RESEARCH ARTICLEReliability assessment of drag embedment anchors in sandand the effect of idealized anchor geometryAmin Aslkhalili 1 & Hodjat Shiri 1 & Sohrab Zendehboudi 2Received: 25 June 2019 /Revised: 21 September 2019 /Accepted: 25 September 2019# Springer Nature Switzerland AG 2019AbstractInthisstudy, the reliability ofdragembedment anchors inthe sandwas assessedand the effectofanchorgeometrical idealizationon reliability indices was investigated as an inherited characteristic of analytical approaches. The anchor holding capacity wasobtained by performing a series of iterative limit state analyses and a probabilistic model was developed for the selected anchorfamilies. The tensions of the mooring lines connected to a semisubmersible platform were obtained by performing a series oftime-domain dynamic mooring analyses using the OrcaFlex software. The uncertainties in environmental loads, metoceanvariables, and stress distribution along the catenary mooring lines were incorporated into the line tensions through the responsesurfaces. An iterative procedure was performed by adopting the first-order reliability method (FORM) to calculate the compar-ative failure probabilities in sand and clay. The study showed significant dependence of the anchoring system reliability ongeometrical configuration of anchors, the seabed soil properties, and the environmental loads. It was observed that the imple-mentation of the reliability-based design into the existing in-filed trial procedures could significantly improve the efficiency andcost-effectiveness of the design practice.Keywords Reliability analysis . Drag embedment anchor . Catenary mooring . Response surface . Numerical method . SandseabedNomenclatureA s area of shank.d nominal chain diameter.D pad-eye embedment depth.d f fluke thickness.d s average depth of the shank.du a the absolute displacement of the anchor.du s soil wedge displacement.du sa displacement of the soil relative to the anchor.d w wave direction.d wc current direction relative to wave.d ww wind direction relative to wave.E n normal circumference parameter.E n tangential circumference parameter.f form factor (Neubecker and Randolph, 1996a).F friction force.F f the fluke force.F fb the force on the back of the fluke.F s the shank force.h back edge of the fluke.H depth of fluke tips.H s significant wave height.L f fluke length.L f caisson length (Silva-González et al., 2013)L s shank length.N q standard bearing capacity factor.N qs shank bearing factor.p F probability of failure.p Fa annual probability of failure.q bearing pressure.Q normal soil reaction on chain segment.Q average bearing resistance per unit lengthof chain over embedment depth.R anchor capacity at mudline.* Hodjat Shirihshiri@mun.ca1Civil Engineering Department, Faculty of Engineering and AppliedScience, Memorial University of Newfoundland, A1B 3X5, St.John’s, NL, Canada2Process Engineering Department, Faculty of Engineering andApplied Science, Memorial University of Newfoundland, A1B 3X5,St. John’s, NL, CanadaSafety in Extreme Environmentshttps://doi.org/10.1007/s42797-019-00006-5
R soil reaction.R a anchor capacity at pad-eye.R d design anchor capacity at mudline.R d,a design resistances at the pad-eye.r i distance between point i and anchor shackle.s length of chain.SF side friction.T line tension.T a line tension at the pad-eye.T d design line tension at mudline.T d,a design tensions at the pad-eye.T dyn,max mean maximum dynamic line tension.T dyn,max-C characteristic mean maximumdynamic tension.T mean mean line tension.T mean-C characteristic mean line tension.T o Chain tension at mudline.T p spectral peak period.T* normalized tension.Δt extreme sea state durationU 10wind velocity.U c surface current velocity direction.w chain self-weight per unit length.W a anchor dry weight.W s the mobilized soil mass.x a anchor horizontal displacement.X absolute displacement of point i.x* horizontal distance normalised by DΔx absolute penetration incrementof the origin Yabsolute displacement of point iΔy absolute penetrationincrement of the origin zdepth below mudlinez* depth normalised by Dβ inclination of flukeβ reliability indexβ annual annual reliability index∅ ′ soil friction angle∅ p sand peak friction angleγ′ effective unit weight of soilγ dyn partial safety factor on dynamic line tensionγ mean partial safety factor on mean line tensionλ failure wedge angleλ mean annual rate of extreme sea statesη a anchor efficiencyμ chain-soil friction coefficientΘ vector of environmental variablesθ line tension angleθ a line tension angle at the pad-eyeθ i polar coordinate angle of point iθ fs fluke-shank angleθ o line tension angle at mudlineΔθ rotation increment of the originψ dilation angleIntroductionDrag embedment anchors are widely used as a cost-effectivesolution for temporary and permeant station keeping of float-ing structures. By growing offshore exploration and produc-tions, the number of incidents in floating facilities induced bythe failure of mooring system has been increased, subsequent-ly (Wang et al., 2010; Ma et al., 2013). This has caused theindustry to further emphasize on reliability assessment of themooringsystemsandtheirkeycomponentsinvarioustypesofseabed sediments. Drag embedment anchors are amongst thecrucial components of the mooring systems that are used withcatenary and taut leg mooring systems.Different anchoring solutions might be used to provide anefficient and reliable mooring system such as suction anchors,propellant embedded anchors, screw-in anchors, plate an-chors,deadweightanchors,pile anchors,anddragembedmentanchors. However, the latter one is one of the most attractiveoptions that are simple and cheap to install but challenging toevaluate the holding capacity (Neubecker and Randolph,1996a; Aubeny and Chi, 2010) due to complex and uncertaininteraction with the seabed. (see Fig. 1).There are several studies in the literature that have consid-ered the reliability assessment of various anchor families suchas suction anchors (Choi, 2007; Valle-molina et al., 2008;Clukey et al., 2013; Silva-González et al., 2013; Montes-Iturrizaga and Heredia-Zavoni, 2016; Rendón-Conde andHeredia-Zavoni, 2016). The high level of certainties in theevaluation of the holding capacity of suction anchors and thepromising results obtained in aforementioned reliability stud-ies has provided confidence about no need for field trials inassessment of holding capacities of these anchors. However,the situation is totally different in drag embedment anchors.Difficulties in collecting in-field holding capacity databases,the complicated interaction between the anchor and the sea-bed, the unknown ultimate depth and location of the anchor,andthe needfor the extensive amountofcostlycomputationalanalyses have resulted in limitations to assess the reliability ofthese important anchor families. Therefore, the current designpractice includes performing costly filed trials (e.g., API RP2SK, 2008). Moharrami and Shiri (2018) published the firststudy on the reliability of drag embedment anchors and initi-ated a reliability assessment approach that could potentiallyresult in elimination or mitigation of the filed trial expenses infuture.However,thestudywaslimitedtoclayandusedplasticyield loci to be obtained from a series of time-consumingfinite element analyses to characterize the fluke-soil interac-tionand failure states. Also, the authors did not investigatetheSaf. Extreme Environ.
impact of anchor geometrical idealization of the reliabilityindices, which is an inherited consequence of the numericaland analytical approaches.In this study, the reliability of the drag embedment anchorswas assessedinsandthathas not beeninvestigatedinthe past.A limit state approach, with no need to time-consuming finiteelement analysis was adopted to characterize the anchor fail-ure state.Theinfluenceofusingidealizedanchorgeometryonreliability indices was also examined and comparative studieswere conducted between the sand and clay to obtain the effectof different uncertainties in shear strength parameters on reli-ability indices.The holding capacity of anchors was calculated by devel-opinganExcel spreadsheetandincorporationofthe limitstateanalysis proposed by Neubecker and Randolph (1996a).There are several studies on the prediction of drag anchorscapacity by analytical and empirical solutions (Neubeckerand Randolph, 1996a; Thorne, 2002; O’Neill et al., 2003;Aubeny and Chi, 2010). However, the adopted solution(Neubecker and Randolph, 1996a) benefits from several ad-vantages such as simplified prediction of the anchor capacityand trajectory, incorporation of chain-sand interaction, andcomprehensive validation against the experimental studies(Neubecker and Randolph, 1996a; Neubecker andRandolph, 1996b; O’Neill et al., 1997). This model has beenwidely used in several studies in the literature (Neubecker andRandolph, 1996b; Neubecker and Randolph, 1996c; O’Neillet al., 2003) and recommended by design codes (e.g., API RP2SK, 2008). The mooring line tensions were obtained byperforming dynamic mooring analysis using OrcaFelx soft-ware and a generic semisubmersible platform. Reliability as-sessment was performed by using the first-order reliabilitymethod (FORM) through developing a probability model foranchor holding capacities.The study further prepared the ground for improvement ofthe anchor design codes (e.g., API RP 2SK, 2008), where theFig. 1 Configuration of dragembedment anchor and thecatenary mooring systemFig. 2 Force equilibrium of chainelementSaf. Extreme Environ.
effect of different reliability indices on proposing an optimizedfiled trials is currently neglected, and an identical holding capac-ity evaluation procedure is recommended in both sand and clay.MethodologyThe reliability analysis was conducted by calculation of the an-chor capacity against the mooring line tensions. The model pro-posed by Neubecker and Randolph (1996a) was used to analyzechain-soilandanchor-soilinteractionsinthesandandpredicttheanchor capacity at the mudline and shank pad-eye. The anchormodel was programmed in an Excel spreadsheet macro usingVisual Basic Application (VBA) (see Appendix 1 (Table 17)).OrcaFlex software package was employed to model a genericsemisubmersible platform in the Caspian Sea to obtain the char-acteristic mean and maximum dynamic line tensions for a100 years return period sea states. Various key parameters wereincorporated in the estimation of anchor capacities includingpeak friction at the seabed, dilation angle, soil density, flukeand shank bearing capacity factors, anchor geometrical configu-rations,linetensionangleatmudline,andsidefrictionfactor.Theresponsesurfaceswereusedtodeterminethemeanandexpectedmaximum dynamic line tensions. First-order reliability method(FORM) was used to assess the reliability of anchors connectedto the catenary mooring line. The DNV design code (DNV-RP-E301, 2012) was used to define the partial design factors on themean and maximum dynamic line tensions and capacities.Anchor-seabed interactionThe anchor system mobilizing the ultimate holding capacitycomprises of the anchor and the connected chain, both ofwhich were modeled in this study. The Stevpris MK5 andMK6 anchors were used, as the most popular choices in theindustry.Frictional capacity of chainThe frictional capacity between the chain and the soil cansignificantly contribute to the ultimate anchor capacity. Also,the angle between the anchor and the chain at the pad eye hasan important effectonthe soil-chaininteraction. Inthe presentstudy, a stud chain with a free body diagram of its differentialsegment shown in Fig. 2 was considered, and themethodology proposed by Neubecker and Randolph (1995)was adopted to implement the frictional chain capacity.TheparameterTisthelinetension;θistheinclinationfromthe horizontal; F is the friction force, and Q is the typical soilreaction on the chain segment.Fig. 3 The three-dimensional failure wedge in plan & side view and force system of the anchorSaf. Extreme Environ.
AccordingtoFig.2,thetangentialandnormalequilibriumscan be written as:dTdS¼ F þ WSinθ ð1ÞTdθdS¼ −Q þ WCosθ ð2ÞIt is possible to describe the normal (Q) and tangential (F)soil resistances acting on the chain as soil pressures:Q ¼ E n d ð Þq ð3ÞMK5MK6BHCAEFsandmudsandmiddlemudBCS SAH EFSFig. 4 Schematic of the modeled anchor in the present study (Vryhof Anchors, 2010)Table 1 Main dimensions for 12 t anchors (Vryhof Anchors, 2010)Dimension Mk5 (L f /d f =6.67) Mk6 (L f /d f =3.09)A (mm) 5908 5593B (mm) 6368 6171C (L f ), (mm) 3624 3961E (mm) 3010 2642F (d f ), (mm) 543 1282H (mm) 2460 2394S (mm) 150 140Fluke-shank angle(θ fs ), (°) 32.00 32Table 2 Properties of the modeled drag anchors and the correspondingresistance and line anglesAnchor type L f /d f L f (mm) d f (mm) R d,a (kN) θ a (°)Mk5 6.67 4297 644 2275 13.0Mk6 3.09 4534 1468 2267 12.9Saf. Extreme Environ.
F ¼ E t d ð Þf ð4Þwhere d is the nominal chain diameter, E n and E t are circum-ference parameters. In non-cohesive soils, the bearing pres-sure q can be expressed by:q ¼ N q γ0 zð5Þwhereqisbearingpressure;N q isthestandardbearingcapacityfactor; γ ′ is the effective unit weight of the soil; z is depth.These governing equilibrium equations are non-linear, whichmakes difficulties in finding the solution. Therefore, to sim-plify the equation, the chain was assumed to be weightless.Although, it is possible to account for the chain weight by asecondaryeffect,i.e.,reducingthe profile ofnormalresistanceperunitlengthbyanamountequaltothechainweightperunitlength. However, Neubecker and Randolph (1995) showedthat the contribution of the chain weight has a minor effecton ultimatecapacity. The governing equilibrium equations forweightless chain now become:dTdS¼ FTdθdS¼ −Q ð7Þwhere the relations ship between F and Q can be written as:F ¼ μQ ð8ÞFig. 5 Flow chart for analysis of anchor embedment historyFig. 6 Validation of theperformance of developed VBAmacroSaf. Extreme Environ.
where μ is the frictional coefficient which is between 0.4 and0.6. By substitution of the eqs. (6) and (7) into eq. (8), thegoverning formula can be obtained:dTdSþ μTdθdS¼ 0 ð9ÞEquation (9) can be written in the following form to givethe expression for the load development along the chain:T ¼ T a e μ θ a −θð Þð10ÞNow substituting eq. (10) into eq. (7) and considering thesmall values of θ leads to:T a2θ 2a −θ2? ? ≈∫ Dz Qdz ¼ D−zð ÞQ ð11Þwhere Q is the average bearing resistance (per unit length ofchain)overthe depthrange ofz toD.Equation(11) allowsthechange in chain angle to be estimated directly regarding thechain tension at the attachment point, T a and the average bear-ing resistance. Since the chain angle is close to the zero at theseabed, the eq. (11) can be simplified as below:T a θ 2a2¼ DQ ð12ÞCombining eq. (10) with eq.(12) results inanequationthatdescribes frictional development along the chain:T oT a¼ e μ θ að Þ¼ e μffiffiffiffiffiffiffiT * =2pð13Þwhere T o ischain tension atmudline;T * isnormalized tensionthat is given by:T * ¼T aDQð14ÞFig.7 Schematicplanviewofthemooring line arrangementTable 3 Soil and anchor input parameters in the current analysisParameter ValueAnchor dry weight, W a (kN) 98.06Fluke length, L f (m) 3.41Fluke width, b f (m) 5.99Fluke thickness, d f (m) 0.51Shank length, L s (m) 5.55Shank width, b s (m) 2.31Fluke-Shank angle, θ fs (°) 32Effective chain width, b c (m) 0.24Chain self-weight, w c (kN/m) 2Chain soil friction coefficient, μ 0.4Peak friction angle, ϕ p (°) 35Residual friction angle, ϕ r (°) 25Dilation angle, ψ (°) 8.5Effective unit weight, γ ′ (kN/m3) 10Saf. Extreme Environ.
Assuminga soillayer withbearingcapacityproportionaltodepth, for a surface chain angle equal to zero, Neubecker andRandolph (1995) proposed the following equation for chainprofile:z * ¼ e −x*ffiffiffiffiffiffiffi2=T *p ? ?¼ e −x* θ að15Þwhere z* and x* are depth and horizontal distance normalizedby D, respectively.Incorporating the anchor chain weight into the formulationto obtain a higher accuracy for general tension capacity, thefollowing formulation was obtained:T ¼ T a e μ θ a −θð Þþ μws ð16Þwherewischainself-weightperunitlength;andsisthelengthof chain.Anchor holding capacityIn the present study, the drag anchor was assumed tomove through the soil in a quasi-static condition.Although the anchor has some finite velocity, the mag-nitude of this velocity is small so that the inertial con-siderations can be neglected. To obtain the anchor hold-ing capacity, the limit state model proposed byNeubecker and Randolph (1996a) was adopted.Compared with the plastic loci approach adopted byMoharrami and Shiri (2018), incorporation of the limitstate approach eliminated the need for time-consumingfinite element analysis. Figure 3 shows the three-dimensional wedge failure mechanism for calculationof the anchor capacity at pad eye (T a ).Using the force equilibrium system shown in Fig. 3, thefirst step is to calculate the cross-section area of the wedge:A ¼H 2 −h 22tanβþH 2 tanλ2ð17Þwhere H is the depth of fluke tips; h is the back edge of thefluke; β is the inclination of the fluke, and λ is the failurewedge angle. The lateral extent of failure wedge can be cal-culated by:Table 4 Catenary mooringsystem characteristicH s (m) T P (s) U 10 (m/s) T mean-C (kN) T dyn,max-C (kN) T d (kN) θ o (°)9.5 12.8 29 846 623 2493 1.3Fig. 8 Histograms of simulated and fitted capacities at mudline, (a) absolute frequency, (b) Cumulative frequencySaf. Extreme Environ.
X ¼Htanψcos λ−ψ ð Þð18Þwhere ψ is the dilation angle. Now, the mobilized soil masscan be obtained based on the known values of X and A:W s ¼ γBA þ23γXA ð19Þwhere W s is the mobilized soil mass; B is the width of thefluke. The side friction (SF) should be determined to satisfylimit equilibrium formulation:SF ¼γL H þ h ð Þ 2 sin∅0 −sinψ? ?4cosψ 1−sin∅0 sinψ? ? ð20Þwhere ∅ ′ is the soil friction angle.Using the force equilibrium system shown in Fig. 3, theshank force could be driven from the standard bearing capac-ity as below:F s ¼ A s γd s N qs ð21Þwhere F s is the shank force; A s is the area of the shank; d s isthe average depth of the shank; and N qs is the bearing factorfortheshank.Therearestilltwounknownforcesactingonthesoil wedge, i.e., the fluke force (F f ) and soil reaction (R). Byconsidering horizontal and vertical force equilibrium, the un-known forces can be simply determined. Now, using the forceequilibrium of the anchor alone, the unknown forces in theback of the fluke (F fb ) and the chain tension (T a ) can be cal-culated based on horizontal and vertical force equilibrium.This procedure was iteratively continued with different valuesof the failure wedge angle (λ) to calculate the minimum upperbound estimate of the anchor holding capacity (T a ).The MK5 and MK6 anchors that were considered in thisstudy have a fluke length to fluke thickness ratios (L f /d f ) of6.67 and 3.09 respectively (see Fig. 4).ThegeometricalpropertiesoftheseanchorsareprovidedinTable 1.Table 2 shows the calculated values of the holding capac-ities or design resistances (R d,a ) and the corresponding linetension angles (θ a ) at the pad-eyes for the selected MK5 andMK6 anchors.The selection of these anchor families facilitated makingcomparisons between the current study and the results obtain-ed in earlier investigations in clay.Anchor kinematicsThe anchor trajectory is a key parameter that can be used forinterpretationofthe obtainedholdingcapacitiesandreliabilityindexes in the later stages. The solution proposed byNeubecker and Randolph (1996c) for prediction of the anchortrajectory was adopted, where three main conditions were setto ensure kinematic admissibility of the anchor model. Theseconditions put constraints on the absolute and relative dis-placements of the anchor and the soil wedge and hence arehelpful in defining the kinematics of the system. First,the soil wedge will move at the dilation angle to thefailure surface. Second, displacement of the soil relativeto the anchor (du sa ) must be parallel to the upper faceof the flukes. Third, the anchor must maintain contactwith the soil behind it by traveling in a direction paral-lel to the back of the fluke. The third condition applieswhen there is a force on the rear of the flukes so thatwhenthisforcebecomeszero,theanchorisfreetotravelawayfrom the soil behind it and this condition is meaningless.These three conditions for anchor and soil displacements fullydescribe the kinematics of the system so that for a given an-chor displacement the magnitudes and directions of the soildisplacement and the relative anchor-soil displacement can beeasily calculated. Figure 5 shows the main flowchart used forincorporation of the anchor kinematics.The minimum work approach was applied and the penetra-tion Δy and rotation Δθ were considered to obtain the incre-mental anchor displacements.Table 5 Statistical properties of anchor capacity at pad-eye and mudlineModel L f /d fL f (m) Padeye Mudline μ Ra /μ Rμ Ra (KN) σ Ra(KN)δ Ra m Ra (KN) μ R(KN)σ R(KN)δ R m Ra(KN)MK5 6.67 2.707 2283.1 506.6 0.222 2419.0 2650.5 618.8 0.233 2801.0 0.86MK5 6.67 3.166 3754.2 978.8 0.260 4001.0 4314.8 1174.1 0.272 4620.0 0.87MK5 6.67 3.410 4874.1 1273.2 0.261 5183.5 5590.7 1522.7 0.272 5970.0 0.87MK5 6.67 3.624 6093.6 1636.5 0.268 6506.0 6978.8 1949.2 0.279 7485.0 0.87MK6 3.09 2.958 2876.4 429.2 0.150 2917.0 3357.1 524.4 0.156 3411.5 0.86MK6 3.09 3.460 5149.2 885.5 0.172 5246.0 5983.4 1070.7 0.178 6095.0 0.86MK6 3.09 3.728 6702.5 1101.0 0.164 6822.2 7768.5 1327.9 0.170 7434.0 0.90MK6 3.09 3.961 8451.4 1505.3 0.178 8588.0 9779.6 1805.6 0.184 9958.5 0.86Saf. Extreme Environ.
Developing iterative macro for prediction of anchorperformanceThe staticlimit state and kinematic modelswerecoded intoanExcel spreadsheetusingVBA macrostocalculate the ultimateholding capacity of the anchor-chain system and the anchortrajectory. The developed spreadsheet performed a series ofiterative analyses with the calculation procedure outlined inFig. 5.The proper performance of developed Excel spreadsheetwas validated against the published experimental and analyt-ical studies (Neubecker and Randolph, 1996c), the sampledesigncodes(NCEL,1987),andthereferencedmanufacturersdatasheets (Vryhof Anchors, 2010) (see Fig. 6).The input parameters of the validation case study are givenin Table 3.Figure 6 shows a perfect agreement between the developedVBA macro and the results published by the developer of theoriginal limit state anchor solution.Time-domain mooring analysis of semisubmersibleplatformA generic semisubmersible platform located in the CaspianSea was considered with eight leg catenary spread mooringsystem for dynamic mooring analysis (see Fig. 7). Similarconfiguration with earlier studies (Moharrami and Shiri(2018)) was adopted to enable a comparison of the results.Each mooring line comprised of three different parts, i.e., theupper, middle and lower segments. The upper and lower seg-mentsweremadeofchain,whilethecentralsegmentwaswirerope. A water depth of 700 m was assumed and a finite ele-ment model was developed using OrcaFlex software to obtainthe dynamic line tensions at the touchdown points (TDP).Performing a three hours’ time-domain simulation, themost critically loaded line was detected for the environmentalloads with a 100 years return period (i.e., H s =9.5 m, T P =12.8 s, and U 10 =29 m/s). A similar head sea response ampli-tude operator (RAO) of the platform published by Moharramiand Shiri (2018) was adopted to facilitate comparison of theresults.The key outcome of dynamic mooring analysis is summa-rized in Table 4.The main output of the analysis includes the parameters T d(designlinetension),θ o (line angle atmudline),T mean-C (char-acteristic mean tension), and T dyn,max-C (characteristic meanmaximum dynamic tension) that will be used for reliabilityassessment in the next section.First-order reliability analysisFirst-order reliability method (FORM) was adopted throughan iterative procedure to obtain the probabilistic results byincorporation of uncertainties in seabed soil properties andenvironmental loads. The probabilistic modeling of anchorFig. 10 Response surfaces forT mean and T dyn, maxFig. 9 The mean and standarddeviation of anchor capacityversus fluke length; MK6Saf. Extreme Environ.
capacity was conducted by using the limit equilibrium meth-od. The embedment profile and the frictional capacity of thechain were also accounted for in the calculation of ultimateholding capacities. The response surface approach and appro-priate probability density functions were used to take intoconsideration the uncertainties of the environmental loadsand metocean variables including significant wave height,spectral peak period, wind velocity, and consequently thestress distribution throughout the catenary lines. A target fail-ure probability of 10E-5 was set assuming a consequenceclass of 2 as per recommendations made by DNV-RP-E301(2012). Further details are provided in the coming sections.Limit state functionIn order to establish the limit state function, care should betaken on considering the contribution of the frictional chaincapacity and its effect of the complexity of the reliability anal-ysis.Ifthe limit state functionisformulatedatthe pad-eye,thestatisticaldependencebetweentheappliedloadandthecapac-ity of the anchor must be determined, and the complexity ofthe reliability analysis will be significantly increased. On theother hand, the current study aims to focus on uncertaintiesexisted in the evaluation of anchor capacity rather than thechaincapacity.Therefore,analternativeapproachthathasalsobeen used by other researchers (Choi, 2007; Silva-González et al., 2013) was adopted to prevent unneces-sary complication in the reliability analysis. The limitstate function was formulated at mudline, but thechain-soil interaction impacts were considered in thecalculation of the ultimate holding capacity. This ap-proach facilitated the reliability analysis by keeping thevariables independence between the line tension and the ca-pacity of the anchor at the mudline. Therefore, the limit statefunction was written as follows (DNV-RP-E301, 2012):M ¼ R d −T d ð22Þwhere R d is the design anchor and chain system capacity atmudline.The design line tension at mudline (T d ) was defined as(DNV-RP-E301, 2012):T d ¼ T mean−C ? γ mean þ T dyn−C ? γ dyn ð23Þwhere T mean− C is the mean line tension due to pretension andmeanenvironmentalloads;T dyn− C isthedynamiclinetensiondue to low frequency and wave frequency motions; γ mean isthe partial safety factor for the mean line tension; and γ dyn-c isthe partial safety factor for the dynamic line tension. Thevalues of γ mean and γ dyn-c for consequence class 2 and thedynamic analysis were taken as 1.40 and 2.10, respectively(DNV-RP-E301, 2012). Both T mean-C and T dyn,max-C areexpressed at the mudline as functions of the significant waveheight (H s ), peak period (T p ), and wind velocity (U 10 )representing an extreme sea-state. Consequently, the limitstate function can be written as:M R;H s ;T p ;U 10? ?¼ R d −T mean−C ? γ mean −T dyn;max−C ? γ dynð24ÞThe anchor capacity and load tensions are evaluated in thedirection of the mooring line at the touchdown point, wherethe anchor line starts to embed (i.e., at an angle θ o with thehorizontal direction). The probability of failure P F during agiven extreme sea state was defined as:p F ¼ P M R;H s ;T p ;U 10? ? ≤0 ? ?ð25ÞByusingaPoissonmodelfortheoccurrenceofextremeseastates (Silva-González et al., 2013), the annual probability offailure P Fa was written as an exponential function of the prob-ability of failure P F :p Fa ¼ 1−exp −λp F ð Þ ð26Þwhere λistheratioofthenumberofextremesea statestotheirobservation period (in years); for small values of λ P F , theannual probability of failure is P Fa ≈ λ P F .Probabilistic modelling of anchor capacityThe crucial factors that were used to construct the anchorcapacities database were including the peak friction angle(ϕ p ), the dilation angle (ψ), and the soil density (γ′). Themean value of peak friction angle (μ ∅ p ) for lognormal distri-bution was set to 35° with a coefficient of variation (δ ∅ p )equal to 0.05 to take into consideration the uncertainty dueto systematic test variations and spatial variations of the soilTable 6 Distribution parameters of environmental variablesVariable Probability distribution Distribution parametersH s (m) Weibull Scale 9.5351Shape 10.1552T p (s) Lognormal μ lnTp 2.4966σ lnTp 0.1196U 10 (m/s) Lognormal μ lnU10 3.4827σ lnU10 0.1095Table 7 Estimated correlation coefficientsH s (m) T p (s) U 10 (m/s)H s (m) 1.0 0.9728 0.9905T p (s) 0.9728 1.0 0.9935U 10 (m/s) 0.9905 0.9935 1.0Saf. Extreme Environ.
properties (Basha and Babu, 2008; Anchor manual, 2010). Anormal distribution with a mean value (μ ψ ) of 8.49°and a coefficient variance (δ ψ ) of 0.28 was adoptedfor the sand dilation angle (ψ) that was calculated byusing Bolton’s empirical equation for sand (Bolton,1986; Phoon, 1999; Simoni and Houlsby, 2006). Thesoil density was represented by a normal distributionwith a mean value (μ γ 0 ) of 10.07 and a coefficientvariance (δ γ 0 ) of 0.02 (Neubecker, 1995; Phoon,1999; Simoni and Houlsby, 2006). To construct the ca-pacity database, 5000 simulations were conducted byadopting different values of ϕ p , ψ and γ′.Figure 8 shows the fitted distribution and the histo-grams of the anchor capacities at mudline for MK5 andMK6 anchors with L f equal to 3.624 (Left) and 3.961(Right).Table 5 shows the mean (μ), standard deviation (σ), medi-an value (m), and coefficient of variation (δ) of anchor capac-ities at pad-eye and mudline for MK5 (with fluke lengths of2.707, 3.166, 3.41 and 3.624 m) and MK6 (with fluke lengthsof 2.958, 3.46, 3.728 and 3.961 m).The variation of the mean and standard deviation of anchorcapacity versus the fluke length for the MK6 anchor family atpad-eye and mudline are illustrated in Fig. 9 to show thecapacity distribution.The mean capacity at mudline is 10 - 14% higher than themean capacity at pad-eye. Commonly in all anchor models,when the fluke length and fluke thickness increase, the differ-ences between capacity at the pad-eye and mudline increase.The same conclusion can be driven for differences betweenmedian capacities at the pad-eye and the mudline, but in someanchor models (MK6 with L f = 3.961 m) the differenceFig. 11 Annual reliability indexversus (a) fluke length, and (b)anchor weightSaf. Extreme Environ.
between median at the mudline and pad-eye decreases by anincrement of fluke length and thickness. The coefficients ofvariation of the capacity at pad-eye and mudline are about 23-27% for all MK5anchorfamiliesandare about 16-18%for allMK6 anchor families.Probabilistic modelling of line tensionThe responsesurfaceswere developed using an approach pro-posed by Silva-González et al. (2013), where aGaussian process was adopted to define the dynamicline tensions (Sarkar and Eatock Taylor, 2000; Choi,2007). The maximum expected dynamic line tensionduring the extreme sea state (presented by a randomvector of r uncertain environmental variables (Θ)) wasexpressed based on the model proposed by Davenport(1964):E T dyn;max? ?θ¼ μ Tdyn;max¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ln ν Θ Δt=2 ð Þpþ0:5772ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ln ν Θ Δt=2 ð Þp" #σ T;Θ ð27Þwhere Δt is the duration, ν Θ = ν(Θ) and σ T,Θ = σ(Θ) arethe mean crossing rate and the standard deviation of thedynamic line tension, respectively. A second order poly-nomial expansion was used to represent both the linetension T mean , and the predicted maximum dynamic linetension at mudline T dyn, max by using Θ:Y Θ ð Þ ¼ c þ a T Θ þ Θ T bΘ ð28ÞWhere Y(Θ) is the response of interest, and Θ is the r×1vector of environmental variables. The following unknowncoefficients c, a (r × 1) and b (r × r) were determined byFig. 12 The logarithm of failureprobability versus (a) flukelength, and (b) anchor weightSaf. Extreme Environ.
response analysis. To develop response surfaces, seven keyenvironmental parameters were investigated on the mooringsystem in the Sardar-e-Jangal gas field in the Caspian Sea. Adatabase of 8100 different combinations was built using di-vergent environmental variables such as significant waveheight (H s ), the direction ...