Improved theory for wave induced loads on twin hull semi-submersibles

ABSTRACT

Qr-i

Wave-induced loads on semisubmersible platforms in

operational and survival conditions are usually

evalu-ated from the Morison equation.

If the pontoons are

unusually large, closely-spaced, or near the

sea

sur-face, then interaction effects that are not included in

the Morison equation approach become significant.

In

the transit condition, when the pontoons are floating

in the sea surface, the wave loads must be calculated

by other methods.

An improved strip theory approach is presented

uti-using "opposed motion potentials't.

_Interaction

effec-ts between the pontoons are included.

The present

im-provement provides additional information about the

spatial distribution of tue diffraction forces,

en-abling the determination cf the internal forces

between

the pontoons.

Comparisons are shown between results from the

Morison equation approach, the improved strip theory,

and model tests for a recent semisubmersible design at

operational draught. The improved strip theory is

also compared to 3-D sink-source calculations for the

same rig at transit draught.

NOMENCLATURE

A

added-mass matrix

B

damping matrix

B

breadth between pontoon centerlines

C

restoring coefficient matrix

D

centerline sectional diffraction force

F

_{excitation force (equations l-3)}

F

_{centerline sectional force (figures)}

G

diffraction force on starboard hull

H

diffraction force on port hull

H

wave height

g

_{acceleration due to gravity}

J

imaginary unit

k

wave number

L

_{characteristic length of semisubmersible}

M

generalised inertia matrix

N

_{unit normal vector (toward vessel surface)}

n

_{components of N}

P

_{dynamic pressure}

B

_{position vector for a point on the vessel surface}

S

_{surface region}

t

t i me

u

_{coordinate axes/coordinates of a point}

V

_{volume region}

X

_{motion amplitude}

O

incident wave amplitude

B

_{heating angle of waves (o degrees = head seas)}

fl

motion vector

O

diffraction potential

À

wave length

P

density of fluid

O

_{opposed motion potential}

incident wave potential

IMPROVED STRIP THEORY FOR WAVE-INDUCED LOADS ON TWIN HULL SEMISUBMERSIBLES

Subscriots

D

diffraction

I

incident wave

i

direction, il,..

.6

corresponds to axes in fig.2

surge,sway, heave, roll, pitch, yaw

s

_{significant crest-to-trough response in irregular}

seas

Conventions. Very many of the quantities discussed in

this paper are harmonic functions involving exp(-jwt).

This function is generally omitted in the text, with

the remaining symbols indicating complex amplitudes

(of the motions, potertials, etc.).

When finally

cal-culating the magnitude of physical quantities, it is

understood that the real part is to be taken.

INTRODUCTION

During the last few yesr

there has been a great deal of

activity in the design and construction of

semisubmer-sible platforms for many different types of offshore

duty.

The present generation of semisubmersibles

de-parts to some extent from the archetype of the early

seventies, generally having greater payloads

on deck,

larger pontoons , and different arrangements of columns

or bracings.

The basic twin pontoon concept still

re-mainS very popular.

Some development has also been carried Out on the

theoretical methods and computer programs used as

de-sign tools.

Many of the available methods for the

cal-culation of wave-induced motions and loads on

semisub-mersibles were also initially developed in the early

seventies.

It is necessary to ensure that these

tech-niques can adequately handle the new generation of

rigs, or alternatively develop improved methods.

The

transit condition is now being paid more attention,

placing somewhat greater requirements on the

calcul-ation methbds.

Existing methods for wave-induced motions and, loads

have mainly been based on a HorizOnS equation approach

(eg. [l-21), or on a strip theory approach t3-51.

3-dimensional sink-source techniques [6-'?j are also

sometimes used, but these generally require

consider-ably more computer resources.

The Morison equation

approach has been very successful for survival and

operational draughts, when experience has shown that

hydrodynamic interaction effects between different

parts of the structure can be neglected with respect

to the motions and overall loads.

The strip theory

approach allows interaction between the pontoons to be

taken into account in the calculation of the motions.

This is useful if the pontoons become very large, are

closely spaced or near the surface.

However, recent

investigations have shown serious shortcomings in the

normal strip theory approach, when internal loads

acting on the structure are required [8-9]

.

Results

obtained using an improved strip theory to overcome

TECHNISCHE WIIVERSITEIT

Laborum voor

Scheepehydmmschaa

Archief

Mekelweg 2,2628 CD Deift

-K. Lindbrg

'& 015-786813- Fax 018.781es8

Gotaverken Arendal AB

Gothenburg, Sweden

radiation (rigid body motion) potential

w

circular frequency of encounter

J. Mathisen

Det norske Ventas Hovik. Norway

A. Borrasen

Det norske Ventas

this problem are presented in the following. THEOBY

The theoretical basis for the calculation of

wave-induced motions and loads on semisubmersìbles is well known

and will not be repeated here.

Nef.

[8] provides

a

review of the available methods. In this paper, an explanation of the present improvement to the usual strip theory for application to semisubmersibles will be given.

The major assumptions applied in the theory are: The structure is treated as a rigid body; i.e. in-ertia forces due to wave-induced motions of the whole structure are included, but inertia forces due to elas-tic deformation of the structure are not included;

Linear

(Airy) wave theory is applied; Cc) Both the incoming waves and the

platform

motion are of small amplitude; (d) The pontoons are long compared to their

beam,

without abrupt changes in cross-section; (e) The fluid is assumed irrotational, allowing potential flow theory to be applied.

The equations of motion for the structure may then he written:

-jut

(A+M) + B + C = e (i)

where A is the added mass matrix, M is the generalised inertia matrix, B is the linear damping matrix, C is the restoring coefficient matrix, including both hydrostatic and linearised anchor line effects, F is the excitation force vector, and r is the body motion vector. The origin of the coordinate system is located in the inter-section of the stillvater and centerline planes, with direction i1 (surge) pointing aft, i=2 (sway) pointing to starboard, and i=3 (heave) pointing upwards, passing through the center of gravity. Directions i1 to 6 (roll, pith, yaw) are rotations about the correspond-ing axes. The axes system is indicated in Fig.2.

It is usual practice to split the excitation force into two parts:

F=F1F

(2)

The force

(F1)

due to the pressure distribution of the undistrubed, incident wave is referred to as the Froude-Kriloff force. The diffraction force (Fn) corresponds to the scattering of

the

incident waves By the

struc-ture.

Note that

both forces

are calculated without

considering motion of the structure.

Hydrodynamic

effects of the motions are taken care of by the

added-nass and damping terms.

Potential flow theory allows the use of

three

poten-tial fields to provide complete information about the fluid flow related to the three corresponding wave syst-ems. The potential due to the incident wave () is

known from classical wave theory. The potential due to the radiated waves ()

is usually determined by

2-D, numerical, sink-source techniques f10]. The diffraction

potential is more difficult to handle directly, but it

can be eliminated to provide

an expression (often re-ferred to as the Haskind relation) for the overall diff-raction forces based on the other two potentials:

ds (3)

where the i-index indicates direction no.i, _{p is the}

density of the fluid,

SV indicates a field of

integration

over the entire wetted surface of the semisubmersible, N is the

_{unit normal}

vector pointing toward the vessel, and ds is the element of surface area for the integration.

Equation 3 supplies the diffraction force acting on

the whole structure. This is sufficient information-for the solution of the equations of motion. However,

more information about the distribution of the

diff-raction force over the various parts of the structure is required for the structural analysis of a semisub-mers ible. Equation 3 cannot be used to provide such information. It is based on a mathematical identity requiring integration over the whole wetted surface of the structure. Application of this identity is not valid for parts of the structure. Attempts to ig-nore this theoretical limitation have been shown to produce erroneous sectional forces in [8,9].

This problem can be overcome be employing an alterna-tive radiation potential

(a)

instead of the normal radiation potential for rigid body motion (). The boundary conditions for this alternative potential are chosen in such a way that integration over the whole structure will provide the contribution of the diff-raction force to the oenterline sectional force. In this case opposed motion of the two pontoons is chosen to give the body boundary condition; eg. when the starboard pontoon heaves upwards, the port pontoon heaves downwards. This does not imply that the actual seniisubmersible behaves in this way, but that the sol-ution of the radiation problem for two pontoons in opposed

motion enables

us to obtain more information about the distribution of the diffraction force on the rigid semisubmersible. The alternative radiation po-tentials are referred to as Opposed Motion Potentials'. This technique was originally suggested by Ogilvie [11]. Mathematical details are given in the appendix.

Numerical Methods. The numerical results presented in this paper have been obtained using computer programs

developed at Det norske Ventas.

The same program [2]

was used to obtain results from application of both Morison's equation and improved strip theory. Further

details of the methods employed are given in f8] , with recent implementation of the opposed motion potentials documented in ref.[l2j . 3-D sink-source results have been obtained using L6]

Nesults in long-crested irregular seas were calcul-ated by the usual linear combination of the transfer function in regular waves with the wave spectrum. The short term response operator presented in the figures is the significant crest-to-trough value

of the response

variable. This is equal to four times the standard deviation of the response.

MODEL TESTS

The

test rig is a new type of semisubmersible rig

designated the GVA O00, designed by Gøtawerken Arendal

AB, Sweden, intended for drilling operations

in the North Sea above the 62nd parallel, or in similar regions. The structural approach utilizes a strong double deck struc-ture to replace the usual complicated configuration of bracings. Figure 1 gives an overall view of the rig,

and its main data are given in

Table 1.

Design Load Cases.

The structural analysis of the rig

was carried out for a number of lomi cases. Each dyna-mic load case was chosen to provide the extreme value of one parameter critical for the structural response,

together with simultaneously occurring values of all

other loads. These load cases may be briefly character-ised as follows:

Static - all stillwater loads.

Longitudinal racking -

maximum longitudinal shear

forces transmitted from the deck to the rest of the structure, due to longitudinal acceleration of the deck including the component of weight due to pitch angle.

trans-Fig.i Cenerai view of semisubmersible GVAOOO The Model. _{Table 1 gives main dimensions of the rig} model. _{Accelerometers were used to measure the motions} of the rig. _{Figure 2 indicates how the model was split} into parts and the sectional forces were measured. The following forces and moments were recorded at the center-line section:

F1 - shear force in the longitudinal direction

F2 - split force in the transverse direction between the two halves of the structure.

F3 - vertical shear force, closely related to the trans-verse racking force (case (e) above).

- bending moment about a longitudinal axis (splitting moment).

3

F5 - torsional moment about a transverse axis (pitch-connecting moment).

- bending moment about a vertical axis.

The horizontal moment axes were located at deck level, with origin on a vertical through the centre of gravity.

At the section through one column at the lower, deck level, forces were measured in order to study the stress in the connection between the column and the deck. Thes forces have not been included in the present paper.

The model was built to a scale of 1:65 in glass-fibr reinforced plastic and aluminium. It ws anchored by 8 chains arranged in a symmetrical pattern with 5 degrees between each anchor line, and with chain weight to the

appropriate model scale.

ig.2 Location of Force Sensors (0U1U2 plane is at stillwater level).

Table 1. Main Data for Semisubmersible at Operating Draught.

Model calculation Test

scale 1:65

-anchor lines (3 inch chain) 8 o

anchor pretension 1280 kN O water depth 195 m 300 draught 20.5 n 20.5 metacentric height 2.1 m 2.76 pontoon length 80.6 n 80.6 pontoon depth 7.5 m 7.5 pontoon width 16.0 n 15.7

distance between pontoon 51.7 m 5I.7 centerlines

longitudinal distance between 514.7 m 514.7

column centers

column diameter 12.9 m 12.9

bracing diameter 3.014 m 3.014

truss height over baseline 11.2 m 11.5 main deck height over baseline 141.0 n ' 37.0

roll 'radius about waterline 29.6 n 29.3 pitch gyradius about waterline 27.3 n 26.9

yaw gyradius 31.2 n '31.2

center of gravity above baseline 20.0 m 19.5

displacement 25307 m3 25320

height of force measurement in

deck 37.0 m 37.0

motions me,sured at centre of gravity water plan verse acceleration and roll.

(t) vertical acceleration - maximum vertical inertia forces acting on the deck structure.

split force - maximum axial forces in the trans-verse bracings due to hydrodynamic forces, on the pontoons. The transverse force acting between the two halves of the structure at a section along the centerline is used as be parameter for this case. pitch connecting moment - causing large bending moments in the deck girders at the connection with the columns. The torsional moment about a trans-verse axis at the centerline section is the critical parameter for this case.

Determination of this type of design load cases is further discussed in [9,13]. The r.odel tests vere designed to

check the values of the critical parameters used for the design load cases.

ce Te5ts

The model tests were carried out in the

_{Laboratory of the Swedish Maritime}

Research Center.

This tank is 88 m long, 39 n wide,

and was filled to a depth

of 3 m, corresponding to

a full scale

water depth of 195 n.

Regular wave tests

were performed

in head, and beam seas for wave periods

of 7,iO,1,l9,21 and 25 seconds

with a Wave height of

7 n.

Irre&'lar wave tests Were carriel out in head,

quarterir.g and beam Seas with both Jonswap and ISSC

type spectra.

Long-crested seas were employed.

The

wave spectra used for

the results presented in this

paper are listed in Table

2.

Draught, anchor

preten-sion and

etacentric height were varied in the tests,

but only the values of these parameters given in Table

1 are employed herein.

The original reports of the

tests are given in

[l!].

Table 2.

Irregular Seas - ISSC spectra

significant wave height (n)

3 6 ' 9

RESULTS

Transfer F-nction - ODerational Draught. Figures 3-5 and

7-9 show coparison5 of results obtained using the

Mori-son eouation approach, improved strip theory(indicated

as

2-D sink-source in the figures) and model tests.

Good agreement is shown between the two theoretical

aptrcaches in all cases.

For very long waves, the heave

response shcws some deviation.

This is probably

intro-duced by the increased importance of damping near

reso-nance, where one method applies drag forces and the other

applies wave-making damping only.

Fair agreement is generally shown between model tests

and calculations at short and moderate wave-lengths,

while large discrepancies Occur in long waves.

Diff-erent water depths in model tests and calculations have

influenced, the results in long waves to some extent.

Anchor line forces are another source for this

discre-pancy.

Anchors are present in the model tests, but not

in the calculations.

In regular waves the wave-induced

drift force

is relatively constant, causing a mean

dis-placement of the model from the stillwater position.

This, in turn, affects the anchor line coefficients,

making then unsymmetrical and increasing the restoring

farces on the side of the incoming waves.

The

unsymme-trical restoring forces induce coupling effects between

the mctions.

Figures 3-5 appear to indicate such

coup-ling of heave Into pitch and surge.

A small hump is shown in the model test results for

the vertical shear force (Fig.8) in long waves.

This

hump corresponds to the roll resonance (not shown)

ob-served in the model tests.

The split force (Fig.7) and bending moment about a

long-itudinal axis (Fig.9) are exceptions to the general trend

of agreement.

The calculated split force has been

checked against a simple, reliable formula with good

agreement.

Sway and roll motion results (not shown here)

at moderate wavelengths also indicate good agreement for

the excitation forces.

The three horizontal force

trans-ducers at the centerline section were arranged in such a

manner that a faulty signal form one of these could have

affected F2 and F9 without affecting F6, corresponding

to the trend in Figures 7,9,11.

Short Term ResDonse - ODerational Draught.

Short term

response operators are presented in Fig.6,lO,ll for

re-sponse parameters whose maxima occur in oblique seas.

Model tests and theory agree well for the longitudinal

force (Fig.6) and the bending moment about the vertical

axes (Fig.11).

The poor agreement for the torsional

moment (Fig.lO) is surprising, since much better

agree-4

ment is shown for the underlying forces, longitudinal

shear force (Fig.6) and vertical shear force (Fig.8).

Alt.

Fig.3

Transfer Function for Surge - Operational Draught

1.5 a 10 0.5 Oc. 2.0 4.0 60 135 A IL Fig.14

Transfer Function for Heave - Operating Draught.

z, a/A .o. MORISON 20-S NK-SOURE ® MODEL TEST

p

® MOSEL TEST - MORISOM ---20 5lNK-5OU(E

Fig. 5

Transfer Function for

Draug

e

e

e

001 -4S LOMO-CRESTES SEAS

e

MOSEL TESTS

- MORISOM

20 SINK-SOURCE s0 7.'S - T, (s

.6

Short Tern Response, Longitudinal Shear Force at Centerline Section - Operating Draught.

0 15 F, BL. 0 10 005 003 F, BL, 0 02 001 ,9O. - MORISON 20 SINK- SOURCE

e

MOSEL TEST

.1 Transfer Function for Split Force at Centerline Section Operating Draught.

90 MCeISON ----20 SINK-SOURCE e MODEL TEST

e

e e

04 1.0 2.0 4.0 13.5

B _{Tranfer Function for Vertical Shear Force at}

Centerline Section - Operating Draught.

Fig.9 Transfer Function for Bending Moment about Long-itudinal Axis at Centerline Section - Operating Draught 0.02 -001 - RO' - MORISON - 20 -SINK- SOURCE MODEL TEST 45 LONG -CRESTED SEAS ® MODEL TESTS - EIORISON 20 SINK-SOURCE LONG - CRESTED SEAS MODEL TESTS M OR SON 20 SINK-0000CI

Fig.11 Short Tern Response, Bending Moment about Verti-cal Axis at Centerline Section - Operating Draught. Transfer Functions - Transit DrauRht. Figures 12-17

show comparisons of results from the improved strip theory (2-D sink-source) and 3-D sink-source

calculat-ions. Reasonable agreement is exhibited in most cases, though the points are a little sparse. The motions at the transit draught (Table 3) are naturally much larger than at the operational draught.

The sway transfer function (Fig.l2)shows a peak at a wavelength (AIL = O.61s) corresponding to the distance between the pontoons.

Pitch resonance (Fig.11s) is slightly displaced in frequency and less severe in the 3-D sink-source calcul-ation. This is due to 3-dimensional effects: the fluid

i

38 5.0

T,(s) 15 0

ig.lO Short Term Response, Torsional Moment about Trans-verse Axis at Centerline Section - Operational Draught.

1O° 15.0

o03

F,, BLM.

notion between the pontoons is allowed to transmit Dut beyond the ends of th pontoons in the 3-D Calculation. This effect is more important for pitch (and yaw) than for the other motions, due to

he long pitch moment arm applicable at the pontoon onds.

At the transit draught, the freeboard is very sml1 'otiOns or waves of moderate amplItude will easily sub-aerge the top of the pontoons. This introduces

signi-icant non-linear effects (e.g. in the buoyancy forces) oat taken into account In the lìnear formulation of the oquations of motion. The results of linear calculations

or transIt are therefore accurate only for relatively omall wave heights. A rough correction to apply these

esults in Somewhat larger amplitude waves may be offected by modifying the notion transfer functions at :he resonant wavelengths, based on experience from iodel tests (cf. _[8,9]).

able 3. Altered Data for Semjsubmersible at Transit Draught.

Lraught 7.2 n

Lisplacement 17200 m3

laterplane area 238 n2

;ransverse metacentric height

8.7 n

ongitudinal metacentric height 186.9 n enter of gravity above baseline 25.3 m oll 'radius about waterplane 33.3 n itch rradius about waterplane 32.3 n

av rradius 31.5 m C O OS 20 -X, 1.0 A - 9O - 20 SINK-SOURCE , 3D SINK-SOURCE 0.0 0.4 10 20 4.0 _AIL NO

Lg.12 Transfer Function Sway - Transit Draught

p go.

-

20 SINK-SOURCE A 32 SINK-SOURCE

6

Fig.11o. Transfer Function Pitch - Transit Draught.

Fig.l5. Transfer Function for Split Force at Center-line Section-Transit Draught.

03 F) Pg BL, 020.1 -o

£

-. 1.0

p

- 20 SINK-SOURCE

£

3D SINK-SOURCE 20 p.90 - 20 SINK-SOURCE A 3D SINK-SOURCE

-80

1O 20 Fig.16. .0

Lg 13 Transfer Function Heave - rÇansit Draught

Centerline Section

Trsit Drught

PgRL 01 00 O,' 2,0 40 _{A /1} 8U p. 0' -20 SINK-SOURCE 15 L 30 SINK-SOURCE 10-A 04 'U 2.0 40 0. 0 A /L

00

10 20 O

Fig.lT.

Transfer Function for Berdin

Mo=ent about

Longitudinal Axis at Centerline Section

-Transit draught.

CONCLUSION

For operational draught:

Good agreement has been shown between the improved

strip theory and the Morison equation ancroach.

Fair agreement has been shown b- model test

results and theoretical calculations.

Peasonable

hypotheses have been suggested to explain the cases

with larger discrepancies.

The Morison equation approach is recormended for

operational and survival draughts, unless there is

reason to expect more hydrodynsmic interaction between

the elements of the structure than in the present case.

For transit draught:

Fair agreement has been shown between the improved

strip theory and 3-dimensional sink-source calculations.

Sectional loads obtained using strip-theory with

oppos-ed motion potentials provide much better agreement than

the usual strip theory in

[8]

.

Pitch motion is affected

by 3-D effects at the ends of the

ontoons and shows

some discrepancy.

Assuming that the 3-D sink-source calculations

pro-vide a valid basis for comparison,

the present results

indicate that the improved strip theory nay be used for

the calculation of motions and loads acting on

semisub-mersibles at transit draught.

Pitch motion is

calcul-ated less accurately than the other motions.

Further

comparison with model tests is desirable.

The

limitat-ions of linear theory are, however, particularly

re-strictive for these conditions, with significant

non-linearities in the buoyancy forces being introduced at

moderate wave heights.

APPENDIX - CTERLINE SECTION DIFFRACTION FORCE

This derivation is only concerned with the

appli-eatlon of opposed motion potentials to determine

_the

Contribution of the diffraction force to the sectional

force at the centerline between two pontoons.

More

details about the other hydrodynamic

forces may be found

in

[8J,

_{where the relevant potentials}

_{are treated in a}

similar mamner.

We consider two geometrically similar vessels, each

having two pontoons, floating at zero speed.

Vessel

A is rigidly fixed in incoming

waves.

The diffraction

7

forces on this vessel will be equal to the diffraction

forces acting on the semisubmersible we are actually

interested in.

Vessel B is forced to perform harmonic

motion with the two pontoons moving in opposite

direc-tions.

Vessel B is not subject to incoming waves.

The

fluid motion induced by the motionofvesse0 B is

describ-ed by the opposdescrib-ed motion potentials.

The assumptions

previously listed are considered to apply.

The incident

(), diffraction (0), and opposed motion (o) potentials

all must satisfy the Laplace equation.

Boundary Conditions. At the free surface, kinematic and

dynamic considerations lead to the classical linearised

free surface condition, which must be satisfied by all

the potentials (,O,o):

+ g

= O

on u

= O (lo)

On the surface of vessel A, the incident and diffracted

potentials must satisfy a kinematic boundary condition

giving zero velocity through the body surface at the

mean position:

on vessel A

(5)

The opposed motion potentials must satisfy a kinematic

boundary condition on the mean position of the surface

of vessel B, giving a normal fluid velocity equal to the

velocity of the hull surface due to the motions:

3h

(x-a.) = - jx.n.

ii

i].

ori vessel B

starboard (SS)

(6)

(xe.) = + joix.n.

on vessel B

3N

ii

ii

port (SP)

(T)

where the generalised normal (n.) is defined by:

(n1,n2,n3) = N and (n,,n5,n)

x N

The bottom condition gives:

(8)

asu

-3u3 3

At large distance from the vessels, both the diffraction

and opposed notion potentials must fulfill a radiation

condition for outgoing waves [151:

hm r (- -jk0) = O

on r = v'u1 + u

(9)

30

where

_{is the velocity normal to the control surface}

in the outward direction.

Determination of the Potentials. The incident wave

pa--tential

may be written:

exp Ck(Ju1 cos

+ ju2sinß +

u3))

The opposed motion potentials may be determined by the

same 2-D sink-source techniques as are normally used

within strip theory [loi.

The body boundary condition

must be appropriately modified, and the symmetry of the

problem is changed.

_{The diffraction potential is not}

determined directly, but is eliminated as imdicated

be-low.

Expression for Sectional Force. The dynamic pressure

due to the diffraction potential may be expressed using

the hinearised Bernoulli equation:

(11)

= j w p e

(12)

A 0

08-- 2

-S..oUeCE A

SX-SuCE

Pg3L 006-00 002 A

['he corresponding forces on the starboard (G) and port :H) pontoons of vessel A are found by integrating the Dressure over the aporopriate areas:

f

Pn1 ds = jwp

f

Sn. ds (13) J.

SS SS

= jwp

f

Sn. ds (la)

1 J.

y consideration of the even and odd portions of the liffraction force acting on either side of the center-Line of vessel A, it is easily shown that the centri-)utiOfl of the diffraction force to the sectional force is given by: D. = (G. - H-) J. 1 1 =

jup[ f

On ds -

f

Sn ds] (15) SS SP

['he opposed motion potential for vessel B may be sub-tituted for the unit normal on vessel A using the body Doundary condition (eq.(6,7)), since the two vessels

re geometrically similar: =

pf

ds -® ds

= -p5 S

ds (16)

result obtainable from Green's second identity is now ieeded. This identity relates volume and surface inte-rals over a closed region for two potential functions

e ,o. ):

f (OV2a

-

e.V20)dv

= f

(5 i - ) ds (17)

y 1 j.

SC 3M

iBN

['he region of integration is indicated in Fig.l8. The murface (SC) enclosing the volume (V) includes the ressel surface (SV = SS + SP), the free surface (SF) [istant control surfaces (SR), and the sea bottom (SB).

q.(l7) is then simplified by making use of the proper-;ies of the potentials. Both potentials satisfy

aplace's equation in the fluid domain, thus eliminat-ng the left-hand side of eq. (17). By invoking the ottom, radiation, and free surface conditions, it can e shown that the surface integral of the right-hand ide of eq.(l7) reduces to an integral over the sur-'ace of the vessel:

f (o

-

0. (18)

SV

ubstituting in the expression for the sectional diff-action force gives:

D. ₌

-p

f

a. ds (19)

1

sv

inally, the body boundary condition (eq.(5)) is applied o eliminate the diffraction potential

D. = sP

f cr

30

as

(20)

SV

C10WLEDGNTS

Permission given by Gøtaverken Arendal AB to present he contents of this paper is gratefully acknowledged.

8

Fig.l8. Region of Integratic:. fcr Application of Green's 2nd Identity.

R'EH EN CES

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Wichers, J.E.W., and De 3com, W.C., 'The Dynamic Loads for the Strength Design of Moored Offshore Struc-tures under Storm Conditions," Offshore Technolo' Conference, paper No. OTO 32L9, Houston, 1978, Vol.111, p.1701.

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FREE oT s

c-o

SURAC POTOOM

BØrresen, R., Mathisen, J., "Strip Theory Methods for the Prediction of Wave Loads on Twin Pontoon Structures Floating Near the Sea Surface," Det norske Ventas, Report No. 81-0571, Høvik, Norvay _Jan.1981. Cansen, C.A., Gundersen, N.B., Gran.'S., 'Envi-ronniental Data jo Operation and Design. Design Case-Mobile Rig," Norwegian Petroleum Society, RØros, Norway, Feb. 1981.

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