8 JAN. 1916

kRCHFee note inside cover

L Se.4 ¿w,2 j4I43. LJ4,. .o4L

Lab. v Scheepsbouwkunde

Technische EhpjcieL,rt Ship 192

Delft September 1975

NationalPhysicalLaboratory

Ship Divisionu e u

EXPERIENCE IN COMPUTING WAVE

LOADS ON LARGE BODIES

Reprint of paper presented at theSeventh Annual Offshore Technology

Conference, Houston, Texas. May 1975

by NG Hogben and RG Standing

Department of Industry

SMTR-7523

Extracts from this report may be reproducedproviding the source is acknowledged.

11PL Ship Report 192

September 1975

NATIONAL PHYSICAL LABORATORY

PERIENCE IN COMPUTING WAVE LOA])S ON LARGE BODIES

Reprint of paper presented at the Sevénth iva Offshore

Technology Conference, Hpuston, Texas. May 1975.

by

N G Hogben arid R G Standing

OFFSHORE TECÑNCLGCY CONFERENCE6200 North Central ExpresswayDallas,.Texas 75206

THIS PRESENTATION IS SUBJECT TO CORRECTION

Experience in Computing Wave Loads on Large Bodies

N. Hogben and R. G. Standing, Ship Div., National Physical Laboratory

©Copyright 1975

Offshore Technology Conference on. behalf of the American Institute of Mining, Metallurgical, andPetroleurr Engineers, Inc (Society of Mining Engineers, The Metallurgical Society and Societ1 ofPetroleum Engineers), American Association of Petroleuir Geologists, American Institute of Cheffi-cal Engineers, American Society of Civil Engineers, American Society of Mechanical Engineers,Institute of Electrical and Electronics Engineers, Marine Technology Society, Society of Explor-ation Geophysicists, and Society of Naval Architects and Marine Engineers.

This paper was prepared for presentation at the Seventh Annual Offshore TechnologyConference to be held in Houston, Tex , May 5-8, 1975 Permission to copy is restrictedto an abstract of not nore than 300 words Illustrations tray not be copied Such use cfan abstract should contain conspicuous acknowledonient of where and by hoir the paper ispresented.

PAPER

NUMBEROTO 2189

ABSTRACT.

Designers of offshore installations for theNorth Sea and adjacent waters face a number ofexceptional demands specially regarding theextreme and continuing severity of the weatherand the large watér depths. In response to thischallenge, there has been rapid development ofnew design concepts and a major trend alreadywell established, is f 9r the emergence of newtypes of large monolithic structure. These posemany difficult design problems and, in particu-lar, are. not genér1ly amenable to conventionalmethods for estimating wave loads. For suchstructures, prediction methods based ondiffraction theory are needed.

INTRODUCTION

This paper describes eerience in shipDivision. of the National Physical Laboratory inthe development, validation and application of acomputer program written for this purpose thathas been supplied to a number of computerbureaus and to Lioyd Register of Shipping andhas been extensively used by.the offshoreindustry. This work has been undertaken as partof a research program approved by the Ship andMarine Technology Requiraments Board. Someearlier aòcounts have already been published inRef5. 1 and 2, the first being mainly concerned

References and illustrations at end of paper.

with the computer program development and, thesecond with experimental validation andpreliminary operational experience. The presentpaper recapitulates the salient results of theearlier' publications and reports recent progressincluding some further experimental results, butemphasizing the more .extensive experience sincegained in practical applications of thecomputer program.

Before discussing this experience indetail it may be helpful to be mo±e specificabout the nature of the structures concerned.The term monolithic used above is intended inthis context to refer rather loosely to a. rangeof structure tpes that are 'very diverse inconfiguration. All, however, contain at leastone component' which is very much larger indiameter or equivalent section dimension andoften quite complex in shapè compared with thenormal tubular components of a steel jacket, It

is because of these features that diffractiontheory methods of wave loading analysis aregenerily needed in fact for reasons that areexplained iii detail in Ref. 3 and will bebriefly indicated in a later section of thispaper. 'In many cases construction is primarily'of concrete and diameters of the main componentsmay sometimes range up to 100 or 200 mo

There are currently sverai bro1 classes

L114 XPERIFNCE IN COMPUTING WAVE LOADS ON LARGE BODIES OTC 21

of structure to which this description applies,and these include gravity platforms, tensionleg and tension stay structures and storagetanks. At the time of writing the program hasmainly been used for gravity platforms forwhich a substantial number of orders have nowbeen placed. The level of demand may in factbe judged from a recent list of UK North SeaPlatform Orders4 showing a count of 11 con-.crete gravity structures in comparison with 13steel jackets. The definitive feature of thesetypes of platform is that they rest on the seabed under their own weight without piling.They commonly consist of a single massive baseon which stand a group of three or four towerscarrying the platform clear of the sea surface.There is, however, a diversity of actualgeometry as is well illustrated by an inter-esting pictorial review of more than a dozencurrent designs in Ref. 5. This includes somehybrids involving combination of jacket-typesteel structures on concrete bases.

The program has also been used for atleast one tension stay platform, at least onestorage tank and for studying the forces onone large complex-shaped component of a semi-submersible. For all these types of structurethere is again great divrsity of design.Tension stay platforms, like tension ones reessentii1y floating structures anëhored to theseabed by cables kept in tension by excess ofbuoyancy over weight. in this case, however,the cables are spread to restrict lateralmotion, whereas those of a tension leg platformare vertical. The review Ref. 5, mentionedabove, includes a picture of a tension stayplatform consisting of a very large anchoredbuoyancy chamber carrying a group of towers onwhich the platform is mounted. Informationabout tension leg platforms may be found inRef. 6.

Storage tanks may be independent unitssuch. as the Dubai installations described inRef. 7, grouped like the Ekofisk tank complex,or incorporated in the base units of avityplatforms or in single buoy .moorings. Re-garding semisubmersibles, Paulling'P andHòoftl'l have successfully estimated wave loadsand motions without using diffraction theory.for a number f conventional designs. Diffrac-tion analysis may be needed in some caseshowever, especially for designs involving verylarge unconentionl ly shaped elements orstrong interactions between neighboringcomponents.

Many other applications are possible, butthe foregoing account covering differentstructure types for which the authors have somecomputing experience should serve to emphasizethe versatility of the program and to assistdiscusion.of some of the special.problems thathave been encountered in practice.

9

INTERPRMATION AND VALIDITY. OF PRO.AN

It is essential for effective applicationof the program that the physical basis of thecalculations and the meaning of the output isclearly understood and that reliability is wellestablished. These questions of interpretationand válidation have been discussed in somedetail in the two previous papers, but forcompleteness will be briefly recapitulated inthis section.

Accounts of diffraction theory may befound in Ref s. 12 through 15. . The NPLcomputer program uses method of analysisdeveloped by Hess and .Smith-°' and by Garrisonand Chow, 17 which are applicable to fixedbodies or arbitrary shape placed in 'a regularlinear wave fIeld. The bodies disturb thewave fiéld and èxperience forces 'that combineconventional added mass änd diffracted waveeffects.' The computing proáess involves firstreplacing the body urfaces by a large numberof mál]. plane facets. Computer storage andrun timè limit the number of facets and,therefore, the resolution of the structure sur-

face. A fluid source placed at the center ofeach facet pulsates with the frequency of theincident wave. The ource strengths are. calcu-lated by matrix inversion technique to makethé'volocity normal to the body surface zeroat each source point. The oscillatory disturb-ance created by the sources represents thescattered wave, which includes the added massand is superposed linearly on the incident wavepattern. The resulting pressure 'fiêld isintérated over the body surface to' give .theover-all forcés arid moments.

Physical Inteì'pretatjôn'

The physical interprétation of this pro-cess has been discussed in the previouspapers12 and à fuller' account may be found inRef. 3. For the present a very brief explana-tion will' be given with emphasis on 'showing howdiffraction forces relate to conventionalinertia forces 'as define& by Morison' sequation.18 ' .

.

'It is convenient to regard the pressuredistribution on 'the body surface and theresulting forces and moments as being the sumof two componénts. The first is the pressurein the undisturbed incident 'wave th'at inte-grates th the force Fj., generlly Ithown as theFroude Krylov force. By an analogy with theArchimedes principle, this force may' be equatedto that which would be eeziéncèd by a fluidreplacement of the body volume V. Hence whenthe body spans a small enough fraction of wavelength such that the ambient acceleration U isaprodmately constant, it foUow that Fk =pVU (where p is the' density of water) and maybe. regarded as a dynamic analog of. static

buoyancy.:

The second component may be referred toas the disturbance pressure and integrates tothe force Fd, which is that due to the disturb-ance of the ambient pressure in the incidentwave by the presence of the:: body. This

includes the effect both of the local disturb-ance, normally déscribed in terms of an addedmass term kpVtJ, that edsts even when there isno free surface,:asw]l as: that of thescattering or diffraction of the incident wave.In fact, when the body spans a small fractionof wave length (less than a fifth say), thescattering term is negligible so that

F :.:kpVU,

It may. thus be found that a diffractioncoefficient Ch defined as

=: (Fk +

may be used to determine the total force F(neglecting drag) on: a fixed body by the for-mula

:

:

F ChFk

and for bodies spanning a small fraçtion ofwave length

V: :

C: 1+k Ch::. :: m

and:::

CpV(J:

: -

where Cm is the conventional mass coefficientas used in.Morison's equation.

For t:he general case of a boy of mass M,which has itself an acceleration U0 and is sub-.ject to an external force Fe, it may be shownas explained in Ref. 3 t:hat:

F CpV (ir - j) + (M- pV) TI

It shoiild be noted that some authors.identify mass coefficient with the added massfactor k defined, above and., if this is: now:denoted by C, the foregoing equation becomes

MtJ-.pV_CpV(U_tJ).

These twò more general forms for theinertia force equation become siificant whenanalyzing cases such as tension leg and tensionstay platforms where there is some dynai.cresponse of the body to :the action of the waveforces. -

Validation of the Proam

Ref. 1 describes comparisons with somevery preliminary :eg)eriments, and with data

from other published work including theoreticalresults of MacCarny and Fuchs'4 for a verticalcylinder, Garrison and Chow7 for a floatinghemisphere, and Gran19 for a gravity platformbase, the last named including some comparisonswith the results of model experiments. Thecomparisons with published data involvingpressure distributions as well as forces andmoments established confidence in the theoret-ical reliability of the program.: Regardingthe author' s own experiments these could onlybe discussed in a rather preliminary way inRef.: i and a much fuller acöount is given inRef. 2 prefaced by a shor:t review of otherpublished work reporting comparisons betweentheory and experiment. Even in this laterpaper, the eerimental results could not bereported in full since the test schedule wasnot finishéd in time. A more extensive setof results is however now available, and forcompleteness the following brief review summa-rizes: the over-all picture including recapitu-lation of salient material from Ref s, 1 and 2.

This may be introduced by again quotingsome of the references to other publishedcomparisons of theory and experiment recitedin Ref. 2. These broadly indicate remarkablygood agreement between theory and the resultsof model experiments but there is little gooddata for surface piercing bodies with diameterslarge in comparison with wave length and theonly data from full-scale tésts in the diffrac-tion regime biown to the authors is t:hat ofBrôgren As :mentioned:in Ref. 2, acomprehensive review of published comparisonswith model e,eriments is given byCh1OEabarti.2U This presents results for awide- range of simple geometric shapes takenfrom. other references, as indicated by thefollowing list using reference ntimbers appli-cable to the present paper, sown beside eachshape: Hemisphere,2- Sphere, Horizontal HalfCylinder, Z3 Horizontal Cylinder,24 RectangularBlock25. and Vertical Cylinder.6

Mention should also be made of a paper byLebreton énd Cormault2? containing results fortruncated vertical cylinders and 'to more recentpapers by van Oortmerssen28 and by Boreel29which contain comparisons including pressuredistributions'on a square sectiOn surfacepiercing column on a pyramidal base.

The authors' experiments invòlved measure-ment: of forces, moments and pressures on thefamily of four circular and four square-sectionvertical columns of varying height Shown inFig. i in water depth d = 2.3 m. They broadlyconfirm the favoráble comparisons with theoryreported by other investigators, but have drawnattentiön to some discrepancies occurring incertain cases.

OTC 2189 N. HOGB and R. G. STANDING 1415

The forces and moments were. measured bymounting the test columns on the five componentstrain gauge dynaniometer illustrated diagram-matically in Fig. 2, which includes a photo-graph of a test cólumn ready for mounting.The dinaniometer was designed by H. Ritterassisted by R. A. Browne, and consists essenti-

V1y of two vertical conáentric cylinders. Theinner one is bolted to the tank bottom and theouter one carrying the test column is connectedto the inner by a system of strain gaugedflexures. Pressures were measured at points onthe cylinder surface using flush diaphragmsilicon strain gauge transducers, and onetransducer was mounted internally on top ofthe dynamometer to monitor the pressure insideso that the measurements of vertical forcecould be adjusted accordingly. Some internalpressure variation resulted from the small gaparound the base needed to allow freedom ofdeflection for the dynamometer. The datà-handling equipment provided magnetic taperecords of all signals for subsequent computeranalysis and pen records for display duringthe experiments.

Measurementsof forces and moments fromtests in regular waves over a range of wavelengths for various wave heights are plottedfor all the eight coluims in comparison withcurves derived from the diffraction theoryprogram in Fig. 3.

The measurements of vertical force shownhave been corrected to remove the effect ofinternal pressure variation. The results forthe four circular columns have already beenpublished,2 but are recapitulated here forcompleteness with the perniission of the RoyalInstitution of Naval Architects. The resultsfor the square section. columns have not beenpreviously published.

It is, unfortunately, not possible to givehère corresponding details of the comparison ofpressure measurements with theory due to thelarge amount of space that would be needed.It is hoped that in due course a fuJ4er acçountwill be published, but meanwhile those inter-ested in comparisons of measured pressure withdiffraction theory predictions will findvaluable data in the paper by Boreeì29 men-tioned above.

The main conclusions from the experimentalinvestigations may be summarized as follows:

1. The results broadly confirm theexperience of other investigators who havefound good agreement between diffraction theoryand experiment. They are reassuring todesigners regarding the reliability of thecomputer program since nearly all the experi-mental values for force and moment lie below

the theoretical lines. No systematiò trend hasbeen identified in the scatter of variationwith wave height, and it seemS probable thatthis is a measure of the degree of eerimentalerror.

2, In certain cases it rna be seen thatthe experimental results for the verticalforces lie well below the corresponding theo-retical lines. This is believed to be due towavebreaking on the column top causing someloss ofwave height and, hence, reduction ofvertical force. This explanation is consistentwith the trend of the results, the defectbeing greatest for the coluimis that caused themost wavebreaking. As might be expected, thesquare columns are the most strongly affected,especially for the cases h/d = O. and 0.9 whenthe coluimi top is nearest to the surface, asmay be seen in Figs. 3f and g. Regardin thecircular columns only, the results for h/d =0.9 are affected and the gap is much smaller asshown in Fig. 3c.

The experimental measurements deviatedfrom linear theory in some other minor respectsthat did not noticeably affect the over-allforces and momènts. In Ref. i it is noted thatin shorter waves the wave profile on the frontface of the square section surface piercingcolumn rises to a central crest that is muchmore sharply peàked than the prediction oflinear theory, as demonstrated by a comparisonusing measurements from a photograph of suòh awave. This is a phenomenon to be expected forsteep nonlinear wàves and is a form of"clapotis". It haS subsequently been exten-sively observed and filmed on front and rearfaces of both square and circular sectioncolumns.

Irregularities noted in pressurerecords from the preliminary expeï'iménts de-scribed in Ref. i were subsequently found to bedue partly to spurious thermal effects on thetransducers since corrected and also to genuinesecond harmonic components'. These are' a Imownnonlinear phenomena occurring in steep reflec-ted waves and hàve the special property thatthey do not decay in the usual way with depth.Detailed accounts of the phenomena may befound in Ref s. 30 through 32 and some commentsrelating to the present investigation areincluded in Ref. 2. They do not significantlyaffect forces ànd moments, but unless theiredstence is recognized, they may wrongly'suggest faulty instrumentation.

in the case of the square columns,large vortices were observed at the cornersas illustrated by photographs in Ref. 1.0riginJ.1y, as indicated in the reference,' itwas thought that these might significantlyafféct the experimental results. Subsequent

ECPERIEICE IN COMF(JTING WAVE LOADS ON LARGE BODIES OTC 2i9

417

investigations, however, inclding some highlysimplified theoretical. estimates of the influ-ence. of the vortices on the pressure distribu-tion, persuaded the authors that the effectsshould at least theoretic11y be confined tovery small regions at the. corners. This viewseems to be con.firmed by the good agreementbetweer theoretical and eerimental pressuresreported by Boreel, 29 even, near the corners ofa square.sectioned column.

COtJTING EaIErCE

The NFL wave diffraction. program is docu-mented more fully in Ref s, 1 2 andresembles other programs described in Ref s. 17,28, 33 and 34. It is currently being operatedwithin NFL, by two computer bureaus and byUo,ds Register of Shipping. As noted in theintroduction, it has been applied mainly togravity structures including storage tanks andproduction platforms, but also to tension stayand semisubmersible pontoons. Some of thequestions that have arisen are of generalinterest. The remainder of this paper surnina-rizes the authors' eerience of running theprogram and answering the questions.

Section Shape Study

Comparisons were made in Ref. i of tuehorizontal force, vertical force and over-turning moment on simple gravity structuresresting on the sea bed.. These were simplevertical columns of circular, hexagonal andsquare section. The ratio of column height hto water depth d. was h/d = 0.3. The sectionareas were all equal to ita2, where a/d = 0.3.Section shape had little effect on the verticalforòe,. The horizontal force and moment on thesquare column was up to 8 percent 'higher thanon the circular, the hexagonal column resultsbeing intermediate.

Parametrisation of Forceson Circular Cälumns

For the reasons discussed in an earliersection, it s convenient to regard the waveforce on a body as the product of the F±'oude-Krylov force on that body and a diffractioncoefficient that takes accoimt of added masseas well as wave scattering. This becomes theconventional mass coefficient Cm in very longwaves. For models synùnetric abOut the twoplanés x = O and y = 0, threç diffractioncoefficients Ch, C and Cy are defined as theratios of the mad.rnuin total to ma,thnum Fronde-Krylov horizontal force, vertical force andoverturning moment, respectively. Values ofC, C. and C have been computed for a range ofcircular columns, of different aspect 'ratios,typified by their height-to-diameter ratioh/2a, resting on the sea bed. The results,

tabulated and shown graphicUy in Ref. 2,Table 1 and Fig..7, are relatively insensitiveto wave length X and water depth d over a rangeof large X/a and d/h values. These casesinclude many of interest tO designers ofgravity structures, particularly because sec-tion shape is also comparatively unimportänt(see section, Shape Study). The results arereprinted here in Table 1. The original tablealso included a phase añgle associated Witheach force and moment, but these differedlittle. from t'he corresponding Froude-Krylovvalues. Over most of the range the coeffi-cients can be fitted quite well to the approd.-mate formulae:

Ch 1 0.75 (h)1/3(i -0.3

C 1 + 0.74 k2a2.(--) for 1.48 ka (_!L)V 2a' 2a

1 + 0.5 ka for 1.48 ka (1;) > 1,

C = 1.9 -0.35 ka.

The range of vàlidity, by comparison with theresults of the full diffraction analysis, isroughly

h/d <0.6

0.3 < h/2a <:2.3 for Ch and

0.6 < h/2a c 2.3 for C,,.

Over these ranges the approd.mate formulas,shown in Table '1, are nearly lJ. less than. 5percent in error. Over most of the range, theerror is only 1 - 2 percento The formulas maybe italid over a greater range of h/2a, but.notests have been carried out. When h/2a islarge, both Ch and C tend to a constant valueclose to that predicted by the theory ofMacCamy and Fuchs and shown as in Fig. 2 ofRef. 35.

It must be emphasized that these resultsare pplicable only to columns resting on thesea bed. The horizontal force on a suspendedor buoyant cylinder, such as a tension-stayplatform or spar buòy, is probably similar, but'the vertical force iS completély different,depending on the vértical pressure gradient inthe fluid rathei than on the pressuré itself.The corresponding C. differ.s accordingly, andin particular becomés very much greater than iif h/2a is small. The analogy is with a thin.disc moving at right angles to its plane,entraining a large mass of fluid.

Interaction Effects

The velocity potential at angular positione and distance R from the añs of a circularcylinder of radius a in a uniform unbounded

< 1,

OTC 2189 N. HOGBEN and R. G. STAIWING

stream, speed U, is from6

U (. + a2/R) cose

This means that the disturbance velocity decayswith distance as a2/R2 and is small for Rgreater than two cylinder diameters. If a freesurface is introduced. and th cylinder passesvertically through it, a similar decay rate isexpected both in currents far below the. surfaceand in waves that are long compared with R anda, and can be treated instantaneously assteady.

The cylinder scatters shorter waves how-ever, and the disturbance field extends muchfurther. A diffraction program of the typedescribed Is useful for studying interactionsbetween two or more members of a structure.Lebreton and Cormault,27 their Fig. 9, usingthe program described in Ref. 33, calculatedthe forces on two vertical piles 5 diameters

apart. The inaxintum effect was felt upstreamwhere waves reflected from the downstream pileface combined with the incident wave to form

partially' standing wave pattern, as in Ref.

28, Fig. 9. The upstream pile experiençéd amadrnurn increase or decrease of force of about20 percent near ka = 0.6, depending on itsposition in the other's standing wave pattern.Smaller effects were noted in both longer andshorter waves, with for example changes ofabout 10 percent near ka = 0.4. The down-stream pile was less affected, correspondingchanges beingless than 5 percent. Pilesarranged parallel to the wave crests affectedeach other by only 2 percent.

Ref. 2 confirmed the extent of the up-stream disturbance at ka = 0.5e The waveelevation, according to MacCainy and 'Fuchs'theory, 14 was stil]. 15 percent of the 'incidentwave at R/a = 10. The conclusion, therefore,is that in long waves and deep currents, inter-actions are unimportant at radial distancesgreater than two colurrni diameters, but thereseems to be considerable upstream influence at5 diameters near ka = 0.5 and shorter wave--lengths.

There may also be interactions betweencoluimis placed on top of each other. Calcu-lations in Ref. L showed less than 10 percentdifference between the forces on a completetwo-coluzini structure and the sum of foròs onthe separate components. But in that case theupper coluxmi diameter O.2d was small enòugh incomparison with the base diameter 0.6d toleave the'flow around the base substantil1yunaffected. The difference is probably greaterif the coluni diameters are similar or if thesuperstructure consists, of several towersinstead of'one.

The Effect of Diffraction,on Wave Elevation

When a wave travels into shallow water, itsteepens and its length decreases. A wave'passing over an obstacle experiences a similareffect, intensified by focusing of the waverays as they change direction.i7 This meansthat the wave elevation over the top of asubmerged structure maybe greater 'than thaton either side. Desi'iers should take accountof this (i) to ensure that thedéck is highenough to clear the highest waves expected, (2)when computing wave forces because members nearthe surface are immersed more deeply (3) inconsidering impact . loads where the wave steepenenough to break. . '

The NPL program can give designers guid-ance because it can provide optionally thelinearized diffracted wavè elevations at any'point. This is done by requesting pressure'output on the plane z = O. The linearizedBernoulJ.i equation. for the subsurface pressure

p is' '

p = p ò/òt,

where '1' is the velocity potential and t istime. On z = 0, the Bernoulli equation givesthe surface wave elevation instead,

= 1à/òtonzO.

Thus, requesting dimensionless pressure p/pgHon z = o in fact gives the ratio of waveelevation to indident wave amplitude /%H.Fig. 4 shows aprofile 'along the 'centerliney = O of the 'total wave amplitude over thesubmerged square-section colunri shown in 'Fig. 1.

Here h/d 0.9 and ka = 1.0. The wave amplitudEoscillates upstream where the reflected andincident waves interfere to form a partiallystanding wave pattern. It decays gradlly to'the incident wavé level downstream. Alsoshown is the actual wave profile at the instantwhen the total downward force is a mad.mum(phase= 2.22 rad)., Over the top of the coluim,the wave is both higher and. shorter than 'theincident wave. In practice, quite steep inci-dent waves broke at this point.

The Effect of Óurrents

Currents have three distinct effects.First, by changing the fluid particle velocitythey change the fluid drag. Because dragdepends on the square 'of the.velocity, and 'thecurrent velocity, decreases slowly with depth,a comparatively small current can increase dragsiiificant'ly. Because this paper is concernedmainly 'with inertia-dominated situations, thiseffect will not be discussed further.

BERIENCE IN COtJTING WAVE I.ÛADS ON LARGE BODIES OTC 2189

The second effect is that of changing thewave speed, the wave propagating over a movingrather than stationary fluid. This may beassociated with wave steepening.3 Thiseffect is small in a North Sea desi wavesituation, where, for example, wave period T =15s, wave speed x/T = 23 m/s, but the madmumcurrent speed is only 1.5 rn/s.

The third effect of a current i8 to makethe structure itself ge±erate waves. A bodyin a uniform current causes a stàtionary wavepattern to form on the free surface andexperiences a corresponding net force, whichmay be regarded as a special type of diffrac-tion force. It is also directly equivalentto the so-called wave-making resistanceexperienced by a ship or other body in uniformmotion through calm water. This fOrce can becalculated by basici ly the same method as thediffraction analysis used in the edstingprogram, but with a different form of Greentfunction that now describes the potential dueto a unit source in a tthiform current ratherthan a pulsating soUrce in still water. Theauthors have examined the possibility ofmodifying the computer program so that it cancalculate these uniform current forces butdecided against it for reasons that are dis-cussed in detail in Ref. 39 and can only bebriefly summarized here.

Although the revised fOrm of Green' sfunction is laiown and may be foUnd for examplein Refs. 15 and 40, it is not simple, andsubstantial effort would be nèeded to make thenecessary programming changes. This effortmoreover would not be justified because forpractical offshore structures in realisticcurrents conventional drag forces due to flowseparation and wake formation effects aredominant and the diffraction forces are notonly negligible in comparison, but also deviategrose]iy from the predictions of potential flowtheory. This point is well substantiated bycomparisons for the case of a surface piercingvertical cylinder, between wave resistancescomputed from theory of Ref. 41 and experi-mental values derived from measured wävepatterns by the methods of Ref. 42.

Fig. 5 summarizes the results. As dis-cussed in the references, the theory lacksi.iniqueness, but using the favored sOlution forcylinders of elliptical section with = b/a asshown, it was found that the ratio Cw/ 2.4independent of e. when plotted to a base of

Froude number je F = c// , where

Cw = wave rsistance/(pAc2)

A = projected frontal areac = current spéed

F = c/f.

The figure shows experimental results for acircular cylinder, a = 1.0, but to achieve evenapprodmate correspondence with theory it wasnecessary to assume, due to flow separation andwake formation, an effectively ellipticalsection with 6 = 0,2. 1Ìi Ref. 39 the relevanceof these results to offshore structures isdiscussed, and it is shown that even forexception11y strong North Sea currents, Cw isnegligible in comparison with Cd.

Nonlinearity of the Waves

The NPL wave-diffraction program useslinear wave theory even under extreme designwave conditions. Designers have expressed con-cern about two nonlinear effects. First, arethe differences between linear and higher-ordertheories significant for a typical structurein a typical design wave? Forces F on a con-ventional jacket-type structure are usuallypredicted using Morison' s formula18

F = CpV(J+CapAUjUI

with velocities U and acceleratiOns U givenby higher order StOkes43 or stream functiontheory.44 Skjelbreia and Hendrickson's.Fig.343 shows linear and Stokes V velocity profilesdiffering by 20 percent or more at the freesurface. This would indicate differences indrag of over 30 percent. Is this an extremeexample?

Second, where the structure pierces thesurface, is it necessary to integrate theforces up to the actual free surface, ratherthan to mean water level? Conventional lineartheory, in particular the NPL program, computeswave forces over a constant immersed depth, thewave pressure field being continued up to orcut off at mean water level.

Comparisons were made first between linearand Stokes V43 regülar waves of the sarnê heightH and period T in the same méan water depth d.Some authors prefer to use still water depthinstead, the difference being terms propor-tional to H2 and 4, according to Stokes Vtheory. The ratios chosen H/gT2 = 0.015 andH/d = 0.2, are typical 6f a northern North Seadesign wave situation, where for example d =150 m H = 30 m T 14 s. Comparisons werealso made using a less steep wave H/gT2 = 0.01àt H/d = 0.2. The conclusions were similar,but the differences smaller. Dean45 comparedvarious anaJ.ytical theories to find the bestfit to the free surface boundary conditions.He recommended Stokes V Or stream functiontheory forthe cases described here. WhenH/gT2 = 0.015, the linear and Stokes V wave-lengths X are given by x/gT2 = 0.158 and 0.170,respectively. The difference, of some 7 per-cent might be important for a structure

0C 219 N. HO and R. G. STMDDG )#19

spanning several wavelengths, though norm1 lysuch short waves would contribute little tothe over-ailforces. Velocity and accelerationprofiles were also computed. Fig. 6 shows thethese as dimensionless Morison drag an4 inertiaterms, uIup/-H, wPwl/gh, /g and */g, whereu and w are the horizontal and vertical veloc-ity components, and and * are the corre-sponding accelerations. Each profile is shownat its mañmum in the wave cycle. The differ-ences between the linear and Stokes V termsare roughly 20 percent in wßwj, 9 percent in uwhich are in phase with wavé slope, but 13 per-cent in uf uJ, 3 percent in *, which. are inphase with wave elevation. This suggests thatthe most important effect of nonlinearity isto steepen the waves rather than change itselevation. The differences decrease withdepth, the theories being identical near z/d =0.4. There are qmVI differences of oppositesign below this depth.

Fig. 7 shows the resulting force and over-turning moment on a slender vertical column ofradius a extending fróm the sea bed below theorigin of coordinates up through th freesurface. The linear and Stokes V inertiacontributions, both integrated up to meanwater level z = 0, differ by less than 7 per- -

cent of the madmum; the drag contributions byless than 13 percent. The load on a typicalgravity structure in a North Sea design wave ismainly inertial and most of the structure isquite deeply submerged. Linear theory istherefore quite adequate to describe the over-all loads. A higher-order theory may beneeded for loca:I loads on drag-dominatedmembers such as conductor tubes, particularlynear the free surface.

There the structure pierces the freesurface, the error in using linear rather thanhigher-order Stokes theory may be rather lessthan the error in regarding the immersed depthas constant. Fig. 7 compares calculationsof the Stokes V force integrated up to theactual free surface z = with the forôe tip tomean water level z = 0. A similar calculationusing linear theory showed only small dif f er-ences from Stokes V theory. In Fig. 7, theinertia force and moment are skewed sidewaysso that the madma occt2r closer to the wavecrest, but their magnitudes are only slightlyincreased. Thus, the main effect f the extrawave immersion on the inertial load is a phaseshift of the mañmum. If the wave is. distortedby diffraction, the effect is less predictable.The wave elevation may be increased as well asthè phase changed (see Fig. 4 for exaple)o.If the linear diffracted wave elevation isrequired this can be output optionily by theNPL program as elained in the effect o.fdiffraction on wave elevation.

Tald.ng the actual rather than meanimmersed depth has a dramatic effect on drag.The madmwn in Fig. 7 still occurs at the wavecrest, but its maitude is almost doubled.This effect is most important, particularly inrespect of local loads where drag-dominatedmembers pierce the free surface.

Ai-i Approdmate Method for Bodies With Wells

Some recently proposed gravity structureshave featured either a protective wafl aroundthe base or a vertical well extending part orall of the way. down to the sea bed. Thesepresent similar problems to the computer, inthe first case if the wall is too thin, inthe second if the well is too narrow and deep.-As noted in Ref s. 16 and 2, a satisfactorycomputer model of the inner surface arid lipmay require a large number of facets at a I

correspondingly high cost. An inadequatedeséription results in program failure duringmatrix inversion or inaccurate results.Fortunately, it is often possible to cap thewell, correcting for it later. Numericaltests .to find a satisfactory correction pro-cedure were carried out on a simple circularbase with a vertical central well extendingdown to the sea bed. Two approd.znate calcu-lations of the forces and moments were com-pared with results from the actual modelincluding the-well. The first approdmationmakes the assumption thàt thè water inside thewell is dead and at a uniform but time-varyingpressure. This means that the well is effec-tively cápped and the actual body experiencesthe same forces as the capped body less thevertical forces on the cap itself. The secondestimate comes from multiplying the forces onthe capped body by the ratio of the displace-

-

ment volume of the actual and capped bodiesoThis approdmation roughly represents the otherextreme where the external pressure gradientpenetrates right inside the otherwise deadwell. The analor here is with the buoyancyforce on the body turned on its side in a.uniform vertical gravitational pressuregradient. It may be expected that the forceson the actual body should lie somewhere betweenthe two. approximations.

'Fig. shows the horizontal and verticalforce and the ovei'turriing moment on typicalgravity bases of height h, radius a, well radiusr in water of depth d. Here the ratios dfa'=3, h/a 0. and the two wavelengths chosen àreka = 0.5 and 1.0, where k = 2tJX. The resultsare plotted as functions of r/à, small r/arepresenting a narrow deep well that hardlyaffects the forces, r/a close to 1 representinga thin protective wall on its own. As washoped, the forces and moment on the actual modellie between the, two estimates. The actualstructure surface was divided into 220 facets,

20 xitICE IN COI1PJTING WAITE LOADS ON LARGE BODIES OTC 219

80 on the inside and 80 on the outside wallswith 60 on the top. The pressure gradientpenetrated with attenuation roughly 0.5 to 1.0r inside the well. Below that depth the pres-sure was roughly uniform. The small pressuregradient remaining there was probábly theresult of using too coarse a mesh, but thisdid not affect over-all forces appreciably (byless than 1 percent). Assuming the well to bedead gave conservative estimates of horizontalforce and overturning moment but the correctvertical force.

It is therefore recommended that if awell cannot be included in the structuralmodele it should be capped, the force andmoment on the cap being subtracted at the end.The NPL program does this quite simply. Othertypes of indentation can be treated in asimilar way, allowing for pressure variationson the bottom if this.is part of the structure.This procedure should normally providé a con-servative estimate of forces, but the momentmust be treated with care. As noted, in Ref. 2,

the overturning moment on a gravity structurewith h/a= 0,8 is small as a result of thefine balance between forces on the base top and

sides. In the cases described here themoment due tO side force predominates, so thatan overestimate of horizontal force accompaniesone of momênt. But if the base height is re-duced slightly, the top forces predominate.An ovérestimate of horizontal force may thenreduce the overturning moment and give mis-leading results.

The physical effect of flow separation atthe wêfl top probably makes the water in thewell more dead than linear potentiaJ. flowtheory predicts. But if the structure and welldiameter are large, this is not expected toaffect the over-all forces too much.

Moving Bodies

The NPL program was written with gravitystructures in mind and at present works withfixed structures only. Recent developments intension-stay, semisubmersible and spar-buoy-type platforms have raised questions concerningresponse to wave excitatiOn. The theoreticaltreatment of moving bodies is similar ta thatof Í'ied bodies, invelving merely a change inthe body surface boi.mdary conditions. Underthe assumptions of lin?ar and harmonic re-sponse the body motion can be brOken down intooscillations in the six degrees of freedom,three of translation and three of rotation, asdescribed in Ref s, 15 (Section 19), 4.6 arid 34.Refs. 34 arid 47 describe programs that solvethese component problems. They compute addedmass arid damping. coefficients for bodiesosciflating in. surge, sway, heave, roll, pitchand yaw in otherwise calm water, also the

exciting forces on the fixed body in waves.This capability will shortly be added to theNFL program.

Until then it is possible to estimate someof the coefficients. The Hasldnd relationshl°relate the forces on a fixed body lxi t1aves tothose on the same body oscillating in otherwisestill water. Thus for example, the principaldamping coefficients, according to Ref. 46,Eq. 30, are proportional to the integrals overall incident wave directions of the squareC ofcorresponding exciting forces. This integra-tion is not practicable in general, but issimple if the body is añsymmetric abou thevertical axis, It then gives Newman's4 Eqs.31 through 33 for damping coefficients insurges heave and pitch in deep water orGarrison' 47 Eq. 44 for heave in shallow water.

In many design situations the excitingwave is long compared with body dimensions. Ifthe response period is comparably long, thereare several simplifications. First, accordingto Newman' s equations the damping coefficientis proportional to the cube of wave frequencyand is therefOre small for long period oscilla-tions. Second, the added mass fluid f9rcesacting on a body in motion depend only on therelative motion of the body and surroundingfluid, If the body is small compared withwavelength, it creates the same disturbance andexperiences the same added mass forces whetherit is fixed in waves and the surrounding fluidparticles travel around elliptical Orbits or

itself travels around the same elliptical orbitat the same speed but in otherwisé still water.This means incidentally that, if the bodyexcursions are small compared with local waveorbit diameters, the fluid forces are im-affected by body motion. To separate the co-efficients in heave, surge and sway, it isnecessary to ook at the three òompänentmotions separately. Care is needed if the bodyis unsymmetric about the x = O or y = O planebecause the horizontal acceleration phase ofthe elliptical motion causes both horizontaland vertical forces, and similarly the verticalphase. Many proposed structures fortunatelyhave double symmetry so that there are no surge/sway/heave interactions. The separate fOrcecomponents then give corresponding added masscoefficients, In cases where the body movescare must be taken to use the full inertiaforce equations cited in Interpretation andValidity of Program, which includes the effectof the acceleration of the actual mass of thebody.

CONClUSIONS

1. Experimental measurements of over-allforces and moments on colunins gener11y agreewell with the predictions of diffraction

OTO 2189 N. H0EN and R. G. STANDflIG 421

theory, the measured forces being slightlysmaller. This broadly confirms earlierfindings, but the eerimental results for thesquare-section coluiris showed more scatter andgreater deviation from linear theory, espe-ciVIy regarding the effect of wavebreaidng onthe vertical forces, as discussed in the next.paragraph.

Measurements of the vertical force insome cases tend to lie further below thetheoretical line than in others. This isthought to be associate4 with wavebrealdng ob-served over the C luirai top. In the case of thecircular columns, only the results for h/d =0.9 are slightly affected. For the squarecolumns where greater wavebrealdrig occurredas may be eected, the effect is very muchstronger arid extends to h/d = 0.8 as well as0.9.

Approidmate formulas are given for thediffraction coefficients of circular columnsin quite long waves and deep water. Thesediffer from the results of the full diffractionanalysis by less than about 5 percent.

40 The wavemaldng resistance of coluimisin currents is much smaller than the drag. itis not thought worthwhile to include this.effect in the NPL program.

The differences between linear andStokes V theory are small in tipicaJ. North Seadesign wave conditions, especially with regrd.to inertia force. More siificant is the.differnce between integrating fôrces up to themean and actual free surfaces.

If wells inside gravity bases cannotbe includedin the full diffraction analysis,they should be capped and corrected for lateron the assumption that the enclosed water isdead. This seems to give a conservative esti-mate of fòrces but care is. needed ininterpreting the moment0

N0MECLATURE

a = column radius ora,b = major arid minor haJ f-axes of ellipse

A = projected frontal area of bodyc = current speed

Cd = drag coefficientC ,C = diffraction coefficients for hori-y y

. zontal force, vertcai force ndoverturning moment, defined asmadmum total force or moment/corresponding madmum Froude-Krylov force or moment

Cm= mass coefficient2

C = wave resistance/pAcd = water depthF=forée

= disturbance forceF = external forceeFk = Froude-Krylov forceF= Froude numberg = acceleration due to gravityh = column heightH = wave heightk = wave number .2it/X ork C' Cm 1, added mass coefficient

in review sectionM = body massp = pressurer = well radiusR = radial distancet = timeT = wave period

u, .w = fluid velocity components. in x andz directions

u, w = corresponding acceleration conipo-nents

U.= fluid velocityf luid acceleration

ACIÔWLEDGNTS

The authors wish tO acIthowledge their debtto their colleagues who helped with this work,especially to J. Osborne for conduct andanalysis of the experiments, and to H. Ritter,R. A Browne and G. S. Smith for design of theinstrumentation.

RERCES

Högben, N. ad Standing, R. G.: "WaveLoads on Large Bodies," Proc., Inter-national Symposium on Dynaniic of Max'ineVehicles and Structures in Waves at -

University College London, published by theInst. of Mechanical Engineers (April 1974).Hogben, N., Osborne, J. and Standing, R.G.: "Tave Loading on OffshOre StruOtures -Theory and Experiment," Proc., Symposiumon Ocean Engineering at the National Physi-cal Lboratory, London, published by theROyal Inst. of Naval Architects (1974).Hogben, N.: "Fluid Loading o±i OffshàreStructures, A State. of ArtAppraial: WaveLoads," Maritime TechnOlogy Monograph No.1, published by the Royal Institute ofNaval Architects (Nov. 1974).

EflPERIENCE IN COMHJTING WAVE LOADS ON LARGE BODIES OTO 2189

aniilton, A.: "A Low Load for Platforms,"Financial Tithes, London (Nov. 1, 1974).Anon.: "Gravity Platforms: Who isProposing What," New Civil :HgineerSpeciàì Review on North Sea Oil (May1974).Paulling, J. R. and Horton, E. E.,: "Analy-sis of the Tension Leg Platform," .Pet. Eng., J. (Sept. 1971) 285-294.Brogren, E.,.Soderstrom, J., Snider, R.and Stèvens, J.: "Field Data RecoverySystem, Iazzn Dubal No. 3," Paper OTC1943 presented at Sixth Offshore Tech-nology Conference, Houston, May 6-8, 1974.Marion, H. A.: "Ekofisk Storage Tank,"Royal Inst. of Naval Arçhitects Symposiumon Ocean Engineering, Nov. '1974.

'9.' P. E.;Tòwnshend, M. A.: "Offshore Storageand Tanker Loading," Royal Inst. of NavalArchitects Symposium on Ocean Engineering,Nov. 1974.Paulhirg, J. R.: "Elastic Response ofStable Platform Structures to WaveLoading," Prc., International Sympo siuinon Dynamics of 'Marine Véhicles and Struc-tures in Waves at University CollegeLondon, published by the Inst. of Mech-anical Engineers (April 1974).Hooft, J. P.: "A Mathematical Method ofDetermining Hydrodynamically InducedForces on a Semi Submersible," Trans.SNAME (Nov. 1971).Havelock, T.H.: "The Pressure of WaterWaves Upon a Fixed Obstacle," Roy.Soc. A-963 (1940) .John, F.: "On the Motion of FloatingBodies Pért II," Ccrnim. Pure and AppliedMaths (1950) .MacCamy, R. C. and Fuchs, R. A.: "WaveForces on Piles: A Diffraction Theory,"Beach Erosion Board Technical MemorandumNo. 69 (1954).Wehausen, J. V. and Laitone, E. V.:"Surface Waves," Encyclopaedia of Physics,Springer, Brlin (1960).Hess, J. L. and Smith, A.M.0.: "Calcu-lation of Potential Flow About ArbitraryBodies," Progress in AeronauticalSciences (1967) .Garrison, C. J. and Chow, P. Y.: "WaveForces or Submerged Bodies," ASCE Water-ways and Harbors Div. (1972) 2.Morison, J. R., O'Brien, M. P., Johnson,J. W. and Schaaf, S. A.: "The ForceExerted by Surface Waves on Piles,"Trans., AINE (1950) .12.Gran, S.: "Wave Forces on SubmergedCylinders," Paper OTC 1817 presented atFifth 'Offshore Technology Conference,Houston, April 30-May 2, 1973.Chalabarti, S.: "Wave Forces on Sub-merged Objeçts of Symmetry," ASCE Water-ways and Harbors Div. (1972) 2.Garrison, D. J. and Snider, R. H.: "Wave

Forces 0±1 Large Submerged Tanks," TexasAßeN U. , Sea Grant Publication No. 210COE Report No. 117, 1970.O'Brien, M. P. and Morison, J. R. "TheFôrces Exerted by Waves on Objects," Trans,Amer. Geophys.,Union (Feb. 1952) , No. LShank, G E. thd Herbich, J. B. :. "ForcesDue to Waves on Submerged Structures,"Texas MN U. COE Report No. 123, May 1970.Schiller, F. C.: "Wave Forces on a Sub-merged Horizontal' Cylinder " MS thesis,Naval Postgraduate School, Monterey,Calif.,Report No. AD 727 691 (June 1971).Brater, E. F., McNown,J. S. and Stair,L. D. : "Wave Forceé on Subnierged Struc-tures," J. of the Hydraulics Div., ASCE(Nov. '1958) Nó. HY6. '

Jen, Y.: "Wavé Forces on Circülar Cylin-drical Piles Used in 'Coastal Structures,"Hydraulic Engineering Laboratory, Collegeof Engineering, U. òf C1iforriia,Berkeley, HM 9-11 (jàn. 1967).Lebreton, J. C. and Cormault, P.: "Wave'Action on Slightly Immersed Structures,Some Theoretical and Experimental Con-siderations," Proc. Symposium, "Researchon Wave Action," Deift (1969) J.Van Oortmerssen, G. : "Some Aspects ofVery Large Offshore Structures," Ninth ONRSymposium on Naval Hydrodynamics, Paris(1972).Boreel, L. J. : "Wave Action on Large Off-shore Structures,," Proc., Inst. of CivilEngineers Conference on Offshore Struc-tures, London (Oct. 1974).Miche, R.: "Mouvements Ondulatoires desMers en Profondeur Constante ouDScroissante," Annales des Ponts etChaussees (1944).Longuet-Higgins, M. S.: "A Theory, of theOrigin of Micro seisms," PhIl. Trans. A243(1950).Cooper, R.I.B. and Longuet-Higgins, M. S.:"An Experimental Study of the PressureVariations in Standing Water Waves,"Roy. Soc. A2O6 (1951).Lebreton, J. C. and Margnac, A.: "Calculdes Mouvements d'un Navire ou d'unePlateforme Arnarre dans la Houle," LaHouille Blanche (1968) , 379-389.Faltinsen, O. M. and Michelsen, F. C.:"Motions of Large Structures in Waves atZero Froude Number," InternationalSymposium on the Dynamics of MarineVehicles and Structures in Waves, Univer-sity College London (1974).Chakrabarti, S. K. and Tam, W. A.: "Grossand Local Wave Loads on a Large VerticalCylinder - Theory and Experiment," PaperOTC 1818 presented at Fifth Offshore Tech-nology Conference, Houston, April 30-May2, 1973.Lamb, H.: Hydrodynamics, Cambridge U.Press, 5th ed.. (1924) paragraph 68.

OTC 2189 N. HOGB and R. G. STA1DING L2 3

2

EXFERIENCE IN COUTING WAITE LOADS ON LARGE BODIES OTC 2189

37. Stoker, J. J.: WaterWaves, Interscience Tests," Trans., Roy. Inst. Nay. Arch.Publishers (1957) Chap. 5. (issued for written discission as paper

38. Longuet-Higgins, M. S. and Stewart, R. W.: W11, 1974)."The Changes in Amplitude of Short Gravity 43. Skjelbreia, L. and Hendrickson, J.:

Waves on Steady Non-Uniform Currents," J.Fluid Mech. (1961) 10, 529-549.

"Fifth Order Gravity Wave Theory," Proc.,Seventh Cönference oñ Coastal igineering

39. Hogben, N.: "Wave Resistance of Surface (1961) Chap. 10. -

Piercing Vertical Cylinders in UniformCurrents," NPL Ship Div. Report No. 183

44. Dean, R. G.: "Stream Function Representa-tion of Non-Linear Oceàn Waves," J.

(Nov. 1974). Geophys. Res. (1965) 70, 4561-4572.40. Lunde, J. IC: "On the Linearisèd Theory

of Wave Resistance för Displacement Shipsin Steady and Accelerated Motion," Trans.

45. Deañ, R. G. : "Relative Validities ofWater Wave Theories," gïneering Pro-gress at the .U. of Flórida, Technical

SNAME (1951) , 24-76. Progress Report No. 16, 1968.41. Kotik, J. and Morgan, R.: "The Uniqueness 46. Newman, J. N.: "The Exciting Forces on

Problèm for Wave Resistance Calculatedfrom Singularity Distributions Which Are

Fixed Bodies in Waves," J. Ship Rés.(ì962) 6, 10-17. -

Exact at Zero Froude Number," J. Ship Res. 47. Garrisàn, C. J.: "Hydrodynamics of Large

(1969) , 61-68. Objécts in thé Sea - Part 1 - Hydro-42. Hogben, N. and Standing, R. G.: "Wave dynamic Analsis," J. Hydronautics (1974)

Pattern Resistance from Routine Model , 5-12.

TABLE i - COARISON BETWEEN COÌ.PUTED DIFFRACTION COEFFICIETS FOR CIRCULAR CYLITDERS

Ch Ci,, C : DIFFRACTION TIORY AND APPROXIMATE FORMULA

.L'kACTÏONTHEORY FROM REF. 2

APPROXThLATE

FORMULA

ka h/2a h/d Ch C C Ch Cv C

0.1 0.75 0.15 1.70 1.01 1.78 1.68 1.01 1.870.75 0.3 1.70 1.01 1.78 1.68 1.01 1.871.75 0.7 1.89 1.02 1.86 1.90 1.01 1.87

0.2 0.25 0.1 1.44 1.01 0.93 1.47 1.01 .1.830.2 1.60 . 1.02 1.92 1.59 1.01 1.83

0.5 0.4 1.63 1.02 1.94 1.59 1.01 1.830.75 0.3 1.70 1.03 1.78 1.67 1.02 1.830.75 0.6 1.76 1.Ö3 1.83 1.67 1.02 1.83i..O 0.8 1.87 1.06 1.89 1.74 1.03 1.831.25 0.5 1.81 1.05 1.81 1.80 1.04 1.831.75 0.44 1.89 1,08 1.86 1.89 1.05 1.831.75 0.7 1.89 1.08 1.86 1.89 1.05 1.832.25 0.56 1.89 1.09 1.85 1.91 1.07 1.83.?.25 0.9 1.98 1.15 1.94 1.97 1.07 1.83

0.5 .25 0.1: 1.42 1.04 0.98 1.44 1.05 1.730.5 0.2 1.57 1.10 1.86 1.55 loO.9 1.730.5 0.4 1.60 1.12 1.89 1.55 1.09 1.730.75 0.3 .1.66 1.15 1.72 1.63 1.14 1.730.75 0.6 1.71 1.20 1.78 1.63 1.14 1.731.0 0.8 1.85 1.42 1.85 1.69 1.19 1.731.25 0.5 1.74 1.23 1.72 1.75 1.23 1.731.75 044 1.78 1.28 1.74 1.84 1.25 1.7.31.75 0.7 1.78 1.28 1.74 1.84 1.25 1.732.25 0.56 1.80 1.27 1.74 1.91 1.25 1.732.25 0.9 1.90 1.63 1.86 1.91 1.25 1.73

1.0 0.251 0.2 1.34 1 17 .1.17 1.33 1.18 1.550.5 0.4 1.44 1.33 1.54 .1.42 .1.37 1.550.75 0.3 1.54 1.43 1.54 1.48 1.50 1.550.75 0.6 1.49 1.43 1.47 1.48 1.50 1.551.0 0.4 1.58 . 1.48 1.54 1.53 1.50 1.551.0 0.8 154 1.57 1.49 1.53 1.50 1.551.25 0.5 1.59 1.49 1.54 1.57 1.57 1.551.5 043 1.62 . 1.52 1.56 1.60 1.50 1.551.5 0.6 1.59. 1.48 . 1.53 1.60 1.50 1.55

.1.75 Ó.5 1.63. 1.53 . 1.56 1.63 1.50 1.551.75 0.7 1.58 1.46 1.53 1.63 1.50 1.55

STILL WATER LEVEL

ELEVATIONFig. i - Series of circular and square section columns of varying height.

o

c) r

PLAN:

'jSi

S

SS

t

Fig. 2a - Dynamometer and circular column(h/d 0.7) before mounting.

WATE H O9 d SURFACEDEPTH

d

i

O7 ¿08 ¿ PIERCING

TAN KBOTTOI.I

sj I

test

colu

mn

Fig. 2b

- Ctaway

view of te;st

co I ümn mounted on

dynamometer.

b) 2

- c

ompo

nent

horiz

onta

lfo

rce

flexu

re(u

pper

)

f) 2

-com

pone

ntho

rizon

tal

forc

efle

xure

(low

er)

stra

inga

uges

stra

inga

uges

Str

ain

gaug

es

a) v

ertic

alfo

rce

flexu

re

c) s

poke

str

ansm

ittin

gho

rizon

tal

forc

e

e)'fl

oatif

lg'

cylin

der

g) s

poke

str

ansm

ittin

gho

rizon

tal

f orc

e

h) b

ase

plat

e

Fig. 2c - Di.agirammattc sketch of

dynamometer.

HORîZOS1AL FOUC

COPUTED

k.

VERTICAL FORCE

SPOTS DENOTE MEASURESIENTS USING STRAIN GAUGE O H/. - 033DYNAIIjOMETER IN WAVES OF VARIOUS HEISS 0 022

- -- A- 0.17

V 0.11

hid - ös

04 IO 15 0 0.5 1-0 1.5

HORIZONTAL FORCE

h/d og

MOMENT

MOM EÑ T

SPOT! DENOTE MEASUREMENTS USINO STRAIN GAUGE .0 H/P - 0.33ZIVNAMOMEIER IS WAVES OF VARIOUS HEIGHTS 0 022

- £ 0.17V DII

SPOTS DENOTE MEASUREMENTS O 4/. - O-33USING STRAIN GAUGE 0 022DVNAMOMETER IN WAVES .0 0-1OF VARIOUS HEIGHTS V 0.11

112 99 H

2.

SPOTS DENOTE MEASUREMENTS USING STRAIN GAUGEDVNAMOMETER IN WAVES OF VARIOUS HEIGHTS

COMPUTED2

COMPUTED

k. IO I

h/d -07-

HORIZONTAL FORCE - VERTICAL FORCE

COMPUTED

-s-

HORIZONTAL FORCE

COU UT E D

h/d 08SPOTS DENOTE MEASUREMENTS USING STRAIN GAUGE O H/ -0.23DVNAMOMETER IN WAVES OF VARIOUS HEIGHTS 0 022

6 0.17V 0.51

HORIZONTAL FORCE VERTICAL FORCE MOMENTe

COMPUTES--/2pHPd

I I I I

IO IS 0 05 k 1.0 1.5

h/d 09

SPOTS DENOTE MEASUREMENTSUSING STRAIN SAUSEDYNAMOMETER IS WAVES0E VARIOUS HEIGHTS I

MOM E NT

O

0.0

Fig. 3 Comparison of computed and measured force.s and momentson vertical columns.

o R1N-0-33O G-22A 07V Oli

CIRCULAR SECTION SQUARE SECTION

SPOTS DENOTE MEASUREMENTS U5ING STRAIN GAUGE o H/H 033ETNAMOMETER IN WAVES OF VARIOUS HEIGHTS 0 022

L 017V 0-It

SPOTS DENOTE MEASUREMENTS USING STRAIN GAUGE O lI/ooS3DYNAMOMETER IN WAVES OF VARIOUS HEIGHTS o 022

5.17V OIl

SURFACE PIERCING SURFACE PIERCING

HORIZONTAL FORCE VERTICAL FORCE

e0 o

ko ES

VERTICAL FORCE MOUE NT

1-!0 0.5 1.0

Cw

......-.--'----

-'- .- --- - __*__ ,/ %\

-2Fig. 4- Effect of diffraction on the wave

profile over a square section colymn (see fig.h/d 0.9, ka I .0). Phase= 2.22 RAD withmaximum total downward force.

I4

12

19

08

O4

02

TOTAL WAVEINCIDENT ANDSCATTERED

2,e7TOTAL

WAVE AMPLITUDE

INCIDENT WAVE

THEORY

k¡ 'EXPERIMENT

/5

1' \ THEORY gi (CIRCLE)

I I - I

---O--- MEASURED WAVE PATTERNRESISTANCE ASSUMINGEFFECTIVE E = O'2

2b

I. i

= b/a

(_ .

I

- -i2a J

0.3 04. O5 06 O7 08 O9. I-2 13

IFn.Fig. 5 - Comparison of theory. with experiment

assuming effective E 0.2.

-10O 11/4 /2 311/4

211 (t/r._x,À)

Fig..6- Linear änd Stokes V rófuIésofwäveelevation, maximuîi acóeleration añd velocity coth-ponents. . .

050

00.25

wo

-O25Fig; 7- Linear

force and overturn

.IL,/9

O 2

2zw

4

w... o

11/2 :11/2 /4

050

025I-zw

o2

4.025

-04 -02p p i-

o

o W4 W2

and Stokes V, drag and i.hertia,ng moment on a sLendercoiumn.

H/9T2 0015M/d =02

WI WI / !/2 9H-02 O

- LINEAR TO Z-0-- STOKES V TO Z=0

STOKES V TO Z-

U.ILLI/I/2 9H

o 02 04.

o-HORIZONTAL

Z. O-4- FORCE

HIL

ea

DI

O-2o

l-5

1-0o-

N(L 05O

O-06-

g: O-04

O-O2-

00

0-2 0-4 0-6 08

OVERTURNINGI I I d

MOMENT

rfa02 0-4 0-6 0-8

0-5

i::Tz o-o; -

I I I

r 0'r/a i

0-2 0-4 0-6 08

MODEL WITH WELL-- FORCE ON CAPPED

MODEL - FORCEON CAPFORCE ON CAPPEDMODEL X VOLUME

RATIO.

PLAN

d/a = 3h/a = O-8

Fig. 8 - Estimates of linear forces and overturning moment on circular bases withcentral wells.

0-2 0-4 O-6 080-2 0-4. 06 0-8

(b) ka=i-0(a) ka =0-5