Proceedings of the Symposium on Offshore Hydrodynamics, NSMB, Wageningen, Publication No. 375, The Netherlands

SYMPOSIUM ON

"OFFSHORE HYDRODYNAMICS"

ARCH1EF

4-4

Edited by

M.W.C. Oosterveld

Publication No. 375

Netherlands Ship Model Basin

SYMPOSIUM ON "OFFSHORE HYDRODYNAMICS"

lab. v. Scheepsbouwkunde

Technische Hogeschool

August 25 - 26, 1971

_Delft

Wageningen, the Netherlands

JDOCU.FNTATIEI

sche hkvmmdlool,

the industrial service possibilities of hydro-dynamic laboratories to solve problems in the field of offshore industry.

In order to come to a fruitful exchange of knowledge, papers are presented by specialists of invited companies and organizations and by specialists of the N.S.M.B.

A meeting of this nature may be valuable to all concerned. The following major themes were

se-lected for the technical program of the symposium:

behaviour of floating structures in waves offshore engineering

special projects manoeuvring of ships

This symposium contains the results of the efforts of many. On behalf of the management of the N.S.M.B. we like to extend our gratitude and appreciation to all who have contributed to the realization of this symposium.

Appreciation is especially expressed to the companies and organizations which have contributed to the tech-nical progam.

Preface

Behaviour of floating structures in waves

Distribution of wave forces on structural parts of ocean structures.

by J.P. Hooft, Netherlands Ship Model Basin, Wageningen.

The role of model tests and their correlation with II

full scale observations.

by Dr. J.H. Vugts, Shell Internationale Petro-leum Maatschappij, Den Haag, the Netherlands.

, Model testing for the design of offshore structures. III

by G.F.M. Remery, Netherlands Ship Model Basin, Wageningen.

Offshore engineering

The single anchor leg mooring for tankers. IV

by J.F. Flory, Esso Research and Engineering Company, Florham Park, N.J., U.S.A.

Dynamic positioning of the drilling ship "Pelican". V by G.H.G. Lagers, I.H.C. Holland, Offshore

Di-vision, Schiedam, the Netherlands.

Experimental and theoretical motion correlation of VI

a pipe-laying barge.

by M.F. van Sluys and Tan Seng Gie, Netherlands Ship Model Basin, Wageningen.

Elastic similarity models as a tool for offshore VII

engineering development.

Hy-nology, Delft, the Netherlands.

/Wave

excitation and structural response. IX

by S. Hylarides, Netherlands Ship Model Basin, Wageningen.

Local scour near offshore structures. X

by H.N.C. Breusers, Delft Hydraulics Laboratory, Delft, the Netherlands.

The interaction between a vertical cylinder and re- XI

gular waves.

by G. van Oortmerssen, Netherlands Ship Model Basin, Wageningen.

Manoeuvring of ships

Behaviour of large tankers in shallow water in rela- XII

tion to the dimensions of an approach channel.

by L.A. Koele, Rijkswaterstaat, Harbour Entrance Division, Ministry of Transport, Water-Control and Public Works, Hoek van Holland, the Nether-lands.

On the mathematical description of ship manoeuvring. XIII

by D.L. Kettenis, Netherlands Ship Model Basin, Wageningen.

Construction, operation and capabilities of the NSMB XIV

BEHAVIOUR OF FLOATING STRUCTURES IN WAVES

Chairman Prof.Dr. R.Timman, University of

Tech-nology, Delft, the Netherlands.

Distribution of wave forces on structural parts of ocean structures,

by J.P. Hooft, Netherlands Ship Model Basin, Wageningen.

The role of model tests and their correlation with full scale observations,

by Dr. J.H. Vugts, Shell Internationale Petroleum Maatschappij N.V., Den Haag, the Netherlands.

Model testing for the design of offshore structures, by G.F.M. Remery, Netherlands Ship Model Basin, Wageningen.

DISTRIBUTION OF WAVE FORCES ON STRUCTURAL PARTS OF OCEAN STRUCTURES

by J.P. Hooft

Netherlands Ship Model Basin Wageningen

Summary

The wave excited force on a structural part of an ocean structure can be approximated by three components which will be discussed in this paper:

Component 1 The undisturbed pressure force F1 which is the force that arises from the pressure over the structural part in a wave that is not disturbed by that part (Froude-Krilov force).

Component 2 The inertia force F21 which is the force that arises from the acceleration of the added mass of the structural part in a wave that is not disturbed by that part.

Component 3 The damping force F22 which is the force that arises from the damping, due to the structural part, of the velocity of water particles in a wave that is not disturbed by that part.

The influence of each component on the total force will de-pend on the wave frequency and the form of the body. Also

It becomes ever more evident that not only a good estimation of the

motions of and forces on ocean structures in waves is of extreme

importance but that also more insight is needed of the manner in which these motions or forces are set up.

This means that not only the overall result is important but also the elements which have led to this result;

especially in the design stage when the structure has to be adapted in finding a compromise.

In Ref. 1 the forces on small components of a structure have been discussed. As a result of that paper an approximation has been ob-tained for an easy and quick determination of wave excited forces which mostly only differs 5% with exact calculations or model test

results when the diameter of the body is less than 1/5 of the wave length.

In this approximation the total force is split up in three parts which will be discussed in section 2 of the present paper. The re-sults of the wave excited forces on the motions are discussed in section 3.

Description of waves

The motions of the water particles in waves are defined relative

to a system of coordinates (F,

n,

c) in which the direction

coin-cides with the direction of propagation of the waves, the ri direct-ion is parallel to the wave crest and the cdirectdirect-ion is vertically downward. The centre of the system lids in the still water plane

(see Fig. 1).

The characteristics of a regular long crested wave can be deduced

from the velocity potential (see e.g. Wehausen and Stoker Refs.

2,3)

in which

g = acceleration due to gravity

t = time

gCa

= wave frequency = 27/T T = wave period ca = wave amplitude -cosh K(h - c) VI _{cosh K h} h = waterdepth,

while also the following relation between the wave number and the wave frequency exists:

w2 = g K tanh K h (2)

From equation (1) the velocity and acceleration of the water par-ticles can be deduced. From this also the motion of the water sur-face and therefore the variation of the pressure can be gathered:

surface =ca sin (wt -

K)

p -

pl

= pipgC sin (wt -

K)

(3b)

in which

p = pressure as a function of time and location (E, n,

c).

pl

= static pressure force as a function of the depth below

the still water surface. p = mass density of water.

cosh

K(h

-cosh

c)

111

K h (see Fig. 2)

The velocity of the water particles follows from:

=--=

c

dt

sin (wt - KC) (4a)

. 11

dt = 3C

=wca

" cos (wt - KC) (4b) in which = (3a)

are deduced: d2

_d

s = dt

--7 -

= P3

W2 C a COS(Wt - KE) dt

-dc

₌

aT7 = dt =

-2"

w2 C a sin(wt - KE)

As the wave excited forces are generated by the pressure variations and particle velocities and accelerations, these quantities will be considered here in detail:

I. Wave pressure

From equation (3b) it can be concluded that:

The pressure varies in the same way as the water level variation. This means that under a wave top the pressure is increased while

under a wave trough the pressure is decreased. (see Fig. 5).

The pressure variation changes from p g Ca on the still water surface to pi pg ca on the bottom of the sea (see Fig. 2). This means that in Fig. 5 the pressure variation underneath the

floating body is larger than on top of the body on the sea bottom.

C. This change of pressure variation over the depth also means that on top of a submerged body the pressure variation will be larger than underneath this body which results in a net force, due to the static pressure variation, which is 180° out of phase with the wave height. As can be seen in Fig. 6 the static pressure force on the submerged body will be down under a wave top and

upwards under a wave trough.

As a result of Fig. 5 and 6 it will be clear that due to the static pressure variation a submerged body under a floating body will be moved 1800 out of phase with the floating body (see Fig.7

When the dimensions of a submerged body are small relative to the wave length (say the diameter is less than 1/5 of the wave length) the static pressure force on the submerged body can be approximated by the following equation (see Fig. 8a):

A

pl

- Ap2 =

by which:

X = + pi

KgCa

cos(wt - M = 1J3 W2Ca M COS(Wt xE) (6a)

in which:

M = p II AdS = mass of water displaced by the body.

In the same way (see Fig. 8b) the vertical force can be transformed:

12

z = I I (A

p1 - Ap2) dS - ff,

tic

ds

by which:

Z = - py Kg

ca

M sin(wt - = -P2 W2

Ca

sin(wt - xE) (6h)

When comparing the equations (6a) and (6b) with the equations (5a) and (5b) it follows that the static pressure force on a submerged body of which the dimensions are small relative to the wave length can be approximated by the product of the accelerations of the waterparticles and the mass of water displaced by the body:

X =M

ILE

dt2

= M d2

1

aFT

f. The change of pressure variation over a distance 1 in the direc-tion of wave propagadirec-tion for waves with a great length relative to the distance 1 amounts to:

(2) -

22

= a -K 111 P g _cos(wt-x) 1 _A>>1- ac =

_{i3 w2ca}

_cos(wt-KE) (8) (7)

the wave frequency decreases as can be seen from Fig. 2 in which for

= 0,1

rad/sec (wave period = 62,8 sec) the pressure varia-tion at 30 m underneath the still water suface is only 3% less than the pressure variation at the water surface. This means that the static pressure force at small wave frequencies equals the force due to the increase of immersion of the floating body in the wave by which the force variation on submerged bodies will become negligible in long waves.

II. Velocities of water particles

From equations (4a) and 4b) it can be concluded that:

The water particles have a maximum horizontal velocity in a wave top and then are directed in the direction of propagation of the waves (see Fig. 9a) while the water particles in a wave trough have a maximum horizontal velocity in an opposite direc-tion.

The water particles have a maximum vertical velocity when the wave elevation is zero. The direction is upward in a transition from a trough to a top and vice versa.

The maximum horizontal velocity will decrease at deeper loca-tions according to Fig. 4 while the maximum vertical velocity

will decrease at deeper locations according to Fig. 5.

The maximum horizontal velocity of the water particles will be-come constant (see Fig. 5) over the water depth for extremely long waves (wave frequency approaches zero):

( dt ) =

Eaa

/71'

when w4-o

(9)

In extremely long waves the maximum vertical velocity of the water particles will become:

1-1

when w,o (10)

dt

max = w a h

From equation (10) it follows that the maximum vertical velocity

at the water surface

wca,

o when w o will become zero at the

still water level.

The wave excited forces due to the velocity of the water par-ticles are called damping forces. This damping can be due to potential effects and viscous effects. The potential effects

F22 potential = b.

. .

F22 viscous = I

As also the potential damping coefficient will become negligible for low wave frequencies it will be obvious that at higher wave frequencies potential effects play a more important role than the viscous damping. However, at low wave frequencies the dam-ping due to viscous effects will be more important than potent-ial damping.

III. Acceleration of water particles

From equations (5a) and (5b) it can be concluded that:

The moment at which the accelerations of the water particles become maximum are indicated in Fig. 9b.

At increasing depths under the free water surface the amplitude of the accelerations decreases in the same ratio as the

veloci-ties.

C. The maximum accelerations of the water particles for extremely

long waves become negligible:

d2E,

_77i-'

= _max a when w o a dt ,d2c, C = W2c

(h-0

max a h when w o a dt

d. The inertia forces in waves amount to the product of the added mass of a body and the acceleration of the water particles in waves:

X21 =a

. XX Z21 = a . zz in which

a.. = added mass of body

11

= damping force due to velocity of water particles along a body in waves

Fromequation(14)themaximummotions.3a in

regular waves with an

amplitude Ca and a frequency co can be deduced:

s.

Fj/C

3a

Ca ir(M+a) 1.42-c12 + bzw2 '

Using equation (15) the following remarks can be made:

I. Horizontal motions

When in addition to the information given in the previous section all the hydrodynamic and hydrostatic forces on a body are known, the motion of the body in waves can be deduced from the following elementary equation of motion:

b.ds.

(M + a.) ---1 + 2--/ + c.s. = F.

at' dt 3 3

in which:

s. = motion in direction j

= mass of the body

a. = added mass of the body moving in direction j

bj = damping coefficient of the body moving in direction j

c. = restoring coefficient of the body moving in the j

di-rection

F.

7

= wave excited force in direction j = Fj, + F121 _Fj22

F11 = wave excited force due to static pressure variations

(Froude-Krilov force)

Fj21 = inertia force due to acceleration of the added mass in

waves

Fj22

By adding the static pressure force (see equation 6a) to the iner-tia force (see equation (13)) and neglecting the damping force, the total horizontal wave excited force in the direction of propagation of the waves is found. When the dimensions of the body are small relative to the wave length one then finds:

xa (M+a) Pa

the wave height.

xa

p3 0, when w o

Ca

This is due to the fact that the body will move in the same way as the water particles when the damping is neglected. This can be seen when the velocity of the body which is not anchored

(cx = o):

ka T;

P3 for bx o

is compared with equation 4a.

When the body is anchored in a soft mooring system (natural period of oscillation about 1 minute) it will be obvious that the damping will be negligible with respect to the inertia of

the body. From this it will be obvious that the motion of the

body then will be:

xacosh K(h - C)

₌

1_13-Ca sin K h

The horizontal motion being 900 out of phase behind the motion of the wave elevation.

When the structure consists of two bodies each with a mass M and an added mass ax, one finds

xa

= P3H0S

K 1 I (18)

Ca

in which

1 = distance between the two bodies in the direction of propagation of the waves.

When the length of the body in any direction is not negligible with respect to the wave length, the static pressure force is

Nearly analog conclusions can be found for the heave of a submerged body as were found for the horizontal motions.

However, on a semi-submersible different phenomena are observed (see Fig. 10).

From equation(3b) one finds that the oscillating part of the static pressure force amounts to:

Z1

_{= B L}

pg cosh K (h - T) sin (wt

-

7-'a cosh K h

Combining equation (5b) and (13) one finds the inertia force Z2I:

Z21sinh K(h-T)

.

= - a w2 sin (wt

-z KE )

'a sinh K h

in which _{az can} be approximated by:

az = pf

B2 L

(20)

f = factor depending on ratio L/B

Substituting equation (19a) and (19b) in equation 15 one finds

Za (Pg

PI 0 -

a, a2 112) (21)

a

/f(p

0 T + a )(4)2

-

0 p g12 + b2 w2

in which O = B L

For very long waves (w + 0 it will be obvious that the influ-ence of the terms multiplied by the frequency will become negli-gible while the factor ulaPproaches unity:

Za

1 if w + o

a

The natural frequency follows from

(p OT+

az)w2z

-Opg= o

by which wz = 1 a 1+ --Z-p0 T

the wave exciting force as small as possible. The wave exciting force will be negligible for a frequency wmz for which:

Pg PIO - a w2 1.12= o

z mz

w

mz = / pg 0'

777'

az _P2

The wave exciting force will be small at the natural heave fre-quency wz when the frefre-quency wmz will be close to wz.

When instead of the single cylinder through the water surface also a footing (submerged body) has been connected to the cylin-der (see equation (10b)) the same equation as in (21) will be found except for the fact that az then is the virtual mass of the submerged body instead of having a value according to equation (20).

When the construction is composed of two columns (see Fig. 10c) one finds:

Za (pg 111 0-a7 w2 1_12) cos 1/2 K 11 Ca _{(p OT + az) w-0}

_pg},

+b`z

in which

0 = waterplane area of one column az= virtual mass of one footing II= distance between two footings

III. Rotation of structure

Za 11 sin 1/2K 1

1

2X

a 12

K Ca K a

When the rotation (I) of the structure which is not anchored,

indica-ted in Fig. 10c is looked at, equation (15) can be rewritten in the following form:

Xa = amplitude of horizontal wave excited force on one column

11 = horizontal distance between the two columns

12 = vertical distance between centre of application of

ho-rizontal wave force X and centre of gravity = mass inertia moment of construction

4)4)

a = inertia moment of added mass

= restoring force due to buoyancy forces

(Pcib

= 1/2 Pg 0 12 - BG A (when 0 is small relative to 12)

= damping

= frequency of waves = 27 /wave period = wave number = 27/wave length

In equation (26) the vertical force follows from the addition of equation (19a) and (19b) while the horizontal force follows from numerator of equation (16). The question now arises how much the rotational motion will be for very long waves.

In that case (w o) one finds:

2 Xa 12 Ta

e

a ,a 1

Ka

Ka

_{k pg 0 12/ - BG A} in which za

7-

Pg 0 when w o (see remark I g in section 2)

X a

. 1/2 (A + 2 g.ax) when w. o (while the centre of

appli-Ka

_{cation of this force then}

lies under the centre of buoyancy B).

Insertion of the forces in equation (27) gives:

(pa pg 0 121.-(A + 2 g ax) 12

(28)

a pg 0 12i_ BG . A

when w o

Since the product BG.A differs from (A + 2 g ax) 12 one finds that also for very long waves the angle of rotation of the

structure will not be equal to the wave slope.

with application"

New York, London Interscience Publi.shers (1957).

1 Hooft, J.P. "Wave excited forces on small bodies"

I.S.P. (1970).

2 Wehausen, et al : "Encyclopedia of Physics"

Berlin, GOttingen, Heidelberg. Springer Verlag. (1960).

z

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t--_,--,,,,am NM ,...,.,..;,,,

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wave top

force variation Z1 due to static pressure variation

Fig. 5 Schematic indication of static pressure force in a wave train.

wave top _{wave trough}

ZA= force on plane A

Z

_{B. force on plane B}

Z1 . total force on the body

Fig. 6 Schematic indication of static pressure force on a submerged body.

plane A

force variation Z due

plane B

Z.i

to static pressure

variation

wave top wave trough

21

bodies are forced

away from each other

bodies are forced

towards each other

Fig. 7 Schematic indication of static pressure forces

on floating and submerged body.

1

t

a

direction of

propagation of waves

Ap.p.ipgC, sin(wt-K)E

area ci5

G

at

G= centre of gravity =G(TIE)

Fig. 8 E2

Am2ilpgCasin(wt-Kt2)

right side S2

upper side Si

under side S2

Left side Si

t2.( dd

)m2x dd )

direction

of propagation of waves

i2.(d2E

)max d t2 t2 d2 10)m2x

7-Fig. 9 Review of maximum velocities and accelerations of

a

Li

G = centre of gravity B = centre of buoyancy

Fig. 10 Sketch of semi-submersible structures.

X 0=B L

rioating Structures in Waves

THE ROLE OF MODEL TESTS AND THEIR CORRELATION WITH FULL SCALE OBSERVATIONS

by Dr.Ir. J.H. Vugts

Shell Internationale Petroleum Maatschappij, Den Haag, the Netherlands.

Summary

The behaviour of offshore floating structures is partly static and partly dynamic. To benefit fully from the re-sults of experiments or calculations, model tests and an analysis by computational methods should be conducted in accordance with a theoretical model of the occurrences. To construct a theoretical model of this type it is ad-visable to deal with the two parts separately. This pro-cedure is justified by considering the environmental conditions more closely.

The statics of the performance can easily be determined by calculations or by model experiments. The cyclic os-cillations of a quantity produced by waves should be stu-died by spectral techniques and statistical methods. The ultimate result is a stochastic variation around a cer-tain non-zero level. Due to the random character of sea waves a deterministic description cannot be given.

Introduction

Offshore floating structures are generally complicated construct-ions placed in a complex environment. They are subjected to the complete combination of environmental conditions: wind, current and waves, possibly together with low temperatures and icing. To resist the environmental conditions the structures are, in the great majority of cases, anchored to the sea bottom. Various en-vironmental conditions and the characteristics of the anchoring

system are non-linear in nature. Furthermore, circumstance require that the problems of both motions and strength in floating struc-tures must be studied dynamically.

In this situation model tests can play a very important role,either as a verification of calculation methods, or in providing the

ne-cessary information for design or evaluation of structures when

calculation methods do not exist, or in studying special features in more detail.

Sometimes, however, model tests appear to become a kind of black magic: imitate all the conditions as realistically as possible in

the experiment and it will provide the answers for all the questions and all the unknowns. Dnfortunately this is far from being the truth Since the early days when model experiments began to be used, the basic law of model tests has been the complete similarity between model and prototype and the basic problem has been the

extrapola-tion of model results to full scale values. This still holds.

In this paper an attempt will be made to analyze the problem of the behaviour of floating structures. Questions of linearity,

super-imposition, calculations and experiments in regular or irregular waves will be discussed briefly and a few examples will be given.

current, wind and waves. Each of them will be discussed brief-ly in the following paragraphs.

2.2. Current

Current is not one general flow of water. It consists of seve-ral components which develop independently and therefore have their own characteristics. The prevailing ocean current is a part of the ocean circulation system and a very steady compo-nent. The tidal current has one or two cycles a day. Its magni-tude may be especially important in coastal areas. The density current is produced by an unbalance due to differences in water temperature and salinity. The wind or drift current results from the drag of the wind on the water surface. It will be ob-served near the surface and its magnitude will be in the order of 11/2 to 2% of the wind speed for winds blowing from one

di-rection for a sufficiently long time.

The relative importance of the above four components varies from location to location. They may also have different direct-ions. In any case, however, the variations in current velocity will be slow compared with the variations apparent at the water surface in the form of waves. Therefore current may be consider-ed as a steady phenomenon and current forces become steady forces.

2.3. Wind

Wind is a usually turbulent flow of air with fairly large fluc-tuations in velocity and direction. It is common meteorological practice to give the wind velocity in terms of the average of a one-hour or 10-minute interval, indicating that the variation in mean velocity is again slow compared with the wave periods. The fluctuations around the mean values will no doubt impose dynamic forces on the structure. However, in view of the fact that the density of water is about 800 times that of air, the aerodynamic forces may generally be neglected with respect to the hydrodynamic forces, when considering the dynamic behaviour of the floating body as a whole. It is, therefore,an acceptable simplification to take the windspeed as a mean and constant value, so that wind forces will be steady forces like the cur-rent forces.

of the structure, but they will hardly influence the overall behaviour.

2.4. Waves

The term waves is applied to the continuous variation of the surface of the sea. This surface is irregular and it can only be described adequately in statistical terms, so any select-ion of a regular wave to represent a sea conditselect-ion is mislead-ing. We must always consider the complete and complex picture of the sea. See also Ref. 1.

Regular waves, such as those artificially generated in labora-tories, do not occur in nature. However, a comprehensive des-cription of the irregular sea surface is possible by super-imposing a very large number of regular sine waves, the so-called component waves, each with its own direction and period and each of a very small height. It is usually assumed that the component waves are unidirectional. In fact this is more a simplification, the selection of a specific case, than a limitation of the theory applied.

By pure summation of a great number of component waves we will obtain a realistic wave record. Or conversely, given a record, methods exist to break it down into its component waves. It is the nature of the sea that the position of the component waves,

one with respect to the other, is not fixed but arbitrary. So

the phase relationship between the component waves is random. This has some very important consequences, for it means that:

a wave record is not specific to a certain sea condition; it is a unique occurrence which will never happen again. the maximum wave height of a record is not typical for a sea condition.

the irregular surface profile, with its surface elevations at various instants and its apparent wave heights can only be characterized quantitatively in statistical terms; they

dis-developed for a stationary sea state, which means that the to-tal amount of energy contained in the sea is constant. The energy supplied by the wind is equal to the energy loss due to internal friction in the water and to wave propagation into more quiet water outside the area considered. A stationary sea

state is completely described by its energy spectrum or wave spectrum, specifying the wave energy, or equivalently the wave height of the component waves over the periods of the compo-nent waves. The wave spectral density function is defined by the equation:

w+dw

w(w) dw = E 11{Ca(0.0}2

or more simply: _{Sw(w) dw}

_{= kca;}

In this equation Ca is the amplitude of the component wave with circular frequency w. So by definition S(a) is equal to the sum of half the squared amplitudes of the component waves in the frequency interval from a to (w + dw), divided by the width of the interval.

The whole technique making use of wave spectra as input to a problem and leading to corresponding motion spectra, force spectra or stress spectra, is called spectral analysis. With this method a realistic and dynamic analysis becomes possible.

The summation or superimposition described above implies that the system must be linear. The wave theory used for the compo-nent waves is therefore the simple Airy sine wave theory. Higher order wave theories, as often used for a more precise description of a high, single wave and used for the calculat-ion of wave forces on fixed platforms, are related to regular waves. They are associated with a concept of selecting a maxi-mum single wave, producing a maximaxi-mum loading on the platform,

after which the structure is analyzed statically. It bears no

relation at all to irregular waves or a dynamic analysis. In the opinion of the author a proper dynamic analysis can only be made by spectral techniques, whereby full attention is given to the actual nature of the sea.

statistical theory predicts that the significant wave height H1/3 (being the average of the one third highest part of all apparent wave heights) is equal to 4,4T-c-;, where mo is the area of the wave spectrum. Fig. 1 shows this relation in 27 wave records taken in various parts of the world. The confirmation is quite satisfactory. In a more refined version of the theory the factor 4 depends on the width of the wave spectrum and is smaller, e.g. 3.9 to 3.6. When this is taken into account the agreement is excellent.

-Fig. 2 shows the measured maximum apparent wave height Hmax

vs. the measured significant wave height H of the same 27

records. This graph also shows the lines predicted by the statistical theory for a probability of 95, 63 and 5 per cent exceedance. It will be clear that the ratio Hmax/H1/3 will depend on the length of the record, that is on the number of waves N. The duration of the records varied from 10 to 30 minutes and N varied from 50 to 200. Therefore lines for N = 50 and N = 200 are shown. All measurements fall within the zone indicated. A lower or higher Hmax is indeed an unlikely occurrence, but it can happen. A more definite relation be-tween the significant and the maximum wave height than that shown in this graph cannot be given, however.

3. The analysis of the behaviour of offshore structures

The discussion of the environmental conditions in the preceding chapter has shown that a part of the environmental forces may be considered as steady and that another part is really to be looked at as unsteady; the distinction between steady and unsteady being made by using the period of the variation in relation to the wave periods as a yardstick.

It is therefore proposed to split the behaviour of the structure into two parts, namely a static problem under the action of the

linearity. Strictly speaking this is, of course, not true. What matters, however, is whether a sufficiently accurate engineering solution can be obtained by a linearization. For the time being

linearity only means that the steady loads and the static

beha-viour can be separated from the unsteady loads and the dynamic behaviour. If one is in fact influenced by the other this influ-ence should be negligible. However, the superimposition of a static and a dynamic problem is a very reasonable approach in many technical fields. In the opinion of the author there is no reason to depart from this practice when considering the behaviour of offshore structures. And following a more pragmatic approach: it is often the best we can do. So we will accept this proposition,

but it is well worth checking it as far as possible by adequate

model tests when a specific problem is studied.

The static problem is controlled by the weight and buoyancy of the structure, by the environmental loads of wind and current, and the anchor line restraints which keep the structure on location. The static equilibrium of the forces determines the position of the unit and a strength analysis produces a constant stress level in each member. Within the static sub-problem it is very well possiEle

to include, for example, the fact that the anchor line restraints are non-linear with displacement. The equilibrium is then simply found from a combined analytical and graphical solution.

In the dynamic problem only the cyclic oscillations of the

quanti-ty under consideration about a zero mean value are taken into

ac-count as these are generated by the waves. They are controlled by the unsteady loads, primarily the wave forces, and the dynamic response of the structure. Additional external loads may be varia-tions in the anchor line restraints due to a change in position of

the unit. In many cases the anchor line forces are very small

com-pared with the wave forces and all the restraints may be neglected.

The unit can then be analyzed in a freely floating condition. _The

influence of the anchoring system can also be included, _however,

in the form of a linearized spring. The spring constant _{may change}

the dynamic response of the system to _{some extent, although}

reson-ance periods in the horizontal plane lie generally well outside the range of wave periods.

considered is linearly related to the wave amplitude. As discussed in the preceding chapter the sea may be thought of as consisting of a summation of a large number of regular sine waves of small ampli-tude and random phase. The response to each separate wave component is expressed in the transfer function. This is the ratio of the amplitude of the quantity under consideration to the wave amplitude, together with the phase difference between the output quantity and the input wave. The ultimate behaviour is obtained by summing the separate responses again.

In practice this will be done in the following way. Consider the quantity q. Define a spectral density function by:

cla

S (w) dw =

qa2

=c)2 Sw

(W) dw

or _{Sq(w) =}

()2. Sw(w)

(la

where -- is the magnitude of the transfer function or the response

a

amplitude operator. Using the same theory as for sea waves, a sig-nificant value of the quantity q

ing a certain 4a and the maximum tam n number of cycles and with a be determined from the spectrum theory may be found in the Refs.

(q

1/3) ' a probability of

exceed-a

4max of q to be expected in a cer-certain chance of occurrence can S (w). Details of the statistical

2, 3 and 4.

The quantity q may be any quantity linearly depending on the wave amplitude, i.e. a body motion, a force or a stress at some locat-ion in the structure. The linearity required is generally satis-fied much better than assumed. It must be remembered that in the dynamic problem only the cyclic oscillations of q generated by the waves are considered and not the whole of q. It must also be re-membered that the description given is a theoretical model of

reali-ty, capable of producing a good formulation of what is actually happening. The fact that the motions of the unit in high regular

tic and the dynamic responses. It is a stochastic variation around a certain non-zero level. By applying the technique of spectral analysis to the dynamic part of the problem the nature of sea waves is taken into account, which is absolutely necessary for a real understanding of what is happening. To support the method described the results of a spectral analysis on some full scale measurements of the vertical motion of the rotary table on a semi-submersible is shown in the Figs. 3 and 4. They are compared with fully analytic calculations. These measurements will be discussed further in Chapter 5.

The author is well aware that he has excluded one special phenome-non from the discussion, that is the low frequency oscillations which may occur especially in the horizontal plane. It seems that these are caused by a second order wave force which tends to excitn the system in its natural frequency. The phenomenon is, in fact, a dynamic one. It is, however, outside the scope of the dynamic behaviour of the structure over the range of wave periods contained in an irregular sea and it cannot be dealt with in a normal spec-tral analysis. The phenomenon is far from completely understood. Various investigators are studying it and it is most likely that we will hear more about it during this symposium. Low frequency oscillations have been noticed in model experiments and at full scale. It is very difficult, however, to establish them at full scale for there is hardly any instrument that can reliably record such occurrences, and in the opinion of the author model tests can only be considered as a qualitative indication.

For the time being it is proposed to consider low frequency oscil-lations as a separate phenomenon which, again, can be added to the static and the dynamic behaviour. In several investigations it plays no role at all. If it does it is an unknown, but one which will not prevent the investigation of the static and the normal

dynamic behaviour thoroughly in the way described.

4. Model experiments

From the above it will be obvious that the author is no supporter of model experiments in which all circumstances have been imitated

beforehand how the results will be utilized. In this respect it is of great importance to identify the problem clearly. Is it a question of economics, of safety or purely a technical matter? The purpose of the investigation often determines the type of tests, the quantities to be measured and their analysis.

The various forces influencing the behaviour depend partly on the

VL V

Reynolds-number Re

= 7T

and partly on the Froude-number Fn =

In model experiments it is impossible to satisfy both. Since the wave forces and the dynamics of the structure are very important it is usual in ship model investigations to maintain the Fn-number and consequently to neglect the Re-number. For offshore structures the same practice is followed. This means that if the model scale

is a, the time is proportional to

77

the forces to a'and the

mo-ments to a4. It also means that the Re-number is a factor a3/2

smaller than in reality; see also Table 1.

Table 1

Ratio of Re-number in prototype and model for various model scales

The effect of the Re-number on the various forces depends on the

shape of the body, but in general it may be concluded that for the

usually circular elements in offshore structures very significant scale effects will be experienced.

Of the environmental forces the wave forces depend predominantly

a ratio

25 125

50 350

are desired. When the statics and the dynamics of the problem are separated the above problem is circumvented.

The static problem can be analyzed either by calculations or in model experiments with a larger model, or else in windtunnels at higher speeds. By omitting waves the Fn-number can be abandoned and the Re-number approximated as closely as possible.

In dealing with waves and the dynamic responses of the structure by spectral techniques the purpose of a model experiment should be to determine the transfer function of the quantity under consider-ation. Model tests should further serve the purpose of checking the assumptions underlying the analysis, e.g. a check on calculations, on linearity and on the influence of a static shift on the dynamic response.

By way of example, the investigation of the behaviour of a spar-type storage unit will now be described. This is a floating struc-ture, anchored to the sea bottom. A tanker takes the oil from the spar and brings it ashore in a regular shuttle service. The tanker will be moored to the spar which is therefore a single point moor-ing. This is illustrated in Fig. 5.

In this problem both the behaviour of the spar and of the tanker must be determined. The tanker must be able to moor and remain

connected to the spar in weather conditions _up to a certain

opera-tional limit. In more severe weather the spar must remain safely anchored on location without the tanker being connected. Finding the operational limit is one of the problems to be solved.

The static performance of the spar was calculated. It is influenced by wind and current forces, the tanker pull and the anchor line restraints. In addition a drift force on the spar was calculated, which is a non-linear zero shift of the wave force. The anchoring

system was designed from the static part of the problem to balance the external forces. There is, of course, an influence from the

dynamics of the system in that the oscillatory motions of the spar

ave-For the dynamic performance of the spar, calculations were made by a method developed by Mr. J.P. Hooft at the NSMB, while the dynamic behaviour of the tanker was calculated using existing ship motion programs. Both these calculations were made for freely floating bodies in waves, without anchor lines and bow hawser. It was deci-ded to do model experiments on the dynamic behaviour to check the calculations and a number of assumptions made in the analysis.

For quantitative results it is desirable to test as large a model as possible. Since current is not considered in the dynamic problem it was possible to select the seakeeping basin of the NSMB to per-form the tests. For the water depth of 120 in the model scale could be 1:50.

The transfer functions calculated separately for the motions of the spar and tanker were compared with measurements of the combined system of spar and tanker in regular waves. Tests in regular waves were chosen as a basis because of their simplicity of

instrument-ation, analysis and interpretation. Tests in irregular waves were also performed to check the linearity of the system. If linearity is satisfied, if instrumentation is satisfactory and execution of the tests

and analysis of

the results is performed well, experiment: in irregular waves are, in fact, identical to those in regular waves If not, the interpretation of irregular wave experiments becomes difficult.

The quantities measured were:

wave

spar motions (surge, heave, pitch) anchor line forces in three lines

tanker motions (three displacements of the bow and yaw) bow hawser force.

expe-The oscillatory movements of the tanker can be sufficiently accurately calculated neglecting the bow hawser force and the presence of the spar in front of the tanker.

The results of the regular and irregular wave experiments agree well, indicating that the dynamic system is sufficiently linear to use the technique of spectral analysis.

A clear insight into the occurrences has been obtained by the pro-cedure followed and an analytical method for studying the behaviour has been checked. It has therefore become possible to calculate the behaviour in any weather condition with confidence. When circum-stances or design parameters are changed the influence can easily be established without new model tests being necessary.

5. Full scale observations

Full scale observations can be distinguished in actual measurements at sea and in the collection of a large number of routine data, mostly not obtained by instruments, which may serve to abstract statistical information.

Actual measurements are sometimes performed on the motions of the rig, notably heave, and on the stress variations due to waves (see Ref. 5). There is a distinctly increasing trend towards special and more sophisticated measurements under service conditions. These generally require special instrumentation and elaborate analysis,

but it will be clear that such measurements are indispensable _for

evaluating calculation methods and model experiments. For the same

reasons the quantities to be measured should be carefully selected,

and the measurements thoroughly planned, executed and interpreted.

In the Figs. 3,4 and 10, 11, 12 the results of full scale

measure-ments of the vertical motion of the rotary table for two

semi-submersibles when operating in the North Sea are shown. In the

case of Staflo both the motion and the waves were measured with a

waverider buoy. The motion buoy was installed on deck next to the derrick. Thus both quantities were measured with an

accelerometer-on the unit, while the vertical motiaccelerometer-on is measured with a taught wire connected to the marine riser. The vertical motion is necess-arily used to correct the apparent waves as measured by the wave

staff to obtain the true waves, although these may still be dis-turbed by the presence of the unit itself.

The transfer function of the vertical motion is obtained by divi-ding the spectrum for the vertical motion by the wave spectrum. The transfer function has been truncated at both sides. In the long period range very little or no wave energy is present, so that the transfer function will be the result of the division of two very small values. As the natural periods of semi-submersibles are high,

the rig response and thus the motion spectrum in the short period

wave range are very small. Consequently, the transfer function can-not be determined with sufficient accuracy either. It is in the

high period range especially, e.g. 10-20 sec.,that the rig response

is important. Since semi-submersibles are remarkably stable as

com-pared with ships a spectral analysis can only be done properly on appreciable signals when relatively large motions in severe waves

are measured. For these reasons the accuracy of such full scale

measurements does not enable us to determine minor details in the

response of the rig. With these limitations, however, the results

still indicate that a spectral analysis to determine the motions is equally applicable to semi-submersibles and ships alike. The correlation between calculations, model experiments and full scale measurements is also satisfactory.

The data collected on a routine basis are generally very rough and

difficult to evaluate. At best they may give statistical

informat-ion of trends. They are hardly useable for checking calculations

or model tests. If this is attempted nevertheless, one often finds

motion in various wave conditions. It shows the calculated ratio of significant double motion over significant wave height as a function of the mean period of the wave spectrum. The calculations were made using the Pierson-Moskowitz wave spectrum, which is specified by the significant wave height and a mean period. Since the shape of the wave spectrum is eliminated as a parameter and the response of the rig is considered to be linear, a single curve is obtained, depending only on the water depth, the wave direction and the con-dition of the unit.

Fig. 13 certainly gives an indication of what may be expected in

relation to wave

conditions.

Similar diagrams can be constructed

for other motions. Checking this curve by routine data is difficult. Generally "maximum waves" and "maximum vertical motions" are re-ported, which cannot obviously be reduced to significant values. The two maxima as observed may differ from the significant values by very different factors. Estimates of the mean wave period are even more difficult for the crew of a drilling rig.

Finally, the wave spectrum will deviate more or less from the Pierson-Moskowitz spectrum and consequently the actually observed motions will differ from those indicated in Fig. 13. Thus a

con-siderable scatter will occur, from which it may not be concluded, however, that there is a discrepancy between full scale

observat-ions and calculatobservat-ions or model tests.

6. Conclusions

The behaviour of floating structures in waves is investigated by separating the statics and the dynamics. Both parts are studied individually, after which the results are added again. The dyna-mics arise from the wave action and since this action is by its very nature a random phenomenon the result is a stochastic

variat-ion about some mean static value. This variatvariat-ion is dealt with by the technique of spectral analysis and by the same statistical theory as that applied to the waves.

the cyclic oscillations of the quantity considered may be regarded as linearly related to the amplitude of regular waves.

In the majority of cases both assumptions are reasonably satisfied,

so that the results are certainly accurately enough for engineering

applications. However, in any specific problem studied it is worth-while checking their validity as far as possible by suitable

ex-periments.

If low frequency oscillations play a role as well, they should also

be handled independently, as long as this phenomenon is not suffi-ciently understood. It is doubtful whether model experiments can provide quantitative information on this point.

In many cases it is possible to study both the static and the

dy-namic behaviour by calculations. When experiments are performed they can serve to check these calculations. In any case model tests

should be conducted according to a theoretical model of reality as

observed and where necessary and where possible be utilized to verify the theoretical model. Since it is impossible to satisfy

both the Froude and the Reynolds-number at model scale, forces of

viscous origin will generally be greatly overestimated with res-pect to wave forces. Therefore model experiments. under supposedly "realistic" circumstances for the purpose of obtaining direct quantitative information for design or evaluation of structures will generally fail in their objective. The possibility of using and extrapolating their results is limited in most cases.

1 Freudenthal, A.M. : "Probabilistic evaluation of design

criteria for maritime structures" Bulletin of PIANC (Permanent Int. Ass. of Naval Cong.) vol. III/IV ho.2

(1968/1969).

5 Bell, A.O. and

R.C. Walker

seawaves from the highest waves in a record"

Proc. Royal Society of London, A. vol. 247 (1958).

: "Dynamic stresses in an offshore

mobile drilling unit"

Conf. on dynamic waves in civil engineering, Un. of Wales, Swansea, (July 1970).

Also: Offshore Technology Conference Houston, U.S.A. (April 1971).

2 Cartwright, D.E. and

M.S. Longuet-Higgins

: "The statistical distribution of the

maxima of a random function" Proc. Royal Society of London, A. vol. 237 (1956).

3 Longuet-Higgins, M.S. : "On the statistical distribution of

the heights of seawaves"

J. of Marine Research, vol. XI, no.3 (1952).

6 2 7

Fig.

1 H

I

Ill UI

1

1

1,d

rdAND RELATION BETWEEN fly;WM.AS ( WITH DERIVED

m. AREA OF THE MEASURED SPECTRUM) FROM THE WAVE RECORD

plir

4

Fig. 2

RELATION BETWEENRm. AND F41/3 DERIVED FROM THE WAVE RECORDS

/

1

/ ...''

04" ,p

N

,

de %'

IMMIBM11111raan---MilrAra- 1

INIFAIIIPI4MIIM

ENVII__M

F10:01

Iffedffigs-Wo

11 I

Abgartmar r STATISTICAL meony

7.." --- - LINE N 5% EXC. Ma-. 55% EXC.

, .----

--

50 1.55 1.425 1.20 200 2.025 1.64 1.45 3 4 a i41/3

20

II 12 m 16 14 12 10

5 15 15 U2 SEC M2SEC STA FLO

COMPARISON BETWEEN CALCULATIONS AND FULL SCALE MEASUREMENTS

FOR THE VERTICAL MOTION OF THE ROTARY TABLE OF STAFLO

CALCULATED WATERDEPTH 80m MEASURED 4 FEBR. 1971, 02 40-0300 HR WATERDEPTH 85m 05 10 45 RAD./SEC. 10-1.38 1.86 13.23 M M SEC 2-11/5 eel = = 10 05 30° WAVES 05 10 115 05 10 15

1.12SEC 10-5 0 15 1.5 0 STA FLO

COMPARISON BETWEEN CALCULATIONS AND FULL SCALE MEASUREMENTS FOR THE VERTICAL MOTION OF THE ROTARY TABLE OF STAFLO

221/3 r 1.19 M 2 21wo, = 1.80 M Tmoonr 10.30 SEC 2 ivy r 1.03 M 2Z,,, 1.43 M 10 22 SEC 11.5 .SEC WAVES 0° 0 5 0 0 0.5 4/3 = 4.17 H ma. = 6.38 M r 7.16 SEC HI4 /) 3.77 53..8777 1.164 TMO. r 7.16 SEC CALCULATED WATERDEPTH ElOm MEASURED 31JAN.1971,14.40-15.00 HR WATERDEPTH 85rn ic MEASURED 31 JAN. 1971,20.40-2100 HR V/ATERDEPTH 85n, 410 15 RAD./SEC. 10 0.5 0 05 4.0 15

Fig.

5

EXPORT

,C

STORAGE SPAR (BALLAST CONDITION)

HEAVE RESPONSE

(1) IN RAD. SEC.'

THEORETICAL

0 REGULAR WAVES (20'± 5.00m)

REGULAR WAVES ( 2c0=±5.00m)

(WITH SIMULATED WIND AND CURRENT FORCE) IRREGULAR SEA twit, 12 09m, T 125 SEC. IRREGULAR SEA ';,,,./3r 12.09m , T .12.5 SEC

(WITH SIMULATED WIND AND CURRENT FORCE)

SURGE RESPONSE

(s) IN RAD. SEC.-'

Fig. 6a

STORAGE SPAR ( BALLAST CONDITION) WITH A TANKER (LOADED CONDITION)

SURGE RESPONSE

THEORETICAL (WITHOUT TANKER) 0 REGULAR NAVES ( 2 ;. ±5.O0 IC

IN RAD. SEC.-,

STORAGE SPAR (LOADED CONDITION) WITH A TANKER ( BALLAST CONDITION)

SURGE RESPONSE

4.1 IN RAD. SEC.-'

Fig.7a

Fig.7b

IRREGULAR SEA , w1/3= 2.91m,T= 78 SEC.

IRREGULAR SEA : v3. 3.73m,T= 86 SEC.

IRREGULAR SEA 4.77m,T= 9.3 SEC. IRREGULAR SEA : 5.68m,T= 9.8 SEC.

STORAGE SPAR (WADED CONDITION)

WITH A TANKER(BALLAST CONDITION)

FORCE IN MOST HEAVILY LOADED ANCHOR CHAIN 0 MEASURED MAXIMUM Fmax 5% EXCEEDANCE MOST PROBABLE o t'vrips IN M Fig. 8 Frnean PRETENSION 95% EXCEEDANCE

STORAGE SPAR (LOADED CONDITION) WITH A TANKER (BALLAST CONDITION)

HEAVE RESPONSE AT THE BOW OF TANKER

4) I 1.4 RAD. SEC.-'

THEORETICAL (WITHOUT SPAR p.180°)

THEORETICAL (WITHOUT SPAR 1.1=1500)

A BOW HAWSER LENGTH 1

0 BOW HAWSER LENGTH 2 REGULAR WAVES o BOW HAWSER LENGTH 3 2C,.. t 5.00m

BOW HAWSER LENGTH I IRREGULAR WAVES

BOW HAWSER LENGTH 2 cwo= 4.77m.

../.

/

-K

N.,

\

,

\

.---

BOW HAWSER LENGTH 3..;.,.. 9.35E0.

/

\

_\

/

\-

_\

/

.

7

.

/

/

\

J

a

/

\

0

\

\

,\

\\\

. , .s\

\

.---°-. Vs \\

\

\

.

\

\\\

ollks

\

\

\?/.

\

\ \

\..., ...--11

\

_\

\

1.0

0.5

0

SEDNETH I

---- CALCULATIONS FOR p=0° IN 400 WATERDEPTH

Abs. vert. motion on rotary

Wave 1.5 I MEASURED p,-15° IN 13APRIL 1969,15.30-16.00 OFF NORWAY HR 200 WATERDEPTH I 0.5 _{10 15} Ct.) in rod/sec. Fig. 10

1.0

0.5

SEDNETH I

CALCULATIONS FOR IN 400 WATERDEPTH

Abs. vert. motion of rotary Wave

1.5

MEASURED OFF NORWAY

11 FEBR. 1969, 23.30-24.00 HR.

#2,130° IN 285 WATERDEPTH

0.5 10 15

Abs. vert motion of rotary 1.5 Wave 1.0 0.5 0

SEDNETH I

CALCULATIONS FOR} IN 400' WATERDEPTH_{p= 0°}

MEASURED OFF NORWAY 10 MARCH 1969, 05.45-06.15 HR. #S-70° IN 285 WATERDEPTH CD 11 MARCH 1969,06.00-06.30 HR. p-70° IN 285 WATERDEPTH I 1 1 I 1 I I I I I

I°

eit',4

I

/

/

\,,D 0.5 10 45 CO in rod./sec. Fig. 12

as

Fig. 13

THE VERTICAL MOTION OF THE ROTARY TABLE OF STA FLO AT 59 DRAUGHT IN PIERSON-MOSKOWITZ WAVE SPECTRA (WATERDEPTH 120 N)

41. 180° . p 90° /

/

1.1

./

.1

/

10 12 14 46 18 20 SEC.

MODEL TESTING FOR THE DESIGN OF OFFSHORE STRUCTURES

by G.F.M. Remery

Netherlands Ship Model Basin, Wageningen.

Summary

The many problems involved in the rapidly increased demand for unconventional offshore structures, both floating and fixed to the bottom, requires new and special adapted design methods.

In combination with theoretical studies for the

determina-tion of mathematical models to describe the behaviour of a

particular object, model tests may play an important role. They are a useful tool for studying the feasibility and the workability of the structure or for improving the system. Also data required for the structural design of the ultima-tely chosen system may be obtained from modeltests.

Due to the irregular character of waves and weather condi-tions, the behaviour of offshore structures has to be treated in a statistical way. The statistical interpretation of model tests both conducted in regular and irregular waves, and the extrapolation of the results to long term weather conditions,

ration and exploitation of the sea bottom, floating structures or structures resting on the sea floor are required.

Station keeping of the structure above a fixed point on the bottom often appears to be an important requirement. External forces due to wind, current and wave action tend to move a structure from the

desired position, while reaction forces due to inertia, damping

and mooring or attachment arrangement tend to restore the deviation. The resulting motions of and the stresses in the structure and in the mooring arrangement, due to the forementioned external forces, determine the feasibility and workability of the system.

It is important that proper means are available for the determina-tion of the response of a designed structure to the environmental conditions. A powerful tool can be modeltesting in combination with

theoretical studies and empirical calculations. Properly used it may provide the possibility to optimize a particular design and to

compare various types of structures during the design stage.

Because the environmental conditions, at the localities where the structure will have to operate, determine for the greater part

the behaviour, the strength

of

structural members and the

workabi-lity of the designed structure, a clear insight in the wave, wind and current conditions is indespensable.

As wind, wave and current are irregular phenomena, of which the magnitude can only be expressed in terms of statistics, also the life of a structure or the workability of a system is a case of

statistical probability. Therefore, one of the most important aims

of model testing is to provide directly or indirectly information about the statistical behaviour of the investigated quantities.

rents on the behaviour of a structure in order to determine whether it is permissible to study them seperately or to ne-glect one or more of them during model testing.

Laws involved with model testing and extrapolation of measured results to a statistical reality.

2. Environmental conditions

Generally the environmental conditions are the governing factors for the design of an offshore structure. Information about waves, winds, currents and waterdepths on the locations where a particu-lar structure will have to operate is requiredin statistical terms in order to be able to determine the design conditions. Design con-ditions are the maximum concon-ditions which the structure, or a spe-cial part of the structure (e.g. the mooring or attachment system) should be able to withstand, with respect to both motions and structural forces.

They normally are expressed in terms of return period, being the average number of years that passes between the occurrence of a seastate exceeding some given heavy seastate, wind speed or cur-rent speed.

It will be clear that generally more than one critical return period will have to be determined. For example: a single buoy mooring system may be designed for a 50-years wind, wave and cur-rent condition when no tanker is moored to it.

However, the heavy sea state corresponding with this 50-years storm condition will not permit to moore a ship to the buoy at this severe condition. The system will probably be designed to load and discharge ships at maximum weather conditions having a return period in the order of about 1 year at a survival condition of 50 years. However, the use of an operational return period as a design criterion is often not very appropriate. It is better to express the workability criterion in terms of the average number of days per year that the environment has to permit operation. When human life or desastrous polution is involved, the return

period will have to be in the order of 100 years or more.

Wind

Like all environmental phenomena, wind has a statistical nature which greatly depends on time and location. The role which wind plays in the determination of the environmental conditions for the design of offshore structures is twofold:

On the part of the structure above the water surface, wind forces are exerted due to the stream of air around the va-rious parts. For the determination of these forces infor-mation is required about the local winds only.

The forces exerted by the wind on the water surface cause disturbance of the still water level, thus generating waves and currents, which induce forces on the submerged part of the structure. To determine this effect of the wind, infor-mation is required about the wind and storm conditions in a much larger area.

The waves and currents generating character of the wind will be left out of consideration. The effect of both waves and currents will be dealt with seperately.

Local winds are generally defined in terms of mean velocity and mean direction related to a certain height above the still water level.

The standard height above the water surface for which generally the wind velocity data are given amounts to 10 m or 30 ft. A number of empirical and theoretical formulas is available in literature to determine the wind velocity at other heights.

An adequate vertical distribution of the wind speed is represented by:

V(z)

(

z,1/7

V(10) '10'

in which V(z) = wind speed at height z above the water surface V(10) = wind speed at 10 in height above the water surface

treme wind velocities, can only be used properly when information is available about the period over which the reported mean wind velocities were averaged. Dependent on the averaging-time interval destinction is made between the sustained wind speed, being the

long term average of the recorded velocity (averaging-time e.g.: hour), and the instantaneous (short period) velocity or gust speed. The values of both the sustained wind speed and the gust speed are dependent on their averaging-periods.

In determining a design wind speed, long term predictions are made of the extreme sustained wind speed and are generally expressed in

terms of recurrence interval or return period, being the average time that passes between the occurrence of a particular extreme sustained wind speed.

After the selection of the design sustained wind speed, the most probable or mean gust speed, which will occur during the averaging time interval of the sustained wind can be determined in terms of a gust factor. The gust factor is defined as the ratio of the maxi-mum gust speed, during the averaging-time period of the sustained wind speed, over the sustained wind speed. From data given in

Ref. 1, it appears that the most probable gust factor is almost

independent of the absolute magnitude of the sustained windspeed, but dependent, to some extend, on the height above the surface for which the wind velocities have been determined. As wil be clear the gust factor is a function of the ratio of the averaging-time intervals of sustained wind and gust speed, see Fig. 1, which was constructed from data given in Ref. 1. From this Fig. which is

based on a large amount of data (according to Ref. 1) it appears

that the most probable maximum gust factor can be approximated by the empirical relationship:

ts fg = 1 + 0.162 log

7-in which: f = most probable maximum gust factor

ts = averaging-time interval of sustained wind speed tg = averaging-time interval of gust speed

It is evident that the gust factor increases at decreasing averag-ing-time interval or gust speed relative to that of the sustained

P(x) of the extreme values, occurring during certain periods of time T.

Tr(x) - . T

in which Tr(x) = return period = average time (in years) that passes between the occurrence of an extreme sustained wind speed exceeding a particular value x.

P(x) = probability that the maximum sustained wind speed

will exceed the value x during a certain period of time T.

As is usual in wind statistics, T is taken equal to one year.

In literature various methods are available to fit the extreme values, obtained from observations, to a mathematically defined

distribution P(x). (See Refs. 2 and 3).

Gumbel (Refs. 4, 5 and 6) supposes that the initial distribution

of wind speeds is of the exponential type, which leds to a distri-bution function of the extreme annual values described by:

P(x) = 1-e-e-Y

in which y = a(x-u) = reduced largest value.

a en u are coefficients, which can be determined as

function of the observations available.

When the distribution function of the extreme annual values is known, all the information is availbale for determining a proper design wind speed in the following steps:

select average operational and survival return periods select statistical data about sustained wind speeds from weather stations near the location of operation. If the

Select or estimate the minimum gust period to which the de-signed type of structure will respond.

determine the gustfactor corresponding with the ratio of the averaging-time interval of sustained wind speed and minimum gust period (e.g. from Fig. 1)

multiply the design sustained wind speed with the gust factor in order to obtain the design wind speed.

Attention has to be paid to the fact that the thus obtained design wind speed is a static one inducing static forces on the structure. However, the frequency of encountering of gusts may also play a

role in the dynamic behaviour of a structure. Spectral analysis of the wind speed records might be used for the determination of dominant gust frequencies. Especially for dynamic positioned or moored off floating bodies, the dynamic wind behaviour may be of

importance.

Current

As currents induce forces on the submerged part of a structure it is important to make a prediction of the probability of exceeding a particular extreme current speed and to have insight in the cur-rent direction and the pattern of the curcur-rent at the locations where the structure will operate. Although for most structures the surface currents will be the governing ones, the vertical current distribution may also be of importance especially for the operational part of the design (deep-drilling, pipe-laying). The main sources of currents are the wind, causing major ocean currents and storm surge, and the cyclical change in lunar and solar gravity, causing tidel currents.

The free surface steady velocity due to wind amounts to about 3% of the wind velocity (at 30 ft height).

Tidal currents are of main importance in areas of restricted water depth and can attain values up to 10 knots. However, these extreme velocities are rare. A 2 or 3 knots tidal current speed is common. The prediction of the magnitude of tidal currents is a special science. Some predictors take into account up 60 parameters in