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PAPERNUIIBER 0 TC 2040

Wi II the "Re qu l a r Wave Conc ep t " Yield MeaningfulMotion Predictions for Offshore structures?

By

H. L. Minkenberg and Tan Seng Gie, Netherlands Ship Model Basin, Wageningen

©Copydght 19i1

Offshore Technology Conference on behalf of' the American Institute of Mining, Netallurgical, andPetroleum Engineers, Inc., (Society of Min i.ng Engineer s, The Metallur gical Society and Society ofPetroleum Engineer s}, Arner ican Association of Petroleum Geologi st s, Arner ican Institute of ChemicalEngineer s, American Society of Civil Engineer s, Amer ican Society of Mechanical Engineer s, Instituteof Electrical and Electronics Engineers, Marine Technology Society, Society of ExplorationGeophysicists, and Society of Naval Ar chLt.ec t s and Marine Engineers.

This paper wa.s prepared for presentation at the Sixth Annual Offshore Technology Conferenceto be held in Houston, 'I'ex . , Nay 6-8, 197L~" Permission to copy is restricted to an abstract ofnot more than 300 words" Illustrations may not be copied" Such use of an abstract should containconsptcuous acknowledgment of whe r e and by whom the paper is presented"

It has long been recognized that oceanwaves, and therefore their induced motions andforces on structures, are random phenomenawhose exact time cannot be predicted. Today,however, a spectral analysi.s technique i.savailable that can be used to predi ct accuratelyimportant statistical properties, such as thesignificant and maximum values of' waves,motions, and forces. For these predictions thewave energy distribution over the frequency andthe motion, or force responses to regul.ar wavesover a wide range of frequencies, should bemown.

In design procedures for off shore et.ruc-.tures very often the study of motions andforces is made on the basis of one si.ngle regu­lar wave whose properties are believed to beequivalent to those of an irregular sea (regularwave concept).

To investigate the merits of the regularwave concept, the motions of a semisubmersibleand two supply vessels (one conventional and onetwin-hull type), as predicted by the statisticalmethods and the regul.ar wave concept, arecompared. The comparison shOWS that in somecases the regular wave concept under estimates,while in other cases it overestimates, themotions that may be expected. It may be

References and illustrations at end of paper.

concluded that the regul ar wave concept doesnot always lead to a conservative motion pre­diction.

In considering the effect of the wavedirection on the motions of a semisubmersible,both methods gi.ve the same tendency in aBeaufort 8 North Sea condition. However, thedifferences between the motions in various wavedf.r ectd.ons are considerably smaller when pre­dicted by the statistical methods.

Finally, a motion optimization studyreveals that the regular wave concept yieldserroneous results.

INTRODUCTION

Ocean waves and thei.r related phenomenaare randomly varying in time. However in 1905,R. E. Froude remarked at the end of his paper 1that "Lrr-egul.ar wave systems are only a com­pound of a number of' regular wave systems(individually of comparatively small amplitude)of various periods ,,". • And the effect ofsuch a compound wave series would be more orless a compound of the effects proper to theindividual units composing of it."

In fact, these assumptions were the same asthose made by st. Denis and Pi.erson. 2 In 1953,they introduced a statistical method by means

of which the response of a shi.p in a seaway c~

62WILL THE "REGULAR WAVE CONCEPT" YIELD MEANINGFUL

MOTION PREDICTIONS FOR OFFSHORE STRUCTURES? OTC 2040

be pr edicted when the wave (ener gy) spectrumand the response to regular waves are known.The wave spectrum describes the distributionof wave energy over the frequency. Theirapproach, usually called the spectral analysistechnique or the linear superpositi.on principle,has been used frequently to predict shipmotions and performance in irregular seas frommodel test results in regular waves. Theapplicability of the superposition principlehas been verified many times by comparing thepredicted values with those directly measuredin irregular seas, as was done by Lewis, 3Ochi,4 and Gerritsma. 5 For catamarans andother floating offshore structures, the appli­cability of the superpositd.on principle hasbeen shown by Van Slui j s6 and Vugts,'1

analysis technique.

To investigate the merits of the regularwave concept in predicting the motions offloating offshore structures, the followingexamples wer-e examined: (1) Pitch angles ofa 45·--m-long conventional supply vessel oper­ating on the North Sea. (2) Pitch angles of a40-m-l.ong twi.n-hul.l supply vessel operatingon the North Sea. (3) Heave motion of a 120­m-Long semi.submersible pipe-.lay barge operatingon the North Sea. (In this case the effect ofthe wave direction on heave predicted byregul.ar wave concept and spectral analysistechnique was also sbuda.ed , ) (4) Heave motionoptimization study of above barge operating onthe North Sea and off South Australia.

DESCRIPTION OF OCEAN WAVES

The above formula applies for deep waterwhere the waves are not affected by the seabottom. When the wave is studied at a fixedpoint with x = 0, Eq, 1 becomes

• (2)

.. 0)

l;;a cos wt .

wt), •••• (1)

y cos (·"U,t)"a

wave elevationwave amplitudewave number = 21T/Awave lengthhorizontal coordinatewave circular frequencywave periodT

xw

1;(t) = 1; ccs(kxa

where ~ (t)~a

kA

Observing the sea surface, it will. beobvi.ous that the sea can never be replaced byone si.mple, harmonic regular wave. Even aswell ori.ginating from a distant wave fieldwith a fair ly regular appearance woul.d be over···simpli.fied when it is described by one harmonicwave of the following form (moving in thepositive x-df.r-ectd.on},

The relation between the wave length andperi.od for deep water waves is gi.ven by

?_ 0,T-A = "21"[-

In studying the motions of floatingstructures, however, there is still a greatpreference to replace each irregular sea by oneregular wave wi.th a height and a period equalto a certain characteristic height and periodof the irregular sea.

'['hi s preference is partly due to the factthat motion predictions by means of the spectralanalysis technique require i.nformation on themotion response over a wide wave frequencyrange,whereas for the regular' wave concept theresponse should only be lmown for a fewselected frequencies. Furthermore, with theregular' wave concept the prediction is re,,··'stricted to simple, understandable moti.ons.However, this restri.ction may not always bejustified.

For instance, Swaan and Rijken8 showed fora 100-m-long ship sailing at a speed of 15.2lmots in various head seas that, using ar-egul.ar wave (height = si. gnificant height;period = mean per-Led of the irregular sea),practically no pitchi.ng is predi.ct.ed in waveslower than 2.15 m (Beaufor-t 5 on the NorthAtlantic Ocean). This is contrary to theexperience at sea; the 40 significant pitchangle eval.uatsd wi.th the spectral. analysis technique is much more realistic. Moreover, i.tappears that the regular wave concept under-­estimates the pitch angle consi.derably in waveslower than 3.75 m (Beaufort 7), whereas inrougher seas a substantial overesti.mate occurs.

Thus it cannot be concluded beforehandthat the regular wave concept always yields aconservative motion predicti.on. The same state­ments are also valid for the forces and over­t.urm.ng moment acting on a fixed cylindricalpile and on a jack-,up rig, as has been illus­trated by Vugts. 9

Thi.s paper gives a description of oceanwaves, presents the regular wave conceptapplied, and describes briefly the spectr al

The average energy over the wave lengthand per uni t area of sea sur-race, consisting ofkinetic energy associated with the orbitalmotion of water particles and potential energyresulting from the change of water level incrests and troughs, is 1zPg ~a2, as shown byLewi.s.3

According to oceanographers2,3 the ir­regul.ar sea surf'ace can be represented by thesupez-posd.td.on of many small,-·ampli tude regular

'----- --L- ~

63

• • • (8)

• • • • • • • • (7)

"1--13(:)

e-·5A w

The total energy E of the wave system ofall. component energies amounts to

"'n (Fig. 2).

00

E = pg J 81',; (r.iJ) dUJo

The function s~(",) has been proposed by manyLnves'td.gabors, such as Neumann, Darbyshire, andBretschneider, mainly based on wind velocity.3R.ecently, the wave spectrum formulation, asproposed by Pierson and Moskowitz,11 wasrecommended by the International Shipstructures Congress12 and by the Internati analTowing Tank Conference.13

This recommendation was followed in thispaper. The spectral density is formulated byPierson and Moskowi.tz as

H. L. MINKENBERG and TAN SENG GIE

Usually the motion and force analyses arerestricted to the long-crested seas for whichthe direction of propagation of al.L componentwaves is the same. This is, for the timebeing, the best approach since no adequateinformation on short-crested seas is available.In this paper motion predictions will be donefor long-crested seas.

Michel10 ill.ustrated how irregular theresulting wave is as formed by only fourregular waves (Fig. 1). Moreover, it caneasily be shown that even a very rough sea canbe obtained from the summation of a largenumber of small-amplitude regular waves.Mathematicall.y, the long-crested irregular seacan be formulated as

OTC 2040

Iwaves having different amplitudes, periods, anddirections. Each component wave follows thesimple harmonic-wave theory. The phaserelationships between the component waves areconsider ed to be completely random.

00

At a fixed point with x = 0, Eq, 4 becomes

I; (t I = l: 1; cos (OJ t - € ) ..•• (5)n~l an n n

where ( , k 1 W , and Eare the amplitude,wave m.unager, ncirgular f'I'~quency, and phase ofthe n-th wave component, r-espectdvely,

• . • • • • .. • (9)1 N

~

n n=lo

1;

The constants A and B will be rel.ated to theactual wave observations, as will be shownlater.

From oceanographic data it is known thatthe wave elevati.on follows approximately anormal (Gaussian) distribution, which is fullycharacteri.zed by the root-mean square value of'~(t):

(~\ t -+ E ) ,n n

• . • • • (4)

00

i 1 cos (k xn:::.l -'an n

Conversely, a given i.rregular sea, at a certainlocation for which the sea condi.tion i.s sta­tionary for a certain duration (20 to 30minutes) can be split up into a very large(theoretically, infinite) number of regularwaves by means of (Fourier) harmonic analysis.

Then the average energy, being 1zPg ~an2,i.s known for each component wave and, conse­quently, the energy distribution over the wavefrequency. This di.stribution is cal.Led thewave spectral density functi.on, the waveenergy spectrum or, briefly, wave spectrum.

With the normal distribution and the calculatedroot mean square value of (, the frequency wi.thwhich val.ues less than or equal to any givenvalue of ( will occur can be derived.

The distribution of the apparent waveampli tudes depends on the width c of thespectrum. The spectrum wi.dth « can be obtaineddirectly from the recordi.ng by the foUowingformula.

(10

In oceanography usually the followi ngfunction is used.

• • . • (6)

The f'unct.Lon s~(",), designated wave spectraldensity, multi.plied by the frequency intervalf':...waround Wn, represents the total energy ofall component waves in an interval 6.co around

in which Tc = mean apparent crest period; Tz =mean apparent zero up-crossing period (Fig. 3).

For c = 0, which is currently assumed forthe apparent wave ampli.tudes, the distributionis a Raylei.gh distributi.on for which thefollowing relations between the significantwave height, mean period, and wave spectrum arevalid:

. . • • • • • . • • (11

64WILL THE "REGULAR WAVE CONCEPT" YIElD MEANINGFUL

MOTION PREDICTIONS FOR OFFSHORE STRUCTURES? OTC 2040

m1j ::0: ;2 'T __2f

TIl]/';

•.••••. (12)defined as the mean of the zer 0 up-crossingperiods associated with the one-t.hi.rd highestwaves.

[w"St; (t:.\) ,.dwo

When the wave energy spectrum is relatedto actual observations at sea, it is assumedthat the significant wave height conforms tothe average observed wave height (wand thatthe mean w~ve Eeriod equals th~ avera.£8 o£2.-'served per-Led T, thus (wI/3 = 'wand T = T.

From Eqs 8, 11, and 12, it follows thatthe coefficients A and B are equal to

A 172.8 r 2 --4SY.,T T

B = 691 T-4

For the Pierson-Moskowitz spectra it canbe derived that the following relati on holdstrue:

• • • • (16)

• • • • (17)

. • • • • • • • • • (15)

'l~ ~ 1. 14 '1's

'I' = ),14 TP s

This choice is more or less arbitrary, butfor the above comparison it is not importantwhich characteristic wave height and period ofthe Lr-r-egul.ar- sea are used. For instance, thesignif'igant wave height (wlll3 and mean waveperiod T are often chosen. In some cases,the period Tp at which the maximum wave energyoccurs is selected.

Tp = 1.295 ''F

Darbyshire17 developed the following relation­ship between Tp and Ts:

From Eqs. 15 and 16 it follows that• (13)

• (14)

significant wave heightmean of the one-third highest

wavesmean wave period

area of wave spectrum" f S I; ( w) d '"o

first moment of wave spectrum =00

where [Wl/3

The significant wave height and mean waveperiod used in this paper apply to sea con­ditions on the North Sea and offshore ofT SouthAustralia reported by Petri l4, and by Hogbenand Lumb15 respectively. The main differencebetween the sea areas is that at the samesignificant wave height the dominant periods ofthe waves on the North Sea ar:e shorter.

The most probable maximum wave height (; Will

can directly_be obtained from the significantwave height (w1/3 using Table I, whigh is basedon the analysis of Longueb-Hi.ggi.nsv- For1, 000 oscillations, which corresponds to aboutIi to 3 hours duration of a stationary seacondition,

PREDICTION OF MOTION;2 IN IRREGULAR SEAS.= 1.86 ~\d/3 ..•••.• (18)

Wi th the regular wave concept the "maximum"motions 2ua max ' measured from crest to trough,were calculated as

The motion responses per uni.t of waveheight are known on a base of the wave cfrcul.arfrequency.

The motions of various vessels and st.r uct.ure swere predi.cted usi.ng the response f'unctd.onsthat were established either by model tests orcalculati.ons.

, ••• (19)

u"'" ([--~) (

S ',,'I max~a

~ar Wave Concept

Although in 1953, St. Denis and Pierson2

introduced a statistical theory to predl.ot theship motions and performance in irregularseas, many st.udd.es of motions and forces arestill based on the simpler, regul.ar wave theory.With this r egul.ar wave concept it is assumedthat the properties of an irregular sea can bereplaced by the properties of a so-calledequi.val.ent regular wave. The effect of thiswave on motions and forces is believed torepresent accurately the eifects of theirregular sea.

The main purpose of this paper is toinvestigate the merits of the regular waveconcept in predicting the motions of offshorestructures by compari.ng the motions determinedby this concept and by means of the spectralanalysis. For t.hi.s comparison, the height ((;w)of the equivalent regular wave was equaled withthe most probable maximum wave height e::w max.)occurring in 1, 000 wave cycles. The period Tcorresponded to the significant wave period Ts,

with (ual(;a) s motion response at the periodT ~ Ts '

Spectral Analysis Techni~~

Knowing the response of a structure toregular waves and the energy distribution ofthe component waves in an irregular sea, theforces and motions can be predicted by the

OTC 2040 H. L. MINKENBERG and TAN SENG GIE 65

which can also be written as

confirmed by model tests. In this respect itwin be remember ed that in pr edfcting themotions in irregular seas, even in rough seas,the motion responses should be determined inrelatively low, regul.ar waves.

The merits of the regular wave concept inpredicting the motions of offshore structuresin irregular seas were investigated by com­paring these predictions with those obtained bymeans of the spectral analysis technique, ofwhich the applicability has been shown manytimes. 3-7 For this comparison the followingmotions were selected: pitch of a 45-m-longconventi.onal supply vessel and a 40-m-long twin­hull supply vessel, and heave of a 120-m-longsemi submersible barge designed for pipe--layingoperations.

COMPARiSON OF MOTIONS PREDICTED BY REGUIAR WAVEQONCE;ET AND BY SPECTRAL ANALYSIS TECHNIQUE

• • • • • • • • • (20)

u 2{r a

(w)} , ••••••• (21)'a

2u . =a] /3

following formula.

u{/ (OJ)}2 , •••••• (22)

-'a

In Eq, 20 it is seen that the response toregular waves has to be squared and multipliedby the wave spectral density for each circularwave frequency. Thus,

or

Su(",) is the spectral density of the quantity u

The calculations were carrd.ed out forNorth Sea conditions, corresponding to windforce Beaufort 5, 6, 7, and 8; for the barge,the sea area off' South Australia was also con­sidered. The wave spectra wer e of' the Pi erson-·Moskowitz type and are shown in Figs. 4 and 5.

Far the conventi anal. supply vessel, thepitch response in the hove-to condition (Fig. 6)was determi.ned by model tests in regul ar headwaves.

Moti.on responses used are gi.ven in Figs. 6,8, 10, and 11, whereas the usefulness of theregular wave concept can be judged from Figs.7, 9, 12, and 13, and Table II.

• • • (24)

. • • • • • • • (23)

C--~----~-,

r:4\1 f 5 (k)do)o 1..1

Similar to the determination of the signi­ficant wave height from the wave spectrum, Eq.11, the significant double amplitude of themotion 2ua 1!3 is establi shed fr am the heavespectrum:

dw

The combination of Eqs , 21 through 24 gives Eq,20.

Assuming that the apparent amplitudes ofthe motion u are Rayleigh dfstributed,16 themost probable maximum amplitude for 1,000oscillations is obtained from

The predi.cted maximum pitch angle, fromcrest to t.rough , is plotted on a base of themaximum wave height in Fig. 7, illustratingthat for the Beaufort 5, 6, and 7 sea cond.i>­tions the regul.ar wave concept overestimatesthe pitch angle. In Beaufort 6 sea a largediscrepancy Occurs.

This is because for this condftion wi.th aTs = 6.5 seconds, the response function has alarge peak value that can be seen if thedimensionless pitch ea/k(. in Fig. 6 is trans­formed to ea/'a' (See al~o the example givenin the Appendix.)

In the higher sea states the regular waveconcept yields much smaller values than pre­dfcted by the spectral analysis. Moreover, thedecrease of the pitch angle with increasingwave height, when waves higher than those corr-e­spondi.ng to Beaufort 6 are encountered, iscontrary to the experience at sea.

.(25)2uEJ. max:

This most probable maximum ampli tude has aprobabi.lity of 63 percent of being exceeded.For other numbers of' oscillations and othermaxima with another probability of being eJ<­ceeded, reference is made to Table I.

A detailed example of calculations bymeans of the above described spectral. analysistechnique is included in the Appendix at theend of this paper. As pointed out in theintroduction, the applicabi1ity of this tech­nique has been verified many times.3-7 Toapply this technique, the motions should beassumed to be linearly proportional with thewave height, an assumption which is often

A much better predfction as to the magni­tude and trend with wave height was obtained

l-- ---L-____________~

66WILL THE "REGUIAR WAVE CONCEPT" ITEIJl MEANINGFUl

MOTION PREDICTIONS FOR OFFSHORE STRUCTURES? OTC 2040

f or the pitch angle of a 40-m-Iong twin-hullvessel, which was investigated by Van Sluijs.6Fig. 9 shows that for wave heights (mostprobable maximum wave heights) between 3.0 and8.0 m (Beaufort 5 to 7), the pitch motions ofthe twin-hull are well predicted by theregular wave concept. The difference is notmore than 10 percent. For waves higher than8.0 m, the regular wave concept will lead tomuch smaller values than predicted by thespectral analysis. The pitch response used(Fig. 8) is obtai.ned from tests in regularhead waves with a 1:25 scale model lyinghove-to. To this purpose the model waspositioned by four soft springs such that themooring lines hardly affected the pitchcharacteristics of the twin-hull. Thisresponse has ~een confirmed by full-scalemeasurements.

Fr om the above, it may be concluded thatthe regular wave concept does not alwaysyield a conservative moti.on predi ctd.on, 'Itusconclusion is also valid for the heave motionof the 120-m-·long semisubmer sible barge, ascan be seen in Table II.

the response due to a beam wave is smal.Ler thanfor a head wave. This onl.y holds true, however,for waves with frequencies between 0.8 and 0.9rad./second. Outside this region the responseto beam waves is considerably larger. Thus,to determine the most favorable wave direction,deci.sions cannot be made on the basis of a fewselected wave frequencies.

Finally, to investigate the mer1.ts of theregular wave concept for motion optimi.zationstudies, the heave motions of the above semi­submersible barge in beam seas were calculatedto determine the most favorable lower-hull shapfor heaving. For this study, which was alreadyconducted by Minkenberg and Van Sluij s20 bymeans of the spectral analysis, two alternativeLower-shul.L sections were chosen, a verti.calelliptical shape and a square section. Theshape of the Lower--hukl, section was alteredwhi.le the sectional. area had been kept constantand equal to the original circular design.Characteristics of the weight distribution werekept constant throughout. In Fig. 11 the heaveresponse to regular waves, calculated accorcli.ngto Hooft,19 is shown.

Motion predicti.ons in Lr'regul.ar seas werecarried out for North Sea condi tions and forthe South Australian sea area. For SouthAustr ali a the same signi.ficant wave heightswere applied as for the North Sea. The mai.ndifference of the seas in these areas Ls thatdominant wave periods at South Australia arelonger (Figs. 4 and 5).

In Fig. 12 the results are gi.ven for theNorth Sea conditions; in Fig. 13, for the SouthAustralian conditions. It appears that theheave moti.ons predicted by means of the spectralanalysis are mostly much higher than the valuesobtained from the regular wave concept. Onlyfor the square cross-section is the dif'f'erencebetween both methods small when South Australiarconditions are considered.

From the calculati.ons carded out by meansof the spectral analysis, it may be concludedthat in North Sea waves up to 7.5 m a verticalelliptical, shape is superior, whereas in hi.gherseas heave will increase appreciably since thelonger wave components excite the platform inits natural period. From the regul.ar waveconcept it follows that, for all North Seaconditions considered, the vertical. ellipticalshape is superior'.

Both methods showed that no practical.pr ef'erence regarding heave exists between theoriginal design and its Alternative I (squarecross-eectd.on) for the investigated North Seaconditd.ons ,

Table II also shows the effect of thewave di.rection on the heave motion.

In most cases the regular wave conceptunderestimates heave considerably. The greatdifference between heave predictions byregular wave concept and by spectral. analysi.sis to be expected with the fluctuatingcharacter of' the heave response as shown inFig. 10. These response curves were derivedby testing the barge model moored between foursprings in an irregular sea with st.grri fd carrtheight of about 17 ft. The bar-ge consists oftwo horizontally submerged circular cylindricalhulls, each of which is connected to fivesurf'ace-pd.ercf.ng vertical columns. Deck andhulls are connected by bracings such that aful.l symmetry exists with regard to a verticalplane through the center line and thr ough themidship section. This barge has been used byvan Sluij s and Tan18 to demonstrate the appli·-·cability of the calculation method developedby Hooft19 in predicting motion responses offloating structures.

It appears that only for the Beaufort 8sea condition was the same tendency obtainedby both methods. The difference between thewave directions, however, .i.s less pronouncedfor the spectral anal.ysis. According to thismethod, the largest heave motion occurs inbeam seas. Generally, this was also predictedby the regular wave concept except for theBeaufort 7 sea, where the largest heave motd.onswere predicted for head waves. For this seathe significant wave period amounts to 7.6seconds and Fig. 10 shows that, at this period, For the South Austr ali an area the spectral

L- ---L- -.J

OTC 2040 H. L. MINKENBERG and TAN SENG GIE 67

00

total energy of wave systemacceleration due to gravity217/A = wave numberwave number of the n-Lh wave componenwave slope amplitude

e

"'n

Egk

kn =k';a =

oc

mo~ = [s,.... (widJj= area of wave spectrumC> '~

A more realistic motion prediction will beobtained by the spectral analysis technique,which, therefore, is strongly recommended.

NOMENCLATURE

js ; (Cd) 0.;dtcl= first moment of waveo l, spectrumspectral density of quantity u

wave spectral densitywave periodaverage observed wave period

mean wave periodmean apparent crest periodperiod at maximum wave energJTmean of' z8roup-crossing periods

associated with the one--bhi.rdhighest waves

mean apparent zero up--crossing periodtimemaximum double amplitude of quantity

ux horizontal. coordinate

width of the spectrumE n phase of' the n-rth wave component

A wave lengthco 217'/T = wave circular frequency .

wave circular frequency of the n-thwave component

mass densitywave elevationwave amp1i.tudewaiTe amp1i.tude of the n-th wave

component(w wave height,;w = average observed wave height

<:' 1/3 = signi.ficant wave height'wwmax = maximum wave height

REFERENCES

The usefulness of the regular wave conceptwas investigated by comparing the pitch motionof two supply vessels and the heave motion of asemi submersible barge, predicted by this con­cept with the motion values obtained by thespectral anal.yai s technique. For this studyit may be concluded that the regul.ar wave con­cept doea not always yield a conservativemotion prediction. In some cases, such as forthe heave motion of' the semisubmersible, aconsiderable underestdrnatd on r esul,ted.

anal.ysis pr edi.cted that the squar e lower·-hullshape is favorable; a vertical ellipticalsection gives rise to considerable heaving.However, from the results of the regular waveconcept the opposite holds true, so that byapplying this theory an erroneous conclusionwill be drawn. Moreover, Fig. 13 shows that,according to the regular wave concept, thesemisubmersible barge with lower-hulls of thevertical elliptical shape will heave less inwaves with a maximum height larger than 6.0m, This is caused by the character of theresponse curve between the frequency co == 0.5and 0.55 rad./second. This curve was calcu­lated using a method developed by Hooft,19which ignores the effect of damping. Generallyfor floating platforms the influence of dampingis small outside the resonance region and canbe neglected. 1S

For co = 0.5 rad , / second the curve goes tozero because, at this point, the undisturbedwave-pressur'e force (Fr-oude-Er'Ll.of'f hypothesis)will be cancelled by the inertia force. Inreality, however, at this frequency the dampingforce becomes important, whi.ch will lead to aconSiderably larger heave response around thisfrequency than calculated.

Hence, the trend with the wave height willprobably be well predi.ct.ed by the regular waveconcept, although the erroneous conclusion withregard to the best shape still exists when aheave optimization study Ls based on this con­cept.

Moreover, the trend of motion amplitudewith respect to the wave height and wave direc­tion was not always well, predicted. Finally,from a heave optimization study of the Lowerv­hull of the semisubmersible, it appeared thatin some cases the regular wave concept even ledto err oneous conclusi.ons.

Although strictly speaking, the abovefindings apply to the particular motions andoffshore structures studied, it is at leastclear that with the regular wave concept itwill be impossible to determine beforehandwhether or not the motions are overestimated.

1.

2.

3.

Fraude, R. E.: "Model Experiments onHollow Versus Strai,ght Lines in StillWater and Among Artifici.al Waves," Tran£.I.N.A. (1905) bl.St. o'enis, M. and Pierson, W. J.: "On theMotions of Ships in Confused Seas," .Tr~,

SNAME (1953) 61.LeWiS~' E. Vo.: --"The Moti.on of' Ships inWaves " Principals of Naval Architectureedi.ted by J. P. Comstock and published bySNAME (1967).Ochi., M. K.: "Extreme Behavi.our of a Shi.pin Rough Seas- Slamming and Shi.pping ofGreen Waters," !rans. SNAME (1964) 72. ~

68WILL THE "REGULAR WAVE CONCEPT" YIElJJ MEANINGFUL

MOTION PREDICTIONS FOR OFFSHORE STRUCTURES? OTC 2040

APPENDIX

1 .. 74L,9,}J

,14x7,,9

9" } 1 meters

,-~-hi Iila):

TS=1,14f

9,0 seconds.

28a :HA.X.

and

As an example, pitch motions of a 45,-m­long conventional supply vessel in Beaufort 8North Sea conditions were calculated by meansof the regular wave concept and the spectral,analysis technique.

For the maximum wave height of 9.11 mpitch will be

!:rediction of Motions in Irregular Seas.

Fig. 6 shows that the pitch response ofthe vessel to a wave with a period of 9.0seconds, viz., Ws = 0.7 rad.!sec, amounts to0.605, whi.ch equals to ea/i;a = 1.74 degree/m.

The procedure applied is described in thesection entitled "Regul.ar Wave Concept" of thispaper. For the Beaufort 8 sea condi.td.on theequivalent regular wave can be obtained fromEqs. 17 and 18:

~,.•,,1 - ~,. n'd-X = },.86 P /', v" -~'id~ 'J

The wave spect.rum used is shown in Fig. 4,whereas the response of' the vessel due to regu­lar' waves is gi.ven in Fi.g. 6 in a dimensionlessform. For Beaufort 8 the significant waveheight amounts to 4.9 m and the mean waveperiod 7.9 seconds.

The procedure applied is described in thesection on "Spectral Analysi.s Techrri.que;"To predict the ship motions in Lrz-egul.ar seasfrom the response of the supply vessel toregul.ar waves, the following assumptions haveto be made.

1. An irregular sea is composed of thesum of a large number of simple harmonic wavesof various period (superposition principle).

Ger-r-ltsrna, J.: "Behavi.our' of a Ship ina Sea-Way", International Shi!2buildi!!Z.Progress (1966) 13. ---Van Slui.js, M. F.: "Model and Full ScaleMotions of a Twin-Hull Vessel,"Netherlands Ship Research Centre, T.N.O.,Report No. 1315 (1969).Vugts, J. H." "The Role of Model Testsand Their Correlation with Full-ScaleObservations," Symposium on OffshoreHydrodynamics, Wageningen, Publication No.375 of N. S. M. B. (1971).Swaan, W. A. and Rijken, H.: "Speed Lossat Sea as a Function of Longi.tudinalWeight Distribution," Intern~:J:,:!:£nal_Ship­

building Pr2EE~ss (1964) 11.Vugts, J. H.: "The Analysis of StructureSubjected to a Stochastic Wave Load," (inDutch) De Ingernieur, (1972)§!±.Mi.chel, W. H.: "Sea Spectra Si.mplified, nMarine Tech. (1968) 5.Pierson, W. J. and MOskowitz, L.: "AProposed Spectral Form for Fully DevelopedWind Seas Based on Similari ty Theory ofS. A. Kitagarodskii," i[",-of_Geop!!lsi~al

~"arch (1964) 69.Proc , of the International Shl.p StructuresCongress 1964, Report of Committee No. 1on environmental conditions.Report of the Seakeeping Commit tee of theInternational Towing Tank Conference(1969).Petri, 0.: "Statisti.k del' Meer8swell.enin de!' Nordsee," EinzelveroffentlichungNr-, 17, Deutsche Wetterdienst,Seewetteramt, Hamburg (1958).Hogben, N. and Lumb, F. E.: "Ocean WaveStati sties," H. M. Stationary Office,London (1967).Longuet-Higgins: "On the Statistical,Distribution of the Heights of Sea Waves,"J. of Marine Research (1952) XI.Darbyshire, J.: "A Further Investigationof Wind Generated Wav.es," Deutsche Hydl'o­graphische Zeitschrift, Band 12 (1959).Van Sluijs, M. F. and Gie, Tan Seng:"Experimental and 'I'heor-etd.cal.Tlotd.onCorrelation of' a Pi.pelaying Barge , II

Symposium on Offshore Hydrodynami cs,Wageningen, Publication No. 375 of N.S"M.B. (1971).Hoof't , J. P.: "A Mathematical Method ofDetermining Hydrodynamically InducedForces on a Semi.e-Submersd.hl.ej " Trans.SNAME (1971) 79. --­Minkenberg, H:-t. and van Sluij s, M" F.:"Motion Optimization of Semi·_,Submersibles," paper OTC 1627 presented atSPE-AIME 4th Annual Offshore TechnologyConference, Houston, May 1-3, 1972.

9.

7.

8.

6.

5.

10.

11.

12.

14"

15.

16.

17.

18.

20.

19.

2. The sum of' the responses of a ship tothese component waves is equal. to the response;of the vessel to the sum of' these waves. The'--- -'-____________---.J

OTC 2040 H. L. MINKENBERG and TAN SENG GIE 69

response at each wave period is linear proportional to the wave height (linearity principle).

Based on these assumptions, the significant pitch can be calculated according to thefollowing formula, using Figs. 4 and 6.

~-_._-,

il~ S (, .• aO'J'V J • 1; ui , c.a

The calculation is illustrated in the table below; the Simpson's rule was used to determine thearea under the pitch spectrum.

1

'"rad./sec.

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

1.3

1.4

2

8ak(a-

0.66

0.65

0.65

0.61

0.65

0.83

0.51

0.42

0.23

0.11

0.05

38at;

deg./m0.62

0.94

1.36

1.74

2.44

3.92

3.00

3.00

2.00

1.12

0.58

4(8al(a)2

9oeg.2/sg m

0.384

0.884

1.850

3.028

5.953

15.366

9.000

9.000

4.000

1.254

0.336

5

S(89 ill sec.

0.11

2.05

3·49

3.02

2.12

1.39

0.89

0.61

0.42

0.28

0.19

6

4 x 5deg. 2 sec.

0.042

1.811

6.455

9.145

12.620

21.359

8.010

5.490

1.680

0.351

0.064

7

S.M.

1

42

42

42

42

41

8

6 x 7deg. 2 sec.

0.042

7.244

12.910

36.580

25.240

85.436

16.020

21.960

3.360

1.404

0.064

.s = 210.260

Col. 1:

Col. 2:

Col. 3:

Col. 4:

Col. 5:

Col. 6:

Col. 7:

Col. 8:

wave circular frequency f W

dimensionless r-esponse, ea/k~~ at ca

response, ea/~a at co

square of response, (8 aI( a)2 at '"

wave spectral density, S( at co (see Fig. 4)

Cal. 4 x Col. 5 -- (8al(a)2 S( = S8 = pi.t.ch spectral density at co

Simpson's multi.plier

Col. 6 x Col. 7 to obtain the area under the pitch spectrum by: ("'",/3) x.sx 210.260 = 7.009

(0.1/3)

According to the above formula, the si.gnificant pitch amounts to

2iia l /3 = 4";7.009 = 10.60

The most pr'obabl.e maximum. for 1,000 oscillation equals

28a max = 1.86 x 2iia 1/3 = 1.86 x 10.6 = 12.:1~ .

In the above calculation the frequency interval "'''' is equal to 0.1 rad./sec.responses with fluctuating character as the heave motion of' a s emi.submer al.b.Le , 6.0or even 0.025, rad./sec to improve the accuracy of the calculations.

For theshould be 0.05,

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CIRCUL.AR FREQUENCY W

Fig, 2 - gxpjanator y figu.re for wave ener gy dt.strr tbut fon

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Fig 3 - Definition of various wave per-iods

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