493

i) PhD Student, Division of Civil Engineering, University of Dundee, UK.ii) Senior Lecturer, ditto (m.f.bransby＠dundee.ac.uk).

The manuscript for this paper was received for review on January 4, 2006; approved on January 4, 2007.Written discussions on this paper should be submitted before January 1, 2008 to the Japanese Geotechnical Society, 4-38-2, Sengoku, Bunkyo-ku, Tokyo 112-0011, Japan. Upon request the closing date may be extended one month.

493

SOILS AND FOUNDATIONS Vol. 47, No. 3, 493–505, June 2007Japanese Geotechnical Society

THE UNDRAINED VERTICAL BEARING CAPACITY OF SKIRTED FOUNDATIONS

G. YUNi) and M. F. BRANSBYii)

ABSTRACT

Skirted foundations are becoming an increasingly prevalent oŠshore foundation solution for the oil and gas andrenewable energies industries. Their capacity under combined horizontal, vertical and moment loading must be foundin order to ensure their stability under environmental loadings. As part of this process, knowledge of the verticalbearing capacity is required. In this paper, the vertical bearing capacity of skirted foundations on normally consolidat-ed undrained soil was investigated using numerical and physical modelling. Finite element analyses were carried toinvestigate the vertical bearing capacity of foundations with diŠerent geometries for various embedment ratios.Accompanying upper bound plasticity analyses highlighted the mechanistic reasons for the varying response andallowed examination of the eŠect of changing skirt interface friction. Analyses showed that skirted foundationcapacity under vertical load may be considered normally as if the foundation is rigid with an embedment depth equalto the skirt depth. Based on the numerical analysis, a design method is proposed to calculate vertical bearing capacity.Finally, good agreement with results from a series of centrifuge model tests partially validated the design method.

Key words: bearing capacity, centrifuge modelling, ˆnite element method, plasticity, shallow foundation (IGC: E3)

INTRODUCTION

Skirted foundations have become a common oŠshoreengineering foundation type (e.g., Tjelta and Haaland,1993; Andenaes et al., 1996) particularly in soft soils.Skirted foundations consist of a raft-type foundation ofbreadth, B on the soil surface with thin vertical plates orskirts connected to the raft perimeter penetrating thesoil to a depth, D (e.g., Fig. 1(c)). It is believed that thevertical skirts improve the foundation capacity by`trapping' the soil beneath the raft and between the skirtsso that applied load is transferred to the soil at the level ofthe skirt tips (e.g., Tani and Craig, 1995).

For deep water sites, it is quite common to encounterthick, normally consolidated, low permeability sedi-ments. These will have negligible shear strength at the soilsurface and an increasing undrained shear strength ofabout 1 kPa per metre below the soil surface with thegradient depending on the frictional properties of the soiland its self-weight. Consequently, skirted foundationswhich transfer loads beneath the soft soil surface oftenare an attractive foundation solution.

Although oŠshore foundations are subject to verticalloading due to the self-weight of the installation (e.g.,jacket structure, gravity base, wind turbine) the verticalapplied load (V) is often small compared to the verticalcapacity (Vo). Instead, it is most often the combinedvertical, horizontal and moment load capacity that iscritical due to environmental (wind, wave, current) and

man-made (snag, impact) loadings. A common designapproach (e.g., Zaharescu, 1961; Ticof, 1977) deˆnes afailure envelope in V-H-M load space which is used tosize oŠshore foundations under monotonic loading con-ditions. Combined (V-H-M) loading envelopes are oftenscaled by the vertical bearing capacity, Vo (e.g., Ticof,1977; Butterˆeld and Ticof, 1979; Gottardi, 1992;Gottardi et al., 1999) or combinations of the pure vertical(Vo), pure horizontal (Ho) and moment (Mo) capacity(e.g., MurŠ, 1994; Bransby and Randolph, 1998;Gourvenec and Randolph, 2003). Therefore knowledgeof the vertical capacity, Vo is imperative for design evenfor these more complex loading conditions.

Although the undrained vertical bearing capacity ofshallow foundations is a long researched topic (e.g.,Prandtl, 1921; Hill, 1950; Skempton, 1951), for the caseof skirted foundations in normally consolidated depositsthe capacity is di‹cult to calculate accurately. This isbecause of the combination of (i) the non-uniformity ofthe soil strength beneath and around the foundations(e.g., Davis and Booker, 1973; Gourvenec and Ran-dolph, 2003), (ii) the embedded nature of the foundation(e.g., Skempton, 1951), and (iii) the skirted geometry ofthe foundations (e.g., Tani and Craig, 1995; Hu andRandolph, 2002). This combination of di‹culties is in-vestigated in this paper predominantly using numericalanalysis.

In this paper, ˆnite element and upper bound plasticityanalyses have been used to examine the undrained vertical

494

Fig. 1. Geometry of footing types studied Fig. 2. Equivalent shear strength for ˆnite element analysis

494 YUN AND BRANSBY

bearing capacity of skirted strip foundations with a rangeof embedment ratios from DWB＝0.2 to 1.2. First, em-bedded and skirted foundations in uniform strength soilwith geometries shown in Fig. 1 have been examinedbefore the case of a surface foundation on non-uniformstrength soil. Next, the case of skirted foundations innormally consolidated soil has been studied using FE andplasticity analysis. Finally, a design method deducedfrom the above approach has been validated using theresults of a series of centrifuge model tests, and this alsoallows discussion of displacement conditions at failure.

The conditions of a strip (i.e., inˆnitely long) footing isexamined in this paper. Although, this condition may notbe typical of circular caissons investigated by previousresearchers (e.g., Watson et al., 2000; Martin andRandolph, 2001; Byrne and Cassidy, 2002) it allows moredetailed investigation of mechanisms occurring, whilststill retaining the key behavioural aspects of the bound-ary value problem (e.g., Bransby and Randolph, 1999a;Gourvenec and Randolph, 2003). In addition, someskirted foundations such as gravity bases and manifoldbases are rectangular and so the plane strain case may bedirectly applicable to their design.

FINITE ELEMENT ANALYSIS

Geometry and Material PropertiesThe ˆnite element study examined the vertical capacity

of four diŠerent types of strip foundation in uniform andnormally consolidated undrained material. The fourtypes were surface rafts (Fig. 1(a)), embedded founda-tions (Fig. 1(b)), skirted foundations (Fig. 1(c)) and askirted foundation with an additional skirt in the centre(Fig. 1(d)). Foundations were studied with a range ofdiŠerent embedment ratios (DWB＝0.2, 0.3, 0.5, 0.75, 1.0and 1.2) to examine the eŠects of varying skirt length.For all analyses, the breadth, B＝10 m, but results arepresented normalised so the results are independent ofthe footing breadth chosen.

For all analyses, the ABAQUS 6.2 ˆnite element pack-age was used (HKS, 2001) running on a Unix mainframe.Because only the undrained ultimate limit state conditionwas being studied, the soil was modelled as a linearelastic-perfectly plastic material with an associatedTresca undrained shear strength (su) criterion. In theelastic regime, the soil had a Young's modulus (Eu)

deˆned to give a rigidity index, EuWsu＝400 and a Pois-son's ratio, nu＝0.49 (i.e., approaching undrained, con-stant volume conditions). Additional analyses varyingrigidity index conˆrmed that this did not aŠect thefoundation capacity for the range of problems studied.

The shear strength, su varied linearly with depth asshown in Fig. 2 and is given as:

su＝so＋kz (1)

where, so is the shear strength at the soil surface and k isthe strength gradient with depth, z. The undrained shearstrength at the level of the base of the foundationWskirttips is deˆned as suo (e.g., Tani and Craig, 1995).

Finite Element MeshFigure 3 shows the ˆnite element mesh for the skirted

foundation with DWB＝0.5 as a typical example. Themodelled soil zone is 3B deep and 10B wide and theboundary conditions allowed no vertical or lateral soilmovement at the soil base and no horizontal movement atthe vertical boundaries. The positions of these bounda-ries were considered suitably remote not to aŠect thebearing capacity of the foundations. The interactionbetween soil and footing was assumed to be rough withno detachment allowed. The foundation was modeled asa perfectly rigid body. Therefore, it is assumed that thereis no deformation within the foundation.

The element types used were 8-noded plane strainbiquadrilaterals with reduced integration (CPE-8R). Forfoundations with diŠerent DWB, the model height and thefoundation embedment depth were modiˆed appropriate-ly. The mesh for the skirted foundation with an internalskirt (Fig. 1(d)) required more elements.

Loading ConditionsBearing capacity was determined by using displace-

ment control to move the foundation vertically down-wards. This has been found to give more accurate resultsfor failure conditions than using load control (e.g.,Bransby and Randolph, 1997). The foundation transla-tion was performed in small displacement steps to ensurecalculation accuracy. The vertical load applied todiŠerent areas of the foundation (e.g., end-bearing, ex-ternal skirt shear, internal skirt shear) was calculated byexamining stresses in integration points in the `soil'immediately adjacent to the foundation and numericallyintegrating the results. Results using this method were

495

Fig. 3. Example ˆnite element mesh (skirted foundation with 5 m skirt length)

Fig. 4. Upper bound plasticity mechanism

495UNDRAINED VERTICAL BEARING CAPACITY

compared to applied loads during separate load con-trolled events and to nodal reaction forces.

UPPER BOUND PLASTICITY ANALYSIS

Upper bound plasticity calculations were carried out inorder to provide estimates of the failure load of rigidembedded and surface foundations for a range offoundation embedment ratios and diŠerent soil strengthproˆles. The general postulated collapse mechanism isshown in Fig. 4. It has three kinematic variables (a, b, e)and is of a similar form to that used by Kusakabe et al.(1986) for circular surface foundations when e＝459.Typical sub-mechanisms resulting from the genericmechanism (of which both the Prandtl and Hill mecha-nisms are members) are shown later in the results section.

Given the nature of the upper bound approach, thecalculated failure load will be greater or equal to the exactsolution for each foundation and soil condition (e.g.,Chen, 1975). Therefore, for each condition, the kinemat-ics of the mechanism was varied to ˆnd the mechanismgiving the minimum value of vertical load at failure. Thiswas done by setting up the work dissipation equations forthe failure mechanism in a Microsoft Excel spreadsheetand then using the Solver to minimize the load by varyingthe kinematic variables.

When calculating the energy dissipation mechanisms,care was taken to allow for the variation of undrainedshear strength with depth and an allowance for the

interface shear stress, t＝asu. For the particular casewhen e＝pW2, the resulting vertical shear plane along thefoundation sides was given the interface shear stress, asu

and the static triangular blocks at the edge of the founda-tion disappeared.

RESULTS: EMBEDDED FOUNDATION INUNIFORM SOIL

The capacity of a range of foundations with diŠerentembedment ratios, DWB from 0 to 1.2 in uniform strengthundrained soil were investigated by ˆnite element analysisand upper bound plasticity analysis. When DWBÀ0, threefooting geometries were analysed: (i) a rigid embeddedfooting (Fig. 1(b)); (ii) a skirted foundation (Fig. 1(c));and (iii) a skirted foundation with a central skirt(Fig. 1(d)). Unless otherwise stated, all the foundation-soil interfaces are fully rough.

Figure 5 shows the normalised load—displacementresults from the ˆnite element analyses of the skirted andembedded foundations in uniform soil. As the founda-tion is displaced vertically a distance, v into the soil, thenormalised load VWBsu, increases until a plateau at a peakload VoWBsu at failure. Both normalised peak load andthe distance required to mobilise it increase with increas-ing embedment depth as expected. Interestingly, both thebearing capacity and the mobilisation distances for theembedded and skirted foundations are almost identical.

The peak load calculated for each condition is shown inFig. 6 where the results are shown normalised by theundrained shear strength, su and the footing breadth, B.Again, there is very close agreement between the totalbearing capacity of the embedded, skirted and internalskirted foundations for a wide range of embedmentratios. This is examined further in Fig. 7 which shows thecalculated deformation mechanisms at failure for thediŠerent foundation geometries investigated for a typicalintermediate embedment ratio DWB＝0.5. There appearsto be little diŠerence in deformation mechanism (the soil

496

Fig. 5. Normalised load-displacement for skirted (and embedded)foundations in uniform strength soil (DWB＝0.2, 0.5, 1.0)

Fig. 6. Calculated bearing capacity for a range of embedment depths(ˆnite element analysis & UB): Uniform soil

(a) Upper bound-embedded (a＝609; b＝0; e＝609)Vo/WBsuo＝7.61

(b) FE-embedded foundationVo/WBsuo＝7.02

(c) FE: skirted foundationVo/WBsuo＝7.01

(d) FE: internal skirted foundationVo/WBsuo＝7.03

Fig. 7. Calculated failure mechanisms: UB, embedded, skirted and3-skirt (DWB＝0.5)

496 YUN AND BRANSBY

within the skirts is displacing as a rigid body) whichexplains why there is no diŠerence in failure load. Itappears that the vertical capacity of skirted foundationsin uniform soil can be calculated assuming that they areequivalent embedded rigid foundation.

The total capacity, Vo of a skirted foundation may beconsidered to be made up of two components analogousto the axial capacity of a pile: (i) the `end bearing', Vb and(ii) the external skin friction, Vs:

Vo＝Vb＋Vs (2)

This approach was also used by Byrne and Cassidy(2002) and House and Randolph (2001). It is standardpractice to assume that the terms are independent.

For skirted foundations with `trapped' soil within theskirts, the skin-friction term in uniform strength soil for astrip footing will depend on the soil strength, su, the inter-face roughness term, a and the embedment ratio, DWB:

Vs＝2aDsu (3a)

which normalizes to

Vs WBsu＝2a(DWB) (3b)

The skin friction term is thus very dependent on theinterface roughness term, a, which is di‹cult to calculate(e.g., Anderson and Jostad, 2004). It will become anincreasing proportion of the total foundation capacity asthe embedment ratio increases and hence is a key researchtopic for suction caisson foundations (e.g., Andersonand Jostad, 2004).

The `end-bearing' term for skirted foundations refers

to the load component derived from either, (i) the averagevertical pressure applied to the skirt tips, the inside skinfriction and the base of the raft, or (ii) the average verti-cal pressure applied to the soil between the skirts at skirttip level. The value of terms (i) and (ii) are equal. It can becalculated for the ˆnite element analyses (where a＝1)and the normalized end-bearing components are shownalongside the total foundation capacity in Fig. 6.

If the end bearing component can be considered to bethat of an equivalent embedded foundation as suggestedby the results above (and supported by Tani and Craig,1995; Bransby and Randolph, 1999b) then the endresistance follows standard bearing capacity formulaefor undrained soil (e.g., Terzhaghi, 1943):

VbWB＝Ncsu＋gD (4)

where Nc is a bearing capacity factor, which is expected todepend only on the embedment ratio, DWB for relativelyuniform soil deposits, and g is the soil unit weight.

For a skirted foundation, the net bearing capacity of

497

Fig. 8. Variation of end bearing capacity with embedment ratio andcalculated deformation mechanisms: Uniform soil

Fig. 9. Calculated variation of depth factor, dc, with embedmentdepth (ˆnite element analysis): Uniform soil

497UNDRAINED VERTICAL BEARING CAPACITY

the foundation per unit length is reduced by the weight ofsoil between the skirts: W＝gH(D－2t). Given that theskirt thickness, t9D, then the soil self-weight W§gDBand it is convenient to consider this a reduction of the endbearing resistance contribution. Hence, the net endbearing capacity of the foundation, Vb＝(Ncsu＋gD)B－W and so

VbWBsu＝Nc (5)

and the foundation capacity is independent of the unitweight of the soil. To model this condition in the FEanalyses, the foundation below the soil surface was givena unit weight equal to that of the soil.

Skempton (1951) gave solutions for the variation of Nc

which was related to the shape of footing (BWL) anddepth of the foundation (DWB). Whitlow (2001) provideda curve-ˆt to his recommendations which for the case of astrip footing simpliˆes to:

Nc＝5.14 «1＋ 0.0553 ØDB »$. (6)

The total bearing capacity can alternatively be writtenin terms of variations from the exact solution for thebearing capacity of a surface footing on uniform strengthsoil (e.g., Prandtl, 1921; Hill, 1950) by using a range ofmainly empirical factors to allow for changes in shape(sc), load inclination (ic) and embedment depth (dc) (e.g.,Meyerhof, 1953; Hansen, 1970). For the conditionsinvestigated here of an embedded strip foundation (ofneutral weight) resting on uniform strength undrainedsoil with vertical, central loading:

VbWB＝(2＋p)dcsu (7)

where the depth correction factor, dc is assumed to be afunction of embedment ratio (DWB) only.

Clearly, Eqs. (6) and (7) are equivalent when Nc＝(2＋p) dc and the preference of use is dependent only onthe format of design recommendations. Figure 8 plots theend bearing capacity results in terms of Nc and comparesthe results to those of Skempton (1951). Figure 9 showsthe same data in terms of depth correction factor, dc.

Previous researchers (e.g., Meyerhof, 1953; Hansen,1970) suggested that the depth correction factor, dc, maybe approximated as:

dc＝1＋nDB

(8)

where n is in the range 0.2 to 0.4. Clearly, Fig. 9 showsthat there is signiˆcant non-linearity of dc with respect toembedment depth, but the use of 0.2ÃnÃ0.4 (shown onFig. 9) brackets the general behaviour.

The upper bound results are also shown on Figs. 6 to 9.The upper bound calculations overestimate the ˆniteelement results as the embedment ratio increases witha 20z overestimation of end bearing factor Nc forDWB＝1.

Selected calculated optimum deformation mechanismsare shown on Fig. 7 (for the case of DWB＝0.5) and onFig. 8. A mechanism similar to the Prandtl (1943)

mechanisms prevails at low embedment ratios, and thewedge angle (a) increases slowly with increasing DWB.Note that if a single mechanism prevailed over a range ofembedment ratios, the end bearing capacity factor, Nc

would vary linearly with DWB and so the adaptingmechanisms cause the non-linearity with respect to DWB.

From the above results, it appears that the capacity ofskirted foundation in uniform undrained clay should becalculated by adding the end bearing and skin frictionterm (Eq. (2)). The skin friction should be calculated withEq. (3). The base resistance term is independent of soilunit weight (Eq. (5) or Eq. (7)) and the Skempton (1951)relationship will give adequate results for design.

EŠect of Skirt Interface PropertiesThe above approach is only truly valid if the end bear-

ing and skin friction terms can be decoupled. Thisassumption was investigated brie‰y by performing addi-tional upper bound analyses in which the skirt roughnesscondition a was varied (from a＝1 above to a＝0.5 and 0)and examining how this changed the end bearing founda-tion capacity.

Figure 8 shows that the end bearing capacity factor, Nc

for the analyses with smooth skirts (a＝0) is lower thanthe case with a＝1 for embedment ratio, DWBÀ0.3.When DWB＝1, the end-bearing factor, Nc is 10z lowerfor the fully smooth interface than for the fully rough

498

Fig. 10. End bearing capacity factor for surface strip foundation withvarying soil strength non-uniformity

(a) kBWsuo＝0 (a＝45o, b＝09, e＝459) [VoWBsuo＝5.14]

(b) kBWsuo＝2 (a＝149, b＝419, e＝459) [VoWBsuo＝8.07]

(c) kBWsuo＝5 (a＝0, b＝589, e＝459) [VoWBsuo＝10.71]

Fig. 11. Upper bound mechanisms at failure for diŠerent soil non-uniformity (kBWsuo)

498 YUN AND BRANSBY

condition. This is because the soil chooses a deformationmechanism (also shown in Fig. 8) which diŠers from thatshown for the fully rough condition and utilises thefrictionless shear zone on the side of the skirted founda-tion. Hence, the total bearing capacity (Vo) is reduced bymore than just 2asuD (Eq. (3)) which would be the case ifthe mechanism were as for the rough case (and as as-sumed in the above design approach).

However, results from a set of upper bound analyseswith a more typical interface roughness condition (a＝0.5) is also shown in Fig. 8. End bearing capacity factorsare aŠected signiˆcantly less with only a negligible 3zchange in end bearing capacity when DWB＝1. This sug-gests that this eŠect may not be important for typicalroughness conditions and so bearing capacity factorscalculated with rough skirts will be applicable for a rangeof roughness conditions and it is appropriate to decoupleVb and Vs.

RESULTS: SURFACE FOUNDATION ON NON-UNIFORM SOIL

The case of a strip surface footing resting on soil withundrained shear strength varying with depth has beenstudied by many researchers using plasticity approaches(e.g., Davis and Booker, 1973; Houslby and Wroth,1983; Martin, 2003a) and recently using ˆnite elementanalysis (e.g., Bransby, 2001; Gourvenec and Randolph,2003). The soil has shear strength, suo immediatelybeneath the footing and its strength increases with agradient, k with depth, z (Fig. 2). The non-uniformity ofthe soil strength is characterised by the dimensionlessgroup kBWsuo. The foundation capacity is expressed as

VoWB＝Ncsuo (9)

where Nc is a bearing capacity factor which is a functionof kBWsuo only.

Bearing capacity results from ˆnite element analysesand upper bound calculations are shown on Fig. 10. Alsoshown are lower bound results from Davis and Booker(1973), results using the lower bound Analysis of BearingCapacity program written by Martin (2003b) and ˆniteelement results from Gourvenec and Randolph (2003).The lower bound solutions are believed to be very close to

the exact solutions. The relatively simple upper boundmechanism used (Fig. 4) is able to provide bearing capaci-ty factors which are quite similar to the stress-ˆeld (lowerbound) solutions (e.g., 9z larger when kBWsuo＝5) andshow a similar trend. Optimum deformation mechanismsas calculated in the upper bound analysis for threediŠerent soil non-uniformities are shown in Fig. 11.There is a clear change in mechanism from that of thePrandtl mechanism for uniform soil conditions (with arelatively deep mechanism) to that of an Hill typemechanism for very non-uniform (kBWsuoÀ3) soil condi-tions (with a relatively shallow mechanism).

There are two approaches for the calculation of bear-ing capacity in non-uniform soil: (i) use the undrainedshear strength at some ˆxed depth below the skirt tips(together with Nc＝2＋p), or (ii) modify Nc to account forthe non-uniformity (using suo).

Davis and Booker (1973), Houlsby and Wroth (1983)and Martin (2003a) all gave recommendations for thevariation in Nc which vary negligibly for practical designpurposes. Bransby (2001) recommended a curve ˆt tosimplify design:

Nc＝(2＋p)＋1.646 ØkBsuo

»0.662

(10)

which is shown dotted on Fig. 10 and gives good (butslightly unconservative) agreement with results.

Martin (1994) suggested that it was common designpractice in the oŠshore industry for embedded spudcanfoundations that the bearing capacity should be calcu-lated using Skempton's (1951) formula to allow for theeŠect of embedment together with the average shearstrength in one radius below the footing to allow forshear strength variation with depth. Byrne and Cassidy(2002) suggested that this approach was also appropriatefor suction caissons and this gave reasonable agreementwith their centrifuge test results. For the case of linearlyincreasing soil strength, this is equivalent to using the

499

Fig. 12(a). Load-displacement response of skirted foundations innormally consolidated soil

Fig. 12(b). Normalised load-displacement response of skirted founda-tions in normally consolidated soil

Fig. 13(a). Variation of total bearing capacity with kBWsuo: embeddedfoundation in normally consolidated soil

Fig. 13(b). Variation of total bearing capacity with embedment ratio:Embedded foundations in normally consolidated soil

499UNDRAINED VERTICAL BEARING CAPACITY

undrained shear strength at a depth of BW4 below theskirt tips in design. This produces a modiˆed equivalentbearing capacity factor, Nc＝(1＋1W4kBWsuo).(2＋p) andthis curve is shown on Fig. 10. This approach is verysimilar to that recommended by Skempton (1951),who suggested that the average strength over a depth of2W3B beneath the foundation was used unless theshear strength varied signiˆcantly (i.e., kBWsuoÀ1.5).Skempton's (1951) recommendation involved using adepth BW3 below the foundation for the equivalentshear strength and thus Nc＝(1＋1W3kBWsuo).(2＋p) whichis also shown on Fig. 10.

The use of a ˆxed depth for the representative un-drained shear strength with a constant Nc will give alinearly varying bearing capacity factor with kBWsuo andso will only be able to encapsulate the bearing capacityvalues over a small range of kBWsuo. However, theSkempton approach appears suitable for relativelyuniform soil conditions (kBWsuoº1.5) while the Byrneand Cassidy (2002) method appears suitable for 1ºkBWsuoº3 (typical suction caisson conditions).

RESULTS: EMBEDDED FOUNDATION INNORMALLY CONSOLIDATED SOIL

The case of skirted and embedded strip foundationsembedded in heterogenous strength soil has received less

attention. Bransby and Randolph (1999b) performed alimited study for embedded foundations and Hu andRandolph (2002) analysed circular skirted foundations innormally consolidated soil with larger aspect ratios.Martin and Randolph (2001) provided upper and lowerbound solutions for embedded and surface circular foun-dations in non-uniform strength soil deposits. Similarly,circular suction caissons have been studied experimental-ly (e.g., Watson and Randolph, 1997; Byrne and Cassidy,2002; Cassidy et al., 2004) but there has been little investi-gation of the mechanisms occurring at failure and little ofstrip foundation behaviour apart from by Tani and Craig(1995).

Investigation in this paper is restricted to the case withzero undrained shear strength at the soil surface (so＝0)and a uniform increasing strength with depth (viz. su＝kz)as typical of the soil conditions of a normally consolidat-ed deposit. Figure 12(a) shows the load displacementresults of ˆnite element analyses for skirted foundationsfor a range of embedment ratios where the applied load(V) is normalised by the undrained shear strength at onebreadth below the soil surface (kB) multiplied by thefooting breadth (B) and the displacement (v) of nor-malised by foundation breadth. The required displace-ment to mobilise failure and the capacity both increasewith embedment ratio and peak loads are easy to identi-fy. The results are re-plotted with more conventional load

500

Fig. 14(a). End bearing capacity factor with depth: normally consoli-dated soil

Fig. 14(b). End bearing capacity factor with embedment ratio:normally consolidated soil

(a) upper bound embedded (a＝0, b＝54.99; e＝53.49)[VbWBsuo＝11.67]

(b) FE embedded [VbWBsuo＝11.56]

(c) FE skirted [VbWBsuo＝10.93]

(d) FE internal skirt [VbWBsuo＝10.90]

Fig. 15. Calculated failure mechanisms with DWB＝0.2 in normallyconsolidated soil

500 YUN AND BRANSBY

normalisation (VWBsuo; e.g., Tani and Craig, 1995) in Fig.12(b) and the peak capacities are plotted on Fig. 13. Toallow comparison with the equivalent surface foundationcase (Fig. 2), the normalised peak capacities are plottedagainst strength non-uniformity kBWsuo on Fig. 13(a). Forthe normally consolidated case here (so＝0), then kBWsuo

＝1W(DWB) and the results for the embedded case are alsoshown plotted against DWB in Fig. 13(b).

There appears to be small diŠerences between thebearing capacity of the diŠerent foundation types: thecapacity of the embedded foundations appear to beconsistently slightly higher than that of the skirted foun-dation geometry. The end bearing capacities are plottedin Fig. 14. Figures 15 and 16 show the calculated defor-mation mechanisms (and capacities) for foundations withDWB＝0.2 and 1 respectively. There appears to be negligi-ble diŠerence in the deformation mechanism for the casewith DWB＝1 for the three foundation types investigated(Fig. 16) and their capacities are almost equal. However,for the lowest aspect ratio suction caisson (DWB＝0.2)there appears to be soil deformation occurring within thesoil between the skirts in Fig. 15(c) (and 15(d)) explainingthe 5z reduction in calculated foundation end bearingcapacity.

Despite the subtle changes in mechanisms occurringbetween the foundations, the diŠerence in overall foun-dation capacity between the skirted foundation and the

equivalent embedded foundation (i.e., a surface founda-tion with the same kBWsuo as the embedded one; Fig. 2) isless than 5z even for the case with DWB＝0.2. Hence,design calculations for vertical capacity assuming anequivalent embedded foundation may be suitably ac-curate.

Figure 13(a) shows that the embedded foundationshave signiˆcantly more capacity than the equivalentsurface ones with a larger disparity for the deeper foun-dations. This is due both to the increasing skirt area(increasing Vs) and the embedded geometry which mayin‰uence the end-bearing factor, Nc. Figure 14(a) showsthat the end-bearing factor for the embedded foundationis consistently greater than for the surface foundation atthe same kBWsuo. The diŠerence increases for deeperfoundations (with smaller kBWsuo values) and reduces forshallower foundations (with greater kBWsuo). This dis-agrees with the results of Tani and Craig (1995) who statethat Vb is independent of embedment for strip footings.

The results from the upper bound plasticity analysesare also shown on Figs. 13 and 14. To calculate the end-bearing factor from the total capacity calculated by theupper bound analyses, the skirt friction Vs must be sub-tracted (Eq. (2)). For the general case for soil with su＝so

＋kz:

Vs

Bsuo＝2a Ø1－0.5

kBsuo

DB » D

B(11)

501

(a) upper bound embedded (a＝429, b＝59, e＝639) [VbWBsuo＝8.88]

(b) FE embedded [VbWBsuo＝8.58]

(c) FE skirted [VbWBsuo＝8.23]

(d) FE internal skirt [VbWBsuo＝8.21]

Fig. 16. Calculated failure mechanisms with DWB＝1 in normallyconsolidated soil

(a) Embedded (DWB＝2)(Upper bound: a＝0, b＝54.99; e＝53.49)

(b) Equivalent surface (kBWsuo＝5)(Upper bound: a＝0, b＝58.39; e＝459)

(c) Embedded (DWB＝1)(Upper bound: a＝429, b＝59; e＝639)

(d) Equivalent surface (kBWsuo＝1)(Upper bound: a＝27.29, b＝09; e＝459)

Fig. 17. Calculated deformation mechanisms at failure: FEA＋UB,normally consolidated soil

501UNDRAINED VERTICAL BEARING CAPACITY

which simpliˆes to

Vs

Bsuo＝a

DB

(12)

for the case for normally consolidated soil with kBWsuo＝BWD.

Qualitatively the results are as expected: the analysesgive slightly higher end bearing capacities than the ˆniteelement analyses particularly for higher DWB. The opti-mum upper bound mechanism changes as the embedmentratio and soil non-uniformity both changes. Figure 17shows the optimum upper bound soil deformationmechanisms found for the case of DWB＝0.2 (Fig. 17(a)and (b)) and 1.0 (Fig. 17(c) and (d)) for both the embed-ded and equivalent surface foundation. Note that theequivalent surface foundation (Fig. 2(b)) corresponds toa foundation where the soil above the base level of thefoundation is ignored in the analysis as suggested by Taniand Craig (1995). There are diŠerent mechanisms due tothe changing embedment ratio (compare Figs. 15(a) and

(c)), and due to the changing soil non-uniformity (com-pare Figs. 15(b) and (d)). The variation of bearingcapacity factor with DWB or kBWsuo shown in Fig. 14re‰ects these mechanism changes.

The results using Byrne and Cassidy's (2002) recom-mendations, using the strength at a depth BW4 beneaththe foundation to allow for the strength variation, andSkempton's (1951) relationship to allow for embedmentdepth, are shown in Fig. 14. The results show reasonableagreement with the other analyses except when DWBº0.3. The choice of the depth over which to average theundrained shear strength will promote errors due to thechanging mechanisms shown in Fig. 17. The requiredequivalent depth to give agreement with the other ana-lyses will reduce with embedment ratio as the soil non-uniformity increases. This explains why this approachgives an overestimate of Nc for large kBWsuo and anunderestimate for small non-uniformity (Fig. 14(a)).

In summary, the capacity of skirted foundations can beapproximated assuming that the foundation is anequivalent embedded foundation. Thus, Eq. (2) can beused with the skirt friction term calculated by Eq. (11).The end bearing term requires a consideration of the

502

(a) Cross-section of apparatus

(b) soil strength profile from T-bar tests

Fig. 18. Apparatus for centrifuge model testing

502 YUN AND BRANSBY

embedment andWor the non-uniformity (which are relatedfor normally consolidated soil). The approach suggestedby Byrne and Cassidy (2002) gives reasonable results if anappropriate soil strength is chosen. Alternatively, theresults given here can be used either in the form shown(e.g., Fig. 14(b)) or re-expressed in terms of a variation ofdepth correction factor, dc with DWB. The end-bearingcapacity factor, Nc＝[(2＋p)＋1.646(kBWsuo)0.662] dc and acurve ˆt to the calculated variation of dc with embedmentratio for the normally consolidated clay is given as:

dc＝1＋0.25 Ø DB »

0.4

(13)

EŠect of Skirt RoughnessThe upper bound approach allowed a brief investiga-

tion of the eŠect of skirt roughness on the end bearingfactor. The end bearing factor for a range of embedmentratios with a＝0, 0.5 and 1 are shown on Figs. 14(a) and(b). As for the embedded foundation in uniform soil, Nc

is independent of the skirt roughness for low embedmentratios (DWBº0.3 here). However, for greater embedmentthere is a small diŠerence in bearing capacity. For an em-bedment ratio, DWB＝1, Nc reduces by 6z for the fullysmooth condition. Given that the diŠerence is reduced toonly 1z for the a＝0.5 case for DWB＝1, this eŠect maybe considered negligible for design purposes and the Nc

values calculated assuming rough interfaces are appropri-ate.

CENTRIFUGE MODEL TESTS

Seven centrifuge model tests were carried out in twoseries of tests (denoted `A' and `B'). They were designedto provide a validation of the design approaches present-ed above. Skirted foundations of embedment ratios, DWB＝0.2, 0.5 and 1 were installed by jacking and loadedvertically to failure in normally consolidated clay. A briefoutline of the experimental methods and conditions ispresented below before the results are given and com-pared to the design method suggested from the analyticalwork.

Experimental Apparatus and ProcedureA cross-section of the experimental apparatus is shown

schematically in Fig. 18(a). The soil was contained withina strongbox of 770 mm length, 500 mm breadth and 550mm height. Both the front and the back faces aremanufactured from Perspex allowing visualizationthough the sides. The box was divided with a verticalPerspex-lined partition which split the box into twoidentical containers of 235 mm thickness (each with onePerspex face). This box allowed twice the number ofplane strain tests to be carried out on a single samplewithout disturbance between test sites.

The foundations were of breadth 100 mm and length234 mm (so as to give plane strain conditions) and hadaluminium skirts of either 20 mm, 50 mm or 100 mmlength and 2 mm thickness below an aluminium plate

of thickness 6 mm. They were connected to a load cellwhich was itself connected to a linear vertical actuator(Fig. 18(a)). This allowed installation and loading of thefoundation by displacement control whilst the loadapplied was measured with the load cell and the verticaldisplacement by two linearly variable diŠerential trans-formers (LVDT). The tests conditions for each loadingevent are given in Table 1.

The soil consisted of a normally consolidated claydeposit overlying a sand drainage layer. Speswhite kaolinclay was used because of its relatively high permeability,extensive use in laboratory tests (e.g., Almeida et al.,1986; Stewart, 1991; Martin, 1994) and well known soilproperties (e.g., Al-Tabaa, 1987; Stewart, 1991).

The clay slurry at a moisture content of 120z (approx-imately twice the liquid limit) was placed carefully intothe partitioned centrifuge strongbox on top of a 20 mmthick sand drainage layer to a depth of approximately 485mm. Base and top drainage were set up to maintain a

503

Table 1. Summary of centrifuge model test results

Testseries

TestNo.

SkirtlengthD, m

Displacementat failure vf

(m)

Normaliseddisplacement

at failurevfWD

Totalbearingcapacity(kNWm)

Skinfriction(kNWm)

Endbearing(kNWm)

A

A1 2 0.257 0.1285 303.8 3.3 300.5

A2 5 0.549 0.1098 633.8 14.5 619.3

A3 10 1.092 0.1092 1159.0 54.7 1104.3

B

B1 5 0.636 0.1272 661.9 18.3 643.6

B2 2 0.267 0.1335 320.9 3.5 317.4

B3 2 0.252 0.126 323.3 3.2 320.1

B4 2 0.267 0.1335 307.5 3.5 304.0

Fig. 19. Load-displacement response of foundations during penetra-tion and vertical loading (centrifuge model testing)

503UNDRAINED VERTICAL BEARING CAPACITY

water table height above the soil surface at all stages ofthe test. The strongbox was then placed on the centrifugegondola and accelerated to 100 g and left to consolidatefor more than 24 hours until the degree of consolidationreached more than 90z.

The foundation tests were carried out by displacing thefoundation vertically from a position where the skirt tipsdid not touch the soil, in one movement until the founda-tion was fully installed and the fully plastic load capacitywas exceeded. A typical displacement rate, v＝5 mmWmin(model) ensured that the soil remained undrained (vBWcv§17) during this process and there was no stop afterinstallation to allow any set-up time.

Soil Strength ProˆleT-bar tests (Stewart and Randolph, 1994) were used to

measure the undrained shear strength of the clay sample.The strength proˆle during each test was obtained frommeasurement of the resistance of the T-bar as it pene-trated the clay sample in ‰ight (Fig. 18(b)). Four T-bartests were carried out. These were carried out at thebeginning and end of each test series in locations remotefrom the foundation test sites. In test series A the T-barwas positioned below the soil surface at the start of thetest and so the undrained shear strengths at the start ofT-bar penetration should be discounted. The remainderof the results show an undrained shear strength proˆlethat can be approximated with suo＝0 and k＝1.1 kPaWm(Fig. 18(b)).

Test ResultsThe results of the installation and vertical bearing

capacity tests are shown in Fig. 19. Both loads anddisplacements are shown at prototype scale using conven-tional centrifuge scaling relationships (e.g., Schoˆeld,1980). An installation stage with low vertical loads asonly the skirts penetrate the soil is followed by theincreasing capacity as the base of the raft comes intocontact with the soil within the skirts and the foundationapproaches its plastic bearing capacity. On reaching aplastic failure condition, further penetration of thefoundation leads to plastic hardening as the foundationpenetrates into stronger soil and increases its embedment

ratio. Knowledge of the work hardening relationship isrequired for elasto-plastic combined loading foundationmodels (e.g., Tan, 1990; Martin, 1994). If there is nosigniˆcant work softening with increasing strain (e.g., asin the case of a normally consolidated clay) or signiˆcantsoil surface heave, the bearing capacity approaches dis-cussed previously may be able to characterize this plastichardening curve as well as the initial bearing capacity.

During installation, the low loads result from the endbearing resistance of the skirt tips and the increasing skinfriction acting on both sides of each skirt. For soil withsu＝kz, the force per unit width,

Vo＝(2t)Nckz＋2akz2 (14)

where z is the penetration depth at the skirt tips, t is theskirt thickness, Nc is the `deep' end bearing capacityfactor and a is the skirt adhesion factor. Good agreementbetween the calculated installation loads (Eq. (14)) andthe measured experimental results is obtained by usingNc＝9 (as recommended by Houlsby and Byrne, 2005)and a＝0.5 in Eq. (14). This suggests that adhesion fac-tor, a＝0.5 during installation. Because no set up timewas allowed in the centrifuge model tests before loadcapacity testing, this suggests that a＝0.5 should also beappropriate for calculation of the external skin frictionduring the foundation capacity events.

If the above information was not available, design

504

Fig. 20. Comparison between experimental and numerical calculationof bearing capacity: normally consolidated soil

504 YUN AND BRANSBY

practice (e.g., Anderson and Jostad, 2004) is to use a＝1WSt during installation, but a larger a value after set-uptime because of consolidation and thixotropic issues. Forkaolin, sensitivity, St＝2 to 2.5, again suggesting the useof a＝0.4 to 0.5 for the centrifuge bearing capacity testcalculations.

Comparison between Centrifuge and Analytical ResultsDesign calculations were carried out to ˆnd the predict-

ed foundation capacity using the methods presentedearlier and the results are presented in terms of variationof capacity with embedment depth on Fig. 19. Calcula-tions use Eq. (2) with the skirt friction found usingEq. (12) with a＝0.5. End bearing was calculated usingNc＝[(2＋p)＋1.646(kBWsuo)0.662] dc with dc deˆned as inEq. (13). The calculations use su＝1.1 z, assume fullyundrained conditions and the foundation breadth, B＝10m.

The results from the design calculations (plotted as adotted line in Fig. 19) are in generally close agreementwith the centrifuge data. For all foundation tests, thecalculations underestimate slightly the foundationcapacity, with the only signiˆcant under-prediction beingfor the foundation with the longest skirts (DWB＝1). Thetest result for DWB＝1 does not seem to follow the trendof the other model tests perhaps due to the soil's increas-ing undrained strength gradient with depth (Fig. 18(b)).This eŠect is not considered in the calculations and mayexplain why this foundation capacity is under-predicted.

The peak foundation capacity predicted using thedesign method, the ˆnite element results and the upperbound analyses is compared to the centrifuge results for arange of embedment ratios in Fig. 20. The foundationcapacity from the centrifuge results was the load magni-tude at the interception point of a linear plastic hardeningline (post-peak) and a linear `elastic' loading just beforeplastic failure is achieved as shown on Fig. 19. Thetriangular symbols show the measured centrifuge founda-tion capacities at the embedment ratio at which thefoundation capacity was achieved (with good agreement),whereas the circular symbols show the capacities for thenominal embedment ratios (deˆned by the skirt lengths).Because the foundation is penetrating the soil signiˆcant-ly during mobilisation of bearing capacity, the plasticfoundation `capacity' is larger than that would beachieved using an embedment depth equal to the skirtlength. However, this larger capacity comes at the cost ofa signiˆcant mobilisation distance that may exceedserviceability limits.

CONCLUSIONS

A series of ˆnite element and upper bound analyseshave been carried out to investigate the vertical capacityof skirted foundations in normally consolidated clay.Analyses were ˆrst carried out for the simpler case ofskirted and embedded foundation in uniform strengthsoil, and surface foundations on non-uniform soildeposits before the case of skirted and embedded founda-

tions in normally consolidated soil. Finally, the resultsfrom the design methods deduced from the analyses werecompared to the results of centrifuge model tests of skirt-ed foundations in normally consolidated clay.

Firstly, the analyses showed that in most cases thevertical capacity of skirted foundations in undrained soilcould be calculated as if the soil within the skirts was fullyrigid (i.e., as an equivalent embedded foundation).However, both the embedded geometry and the soilstrength variation eŠect must be considered in design.

Results showed that foundation capacity could becalculated by adding the end bearing resistance, Vb,calculated using the strength of soil at skirt tip level to theskirt friction resistance, Vs, as suggested by previousresearchers (e.g., Byrne and Cassidy, 2002; Houlsby andByrne, 2005). The skirt friction term is very dependent onthe choice of roughness coe‹cient, a used and this termbecomes more signiˆcant for foundations with longerskirts (such as suction anchors).

The end bearing resistance is conventionally calculatedusing Vb＝BNc suo, where suo is the undrained shearstrength at the skirt tip level or foundation base (e.g.,Tani and Craig, 1995). The end bearing factor, Nc is de-pendent on both the embedment ratio of the foundationand the soil non-uniformity and these were investigatedseparately. Skempton's (1951) relationship for the varia-tion of Nc or dc with DWB give good results for the casewith uniform soil.

As found previously (e.g., Davis and Booker, 1973) aprogressive increase in undrained shear strength withdepth beneath surface foundations increases the founda-tion capacity (and the required Nc) signiˆcantly. Goodagreement of ˆnite element and upper bound results withprevious researchers suggested that this case is well under-stood.

For the case of both foundation embedment and soilnon-uniformity (as in the case of a skirted foundation innormally consolidated soil), Skempton's method givesreasonable results if the undrained shear strength ata depth of BW4 (Byrne and Cassidy, 2002) or BW3(Skempton, 1951) beneath the skirt tips is used. Al-ternative design methods were suggested based upon theˆnite element and upper bound plasticity calculations. Aspart of this study, a brief investigation of the interactionbetween the end bearing capacity factors and the skirt

505505UNDRAINED VERTICAL BEARING CAPACITY

roughness suggested that although variation of skirtroughness will aŠect the end bearing capacity factor, thisis unlikely to be of signiˆcance in design for the typicalrange of skirt roughness.

Comparison of the results from the design methodsdeveloped using the numerical and analytical methodswith results from centrifuge model tests revealed goodagreement validating them. However, the bearing capaci-ty approaches are conservative because of the workhardening occurring while capacity is being mobilized.Whether these increased capacities can be used in designdepends on the serviceability criteria for the foundation.

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