Dynamic Response of Offshore Articulated Tower-Under Airy and Stokes Theories

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Dynamic Response of Offshore Articulated Tower-Under Airy and Stokes Theories

1Department of Civil Engineering, Aligarh Muslim University, Aligarh ***

Abstract Articulated tower is a compliant offshore structure that oscillates freely with environmental loads of windandwave.Presentstudydealswiththedynamicresponse ofasinglehingeArticulatedtower.Hydrodynamicloadingof thetoweriscomputedusing Morrisonsequation.Acceleration andvelocitytermsusedinMorrisonequationiscomputedfirst byAirytheoryandthenbyStokestheory.Variousresponsesof theArticulatedtowerarestudiedandacomparativeanalysis isdonesoas toseewhichofAirywaveandStokeswavegives better response. Response of this Articulated tower is then studied under wave alone environment and wind+wave correlatedenvironment.Fluctuatingcomponentofthewindis modelled by Simiu’s spectrum, while the sea state is characterized by Pierson Mosokowitz (P M) spectrum. Random waves and wind are simulated by Monte Carlo simulation technique. Response of Articulated tower is determined by time domain iterative method using Newmarks’s βintegrationscheme.ItisconcludedthatStokes wave gives improved response and presence od wind forces along with the waves amplify the responses significantly particularly atlowerfrequencies.

Key Words: P M (Pierson Mosokowitz), A F (Axial Force), A-T (Articulated Tower), (M-E) Morrison Equation, (A-T) Airy Theory, (S-T) Stokes Theory.

1. INTRODUCTION

ArticulatedTowerisaflexiblecomplaintoffshorestructure whichresisttheenvironmentalforcesthroughtheactionof compliancy,whichmeansthatonesinglebuoyantshellwith sufficientbuoyancyisusedtorestoremomentagainstthe lateralloads.Articulatedtowersaredesignedsuchthattheir fundamentalfrequencyiswellbelowthewavefrequencyto avoiddynamicamplification.Dynamicinteractionofthese towers with environmental loads (wind, waves and currents) acts to impart a lesser overall shear and overturningmomentduetocompliancetosuchforces.This compliancyintroducesgeometricnonlinearityduetolarge displacements,whichbecomesanimportantconsideration in the analysis of articulated towers. Wind and wave loadings have a predominant role in the design of an offshore structure for a successful service and survival in harshseaconditions.Theflexibilityofthenewgeneration, Wind and wave loadings have a predominant role in the designofanoffshorestructureforasuccessfulserviceand

survival in harsh sea conditions. compliant structures (articulatedtower,guyedtowerandtensionlegplatforms) giverisetonaturalperiodsrangingfrom1to100seconds. Such structures comply in the direction of environmental loads and results in an increase in their sensitivity to the dynamiceffectsofwind.Fixedstructureswillrespondthem invirtuallystaticfashion.Forfixedplatforms,contributionof lateralwindloadsforthedesignisonly10%oftheglobal loads.Whileincaseofcompliantplatforms,itincreasesto 25%.

1.1 TOWER RESPONSE UNDER AIRY AND STOKES THEORIES

AsinglehingedArticulatedTowerofheight400mstandingin 350m deep water has been chosen for present study. The platformcharacteristicsaregiveninTable1.Twodifferent seastatesi.eModerateseastateandHighseastatehasbeen considered in the study. The characteristics of these two states are given in Table 2. As flexural deformations are negligible, hence re neglected in comparison to rigid body displacement.Responsesofthetowerhavebeenobtainedfor two most widely used wave theories i.e Airy theory and Stokestheory.RandomwavesarerepresentedbyPierson Moskowitz spectrum. The code was developed using FORTRAN software. Time histories and Power Spectral Density Functions (PSDF’s) are obtained for steady state conditionofthetower.Toobtainthesteadystatecondition transitionphaseofoscillationswhichisroughlytentimesthe timeperiodofthestructurehasbeenignored.Timehistories of responses are obtained at an interval of 0.7s using Newmarkβtimeintegrationmethod.Statisticalparameters suchasmaxima,minima,meanandstandarddeviationare obtainedforDeckDisplacement,HingeShearandAxialForce by using Airy and Stokes wave theory. Drag coefficient is takenas0.6andInertiacoefficientistakenas2.0.

Sea state Significant wave Zero crossing Description height, Hs(m) period, TZ (s)

Moderateseastate 4.51 7.38 Highseastate 12.4 12.23

Table 1:Characteristicsofdifferentseastates

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1.2 Advantage of Articulated Tower

• Very large fundamental sway period so dynamic amplification factor is much less than fixed structure.

• Enhanced“turnability“ofperiodsofthesystemfor aparticularsite.Anadjustmentofnaturalfrequency could be assisted by set up of a ballast water chamberpositionedabovetheconnectorjoint.

• Articulateloadingplatformsarere usable.Oncean oilreservoirisdepleted,itcanbeeasilyrelocatedto otherfieldataminimalcost.

• Articulatedloadingplatformsareusedasaportable offshore system for the moored tankers. Such configurationisparticularlysuitableforfieldsthat havealimitedproductioncapability,oraretoofar offfromrefiningtojustifythelayingofpipeline.

Articulated Towers around the Word Ocean

Field Type Install ation Year

Beryl Articulate dloading Tower

Water Depth Oper ator Platform Function

1975 117 BenC. Gerwi ck

discoveryofOilandNaturalgasinthedeeperwater,there arisesa needofeconomical solutionofoffshorestructure. Conventionalfixedlegplatformisfoundtobeeconomicalfor shallow depth. As depth of water increases, size of these platformwill cross theireffective economicsizemakingit unsuitable.Sotherearisesaneedofnewstructuralsystems tobeeffectivelyusedindeeperwater.Onesuchsolutionis articulated Tower which take advantage of the effect of compliance,i.e.,yieldtotheenvironmentalforces.

ReliabilityofArticulationSystem

VariousenvironmentalloadssuchasWindload,waveload, oceancurrentsetcarecontinuouslyappliedonthearticulate towers.Theseoscillatingstressescancausevarioustypeof damage to the structure such as metal fatigue which is causedduetostressconcentrationofvariousshearandaxial forces.

Variousresearchershavedoneremarkableworkincombat theproblemofdesigningarticulatejoint.

Author Year Salient Features

Loading Tower

Statfor d Articulate d Tower 1978 145 EHM Loading and Mooring

Maure en Gravity articulate dtower

North East Frigg.

Garde n Banks

Single Hinge Articulate d Tower

Compliant articulate dtower (CAT)

1982 490 Howa rd Doris Ltd.

1983 150 Total E&P UK Ltd

1998 501 Amer ada Hess Corp.

NeedofArticulatedTowerinIndianScenarios

Drilling and production

Field control station& gasProd

Drilling and production

InIndiaOilhasbeenfoundattheshallowerdepthusuallyup to100 150m.Articulatedtowercanbeadvantageouslyused forsuchashallowdepth.Asdepthofexplorationincreases thepossibilityoffindingoilandgasindeeperwaterincrease. As depth of water increases, size of conventional fixed legplatformwillcrosstheireffectiveeconomicsizemakingit unsuitable.Sotherearisesaneedofnewstructuralsystems tobeeffectivelyusedindeeperwater.Onesuchsolutionis articulated Tower which take advantage of the effect of compliance,i.e.,yieldtotheenvironmentalforces.Withthe

Chassyetal 1971  He gave the details of the universal joint for the ELFOCEANtower. 

The study was carried out consideringtheweartearand risk of the collapse of pipe wallspassingthroughthejoint Sedillotet al 1982 

He designed a ball type universaljoint. 

Fieldinvestigationsforfatigue life were performed for few years. Based on tests, the minimum life of the articulation system was estimatedas200years.

Objective

Todoastudyof dynamicresponseofArticulated Towerbytwowidelyusedwavetheories,i.eLinear WaveTheory(AiryTheory)andStokesTheory(5th ordertheory)toconcludewhichofthetwotheory givesimprovedresponses.

2. METHODOLOGY

Inthepresentworkfirstlyanonlineardynamicanalysisof thesaidstructureunderwaves/earthquakehasbeencarried out for its time domain responses using Langrangian approach which has the capability of equating kinetic and potentialenergiesofthesystemtotherotationaldegreesof freedom.TherandomwaveshavebeensimulatedbyMonte

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Carlo technique represented by Modified PM spectra. ModifiedMorison’sequationhasbeenusedforestimationof hydro dynamicloading.Waterparticlekinematicshasbeen governedbyAiry’slinearwavetheoryandStoke’sfifthorder theory.Resultobtainedfromboththetheoryarecompared with each other. To incorporate variable submergence, Chakraborty’scorrection[20&21]hasbeenapplied.Seismic inputshavebeenappliedusingNorthridge,Imperialvalley CA, Duzce, Turkey spectra. Stability assessment has been carriedoutusingconceptofminimumpotentialenergyand two dimensional phase plots. Analysis of the response of Articulated Tower to fluctuating Wind and WaveForcesis donebyatimedomainiterativeprocedurewhichincludes thestructuralaswellasforcingnonlinearities.

Modelling of the Articulated Tower

ModellingofArticulatedTowerwillbenonlinear.Threetype of nonlinear matrix will be formed. Stiffness matrix which consist of fluctuating Buoyancy component, Mass matrix whichconsistoftwotypeofmass,oneisstructuralmassand otherbeingtheaddedmassduetothemotionofTowerand thirdmatrixwillbetheDampingmatrix.Thetowerstructure isidealizedbyreplacingitsmassdistributionwithdiscrete masseslocatedatthecentroidofaseriesofsmallcylindrical elements of equivalent diameter Di representing inertia, addedmassandbuoyancy.Allforcesareassumedtoactat thesecentroids.Insubmergedparttheforcesactisweight, inertia, buoyancy and fluid force while wind force acts on exposedarea.

Linear Wave Theory

LinearwavetheoryorAirytheoryisusedwheresurfaceof waterisnotchangingoristoosmalltonotice.Letusdefinea potential function Φ which is proportional to velocity of fluid.Consider a progressive wave with water surface elevationdepictedbycosinecurve.Itismandatorytoanalyse theeffectsofsurfacewavesonthestructures,eitherusinga singledesignwavechosentorepresenttheextremestorm conditions in the area of interest, by use of statistical representation of the waves during extreme storm conditions.

FreeWatersurfaceelevationisgivenby η= (1)

Andthecorrespondingvelocitypotentialis (2)

Morison Equation

Wave and current loading can be calculated by Morison equationas: (3) WhereFTisthetotalforce,ρwisthedensityofwater,CD and Cm arethedragandinertiacoefficientrespectively,Disthe diameter of member including marine growth, V is the velocityandaistheacceleration.

Firsttermisthedragcomponentandsecondistheinertia component.

TotalDrag=Dragcomponent+InertiaComponent.

TheforcedF(t)duetowaveondifferentialsectionoflength of the cylinder is made up of two components namely inertiaforcecomponentwhichisproportionaltothenormal componentofthefluidparticleaccelerationanddragforce whichisproportionaltothesquareofthenormalcomponent ofthefluidparticlevelocitythus:

dF(t)= (4)

dF(t)=

dF(t)= (6)

Follows:

Lagrange’s equation

ThegeneralformofLagrange’sequationis: where T, V and Qθi represents the kinetic energy, the potentialenergyandthegeneralizedforce,respectively.

(5) θ (7)

DeckDisplacement

Deckdisplacementundermoderateseastate

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Sea State Statistics Wave only

Airy Stokes

Fig 1:TimehistoryofDeckDisplacementbyAiryand Stokeswavetheoriesundermoderateseastate.

Moderate Maximum 2.33 0.47 Minimum 2.05 0.50 Mean 0.02 0.02 StandardDeviation 0.83 0.25 High Maximum 5.28 3.03 Minimum 5.59 3.01 Mean 0.12 0.01 StandardDeviation 2.08 1.26

Table 2:StatisticalcomparisonofDeckDisplacement valuesbyAiryandStokeswavetheories.

HingeRotation Hingerotationundermoderateseastate

Fig 2:PSDFofDeckDisplacementbyAiryandStokes wavetheoriesundermoderateseastate. DeckDisplacementunderHighSeaState

Fig 5:TimehistoryofHingerotationbyAiryandStokes wavetheoriesundermoderateseastate. Hingerotationunderhighseastate

Fig 3:TimehistoryofDeckDisplacementbyAiryand Stokeswavetheoriesunderhighseastate.

Fig 6:TimehistoryofHingerotationbyAiryandStokes wavetheoriesunderhighseastate.

HingeShear

Fig 4:PSDFofDeckDisplacement byAiryandStokes wavetheoriesunderhighseastate.

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Hingeshearundermoderateseastate

Fig 7:TimehistoryofHingeShearbyAiryandStokes wavetheoriesundermoderateseastate.

Fig 10:PSDFofHingeShearbyAiryandStokeswave theoriesunderhighseastate.

Sea State Statistics Wave only

Airy Stokes

Fig 8:PSDFofHingeShearbyAiryandStokeswave theoriesundermoderateseastate. Hingeshearunderhighseastate

Fig 9:TimehistoryofHingeShearbyAiryandStokes wavetheoriesunderhighseastate.

Moderate

Maximum 1.29E+07 1.25E+05 Minimum 1.25E+07 1.65E+05 Mean 7.26E+03 1.47E+04 Standard Deviation 1.70E+06 3.66E+04 High

Maximum 6.49E+06 9.76E+05 Minimum 7.58E+06 6.41E+05 Mean 1.35E+05 1.93E+04 Standard Deviation 1.39E+06 2.02E+05

Table 3:StatisticalcomparisonofHingeShearvaluesby AiryandStokeswavetheories.

AxialForce:Axialforceundermoderateseastate

Fig 11:TimehistoryofAxialForcebyAiryandStokes wavetheoriesundermoderateseastate.

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Minimum 1.56E+08 1.72E+08

Mean 1.76E+08 1.77E+08

StandardDeviation 7.20E+06 2.10E+06

Table 4:StatisticalcomparisonofAxialForcevaluesby AiryandStokeswavetheories.

3. CONCLUSIONS

Fig 12:PSDFofAxialForcebyAiryandStokeswave theoriesundermoderateseastate.

Axialforceunderhighseastate

Fig 13:TimehistoryofAxialForcebyAiryandStokes wavetheoriesunderhighseastate.

In the present study Articulated offshore tower has been investigated. First the hydrodynamic loads on the Articulated tower is estimated by using the Airy theory, whichisalinearwavetheoryandisfrequentlyusedwhere surfaceisnotchangingorchangeistoosmalltonotice.Then the hydrodynamic loads are estimated by more rational Stokes5thordertheorywhichtakesintoaccountthesurface variation of waves. Results of time histories, PSDF’s and statisticalvaluesshowsthattheresponsesobtainedbyusing Stokestheoryareimprovedresponsesandareinlinetothe actualresponsestoalargeextent.

REFERENCES

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[2]Bithin,G.,Selvam,R.P.,&Sundaravadivelu,R.(2015). Hydrodynamic Responses of a Single Hinged and Double Hinged Articulated Towers May https://doi.org/10.1115/omae2015 41524.(Bithinetal., 2015)

[3]Chakrabarti,S.K.(2014). AnIntroductiononOffshore Engineering and Technology. 1 62. https://www.scribd.com/doc/207457372/Wave Theory Offshore Structures.(Chakrabarti,2014)

Fig 14:PSDFofAxialForcebyAiryandStokeswave theoriesunderhighseastate.

Sea State Statistics Wave only Airy Stokes

Moderate Maximum 2.03E+08 1.80E+08

Minimum 1.53E+08 1.76E+08

Mean 1.77E+08 1.77E+08

StandardDeviation 7.14E+06 1.69E+06

High Maximum 1.97E+08 1.82E+08

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