
˜
H
aþ
˜
Mαþ2b
˜
H˜
M1¼0ð3Þ
where
˜
Hand
˜
Mare composite horizontal and moment loads
defined by
˜
H¼
˜
H=ð1
˜
V469Þand
˜
M¼
˜
M=ð1
˜
V212Þ;
and aand bare parameters (a=2·13 and b=0·09 for
˜
V
05, or a=1·63 and b=0·05 for
˜
V.05).
Formulation of foundation macro-element models
Failure envelope formulations are a convenient basis for
the development of plasticity-based macro-element models
for foundations. For example, based on experimental data,
Martin & Houlsby (2000) develop a modified version of
Model A (termed ‘Model B’, described further in the later
subsection within ‘Example applications’entitled ‘Model B’)
as the yield function for a macro-element model of a spudcan
foundation in clay.
A macro-element model typically incorporates a yield
function and a plastic potential function; these functions
govern the admissible load states and the plastic displace-
ments, respectively. The yield function may also contain
a hardening parameter that governs the evolution of the
yield surface (see Appendix for discussion on the distinction
between a yield surface and a yield function). However, the
current paper is concerned mainly with developing failure
envelope formulations to provide an approximate fit to
discrete failure load data. For simplicity, for the macro-
element formulation described later in this paper, associated
flow (i.e. where the yield function is also the plastic potential
function) and perfect plasticity (i.e. no hardening) are
adopted.
The yield function adopted in macro-element models
should ideally be convex throughout the domain and it is
insufficient to have convexity at just the admissible loads
boundary (i.e. a convex yield surface). Franchi et al. (1990)
has discussed the key differences between the yield function
and the yield surface and the theoretical relevance of the
convexity of the yield function. A summary of the charac-
teristics of convex functions and the difference between a
yield surface and a yield function is given in the Appendix.
The importance of convexity of the yield function has been
largely overlooked, especially in geotechnics, over the past
few decades. This particular issue was raised by Panteghini
& Lagioia (2014, 2018a, 2018b), who reported numerical
problems (e.g. lack of convergence in boundary value
problems) when using implicit integration algorithms with
non-convex yield functions. To avoid such problems, they
propose an approach for providing convexity to the yield
functions. For rate-independent, non-hardening/softening
materials, a convex yield function has the added benefit of
thermodynamic consistency, provided that an associated
flow rule is adopted and the yield surface contains the
origin of the load space (Ottosen & Ristinmaa, 2005).
Yield (and plastic potential) functions employed in macro-
element models must be continuous and real-valued. Ideally,
they should also be differentiable with a continuous gradient
and Hessian throughout the load domain. Furthermore,
they should not have a restricted domain, as it is not known
a priori the load states at which implicit integration algo-
rithms will require the evaluation of the yield function.
Also, functions should be avoided that have singularities
(either in the function itself or its derivatives) at certain load
states. For the purposes of the current paper, a function
with these ideal attributes is referred to as ‘well-behaved’.
It is noted that successful plasticity models can be
developed on the basis of yield functions that are not
well-behaved; the conventional Mohr–Coulomb yield func-
tion, for example, provides a common basis for soil con-
stitutive models despite not being differentiable at the ‘edges’
of the yield surface (although rounding approximations
(e.g. Sloan & Booker, 1986) can be employed to address
the numerical difficulties associated with these edges). In
general, employing non-well-behaved functions is incon-
venient from a numerical implementation perspective.
Failure envelope formulations can be used in two separate
types of practical application: (a) assessment of ultimate
foundation capacity and (b) representing the yield surface
(and the plastic potential for associated flow) in macro-
element models. Failure envelope formulations that are
suitable for both purposes are therefore particularly con-
venient. However, not all of the current failure envelope
formulations are applicable to both applications, either
due to lack of convexity or because they are not well-
behaved. For example, the extensive use of fractional expo-
nents for the HM-based formulations means that the
yield function may not be real-valued in parts of the
domain (e.g. equation (3) is not real-valued for
˜
M,0or
˜
V,0, since raising negative numbers by fractional powers
results in complex values). Furthermore, the composition
approach within this group of formulations results in
singularities in the function and its gradient at
˜
V¼1,
which makes this form of failure envelope inconvenient for
macro-element model implementation. It is noted that the
HM-based formulations were not developed with macro-
element modelling in mind, and thus suitability for this use
should not necessarily be expected. However, failure envel-
opes developed using the formulation framework described
in the current paper are guaranteed to be convex and well-
behaved; they are therefore applicable to a range of practical
applications.
A new framework for convex, well-behaved, failure envelopes
For the purposes of the current paper, a ‘globally convex’
function is defined as a function that is convex throughout
the domain R
n
(real coordinate space of ndimensions, where
nis interpreted in the context of the current work as the
number of individual load and moment components applied
to the foundation). A new framework is proposed for
the formulation of failure envelopes that are guaranteed to
be globally convex and well-behaved for all possible failure
load combinations. Furthermore, the framework is systema-
tic and general. This is significant, as previous researchers
(e.g. Gourvenec, 2007) have suggested that difficulties in
deriving closed-form expressions to approximate failure
envelopes present a practical obstacle to the adoption of
the failure envelope approach in design. The proposed frame-
work provides a systematic process that can be applied
to failure envelope data sets consisting of multiple load
(and moment) components. The framework is also able to
represent various failure envelope shapes that correspond to
different types of shallow foundation. It therefore facilitates
the further development of the failure envelope approach for
practical applications.
In developing this framework, it is recognised that proving
convexity for a general function is typically a difficult task.
For example, a twice differentiable function with a positive
semi-definite Hessian throughout its domain is guaranteed to
be convex. However, a heuristic approach that demonstrates
that a Hessian is positive semi-definite at a finite number of
points is insufficient to prove convexity for the entire domain.
Conversely, demonstrating non-convexity is often relatively
straightforward; a single case where the Hessian is not
positive semi-definite is sufficient.
SURYASENTANA, BURD, BYRNE AND SHONBERG
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The approach employed in the current framework is
therefore to restrict the search for failure envelope formu-
lations to well-behaved polynomial functions that are known
a priori to be globally convex. In developing this procedure,
use is made of a mathematical technique known as ‘sum of
squares’(SOS) programming, which provides a computa-
tionally tractable method for determining globally convex
polynomials.
POLYNOMIAL FAILURE ENVELOPE
FORMULATIONS
General homogeneous polynomials
Ellipsoids are a class of well-behaved functions that are
guaranteed to be convex, as outlined below. A general
ellipsoid can be expressed in the quadratic form
fxðÞ¼ xcðÞ
TAxcðÞ¼1ð4Þ
where x,c[R
n
are vectors and Ais a positive definite square
constant matrix. The Hessian in this case is r2fxðÞ¼2A
and therefore guaranteed to be positive definite. Since posi-
tive definite matrices are also positive semi-definite, it follows
that ellipsoids are convex. As ellipsoids are also well-behaved,
they are candidates for representing failure envelopes, for
which the components of xare the foundation loads (in
an appropriate normalised form) and the components of A
and care selected to provide a best fit with the available
failure load data. However, ellipsoids have the significant
disadvantage in this application that they are not expressive
enough to represent the varied shapes of failure envelopes
that correspond to the range of plausible shallow foundation
configurations.
Ellipsoids are second degree polynomials; higher degree
polynomials provide options for developing failure envelope
functions with increased expressiveness. Since polynomials of
odd degree 3 are guaranteed to be non-convex (Ahmadi
et al., 2013), odd polynomials are excluded as candidates for
the current application. Polynomials of even degree 4 can
either be convex or non-convex, but it is typically computa-
tionally difficult to prove their convexity in the general case
(Ahmadi et al., 2013). Thus, this paper considers a subset of
polynomials of even degree 2, called sum of squares convex
(SOS-convex) polynomials, for which convexity can be
demonstrated in a computationally tractable manner.
Sum of squares polynomials
A polynomial s(x) of degree 2d(where dis a positive
integer) is an SOS polynomial if it can be represented as
sxðÞ¼
X
npoly
j¼1
gjðxÞ2ð5Þ
where g
j
(x) are polynomials of degree 4dand npoly is the
number of individual polynomials, g
j
(x), employed in the
summation. Equivalently, s(x) is SOS if and only if it can be
represented as s(x)¼z
T
Bzwhere Bis a positive semi-definite
matrix and zis a vector of all monomials of degree up
to and including d. This can be shown by noting that
s(x)¼z
T
Bz¼z
T
Q
T
Qz¼(Qz)
T
(Qz)¼g
T
g(where g
T
gis
equation (5) and Q
T
Qis the Cholesky decomposition
of B). It is evident from equation (5) that s(x)0forx[R
n
.
A necessary and sufficient condition for a twice differenti-
able polynomial f(x) to be convex is for its Hessian r2fxðÞto
be positive semi-definite (see Appendix). This requires that
yTr2fðxÞy0 for all x;y[domain of fð6Þ
A subset of polynomials that satisfy equation (6) is defined
by the restricted but more tractable condition
yTr2fðxÞyis SOS for all x;y[domain of fð7Þ
Polynomials f(x) that satisfy equation (7) are convex
(although convex polynomials exist that do not satisfy
equation (7)). The subset of convex polynomials that satisfy
equation (7) is referred to as ‘SOS-convex’polynomials
(Ahmadi & Parrilo, 2012).
A key benefit of employing SOS-convex polynomials
for failure envelope applications is that the requirement
in equation (7) can readily be incorporated within the
search process for failure envelope formulations using
semi-definite programming (Parrilo, 2003). Additionally,
SOS-convex polynomials are well-behaved and support
general n-dimensional loading. Thus, SOS-convex poly-
nomials form the basis of the failure envelope formulations
in the proposed framework.
DETERMINATION OF SOS-CONVEX POLYNOMIAL
FAILURE ENVELOPES
Procedures to determine SOS-convex polynomials to
provide a fit with failure load data sets are outlined below.
Standardise the data
Failure envelope data typically consist of sets of discrete
failure load combinations. These failure load combinations
are standardised by
ˉ
xi¼xixi;c
xi;ref
ð8Þ
where x
i
is a load component, x
i,ref
is a reference load and x
i,c
is a shift parameter. The purpose of equation (8) is to shift
and scale the data such that the values of ˉ
xifor uniaxial
loading in the positive and negative directions (according to
the selected coordinate system) are 1 and 1, respectively.
Geometrically, this ensures that the failure envelope inter-
sects each ˉ
xiaxis at ± 1. Importantly, if fˉ
xðÞis convex, then
convexity in f(x) is preserved as the mapping in equation (8)
is affine (Boyd & Vandenberghe, 2004).
Define the failure envelope functional form
SOS-convex polynomial functions of the standardised
loading variables ˉ
x, of even degree 2dwhere dis a positive
integer, are now sought to represent the failure envelope fˉ
xðÞ.
Although SOS-convex polynomials are in general non-
homogeneous, homogeneous forms are adopted in the
current framework as they have the important advantage
that the coefficients for uniaxial loading can be identified
straightforwardly, as indicated in step 3 below.
As an example, a homogeneous polynomial fˉ
xðÞ with
degree 2d= 4 and number of loading dimensions n¼2, with
coefficients a
i
(that are subsequently determined to ensure
that the polynomial is SOS-convex) is
fðˉ
xÞ¼fˉ
x1;ˉ
x2
ðÞ
¼ˉ
x4
1a1þˉ
x3
1x2a2þˉ
x2
1
ˉ
x2
2a3þˉ
x1
ˉ
x3
2a4þˉ
x4
2a5ð9Þ
The failure envelope is defined by fˉ
xðÞ1¼0.
Apply the uniaxial conditions
The standardisation process in step 1 requires that the
coefficients of all monomials comprising a single loading
A SYSTEMATIC FRAMEWORK FOR FORMULATING CONVEX FAILURE ENVELOPES 3
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variable are 1. (This is to represent failure correctly
for uniaxial loading conditions.) For the example in
equation (9), this condition gives a
1
¼1 and a
5
¼1.
Apply symmetry principles
Other coefficients can be determined by exploiting
any available symmetry conditions. For the example in
equation (9), if the problem is symmetric such that
fˉ
x1;ˉ
x2
ðÞ¼fˉ
x1;ˉ
x2
ðÞ, then the coefficients of all monomials
that are odd in ˉ
x1must be zero; in the current example this
implies a
2
¼0 and a
4
¼0.
Set up a convex optimisation problem
To identify the remaining unknown coefficients (there is
only one unknown a
3
in the current example but a greater
number of coefficients will typically need to be determined at
this stage), a convex optimisation problem is set up. This is
based on the conditions: (a)fˉ
xðÞis SOS-convex and (b)fˉ
xðÞ
provides a best fit with the failure envelope data by
minimising the objective function C
C¼X
ndata
i¼1
fˉ
xdata
i
1
2ð10Þ
where ˉ
xdata
iis a set of failure load combinations and ndata is
the size of the data set. The optimisation problem is therefore
minimise C
subject to ˉ
yTr2fˉ
xðÞ
ˉ
yis SOS for all ˉ
x;ˉ
y[domain of fðˉ
xÞ
ð11Þ
To solve the SOS program defined in equation (11), it must
first be converted into an equivalent semi-definite program.
In the current work, the Matlab toolbox ‘YALMIP’
(Löfberg, 2004) was used to convert SOS problems such as
equation (11) automatically to their equivalent semi-definite
programs (Löfberg, 2009). Thereafter, the unknown coeffi-
cients in fðˉ
xÞcan be readily determined using this toolbox.
EXAMPLE APPLICATIONS
Three examples are provided below to demonstrate the
application of the proposed formulation framework. The first
example concerns the failure envelope for a surface circular
footing on homogeneous clay; the second example is the
failure envelope for a suction caisson foundation embedded
in homogeneous clay; the last example is a new failure
envelope formulation to represent Model B, which demon-
strates that an existing failure envelope formulation can be
approximated within the proposed framework to achieve the
advantages of global convexity and well-behaved function
attributes.
In all the examples, YALMIP was used, in conjunction
with the SeDuMi semi-definite solver (Sturm, 1999), to solve
the SOS programs. All the failure envelopes for the examples
are plotted in the normalised load space (e.g.
˜
H
˜
M). This
enables convenient comparison with figures in previous
publications (for the case of Model B).
Surface and caisson foundations
The first two examples use the VHM failure load
combinations for surface and caisson foundations in homo-
geneous clay, determined as described in Suryasentana et al.
(2019) using the ‘sequential swipe test’in three-dimensional
(3D) finite-element analysis. The soil in these cases was
modelled as a linear elastic–perfectly plastic von Mises
material. Fig. 1 shows the foundation configurations being
considered, including the assumed location of the loading
reference point (LRP) (which is needed to provide a datum
for the definition of the applied moment).
Based on the numerical failure load data, standardised
forms of the failure load data ½
ˉ
H;
ˉ
M;
ˉ
Vare obtained, from
equation (8), with the computed uniaxial capacities [H
0
,M
0
,
V
0
] adopted as the reference values and [H
c
,M
c
,V
c
]¼[0,
0, 0] as the shift parameters. The foundations analysed
in Suryasentana et al. (2019) assumed that the soil and
foundation were fully bonded (i.e. no contact breaking) and
thus, the foundation has the same tension and compression
vertical uniaxial capacities, resulting in V
c
¼0 in the current
example.
Two separate SOS-convex polynomial failure envelopes
have been determined: f
4
(with degree 2d¼4) and f
6
(with
degree 2d¼6). The general forms of these polynomials are
f4
ˉ
H;
ˉ
M;
ˉ
V
¼a1
ˉ
H4þa2
ˉ
H3ˉ
Mþa3
ˉ
H2ˉ
M2þa4
ˉ
H
ˉ
M3
þa5
ˉ
M4þa6
ˉ
H3ˉ
Vþa7
ˉ
H2ˉ
M
ˉ
V
þa8
ˉ
H
ˉ
M2ˉ
Vþa9
ˉ
M3ˉ
Vþa10
ˉ
H2ˉ
V2
þa11
ˉ
H
ˉ
M
ˉ
V2þa12
ˉ
M2ˉ
V2
þa13
ˉ
H
ˉ
V3þa14
ˉ
M
ˉ
V3þa15
ˉ
V4
ð12Þ
f6
ˉ
H;
ˉ
M;
ˉ
V
¼a1
ˉ
H6þa2
ˉ
H5ˉ
Mþa3
ˉ
H4ˉ
M2
þa4
ˉ
H3ˉ
M3þa5
ˉ
H2ˉ
M4þa6
ˉ
H
ˉ
M5
þa7
ˉ
M6þa8
ˉ
H5ˉ
Vþa9
ˉ
H4ˉ
M
ˉ
V
þa10
ˉ
H3ˉ
M2ˉ
Vþa11
ˉ
H2ˉ
M3ˉ
V
þa12
ˉ
H
ˉ
M4ˉ
Vþa13
ˉ
M5ˉ
Vþa14
ˉ
H4ˉ
V2
þa15
ˉ
H3ˉ
M
ˉ
V2þa16
ˉ
H2ˉ
M2ˉ
V2
þa17
ˉ
H
ˉ
M3ˉ
V2þa18
ˉ
M4ˉ
V2þa19
ˉ
H3ˉ
V3
þa20
ˉ
H2ˉ
M
ˉ
V3þa21
ˉ
H
ˉ
M2ˉ
V3þa22
ˉ
M3ˉ
V3
þa23
ˉ
H2ˉ
V4þa24
ˉ
H
ˉ
M
ˉ
V4þa25
ˉ
M2ˉ
V4
þa26
ˉ
H
ˉ
V5þa27
ˉ
M
ˉ
V5þa28
ˉ
V6
ð13Þ
Application of the uniaxial condition gives a
1
,a
5
,a
15
¼1
in f
4
and a
1
,a
7
,a
28
¼1inf
6
. Additionally, symmetry in
ˉ
H
and
ˉ
Mrequires that a
6
,a
7
,a
8
,a
9
,a
13
,a
14
¼0inf
4
and a
8
,a
9
,
a
10
,a
11
,a
12
,a
13
,a
19
,a
20
,a
21
,a
22
,a
26
,a
27
¼0inf
6
. The
remaining coefficients are determined from the optimisation
in equation (11).
MSurface
foundation
LRP H
V
Caisson skirt
(when present)
Fig. 1. VHM loading configuration for surface foundation and caisson
SURYASENTANA, BURD, BYRNE AND SHONBERG4
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Results. The coefficients for f
4
and f
6
determined from this
process are listed in Tables 1 and 2, respectively. Figs 2 and 3
compare the normalised VH,VM and HM failure envelopes
determined by f
4
1¼0 and f
6
1¼0 for the surface and
caisson foundation, respectively; also shown are the corre-
sponding 3D finite-element results from Suryasentana et al.
(2019). Differences between f
4
and f
6
can be seen more clearly
in Fig. 4 (which omits the finite-element results).
Both f
4
and f
6
are seen to provide reasonably good
approximations of the computed data, although there is a
small under-prediction of the VM failure envelope for the
suction caisson (Fig. 3(b)) and f
4
is unconservative compared
to f
6
for the surface foundation at high vertical loads
(Fig. 2(c)). It is seen that f
6
provides a better fit with the
numerical data than f
4
for the surface foundation (Fig. 2(c)),
but f
4
is marginally better for the caisson (Fig. 3(c)). This is
confirmed by the data in Table 3, which shows values of the
objective function (and their root-mean) at the end of the
optimisation process (the smaller the better).
Model B
Model A, introduced in the earlier section ‘Existing failure
envelope formulations’, is representative of a class of func-
tions used by numerous authors to represent yield envelopes.
A modified form of this model termed ‘Model B’, found to
provide an improved fit to experimental data of spudcan
performance on clay, is described in Martin (1994) and
Martin & Houlsby (2000). The Model B formulation is
f
˜
H;
˜
M;
˜
V
¼
˜
H2þ
˜
M22e
˜
H
˜
Mβ2˜
V2β1ð1
˜
VÞ2β2¼0
ð14Þ
Table 1. Best-fit coefficients in f
4
for the failure envelope data from
Suryasentana et al. (2019)
Foundation a
2
a
3
a
4
a
10
a
11
a
12
Surface 0·36 0·9 1·43 0·4 0·84 1·64
Caisson 3·55 5·12 3·51 1·72 3·59 2·07
Table 2. Best-fit coefficients in f
6
for the failure envelope data from Suryasentana et al. (2019)
Foundation a
2
a
3
a
4
a
5
a
6
a
14
a
15
a
16
a
17
a
18
a
23
a
24
a
25
Surface 0·33 1·22 2·17 2·34 1·86 0·03 0·34 1·1 0·29 0·84 1·97 1·52 4·72
Caisson 5·18 11·9 15·45 11·96 5·2 2·72 10·86 17·27 13·23 4·21 2·13 2·93 1·04
1·0
0·5
–0·5
–1·0
–1·0 –0·5 0
(a) (b)
(c)
0·5 1·0
0
H
˜
V
˜
–1·0 –0·5 0 0·5 1·0
V
˜
1·0
0·5
–0·5
–1·0
0
M
˜
–1·0–1·5 –0·5 0 0·5 1·0 1·5
H
˜
1·0
0·5
–0·5
–1·0
0
M
˜
3DFE
f4
f6
3DFE
f4
f6
3DFE
f4
f6
Increasing V
˜
Fig. 2. Comparison of the normalised VH,VM and HM failure envelopes predicted by f
4
and f
6
with 3D finite-element (denoted 3DFE in figure)
results (Suryasentana et al., 2019) for a surface foundation. The HM envelopes correspond to normalised vertical loads:
˜
V= 0, 0·25, 0·5, 0·625,
0·75, 0·875
A SYSTEMATIC FRAMEWORK FOR FORMULATING CONVEX FAILURE ENVELOPES 5
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or, alternatively
fð
˜
H;
˜
M;
˜
VÞ¼ð
˜
H2þ
˜
M22e
˜
H
˜
MÞ1=2β2
ˉ
β1=β2˜
Vβ1=β2ð1
˜
VÞ
¼0
ð15Þ
where e¼0518 þ118
˜
Vð
˜
V1Þ,β¼β1þβ2
ðÞ
β1þβ2
ðÞ
=ββ1
1ββ2
2,
and β
1
¼0·764 and β
2
¼0·882 are parameters, selected to
fit the model to the data. It is instructive to use the new
failure envelope framework to develop an alternative version
1·0
0·5
–0·5
–1·0
0
H
˜
1·0
0·5
–0·5
–1·0
0
M
˜
M
˜
H
˜
–1·0 –0·5 0
(a)
0·5 1·0
V
˜
–1·0 –0·5 0
(b)
(c)
0·5 1·0
V
˜
2
1
0
–1
–2
–2 –1 0 1 2
3DFE
f4
f6
3DFE
f4
f6
3DFE
f4
f6
Increasing V
˜
Fig. 3. Comparison of the normalised VH,VM and HM failure envelopes predicted by f
4
and f
6
with the 3D finite-element results (Suryasentana
et al., 2019) for a suction caisson foundation. The HM envelopes correspond to normalised vertical loads:
˜
V= 0, 0·25, 0·5, 0·625, 0·75, 0·875
1·0
0·5
–0·5
–1·0
0
M
˜
–1·0–1·5 –0·5 0
(a)
0·5 1·0 1·5
M
˜
H
˜
H
˜
(b)
2
1
0
–1
–2
–2 –1 0 1 2
f4
f6
f4
f6
Increasing V
˜
Increasing V
˜
Fig. 4. Comparison of the normalised HM failure envelopes predicted by f
4
and f
6
under increasing vertical loading (i.e.
˜
V= 0, 0·25, 0·5, 0·625,
0·75, 0·875): (a) surface foundation; (b) suction caisson foundation
Table 3. Minimised objective values (and corresponding root-mean
values) for f
4
and f
6
at the end of the optimisation process for the
surface and caisson foundation results from Suryasentana et al. (2019)
Foundation Cfor f
4
Cfor f
6
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
C=ndata
pfor f
4
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
C=ndata
pfor f
6
Surface 36·96 31·05 0·270 0·248
Caisson 20·3 20·35 0·246 0·247
Note: ndata is the number of sets of failure load combinations
(ndata = 506 and ndata = 335 for the surface and caisson foundation,
respectively).
SURYASENTANA, BURD, BYRNE AND SHONBERG6
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of Model B, to demonstrate how the new framework can be
applied to existing formulations.
Failure envelope data were generated using equation (15)
in the form of the
˜
V
˜
Hfailure envelope,
˜
V
˜
Mfailure envelope
and the
˜
H
˜
Mfailure envelopes for
˜
V= 0·125, 0·25, 0·375, 0·5,
0·625, 0·75, 0·875. The data were standardised according
to equation (8), where the adopted shift parameters
were ½
˜
Hc;
˜
Mc;
˜
Vc¼ ½0;0;05. Thus, ½
˜
Href ;
˜
Mref ;
˜
Vref ¼
½0995;0995;05(the failure envelope intersects the
˜
Hand
˜
Maxes at
˜
H= ±0·995 and
˜
M¼+0995 when
˜
V¼
˜
Vc¼
05, from equation (15)).
Failure envelopes have been determined on the basis of f
4
and f
6
(i.e. equations (12) and (13)) incorporating the same
uniaxial and symmetry conditions as adopted in the earlier
section ‘Surface and caisson foundations’examples. The
remaining coefficients were determined using the optimis-
ation in equation (11).
The computed coefficients are listed in Tables 4 and 5. Fig. 5
shows the normalised VH,VM and HM failure envelopes for
f
4
1¼0 and f
6
1¼0, together with comparisons with the
original Model B equation. Evidently, both f
4
and f
6
provide
reasonably good approximations of the Model B equation
(although neither of them predicts the correct value of
˜
Vat
which the peaks in the
˜
V
˜
Hand
˜
V
˜
Menvelopes occur). Table 6
shows the minimised objective function values (and their
root-mean) for f
4
and f
6
at the end of the optimisation; these
data suggest that f
4
provides a better fit than f
6
. Figs 5(a)
and 5(b) show that, at
˜
V¼0and
˜
V¼1, the failure envelopes
implied by f
4
and f
6
are more ‘rounded’than Model B, and the
surface normal is directed along the
ˉ
Vaxis. This means that,
when employed as a plastic potential, the model correctly
predicts vertical plastic displacement for pure vertical loading;
this is significant, as special numerical procedures are required
to achieve such behaviour with Model B (Martin, 1994). In
general, the results show that the SOS-convex Model B
formulation provides a failure envelope that is reasonably
close to the formulation in equation (15), although it is evident
from Fig. 5(c) that the SOS-convex formulation is unconser-
vative for relatively high values of normalised vertical load; the
SOS-convex formulation, however, is guaranteed to be
globally convex and well-behaved.
Table 4. Best-fit coefficients in f
4
for the Model B failure envelope
data
a
2
a
3
a
4
a
10
a
11
a
12
0·86 2 0·86 3·48 2·93 3·48
Table 5. Best-fit coefficients in f
6
for the Model B failure envelope data
a
2
a
3
a
4
a
5
a
6
a
14
a
15
a
16
a
17
a
18
a
23
a
24
a
25
1·35 3·59 2·74 3·59 1·35 3·56 4·94 8·94 4·94 3·56 7·37 6·59 7·37
1·0
0·5
0
–0·5
–1·0
H
˜
1·0
0·5
0
–0·5
–1·0
M
˜
00·2
(a)
0·60·4 0·8 1·0
V
˜
00·2
(b)
0·60·4 0·8 1·0
V
˜
1·0
0
0·5
–0·5
–1·0
M
˜
–0·5–1·0 0
(c)
0·5 1·0
H
˜
Model B
f4
f6
Model B
f4
f6
Model B
f4
f6
Increasing V
˜
Fig. 5. Comparison of the normalised VH,VM and HM failure envelopes predicted by f
4
and f
6
with the original Model B equation. The HM
envelopes correspond to normalised vertical loads:
˜
V= 0·5, 0·75, 0·875
A SYSTEMATIC FRAMEWORK FOR FORMULATING CONVEX FAILURE ENVELOPES 7
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Macro-element model for a surface foundation
To demonstrate the application of the proposed SOS-
convex failure envelope formulations in a macro-element
model, the 4th degree version of the failure envelope
(equation (12)) was employed, using the coefficients for a
surface foundation in Table 1. Failure envelopes of the form
f
ˉ
H;
ˉ
M;
ˉ
V
such as equation (12) are implicit functions of
the loading variables H,M,Vby way of the affine trans-
formation in equation (8) (in this case,
ˉ
H¼H=H0,
ˉ
M¼
M=M0and
ˉ
V¼V=V0). The failure envelope f
4
(H,M,
V)1¼0 was implemented as the yield surface and plastic
potential of a macro-element model, with the elasto-plastic
integration performed using the implicit closest point
projection method (Simo & Hughes, 2006). The uniaxial
capacities (V
0
,H
0
,M
0
) were determined from the finite-
element data in Suryasentana et al. (2019), while the elastic
stiffness matrix for the macro element was calibrated using
pre-failure data obtained from the finite-element results.
The macro-element model is used, as an example, to
simulate a sequential swipe test in the HM load space for
planar HM loading with
˜
V= 0·25, to replicate the corres-
ponding finite-element simulations in Suryasentana et al.
(2019). Fig. 6 provides a comparison of the load–
displacement behaviour and the normalised HM failure
envelope, computed using the macro-element model and the
3D finite-element model (Suryasentana et al., 2019). It is
evident that the macro-element predictions agree well with
the finite-element results.
DISCUSSION
The procedures presented in the paper provide a con-
venient means of generating failure envelopes that are convex
and well-behaved. This is achieved by selecting functions
from a restricted set of SOS-convex polynomials. The process
is demonstrated by applying it to previously published
numerical data on failure load combinations for a surface
footing and a caisson on soil that is modelled as an elastic–
perfectly plastic von Mises material. Additionally, it is used
to reformulate a previously determined failure envelope,
Model B.
The results indicate that an SOS-convex polynomial of
degree 6, f
6
, does not necessarily provide a better fit to a
prescribed data set than an SOS-convex polynomial of
degree 4, f
4
. The results indicate that f
6
is a better fit for the
surface foundation failure envelope than f
4
, but the opposite is
true for the caisson and Model B failure envelopes. This may
seem counterintuitive, but it is a consequence of the fact that
Table 6. Minimised objective values (and corresponding root-mean
values) for f
4
and f
6
at the end of the optimisation process for the
Model B example
Foundation Cfor f
4
Cfor f
6
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
C=ndata
p
for f
4
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
C=ndata
p
for f
6
Model B 8·23 11·22 0·0476 0·0556
Note: ndata = 3634 is the number of sets of failure load combinations.
1·0
0·5
0
–0·5
–1·0
H
˜
V
˜ = 0·25
1·0
0·5
0
–0·5
–1·0
M
˜
1·0
0·5
0
–0·5
–1·0
M
˜
00·025 0 0·02 0·04 0·06 0·08 0·10 0·12
(a)
0·0750·050 0·100 0·125 0·150
Sh/D θm
Macro-element
3DFE
Macro-element
3DFE
Macro-element
3DFE
–0·5–1·0 0
(c)
(b)
0·5 1·0
H
˜
Fig. 6. Comparison of load–displacement and the normalised HM failure envelopes predicted by the macro-element and 3D finite-element results
(Suryasentana et al., 2019), for a sequential swipe test in the HM load space when
˜
V= 0·25. S
h
and θ
m
refer to the horizontal and rotational
displacements of the surface foundation, respectively. Dis the diameter of the surface foundation
SURYASENTANA, BURD, BYRNE AND SHONBERG8
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the current approach is restricted to the use of homogeneous
polynomials. Although a general polynomial can be used to
represent any polynomial of lower degree, this is not the case
for the homogeneous polynomials employed here. For
practical applications of this method, it is recommended that
a relatively low degree homogeneous polynomial (e.g. degree=
4) is initially adopted. Higher degree polynomials would then
be considered if the low degree polynomial is found to provide
an unsatisfactory fit to the data.
The YALMIP toolbox provides a useful SOS decompo-
sition feature to double check that the failure envelopes are,
indeed, convex throughout the entire domain. For example,
for f
4
employed for Model B, the Hessian can be decomposed
into SOS form using YALMIP as
where ˉ
x¼½
ˉ
V;
ˉ
H;
ˉ
MT,ˉ
y¼½
ˉ
V1;
ˉ
H1;
ˉ
M1T[domain of f
4
.
This equation demonstrates that ˉ
yTr2f4
ˉ
x
ðÞ
ˉ
yis SOS, hence
ˉ
yTr2f4
ˉ
xðÞ
ˉ
y0 for its entire domain; f
4
is therefore shown to
be SOS-convex.
The values of the coefficients in the failure envelope
polynomials determined using the proposed framework do
not necessarily have a physical interpretation. This disadvan-
tage is outweighed by the new approach being able to represent
a wide range of failure envelope shapes, while maintaining
desirable theoretical attributes. A simple way to interpret the
polynomial coefficients would be to view their (absolute)
magnitudes as indicative of the strength of coupling between
the respective loading components at failure.
The framework presented in the paper has certain limit-
ations, as follows. The application of the SOS-convex failure
envelope formulation to macro-element modelling has only
been demonstrated for perfect plasticity with associated flow.
Further work is needed to develop the modelling approach for
the case of non-associated flow and hardening plasticity.
Furthermore, the current study only explores the application
of homogeneous polynomials. Non-homogeneous SOS-
convex polynomials are not explored, but they potentially
provide greater expressiveness than homogeneous SOS-convex
polynomials (since homogeneous polynomials are subsets of
non-homogeneous polynomials). However, based on initial
analyses using non-homogeneous polynomials, the SeDuMi
solver employed in the current work encountered convergence
problems for some of the above examples; this contrasts with
the homogeneous polynomials where convergence issues did
not occur. Further work is needed to explore the implemen-
tation of non-homogeneous SOS-convex polynomials and
their potential benefits.
CONCLUSIONS
This paper introduces a novel SOS-convex polynomial-
based framework to systematically formulate failure envel-
opes that are suitable for use in both ultimate capacity
calculations and macro-element modelling. The approach is
applicable to failure envelope data sets of multiple loading
dimensions, and it is demonstrated in the current paper for
VHM loading. The principal advantage of the proposed
framework is that it leads to formulations for failure envelope
functions (and yield functions) that are guaranteed to be
globally convex and well-behaved. As discussed in this paper,
these attributes lead to significant advantages in terms of
theoretical considerations as well as the numerical stability of
macro-element models.
Examples are presented to demonstrate the application of
the framework to three different shallow foundation cases,
with significantly different failure envelope shapes. These
examples demonstrate the expressiveness and general capa-
bilities of the proposed framework. The framework ration-
alises and expedites the formulation of failure envelopes and
therefore supports the development of the failure envelope
approach for design applications.
Although not shown here, this framework may have
potential uses in other fields of solid mechanics, such as
determining convex yield functions from experimentally
determined yield data for different materials.
ACKNOWLEDGEMENT
The first author would like to acknowledge the generous
support of Ørsted Wind Power for funding his DPhil
studentship at the University of Oxford.
APPENDIX
Convex functions
A function fis defined to be convex if its domain is a convex set
and
fθxþ1θðÞy½θfðxÞþ 1θðÞfðyÞð17Þ
for all x,y[domain of fand for 0 θ1 (Boyd & Vandenberghe,
2004). Depending on the differentiability of f, there are two
equivalent conditions to equation (17) that characterise the
convexity of f.
If fis differentiable (i.e. its gradient rfðxÞexists everywhere in its
domain), fis convex if its domain is a convex set and
fðyÞfðxÞþrfðxÞTyxðÞ ð18Þ
for all x,y[domain of f. Equation (18) is known as the first-order
condition (Boyd & Vandenberghe, 2004).
If fis twice differentiable (i.e. its Hessian r2fðxÞexists everywhere
in its domain), fis convex if its domain is a convex set and r2fðxÞis
positive semi-definite everywhere in its domain –that is
yTr2fðxÞy0ð19Þ
ˉ
yTr2f4
ˉ
xðÞ
ˉ
y¼32835
ˉ
V
ˉ
V130493
ˉ
M
ˉ
M1þ09486
ˉ
M
ˉ
H1þ09486
ˉ
H
ˉ
M130493
ˉ
H
ˉ
H1
2
þ20827
ˉ
V
ˉ
M1þ20827
ˉ
V
ˉ
H120827
ˉ
M
ˉ
V1þ20827
ˉ
H
ˉ
V1
2
þ11788
ˉ
V
ˉ
M111788
ˉ
V
ˉ
H111788
ˉ
M
ˉ
V111788
ˉ
H
ˉ
V1
2
þ15127
ˉ
M
ˉ
M1þ15127
ˉ
H
ˉ
H1
2þ13083
ˉ
M
ˉ
H1þ13083
ˉ
H
ˉ
M1
2
þ00816
ˉ
V
ˉ
V1þ03209
ˉ
M
ˉ
M1þ11727
ˉ
M
ˉ
H1þ11727
ˉ
H
ˉ
M1þ03209
ˉ
H
ˉ
H1
2
þ07908
ˉ
V
ˉ
M1þ07908
ˉ
V
ˉ
H107908
ˉ
M
ˉ
V107908
ˉ
H
ˉ
V1
2
þ07792
ˉ
V
ˉ
M107792
ˉ
V
ˉ
H107792
ˉ
M
ˉ
V1þ07792
ˉ
H
ˉ
V1
2
þ11009
ˉ
V
ˉ
V1þ05572
ˉ
M
ˉ
M101142
ˉ
M
ˉ
H101142
ˉ
H
ˉ
M1þ05572
ˉ
H
ˉ
H1
2
ð16Þ
A SYSTEMATIC FRAMEWORK FOR FORMULATING CONVEX FAILURE ENVELOPES 9
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for all x,y[domain of f. This is known as the second-order
condition (Boyd & Vandenberghe, 2004) and is the principal
condition for assessing the convexity of functions in the current
paper. One way to determine whether a matrix is positive
semi-definite is to check whether all of its eigenvalues are
non-negative everywhere in its domain.
An example of a twice differentiable, convex function is the von
Mises yield function, where it can be shown that the eigenvalues of
the Hessian are always non-negative. The Tresca yield function is
convex according to the general condition in equation (17), but since
it is not differentiable over its entire domain (e.g. at the ‘corners’)
then the more restrictive first- and second-order conditions do not
apply.
A yield surface should not be confused with a yield function. A
yield surface is usually defined as the zero level set of ayield function
f(x) (i.e. {x|f(x)¼0}). A convex yield surface does not necessarily
imply convexity of f(x). Conversely, if f(x) is convex, then the
yield surface is necessarily convex. These conditions arise from
convex analysis relating convex functions and their sublevel sets
(Boyd & Vandenberghe, 2004), where a k-sublevel set is defined
as {x|f(x)k}. In continuum plasticity, the 0-sublevel set of a yield
function typically represents the set of admissible stress states.
NOTATION
Apositive definite matrix representing an ellipsoid in quadratic
form
Bpositive semi-definite matrix representing coefficients of an
SOS polynomial
ccentre coordinates of an ellipsoid
Ddiameter of surface foundation
dpositive integer representing half the degree of an SOS
polynomial
g
j
polynomial components of an SOS polynomial
Hhorizontal load
H
0
horizontal uniaxial capacity
˜
Hnormalised horizontal load
ˉ
Hstandardised horizontal load
Mmoment
M
0
moment uniaxial capacity
ˉ
Mstandardised moment load
˜
Mnormalised moment load
nnumber of loading dimensions
Qmatrix from Cholesky decomposition of B
R
n
real coordinate space of ndimensions
S
h
horizontal displacement of surface foundation
sSOS polynomial
Vvertical load
V
0
vertical uniaxial capacity
ˉ
Vstandardised vertical load
˜
Vnormalised vertical load
x
i
load component
x
i,c
shift parameter
x
i,ref
reference load
zvector of monomials up to and including degree d
θ
m
rotational displacement of surface foundation
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