Bransby & ONeill: Drag anchor fluke-soil interaction in clays 1Drag anchor fluke-soil interaction in claysM.F. Bransby & M.P. ONeillCentre for Offshore Foundation Systems, The University of Western Australia, W.A., AustraliaABSTRACT: Finite element analysis of the fluke-soil interaction behaviour of drag anchors in undrained soilhas allowed calculation of plastic yield loci for characterisation of fluke failure states. The yield loci producedareexaminedintermsofsoildeformationmechanismsandkinematicsandareincorporatedintoanovelmethod for drag anchor design.1 INTRODUCTIONThemovementtooffshoredevelopmentsindeeperwaterhasledtoincreasingrelianceonoffshorestructureswhicharetetheredtotheseabedratherthanrestinguponit.Inaddition,theseareoftenre-quired to be anchored in soft, normally consolidatedsilts and clays.Onecommonseabedmooringsystemisthedraganchorandchainsystem(Figure1).Theanchorislowered to the seabed surface and then installed intothe seabed by dragging the chain laterally. Due to theshapeoftheanchor,kinematicsgovernsothatitmustembedduringdisplacementandhencesignifi-cantholdingcapacitymaybeachieved.Mosthold-ing capacity is achieved by soil resistance against thefluke, but the kinematics are controlled by the lengthoftheshankandtheorientationoftheshankwithrespect to the fluke.The use of drag anchors in offshore mooring sys-tems requires knowledge about the anchors holdingcapacity, embedment depth and drag length requiredfor mobilisationoftheworkingload.Theseallvaryfor different soil conditions and drag anchor designs.Historically, drag anchor design has been empiri-cally based (Vryhof Anchors, 1990). More recently,Figure 1. Drag anchor and chain systemapproaches have been introduced which seek to pre-dicttheentiredragtrajectoryofananchorfromin-stallationtomobilisationofworkingloadcapacity(NeubeckerandRandolph,1996;ONeillandRandolph,1997;Thorne,1998).Theseapproachesarebasedoncombinedkinematicandequilibriumanalyses. They use basic geomechanics principles tofindthegeotechnicalresistiveloadsactingontheanchorintheequilibriumsolutionandsimpleas-sumptionsabouttheanchormovementinthekine-matic solution.Theabovedesignapproachesuseempiricismforsolution(forexample,theanchorformfactorf,NeubeckerandRandolph,1996).Thereappearstobe room for non-empirical prediction methods usingbasic soil mechanics.When drag anchor holding capacity is approachedinsoftundrainedsoils,soilfailureconsistsoflocalplasticflowaroundtheflukeandshank.Thus,thebehaviourwillbeindependentoftheorientationoftheanchorwithrespecttothesoilsurfaceandsoanalysisofthesoilaroundtheflukewillleadtoin-sight about the general anchor behaviour.Thispaperpresentsanalysisofthesoilaroundanchorflukesusingfiniteelementanalysis.Theworkallowsinvestigationoftheanchorundergen-eralloadingconditionsandthebehaviourhasbeencharacterisedintermsofaplasticyieldenvelope.The findings can be used to validate previous designmethods(e.g.NeubeckerandRandolph,1996)oralternativelywillleadtonovelnumericaldesignmethodsusingtheyieldenvelopeapproachwhichare introduced briefly.2PLASTICITYCONCEPTSANDTHEYIELDLOCUSRecently,theanalysisofoffshorefoundationssuchBransby & ONeill: Drag anchor fluke-soil interaction in clays 2as spudcan and shallow foundations have used plas-ticityconceptstoexpressthebearingcapacityoffoundationsundercombinedvertical(V),horizontal(H)andmoment(M)loading(Murff,1994;Tan,1990; Martin, 1994; Bransby and Randolph, 1998).Aplasticyieldlocusisintroducedwhichex-pressesthecombinationofV-H-Mloadsthatresultin foundation failure.Thiscanbegivenasamathe-matical expression, f where f(V,H,M) = 0 at yield, orshown graphically (Figure 2). Not only may the yieldlocusbeusedtocalculatefootingcapacityundercombinationsofloads,butmayalsoformaplasticpotential (g(V,H,M)) for description of plastic verti-cal,horizontalandrotationaldisplacementsofthefootingatfailure(Figure2).Indeed,plasticitytheo-rems(e.g.Chen,1975)showthattheconditionofnormality or associated flow will exist for undrainedfailureconditionswhenthesoilalsoremainsat-tached to the footings.Consideradraganchordeepinundrainedsoil(Figure3).Loadsexertedontheanchorcanbeex-pressedintermsofforcesparallel(H)andperpen-dicular (V) to the fluke and moment load (M) aboutanyoneparticularreferencepointonthefluke.Combinationsoftheseloads(H,VandM)willcausefailureoftheanchorwithconsequentanchormovementsparallelandperpendiculartotheflukeand rotationally about the same reference point.Becausetheanchorisdeep,soilfailurewillbefullyconstrainedandlocaltotheanchorwhateverthe direction of the load. Thus, the failure loads H, VandMwillbeindependentofanchororientation,andsothelocalloadanddisplacementdefinitionsintroduced in Figure 3maybeappropriategenerallyforallfailureconditions.Inaddition,thedeepcon-ditionensuresthatthereisnosoil-anchordetach-ment.Thus,plasticdisplacementswillbegovernedby normality to the failure yield locus, f(H,V,M) andso determination of the failure locus also allows pre-diction of anchor displacement directions at failure.Anchor failure loads (H, V and M) will be a resultofverycomplicatedthree-dimensionalsoildis-placements around a complex drag anchor geometry.To examine this in detail, full 3D analysis would berequired and the detailed 3D geometry of the anchorwould have to be incorporated. This would be extre-Figure 2. The yield locus and plastic potential function.Figure3.Loadsanddisplacementsatfailureforasimplifieddrag anchor.mely time consuming. It is believed that much of thecomplexmechanisticbehaviourmaybeunderstoodbyexaminingasinglepartofthedraganchorsys-tem:thefluke.Thiswillprovidealargeproportionoftheholdingcapacityoftheanchorandgovernmuch of the anchor kinematics.Thefluke-soilinteractionbehaviourisexaminedusingthefiniteelementanalysisbelowandthere-sultsareexpressedasplasticyieldloci.Methodsofutilisingtheyieldlocusinanchordesignareintro-duced and discussed in following sections.3 FINITE ELEMENT ANALYSISThe aim of the finite element analyses was to exam-inesoil-flukeinteractionbehaviourduringundrained,deepfailureconditions.Inparticular,theaimwastodeducetheshapeoftheH,V,Myieldloci for flukes of different shapes.Theflukeshapewasidealisedasbeinginfinitelywide to allow planestrainanalysis.Althoughthisisnotnecessarilyagoodapproximationtoreality,itwillallowdetailedanalysisofasimplergeometrywhichmayenablebetterunderstandingofthegen-eral soil-fluke interaction problem.Two simple fluke geometries were investigated: arectangularflukeandaneccentricwedge.Dimen-sionsweredefinedasshowninFigure4.Forbothflukes,alengthoverdepthratioL/d=7wasadopted,asthisissimilartothatofaVryhofStevpris anchor fluke (Vryhof, 1990).The finite element package CRISP, CRItical StateProgram(BrittoandGunn,1987)wasused.Intheanalyses,approximately80015-nodedtriangularelements with 27 degrees of freedom were employed(cubicstraintriangles).Trescaelastic-plasticsoilwasusedwithG/su=500whereG istheshearFigure 4. Simplified fluke geometry and definitions.Bransby & ONeill: Drag anchor fluke-soil interaction in clays 3modulusandsuistheundrainedshearstrength.Thesoil was modelled as undrained by use of total stressanalysis together with a Poissons ratio, = 0.49.Displacementcontrolledanalysesaremostsuit-ablefordeterminationoftheyieldlocus(Bransbyand Randolph, 1997). For these present analyses, thefootingwasdisplacedslowlyinacertaindisplace-mentdirection(i.e.v/,h/ constant)untilaplastic failure load was reached. The full yield locuswascharacterisedbythefinalloadpointsaftercar-ryingoutarangeofthesedisplacementprobeswithdifferent v/ and h/.3.1 Rectangular flukeResultsfromaseriesofdisplacementprobesundervariouscombinationsoftranslationparallel(h)andnormal (v) to the fluke direction are shown in Figure5.Bothparallelandnormalloadisnormalisedbyfluke length and undrained shear strength so that theyieldlocusispresentedindimensionlessH/(Lsu)V/(Lsu)space.Aconvexyieldlocusresultswithmaximum horizontal load, Hmax/(Lsu) = 4.29 (V andM = 0) and Vmax/(Lsu) = 11.87 (H and M = 0).The peak loads Hmax and Vmax can be compared tosolutions with upper bound calculation. Figures 6(a)and6(b)showkinematicmechanismsfortheupperboundplasticitycalculationofVmaxandHmaxre-spectively. The upper bound calculations give:

,_

+ + ,_

+ cos21Ld42tan4LsVumax(1)

,_

+ + ,_

+ cos2142tanLd4LsHumax(2)where is defined as in Figures 6(a) and 6(b) and isvaried to minimise Hmax or Vmax.ThepredictionsofVmaxandHmaxareshownonFigure 5 for the rectangular footing. Good agreement024681012140 1 2 3 4 5 6H / (L su)V / (L su)Upper bound: V / (L su) = 12.10Curve fit (M = 0)Upper bound: H / (L su) = 5.15Vvh HVmaxHmaxFigure 5. Rectangular fluke yield locus in V-H space.Figure 6. Upper bound mechanisms for calculation of Hmax andVmax.isobtainedbetweentheFEandupperboundcalcu-lation of Vmax. However, less good agreement is ob-tained in prediction of Hmax.Examination of soil deformation mechanisms cal-culated at the horizontal and vertical capacities in thefiniteelementanalysisrevealwhyVmaxiswellpre-dicted by upper bound analysis, but Hmax is not (Fig-ure7).ThesoildeformationmechanismintheFEcalculation at Vmax is very similar to that of the upperboundsolution,butthereisasignificantdifferencebetweentheFEcalculatedmechanismandupperboundmechanismforpureparallel(h)movement.FurtherrefinementofthemethodsforHmaxcalcula-tion is required.TheyieldlocuscanbededucedinfullH,VandM space when rotational displacements are includedwithtranslationinthedisplacementprobes(Figure8). The yield locus is symmetrical about the axes andis convex.Figure 7. FE calculated soil displacements at Hmax and Vmax.Bransby & ONeill: Drag anchor fluke-soil interaction in clays 400.40.81.21.60 3 6 9 12V / (L su)Curve fit (H = 0)0369120 1 2 3 4 5V / (L su)Curve fit (M = 0)00.40.81.21.60 1 2 3 4 5H / (L su)M / (L2 su)Curve fit (V = 0)Figure 8. H-V-M yield locus for rectangular fluke.The moment capacity with V, H = 0 was found tobeMmax/(L2su)=1.49.Thiscanbecomparedtoanupperboundsolutionusingarotationalscoopmechanism(Figure9)ofMmax/(L2su)=1.60(ThegeneralsolutionisMmax/(L2su)=/2(1+(d/L)2).ThereisgoodagreementbetweentheFEandupperbound solutionsbecauseofthesimilarityofthesoildisplacement mechanism (Figure 9).3.2 Wedge shape flukeThefullyieldlocusinH-V-Mspacewasobtainedfortheeccentricwedgeshapedfluke(Figure10).Unlikefortherectangularfluke,theyieldlocusisnotsymmetric.Forexample,underpureHloading,theresultantfailuresoildisplacementswillgivepositivehandpositivev;theanchorwillmoveup-wardstotheright,partlyfollowingthebottomsur-faceofthewedge.Thus,duetonormality,themaximum H load will be sustained with negative V.Themaximumhorizontalloadisalsosustainedwithasmallnegativemoment.Adetailoftheyieldlocus in H-M space is shown on Figure 11. The shiftFigure 9. Upper boundmechanismforcalculationofMmaxandFE calculated soil displacements at Mmax.-0.600.61.21.80 1.2 2.4 3.6H / (L su)M / (L2 su)Curve fit (V = V1)-0.600.61.21.8-3 0 3 6 9 12V / (L su)Curve fit (H = H1)-30369120 1.2 2.4 3.6V / (L su)Curve fit (M = M1)Figure 10. H-V-M yield locus for wedge fluke.ofthepeakHpositionisduetothechanged(andasymmetric) kinematics of the wedge fluke.3.3 Curve fittingThe yield loci are shown in full H-V-M space for thetwo fluke shapes in Figures 8 and 10. It is more use-fultoexpresstheyieldlociasseparateyieldfunc-tions. Preliminary curve fits are suggested as an off-set form of the Murff (1994) equation:p1n1 max1m1 max1q1 max1H HH HM MM M1V VV Vf11]1

,_

+

,_

+

,_

(3)wheretheexponents,q,m,nandparechosento-getherwiththeoffsetsV1,M1andH1aftertheFEanalyses. The parameters used in the curve fitting areshown in Table 1, and the fitted curves are shown inFigures 8 and 10.Both equationsgive reasonably good curve fits to-0.600.61.21.80 0.6 1.2 1.8 2.4 3 3.6H / (L su)M / (L2 su)HhMCurve fit (V = V1)h/LM = 0H1, MmaxHmax, M1 ( = 0)Figure 11. H-M yield locus for wedge fluke.Bransby & ONeill: Drag anchor fluke-soil interaction in clays 5Table 1. Yield locus curve fitting parameters.Parameter Rectangular fluke Wedge flukeHmax/(L su) 4.29 3.34Vmax/(L su) 11.87 11.53Mmax/(L2 su) 1.49 1.60H1/(L su) 0 0V1/(L su) 0 -1.25M1/(L2 su) 0 -0.57m 1.26 2.37n 3.72 2.14p 1.09 0.93q 3.16 3.41thefullyieldlocus.However,forthepurposesofanchordesignitistheloadconditionsclosetothepeakHloadwhicharemostimportant.Morecom-plexcurvefittingtoallowforthedetailedshapeofthe yield locus in this region may be required.4 KINEMATIC ANCHOR ANALYSISTheyieldlocicurvefitsproducedbythefiniteele-mentanalysesforthefluke-soilinteractioncanbeusedinthekinematicanalysisofdraganchors.TheapproachissimilartothatbyNeubeckerandRandolph (1996), but plasticity concepts are used todeterminetheforceontheflukeintheequilibriumsolutionandtheassociatedplasticanchordisplace-ments are used to determine the anchor kinematics.First,theanchoriswishedintoplacenearthesoil surface. Next,equilibriumoftheanchoriscon-sideredwhichrequiresfindingtheflukeloadthatwill cause yield. The plastic normal at this load pointwillthendictatetheplasticdisplacementofthefluke.Thisgovernsthedirectionofthenextanchormovement and the anchor is stepped to the nextpo-sitionwheretheprocedureisrepeated.Hence,theentireembedmenttrajectoryoftheanchorcanbecalculatedtogetherwiththechainforceateachin-stant. This is explained in slightly more detail below.4.1 Geometrical simplificationThesoilstratigraphyisconsideredasasinglelayerofsoil withundrained shearstrength, suo at the soilFigure 12. Simplification of the anchor-chain geometry.surface and a shear strength gradient k.The drag anchor is simplified as shown on Figure12. The fluke is wide enough such that it can be ide-alised as being plane strain. The shank is treated as aplateoflengthslandofwidthsw.Thechainofdi-ameterbisconnectedattheanchorpadeye.Thefluke-shank angle, fs, is also defined in Figure 12.At any instant, the padeye is at a depth, zpbelowthesoilsurfaceandthetopfaceoftheflukeisin-clinedatangletothehorizontal.Thechainisin-clined at an angle a to the horizontal at the padeye,whereaisgovernedbythepadeyedepth,zp,thechain width, b, the force on the chain, Ta and the lat-eralcapacityfactorofthechain,Ncusingtheequa-tion of Neubecker and Randolph (1996):( )21a2p p uo caTkz 5 . 0 z s bN 211]1

+ (4)4.2 Equilibrium solutionThe forces acting on the anchor are shown on Figure12.Forthepurposeofthisanalysis,theanchorisassumedtobeweightless.Theshankdragforce,Hsactsatthemidpointoftheshankandparalleltothedirection of fluke travel. It is calculated as the prod-uctoftheprojectedshankareaperpendiculartothedirection of travel, a bearing capacity factor, Nc = 9,andtheundrainedshearstrengthattheshankmid-point. The unknown chain force, Ta is resisted by theshankdragforce,Hsandtheflukedrag(H,V,M).Giventhatf(H,V,M)=0astheflukeisatfailure,there is enough information to calculate all the forcecomponents.4.3 Normality displacementsIfitisassumedthatelasticdisplacementsarenegli-gibleincomparisontoplasticdisplacements,theflukemovementdirectionwillbegivenbythenor-mal to the plastic yield locus at the load state calcu-latedbytheequilibriumsolution.Thus,/(h/L)=(f/(M/L))/(f/H) and v/h = (f/V)/(f/H).Assuming that the anchor fluke moves a distancex in the direction parallel to the top fluke face in adisplacementincrement,thenh=x,v=((f/V)/(f/H))xand=(f/(M/L))/(f/H)x/L.4.4 Re-calculation of anchor positionTheanchorpositionisrecalculatedafterchoiceofxusingthedisplacementequationsabove.Thechainangleisrecalculatedforthenextequilibriumstep and the direction of the last anchor displacementisrecordedsothatthedirectionoftheshankdragforce,Hs is known.The procedure isrepeated fromBransby & ONeill: Drag anchor fluke-soil interaction in clays 602468100 100 200 300 400Draglength (m)Padeye load (MN)01020304050Padeye depth (m)Padeye loadPadeye depthFigure 13. Results from kinematic drag anchor analysis.theequilibriumstepuntilthechainforce,Tabe-comes constant and the trajectory of the anchor flat-tens.4.5 Typical resultsA typical drag trajectory is shown in Figure 13 for ananchorinanundrained,normallyconsolidatedclaywithsuo=0kPaandk=1kPa/m.Theanchorhasdimensionssimilartothoseofa32tonneVryhofStevprisanchorwitha50fluke-shankangle(Vry-hof,1990).Figure13alsoshowstheholdingforceon the chain, Ta as the drag progresses.Boththedragtrajectoryandholdingcapacityre-sultsaresimilartothoseseeninmodelanchortests(ONeillandRandolph,1997)withagradualin-creaseinholdingcapacityastheembedmentdepthincreases.Afinalholdingcapacityof8.7MNequates to an anchor form factor f of 1.7. This com-parestof=1.4derivedexperimentally,suggestingthattheyieldlocusapproachtotheanalysisofan-chors in clays is promising.5 CONCLUSIONSLocalsoilfailurearoundtheflukeofdraganchorshasbeenanalysedusingfiniteelementanalysistoinvestigate how drag anchor capacity and kinematicsareaffectedbythefluke.Fluke-soilbehaviouratfailurehasbeencharacterisedwithaplasticyieldenvelope in terms of loads parallel and perpendiculartotheanchorflukeandmomentload,usingtheframeworkemployedrecentlyforshallowfounda-tion analysis.Yieldenvelopesarepresentedforidealplanestrainflukesofvariousshapesandupperboundre-sultsarepresentedtoverifythesefindings.Thevarying yield envelope shapes are seen to be the con-sequenceofdifferentsoildeformationmechanismsaroundtheanchorgovernedbytheanchorflukege-ometry.However,thesecanbeexpressedsimplyintermsofcurvefitsinH-V-Mloadspace,andtheseyieldlociwillalsoformplasticpotentialsallowingcalculation of fluke displacement direction at failure.Theworkhasleadtotheintroductionofanewnumericaldesignmethodforpredictingdraganchortrajectoryandholdingcapacityusingtheyielden-velopeapproach.Despitethesimplificationsinher-entintheanalysisandthesimplifiedformofthefluke-soilyieldlocuscurve-fitused,theanalysismethodgaveresultssimilartothoseseeninmodeltestsofanchorsinsoftclays.Thissuggeststhatthemethodhaspromisewithfurtherrefinementoftheyieldenvelopeandbetterapproximationtothean-chor geometry.ACKNOWLEDGEMENTSTheworkdescribedinthispaperformspartoftheactivitiesoftheSpecialResearchCentreforOff-shoreFoundationSystems,establishedandsup-portedundertheAustralianResearchCouncilsRe-searchCentresProgram.SpecialthanksareduetoMr.NicholasSpadacciniforhiscontributiontothework presented in this paper.REFERENCESBransby,M.F.,Randolph,M.F.,1997.Shallowfoundationssubject to combined loadings, Proc. of the 9th Int. Conf. onComputerMethodsandAdvancesinGeomechanics,Wuhan, Vol. 3, pp. 1947-1952.Bransby,M.F.,Randolph,M.F.,1998.Combinedloadingofskirted foundations, Geotechnique, Vol. 48, No. 5, pp. 637-655.Britto,A.,Gunn,M.,1987.Criticalstatesoilmechanicsviafinite element analysis, Chichester, Ellis Horwood Ltd.Chen, W.F., 1975. LimitAnalysisandSoilPlasticity,Elsevier,New York.Martin,C.M.,1994.Physicalandnumericalmodellingofoff-shore foundations under combined loads, PhD Thesis, Uni-versity of Oxford.Murff,J.D.,1994.Limitanalysisofmulti-footingfoundationsystems,Proc.ofthe8thInt.Conf.onComputerMethodsandAdvancesinGeomechanics,Morgantown,Vol.1,pp.223-244.Neubecker,S.R.,Randolph,M.F.,1996.Theperformanceofdraganchorandchainsystemsincohesivesoil,MarineGeoresources and Geotechnology, 14: 77-96.ONeill,M.P.,Randolph,M.F.,Neubecker,S.R.,1997.Anovelprocedurefortestingmodeldraganchors,Proc.7thInt.OffshoreandPolarEng.Conf.,ISOPE-97,Honolulu,Hawaii, 939-945.Tan, F., 1990. Centrifuge and theoretical modellingofconicalfootingsonsand,PhDThesis,TheUniversityofCam-bridge.Thorne,C.P,1998.Penetrationandloadcapacityofmarinedrag anchors in soft clay, Journal of Geotech. and Geoenv.Eng., Vol. 124, No. 10, pp 945-953.VryhofAnchors,1990.AnchorManual,KrimpenadYssel,The Netherlands.