failure envelopes for offshore shallow foundations under

Gourvenec, S. (2007). Geotechnique 57, No. 9, 715–728 [doi: 10.1680/geot.2007.57.9.715]

715

Failure envelopes for offshore shallow foundations under general loading

S. GOURVENEC*

The interaction of vertical, horizontal and moment (VHM)loads acting on a shallow foundation is complex, and it isbecoming increasingly popular to represent ultimate limitstates under general VHM loading as failure envelopes inthree-dimensional load space. General loading is of parti-cular interest offshore, where harsh environmental condi-tions lead to large horizontal and moment foundationloads. Shallow foundations with peripheral skirts thatpenetrate into the seabed are used to resist large lateraland overturning forces. During undrained loading, tensileresistance can be mobilised on the underside of the baseplate by suctions developed within the skirt compartment.This paper presents failure envelopes and kinematic me-chanisms for undrained ultimate limit states of circularskirted foundations in uniform and heterogeneous depositsunder general VHM loading based on finite element re-sults. An approximating method is proposed that permitsaccurate prediction of ultimate limit states under a fullrange of general loading.

KEYWORDS: bearing capacity; footings/foundations; numericalmodelling

L’interaction de charges verticales, horizontales et demoment (VHM) agissant sur des fondations superficiellesest complexe, et on a de plus en plus tendance a repre-senter des etats-limites ultimes, dans le cadre de chargesVHM, comme des enveloppes de rupture dans un espacede charge tridimensionnels. Les charges generales presen-tent un interet tout particulier en mer, ou les conditionsambiantes severes appliquent des charges horizontales etde moment elevees sur les fondations. Les fondations peuprofondes, a jupe peripherique, qui penetrent dans lefond marin, sont etudiees pour resister a des forces derenversement elevees. Au cours de charges sans consoli-dation, la resistance a la traction peut etre mobilisee surle dessous de la semelle par des effets d’aspiration qui seproduisent au sein du compartiment de la jupe. Lapresente communication presente des enveloppes de rup-ture et des mecanismes cinetiques pour etats-limitesultimes sans consolidation de fondations a jupe circulairedans des depots uniformes et heterogenes, sous descharges VHM generales, sur la base de resultats auxelements finis. Une methode d’approximation, permettantd’effectuer une prediction precise des etats-limites ul-times sous une gamme integrale de charges generales, estproposee.

INTRODUCTIONGeneral loading is particularly relevant in the design ofshallow foundations for offshore structures, as wind, waveand current forces provide significant lateral load compo-nents of a magnitude that are not commonly encounteredonshore.

Industry-recommended practice for offshore shallow foun-dation design (DNV, 1992; API, 2000; ISO, 2000) has devel-oped from onshore design codes, for example Eurocode 7(1997), as well as the selection of individual Europeancountry guidelines: see Sieffert & Baygress (2000) and thestate-specific military and naval American guidelines (e.g.NAVFAC, 1982; ASCE, 1993; Caltrans, 1997). All theseguidelines recommend essentially the same approach, basedon classical bearing capacity theory (Terzaghi, 1943; Vesic,1975) which has been shown to underpredict ultimate limitstates for a range of foundation, soil and loading conditionsrelevant offshore (Ukritchon et al., 1998; Gourvenec &Randolph, 2003a; Gourvenec, 2004).

Offshore applications of shallow foundations include oiland gas platforms, and various subsea installations, andrecently there has been increasing interest in their use forwind turbines. Some applications of offshore shallow founda-tion systems are illustrated in Fig. 1. Offshore shallowfoundations are frequently equipped with skirts, which pene-trate into the seabed beneath the base plate confining a soilplug. When the foundation is subjected to large moment

loads, suctions are developed within the soil plug, providingshort-term tensile capacity (Fig. 2). Transient suctionsdeveloped beneath the base plate of skirted foundations canbe relied on over the duration for which undrained conditions

Manuscript received 7 September 2004; revised manuscriptaccepted 17 July 2007.Discussion on this paper closes on 1 May 2008, for further detailssee p. ii.* Centre for Offshore Foundation Systems, University of WesternAustralia

Jacket Gravitybase

structure

Subseaframe

Tensionleg

platform

Fig. 1. Applications of offshore shallow foundation system(Courtesy of NGI)

Suctionsdevelop onfooting/bedinterface,preventinguplift

Seabed

M

D

( )d D�

d

Fig. 2. Skirted shallow foundation

prevail. For a large gravity base foundation on a low-per-meability clay with a plan area of several thousand squaremetres and deep skirts, undrained conditions may prevail foryears. Even for small skirted foundations, such as used on thecorners of jacket structures, which may be only 10 or 15 m indiameter with skirts penetrating 4 or 5 m into the seabed,suctions can be relied on for the period of wave loading, evenin relatively pervious deposits (Bye et al., 1995).

The moment component of a foundation load typicallyresults from horizontal environmental forces acting on thesubstructure, such that the horizontal and moment load com-ponents acting on the foundation will act in the samedirection, HM, as shown in Fig. 3(a). Foundations with aneccentric vertical load, due for example to the weight of thesubstructure or subsea structures with multiple pipeline con-nections, can experience combinations of horizontal load andmoment acting in opposition, �HM, as shown in Fig. 3(b).

A three-dimensional failure envelope for undrained ulti-mate limit states under general vertical, horizontal andmoment loading for a shallow circular foundation of dia-meter D with an interface that can sustain tension (as wouldbe appropriate for a skirted foundation) is shown in Fig. 4.The axes are represented in terms of dimensionless loadsV/Asu, H/Asu and M/ADsu, where A is the plan area of thefoundation and su is the soil undrained shear strength. Thekey characteristics of the failure envelope are as follows.

(a) Moment capacity at low vertical loads. If no tensileresistance were available, the foundation would lift upfrom the seabed under applied moment at low verticalloads, and the capacity of the foundation would bereduced (Fig. 5).

(b) Asymmetry, reflecting the physical reality that combi-nations of horizontal and moment load acting in thesame direction (HM) and opposition (�HM) are notphysically equivalent.

(c) Maximum moment capacity is mobilised under acombination of positive horizontal load and moment,rather than pure moment.

The complex interaction of vertical, horizontal and mo-ment loads is evident from Fig. 4, and is further complicatedby a dependence on soil strength profile. As a result, it isnot appropriate simply to scale a failure envelope derivedfor benchmark conditions (e.g. homogeneous soil strength)by ultimate limit states relevant for heterogeneous condi-tions. Gourvenec & Randolph (2003b) showed that the over-all size of a normalised failure envelope reduces as thedegree of strength heterogeneity increases, such that scalingan envelope derived for homogeneous strength conditionswould be unconservative.

Failure envelopes of the type shown in Fig. 4 (or Fig. 5) areuseful for qualitative assessment of ultimate limit states undergeneral loading; however, a considerable obstacle to theiradoption in design arises as their complex shape is notconducive to a closed-form solution. Expressions have beenproposed (e.g. Bransby & Randolph, 1998; Taiebat & Carter,2000), but are limited in terms of foundation geometry and soilproperties, the compromise on accuracy incurred and theirapplication outside the conditions for which they were derived.

In this paper the results of finite element analyses addres-sing undrained failure of shallow foundations with tensilecapacity under general loading, founded on soils with uni-form and linearly increasing shear strength with depth, arereported. Results are presented as failure envelopes in VHMload space and the kinematic mechanisms accompanyingfailure. An alternative approximate procedure is proposedthat permits calculation of ultimate limit states under generalloading, reconstruction of a plane or part of a failureenvelope containing the range of loading relevant to thedesign, or construction of an entire failure envelope, with ahigh degree of accuracy.

FINITE ELEMENT MODELA circular surface seated foundation of diameter D

founded on soil with uniform and linearly increasing shearstrength with depth has been considered. Bonding at thefoundation/soil interface was assumed, as would be appro-priate for a skirted foundation (with tensile resistance inreality mobilised by transient suctions under undrained load-ing), although the embedment was not modelled physically.Fig. 6 illustrates the idealised soil and foundation conditions.

Skirted foundations transfer loads to the level of the skirttips such that the bearing capacity is related to the shearstrength at skirt tip level (e.g. Tani & Craig, 1995; Watson& Randolph, 1997). Tani & Craig (1995) propose that the

M

H

Loadreferencepoint M

–H

(a) (b)

Fig. 3. Combined horizontal and moment loading: (a) HM;(b) 2HM

V A

s/

u

M ADs/ u

H As/ u

0·8

0

4

2

6�1·0 �0·5 0 0·5 1·0

Fig. 4. Three-dimensional failure envelope for general loading ofcircular surface foundation with bonding at foundation/soilinterface; uniform soil profile

1

0

�1

0

0·6

0·4

0·2

M ADs/ u

V As/ u4

2

0

H As/ u

6

Fig. 5. Three-dimensional failure envelope for general loading ofa circular surface foundation with a zero-tension foundation/soil interface; uniform soil profile (Taiebat & Carter, 2002a)

716 GOURVENEC

behaviour of an embedded shallow foundation in an un-drained soil with a linear increase in strength with depth canbe approximated by analysing a surface foundation, but withthe soil strength profile described by

su ¼ su0 þ kz (1)

where su0 is the shear strength at foundation level (i.e. skirt-tip level), and k is the strength gradient with depth z.Centrifuge tests reported by Tani & Craig (1995) andWatson & Randolph (1997) suggest this is reasonable forembedment depths less than around 30% of the foundationwidth (or diameter).

The degree of heterogeneity can be represented by thedimensionless coefficient

k ¼ kD

su0

(2)

For a shallow foundation with skirts of depth d, the shearstrength at foundation level, su0 ¼ sum + kd, where sum is theshear strength at the mudline.

All the finite element analyses were carried out with thesoftware ABAQUS (HKS, 2002).

Material propertiesAn undrained material response of the soil was repre-

sented with a linear elastic perfectly plastic constitutive lawdefined by the undrained Young’s modulus (Eu) and Poisson’sratio (�) and failure according to the Tresca criterion,defining the maximum shear stress in any plane limited tothe undrained shear strength (su).

Uniform shear strength and linearly increasing shearstrength profiles described by k ¼ 0, 2 and 6 according toequations (1) and (2) were considered. The Young’s modulusof the soil has been assumed to vary linearly with depth,maintaining a constant modulus ratio, Eu/su, of 500. Thefoundation has been modelled as a stiff raft of thickness0.2D, with a Young’s modulus seven orders of magnitudehigher than the surface value of Eu and thickness of 0.2D.Poisson’s ratio was taken as 0.49 for the soil, and 0.15 forthe foundation.

Finite element meshThe finite element mesh, shown in Fig. 7, represents a

semi-cylindrical section through a diametrical plane of acircular foundation of diameter D founded on the surface ofa mass of soil. Common nodes connect the foundation at thesoil interface, preventing separation occurring under tension.The radial boundaries of the mesh are positioned threediameters either side of the foundation, and the base of themesh is positioned a distance of 2.5D beneath the founda-tion. The external faces of the mesh are positioned suffi-ciently remote from the foundation that the load–

displacement response is unaffected by the presence of theboundaries. Displacement boundary conditions prevent out-of-plane displacements of the vertical faces (i.e. the flatdiametrical plane on the front of the mesh, and around thecircumference), and the base of the mesh is fixed in all threecoordinate directions.

Load pathsEach analysis followed a single load path to failure in

vertical, horizontal and moment (VHM) load space. A con-stant vertical load was imposed as a direct force, and thehorizontal and moment load components were applied at afixed displacement ratio (u/Ł). Five discrete levels of verticalload were modelled: V ¼ 0, 0.25Vult, 0.5Vult, 0.75Vult and0.9Vult. The ultimate vertical load, Vult, for each of the soilprofiles considered was calculated from a separate finiteelement analysis imposing a uniaxial vertical displacement(w) to failure. The terminating points of the individual loadpaths were used to construct a continuous bounding envelopeof horizontal load and moment (HM) at a constant verticalload (V). The foundation was subjected to controlled displa-cements (u, Ł), as opposed to directly applied loads (H, M),to enable post-failure conditions to be observed. The refer-ence point for the induced moment (M) is taken at themidpoint of the foundation base. Approximately 150 ana-lyses were carried out—50 for each soil strength profileinvestigated.

Sign convention and nomenclatureThe sign convention for displacements and loads pre-

sented in this paper obeys a right-handed axes and clockwisepositive convention as proposed by Butterfield et al. (1997),as shown in Fig. 8. The notation adopted for loads anddisplacements is shown in Table 1. The ultimate loads arethose for pure loading (e.g. H ¼ M ¼ 0 for Vult). This isimportant in respect of moment loading, where the maxi-mum moment is mobilised under conditions where H . 0.

z

d

su0 su

k

D

sum

s

s s kd

kD s

um

u0 um

u0

�

� �

�

undrained shear strength at mudline

undrained shear strength at foundation level =

Heterogeneity coefficient /κ

D

Fig. 6. Idealised soil and foundation conditions

D

6D

2·5D

Fig. 7. Finite element mesh

A D /4� π 2

V

1 1

Section 1–1

Load referencepointD

M

H

1 1

Fig. 8. Sign convention; positive loads shown

FAILURE ENVELOPES FOR OFFSHORE SHALLOW FOUNDATIONS 717

The maximum moment is denoted by Mmax (and correspond-ing mmax ¼ Mmax/Mult).

ValidationTheoretical solutions of ultimate limit states under pure

vertical and moment loading are available for the geometryand soil conditions investigated in this study. Exact solu-tions, or established upper bounds where exact solutions arenot available, are summarised in Table 2 in terms of bearingcapacity factors NcV and NcM for pure vertical load andmoment respectively. Cox et al. (1961) defined an exactsolution for a circular foundation on a uniform Trescamaterial that was extended with lower bound solutions tocover soils with linearly increasing shear strength profiles bySalencon & Matar (1982) and Houlsby & Wroth (1983), andwhich Martin (2001) showed to be exact solutions. Ultimatemoment capacity is based on an upper-bound solution of ascoop mechanism proposed by Murff & Hamilton (1993) forfoundations on uniform soils, which was extended to hetero-geneous soil conditions by Randolph & Puzrin (2003). Theultimate horizontal capacity is equal to the shear strength atfoundation level, as sliding governs failure of a surfacefoundation, and is therefore independent of the shearstrength profile; that is, NcH ¼ 1.

The theoretical solutions provide useful performance in-dicators for the finite element predictions and confirm goodagreement, to within 2% for pure vertical and horizontalcapacity and 6% in the calculation of Mult (Cox et al., 1961;Martin, 2001; Randolph & Puzrin, 2003).

Interaction under combined vertical and horizontal load,vertical load and moment, and horizontal load and momentfor M ¼ 0, H ¼ 0 and V ¼ 0 respectively are reported byGourvenec & Randolph (2003b) and agree well with avail-able solutions (Green, 1954; Taiebat & Carter, 2002b;Randolph & Puzrin, 2003). For fully combined vertical,horizontal and moment loads acting simultaneously no theor-etical solution exists for comparison with the finite elementresults.

FAILURE ENVELOPESThe applied loads and displacements give rise to load

paths that move from the origin across the failure envelope,initially at gradients determined by the elastic stiffness, but

with the gradients changing owing to internal plastic yield-ing as the paths approach the failure envelope. Once thefailure envelope is reached, each loading path travels aroundthe envelope until it reaches a point where the normal to theenvelope matches the prescribed displacement ratio. Fig. 9shows an example of the load paths from the analysis of thefoundation on the uniform deposit (k ¼ 0).

Bounding envelopes of ultimate limit states under combi-nations of vertical (V), horizontal (H) and moment (M)loading predicted by the finite element analyses are pre-sented as failure envelopes in Fig. 10 for each of the threedifferent shear strength profiles considered. The envelopesare plotted in dimensionless load space H/Asu0 againstM/ADsu0 for each of the five vertical load cases considered:V ¼ 0, 0.25Vult, 0.5Vult, 0.75Vult and 0.9Vult.

Three-dimensional failure surfaces such as those shown inFigs 4 and 5 provide a useful qualitative assessment ofultimate limit states under general loading. For quantitativecomparison, two-dimensional slices through the three-dimen-sional surface are more useful for direct determination ofultimate limit states. Representation of the interaction ofhorizontal load and moment (HM) at constant vertical load(V) is convenient since in reality vertical foundation load isquasi-constant, largely due to the self-weight of the super-structure and foundation system, whereas the horizontal andmoment components result from the environmental forcesand are interdependent and variable.

Table 1. Summary of notation for loads and displacements

Vertical Horizontal Rotational

Load V H MUltimate load Vult Hult Mult

Dimensionless load V/Asu0 H/Asu0 M/ADsu0

Dimensionless ultimate load NcV ¼ Vult/Asu0 NcH ¼ Hult/Asu0 NcM ¼ Mult/ADsu0

Normalised load v ¼ V/Vult h ¼ H/Hult m ¼ M/Mult

Displacement w u Ł

Table 2. Exact solutions for ultimate vertical and moment loads of circular shallowfoundations

k ¼ kD/su0 0 2 6

NcV ¼ Vult/Asu0 Cox et al. (1961) 6.05 – –Houlsby & Wroth (1983) 6.05 7.63 9.67Martin (2001) 6.05 7.63 9.69

NcM ¼ Mult/ADsu0 Murff & Hamilton (1993) 0.67 – –Randolph & Puzrin (2003) 0.67 0.88 1.25

�1·0 �0·8 �0·6 �0·4 �0·2 0 0·2 0·4 0·6 0·8 1·00

0·1

0·2

0·3

0·4

0·5

0·6

0·7

0·8V 0�

V V0·25� ult

V V0·5� ult

V V0·75� ult

V V0·9� ult

M ADs/ u

H As/ u

Fig. 9. Example of load paths from finite element analyses;circular foundation, uniform deposit (kk 0)

718 GOURVENEC

The outermost envelopes in Fig. 10 correspond to ultimatelimit states under combined horizontal load and moment inthe absence of vertical load. The horizontal, moment capa-city reduces with increases in the component of vertical loadindicated by contraction of the failure envelope towards theorigin. Initial increases in vertical load component have asmall impact on foundation capacity. As the component ofvertical load increases further, the incremental increase has amore significant effect on foundation capacity: It is clearthat the interaction of horizontal load and moment is com-plex, and the shape of the envelope (not just the size) isinfluenced by the level of vertical load.

Comparing the failure envelopes for the different strengthprofiles shows increased bearing capacity with increaseddegree of heterogeneity. Pure moment capacity is increasedby 85% and maximum moment capacity by 75% for the soilprofile k ¼ 6 compared with the uniform case k ¼ 0. Hor-izontal capacity is unaffected by soil strength profiles, asfailure by sliding is a function of the soil strength at founda-tion level. The failure envelopes for each of the soil strengthprofiles exhibit a similar form, although the eccentricity ofthe envelope diminishes with increasing degree of heteroge-neity (i.e. increasing k), indicating the reduction in horizontalload accompanying mobilisation of the maximum moment.

FAILURE MECHANISMSDisplacement vector fields corresponding to various points

around the failure envelope for the uniform soil case (k ¼0) are shown in Fig. 11. In the absence of vertical load andmoment, a sliding mechanism will govern failure. As themoment component increases (and the horizontal capacitydecreases), the sliding mechanism transforms into a shallowscoop (Fig. 11(a)), increasing in depth with increased mo-ment, up to the maximum moment where a near-semicircularscoop is observed (Fig. 11(b)). A schematic representationof the scoop mechanism is illustrated in Fig. 12(a). As themaximum moment is mobilised in conjunction with a highhorizontal load (seen in Fig. 10(a)), very little horizontaldisplacement occurs in the failure mechanism. Once thepeak moment has been mobilised, side wedges and fan zonesdevelop to either side of the scoop (Fig. 11(c)). The so-called wedge–scoop–wedge mechanism was defined byBransby & Randolph (1998), and is illustrated schematicallyin Fig. 12(b). With further increases in horizontal load (anddecreased moment capacity), the scoop diminishes and thewedges extend across the base of the foundation, transform-ing to a sliding mechanism.

For non-zero vertical load acting on a foundation, failurethat is influenced more by moment than horizontal loadmanifests as a scoop–wedge mechanism (Fig. 12(c)), thescoop component of the mechanism becoming more domi-nant with increased moment (e.g. Figs 11(e), 11(g) and11(h)). Failure that is influenced more by horizontal loadthan moment manifests as an asymmetric wedge mechanism(Figs 11(j) and 11(m), and schematically in Fig. 12(d);Green, 1954). Following mobilisation of maximum moment,a general asymmetric Brinch Hansen mechanism (Fig. 12(e))is developed (Brinch Hansen, 1970) (Figs 11(f), 11(i), 11(l)and 11(o)).

Failure at horizontal loads less than or greater than thevalue corresponding to maximum moment leads respectivelyto backward (in the reverse direction to H) or forwardtranslation of the foundation, in addition to rotation. This isconsistent with the normality of the system (in terms of theplastic strain increment vectors of the work-conjugate dis-placements u and Ł lying in the direction of the outwardnormal to the failure envelope), and leads to envelopes withpositive and negative gradients in the HM plane intersectingat Mmax, as seen in Fig. 10.

The kinematic mechanisms accompanying failure at maxi-mum moment as a function of vertical load are shown forthe two heterogeneous cases (k ¼ 2 and 6) in Fig. 13. Asthe gradient of the shear strength profile increases (i.e. kincreases), the maximum moment capacity under zero verti-cal load is mobilised in conjunction with decreasing horizon-tal load, and lateral displacement at the edges of the footingaccompanies the scoop mechanism (Figs 13(a) and 13(b)).The resulting wedge–scoop–wedge mechanism is similar tothat illustrated in Fig. 12(b). This is in contrast to the purescoop mechanism observed in the uniform soil (k ¼ 0)

�1·0 �0·5 0 0·5 1·0

M ADs/ u0

H As/(a)

u0

�1·0 �0·5 0 0·5 1·0

M ADs/ u0

H As/(b)

u0

�1·0 �0·5 0 0·5 1·0

M ADs/ u0

H As/(c)

u0

0

0·2

0·4

0·6

V V0·5� ult

V V0·75� ult

V V0·9� ult

0·8

V 0� V V0·25� ult

0

0·2

0·4

0·6

0·8

1·0

0·2

0·4

0·6

0·8

1·0

1·2

0

1·4

Fig. 10. Failure envelopes for circular skirted foundationspredicted by finite element analyses: (a) kk 0; (b) kk 2;(c) kk 6


when the maximum moment is mobilised with nearly all thelateral resistance of the soil, and a near-perfect scoopmechanism accompanies failure (Figs 11(b) and 12(a)).Under non-zero vertical loads the wedge–scoop–wedge

mechanism transforms to a scoop–wedge mechanism (asshown in Figs 13(c)–13(j)), the scoop component diminish-ing and the wedge component becoming more dominantwith increasing vertical load.

� Mmax at Mmax � Mmax

V V0·5� ult

V V0·75� ult

(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

(j) (k) (l)

V V0·9� ult

(m) (n) (o)

V V0·25� ult

V 0�

(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

(j) (k) (l)

(m) (n) (o)

Fig. 11. Failure mechanisms at various ultimate limit states for circular foundation on a uniform deposit (kk 0)

L

Centre ofrotation

for scoop

(e)(d)

Centre ofrotation forfoundation

(a)

(c)

(b)

Fig. 12. Sketches of key failure mechanisms under general loading: (a) scoop; (b) wedge–scoop–wedge; (c) scoop–wedge; (d) asymmetric wedge; (e) Brinch Hansen

720 GOURVENEC

The increase in shear strength with depth is reflected by areduction in the depth of all the failure mechanisms, result-ing from soil displacements being confined to the weakersoil nearer the surface.

APPROXIMATING EXPRESSIONS FOR PREDICTION OFULTIMATE LIMIT STATES

Three-dimensional failure envelopes, such as that shownin Fig. 4, and two-dimensional slices through a three-dimen-sional envelope, such as those shown in Fig. 10, illustratehow foundation capacity varies with interaction of the differ-ent load components, and permit direct determination ofultimate limit states under general loading. For routine use itis convenient if the form of the interaction diagram can bedefined explicitly. The complex shape of failure envelopesfor general loading conditions is not conducive to an inter-polating expression, and the dependence of the VHM inter-action on foundation geometry and soil strength profilemakes the nature of finding an approximating expression allthe more challenging.

Approximate expressions to describe VHM failure envel-opes for shallow foundations with bonding on the founda-tion/soil interface have been proposed in the past (e.g.Bransby & Randolph, 1998; Taiebat & Carter, 2000).

Bransby & Randolph (1998) proposed a simplifying trans-formation based on an upper-bound solution from limitedplane strain finite element analyses (equation (3)). Theirexpression was derived for a surface strip foundation and a

soil with a shear strength profile linearly increasing withdepth given by k ¼ 6.

f ¼ V

Vult

� �2:5

� 1 � H

Hult

� �1=3

1 � M�Mult

� �

þ 1

2

M�Mult

� �H

Hult

� �5

¼ 0

(3)

where Vult, Hult and Mult are the capacities under purevertical, horizontal and moment load respectively, and M� isa modified moment parameter given by the expression

M�ADsu0

¼ M

ADsu0

� L

D

H

Asu0

(4)

where L is the height above the foundation of the centre ofrotation of the scoop mechanism (as shown in Fig. 12(a))governing ultimate moment capacity, Mult.

Figure 14(a) shows ultimate limit states predicted byequation (3) compared with finite element results for a stripfoundation on a deposit with an undrained shear strengthprofile given by k ¼ 6 (the same conditions as those fromwhich the approximating expression was derived) for V ¼ 0and V ¼ 0.5Vult. Some asymmetry of the HM envelope iscaptured by the approximating expression, but discrepancybetween the ultimate limit states predicted by equation (3)and the finite element results is evident.

Equation (3) was derived for a shear strength gradient of

κ � 2 κ � 6

V V0·25� ult

V 0�

V V0·5� ult

V V0·75� ult

V V0·9� ult

(a) (b)

(c) (d)

(i)

(g) (h)

(e) (f)

( j)

Fig. 13. Failure mechanisms at maximum moment for circular foundation on depositswith shear strength increasing linearly with depth (kk 2 and 6)


k ¼ 6, such that application of the expression for an alter-native shear strength gradient requires re-optimisation of theexponents. Exponents for use with other soil shear strengthprofiles were not suggested by Bransby & Randolph. Fig.14(b) shows equation (3) applied to uniform soil conditionswithout re-optimisation of the exponents, to illustrate theincreased lack of fit if the expression is applied outside theconditions on which it was based.

Taiebat & Carter (2000) proposed an approximating alge-braic expression based on three-dimensional finite elementanalyses to describe a failure envelope for undrained capa-city of a surface circular footing (bonded to the soil surface)on a homogeneous deposit (equation (5)).

f ¼ V

Vult

� �2

þ M

Mult

� �1 � Æ

HM

Hult Mj j

� �2" #

þ H

Hult

� �3��

�� 1 ¼ 0

(5)

where Æ is a factor to adjust the rotation and eccentricity ofthe failure envelope. For the uniform soil conditions consid-ered, Taiebat & Carter (2000) suggested Æ ¼ 0.3 as a goodfit to the predictions from their finite element analyses.

Figure 15(a) shows ultimate limit states predicted withequation (5) for Æ ¼ 0.3 with the finite element resultsshown in Fig. 10(a), which represent the same foundationand soil conditions. The comparison shows that equation (5)captures some of the asymmetry of the failure envelopes butthere is still a clear discrepancy between the predictionsfrom equation (5) and the finite element results. Fig. 15(b)shows the influence of the magnitude of Æ on the shape ofthe envelope in the HM plane, comparing predictions fromequation (5) for Æ ¼ 0.3, 0.4 and 0.5 with the finite elementresults for vertical loads V ¼ 0 and 0.75Vult. The rotation ofthe envelope associated with additional eccentricity withincreasing Æ causes the gain in accuracy of fit in the HMquadrant to be countered by an increased loss of fit in the –HM quadrant. Also, the dependence of the interaction ofhorizontal and moment load on the level of vertical loadmeans that the increased eccentricity with increased Æ,

0

0·5

1·0

1·5

M ADs/ u0

Eqn (3); Bransby & Randolph (1998)

Finite element prediction

V V0·5� ult

V 0�2·0

V 0�

V V0·5� ult

0

0·5

1·0

M ADs/ u0

�1·0 �0·5 0 0·5 1·0

H As/ u0

(a)

�1·0 �0·5 0 0·5 1·0

H As/(b)

u0

Fig. 14. Comparison of equation (3) (Bransby & Randolph,1998) with finite element results (strip foundation): (a) kk 6;(b) kk 0 (original expression; exponents not re-optimised)

�1·0 �0·5 0 0·5 1·00

0·2

0·4

0·6

0·8

M ADs/ u0


V V0·5� ult

V 0� V V0·25� ult

V V0·75� ult

V V0·9� ult

Eqn (5); Taiebat & Carter (2000), 0·3α �

H As/ u0

α 0·4�

α 0·5�

�1·0 �0·5 0 0·5 1·0H As/ u0

(b)

0

0·2

0·4

0·6

0·8V 0�

M ADs/ u0

V V0·75� ult

α 0·3�

Eqn (5); Taiebat & Carter (2000), varying α


(a)

Fig. 15. Comparison of equation (5) (Taiebat & Carter, 2000)with finite element results; circular foundation on a uniformdeposit (kk 0): (a) Æ 0.3; (b) varying Æ

722 GOURVENEC

although beneficial at zero (or low) vertical load, leads toincreasing disparity at high vertical loads.

Taiebat & Carter’s finite element analyses used a Fourierseries expansion to model the three-dimensional problemusing nodal load incrementation and a tangent stiffnesssolution approach (as opposed to a full three-dimensionalmodel with displacement-controlled load paths, as presentedin this paper). They noted problems were encountered inidentifying the ultimate capacity, which led to overestimationof the ultimate moment capacity Mult of 19% and under-estimation of the uniaxial vertical capacity Vult of 6%. There-fore the finite element results from this study are used forcomparison with their approximating expression in Fig. 15.

The factor Æ in equation (5) could be altered to adjust theshape of the locus to try to fit failure envelopes for non-uniform shear strength profiles, although this has not beenaddressed to date. Indeed, the expression may be bettersuited to describe the shape of failure envelopes for condi-tions of soil strength heterogeneity, which exhibit a moregentle response around the maximum moment. However, itis still questionable whether the expression is sufficientlyversatile to be applicable to a range of soil strength profiles.

As a result of the difficulty of fitting an approximatingexpression to failure envelopes for general loading, existingsolutions represent only limited cases of foundation geome-try and specific soil strength profiles. Available solutionsappear to lack the versatility to be applied outside theconditions for which they were derived, for example alter-native shear strength profiles. There is also, inevitably, acompromise on accuracy in the reconstructed failure envel-opes, even for the conditions for which the solutions areintended.

ALTERNATIVE APPROACH FOR PREDICTINGULTIMATE LIMIT STATES

Figure 16 shows ultimate limit states for a circularshallow foundation on a uniform deposit (k ¼ 0) in terms ofcontours of normalised moment to horizontal load, plotted innormalised vertical and moment load space.

Curves representing ultimate limit states involving mo-ment and horizontal load acting in the same direction and inopposition (according to Fig. 3) are presented on separategraphs in Fig. 16 for clarity.

The benefit of expressing the ultimate limit states in termsof constant m/h is that the horizontal load (H), due to theenvironmental loading, is a known quantity, as is the heightof the line of action of H above the load rotation point, LH .Given M ¼ HLH and the theoretical solutions for Mult andHult, the normalised moment to horizontal load ratio, m/h, isknown. The benefit of representing the interaction as contin-uous functions in vm space is that determination of criticalvertical load at a ratio of m/h is not restricted by discreteintervals of vertical load, as with the slice-style failureenvelopes shown in Fig. 10.

Derivation of the curvesThe curves are derived directly from the failure envel-

opes shown in Fig. 10. Considering the case k ¼ 0 forillustration, the failure envelopes expressed in dimensionlessload space in Fig. 10(a) are first re-plotted in terms ofnormalised loads v ¼ V/Vult, h ¼ H/Hult and m ¼ M/Mult.The resulting envelopes are shown in Fig. 17(a) in terms ofh and m at intervals of v ¼ 0, 0.25, 0.5, 0.75 and 0.9. Thenormalised load space is then divided by planes of constantm/h. Along a particular m/h plane the load state at theintersection with each failure envelope is selected, asillustrated in Fig. 17(a) for m/h ¼ 2 and v ¼ 0, 0.25, 0.5,

0.75 and 0.9. The ultimate limit states are then re-plottedon axes of normalised vertical and moment load v ¼ V/Vult

and m ¼ M/Mult, and the discrete points are joined by asmooth curve (Fig. 17(b)).

This process should be repeated for sufficient ratios ofm/h to adequately encapsulate the shape of the failureenvelope. In Fig. 17(a) 24 planes have been used, approxi-mately equally distributed between m/h ¼ 0 and m/h ¼ 1for both combinations of HM and �HM. For the value ofnormalised horizontal load h at the interception whenm ¼ 0, Green’s exact closed-form solution for undrainedfailure of a shallow foundation under inclined load can beapplied (Green, 1954). Although Green’s original expres-sion was derived for uniform soil conditions, Gourvenec &Randolph (2003b) showed that the normalised shape of theVH failure envelope was independent of soil strengthprofile (for a linear increase in strength with depth).Horizontal capacity for load combinations in conjunctionwith a vertical load but in the absence of moment, for arange of soil strength profiles, can be given by Green’snormalised relationship

�vv ¼ 0:5 þ cos�1 h þffiffiffiffiffiffiffiffiffiffiffiffiffi1 � h

2p

2 þ �

h ¼ 1 for v < 0:5

(6)

The curves can be used to identify a value of verticalload to cause failure acting in conjunction with a knownhorizontal load and moment, for example in order to

1·0

1·0

0·8

0·8

0·6

0·6

0·4

0·4

0·2

0·2

(a)

(b)

�1·5

∞

3

�0·2

�0·4

�0·6

�0·8

0·2

0·4

0·6

0·8

1·25

1·5

2 4 8

0

0·2

0·4

0·6

0·8

1·0

0

0

v V V/� ult

v V V/� ult

mM

M/�

ult

mM

M/�

ult

�1·0

�1·25 �2 �3 �4 �8

0

0·2

0·4

0·6

0·8

1·0

1·2

1·0

Contours of m/h

Contours of /m h

Fig. 16 Alternative representation of failure envelopes; circularfoundation on a uniform deposit (kk 0): (a) V–HM; (b) VHM


determine the critical foundation area. Alternatively, thecurves can be used to reconstruct a failure envelope inhorizontal and moment load space for any discrete valueof vertical load (such as those shown in Fig. 10). Con-struction of a failure envelope enables combinations oflateral load and moment that would cause failure under aworking vertical load to be identified, which is useful, forexample, to indicate the margin of safety to changes inenvironmental loading. If it were desired, a number of HMenvelopes at selected levels of vertical load could becalculated and interpolated to create a continuous surfaceof the form shown in Fig. 4.

Curve fitThe curves shown in Fig. 16 can be described by poly-

nomial functions. Combinations of VHM can be describedby cubic polynomials (equation 7(a)) and combinations ofV–HM by quartic polynomials (equation 7(b)).

m ¼ c1v3 þ c2v

2 þ c3vþ c4 (7a)

or

m ¼ c1v4 þ c2v

3 þ c3v2 þ c4vþ c5 (7b)

Figure 18 compares the approximating polynomial functionswith the finite element results. The close fit of the poly-nomial function indicates that reconstruction of a failureenvelope will incur almost no error. The polynomial coeffi-cients for the ratios of m/h shown in Fig. 16 are tabulated inthe appendix (see Tables 3 and 4 and Fig. 19).

Interaction under general loading of a shallow foundationis a function of soil shear strength profile (as seen in Fig.

10); however, vm curves similar to those illustrated in Fig.16 for homogeneous shear strength are shown to have asimilarly simple form for the heterogeneous cases consid-ered. Figs 19 and 20 respectively show ultimate limit statesfor the two linearly increasing strength profiles considered inthis study (k ¼ 2 and 6). The curves are derived from thefinite element results shown in Figs 10(b) and 10(c), and canalso be accurately described by cubic and quartic polyno-mials for VHM and V–HM combinations respectively. Thepolynomial coefficients are tabulated in the appendix fork ¼ 2 and 6 (Tables 5, 6 and 7, 8 respectively).

CONCLUSIONSFailure envelopes have been presented for general loading

of shallow skirted circular foundations on Tresca soil with

0

8

1·00·5�0·5 0

0 0·2 0·4 0·6 0·8 1·0v V V= / ult

0

0·2

0·4

0·6

0·8

1·0

1·2 Contour of / 2m h �

m h/ 0·2�

0·4

0·6

1·0

246

�0·2

�0·4

�0·6

�1·0

�2 �4 �6 �8

�0·8

�1·25 �1·5 1·5 1·25

0·8

V V0·25� ult

V V0·9� ult0

0·2

0·4

0·6

0·8

1·0

1·2

�1·0

m M M/� ult

mM

M/�

ult

h H/H� ult

V V0·5� ult

V 0�

V V0·75� ult

(a)

(b)

Fig. 17. Derivation of curves; circular foundation, on a uniformdeposit (kk 0): (a) failure envelopes expressed in normalisedload space; (b) example contour of m/h expressions in vm space

1·0

1·0

0·8

0·8

0·6

0·6

0·4

0·4

0·2

0·2

3

� 1·5

Contours of /m h

Contours of /m h

1·25

1·5

�0·2

�0·4

�0·6

�0·8

�2 �3 �4 �8

∞

0

0·2

0·4

0·6

0·8

1·0

1·2

0·2

0·4

0·6

0·8

1·0

2 4 8

0

0·2

0·4

0·6

0·8

1·0

0

0

�0·2

�0·4

�0·6

�0·8

�1

�1·25 �2 �3 �4 �8

∞

v V V/(a)

� ult

v V V/(b)

� ult

mM

M/�

ult

mM

M/�

ult

Fig. 18. Comparison of approximating polynomial function(solid line) with finite element results (data points); circularfoundation, on a uniform deposit (kk 0): (a) V-HM; (b) VHM

mM

M/�

ult

v V/V� ult

m c v c v c v c VHM

m c v c v c v c v c V–HM

for , or

for

� � � �

� � � � �1

32

23 4

14

23

32

4 5

Fig. 19. Sketch of polynomial function describing v–m curve

724 GOURVENEC

uniform or linearly increasing shear strength with depth. Theinteraction of vertical, horizontal and moment loads hasbeen shown to be complex, with the shape of the failureenvelope defining combinations of ultimate horizontal loadand moment changing with the magnitude of vertical loadand degree of shear strength heterogeneity.

Comparison of the failure envelopes predicted by thefinite element analyses with available approximating solu-tions showed that the ultimate limit states predicted bythe approximate methods can deviate (quite considerablyunder some load combinations) from the finite elementpredictions (Bransby & Randolph, 1998; Taiebat & Car-ter, 2000a). The extension of these solutions to a widerrange of soil conditions was also shown to be question-able.

An alternative representation of ultimate limit states hasbeen proposed, presenting contours of ratios of normalisedmoment to horizontal load (m/h) in normalised vertical loadand moment space (vm).

The curves can be used to identify ultimate limit states

under combinations of vertical, horizontal and moment load,to construct any plane or part of a failure envelope contain-ing the range of loading relevant to the design, or toconstruct a complete three-dimensional failure envelope.Cubic and quartic polynomial functions describe the shapeof the curves for deposits with uniform and linearly increas-ing shear strength profiles.

The principal benefits of the proposed procedure overother approximate reconstruction methods (e.g. Bransby &Randolph, 1998; Taiebat & Carter, 2002a) are the mini-mal compromise on accuracy and the versatility of themethod to be applied to a range of soil and foundationconditions.

ACKNOWLEDGEMENTThe work described here forms part of the activities of

the Special Research Centre for Offshore Foundation Sys-tems, established under the Australian Research Council’sResearch Centres Program.

V–HM

VHM

0

0·2

0·4

0·6

0·8

0·2

0·4

0·6

0·8

1·25

4 81·5

0

0·2

0·4

0·6

0·8

1·0

0

0

v V/V� ult

v V/V� ult

mM

/M�

ult

mM

/M�

ult

�0·2

�0·4

�0·6

�0·8�1·0

�1·5 �2 �3 �4 �8

1·0

1·2

1·0

2

Contours of m/h

Contours of m/h

(a)

(b)

0·8

0·8

1·0

1·0

0·6

0·6

0·4

0·4

0·2

0·2

�1·25

∞

3

Fig. 20. Alternative representation of failure envelopes; circularfoundation on heterogeneous deposit (kk 2): (a) V-HM; (b)VHM

V–HM

VHM

0

0

v V/V� ult

v V/V� ult

mM

/M�

ult

mM

/M�

ult

(a)

�2 �3 �4 �8

4 8

1·5

0

0·2

0·4

0·6

0·8

1·0

�0·2

�0·4

�0·6�0·8�1·0

�1·25

4 8

0

0·2

0·4

0·6

0·8

1·0

1·2

0·2

0·4

0·6

0·8

1·0

1·25

4 8

Contours of m/h

Contours of m/h

(b)

0·8

0·8

1·0

1·0

0·6

0·6

0·4

0·4

0·2

0·2

∞

�1·5

32

Fig. 21. Alternative representation of failure envelopes; circularfoundation on heterogeneous deposit (kk 6): (a) V-HM; (b)VHM


APPENDIX: POLYNOMIAL COEFFICIENTS DESCRIBING v–m CURVES

Table 3. kk 0, VMH, m c1v3 c2v

2 c3v c4

m/h c1 c2 c3 c4

0.2 �0.6771 0.6196 �0.1399 0.20140.4 �0.9013 0.6257 �0.121 0.40110.6 �1.0205 0.5429 �0.1182 0.60040.8 �0.8119 0.042 �0.0155 0.79091 �0.733 �0.1992 �0.0351 0.97061.25 �0.7196 �0.2893 �0.1193 1.13191.5 �0.9266 �0.2022 0.0065 1.1192 �1.3989 0.4425 �0.1465 1.10013 �1.6475 0.7689 �0.1781 1.06054 �1.6645 0.7688 �0.1398 1.04078 �1.7847 0.9597 �0.1896 1.0212

Table 4. kk 0, V–HM, m c1v4 c2v

3 c3v2 c4v c5

m/h c1 c2 c3 c4 c5

1 �1.132 0.5228 �0.5062 0.1179 0.9998�8 �0.9186 �0.0257 �0.0496 0.0374 0.9597�4 �0.9861 0.0128 0.0457 0.0104 0.9197�3 �0.6575 �0.583 0.4003 �0.0477 0.8898�2 �1.3377 0.5988 �0.099 0.0076 0.83�1.5 �2.4101 2.7125 �1.2635 0.1793 0.7802�1.25 �2.5806 3.1482 �1.5328 0.2243 0.7401�1 �3.05 4.1755 �2.1236 0.3184 0.68�0.8 �2.7243 3.6245 �1.7779 0.2722 0.6096�0.6 �2.6204 3.5688 �1.7126 0.2591 0.5095�0.4 �2.3486 3.296 �1.5411 0.2283 0.3695�0.2 �1.0971 1.5069 �0.7175 0.1109 0.1997

Table 5. kk 2, VHM, m c1v3 c2v

2 c3v c4

m/h c1 c2 c3 c4

0.2 �0.6339 0.5616 �0.1244 0.20130.4 �0.8313 0.5589 �0.1254 0.40090.6 �0.8804 0.4093 �0.127 0.60.8 �0.8164 0.1578 �0.1165 0.78061 �0.6093 �0.2665 �0.0589 0.94031.25 �0.5721 �0.3835 �0.1218 1.08081.5 �0.4782 �0.6514 �0.0136 1.14442 �0.7836 �0.2721 �0.0618 1.1213 �1.0845 0.1916 �0.1989 1.0974 �1.13 0.2618 �0.2074 1.08288 �1.1345 0.2347 �0.1355 1.0425


3 c3v2 c4v c5

m/h c1 c2 c3 c4 c5

1 �1.5647 1.9743 �1.6461 0.239 0.9998�8 �0.9457 0.5598 �0.6396 0.0691 0.9596�4 �0.844 0.272 �0.3566 0.0107 0.9198�2 �0.7176 �0.0328 �0.1013 �0.0363 0.8898�2 �0.9789 0.349 �0.1661 �0.0434 0.8399�1.5 �1.1477 0.4913 �0.0131 �0.1312 0.8001�1.25 �1.5675 1.484 �0.7439 0.0773 0.75�1 �1.6634 1.684 �0.8402 0.1289 0.6901�0.8 �1.7405 1.7795 �0.7659 0.1065 0.62�0.6 �1.6893 1.719 �0.6391 0.0799 0.53�0.4 �1.567 1.742 �0.6284 0.0736 0.38�0.2 �1.2662 1.8333 �0.9053 0.1423 0.1996

726 GOURVENEC

NOTATIONA cross-sectional area of foundationD footing diameterE Young’s modulusk gradient of undrained shear strength profileL height of centre of rotation of scoop mechanism

LH height of line of action of horizontal loadsu undrained shear strength

su0 undrained shear strength at foundation levelsum undrained shear strength at mudline

z depth below foundation levelk strength non-homogeneity ratio, kD/su0

� Poisson’s ratio

Summary of load nomenclatureVHM vertical, horizontal and moment load

Vult, Hult, Mult ultimate loadsv, h, m normalised loads (V/Vult, H/Hult, M/Mult)

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3 c3v2 c4v c5

m/h c1 c2 c3 c4 c5

1 �1.1634 1.5282 �1.5111 0.1499 0.9996�8 �1.2194 1.4839 �1.3283 0.1072 0.9597�4 �1.1375 1.2265 �1.0722 0.0555 0.9298�3 �0.7842 0.3928 �0.4327 �0.0788 0.9048�2 �1.0758 0.7417 �0.3587 �0.1757 0.8698�1.5 �0.8427 0.1627 0.1249 �0.2629 0.8198�1.25 �0.9138 0.3458 0.0009 �0.1999 0.7697�1 �0.8282 0.2114 0.1207 �0.2198 0.7196�0.8 �0.9241 0.4114 0.0244 �0.1683 0.6597�0.6 �0.9132 0.6038 �0.2836 0.0564 0.5397�0.4 �1.3731 1.3987 �0.4684 0.0537 0.3899�0.2 �1.6045 2.4861 �1.2808 0.205 0.1994

Table 7. kk 6, VHM, m c1v3 c2v

2 c3v c4

m/h c1 c2 c3 c4

0.2 �0.5929 0.5186 �0.1216 0.20160.4 �1.016 0.8557 �0.2258 0.40040.6 �0.8929 0.4422 �0.1256 0.57960.8 �0.7449 0.0657 �0.0525 0.74041 �0.6168 �0.1974 �0.0601 0.87961.25 �0.4171 �0.4952 �0.0808 1.00171.5 �0.5557 �0.2469 �0.1421 1.05122 �0.8909 0.1175 �0.224 1.09962 �0.6622 �0.2672 �0.1554 1.09124 �0.6261 �0.2148 �0.1019 1.06228 �0.8819 0.0517 �0.194 1.022


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728 GOURVENEC

failure envelopes for offshore shallow foundations under

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