Experience in computing wave loads on large bodies

8 JAN. 1916

kRCHF

ee note inside cover

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_{v Scheepsbouwkunde}

Technische EhpjcieL,rt

Ship 192

Delft

September 1975

National

Physical

Laboratory

Ship Division

u e u

EXPERIENCE IN COMPUTING WAVE

LOADS ON LARGE BODIES

Reprint of paper presented at the

Seventh Annual Offshore Technology

Conference, Houston, Texas. May 1975

by NG Hogben and RG Standing

SMTR-7523

Extracts from this report may be reproduced

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

OFFSHORE TECÑNCLGCY CONFERENCE

6200 North Central Expressway

Dallas,.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, and

Petroleurr Engineers, Inc

(Society of Mining Engineers, The Metallurgical Society and Societ1 of

Petroleum Engineers), American Association of Petroleuir Geologists, American Institute of

Cheffi-cal Engineers, American Society of Civil Engineers, American Society of MechaniCheffi-cal 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 Technology

Conference to be held in Houston, Tex

,

May 5-8, 1975

Permission to copy is restricted

to an abstract of not nore than 300 words

Illustrations tray not be copied

Such use cf

an abstract should contain conspicuous acknowledonient of where and by

hoir the paper is

presented.

PAPER

NUMBEROTO 2189

ABSTRACT.

Designers of offshore installations for the North Sea and adjacent waters face a number of

exceptional demands specially regarding the extreme and continuing severity of the weather and the large watér depths. In response to this challenge, there has been rapid development of new design concepts and a major trend already well established, is f 9r the emergence of new types of large monolithic structure. These pose many difficult design problems and, in

particu-lar, are. not genér1ly amenable to conventional methods for estimating wave loads. For such

structures, prediction methods based on diffraction theory are needed.

INTRODUCTION

This paper describes eerience in ship

Division. of the National Physical Laboratory in the development, validation and application of a computer program written for this purpose that has been supplied to a number of computer bureaus and to Lioyd Register of Shipping and has been extensively used by.the offshore

industry. This work has been undertaken as part

of a research program approved by the Ship and Marine Technology Requiraments Board. Some

earlier aòcounts have already been published in Ref

5.

1 and 2, the first being mainly concerned References and illustrations at end of paper.

with the computer program development and, the second with experimental validation and

preliminary operational experience. The present paper recapitulates the salient results of the

earlier' publications and reports recent progress including some further experimental results, but emphasizing the more .extensive experience since

gained in practical applications of the computer program.

Before discussing this experience in detail it may be helpful to be mo±e specific

about the nature of the structures concerned. The term monolithic used above is intended in this context to refer rather loosely to a. range of structure tpes that are 'very diverse in

configuration. All, however, contain at least one component' which is very much larger in diameter or equivalent section dimension and often quite complex

in

shapè compared with the normal tubular components of a steel jacket, It is because of these features that diffraction theory methods of wave loading analysis are generily needed in fact for reasons that are

explained iii detail

in

Ref. 3 and will be

briefly indicated in a later section of this paper. 'In many cases construction is primarily 'of concrete and diameters of the main components may sometimes range up to 100 or 200 mo

L114 _{XPERIFNCE IN COMPUTING WAVE LOADS ON LARGE BODIES OTC 21}

of structure to which this description applies, and these include gravity platforms, tension leg and tension stay structures and storage

tanks. At the time of writing the program has

mainly been used for gravity platforms for which a substantial number of orders have now been placed. The level of demand may in fact be judged from a recent list of UK North Sea Platform Orders4 showing a count of 11 con-. crete gravity structures in comparison with 13

steel jackets. The definitive feature of these

types of platform is that they rest on the sea bed under their own weight without piling.

They commonly consist of a single massive base on which stand a group of three or four towers carrying the platform clear of the sea surface. There is, however, a diversity of actual

geometry as is well illustrated by

_an

inter-esting pictorial review of more than a dozen current designs

in

Ref.

_5.

This includes some hybrids involving combination of jacket-type

steel structures on concrete bases.

The program has also been used for at least one tension stay platform, at least one

storage tank and for studying the forces on one large complex-shaped component of a

semi-submersible. For all these types of structure

there is again great divrsity of design. Tension stay platforms, like tension ones re essentii1y floating structures anëhored to the seabed by cables kept in tension by excess of buoyancy over weight. in this case, however, the cables are spread to restrict lateral motion, whereas those of a tension leg platform are vertical. The review Ref.

_5,

mentioned above, includes a picture of a tension stay platform consisting of a very large anchored buoyancy chamber carrying a group of towers on which the platform is mounted. Information about tension leg platforms may be found in

Ref.

6.

Storage tanks may be independent units such. as the Dubai installations described in Ref. 7, grouped like the Ekofisk tank complex,

or incorporated in the base units of avity

platforms or in single buoy .moorings.

Re-garding semisubmersibles, Paulling'P and Hòoftl'l have successfully estimated wave loads

and motions without using diffraction theory. for a number f conventional designs. Diffrac-tion analysis may be needed in some cases however, especially for designs involving very

large unconentionl ly shaped elements or strong interactions between neighboring components.

Many other applications are possible, but the foregoing account covering different

structure types for which the authors have some computing experience should serve to emphasize the versatility of the program and to assist discusion.of some of the special.problems that have been encountered in practice.

9

INTERPRMATION AND VALIDITY. OF PRO.AN

It is essential for effective application of the program that the physical basis of the calculations and the meaning of the output is clearly understood and that reliability is well established. These questions of interpretation and válidation have been discussed in some detail in the two previous papers, but for completeness will be briefly recapitulated in this section.

Accounts of diffraction theory may be

found in Ref s. 12 through 15. . The NPL

computer program uses method of analysis developed by Hess and .Smith-°' and by Garrison

and Chow, 17 which are applicable

to

fixed

bodies or arbitrary shape placed in 'a regular linear wave fIeld. The bodies disturb the wave fiéld and èxperience forces 'that combine conventional added mass änd diffracted wave effects.' The computing proáess involves first replacing the body urfaces by a large number

of mál]. plane facets. Computer storage and

run timè limit the number of facets and,

therefore, the resolution of the structure

sur-face. A fluid source placed at the center of

each facet pulsates with the frequency of the incident wave. The ource strengths are. calcu-lated by matrix inversion technique to make thé'volocity normal

to

the body surface zero at each source point. The oscillatory disturb-ance created by the sources represents the scattered wave, which includes the added mass and is superposed linearly on the incident wave pattern. The resulting pressure 'fiêld is

intérated over the body surface to' give .the over-all forcés arid moments.

Physical Inteì'pretatjôn'

The physical interprétation of this pro-cess has been discussed in the previous

papers12 and à fuller' account may be found in Ref. 3. For the present a very brief explana-tion will' be given with emphasis on 'showing how diffraction forces relate to conventional

inertia forces 'as define& by Morison' s

equation.18 ' . .

'It is convenient to regard the pressure distribution on 'the body surface and the resulting forces and moments as being the sum of two componénts. The first is the pressure in the undisturbed incident 'wave th'at inte-grates th the force Fj., generlly Ithown as the Froude Krylov force. By an analogy with the Archimedes principle, this force may' be equated to that which would be eeziéncèd by a fluid replacement of the body volume V. Hence when the body spans a small enough fraction of wave length such that the ambient acceleration U is

aprodmately constant, it foUow that Fk = pVU (where p is the' density of water) and may be. regarded as a dynamic analog of. static

buoyancy.:

The second component may be referred to as the disturbance pressure and integrates to the force Fd, which is that due to the disturb-ance of the ambient pressure in the incident

wave by the presence of the:: body. This

includes the effect both of the local disturb-ance, normally déscribed in terms of an added mass term kpVtJ, that edsts even when there is no free surface,:asw]l as: that of the

scattering or diffraction of the incident wave. In fact, when the body spans a small fraction of wave length (less than a fifth say), the scattering term is negligible so that

F :.:kpVU,

It may. thus be found that a diffraction coefficient 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 of wave length V: : C:

_{h::. ::}

1+k

C m and:: : CpV(J : :

-where Cm is the conventional mass coefficient as 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 shown as 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 mass factor 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 the inertia force equation become siificant when analyzing cases such as tension leg and tension stay platforms where there is some dynai.c response of the body to :the action of the wave

forces.

-Validation of the Proam

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

from other published work including theoretical results of MacCarny and Fuchs'4 for a vertical cylinder, Garrison and Chow7 for a floating hemisphere, and Gran19 for a gravity platform base, the last named including some comparisons with the results of model experiments. The

comparisons with published data involving pressure distributions as well as forces and moments established confidence

in

the theoret-ical reliability of the program.: Regarding the author' s own experiments these could only be discussed in a rather preliminary way

in

Ref.: i and a much fuller acöount is given in

Ref.

2

prefaced by a shor:t review of other

published work reporting comparisons between theory and experiment. Even in this later paper, the eerimental results could not be reported in full since the test schedule was not finishéd in time. A more extensive set of results is however now available, and for

completeness 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 quoting some of the references to other published comparisons of theory and experiment recited in Ref. 2. These broadly indicate remarkably good agreement between theory and the results of model experiments but there is little good data for surface piercing bodies with diameters large in comparison with wave length and the only data from full-scale tésts

in

the diffrac-tion regime biown to the authors is t:hat of Brôgren As :mentioned:in Ref. 2, a comprehensive review of published comparisons with model e,eriments is given by

Ch1OEabarti.2U This presents results for a wide- range of simple geometric shapes taken from. other references, as indicated by the following list using reference ntimbers appli-cable to the present paper, sown beside each shape: Hemisphere,2- Sphere, Horizontal Half Cylinder, Z3 Horizontal Cylinder,24 Rectangular Block25. and Vertical Cylinder.6

Mention should also be made of a paper by Lebreton énd Cormault2? containing results for truncated vertical cylinders and 'to more recent papers by van Oortmerssen28 and by Boreel29 which contain comparisons including pressure distributions'on a square sectiOn surface piercing column on a pyramidal base.

The authors' experiments invòlved measure-ment: of forces, moments and pressures on the family of four circular and four square-section vertical columns of varying height Shown in Fig. i in water depth d =

_2.3

m. They broadly confirm the favoráble comparisons with theory reported by other investigators, but have drawn attentiön to some discrepancies occurring in certain cases.

The forces and moments were. measured by mounting the test columns on the five component strain 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. Ritter assisted by R. A. Browne, and consists

essenti-V1y of two vertical conáentric cylinders. The inner one is bolted to the tank bottom and the outer one carrying the test column is connected to the inner by a system of strain gauged flexures. Pressures were measured at points on the cylinder surface using flush diaphragm silicon strain gauge transducers, and one transducer was mounted internally on top of the dynamometer to monitor the pressure inside so that the measurements of vertical force could be adjusted accordingly. Some internal pressure variation resulted from the small gap

around the base needed to allow freedom of deflection for the dynamometer. The datà-handling equipment provided magnetic tape records of all signals for subsequent computer

analysis and pen records for display during the experiments.

Measurementsof forces and moments from tests in regular waves over a range of wave lengths for various wave heights are plotted for all the eight coluims in comparison with

curves derived from the diffraction theory program in Fig. 3.

The measurements of vertical force shown have been corrected to remove the effect of internal pressure variation. The results for the four circular columns have already been published,2 but are recapitulated here for

completeness with the perniission of the Royal Institution of Naval Architects. The results for the square section. columns have not been previously published.

It is, unfortunately, not possible to give hère corresponding details of the comparison of pressure measurements with theory due to the large amount of space that would be needed. It is hoped that in due course a fuJ4er acçount will be published, but meanwhile those inter-ested in comparisons of measured pressure with diffraction theory predictions will find valuable data

in

the paper by Boreeì29 men-tioned above.

The main conclusions from the experimental investigations may be summarized as follows:

1. The results broadly confirm the experience of other investigators who have found good agreement between diffraction theory and experiment. They are reassuring to

designers regarding the reliability of the computer program since nearly all the experi-mental values for force and moment lie below

the theoretical lines. No systematiò trend has been identified in the scatter of variation with wave height, and it seemS probable that this is a measure of the degree of eerimental

error.

2, In certain cases it rna be seen that

the experimental results for the vertical forces lie well below the corresponding theo-retical lines. This is believed to be due to wavebreaking on the column top causing some loss ofwave height and, hence, reduction of vertical force. This explanation is consistent with the trend of the results, the defect being greatest for the coluimis that caused the most wavebreaking. As might be expected, the square columns are the most strongly affected, especially for the cases h/d = O. and 0.9 when the coluimi top is nearest to the surface, as may be seen in Figs. 3f and g. Regardin the

circular columns only, the results for h/d = 0.9 are affected and the gap is much smaller as shown in Fig. 3c.

The experimental measurements deviated from linear theory in some other minor respects that did not noticeably affect the over-all forces and momènts. In Ref. i it is noted that in shorter waves the wave profile on the front face of the square section surface piercing column rises to a central crest that is much more sharply peàked than the prediction of linear theory, as demonstrated by a comparison using measurements from a photograph of suòh a

wave. This is a phenomenon to be expected for

steep nonlinear wàves and is a form of "clapotis". It haS subsequently been exten-sively observed and filmed on front and rear faces of both square and circular section columns.

Irregularities noted in pressure records from the preliminary expeï'iménts de-scribed in Ref. i were subsequently found to be due partly to spurious thermal effects on the transducers since corrected and also to genuine

second harmonic components'. These are' a Imown

nonlinear phenomena occurring in steep reflec-ted waves and hàve the special property that they do not decay in the usual way with depth. Detailed accounts of the phenomena may be found in Ref s. 30 through 32 and some comments relating to the present investigation are included in Ref. 2. They do not significantly

affect forces ànd moments, but unless their edstence is recognized, they may wrongly' suggest faulty instrumentation.

in the case of the square columns, large vortices were observed at the corners

as illustrated by photographs in Ref. 1. 0riginJ.1y, as indicated in the reference,' it was thought that these might significantly

afféct the experimental results. Subsequent ECPERIEICE IN COMF(JTING WAVE LOADS ON LARGE BODIES

OTC 2i9

417

investigations, however,

inclding some

highly

simplified theoretical. estimates

of

the

influ-ence. of the vortices on the pressure

distribu-tion, persuaded the authors that the effects should at least

theoretic11y be

confined to

very

small regions at the. corners. This view

seems to be con.firmed by the good agreement

betweer theoretical and eerimental pressures

reported

by

Boreel, 29 even,

near

the corners of

a square.sectioned column.

COtJTING

EaIErCE

The NFL wave diffraction. program is

docu-mented more fully in Ref s, 1 2 and

resembles other programs described in Ref s. 17,

28, ₃₃ and

_34.

It is currently being operated within NFL, by two computer bureaus and by Uo,ds Register of Shipping. As noted in the introduction, it has been applied mainly to gravity structures including storage tanks and production platforms, but also to tension stay and semisubmersible pontoons. Some of the questions that have arisen are of general interest. The remainder of this paper surnina-rizes the authors' eerience of running the

program

and answering the questions.

Section Shape Study

Comparisons were made

in

Ref. i of tue

horizontal

force, vertical force

and

over-turning moment

on

simple

gravity

structures

resting

on

the sea bed..

These were simple

vertical

columns of

circular,

hexagonal and

square

section. The ratio of column height h

to water depth d. was h/d = _0.3. The section

areas were all equal to ita2, where a/d = 0.3. Section shape had little effect on the vertical

foròe,. The horizontal force and moment on the square column was up to 8 percent 'higher than on the circular, the hexagonal column results

being intermediate.

Parametrisation of Forces on Circular Cälumns

For the reasons discussed in an earlier

section, it

s convenient to regard the wave

force on a body as the product of the

F±'oude-Krylov force on that body and a diffraction

coefficient that

takes

accoimt of added masse

as well as wave scattering. This

becomes the

conventional mass coefficient Cm in very long waves. For models synùnetric abOut the

two

planés x = O and y = 0, threç diffraction

coefficients Ch, C

and Cy are defined as the

ratios of the mad.rnuin total to ma,thnum

Fronde-Krylov horizontal force, vertical force and

overturning moment, respectively.

Values of

C, C. and C

have been computed for a range of

circular columns, of different aspect 'ratios,

typified by their height-to-diameter ratio

h/2a,

resting on the sea bed. The results,

tabulated and shown graphicUy in Ref. 2,

Table 1 and Fig..7, are relatively insensitive

to wave length X and water depth d over a range

of large X/a and d/h values.

These cases

include many of interest tO designers of gravity structures, particularly because

sec-tion shape is also comparatively unimportänt

(see section, Shape Study). The results are

reprinted here in Table 1. The original table

also included a phase añgle associated With each force and moment, but these differed little. from t'he corresponding Froude-Krylov

values.

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

the

results of the full diffraction

analysis, is

roughly

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. 5

percent

in error.

Over

most of the range, the

error is only

1 -

2 percento

The formulas may

be italid

over a greater range of

h/2a, but.no

tests have been carried out.

When h/2a

is

large, both Ch and C

tend to a constant value

close to that predicted by the theory of

MacC

_{amy and Fuchs and shown as}

in Fig. 2 of

Ref.

35.

It must be emphasized that these results are pplicable only to columns resting on the

sea bed. The horizontal force on a suspended

or buoyant cylinder, such as a tension-stay

platform or spar

buòy, is probably similar, but'

the vertical force iS completély different, depending on the vértical pressure gradient in the fluid rathei than on the pressuré itself.

The corresponding C. differ.s accordingly, and

in particular becomés very much greater than i

if

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 position e and distance R from the

añs

of a circular

cylinder of radius a

in a

uniform

unbounded < 1,

stream, speed U, is

from6

U (. + a2/R) cose

This means that the disturbance velocity decays with distance as a2/R2 and is small for R greater than two cylinder diameters. If a free

surface is introduced. and th cylinder passes

vertically through it, a similar decay rate is expected both in currents far below the. surface and in waves that are long compared with R and a, and can be treated instantaneously as

steady.

The cylinder scatters shorter waves how-ever, and the disturbance field extends much

further. A diffraction program of the type

described Is useful for studying interactions between two or more members of a structure. Lebreton and Cormault,27 their Fig. 9, using the program described in Ref.

33,

calculated the forces on two vertical piles 5 diameters

apart. The inaxintum effect was felt upstream

where waves reflected from the downstream pile face combined with the incident wave to form

partially' standing wave pattern, as

in

Ref.

28, Fig. 9. The upstream pile experiençéd a madrnurn increase or decrease of force of about

20 percent near ka = 0.6, depending on its position in the other's standing wave pattern. Smaller effects were noted in both longer and

shorter waves, with for example changes of about 10 percent near ka = 0.4. The down-stream pile was less affected, corresponding changes beingless than 5 percent. Piles arranged parallel to the wave crests affected

each other by only 2 percent.

Ref. 2 confirmed the extent of the up-stream disturbance at ka =

_0.5e

The wave elevation, according to MacCainy and 'Fuchs'

theory, 14 was stil]. 15 percent of the 'incident

wave at R/a = 10. The conclusion, therefore, is that

in

long waves and deep currents,

inter-actions are unimportant at radial distances greater than two colurrni diameters, but there seems to be considerable upstream influence at 5 diameters near ka = 0.5 and shorter wave--lengths.

There may also be interactions between

coluimis placed on top of each other.

Calcu-lations in Ref. L showed less than 10 percent difference between the forces on a complete two-coluzini structure and the sum of foròs on the separate components. But in that case the upper coluxmi diameter O.2d was small enòugh in

comparison with the base diameter 0.6d to leave the'flow around the base substantil1y unaffected. The difference is probably greater

if the coluni diameters are similar or if the superstructure consists, of several towers instead of'one.

The Effect of Diffraction, on Wave Elevation

When a wave travels into shallow water, it steepens and its length decreases. A wave' passing over an obstacle experiences a similar

effect, intensified by focusing of the wave rays as they change direction.i7 This means that the wave elevation over the top of a

submerged structure maybe greater 'than that on either side. Desi'iers should take account of this (i) to ensure that thedéck is high

enough to clear the highest waves expected, (2) when computing wave forces because members near

the surface are immersed more deeply

₍₃₎

in considering impact . loads where the wave steepen

enough to break. . '

The NPL program can give designers guid-ance because it can provide optionally the linearized diffracted wavè elevations at any' point. This is done by requesting pressure'

output on the plane z = O. The linearized BernoulJ.i equation. for the subsurface pressure

p is' '

p = p ò/òt,

where '1' is the velocity potential and t is

time. On z = 0, the Bernoulli equation gives

the surface wave elevation instead,

=

1à/òtonzO.

Thus, requesting dimensionless pressure p/pgH on z = o in fact gives the ratio of wave elevation to indident wave amplitude _/%H. Fig. 4 shows aprofile 'along the 'centerline

y = O of the 'total wave amplitude over the submerged square-section colunri shown

in

'Fig. 1.

Here h/d 0.9 and ka = 1.0. The wave amplitudE oscillates upstream where the reflected and incident waves interfere to form a partially standing wave pattern. It decays gradlly to'

the incident wavé level downstream. Also shown is the actual wave profile at the instant when 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 'the incident 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 velocity they change the fluid drag. Because drag depends on the square 'of the.velocity, and 'the current velocity, decreases slowly with depth, a comparatively small current can increase drag siiificant'ly. Because this paper is concerned mainly 'with inertia-dominated situations, this effect will not be discussed further.

The second effect is that of changing the wave speed, the wave propagating over a moving rather than stationary fluid. This may be

associated with wave steepening.3 This effect is small in a North Sea desi wave situation, where, for example, wave period T = 15s, wave speed x/T = 23 m/s, but the madmum current speed is only 1.5 rn/s.

The third effect of a current

i8

to make the structure itself ge±erate waves. A body

in

a uniform current causes a stàtionary wave pattern to form on the free surface and

experiences a corresponding net force, which may be regarded as a special type of

diffrac-tion force. It is also directly equivalent to the so-called wave-making resistance

experienced by a ship or other body

in

uniform motion through calm water. This fOrce can be

calculated by basici ly the same method as the diffraction analysis used in the edsting program, but with a different form of Greent

function that now describes the potential due to a unit source

in

a tthiform current rather than a pulsating soUrce

in

still water. The

authors have examined the possibility of modifying the computer program so that it can

calculate these uniform current forces but decided against it for reasons that are

dis-cussed in detail in Ref. 39 and can only be briefly summarized here.

Although the revised fOrm of Green' s function is laiown and may be foUnd for example

in

Refs. 15 and 40, it is not simple, and substantial effort would be nèeded to make the necessary programming changes. This effort moreover would not be justified because for practical offshore structures

in

realistic currents conventional drag forces due to flow

separation and wake formation effects are dominant and the diffraction forces are not only negligible in comparison, but also deviate grose]iy from the predictions of potential flow

theory. This point is well substantiated by

comparisons for the case of a surface piercing vertical cylinder, between wave resistances computed from theory of Ref. 41 and experi-mental values derived from measured wäve patterns by the methods of Ref. 42.

Fig. 5 summarizes the results. As dis-cussed in the references, the theory lacks i.iniqueness, but using the favored sOlution for cylinders of elliptical section with = b/a as shown, it was found that the ratio Cw/ 2.4 independent of e. when plotted to a base of Froude number je F = c// , where

Cw = wave rsistance/(pAc2) A = projected frontal area c = current spéed

F =

c/f.

The figure shows experimental results for a circular cylinder, a = 1.0, but to achieve even approdmate correspondence with theory it was necessary to assume, due to flow separation and wake formation, an effectively elliptical section with 6 = 0,2. 1Ìi Ref. 39 the relevance of these results to offshore structures is discussed, and it is shown that even for exception11y strong North Sea currents, Cw is negligible in comparison with Cd.

Nonlinearity of the Waves

The NPL wave-diffraction program uses linear wave theory even under extreme design wave conditions. Designers have expressed con-cern about two nonlinear effects. First, are the differences between linear and higher-order theories significant for a typical structure

in

a typical design wave? Forces F on a con-ventional jacket-type structure are usually predicted using Morison' s formula18

F =

CpV(J+CapAUjUI

with velocities U and acceleratiOns U given by higher order StOkes43 or stream function theory.44 Skjelbreia and Hendrickson's.Fig. 343 shows linear and Stokes V velocity profiles differing by 20 percent or more at the free

surface. This would indicate differences in

drag of over 30 percent. Is this an extreme example?

Second, where the structure pierces the surface, is it necessary to integrate the forces up to the actual free surface, rather than to mean water level? Conventional linear theory, in particular the NPL program, computes wave forces over a constant immersed depth, the wave pressure field being continued up to or

cut off at mean water level.

Comparisons were made first between linear and Stokes V43 regülar waves

of

the sarnê height H and period T in the same méan water depth d. Some authors prefer to use still water depth instead, the difference being terms propor-tional to H2 and 4, according to Stokes V

theory. The ratios chosen H/gT2 = 0.015 and

H/d = 0.2, are typical 6f a northern North Sea design wave situation, where for example d =

150 m H = 30 m T 14 s. Comparisons were also made using a less steep wave H/gT2 = 0.01

àt H/d = 0.2. The conclusions were similar,

but the differences smaller. Dean45 compared various anaJ.ytical theories to find the best fit to the free surface boundary conditions. He recommended Stokes V Or stream function theory forthe cases described here. When H/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

spanning several wavelengths, though norm1 ly

such short waves would contribute little to

the over-ailforces.

Velocity and acceleration

profiles were also computed.

Fig. 6 shows the

these as dimensionless Morison drag an4 inertia

terms, uIup/-H, wPwl/gh,

/g and */g, where

u and w are the horizontal and vertical

veloc-ity components, and

and * are the

corre-sponding accelerations.

Each profile is shown

at its mañmum in the wave cycle.

The

differ-ences between the linear and Stokes V terms

are roughly 20 percent in wßwj, 9 percent in u

which are in phase with wavé slope, but 13

per-cent in uf uJ, 3 perper-cent

in

*, which. are in

phase with wave elevation.

This suggests that

the most important effect of nonlinearity is

to steepen the waves rather than change its

elevation.

The differences decrease with

depth, the theories being identical near z/d =

0.4.

There are qmVI differences of opposite

sign

below this depth.

Fig.

7

shows the resulting force and

over-turning moment on a slender vertical column of

radius a extending fróm the sea bed below the

origin of coordinates up through th

free

surface.

The linear and Stokes V inertia

contributions, both integrated up to mean

water level z = 0, differ by less than

7

per-

-cent of the madmum; the drag contributions by

less than 13 percent.

The load on a typical

gravity structure in a North Sea design wave is

mainly inertial and most of the structure is

quite deeply submerged.

Linear theory is

therefore quite adequate to describe the

over-all loads.

A higher-order theory may be

needed for loca:I loads on drag-dominated

members such as conductor tubes, particularly

near the free surface.

There the structure pierces the free

surface, the error in using linear rather than

higher-order Stokes theory may be rather less

than the error in regarding the immersed depth

as constant.

Fig.

7

compares calculations

of the Stokes V force integrated up to the

actual free surface z =

with the forôe tip to

mean water level z = 0.

A similar calculation

using linear theory showed only small dif f

er-ences from Stokes V theory.

In Fig.

7,

the

inertia force and moment are skewed sideways

so that the madma occt2r closer to the wave

crest, but their magnitudes are only slightly

increased. Thus, the main effect

f the extra

wave immersion on the inertial load is a phase

shift of the mañmum.

If the wave is. distorted

by diffraction, the effect is less predictable.

The wave elevation may be increased as well as

thè phase changed (see Fig. 4

for exaple)o.

If the linear diffracted wave elevation is

required this can be output optionily by the

NPL program as elained in the effect o.f

diffraction on wave elevation.

Tald.ng the actual rather than mean

immersed depth has a dramatic effect on drag.

The madmwn in Fig.

7

still occurs at the wave

crest, but its maitude is almost doubled.

This effect is most important, particularly in

respect of local loads where drag-dominated

members pierce the free surface.

Ai-i Approdmate Method for Bodies With Wells

Some recently proposed gravity structures

have featured either a protective wafl around

the base or a vertical well extending part or

all of the way. down to the sea bed.

These

present similar problems to the computer,

in

the first case if the wall is too thin, in

the second if the well is too narrow and

deep.-As noted in Ref s. 16 and 2, a satisfactory

computer model of the inner surface arid lip

may require a large number of facets at a

I

correspondingly high cost.

An inadequate

deséription results in program failure during

matrix inversion or inaccurate results.

Fortunately, it is often possible to cap the

well, correcting for it later.

Numerical

tests .to find a satisfactory correction

pro-cedure were carried out on a simple circular

base with a vertical central well extending

down to the sea bed.

Two approd.znate

calcu-lations of the forces and moments were

com-pared with results from the actual model

including the-well.

The first approdmation

makes the assumption thàt thè water inside the

well is dead and at a uniform but time-varying

pressure.

This means that the well is

effec-tively cápped and the actual body experiences

the same forces as the capped body less the

vertical forces on the cap itself.

The second

estimate comes from multiplying the forces on

the capped body by the ratio of the displace-

-ment volume of the actual and capped bodieso

This approdmation roughly represents the other

extreme where the external pressure gradient

penetrates right inside the otherwise dead

well.

The analor here is with the buoyancy

force on the body turned on its side

in

a.

uniform vertical gravitational pressure

gradient.

It may be expected that the forces

on the actual body should lie somewhere between

the two. approximations.

'Fig.

shows the horizontal and vertical

force and the ovei'turriing moment on typical

gravity bases of height h, radius a, well radius

r in water of depth d. Here the ratios dfa'=

3, h/a

0.

and the two wavelengths chosen àre

ka = 0.5 and 1.0, where k

_{= 2tJX.}

The results

are plotted as functions of r/à, small r/a

representing a narrow deep well that hardly

affects the forces, r/a close to 1 representing

a thin protective wall on its own.

As was

hoped, the forces and moment on the actual model

lie between the, two estimates.

The actual

structure surface was divided into 220 facets,

80 on the inside and 80 on the outside walls with

60

on the top. The pressure gradient penetrated with attenuation roughly 0.5 to 1.0 r inside the well. Below that depth the

pres-sure was roughly uniform. The small pressure gradient remaining there was probábly the result of using too coarse a mesh, but this did not affect over-all forces appreciably (by less than 1 percent). Assuming the well to be dead gave conservative estimates of horizontal force and overturning moment but the correct vertical force.

It is therefore recommended that if a well cannot be included in the structural modele it should be capped, the force and moment on the cap being subtracted at the end. The NPL program does this quite simply. Other types of indentation can be treated in a similar way, allowing for pressure variations on the bottom if this.is part of the structure. This procedure should normally providé a con-servative estimate of forces, but the moment must be treated with care. As noted, in Ref. 2, the overturning moment on a gravity structure with h/a= 0,8 is small as a result of the fine balance between forces on the base top and

sides. In the cases described here the

moment due tO side force predominates, so that an overestimate of horizontal force accompanies one of momênt. But if the base height is re-duced slightly, the top forces predominate. An ovérestimate of horizontal force may then reduce the overturning moment and give mis-leading results.

The physical effect of flow separation at the wêfl top probably makes the water in the well more dead than linear potentiaJ. flow theory predicts. But if the structure and well diameter are large, this is not expected to affect the over-all forces too much.

Moving Bodies

The NPL program was written with gravity structures in mind and at present works with fixed structures only. Recent developments in tension-stay, semisubmersible and spar-buoy-type platforms have raised questions concerning response to wave excitatiOn. The theoretical treatment of moving bodies is similar ta that of Í'ied bodies, invelving merely a change in

the body surface boi.mdary conditions. Under

the assumptions of lin?ar and harmonic re-sponse the body motion can be brOken down into oscillations in the six degrees of freedom, three of translation and three of rotation, as

described in Ref s, 15 (Section

19), 4.6

arid

34.

Refs.

₃₄

arid

47

describe programs that solve

these component problems. They compute added mass arid damping. coefficients for bodies osciflating in. surge, sway, heave, roll, pitch and yaw in otherwise calm water, also the

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

Until then it is possible to estimate some of the coefficients. The Hasldnd relationshl° relate the forces on a fixed body lxi t1aves to

those on the same body oscillating in otherwise

still water. Thus for example, the principal

damping coefficients, according to Ref.

46,

Eq. 30, are proportional to the integrals over

all incident wave directions of the squareC of corresponding exciting forces. This integra-tion is not practicable in general, but is simple if the body is añsymmetric abou the vertical axis, It then gives Newman's4 Eqs.

31

through

₃₃

for damping coefficients in surges heave and pitch in deep water or

Garrison'

47 Eq. 44

for heave in shallow water.

In many design situations the exciting wave is long compared with body dimensions. If

the response period is comparably long, there are several simplifications. First, according to Newman' s equations the damping coefficient is proportional to the cube of wave frequency and is therefOre small for long period

oscilla-tions. Second, the added mass fluid f9rces

acting on a body in motion depend only on the relative motion of the body and surrounding

fluid, If the body is small compared with

wavelength, it creates the same disturbance and experiences the same added mass forces whether it is fixed in waves and the surrounding fluid particles travel around elliptical Orbits or

itself travels around the same elliptical orbit at the same speed but in otherwisé still water. This means incidentally that, if the body excursions are small compared with local wave orbit diameters, the fluid forces are im-affected by body motion. To separate the co-efficients in heave, surge and sway, it is necessary to ook at the three òompänent

motions separately. Care is needed if the body is unsymmetric about the x = O or y = O plane because the horizontal acceleration phase of

the elliptical motion causes both horizontal and vertical forces, and similarly the vertical

phase. Many proposed structures fortunately

have double symmetry so that there are no surge/ sway/heave interactions. The separate fOrce components then give corresponding added mass coefficients, In cases where the body moves

care must be taken to use the full inertia force equations cited in Interpretation and Validity of Program, which includes the effect

of the acceleration of the actual mass of the body.

CONClUSIONS

1. Experimental measurements of over-all forces and moments on colunins gener11y agree well with the predictions of diffraction

theory, the measured forces being slightly

smaller.

This broadly confirms earlier

findings, but the eerimental results for the

square-section coluiris showed more scatter and

greater deviation from linear theory,

espe-ciVIy regarding the effect of wavebreaidng on

the vertical forces, as discussed in the next.

paragraph.

Measurements of the vertical force in

some cases tend to lie further below the

theoretical line than

in

others.

This is

thought to be associate4 with wavebrealdng

ob-served over the C luirai top.

In the case of the

circular columns, only the results for h/d =

0.9 are slightly affected.

For the square

columns where greater wavebrealdrig occurred

as may be eected, the effect is very much

stronger arid extends to h/d = 0.8 as well as

0.9.

Approidmate formulas are given for the

diffraction coefficients of circular columns

in quite long waves and deep water.

These

differ from the results of the full diffraction

analysis by less than about 5 percent.

40

The wavemaldng resistance of coluimis

in currents is much smaller than the drag.

it

is not thought worthwhile to include this.

effect in the NPL program.

The differences between linear and

Stokes V theory are small in tipicaJ. North Sea

design wave conditions, especially with regrd.

to inertia force.

More siificant is the

.differnce between integrating fôrces up to the

mean and actual free surfaces.

If wells inside gravity bases cannot

be includedin the full diffraction analysis,

they should be capped and corrected for later

on the assumption that the enclosed water is

dead.

This seems to give a conservative

esti-mate of fòrces but care is. needed in

interpreting the moment0

N0MECLATURE

a = column radius or

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

A = projected frontal area of body

c = current speed

Cd = drag coefficient

C ,C

_{y y}

= diffraction coefficients for

hori-.

zontal force, vertcai force

nd

overturning moment, defined as

madmum total force or moment/

corresponding madmum

Froude-Krylov force or moment

Cm= mass coefficient

2 C

= wave resistance/pAc

d = water depth

F=forée

= disturbance force

F

= external force

e

Fk = Froude-Krylov force

F= Froude number

g = acceleration due to gravity

h = column height

H = wave height

k = wave number .2it/X or

k

C' Cm

1, added mass coefficient

in review section

M = body mass

p = pressure

r = well radius

R = radial distance

t = time

T = wave period

u, .w = fluid velocity components. in x and

z directions

u, w = corresponding acceleration

conipo-nents

U.= fluid velocity

f luid acceleration

ACIÔWLEDGNTS

The authors wish tO acIthowledge their debt

to their colleagues who helped with this work,

especially to J. Osborne for conduct and

analysis of the experiments, and to H. Ritter,

R. A Browne and G. S. Smith for design of the

instrumentation.

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of Max'ine

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Hogben, N., Osborne, J. and Standing, R.

G.:

"Tave Loading on OffshOre StruOtures

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on Ocean Engineering at the National

Physi-cal Lboratory, London, published by the

ROyal Inst. of Naval Architects (1974).

Hogben, N.:

"Fluid Loading o±i Offshàre

Structures, A State. of ArtAppraial: Wave

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Naval Architects (Nov. 1974).

aniilton, A.:

"A Low Load for Platforms,"

Financial Tithes, London (Nov. 1, 1974).

Anon.:

"Gravity Platforms:

Who is

Proposing What," New Civil :Hgineer

Speciàì Review on North Sea Oil (May

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Paulling, J. R. and Horton, E. E.,:

"Analy-sis of the Tension Leg Platform,"

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Pet. Eng., J.

(Sept. 1971) 285-294.

Brogren, E.,.Soderstrom, J., Snider, R.

and Stèvens, J.:

"Field Data Recovery

System, Iazzn Dubal No. 3," Paper OTC

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Tech-nology Conference, Houston, May 6-8, 1974.

Marion, H. A.:

"Ekofisk Storage Tank,"

Royal Inst. of Naval Arçhitects Symposium

on Ocean Engineering, Nov. '1974.

'9.'

P. E.;Tòwnshend, M. A.:

"Offshore Storage

and Tanker Loading," Royal Inst. of Naval

Architects Symposium on Ocean Engineering,

Nov. 1974.

Paulhirg, J. R.:

"Elastic Response of

Stable Platform Structures to Wave

Loading," Prc., International Sympo siuin

on Dynamics of 'Marine Véhicles and

Struc-tures in Waves at University College

London, published by the Inst. of

Mech-anical Engineers (April 1974).

Hooft, J. P.:

"A Mathematical Method of

Determining Hydrodynamically Induced

Forces on a Semi Submersible," Trans.

SNAME (Nov. 1971).

Havelock, T.H.:

"The Pressure of Water

Waves Upon a Fixed Obstacle,"

Roy.

Soc. A-963 (1940)

.

John, F.:

"On the Motion of Floating

Bodies Pért II," Ccrnim. Pure and Applied

Maths (1950)

.

MacCamy, R. C. and Fuchs, R. A.:

"Wave

Forces on Piles:

A Diffraction Theory,"

Beach Erosion Board Technical Memorandum

No. 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 Arbitrary

Bodies," Progress in Aeronautical

Sciences (1967)

.

Garrison, C. J. and Chow, P. Y.:

"Wave

Forces 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 Force

Exerted by Surface Waves on Piles,"

Trans., AINE (1950) .12.

Gran, S.:

"Wave Forces on Submerged

Cylinders," Paper OTC 1817 presented at

Fifth '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," Texas

AßeN U. , Sea Grant Publication No. 210

COE Report No. 117, 1970.

O'Brien, M. P. and Morison, J. R.

"The

Fôrces Exerted by Waves on Objects," Trans,

Amer. Geophys.,Union (Feb. 1952)

, No. L

Shank, G E. thd Herbich, J. B. :.

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Due to Waves on Submerged Structures,"

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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.:

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Cylin-drical Piles Used in 'Coastal Structures,"

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'Action on Slightly Immersed Structures,

Some Theoretical and Experimental

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on Wave Action," Deift (1969) J.

Van Oortmerssen, G. :

"Some Aspects of

Very Large Offshore Structures," Ninth ONR

Symposium on Naval Hydrodynamics, Paris

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Off-shore Structures,," Proc., Inst. of Civil

Engineers Conference on Offshore

Struc-tures, London (Oct. 1974).

Miche, R.:

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Mers en Profondeur Constante ou

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Chaussees (1944).

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(1950).

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"An Experimental Study of the Pressure

Variations in Standing Water Waves,"

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"Calcul

des Mouvements d'un Navire ou d'une

Plateforme Arnarre dans la Houle," La

Houille Blanche (1968)

, 379-389.

Faltinsen, O. M. and Michelsen, F. C.:

"Motions of Large Structures in Waves at

Zero Froude Number," International

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Vehicles and Structures in Waves,

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"Gross

and Local Wave Loads on a Large Vertical

Cylinder - Theory and Experiment," Paper

OTC 1818 presented at Fifth Offshore

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₍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'kACTÏON

THEORY 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

1.87

0.75

0.3

1.70

1.01

_1.78

1.68

1.01

1.87

1.75

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

.1.83

0.2

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

_1.83

0.75

0.3

1.70

1.03

1.78

1.67

1.02

_1.83

0.75

0.6

_1.76

1.Ö3

1.83

1.67

1.02

1.83

i..O

0.8

1.87

1.06

1.89

1.74

1.03

1.83

1.25

0.5

1.81

1.05

1.81

1.80

1.04

1.83

1.75

0.44

1.89

1,08

1.86

1.89

1.05

_1.83

1.75

0.7

1.89

1.08

1.86

1.89

1.05

1.83

2.25

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

1.73

0.5

0.2

1.57

1.10

1.86

1.55

loO.9

_1.73

0.5

0.4

1.60

1.12

1.89

1.55

1.09

1.73

0.75

0.3

.1.66

1.15

1.72

1.63

1.14

1.73

0.75

0.6

1.71

1.20

1.78

1.63

1.14

1.73

1.0

0.8

1.85

1.42

1.85

1.69

1.19

1.73

1.25

0.5

1.74

1.23

1.72

1.75

1.23

1.73

1.75

044

1.78

1.28

1.74

1.84

_1.25

1.7.3

1.75

0.7

1.78

1.28

1.74

1.84

1.25

1.73

2.25

0.56

1.80

1.27

1.74

1.91

1.25

1.73

2.25

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

1.55

0.5

0.4

1.44

1.33

1.54

.1.42

.1.37

1.55

0.75

0.3

1.54

1.43

1.54

1.48

1.50

1.55

0.75

0.6

1.49

1.43

1.47

1.48

1.50

1.55

1.0

0.4

1.58

.

1.48

1.54

1.53

1.50

1.55

1.0

0.8

₁₅₄

1.57

1.49

1.53

1.50

1.55

1.25

0.5

1.59

1.49

1.54

1.57

1.57

1.55

1.5

043

1.62

.

1.52

1.56

1.60

1.50

1.55

1.5

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

1.55

1.75

0.7

1.58

1.46

1.53

1.63

1.50

1.55

STILL WATER LEVEL

ELEVATION

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

o

c)

r

PLAN:

'j

Si

S

S

S

t

Fig. 2a - Dynamometer and circular column

(h/d

0.7) before mounting.

WATE H _{O9d SURFACE} DEPTH d

i

O7 ¿ 08 ¿ PIERCING TAN K BOTTOI.I

s

j

I

test

column

Fig. 2b

- Ctaway

view of te;st

co I ümn mounted on

dynamometer.

b) 2 - component

horizontal

force

flexure

(upper)

f) 2-component

horizontal

force

flexure

(lower)

strain

gauges

strain

gauges

Strain

gauges

a) vertical

force

flexure

c) spokes

transmitting

horizontal

force

e)'floatiflg'

cylinder

g) spokes

transmitting

horizontal

f orce

h) base

plate

Fig. 2c - Di.agirammattc sketch of

HORîZOS1AL FOUC COPUTED

k.

VERTICAL FORCE

SPOTS DENOTE MEASURESIENTS USING STRAIN GAUGE O H/. - 033

DYNAIIjOMETER- - 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 - 0.33

ZIVNAMOMEIER IS WAVES OF VARIOUS HEIGHTS 0 022

- £ 0.17

V DII

SPOTS DENOTE MEASUREMENTS O 4/. - O-33 USING STRAIN GAUGE 0 022

DVNAMOMETER IN WAVES .0 0-1

OF VARIOUS HEIGHTS V 0.11

112 99 H

2.

SPOTS DENOTE MEASUREMENTS USING STRAIN GAUGE DVNAMOMETER IN WAVES OF VARIOUS HEIGHTS

COMPUTED

2 COMPUTED

k. IO I

h/d

-07-HORIZONTAL FORCE - VERTICAL FORCE

COMPUTED

-s-HORIZONTAL FORCE

COU UT E D

h/d

08

SPOTS DENOTE MEASUREMENTS USING STRAIN GAUGE O H/ -0.23

DVNAMOMETER IN WAVES OF VARIOUS HEIGHTS 0 022

6 0.17 V 0.51

HORIZONTAL FORCE VERTICAL FORCE MOMENT

e COMPUTES

--/2pHPd I I I I IO IS 0 05 k 1.0 1.5

h/d

09

SPOTS DENOTE MEASUREMENTS USING STRAIN SAUSE DYNAMOMETER IS WAVES 0E VARIOUS HEIGHTS I

MOM E NT

O

0.0

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

o R1N-0-33

O G-22

A 07

V Oli

CIRCULAR SECTION

SQUARE SECTION

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

L 017

V 0-It

SPOTS DENOTE MEASUREMENTS USING STRAIN GAUGE O lI/ooS3

DYNAMOMETER IN WAVES OF VARIOUS HEIGHTS o 022

5.17

V OIl

SURFACE PIERCING _{SURFACE PIERCING}

HORIZONTAL FORCE VERTICAL FORCE

e

0 o

k o ES

VERTICAL FORCE MOUE NT

1-!

Cw

...-.--

'---'- .- -

--

-

__*__ ,/

%\

-2

Fig. 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 with

maximum total downward force.

I4

12

19

08

O4

02

TOTAL WAVE INCIDENT AND SCATTERED 2

,e7TOTAL

WAVE AMPLITUDE

INCIDENT WAVE THEORY

k

¡

'EXPERIMENT

/5

1'

\

THEORY gi (CIRCLE)

I I _{- I}

---O--- MEASURED WAVE PATTERN

RESISTANCE ASSUMING

EFFECTIVE E = O'2

2b

I. i

= b/a

(_ . I

- -i

2a

_J

0.3

04. O5

06

_O7

₀₈

_O9

. I-2

13

IFn.

Fig. 5

- Comparison of theory. with experiment

-10

O 11/4

_/2

311/4

211 (t/r._x,À)

Fig..6- Linear änd Stokes V

rófuIésofwäve

elevation, maximuîi acóeleration añd velocity

coth-ponents.

. .

050

00.25

w

o

-O25

Fig; 7- Linear

force and overturn

.IL,/9 O

2

2

z

w

4

w..

.

o

11/2 :11/2 /4

050

025

I-z

w

o

2 4.

025

-04

-02

p p

i-o

o

W4

W2

and Stokes V, drag and i.hertia,

ng moment on a sLendercoiumn.

H/9T2

0015

M/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 H IL ea D

I

O-2

o

l-5

1-0

o-N

(L 05

O

O-06-g: O-04

O-O2-00

0-2 0-4 0-6

08

OVERTURNING I I I _d MOMENT

rfa

02

0-4

0-6 0-8

0-5

i::T

z

oo;

-I I I

r 0'r/a

i 0-2 0-4 0-6

08

MODEL WITH WELL

-- FORCE ON CAPPED

MODEL - FORCE ON CAP FORCE ON CAPPED MODEL X VOLUME RATIO. PLAN

d/a = 3

h/a = O-8

Fig.

8 -

Estimates of linear forces and overturning moment on circular bases with

central wells.

0-2 0-4 O-6

08

0-2

0-4.

06

0-8

(b)

ka=i-0

(a)

ka =0-5