Content uploaded by Stephen Suryasentana

Author content

All content in this area was uploaded by Stephen Suryasentana on Mar 05, 2019

Content may be subject to copyright.

Assessment of numerical procedures for determining

shallow foundation failure envelopes

STEPHEN K. SURYASENTANA, HELEN P. DUNNE†, CHRISTOPHER M. MARTIN†,

HARVEY J. BURD†, BYRON W. BYRNE†and 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

Downloaded by [ UNIVERSITY OF OXFORD] on [05/03/19]. Published with permission by the ICE under the CC-BY license

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 soil–foundation

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

Downloaded by [ UNIVERSITY OF OXFORD] on [05/03/19]. Published with permission by the ICE under the CC-BY license

(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 ‘swiped’horizontally. 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 ‘discrete’version

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 w–u

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

Downloaded by [ UNIVERSITY OF OXFORD] on [05/03/19]. Published with permission by the ICE under the CC-BY license

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 Poisson’s ratio of the soil, ν, was set as

0·49, while its Young’smodulus,E, was set as 1000√3s

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

(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= 1000√3s

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

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 7–9 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 7–9 for the sequential swipe test

results represent intermediate equilibrium load states during

each discrete stage of the sequential swipe, which are

determined by Abaqus’s 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

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

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

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 7–10. 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 7–10

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

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 7–9 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 Mohr–Coulomb

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

REFERENCES

Bell, R. W. (1991). The analysis of offshore foundations subjected

to combined loading. MSc thesis, University of Oxford,

Oxford, UK.

Bransby, M. F. & Randolph, M. F. (1998). Combined loading

of skirted foundations. Géotechnique 48, No. 5, 637–655,

https://doi.org/10.1680/geot.1998.48.5.637.

Bransby, M. F. & Yun, G. J. (2009). The undrained capacity of

skirted strip foundations under combined loading. Géotechnique

59, No. 2, 115–125, https://doi.org/10.1680/geot.2007.00098.

Byrne, B. W. (2000). Investigations of suction caissons in dense sand.

DPhil thesis, University of Oxford, Oxford, UK.

Cassidy, M. J., Byrne, B. W. & Randolph, M. F. (2004). A

comparison of the combined load behaviour of spudcan and

caisson foundations on soft normally consolidated clay.

Géotechnique 54, No. 2, 91–106, https://doi.org/10.1680/geot.

2004.54.2.91.

Dassault Systèmes (2014). Abaqus user manual, version 6.13.

Providence, RI, USA: Simulia Corp.

Dunne, H. P. & Martin, C. M. (2017). Capacity of rectangular

mudmat foundations on clay under combined loading.

Géotechnique 67, No. 2, 168–180, https://doi.org/10.1680/jgeot.

16.P.079.

SURYASENTANA, DUNNE, MARTIN, BURD, BYRNE AND SHONBERG10

Dunne, H. P., Martin, C. M., Muir, L., Brown, N. & Wallerand, R.

(2015). Undrained bearing capacity of skirted mudmats on

inclined seabeds. In Frontiers in Offshore Geotechnics III

(ed. V. Meyer), pp. 783–788. Boca Raton, FL, USA: CRC Press.

Feng, X., Randolph, M. F., Gourvenec, S. & Wallerand, R. (2014).

Design approach for rectangular mudmats under fully

three-dimensional loading. Géotechnique 64, No. 1, 51–63,

https://doi.org/10.1680/geot.13.P.051.

Fu, D., Bienen, B., Gaudin, C. & Cassidy, M. J. (2014). Undrained

capacity of a hybrid subsea skirted mat with caissons under

combined loading. Can. Geotech. J. 51, No. 8, 934–949.

Gerolymos, N., Zafeirakos, A. & Karapiperis, K. (2015).

Generalized failure envelope for caisson foundations in cohesive

soil: static and dynamic loading. Soil Dynamics Earthquake

Engng 78, 154–174.

Gottardi, G. & Butterfield, R. (1993). On the bearing capacity of

surface footings on sand under general planar loads. Soils

Found. 33, No. 3, 68–79.

Gottardi, G., Houlsby, G. T. & Butterfield, R. (1999). Plastic

response of circular footings on sand under general planar

loading. Géotechnique 49, No. 4, 453–469, https://doi.org/

10.1680/geot.1999.49.4.453.

Gourvenec, S. (2007). Failure envelopes for offshore shallow

foundations under general loading. Géotechnique 57,No.9,

715–728, https://doi.org/10.1680/geot.2007.57.9.715.

Gourvenec, S. & Barnett, S. (2011). Undrained failure envelope for

skirted foundations under general loading. Géotechnique 61,

No. 3, 263–270, https://doi.org/10.1680/geot.9.T.027.

Gourvenec, S. & Randolph, M. F. (2003). Effect of strength

nonhomogeneity on the shape of failure envelopes for combined

loading of strip and circular foundations on clay. Géotechnique

53, No. 6, 575–586, https://doi.org/10.1680/geot.2003.53.6.575.

Gourvenec, S., Randolph, M. & Kingsnorth, O. (2006). Undrained

bearing capacity of square and rectangular footings.

Int. J. Geomech. 6, No. 3, 147–157.

Green, A. P. (1954). The plastic yielding of metal junctions due to

combined shear and pressure. J. Mech. Phys. Solids 2,No.3,

197–211.

Houlsby, G. T. & Byrne, B. W. (2001). Discussion: Comparison of

European bearing capacity calculation methods for shallow

foundations. Proc. Instn Civ. Engrs –Geotech. Engng 149,No.1,

63–64, https://doi.org/10.1680/geng.2001.149.1.63.

Hung, L. C. & Kim, S. R. (2014). Evaluation of undrained bearing

capacities of bucket foundations under combined loads. Mar.

Georesour. Geotechnol. 32, No. 1, 76–92.

Karapiperis, K. & Gerolymos, N. (2014). Combined loading of

caisson foundations in cohesive soil: finite element versus

Winkler modeling. Comput. Geotech. 56, 100–120.

Makrodimopoulos, A. & Martin, C. M. (2006). Lower bound limit

analysis of cohesive-frictional materials using second-order

cone programming. Int. J. Numer. Methods Engng 66,No.4,

604–634.

Makrodimopoulos, A. & Martin, C. M. (2007). Upper bound limit

analysis using simplex strain elements and second-order cone

programming. Int. J. Numer. Analyt. Methods Geomech. 31,

No. 6, 835–865.

Mana, D. S., Gourvenec, S. & Martin, C. M. (2013). Critical skirt

spacing for shallow foundations under general loading.

J. Geotech. Geoenviron. Engng 139, No. 9, 1554–1566.

Martin, C. M. (1994). Physical and numerical modelling of offshore

foundations under combined loads. DPhil thesis, University of

Oxford, Oxford, UK.

Martin, C. M. (2011). The use of adaptive finite-element limit

analysis to reveal slip-line fields. Géotechnique Lett. 1,No.2,

23–29, https://doi.org/10.1680/geolett.11.00018.

Martin, C. M. & Houlsby, G. T. (2000). Combined loading of

spudcan foundations on clay: laboratory tests. Géotechnique 50,

No. 4, 325–338, https://doi.org/10.1680/geot.2000.50.4.325.

Martin, C. M. & Houlsby, G. T. (2001). Combined loading

of spudcan foundations on clay: numerical modelling.

Géotechnique 51, No. 8, 687–699, https://doi.org/10.1680/geot.

2001.51.8.687.

Martin, C. M. & White, D. J. (2012). Limit analysis of the undrained

bearing capacity of offshore pipelines. Géotechnique 62,No.9,

847–863, https://doi.org/10.1680/geot.12.OG.016.

Mehravar, M., Harireche, O. & Faramarzi, A. (2016). Evaluation

of undrained failure envelopes of caisson foundations under

combined loading. Appl. Ocean Res. 59, 129–137.

Meyerhof, G. G. (1951). The ultimate bearing capacity of foun-

dations. Géotechnique 2, No. 4, 301–332, https://doi.org/10.

1680/geot.1951.2.4.301.

Mosek (2014). The MOSEK C optimizer API manual, Version 7.1.

Copenhagen, Denmark: MOSEK ApS.

Nouri, H., Biscontin, G. & Aubeny, C. P. (2014). Undrained sliding

resistance of shallow foundations subject to torsion. J. Geotech.

Geoenviron. Engng 140, No. 8, 04014042.

Nova, R. & Montrasio, L. (1991). Settlements of shallow

foundations on sand. Géotechnique 41, No. 2, 243–256,

https://doi.org/10.1680/geot.1991.41.2.243.

Roscoe, K. & Schofield, A. N. (1957). The stability of short pier

foundations in sand. Br. Weld. J. 4, No. 1, 12–18.

Schotman, G. J. M. (1989). The effects of displacements on

the stability of jackup spud-can foundations. Proceedings

of the offshore technology conference, Houston, TX, USA,

paper OTC 6026.

Shen, Z., Bie, S. & Guo, L. (2017). Undrained capacity of a surface

circular foundation under fully three-dimensional loading.

Comput. Geotech. 92,57–67.

Shen, Z., Feng, X. & Gourvenec, S. (2016). Undrained capacity of

surface foundations with zero-tension interface under planar

V-H-M loading. Comput. Geotech. 73,47–57.

Si, H. (2015). TetGen, a Delaunay-based quality tetrahedral

mesh generator. ACM Trans. Math. Software 41,No.2,

article no. 11.

Taiebat, H. A. & Carter, J. P. (2000). Numerical studies of the

bearing capacity of shallow foundations on cohesive soil

subjected to combined loading. Géotechnique 50,No.4,

409–418, https://doi.org/10.1680/geot.2000.50.4.409.

Taiebat, H. A. & Carter, J. P. (2010). A failure surface for circular

footings on cohesive soils. Géotechnique 60, No. 4, 265–273,

https://doi.org/10.1680/geot.7.00062.

Tan, F. (1990). Centrifuge and theoretical modelling of conical

footings on sand. PhD thesis, University of Cambridge,

Cambridge, UK.

Terzaghi, K. (1943). Theoretical soil mechanics. New York, NY,

USA: John Wiley and Sons.

Vesic

´, A. (1973). Analysis of ultimate loads of shallow foundations.

J. Soil Mech. Found. Div., ASCE 99, No. 1, 45–73.

Vulpe, C. (2015). Design method for the undrained capacity of

skirted circular foundations under combined loading: effect of

deformable soil plug. Géotechnique 65, No. 8, 669–683,

https://doi.org/10.1680/geot.14.P.200.

Vulpe, C., Gourvenec, S. & Power, M. (2014). A generalised failure

envelope for undrained capacity of circular shallow foundations

under general loading. Géotechnique Lett. 4, No. 3, 187–196,

https://doi.org/10.1680/geolett.14.00010.

Zhang, Y., Bienen, B., Cassidy, M. J. & Gourvenec, S. (2011).

The undrained bearing capacity of a spudcan foundation

under combined loading in soft clay. Mar. Structs 24,No.4,

459–477.

NUMERICAL PROCEDURES TO DETERMINE FOUNDATION FAILURE ENVELOPES 11