SSC 347 NonLinear Analysis of Marine Structures

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SSC-347STRATEGIESFOR

NONLINEAR ANALYSISOFMARINE STRUCTURES

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SHIP STRUCTURE COMMITTEE1991
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S-W STRL!!2TUWQMWUEE11 reSHIP STRUCTURE COMMl llEE isconsti tu ted to prowcute a research program to improve the hul lstructures of ships and other marine structures by an extension of knowledge pertaining to design,materials, and methods of conatruotion.RADM J. D. Sips, USCG, (Chai rman)Chief, Office of Marine Safety, Smmityand Environmental ProtectionU.S. Coast GuardMr. Alexander MalakhoffDirector, Structural lntqr~Subgroup (SEA 55Y)Naval Sea Systems CommandDr. Donald LiuSenior WEe Pria5identAmerican Bureau of Shipping

Mr. H. T, Hai lerAsscmiate Administrator for Ship-building and Ship OperationsMaritime AdministrationMr. Thomas W. AllenEngineering Officer (N7)Military Sealift CommandCDR Michael K. Parrnela, USCG,Secretary, Ship Structure CommitteeU.S. Coast Guard

CONTRACTING OFFICFR TFCHNICAI RFPRF=FNTATIVFSMr. William J. SiekierkaSEA55Y3Naval Sea Systems Command

Mr, Gre D. WoodsfEA553Naval Sea Systems Command

SHIP STRUCTURF SUBCOMMITIEEThe SHIP STRUCTURE SUBCOMMl llEE acts for the Ship Structure Committee on technical mat ters byproviding t~hnical coordinat ion for determinating the goals and objectives of the program and byevaluating and interpreting the results in terms of structural design, construction, and operation.AMERICAN BUREAU OF SHIPPING NAVAL SEA SYSTEMS COMMANDMr. Stephen G. Avntson (Chairman) Mr. Ro lmt A Siel sk iMr. John F. Corr lon Mr. Char les L NullDr. John S. Spenoer Mr. W. Thomas PackardMr. Glenn M. *he Mr. Allen H. EngleMll ITARY SFAI IIT ~ U.S. COA5T GI IAIU!Mr. Albert J. Attermeyer CAPT T. E. ThompsonMr. Michael W. Touma CAPT Donatd S. JensenMr. Jeffery E. Beach CDR Mark E. tdollMARITIME ADMINISTRATIONMr. Frederick SeiboldMr. Norman O. HammerMr . Chao H. L inDr. Walter M. Maolean

SHIP STRUCTURF SUBCOMMITTFF I IAISON [email protected]. COAST GUARD ACADFMYLT Bruce Mustainu.s MERCHANT M4RINE ACADEMYDr. C. B. KmU.S.NAAL ACADEMYDr. Ramswar Bhattacharyya

Dr. W. R. PorterWEl OING RESEARCH COUNCILDr. Mantin Prager

Mr. Afexander B. StavovyNATIONAL A-Mr. Stanley G. StiinaenSOCIEIY OF NAVAI ARCHITECTS ANIIMARINE ENGINEERS - 11-rEEDr. William SandbargAMERICAN IRON AND STEEL INSTITUTEMr. Alexander D. Wilson


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Member Agenci es:United States Coast GuatrlNaval Sea Systems timmandMaritime AdministrationAmerican Bureau of Shi~ingMl i tatyeal RCommand

&ShipStructureCommittee

Address Correspondence to:Secretary,hip Structure CommtteU. S.CoastGuard(G-MTH)2100 SecondStreet.WWashi ngton,. C. 20593-0001PH: (202)267-0003FAX: (202)267-0025

An I nteragencydvi soryCommtteeDedi catedotheImprovementofMati neStructuresSSC-347January 31, 1991 SR-1304

STRATEGIES FOR NONLINEAR ANALYSISOF MARINE STRUCTURES

Marine structures are designed to withstand the extreme responsescaused by environmental factors during its lifetime. This reportcovers available and emerging techniques to determine extremeresponses of nonlinear marine structures and systems. Nonlinearcharacteristics of wave induced forces and extreme value analysesof nonlinear systems are addressed. The review of procedures for+ nonlinear analysis and how they may be applied to the probabilityanalysis of extreme events should prove to be useful.

Rear Admiral, U. S. Coast GuardChairman, Ship Structure Committee


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Techni calkeportDocumentation Pa1. Report No. 2. Government Acc.sssion Ne. 3. Recipients Catalog No.SSC-347

14. Title ad Svbtitle 5. Report DateStrategies for Nonlinear Analysis of AUGUST 1988Marine Structures 6. Performing Organization Cod=

7. Authods) 8. Performing Organization Rmport No.I SR-1304Subrata K. Chakrabarti~.Performing Organizat ion Name and Address 10. Work Unit No. (TRAIS)CBI Research Corporation1501 North Division Street 11. Contract or Grant No.Plainfield, IL 60544-8929 DTCG23-85-C-20069

13. Type of Report ond P*riod Covered12. Sponsoring Ag~nGY Name and Address ! Final ReportCommandantU.S. Coast Guard2100 Second Street, SW 14. Sp~sorig Agency CodeWashington, DC 20593 G-MI

15. Supplementary Notes

Sponsored by the Ship Structure Committee and its member agencies.16. AbstruetThis report extensively reviews the state-of-the-art of available anemerging techniques for determining extreme responses of nonlinearmarine structures and systems. The contents may be categorized intotwo parts. The first presents the nonlinear characteristics of waveinduced forces and corresponding structural responses. The otherdiscusses the extreme value analysis of nonlinear systems relevant toffshore and marine structural design. Generic procedures for thenonlinear analysis of marine structures were reviewed. Discussionsof nonGaussian random processes and the use of extreme valuestatistics are included. Methods that may be applied to theprobability analysis of extreme events were investigated. Differenttypes of nonlinear behavior of interest for various classes ofoffshore structures were studied. Various statistical analysismethods are summarized and their applicability, assumptions andlimitations are discussed.

17. Key ~ords 18. Distr ibution Statement avdLLcl vl e LLUlll -Mar~ne Structures .Nat1 Technical Information ServiceNonlinear Analysis Springfield, VA 22161 orProbabilistic Methods Marine Tech. Information FacilityExtreme Value Analysis National Maritime Research CenterStructural Analysis Kings Point, NY 10024-169919. Security Clossif. (of this report) I 20.Security Classif. (of this poge) 21. No. of Pages I 22. PriceUnclassified Unclassified 380

FormDOT F 1700. 7(B-72) Reproduction of completed pogc authorized


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TABLEOF CONTENTS

sutwARYLIST OF SYNBOLS

Page1.0 INTRODUCTIONw--9*.*..* q *********9 q ***mmmm9m* q ********** ..*m.=...2.0 TYPES OF NONLINEARITIES *m.m*.m.m* .9*99*9**** q *..m.*mmm. q .m9*m***

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DEFINITION OF NONLI NEAR SYSTEMS. *w*****D. * q *m*.m***9* q **9***NONLI NEAR WAVES AND WAVE SIMULATI ON. . . . . . . . . . . . . . . . . . . . . . . . .WAVES PLUS CURRENT q , **w ***mm* q 000000999098***=******* q *.***NONLI NEAR WAVE FORCEq - u*- *****.* q ***mmmm..m q m*-m****** q ****.2. 4. 1 Mori $on Equat i on . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2.4.2 Fi xed Cyl i nder i n Waves and Current . . . . . . . . . . . . . . . . . .2. 4. 3 Osci l l at i ng Cyl i nder i n Waves . . . . . . . . . . . . . . . . . . . . . . . .2.4.4 Osci l l at ing Cyl i nder i n Waves and Current . . . . . . . . . . . .STEADY DRI FT FORCEq *****. *~**~q mm*m*mmmmm* . ...8*****.* q **...2.5.1 Steady Dr i f t For ce Due to Vi scous Fl ow. . . . . . . . . . . . . . .2. 5.2 Steady Dr i f t For ce Due to Potent ial Fl ow. . . . . . . . . . . . .2. 5.2.1 Wave El evat ion Dr i f t For ce. . . . . . . . . . . . . . . . .2. 5. 2. 2 Vel oci t y Head Dri f t Force. . . . . . . . . . . . . . . . . .2. 5.2.3 Body Mot ion Dr i f t For ce. . . . . . . . . . . . . . . . . . . .2. 5.2.4 Rotat ional Iner ti a Dr i f t For ce. . . . . . . . . . . . .NONLI NEAR MOTI ON RESPONSE q . *. m******* 9m..*m.mm.m* q =**=******2.6.1 Fi rst Order Mot i ons w th Nonl i near Dampi ng. . . . . . . . . . .LOWFREQUENCY OSCI LLATION q mm.mmmmmm. q m.*mm*9*** q *******99* q *HI GH FREQUENCY SPRI NGI NG FORCEq . 9*9****** q *9mmm**9mm q mmm.mmm2. 8. 1 Dampi ng at Low and Hi gh Frequency Responses . . . . . . . . . .MATERI AL PROPERTI ES 9*.89******* q **.m.m*.mmm q mmmm*9..*** q ****2. 9. 1 Cat enary Syst em. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2. 9. 2 Fl exi bl e Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3. 0 PROBABILISTIC METHODSFOR EXTREMEVALUE. . . . . . . . . . . . . . . . . . . . . . . .3. 1 SOME COMMON TYPES OF PROBABI LITY DI STRI BUTI ON FUNCTI ONS. . . . .3.1.1 Normal or Gaussi an Di st ri but ion . . . . . . . . . . . . . . . . - . . . . .3. 1.2 Rayl ei gh Di st ri but ion . . . . . . . . . . . . . . . . . . . . . . . . . . .=. . . .3. 1. 3 Gumbel Di st ri but i on . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3. 1. 4 Weibul l Di st ri but i on . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3. 1. 5 Frechet Di st ri but i on . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.1.6 Cumul ants and GramChar l i er Ser ies . . . . . . . . . . . . . . . . . . .3. 2 DI STRI BUTI ON OF SHORT-TERM WAVE PARAMETERS . . . . . . . . . . . . . . . . . .3. 2.1 Wave El evat ion Di st ri but ion . . . . . . . . . . . . . . . . . . . . . . . . . .3. 2.2 Wave Hei ght Di st ri but ion . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3. 2. 3 Wde Band Spect rum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.2.4 Nonl i near Gaussi an Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.2.5 Nonl i near Non-Gaussi an Waves . . . . . . . . . . . . . . . . . . . . . . . . .3. 2.6 Wave Per iod Di st ri but ion . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.2.7 Wave Hei ght -Per iod Di str ibut i on . . . . . . . . . . . . . . . . . . . . . .3. 2. 8 Extreme Wave Hei ght- Steepness Di stri buti on . . . . . . . . . . .

1334810101415181919;:;:252828323639:;4547505051%5456575758586263777983

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3. 3 SHORT TERM RESPONSE PREDI CTI ONq m..m.m.a.m q * m* . * m. * m. .* * * * * * *3. 3. 1 Li near Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . : ;3. 3. 2 Nonl i near Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 903. 3. 2. 1 Wave Drag. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 903. 3. 2. 2 Wave-Pl us-Current Drag. . . . . . . . . . . . . . . . . . . . . 913.3.2.3 St ructural Dynamc Response . . . . . . . . . . . . . . . . 923.3.2.4 General Li near i zat i on Techni que . . . . . . . . . . . . 983.3.2.5 Nonl i near Response Spect ra . . . . . . . . . . . . . . . . . 1003.3.2. 6 Stat i st i cs of Narrow Band Mor ison Force. . . . 1043.3.2.7 Stat ist i cs of Wde Band Mor ison Force. . . . . . 1103.3.2.8 Statistics of Wave-Current Force. . . . . . . . . . . 1183. 3. 2. 8. 1 Narrow-Band Gaussi an Wave andSmal l Cur rent . . . . . . . . . . . . . . . . . . . 1183. 3. 2. 8. 2 Fi ni t e Current . . . . . . . . . . . . . . . . . . 1213. 3. 2. 9 Stat i st i cs of Nonl i nearl y Damped System. . . . 1253.3.2.10 Stat i st i cs of Dr i f t For ce Response . . . . . . . . . 1293.3.2.11 Stat ist i cs of Low Frequency Mot ion . . . . . . . . . 1323.4 SHORT TERM RESPONSE MEASUREMENTS .m .m* * * * * .m q . .m.m. . .mm q mme.. 1373. 4. 1 Random Wave Load Test s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1383. 4. 1. 1 Vert i cal Cyl i nder . . . . . . . . . . . . . . . . . . . . . . . . . . 1383. 4. 1. 2 I ncl i ned Cyl i nder . . . . . . . . . . . . . . . . . . . . . . . . . . 1403. 4. 1. 3 Force Di stri buti ons . . . . . . . . . . . . . . . . . . . . . . . . 1413.4.2 Response of an Ar ti cul ated Tower . . . . . . . . . . . . . . . . . . . . . 1453. 4. 3 Response of a Moored Tanker . . . . . . . . . . . . . . . . . . . . . . . . . . 1463. 4. 4 Response of a Barge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1463.4.5 Response of a $emsubmersi bl e . . . . . . . . . . . . . . . . . . . . . . . . 1473. 5 LONG TERM RESPONSE PREDI CTI ONq **O. *9*0** q W*******, W***, , , * 1473.5.1 Bi var iate Shor t- and Long-Term Predi ct ion . . . . . . . . . . . . 1503. 5. 1. 1 Short Term. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1503. 5. 1. 2 Long Term. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1523. 5. 2 Ti me and Frequency Domai n Long Term Predi cti ons . . . . . . 1543. 5. 3 Extrapol ati on of Wave Scatter Di agram to LongerDurati on q O*Om*mm**Oq 0mmm*99*0mq mOmm**m*** q *m*e***** 1573. 6 EXTREME VALUE STATI STI CS q mmm* *m* * * . q DD * * * * * * * * q * * * * * * * * * * . . . 159

4. 0 EVALUATIONOF PROBABILISTIC APPROACHES* .mm9m9m.9 q mm*mmm* m* m .* .* . 1624. 1 DI SCUSSI ON OF LI NEARI ZATI ON TECHNI QUE. , . . . . . . . . . . . . . . . . . . . . . 1624. 2 DI SCUSSI ON OF NONLI NEAR EXCI TATI ON STATI STI CS. . . . . . . . . . . . . . . 1644. 3 DI SCUSSI ON OF NONLI NEAR RESPONSE STATI STI CS . . . . . . . . . . . . . . . . . 165

5. 0 CONSISTENTMETHODOLOGYm..m. . .am. q * * * * . * * * * . q m . . . . . . . . . . . . . . . . . . . 1705. 1 EXTREME VALUE PREDI CTI ON FOR NONLI NEAR SYSTEMS. . . . . . . . . . . . . . 1715. 2 A CONSI STENT LINEARI ZATI ON METHODq mm.mmmmmmm .m* *m* * m* m* q * . * * 1745.3 RESPONSE SPECTRUM COMPUTATION q amm.m.mm.. q * m. * 9* * * . * . . . . . . . . . 1755. 4 RESPONSE PROBABI LI TY DENSI TY FUNCTI ON. . . . . . . . . . . . . . . . . . . . . . . 1785. 5 RESPONSE EXTREMES BY ORDER STATI STI CS . . . . . . . . . . . . . . . . . . . . . . . 1845. 6 LONG-TERM RESPONSE PREDI CTI ON FOR NONLI NEAR SYSTEMS. . . . . . . . . 186

6. 0 CONCLUDINGREMARKS.mmmm.mm.. q mm.mm.m**m q * * 9 * * * * * * * q * * * * * * * . . . . . . 1917. 0 RECOMMENDATIONSOR FUTUREWORKq *m9**mmmmm q mmmmm . . . .m q 9* * * . * * . . . . 1958. 0 REFERENCES q mmm.mm.mmm . .* m. .* * * * * q * . * * * * * * * . q . . . . . . . . . . . . . . . . . . . . . 198

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SUMMARY

Mari ne St ruct ures are employed i n t he expl orat i on, producti on andt ranspor tat ion of of f shore mneral s as wel l as for t ranspor tat ion of peopl eand products across nati ons and f or the def ense of the country. Thestructures used for the producti on of oil and gas are general l y l ocated at apart icul ar si te of fshore whi l e others are mobi l e. These structures are oftenat the mercy of the harsh envi ronment of the ocean i n the form of waves, w nd,current and earthquake and must survi ve the severest storm encountered duri ngi ts l i fet i me.

Desi gn of an of f shore st ruct ure i s based on the ext reme responsesexper ienced by the components of the st ructure under the i nf l uence of theenvi r onment f aced by the structure i n i t s l i f etime. If the structuralcomponents may be treated as a l i near system the der ivat ion of the extremeresponses i s rel ati vel y strai ghtforward. However, most practi cal off shoresystems have nonl i near responses, and these desi gn tool s are not appl i cabl e.

The purpose of thi s repor t i s to per form an extensi ve state-of - the-ar treview of the avai l abl e and emergi ng techni ques for the determnat ion ofextreme responses of a nonl i near mari ne structure and system The contents ofthi s repor t may be categor ized i nto two par ts; one presents the nonl i nearcharacteri sti cs of wave- i nduced forces and correspondi ng structural responses,and the other di scusses the ext reme value anal ysi s of nonl i near systemsrelevant to of f shore and mari ne st ructure desi gn. The repor t revi ews thegener ic procedures for the nonl i near anal ysi s of mari ne st ruct ures andinvest i gat es t he method in whi ch t hey may be appl i ed t o t he probabi l i t yanal ysi s of extreme events.

Di f ferent types of nonl i near behavi or of i nterest for var ious cl asses ofoff shore structures have been studi ed. Nonl i near i t i es i n the anal ysi s ofthese structures appear at vari ous stages, e. g. , waves, materi al properti es,forci ng functi on and moti on response. The sol ut ions of the dynamc probl emw th t i me dependent l oads fal l i nto two mai n categor ies: determni sti c andnondetermni sti c (stochasti c). Determni sti c sol uti ons i ncl ude both the ti medomai n (t i me hi story anal ysi s) and the frequency domai n anal ysi s. Whi l e thetime hi story anal ysi s can retai n most of the nonl i neari t i es i n a mari nesystem frequency domai n sol ut ions are necessar i l y l i near i zed. Revi ew of

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these vari ous generi c procedures for nonl i near anal ysi s has been made.In the case of probabi l i si tc method of obtai ni ng extremes of a response

of an offshore structure, di st i nct i ons are made regardi ng short-term versusl ong- term

A short termmeans a per iod of t i me whi ch i s short enough to descr ibe thesea and the response as a stati onary random process. Thi s peri od of time i son the order of 30 mnutes to 3 hours. I t i s a general pract i ce t o assumethat the short -term stat i st i cal di str ibut i on of response ampl i tudes fol l owsthe Rayl ei gh di stri buti on functi on. Based on thi s functi on, the probabi l i t i esof certai n extremes over a gi ven short term may be predi cted. The waves areassumed Gaussi an for thi s purpose. For responses, the narrow handedness andl i neari ty are i nherent assumpti ons. However , the response (output ) of anonl i near system i s a non-Gaussi an random process even though the waves(i nput) i s Gaussi an. Thi s fundamental pri nci pl e has been addressed i n therepor t. The predi ct i on of the stat i st i cal proper ti es of mar ine systems w thstrong nonl i near characteri st i cs i s not possi bl e usi ng a l i near anal ysi s. Forstat i st i cal anal ysi s of nonl i near systems, the probabi l i st i c predi ct i on ofnon-Gaussi an random process i s essenti al . Thi s area has been di scussed i n thereport, and several non-Gaussi an random processes have been i ncl uded.

For extreme val ue stat i st i cs, a l ong-term (of the order of 20-100 years)di stri buti on of the response parameters i s often requi red. The l ong-term non-stat i onary random process i s someti mes wri tten as a sum of a l arge number ofshort- term stati onary process. The ext reme val ues of a gi ven probabi l i t yl evel are al so obtai ned by order-stati sti cs. Thi s area i s bri ef l y revi ewed.

Vari ous avai l abl e methods i n the above areas have been summari zed, andthei r appl i cabi l i ty, assumpti ons and l i mtati ons have been di scussed. Basedon thi s di scussi on, several concl usi ons and recommendati ons have been drawn.

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LIST OF SYWOLS

a:cCDCDC~CMDdEE[. ]FffDf~fmax9HHnHrmsH5iIiKkkAKCkDkjkML1M kIllj k

ampl i tude of waverms wave ampl i tudewave cel eri tydrag coef f i ci entsteady drag coef f i ci entrestori ng force matri xi nerti a (or mass) coeff i ci entcyl i nder di ameterwater depthYoung s modul usexpected val uemean forceforce per uni t l ength of structuredrag force per uni t l ength of structurei nert ia force per uni t l ength of structuremaxi mum force per uni t l ength of structureaccel erati on due to gravi tywave hei ghtHermte pol ynomal of order nrms val ue of wave hei ghtsi gni f i cant wave hei ghtH / Hrms moment of i nerti ai magi nary quanti ty ( = = )rat i o of rms drag to rms i ner ti a for cewave number (=&)PCA:D2Keul egan-Carpenter number ( = UOT / D )l ~pCDDcumul ants

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nNN(p, a2)N?J kN!J kxP(x)PP(x)Qc!RReRXyrs(u)s* ( fJ))s+)Su(m)TToTMTRtuUwui ov~vvR j rv(t)

nth moment of wave spectral densi ty (n = 0, 1,2, . . . )TR / Tz ; al so, number of wave componentsnormal di str i buti on of mean, p , and standard devi at i ons G

l i near danpi ng matri x

nonl i near dampi ng matr i xdi recti on cosi necumul at i ve probabi l i ty or probabi l i ty di str i buti on of xdynamc pressure or sl opeprobabi l i ty densi ty of xl ong- term probabi l i ty of exceedanceshort- term probabi l i ty of exceedancecyl i nder (structure) radi usReynol ds number ( = UOD / u )

or probabi l i ty densi ty

correl ati on functi on between functi ons x(t) andy(t)cosh ky/ si nh kdspectral energy densi ty of wavespectral energy densi ty of wave i n currentforce spectral densi tyspectral energy densi ty of hori zontal vel oci tywave peri odperi od of cyl i nder osci l l at i onexpected ti me between two successi ve crestsrecord l ength of ti meti mecurrent vel oci tyw nd speedhori zontal water parti cl e vel oci tyhori zontal water parti cl e accel erati onampl i tude of hori zontal vel oci tyreduced vel oci t y ( = o / D )vert i cal vel oci ty componentu - iei genvectorsrel ati ve vel oci ty ( = u - ~ )u(t) - u

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P

xT

i( l J

r

normal vel oci ty vectorampl i tudes of moti oncoordi nate i n the hori zontal di recti onmoti on components (k = 1, 2, . . . , 6)coordi nate i n the vert i cal di rect i onphase angl e of force or empi ri cal parameter~ or probabi l i t y l evelphase angl e of moti ons or empi ri cal parameterstretch i n moori ng l i necrest fr ont steepnessspectral w dth parameterE/E rmsi nci dent wave ampl i tudewave profi l e or el evati onangl e of mean wave di recti onl / l nNverti cal asymmetry factorcurrent strength ( U / U. )skewnesskur tosi s mnus 3ei genval uespeak rate densi tyhori zontal asymmetry factormean val ue of xki nemati c vi scosi ty of watercorrel ati on coeff i ci entmass densi ty of waterstandard devi at i on of xt i me l agdegrees of f reedomwave f requency ( = 2W/ T )peak f requency or undamped natural f requencyvel oci ty potenti alstandard normal di stri buti onvector si gngamma functi on

vi i


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TABLE 1.1 - TYPES OF OFFSHORESTRUCTURES

MARI NE ANDOFFSHORE STRUCTURES

QI XED1*2 clLOATING2s3Dri l l i ng J acketsProducti on Pl atformsOi l Storage TanksCai ssons

I

Shi psBargesSerni submersi bl esBuoysTLPsArt i cul ated Buoyant TowersGuyed Towers4

rLEXI BLE 1Catenary Cabl esChai nsRi sersTLP TendonsMari ne Hoses

NOTES:1. Fi xed structures may be pi l ed or gravi ty type.2. Fi xed or f l oati ng struct ures may be ri gi d or non- ri gi d. The non-r igi d st ructures w l l undergo smal l def l ect ions or di spl acements

under envi ronmental l oads.3. Fl oat i ng structures are usual l y moored i n pl ace i n operat i onal mode.4. St ri ct l y speaki ng, guyed towers do not bel ong to thi s category; but

i t s anal ysi s i s si m l ar t o t he others i n t hi s cat egory.


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1.0 INTRODUCTION

The mari ne and off shore structures and thei r components may be cl assi fi edi nto three broad categor ies: fi xed structures, f l oat ing st ructures andf l exi bl e structures. Tabl e 1.1 shows these three cl asses of structures.

The f i xed st ructures i n the open ocean are hel d i n pl ace by thei r wei ghtor by pi l i ng. General l y, j acket t ype st ructures consi st ing of a l arge number :of tubul ar members i n var ious pl anes are hel d i n-pl ace by pi l i ng. Many suchstructures may be seen i n the Gul f of Mexi co. On the other hand, l arge-vol umed product ion st ructures made of concrete and steel that exi st i n theNorth Sea are gravi ty-type structures. The weight of t hese st ruct uresprovi des suff i ci ent beari ng pressure to overcome sl i di ng or overturni ng duetoenvi ronmental l oads, thus fi xi ng thei r posi ti on.

There are two pr imary types of f l oat i ng structures. One type i s poweredto move from one l ocati on to another and i s used to transport materi al s acrossbodi es of water. Exampl es of thi s type of . structures are shi ps and barges.The other type of f l oat i ng structures i s mechani cal l y connected to the oceanbottom or moored i n pl ace for use i n of fshore operati on such as i n theproducti on, processi ng and stori ng of oi l . Such structures may be arti cul atedtowers, semsubmersi bl es, tensi on l eg pl atforms (TLPs), etc.

Fi xed and f l oat ing st ructures may be r igi d or non-r i gi d. Large struc-tural components are consi dered r igi d for the anal ysi s of wave for ces andmoti ons. Long members of smal l cross secti ons, e. g. , i n j acket pl at forms mayundergo def l ect ions or di spl acements whi ch are substant ial and shoul d beconsi dered non- ri gi d. An arti cul ated tower may al so experi ence natural peri odvi brati ons i n hi gher vi brati on modes than the ri gi d body moti on. Thesemembers are treated as non- ri gi di n the response cal cul ati on.

The thi rd type of st ructures, namel y, the f l exi bl e st ructures undergol arge deformati ons whi ch must be taken i nto account when bei ng anal yzed. I tmay be i mportant to update the external forci ng functi on on these structuresbased on thei r di spl aced confi gurat i on. Exampl es of these structures areri sers, TLP tendons, catenary l i nes, hoses, etc.

Because of the nature of these f l exi bl e structures, the nonl i near i t i es i nthe desi gn anal yses appear i n di f ferent phases and are somet i mes typi cal of


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the structures i n questi on. On the other hand, certai n types ofnonl i near i t i es are common to most of these st ructures, dependi ng on theenvi ronment experi enced by them

Thi s repor t di scusses the common types of nonl i near i t i es (Chapter 2)encountered i n the desi gn of offshore structures. The types of nonl i neari t i esare arranged i n the order i n whi ch they may enter i nto the anal ysi s of ast ructure. Appl i cabi l i t y of these nonl i near i t i es to the types of st ructuresi ncl uded i n Tabl e 1.1 i s di scussed. Exampl es of the nonl i neari t i es arepresented fr omwhi ch the i mportance of the nonl i near terms may be assessed.

The mai n thrust of the present repor t i s the ext reme val ue anal ysi s ofnonl i near systems rel evant to offshore or mari ne structures. Thi s area i srel at i vel y new but progress i n t hi s ar ea i n t he l ast several years has beenrapi d and steady. Because of the compl exi ty of the probl em the extreme val ueanal ysi s of nonl i near systems makes approxi mate assumpti ons i n order to makethe mathemat i cal probl ems tractabl e and f i t one of the known extreme val ueanal ysi s methods. Chapter 3 di scusses var ious probabi l i st i c methods anddi st ri but i on f unct i ons used i n predi ct i ng short - and l ong- t erm ext remeresponse val ues for an of f shore st ructure. Most of the nonl i near systemswhi ch appeared i n Chapter 2 are addressed here.

The appl i cabi l i ty of the vari ous approxi mate methods i n nonl i near extremeval ue anal ysi s i s di scussed i n Chapter 4. Some of the assumpti ons andl i mtat ions of these techni ques are summar ized. Based on thi s eval uat ion,consi stent methodol ogy appl i cabl e to the probabi l i sti c approaches i s provi dedi n Chapt er 5. As w l l be cl ear f rom the di scussi ons i n t he ear l i er chapt ers,a si ngl e methodol ogy may not be appropr iate for eval uat ion of al l systems.Theref ore, based on cert ai n i nput parameters depending on the t ypes ofnonl i neari ti es, di ff erent methodol ogi es and formul ati ons are recommended.Moreover , because of the cost and schedul e const rai nt s of thi s cont ract ,several recommendati ons are made for possi bl e future work i n thi s area.


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2.0 TYPES OF NONLINEARITIES

The nonl i near i t i es enter i nto an offshore structure anal ysi s at var iousphases. The f i rst and f oremost of these i s the envi r onment i t sel f . Indescri bi ng the envi ronmental condi t i ons that i nf l uence the of f shorestructures, nonl i near theori es are often needed. For exampl e, waves are oftennonl i near and requi re a mathemati cal ser i es expressi on whi ch depends onvari ous wave parameters (e. g. wave hei ght) i n a nonl i near fashi on. I ndescri bi ng the ef fect of t he envi r onment on t he struct ure, the externall oadi ng may become nonl i near. Exampl es of such nonl i neari t i es are currentl oad, w nd l oad and wave drag l oad. The response of the structure resul t i ngfrom the envi ronmental l oads may be nonl i near due to nonl i near dampi ng, forexampl e.

Let us, at thi s poi nt, expl ai n what i s underst ood about a nonl i nearsystem and how i t di f fers from a l i near system

2. 1

thatt i me,

DEFI NI TION OF NONLINEAR SYSTEMSConsi der a nonl i near system If y( t) i s the response at a gi ven time, t,i s si ngl e val ued and nonl i near due to an exci t ati on x( t) at the same, t, then

y(t) = g[x( t)] (2. 1),where g( x) i s a si ngl e ~al ued nonl i near f uncti on. of x. The system g( x) i s

nonl i near i f . ,

9(al X1 + az x2) * al 9(X1) + az 9(X2) (2. 2)

where al and a2 are arbi t rary constant s. Thi s system i s consi dered a zeromemory system meani ng that the response of the system does not depend on thepast val ue of the exci tat i on. If the system i s a constant parameter nonl i nearsystem and i f the exci tat i on x(t) represents a stat i onary random process, thenthe response y(t ) w l l al so be a stat i onary random process. In thi s case, thecorrel at i on functi on of the output, and between i nput and output are gi ven by


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RYY(T) = E[y(t)y(t + T)]= E[g{x(t)} g{x(t + T)}] (2. 3)

RxY(T) = E[ x( t) y( t + T) ] = E[g{x( t) } g{x( t + T)}] (2.4)

where R refers to the correl ati on at a ti me l ag, T, and E refers to theexpected val ue. Exampl es of zero memory nonl i near systems that are oftenfound i n the offshore structure anal ysi s are

q Square- Law System y = X2q Cubi c System y = X3q Square- Law Systemw th Si gn: y = 1X1X

types of nonl i neari ti esstructure desi gn. The

In thi s sect i on, we shal l descr ibe the var ioust hat are encount ered i n t he mari ne and of f shoresubj ects are i ntroduced i n the order menti oned at the begi nni ng of thi ssecti on. Fi rst of al l , nonl i near i t i es encountered i n descr ibi ng theenvi r onment w l l be descri bed. Then the external l oadi ng f rom theseenvi ronments that are nonl i near w l l be di scussed. Fi nal l y, t he responsesf rom external l oadi ngs that are nonl i near w l l be addressed. I t shoul d benoted that the desi gn extreme val ue anal ysi s shoul d properl y account for thesenonl i neari ti es.

2.2 NONLI NEAR WAVES AND WAVE SIMULATIONI n computi ng the wave l oads on the components of an off shore structure, a

sui tabl e wave theory must be chosen based on the wave parameters. Numerouswater wave theor ies have been devel oped whi ch descr ibe the ki nemat i c anddynamc proper ti es of the water par ti cl es at or bel ow the f ree sur face of thewave profi l e. Al though the ocean waves are random i n nature, the wavetheori es descri be wave prof i l es t hat are regul ar and peri odi c i n nature.There are three basi c parameters that are used i n descri bi ng al l wavetheori es: water depth, wave hei ght and wave peri od.

The l i neari t y of waves i s determned by the wave hei ght or the wavesl ope. The si mpl est and most commonl y used wave theory i s known as Ai r ytheory whi ch i s l i near w t h the wave hei ght ( hence, al so cal l ed the l i neartheory). Because of the l i near i t y of the Ai r y theory w th the wave hei ght ,

4


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the st ructural response obtai ned usi ng thi s theory i s of ten qui te st rai ght -f orward, even though not necessari l y l i near. Thi s i s the theory that i sal most excl usi vel y used i n the extreme val ue anal ysi s of responses, and formsthe basi s for the l atter chapters.

The f ree sur face boundary condi t ions are l i near ized i n descr ibi ng thel i near Ai ry wave theory. Therefore, i t i s not possi bl e to accuratel y predi ctt he st at i st i cal and spect ral propert i es of part i cl e ki nemat i cs i n t he f reesurface zone. Anastasi ou, et al . ( 1982) deri ved the probabi l i t y densi t yfunct ion of par ti cl e ki nemat i cs i n the f ree sur face regi on, whi ch i s cor rectup to the second order . The wave l oads i n the f ree sur face zone on a ver ti calcyl i nder were computed to demonstrate the nonl i near properti es of the parti cl eki nemati cs.

However, i n many physi cal si tuati ons the l i near theory i s not adequate toaccuratel y descr ibe waves. In thi s case one has to resor t to other theor iesto match or atother commonl yare (1) Stokescnoi dal theory

l east approach the physi cal data. Besi des the l i near theory,used nonl i near theor ies i n thehi gher order theory [Skel brei a[Wei gel (1960)] and (3) stream

desi gn of off shore structuresand Hendri ckson (1960)], (2)functi on theory [Dean (1965),

(1970)].To offer an exampl e of the di fferences among theori es, Ai ry l i near theory

provi des an expressi on for the hori zontal water part i cl e vel oci ty as

u mH cosh ky Cos (kx ~si nh kd - &) (2. 5)

whereas Stokes second order nonl i near theory expresses the same parameter as

U=%-wrcoskx- (2. 6)d++(+) 2=COS 2(kx-ut)where H = wave hei ght, T = wave peri od, d = water depth, y = verti cacoordi nate of part i cl e, k = wave number (= 2T/ L, L = wave l ength)x = hor i zontal coordi nate of part icl e, u = wave frequency, c = wave cel er i ty,and t = t ime. The f i r st t erm on the ri ght hand si de of Eq. 2. 6 corresponds t othe f i rst order theory and i s l i near w t h the wave hei ght. However, thesecond term i s proport i onal to the square of the wave hei ght (or wave sl ope).

Si ml ar ly, the hor izontal water par ti cl e vel oci ty of an Nth order st ream

5


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funct ion theory i s gi ven i n a ser ies formw th terms Up to N as fol l ows:Nu =. z nk cosh (2n- l )ky [X(2n-1) cos(2n- l )kx + X(2n) si n(2n)kx] (2. 7)n=l

i n whi ch X(n) are t he coef f i ci ent s of t he st ream funct i on. The st at i st i caldi str ibut i on propert ies of nonl i near waves have recei ved some attent i on i nrecent years whi ch have been di scussed i n Chapter 3.

The appl i cabi l i ty of the wave theori es may be descri bed by twonondi mensi onal parameters, d/ gT2 and H/gT2 based on the three basi c waveparameters, d, H and T. Thi s i s descri bed by the regi ons shown in Fi g. 2. 1.The l i m ts of val i di t y of t he vari ous t heori es are based on how wel l t he f reesurface boundary condi ti ons are sati sfi ed, al though there has been l i mtedexper imental ver i f i cat i on. For thi s reason, i n usi ng thi s chart , one need notst ri ct l y adhere to the boundary l i nes i n sel ect ing a theory. I n fact, thel i near theory has been shown to work qui t e wel l i n predi cti ng st ructureresponses wel l beyond i ts anal yti c l i mtati ons.

These wave theor ies are used i n comput ing the response funct ion of anoff shore structure. Hi gh order determni st i c wave theori es are usedextensi vel y i n the desi gn of of f shore st ructures despi t e thei r i nabi l i t y tomodel the randomness of a w nd generated sea. The extreme val ues of responses( l i near and nonl i near) are predi cted i nvar iabl y i n l i near randomwaves. For al i near system thi s procedure i s strai ght-forward w th the use of a wave energyspect rum model as w l l be descri bed i n t he next chapt er. For a nonl i nearresponse functi on, the sol ut i ons are often obtai ned i n the, t i me domai n. Thi srequi res the si mul ati on of a ti me seri es from the energy spectrum model .

The r andom waves i n t he ocean cannot be descri bed by a theoret i calmodel . They are general l y descr ibed by thei r energy densi ty spectrum Oftena mathemati cal formul a i s used to descri be the energy densi ty spectrum of anocean wave. A commonl y used form i s the Pi erson-Moskow tz spectrum gi ven by

(2. 8)u

i n whi ch S(w) i s one- si ded ( i . e. O < w < =) energy spectral densi ty,Hs = si gni f i cant wave hei ght and ~ = peak f requency cor respondi ng to theenergy spectral peak.


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0. 05 I 1 I I 1 I II ! I0.020.01

I

0. 0002} f ~ i /1= I

d-gT2

FI GURE 2. 1 REGI ONS OF VALIDI TY FORVARI OUS WAVE THEORI ES [AFTER LEMEHAUTF(1976)]


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Whi l e t he above f orm of t he energy densi t y spect rum may be used as amodi fi ed two parameter spectrum havi ng Hs and ~ as i ndependent Parameters-the P-M spectrum i s a one-parameter spectrum of a ful l y-devel oped sea i n whi chHs i s rel ated to ~ by the rel ati onshi p

2o s= 0. 1619 (2. 9)

Thus, gi ven a si gni f i cant wave hei ght , the peak frequency can be determnedand vi se versa. Recent l y, Buckl ey (1986) anal yzed ocean wave data obtai nedfrom measurements at a pl atform i n the Gul f of Mexi co, NOAA data buoys, NavySOWM data and dat a f rom Canadi an and Great Lakes water s. Based on thesi gni f i cant wave hei ght and peak peri od of t hi s set of dat a, an empi r i calboundary descri bi ng the l i mti ng steepness was obtai ned as

2o s= 0.3069whi ch i s about twce

For a frequencyare used di rect l y.

(2.10)

that prescri bed by the P-M spectrumdomai n anal ysi s, these spectrum formul as (e. g. , Eq. 2. 8)A wave prof i l e i s simul at ed f rom such a spect rum for a

t ime domai n anal ysi s. One of the st rai ght forward methods of si mul at ion oft he t ime seri es i s t he l i near superposi t i on method of di vi di ng the energydensi t y spect rum int o several sl i ces of w dth, Am. Then the wave hei ghtrepresenti ng the energy under these sl i ces i s gi ven by the formul a

Hi (ui ) =2 f7qq%i (2.11)

where ~ corresponds to the central f requency of the sl i ce. The correspondi ngper iod i s gi ven by

(2. 12)

Thi s, then, gi ves the component of the wave represent ing the f requencyi nterval , i , gi ven by the wave hei ght - peri od pai r, (Hi , Ti ) . The phaseangl e i s assumed uni formy di str ibuted over (O, 21T) and i s chosen randomy.Then the prof i l e of the random wave i s obtai ned by addi ng al l the componentsof the wave thus generated


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N H.n(x, t) = E ~cos(ki x - w t + vi ) (2. 13)j =lwhere ki = 2~/Li , Li = wave l ength at a f requency O+ and N = number of sl i cesmade i n the wave spectrum (typi cal l y 50-200).

The random wave prof i l e produced by thi s method i s Gaussi an onl y i n thel i m t as N extends to i nf i ni t y. In order to avoi d thi s probl em [Tucker , etal . ( 1984) ], Eq. 2. 13 may be rewri t ten i n terms of two coef fi ci ents, ai , bi( i nstead of vi ) whi ch are funct ions of the cosi ne and si ne component of (ki x -Wt). These coef f i ci ent s are then assumed to be randomy di st ri buted i n aGaussi an form to ensure n to be Gaussi an. I t has been shown [El gar, et al . ,( 1985) ] t hat t he group st at i st i cs of t he wave prof i l es by ei t her of t he twomethods produce si ml ar resul ts.

I n a determni sti c approach the maxi mum wave cycl e i n a randomwave fi el di s often used to obtai n the desi gn response val ue. Such cycl es are general l yhi ghl y nonl i near w th sharper crast s and requi re hi gher order wave theory(e. g. stream functi on theory) to descri be the wave cycl e.

A method of si mul ati on of nonl i near random seas was provi ded by Hudspeth(1975). The methodcomponents by a Fastthe l i near randomfrequenci es that arethe product of the

i nvol ves i nvert i ng the Fouri er wave ampl i tude spectralFouri er al gori thm Second order correct i ons are made tosea surf ace. These nonl i near components appear atsums or di fferences of the l i near frequenci es and i ncl udel i near spectral components. The nonl i near random sea

sur face i s der i ved f rom a l i near si mul at i on. The ki nemat i c f i el d, i . e. waterpar ti cl e vel oci ty and accel erat ion are obtai ned by a di gi tal l i near f i l ter ingtechni que.

2. 3 WAVES PLUS CURRENTWhen current i s present al ong w t h the waves, the current i s of ten

consi dered steady and i ts ef fect i s l i near l y super imposed on the ef fect of thewaves on responses. I t i s somet imes found that the combi ned effect of wavesand current on the responses may be di f ferent f rom thei r i ndi vi dual ef fectsl i nearl y superi mposed because of wave- current i nteracti on. Thi s i sparti cul arl y true for a movi ng structure for whi ch the moti on may become qui tecompl ex and nonl i near even for l i near waves. I n addi t i on, however, when

8


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current i s i n the di rect i on of waves there are addi t i onal changes experi encedby the waves.

On encounteri ng a current, the characteri sti cs of a wave change. I npar ti cul ar , i n the presence of cur rent the wave hei ght and the wave l engthexperi ence modi f i cati on. I f the current i s i n the di recti on of wavepropagati on, the wave sl ope decreases and i ts l ength i ncreases. On the otherhand, i f the current opposes the wave, the wave sl ope i ncreases i n magni tudeand the wave l ength shortens. These changes take pl ace due to the i nteracti onbetween the waves and current.

I n deepwater i n the presence of a uni form current the wave number, k, i srel ated to the wave frequency, u, by the general i zed di spersi on rel ati onshi p

k= 4&g[1+ (l +4uMg)VJ * (2. 14)where U may be posi t ive ( i n the di rect ion of wave propagat ion) or negati ve(opposi te to the wave di recti on) . Note that the expressi on i n Eq. 2. 12reduces t o t he deepwater di spersi on rel at i on ( k = u2/ g) i n t he absence ofcurrent (U = O). When U i s posi t i ve the val ue of k i s smal l er so that the+ wave l ength i s l arger. Li kew se when U i s negat ive, the val ue of k i ncreasesand the wave l ength i s smal l er than the no-current case.

The wave-current i nteract i ons i n a random wave fi el d show that the waveenergy densi ty spectrum l i kew se undergoes profound changes. Under the acti onof a steady current i n deepwater, the wave ener~ spectrum takes the form

S*(U) = 4 S(LJ(l +!A : 1 2 [1+(1+ ~)1212(2. 15)

When the current speed i s negat i ve there i s a cut -of f f requency i n the surfacewave spect rum gi ven by the condi t i on 1 + 4Umg = O beyond whi ch no wavesexi st . Si nce the phase speed, c (= w k) , of gravi t y waves i s a monotoni cal l ydecreasi ng funct i on of wave number and frequency, the i nf l uence of currentw l l be predomnant at the hi gher wave number range. Furthermore, thecontr ibut i on from the hi gher wave number range domnates the wave surfacesl ope whereas the current changes the surface sl ope pattern drasti cal l y. Thi si su

demonstrated i nfor di f f erent

Fi g. 2. 2 i nval ues of

whi ch- the rat io of S*(W/S(m i s pl ot ted versussteady current w th and agai nst the wave

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I NLI NE CURRENT. - OPPOSING CURRENT

105

Io- -.& --

0.5?z>3~ 0.1

0.05 .,-

,

0

U IN FT I SEC.

~, RAD./ SEC.

FIGURE 2.2RATl O OF ENERGY SPECTRA WTH AND WITHOUT CURRENT UNDERDI FFERENT CURRENT CONDI TI ONS ; U I NM SEC [ HUANG ET AL. (1972) ]


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o-

a

I NLI NE CURRENT OPPOSING CURRENT

U IN FT/SEC./ \ Uw=ML. / HR./ \

I A/ \/ = -3.0 FT / SEC.I/II//\ \ . - -

/ .6. 5 1: 5 2:0 2.5~: RAD / SEC.

FIGURE 2.3SURFACE WAVE ENERGY SPECTRA UNDER DI FFERENT CURRENT CONDI TIONS

[ TUNG AND HUANG (1972)]


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di recti on. I t i s seen that the ef fect of current at the l ow frequency i ssmal l . At hi gher f requenci es the ef fect of cur rent i ncreases energy l evelwhen i t opposes waves and decreases when i t i s w th the waves. Thi s i sf urther i l l ustrated i n Fi g. 2. 3 where S*( U) i s shown versus m The waveenergy densi ty spect rum S(W represents a P-M spect rum for a w nd speed ofUw = 20 mles/hr . The spect ramean water l evel are gi ven by

su*(m) = (? S*(U)and

s** (m) = #s*((ll)

Fi gure 2.4 shows the ef fect ofvel oci ty spect rum S*U(U) forSpectra of water parti cl e acce

2. 4 NONLINEAR FORCEI t i s cl ear f rom the prev

of f l ui d par ti cl e vel oci ty and accel erat ion at

(2. 16)

(2. 17)

wave current i nteracti on on the water parti cl edi f ferent current speeds of U = t3 f t/ sec.

erati on exhi bi t si ml ar characteri sti cs.

i ous sect i ons that nonl i near waves w l l producenonl i near responses even i f the transfer mechani sm i s l i near. On the otherhand, for a l i near wave the responses are nonl i near i f the t ransfer funct ioni s nonl i near. Thus t he responses of a mari ne st ruct ure w l l be nonl i near i fthe exci t ing forces ar isi ng f rom ( l i near ) waves are nonl i near . One of themost common types of dynamc nonl i neari ty encountered i n the exci ti ng forcesi s due to the drag f orce. The nonl i near steady drag force due to w nd andcurrent i s wel l - known. Extendi ng thi s form to the case of waves, addi ng thei ner ti a component and taki ng i nto account of the reversal of force i n a wavecycl e, an empi ri cal formul a was proposed about 25 years ago whi ch i s commonl yknown as the Mori son equati on.

2. 4. 1 Mori son Equati onThe Mori son equati on was devel oped by Mori son, et. al . ( 1950) f or

descri bi ng the hori zontal f orces on a verti cal pi l e. I t i s wri t ten i n termsof the water parti cl e vel oci ty and accel erati on components as

f = kM O + kDl ul u (2. 18)

10


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. ..-. . .

.

INLINE CURRENT- - - OPPOSI NG CURRENTU I NFT./ SEC.Uw=ML. / HR.

-u= -3.0 FT. /SEC .

o@

oi i iA

/I.: p Iv /% IINo . /pmI L /

/o .N III

o- -

~0 0.5 1. 0 1.5 2,0

. u=3FT. / sEc.

O, RAD. /SEC.

F-5

FIGURE 2.4VELOCI TY SPECTRA UNDER DI FFERENT CURRENT CONDI TI ONS

[ TUNG AND HUANG (1972)]


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KC= 20:M ~

;~6:I-II 0,5 I.0 5-..CD Rex 10-5Re x 10-5

LEGEND K/D~ 1/50~ 1/100~ 1/200~ 1/400. i/coo---- -- 0

KC= 40 KC= 100CM CM2.0 2.0 /1.8 -I .6 -I .4 -1.2r I .2 - // I t @1,,,,1/0 I I 1 1 I 1 1 I1 I1.0Ml 1 I I I 1I 1 I I 1 1 I I I 1* /1015 0.5 5 10 u Q5 Lo 2 5 IolRe~ l@2 Rex 1(35c.iiiI .4

~1.2 -1.0 -0.s -0. 6- x ---1 1 I 1 1 I ll 1 I I I 11

a15 051a25 nR@ x 10-5 1

.61.1.21.0.60.6 =--0.4L 1 I I I I I II I 1 I I 1 I I Ill020.5 1.0 2 5 10Re x 10-5

FI GURE 2.5 VALUES OF CM AND CD AS FUNCTI ON OF Re, KC AND ROUGHNESSCOEFFI CI ENT (K/ D) [FROM SARPKAYA ET AL. (1976) ]


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i n whi ch

M =pCM D2 (2. 19)

andkD=; pCDD (2. 20)

and f = hydrodynamc force per uni t l ength of the vert ical cyl i nder , P = massdensi ty of water , D = cyl i nder di ameter , u and d = water part ic le vel oci ty andaccel erati on, and CM CD = i nerti a and drag force coeff i ci ents respecti vel y.

Thi s empi r i cal f orce model has been the most wdel y used method i ndetermning f orces on smal l di ameter vert i cal cyl i ndri cal members i n anof fshore structure. The computat i on depends on a know edge of the waterpart icl e ki nemat i cs and empi r ical l y determned coef f i ci ents. Extensi veresearch ef for t has been expended i n the past i n obtai ni ng the val ues of theforce coeff i ci ents, CM and CD. I n thi s area, the most noteworthy l aboratoryresul ts on CM and CD were produced by $arpkaya [see Sarpkaya and I saacson(1981) ] fr om hi s U- tube experi ments. Hi s data show that these coeff i ci entsare functi ons of the Keul egan-Carpenter number (KC), Reynol ds number (Re) androughness parameter of the cyl i nder. The Keul egan-Carpenter number i s ameasure of the water part icl e orbi tal ampl i tude w th respect to the cyl i nderdi ameter and i s def i ned as KC = oT/D where U. i s the ampl i tude of the waterpart ic le vel oci ty. Typi cal resul ts for CM and CD from Sarpkaya s experi mentsfor di f f erent values of KC are shown i n Fi g. 2.5. Hi s anal ysi s shows that forsmooth cyl i nders, the val ue of CD approaches 0.65 and CM approaches 1.8 athi gher val ues of Re. In waves, these val ues from pure 2-D osci l l atory f l owshoul d probabl y be consi dered an upper l i mt. Li mted correl at i on of thesedat a i n waves has been made. One such correl at i on i n a l i m ted range of Rewas made by Chakrabar ti (1981) i n Fi g. 2.6. Not e t hat t he correl at i on i squi te good except f or CM near KC = 10 and at hi gher val ues of KC whereChakrabarti s data are sparse and need further veri fi cati on.

The Mor ison equat ion has been used i n the appl i cat ion of both regul arwaves and randomwaves. I n a desi gn, the coeff i ci ents i n the randomwaves areof ten chosen f rom the regul ar wave tests and assumed constant w t h

11


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3. 0- LEGEND2. 5- CHAKRABARTl- - - - - SARPKAYA2. 0- -

CM 15 - - - - - - - - - - - - - - - - - -1. o- - - - - - - - - - - - - - - -\. . -0. 54

-----+ ------ --- -Re = 2X104I .5- ------- 3------ -----

0.5-

0 10 20 30 40 50 60 70u OT/D

FI GURE 2. 6 COMPARI SON OF SARPKAYA S TWO-DIMENSI ONAL FLOWTEST RESULTS WTHWAVE TANK TEST RESULTS


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1000

100

m

100- 90% DRAG

50

xINERTIA10- AND DRAGcm# o

CD #Oy 5.0 -

10 % DRAG

mLARGE0.5 INERTIA

1 % DRAG

o. 1- NEGLIGIBLE-. DIFFRACTION0.05

11ALLINERTIA

c~=o0. 01! 10. 01 0D5 0.1

DEEP WATER--- BREAKING WAVE

m

---------------- --

IrDIFFRACTIONREGION

s5 1. 0 50 )kR

FI GURE 2. 7 REGI ONS OF APPLI CATION OF WAVE FORCE FORMULAS


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f requency. The coef f i ci ents have general l y been der ived i n the l aboratory i nosci l l at ing mot ion or i n regul ar waves. The dat a f rom ocean t est s haveproduced l arge scatter whi ch does not val i dat e the appl i cabi l i t y of theMori son equati on. I n a test w thand Bouquet (1985) measured theki nemat i cs at a smal l sect i on ofconsi dered these si gnal s as output

a vert i cal cyl i nder i n a wave t ank, Vugt sforces and correspondi ng water part icl ethe cyl i nder i n random waves. Then theyand i nput si gnal s respecti vel y, and appl i ed

the measurements to a general transformati on model consi st i ng of l i near andnonl i near paths. They chose four model s, one of whi ch corresponded to theMori son equati on. The Mor i son equat ion was found to be the best sui t ed,gi vi ng a good match between the two si gnal s. The i ner ti a coef f i ci ent wasfound to be reasonabl y constant for a gi ven frequency spectra whi l e the dragcoeff i ci ent decreased i n val ue w th frequency.

The Mori son equati on has been extended to i ncl i ned cyl i ndri cal members ofan offshore structure i n terms of a normal vel oci ty component, ~, and a normalaccel erati on component, ~. I n thi s case, the force i s wri tten as a vectorquanti ty

~=kM~+kDl~

The force vector per

w (2. 21)

[ni t l ength of cyl i nder may be decomposed i nto i ts threecomponents al ong 3 axes XYZ by wri ti ng

Y u~t + Uyi + u~k

and

(2. 22)

(2. 23)

I t has been shown through experi ments [Sarpkaya (1984), Garri son (1985)] thatthe coef f i ci ents CM and CD for an i ncl i ned cyl i nder may be obtai ned as thoseval ues from the vert i cal cyl i nder tests.

The expressi on f or t he i ncl i ned cyl i nder , Eq. 2. 21, i s general enoughthat the f orces on a smal l cyl i nder i n any pl ane may be obtai ned f rom i t .Thi s f ormul a i s appl i cabl e t o der i ve f orces f rom vari ous t ypes of of f shore

12


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st ructures and st ructure components. Some of these are j acket st ructure,ri sers, tendons, arti cul ated- tower, l egs of semsubmersi bl e and guyed tower.I t shoul d be noted, however , that the appl i cabi l i t y of Eq. 2.21 to a randomyori ented cyl i nder needs further i nvesti gati on.

The regi ons of appl i cabi l i ty of the Mor ison equat i on and, i n part icul ar ,the areas of drag and i nert ia force predomnance may be di scussed i n terms ofthe chart i n Fi g. 2. 7. The chart has been obtai ned by examni ng the rat i o ofthe maxi mum drag force, fDO, to the maxi mum i nert ia force, f l O, for a cyl i nderi n l i near waves. Note that

DO CD = ~ (KC) (2. 24) 10 M

where KC i s the Keul egan-Carpenter number. Assumng CD = 1 and CM = 2, thepercentage of drag to i nerti a may be establ i shed. The l i m ts are stated i nterms of the KC number, and the di ffract i on parameter, kR = mD/ L, R = cyl i nderradi us. Accordi ng to thi s char t the nonl i near for ce due to the drag ef f ecttends to become i mportant when KC becomes greater than 2. The wave force fromthe Mori son equati on becomes mostl y drag for KC >90.

4 By vi rtue of the form of the drag term the drag force component i snonl i near i n the ti me seri es even i f the water parti cl e vel oci ty i ssi nusoi dal . On the other hand, t he i nert i a t erm i s l i near i f t he si nusoi dalwat er part i cl e vel oci t y ( e. g. by l i near wave theory) i s used f or t he ( l ocal )accel erati on. I f t he l ocal hor i zontal accel erat i on i s repl aced by the totalhor i zontal accel erat ion i ncl udi ng convect ive terms, the i ner ti a term has anonl i near form

DU _ au QL+v M+WDt %+uax ay (2. 25)

i n whi ch u, v and w are the components of the water par ti cl e vel oci t y vectori n a rectangul ar Cartesi an coordi nate system Wave force data reduced on thebasi s of nonl i near ( i rregul ar) stream f uncti on theory dependent on themeasured wave prof i l e and l ocal measured forces have shown sat isfactorycorrel ati on w th measured total forces [Chakrabarti (1980)] .

I n addi t i on to the extensi ons, of the Mor i son equat ion stated above,several modi f i ed f orms of t he f ormul a are used i n t he of f shore st ruct ure

13


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desi gn. These have to do w th combi ni ng di fferent envi ronmental effects i ntothe formul a. The two most i mpor tant of these are cur rent and st ructuremoti on. Cur rent can be appl i cabl e to al l three types of st ructures whi l e thestructure mot i on i s i mportant onl y for a f l oat i ng or f l exi bl e structure.

2. 4. 2 Fi xed Cyl i nder i n Waves and CurrentWhen current i s present w th waves, the formul a for a f i xed st ructure i s

wr i t ten i n terms of the total vel oci t y i ncl udi ng a steady cur rent , 11, and anosci l l atory component, u as

f =kM~+kD

where -U representsti on.

However , i t i sadequatel y express

I utul (U. tu)

uni form current

someti mes arguedthe total drag

(2. 26)

opposi ng the di recti on of wave propaga-

that a si ngl e drag coef f i ci ent does notforce i n the presence of waves and

current. An al ternate form of the Mori son equati on has been suggested

f=kMi +kD

where ED i s defi ned

~D=~P~DD

I ul u +rD U2

i n terms of a steady drag coef f i ci ent , ~D as

The Keul egan-Carpenter number and the Reynol ds number i n af i el d are def i ned as

(uOt U)TKC= D~e (uOf U) D=

vwhere U. = ampl i t ude of u and v = ki nemati c vi scosi ty of water.

I t shoul d be emphasi zed that the val ues of the hydrodynamc

(2. 27)

(2. 28)

wave- current

(2. 29)

(2. 30)

coef f i ci entsi n the wave-cur rent f i el d are expected to be di f f erent f rom those i n wavesal one. Unfortunatel y, such data are l i m ted. I wagaki , et al . (1983)

14


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presented val ues of CM CD versus KC from a combi ned wave-current test. Theseval ues shown i n Fi g. 2. 8 are not much di fferent from the wave al one data.

Because of t he di f f i cul t y of generat i ng waves on a st eady current anal ternat ive and of ten consi dered equi val ent approach i s taken. Sarpkayaet al . (1984) had adopted one such method i n hi s U-tube i n whi ch the cyl i nderwas moved st eadi l y i n an osci l l ati ng f l ow f i el d. They used the rel at ivevel oci t y model (Eq. 2.26) to der i ve CM and CD. Resul t s obtai ned f rom such atest on CM and CD are shown i n Fi gs. 2. 9- 2. 10. Reference may be made toSarpkaya s (1984) paper for other si ml ar data.

Moe and Verl ey ( 1980) took a sl i ghtl y di f f erent approach. Theyosci l l at ed a hor i zontal cyl i nder si nusoi dal l y i n a uni f orm cur rent f i el d andmeasured forces on the cyl i nder. They used the three- term Mori son equati on,si m l ar to Eq. 2. 27, w th the excepti on that u i s repl aced by ~, J by ~ andkM by kA, where x i s the ampl i tude of the osci l l ati ng cyl i nder and A = PCAmD2/ 4. . The coeff i ci ent, CA, i s def i ned as the added masscoef fi ci ent f or the osci l l ati ng cyl i nder and i s rel ated to CM by CM = 1 + CAfor a buoyant cyl i nder. They deri ved the val ues of~D , and Fouri er averagedCA and CD. The coef f i ci ents CA and CD showed compl ex dependenci es on the*ampl i tude parameter x = xo/ D and the reduced vel oci ty,

v~ = UTO/ D (2. 31)

where To = peri od of cyl i nder osci l l at i on. The pl ot of ~D vs. ; f or vari ousval ues of VR i s shown i n Fi g. 2.11.

From the above test s i t i s obvi ous that the val ues of the hydrodynamccoef fi ci ents are di r ectl y rel ated to the f orm of the f orce equati on used,e.g. , i ndependent f l ow f i el d or rel at i ve vel oci ty model . The advantage of thethree- term Mori son equat i on i s that the steady drag f orce may be easi l yseparated from the osci l l at i ng part , e. g. , i n the anal ysi s of a structuraldynamc probl em However, i t seems si mpl erto use the rel at i ve vel oci ty modelsi nce i t means choosi ng and worki ng w th one l ess coeff i ci ent.

2. 4. 3 Osci l l ati ng Cyl i nder i n Waves

15

When a ri gi d structure i s f ree to move i n waves, the ef fect of thestructure moti on can be combi ned w th the wave effects to form


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3

2CD

1

0

2CM

1

CYLI NDERSYMBOL DI AMETERq 30rnmo 60mm

MEAN CURVES

fi di oOv

o0

q o qo q

o

0

-t--

I 1 1 I01~o 10 20 KC

I

--i-

qq

q

FI GURE 2. 8 I NERTI A AND DRAG COEFFI CI ENTS OF FI XED VERTI CAL CYLI NDER I N ACOMBI NED WAVE-CURRENT FI ELD [ IWAGAKI , ET AL. (1983)]


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. . ,.

Re\KC =1594VR =0. 00 VR =3, 63

x VR=6. 170N~ \

~= ~ /

ko

; 1 I 1

Re/KC =2487VR=0. 00

vR=2. 91x VR 5. 0[od

m- - - , - .Wz00 -~

komd5 7 10 20 30 50

KC - .*

5 7 10 20 30 50KC

FI GURE 2. 9 I NERTI A COEFFI CI ENT FOR A SMOOTH OSCI LLATI NG CYLI NDER I NUNI FORM COLLI NEAR CURRENT (A) Re/ KC = 1594 AND (R) Re/ KC = 2487[SARPKAYA ET AL. (1984)]


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...--.--... ~. .---..

Re/KC =1594VR =0. 00

- - - - VR 3. 63I x VR =6. 170 I. Qgv \

N.0 i

lod 5 7 10 20 30 50KC

Re\KC =24%7VR=0. 00- vR=2. 91 x vR=5. 01

KC

FI GURE 2. 10 DRAG COEFFI CI ENT FOR A SMOOTH OSCI LLATI NG CYLI NDER IN UNI FORMCOLLI NEAR CURRENT FOR (A) Re/ KC = 1594 AND (R) Re/ KC = 2487[S, ARPKAYAET AL. (1984)]


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-c

TESTCYLINDER END

WEIGHTS c!

PLATES. .,mmmr/i r-l

TEST FLUME

I I I I

II I 1 1 I I

I-O*4J 1 I I 10 1.0 2.0 3.0 4.0 5.0Xo/D

FI GURE 2. 11 OSCI LLATORY DRAG COEFFI CI ENT FOR A SI NUSOI DALLY OSCI LLATI NGCYLI NDER IN UNI FORM COLLINEAR CURRENT [MOE AND VERLEY (1980)]


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f =kM8- kA~+kDIUl u -k~l l l x (2. 32)

where kA=pCA~D2/4, k; ~ P C~ D, CA = added mass coeffi ci ent and C~ = dragcoeffi ci ent due to structure moti on defi ned separatel y from the fl ui d dynamcdrag. Thi s f orm i s known as the i ndependent f l ow f iel ds; a far f i el d due tothe wave moti on and rel ati vel y unaff ected by the structure moti on, and a nearf i el d resul t i ng from the structure moti on. The values of CM and CD may beobtai ned from wave experi ments whi l e the coeff i ci ents CA and CL are deri vedfrom the experiments of an osci l l at i ng cyl i nder i n otherw se cal m water. Theval ues of the KC and Re numbers are obtai ned from the respect ive vel oci t iesand peri ods.

When the f orces are wri t ten i n terms of the rel ati ve moti on, si ngl ecoeff i ci ents for the i nert ia and drag are assumed to appl y. Thus, the form ofthe force term i ncl udi ng the structure i nert ia due to i ts accel erat i on,(m term i s

f =kMti - ; ) + kDl u - ~l (u - ~) (2. 33)

Thi s model i s known as the rel at ive vel oci ty model . I t requi res fewercoef f i ci ents than Eq. 2.32, and has been used extensi vel y i n the past , e.g. ,to eval uate the stochasti c dynamc response of offshore pl atforms, moti ons ofart icul ated towers, etc. I n thi s case, the Reynol ds number and KC number aredef i ned i n terms of thd rel at ive vel oci ty, Vr , as

rOTrKC=- Re _ rOD. v (2. 34)

where Vro = ampl i tude of Vr and Tr = combi ned per iod of Vr . Note that Vr neednot be si nusoi dal even i f u and X are.

I t i s somet imes conveni ent to separate the i nerti a coef f i ci ent f rom theadded mass coef f i ci ent . As an exampl e, the di f f ract i on- radi at i on theoryprovi des di f ferent val ues for the force and added mass coef f i ci ents. There-fore, a thi rd al ternat ive form of the modi f i ed Mor ison equat ion i s wr i t ten i nterms of the rel ati ve vel oci ty, and the accel erati on terms are. separated.

16


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f =kMi - kA; +kDl u - x1(u - x) (2. 35)

The questi on ari ses as to whi ch i s the more appropri ate f orm of themodi fi ed Mori son equati on for a structure movi ng i n waves. Si nce there i s avari ety of off shore structures, e. g. , j acket pl atforms, arti cul ated col umns,r isers, tensi on l eg pl at forms, that fal l under thi s category i n whi ch the drageffect i s i mportant, i t i s wor thwhi l e to di scuss the appropr iate and usefulappl i cati ons of Eqs. 2. 32, 2. 33 and 2. 35. Because the or igi nal Mor isonequati on i s empi ri cal , i t i s not possi bl e to j usti f y i t s extensi on to othercases and to di scuss whi ch one i s more correct. Obvi ousl y, coef f i ci ent sderi ved f rom one of these formul ati ons can be j usti f i abl y used i n theappl i cat i on of t hat f orm onl y. However, experiment al dat a i n t hi s area i sscarce. An at tempt to i nvest igate thi s area was made by Chakrabar ti et al .(1983-1984) through model testi ng. An arti cul ated col umn was tested i n threemodes w th the same setup: (1) f i xed i n waves, (2) harmoni ca l y osci l l ated i nst i l l water , and (3) f ree to move i n the pl ane of the waves. The ampl i tude ofvel oci ty of the structure was comparabl e to the water part icl e vel oci ty. Thetest showed that the rel ati ve vel oci ty f orm of the Mori son equati on i sappropri ate, even though the observati ons were l i mted. Consi derabl e work i sneeded to determne the appropri ate val ues of CM and CD i n the rel ati vevel oci ty model i n waves.

An experi ment w th a submerged arti cul ated tower was performed recentl yby Dahong, et al . ( 1982) i n whi ch t he mot i ons of t he t ower both i n- l i ne andtransverse to the di recti on of waves were measured. From these measurementsthe val ues of CM CD f rom a rel ati ve vel oci ty model (Eq. 2. 33) and l i f tcoeff i ci ent, CL, were deri ved. The mean val ues of these coeff i ci ents versusKC are presented i n Fi g. 2. 12.

The regi on of appl i cabi l i ty of the rel at i ve vel oci ty and i ndependent f l owfi el ds model may be di scussed i n terms of reduced vel oci ty, VR (defi ned by U.i nstead of U i n Etappl i cabi whi l e the

For

q 2. 31) and an ampl i tude parameter, ~ . The l im ts ofi t y are g ven i n Fi g. 2. 13. The x-axi s i s the reduced vel oci ty, VRy-axi s i s the KC number based on the water part i cl e vel oci ty.compl i ant structures, e. g. arti cul ated towers, KC, VR and i are

rel at ivel y l arge. For convent ional j ackets,smal l . I n both cases the f l ow i s quasi -steady

17

KC and VR are l arge, but ~ i sand the per iods of osci l l at i on


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d

CMI

CD

1,0

CL o004

I

.

.\

.I I 1 I 1 1 I I 1 1 1 I 1 I I

) 2 4 6 8 10 12 14 16 i s 2KC

FI GURE 2. 12 HYDRODYNAMC COEFFI CI ENTS FOR A SUBMERGED ARTI CULATED TOWER I NTWO DEGREES OF MOTI ON [DAHONG, ET AL. (1982)]


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KC

10-15

0 0 10:15

FI GURE 2. 13 QUALI TATI VE REGI ON OF APPLI CABI LI TY OF I NDEPENDENT FLOl i FI ELDAND RELATIVE VELOCI TY MODELS


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of the st ructure general l y coi nci de w th the per iods of the i nci dent waves.The use of the rel at i ve vel oci ty term i n the computat i on of the drag force maybe appropri ate i n these cases.

Two other cases may be consi dered. High KC and smal l VR val ues cor-respond to a resonat i ng drag domnated st ruct ur e, i . e. a hi gh- f requencycyl i nder osci l l at i on i n a sl ow y osci l l at ing external f l ow Exampl es of thi scase are vi brat i ng structures at hi gh resonant frequency such as ri ser cabl es,TLP tendons, etc. Si ml ar ly, l ow KC and hi gh VR val ues mean a l ow frequencycyl i nder osci l l at ion i n a hi gh f requency f l ow osci l l at ion. Thi s second casei ncl udes the sl ow y-osci l l at ing dr i f t mot ions of a moored st ructure, e.g. ,shi ps, semsubmersi bl es, TLP surge, f l oat ing cai ssons, et c. I n t hese twocases, the concept of rel ati ve vel oci ty appl i ed i n Eqs. 2. 33 and 2. 35 i shi ghl y suspect and the i ndependent f l ow f i el ds model , Eq. 2. 32, i s ap-pl i cabl e. The mai n reason f or t hi s choi ce i s t hat t he two mot i ons are qui t edi fferent and rel at i vel y i ndependent of each other. Thus, the smal l er moti onsare capabl e of creat ing l ocal wakes i ndependent of the l arger mot ions. Therel at i ve vel oci ty model accounts for thei r combi ned effect, thus i gnori ng thesmal l er moti ons. The two drag coeff i ci ents may be chosen from the two typesof t est dat a, one f rom a f i xed cyl i nder i n waves and one f rom an osci l l at i ngcyl i nder i n st i l l water (or al t ernatel y, osci l l at ing water past a stat i onarycyl i nder) . The KC and Re numbers are computed from the i ndi vi dual vel oci ti esfor thi s purpose.

A simpl e techni que may be empl oyed i n determni ng whi ch of the twomodel s, rel ati ve vel oci t y and i ndependent f l ow f i el d, i s appl i cabl e i n aparti cul ar appl i cati on. When the two fl ows are comparabl e, one i nfl uences theother and the rel at i ve vel oci t y model i s appl i cabl e. The i ndependent f l owf i el ds model may be used when one of the vel oci t ies i s l arge compared to theot her. The appl i cabl e coef fi ci ents are chosen based on the test resul t sobtai ned f rom the correspondi ng model s.

2. 4. 4 Osci l l ati ng Cyl i nder i n Waves and CurrentFor a st ructure f ree to osci l l at e i n the presence of waves and cur rent ,

the Mori son equati on i s modi fi ed as

f kM i - kA ; +kDI Ut U- kl (Ut U-A) (2. 36)

18


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Other f orms of Eq. 2. 36 may be wri t ten as bef ore and have been used in t hepast. These forms are appl i cabl e to movi ng structures i n waves and currentwhose member si zes are such that the hydrodynamc drag force i s si gni f i cant.Even though the equati on i s wri t ten t oanal ysi s, the terms f rom Eq. 2.36 appearmoti on. For example, the f i r st term onf uncti on. The second term i s an i nerti asi de of the equati on of moti on. The thi rd

def i ne a force term i n a moti onon both si des of the equati on ofthe ri ght hand si de i s a forci ngterm and bel ongs to the l ef t handterm i ncl udes both a force term and

a dampi ng term coupl ed together. I f thi s term i s l i neari zed, then the twocomponents may be uncoupl ed i nto two terms bel ongi ng to the two si des of theequat ion. In a t ime domai n anal ysi s i t i s t reated as a dampi ng term

Test resul t s under t hi s condi t i on, however, are al most nonexi stent.Consi derabl e work i s needed to achi eve i nsi ght i nto thi s most compl exprobl em The force and the mot ion are dependent upon the water par ti cl eki nemat ics as wel l as the vel oci ty and accel erat ion of the st ructure i t sel f .Because of the lack of data in thi s area, the hydrodynamc coef f i ci ent s forthe anal ysi s of such probl ems are chosen f rom studi es sim l ar to thosedescri bed i n the precedi ng secti ons.

2. 5 STEADY DRI FT FORCEThe second-order theory for the steady dr i f t force i s based on the f i rst -order di ffract i on-radi at i on theory and i s appl i cabl e to regul ar waves. The

regul ar wave resul ts are then appl i ed to wave groups and i rregul ar waves. I naddi ti on, a steady dr i f t f or ce develops f rom the drag for ce term at the f reesurface as wel l as i n current .

I n the f ol l owng secti on, the steady dri f t f orce due to vi scous f l ow i sdi scussed. I t i s general l y appl i cabl e to structures that have members i n thedrag- domnated areas ( ref er to Fi g. 2. 7) , e. g. , i n j acket structures, TLPtendons, etc.

2.5. 1 Steady Dri ft Force Due to Vi scous Fl owThe forces on a smal l vert ical cyl i nder due to l i near waves may be obtai -

ned f rom the Mori son f ormul a by substi t uti ng u( t) = U. cosut i n Eq. 2. 18.Then, noti ng that ; (t) = - uuo Si nm ,

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f=- kMuuosi nut+kDu~

Thi s f orm of the wave f orce at awave cycl e. If the cyl i nder i s

Ices m l cos ( l )t (2. 37)

submerged l ocati on has a zero mean over oneal l owed to osci l l ate harmoni cal l y i n waves

w th a di spl acement ampl i t ude of X. at a phase angl e of a w th respect t o t hewave so that x(t) = X. Cos(ti + a) , then the rel at i ve vel oci t y model(Eq. 2. 33) gi ves

f = - kM AI si n(ti + ~) + kD V2 I cos(ut + $) l cos(m + $) (2*38j

i n whi ch the quant i t i es V and $ are def i ned as

v [u: + (Wo)z - 2 MU. X. Cosl a] l l z

andwxo si n a

+ = tan- l ( u - uxo cos a )o

(2. 39)

(2. 40)

The expressi on i n Eq. 2. 38 al so has a zero mean. Compari ng Eqs. 2. 37 and2.38, i t i s cl ear that for a movi ng st ructure i n waves, U. shoul d be repl acedby V and ut by wt+ $. pence, the subsequent deri vati ons are done onl y for afi xed cyl i nder.

Note that there a re two areas that w l l prod, ucea non-zero mean vi scousdr i f t force. When current i s present al ong w t h waves, a mean dri f t f orce i sgenerated from the drag force at any el evati on of the cyl i nder. Moreover, dueto the changi ng f ree surf ace of the waves at the cyl i nder, the f orce w l lproduce a mean dr i f t at the st i l 1 water l evel (SWL).

I n the presence of current , U, t he rel at i ve vel oci t y drag f orce may bebroken up i nto two si mpl erbetween U and Uo. For 1111> expressi ons dependi ng on the rel ati ve magni tudeUO*

+ cos 2ti ) + 2 U U. COS d] (2. 41)where the negati ve si gn appi es t o t he case of current opposi t e t o t he wave

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di rect i on, and for

fD = kD [U* +

UI


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O EXE DATA THEORYo.w

30d

m00~oGo>+I - L

y~- 2. 0 - 1. 2 - 0. 4 0. 4 1, 2 :

@~

FIGURE 2.14MEAN VI SCOUS DRI FT FORCE ON A FI XEDVERTI CAL CYLINDER I NAWAVE-CURRENT FI ELD[CHAKRABARTI (1984)]


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TABLE 2.1

kH0. 050. 150. 250. 350. 450. 550. 650. 750. 850. 95

VALUES OF c1 AS FUNCTIONW kH

c11.00031.00301.00841.01641.02721.04091.05741.07681.09941.1251

kH0.100. 200. 300. 400. 500. 600. 700. 800. 901. 00

c11.00131.00531.01201.02151.03371.04871.06671.08771.11181.1392


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i n whi ch g = accel erat i on due to gravi ty, k = wave number, H = wave hei ght ,d= water depth and y = el evat i on from bottom Assumng that the l i nearth@ory can be appl i ed up to the free surface, the total force is obtai ned f romthe i ntegral

whi ch provi des

F . ~M 9H si nh k( d + ~)= l k gkH2 cosh kd si n m +7 D( 2ucosh kd )2

[ (d + ~) + si nh 2k (d + ~)2k ] ]Cos ut ICos W

(2. 47)

(2. 48)

The i nerti a part of the f orce has a zero mean. The drag f orce yi el ds anaverage val ue over one wave cycl e.

F =J-#[& +C,(W COth Zkd 1q (2. 49)The numeri cal val ues of Cl are shown i n Tabl e 2. 1 as f uncti ons of kH. Notethat Eq. 2. 49 i s due onl y to the f ree surface vari ati on even though thei ntegrat i on i n Eq. 2.47 i s carri ed out over the ent i re submerged l ength. Themean val ue, however, i s a funct i on of the water depth.

Si nce the l i near theory i s appl i cabl e for i nf i ni tesi mal wave ampl i tudesand i s val i d up to the SWL, use of the expressi ons f or the water parti cl eki nemati cs up to the f ree surface of a f i ni te wave i s questi onabl e.Therefore, stretchi ng formul as have been suggested i n fi ni te water depth bywhi ch the water par ti cl e ki nemat i cs at the wave crest and the wave t roughassume the same val ues. If one of these stretchi ng formul as i s appl i ed, thewater par ti cl e vel oci ty i s wr i t ten as

= gkH cosh kY (~) cos utu z cosh kd (2. 50)

and ~ i s expressed i n terms of U. and u as done earl i er, t hen the t ot al f orceup t o t he f ree surf ace i s gi ven by

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(d + n) [1 ++ ]ICOS ut l cos t i t (2. 51)The mean val ue of F i s obtai ned on i ntegrat i on as before.

T (kH)32kD g/ k2 ~ [ si n; 2kd +* I (2. 52)

Thus, the mean force from the free surface ef fect of a smal l vert ical cyl i nder(where vi scous ef fect i s important) i s a f uncti on of the cube of the wavehei ght as opposed to the square of i t f or the potenti al dri f t f orce, as w l lbe found i n the fol l ow ng sect i on.

Note that i n deep water , M2 = gk and the expressi on i n Eq. 2. 49 approxi -mates as

T c1Zk g, k2 ~(kH)3 (2. 53)D

However, the mean f ree sur face for ce f rom Eq. 2.52 approaches zero as thewater depth approaches i nf i ni ty. The mean dri ft force from Eq. 2. 53 have beenpl otted i n Fi g. 2. 15. Note that the normal i zed f orce depends on kHapproxi matel y as i ts cube [Chakrabarti (1984)].

2.5.2 Steady Dr i f t Force Due to Potent i al Fl owFor structures that are l arge, the force i s mai nl y i nert ial and potent ial

theory i s appl i cabl e. The steady dr i f t force i s second order and can be shownto ar ise f rom the f i r st -order potent ial . The contr i buti on due to the steadydri ft force from the potenti al f l ow about a f l oat i ng body has four componentswhi ch are addressed i n the fol l ow ng secti on.

2. 5. 2. 1 Wave El evati on Dri ftConsi der the extensi on

Forceof pressures above the mean water l evel t o the

i nstantaneous free surface at the body whi l e the body i s i n mot i on. Then thei ntegrat ion of thi s pressure around the obj ect at the water l i ne gi ves r ise toa steady second-order force whose component, Tl , i n the hori zontal di rect i oni s gi ven by

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0.0 , 62 0.4 0.6 0.8 Lo kH

FI GURE 2. 15 MEAN VI SCOUS DRI FT FORCE FROM FREE SURFACE EFFECT ON A FI XEDVERTI CAL CYLI NDER I N DEEPWATER (kd < T)


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d

n

m

LEGENDWAVE ELEVATION- - - - - VELOCI TY HEAD VI SCOUS DRI FT

.

- - - - - - - - -

1 I 1 I 11 2 3 4 5m

FI GURE 2. 16 MEAN POTENTI AL DRI FT FORCE ON A FI XED VERTI CAL CYLI NDER


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i n whi ch the bar denotes mean val ue, g = accel erati on due to gravi ty,C1 = f i r st order wave ampl i t ude at a poi nt on thecosi ne, and WL = water l i ne at the body surface.

For a f i xed vert i cal cyl i nder t hat ext endssubmerged poi nt where there i s no wave acti on, the

movi ng body, nx = di recti on

f rom the f ree surf ace to abody may be treated as two-

di mensi onal and the MacCa~-Fuchs theory i s appl i cabl e. I n thi s case, thehor i zontal wave el evat ion dr i f t f or ce may be obtai ned i n a cl osed form Theti me- i ndependent steady force component may be wri tten as

~l =4pg~2D ~ ~l -n(n+l )l~2(kR)3 n=O (kR)2

1An(kR) An+l (kR)

where D = cyl i nder di ameterk = wave number, and An(kR) =order n of the f i rst andder ivat ive w th respect to the

( = 2R) , g = i nci dent waveJ : 2( kR) + Y~2(kR) , J n, Yn =second ki nd respecti vel y,arguments.

(2. 55)

ampl i tude (= H/ 2),Bessel funct i on ofand pri me denotes

2. 5. 2. 2 Vel oci ty Head Dri ft ForceThe second term of Bernoul l i s equati on provi des a steady second-order

component when the f i rst-order vel oci ty potenti al i ncl udi ng the di ffract i on-radi at i on effect i s used to compute the pressure. Then the steady hori zontalforce component may be obtai ned f rom the i ntegral

(2. 56)

i n whi ch s = surface of the body and V@ = f i rst-order vel oci ty vector.For t he f i xed vert i cal cyl i nder i n deep water, t he hori zont al vel oci t y

head dri ft force may be cal cul ated usi ng the total vel oci ty potenti al .(2. 57)

Combi ni ng Eqs. 2. 55 and 2. 57 we obtai n the total steady f orce on t he f i xedverti cal cyl i nder.

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I n i ntermedi ate water depths, thi s force i s a

1AnAn+l (2. 58)

so a funct ion of the depth.Numer i cal val ues of the wave el evat i on dr i f t force, ~1 , and the vel oci tyhead dri f t f orce, ~2 , as wel l as the total steady for ce due to the total wave

potenti al i n the presence of a f i xed verti cal cyl i nder are shown i nFi g. 2. 16. These quanti ti es are normal i zed w th respect to pgG2D and pl ottedversus the di f f ract i on parameter , kR. The numeri cal val ues correspond towater depths rangi ng f rom d/R = 1 to reasonabl y deep water, d/ R = 5. Notethat the quanti t y~l i s posi t i ve whi l eT2 i s negati ve over the range of kRconsi dered. Al so, the nondi mensi onal steady dri ft force (~1 +~2 ) approachesa constant val ue of 1/3 i n deep water at hi gher val ues of kR.

For a movi ng cyl i nder there are two other components due to the moti on ofthe body contr ibut i ng to the total steady dr i f t force.

2. 5. 2. 3 Body Moti on Dri ft ForceThe f i rst- order wave force on the body i s computed at i ts mean

posi ti on. However , the body undergoes mot i on due to waves and assumes adi f f erent or i entat i on at the i nstant thi s f or ce i s cal cul ated. Therefore, i fa Tayl or ser ies expansi on about the mean body posi t i on i s made, the second-.order steady hori zontal force term takes the fol l ow ng form

(2. 59)

where ~ = moti on vector.

2. 5. 2. 4 Rotat i onal I nert i a Dri ft ForceThi s term ar ises because the f i r st -order forces due to the pressure are

al ways normal to the sur face. As the vessel osci l l ates the di rect ion of thesenormal s rotate. I f the components of these normal s i n the di rect ions of thef i xed coordi nate system are consi dered, a nonl i near dr i f t l oad devel ops.Then, mathemati cal l y, the second order dr i f t force contr ibut i on assumes theform

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oti

oN-

0- -In0

0

BREAKI NGWAVE LI MT Z. o. [x z= Io Z: l o

o

0+

o

0 0.2 0.4 0.6 0. 8 ! . 0kR

FI GURE 2. 17 REGIMES OF POTENTI AL VERSUS VI SCOUS DRI FT FORCE ON A FI XEDVERTI CAL CYLINDER I N DEEPWATER [CHAKRABARTI (1984)1


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.,

.. .-..

n

2.0

1.0

0

113 0 MEASUREDWAVE -- - CALCULATED( Potential & Viscous)~ CALCULATED( Potential)o

/u f/I

\o 2.5 5.0 7.5 10.0 12.5

WAVE PERIOD , SEC.

FI GURE 2. 18 WAVE DRI FT FORCE ON A TLP I N REGULAR WAVES [KOBSYASHI , ET AL.(1985)]


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4 =- (2. 60)

where F= force vector.Thus, onl y the resul t s based on the f i r st -order vel oci t y potent ial are

needed to obtai n the steady second order force. The total steady dr i f t forcei s obtai ned by addi ng the four components

(2.61)

i n the di rect i on of surge.The precedi ng four contri buti ons ari se from the assumpti on that the fl ui d

f l ow i s i r rot at i onal and the pot ent i al t heory i s appl i cabl e. I n order todetermne whether the vi scous eff ect i s i mportant, cal cul ati ons may be made tocompute the vi scous drag force on the movi ng body f rom the drag par t ofMori son equati on. For a movi ng cyl i nder i n surge, thi s cal cul at i on takesform

D =kDl u- ; l (u -~)

i n whi ch ~ = surge vel oci ty of the cyl i nder . When thi s termthe f ree surface- above the SWL, a steady componentproporti onal to the thtrd power of the wave ampl i tude.

thethe

approxi mati on of thi s expressi on i n the absence of moti on gi ves,~= &kD (gk) L3

(2. 62)

i s extended up toari ses whi ch i s

A deep water

, (2. 63)

whi ch i s comparabl e to Eq. 2. 53 except f or the constant Cl ( = 1 here) andwhi ch may be wri t t en i n a f orm sim l ar t o Eq. 2. 58 as

~ _ 2CD~ (kR)( c/ R)PgG2D (2. 64)

The ef fect of the vi scous dri f t f orce on the cyl i nder i n rel ati on to thedr i f t f or ce cont ri but ion f rom the potent ial f l ow i s shown i n Fi g. 2.16. Con-si der the radi us of the cyl i nder to be 2 f t. and a wave hei ght of 0. 5 f t. f oral l wave peri ods so that L/ R = 0. 125. The val ue of CD i s consi dered t o be

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1. 0. Then, t he val ues of t he vi scous dri f t f orce are as shown in Fi g. 2. 16.Note that , i n general , the vi scous dr i f t force i s smal l compared to the poten-ti al force for al l val ues of kR and i ncreases l i nearl y w th kR. At kR = 2,the vi scous ef fect i s about 15 percent of the potenti al dri f t f orce. Forsmal l er di ameter cyl i nders and hi gher waves the vi scous eff ect can become morepredomnant because of i ts thi rd order dependence on the wave hei ght.

For a general f l oati ng body, i t i s di f f i cul t t o di scuss t he ranges of t hedi ffract i on parameter, kR and vi scosi tyvi scous or the potent ial dr i f t forces areassessment of thei r rel ati ve i mportanceverti cal cyl i nder i n deepwater. I f Z i s

parameter , H/D (or q/R) where thepredomnant. However, a qual i tati vemay be made by consi der ing a f i xedconsi dered the rat i o of the vi scous

to the potent ial dri f t f or ce, then the regi on may be constructed as shown i nFi g. 2. 17 f or di f f erent val ues of Z = 0. 1, 1 and 10. The mddl e curve ( Z = 1)represents equal contr ibut i on from the vi scous and potent i al dr i f t forces.For Z = 10, the pot ent i al dri f t may be negl ect ed j ust as t he vi scous dri f t atz = 0.1.

An exampl e of the ef f ect of a nonl i near vi scous term on the mot ion of aTLP [Kobsyashi , et al . ( 1985)] i s shown i n Fi g. 2. 18. In thi s case, steadydri f t f orce as a f uncti on of the wave peri od i n regul ar waves i s gi ven. Thesol i d l i ne represents the computed val ues based on potent i al theory. Thedot ted l ine incl udes the ef f ect of the vi scous dri f t f orce f rom the rel at i vevel oci ty model . The correl at ion of the exper imental data i s much better w ththe l at ter resul t s. Moreover , the cont ribut ion of the vi scous dri f t f or ce i smuch l arger than the potenti al dri f t f orce at hi gher peri ods ( beyond 7. 5see

SSC 347 NonLinear Analysis of Marine Structures

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