OTC 7190

Hydrodynamic Aspects of Moored Semisubmersibles and TLP's J.A. Pinkster, Delft U. of Technology; Albertus Dercksen, MARIN; and A.K. Dev, Delft U. of Technology

Copyright 1993, Offshore Technology Conference

This Paper was presented at the 25th Annual OTC in Houston, Texas, U.S.A., 3-6 May 1993.

This paper wes selected for presentation by the OTC Program Committee followlng revlew of information contained in an abstract submitted by the author(s). Contents of the paper, as presented, have not been reviewed by the Offshore Technology Conference and are subject to correction by the author(s). The material, as presented, does not necessarily reflect any position of the Offshore technology Conference or its officers. Permission to copy is restricted to an abstract of not more than 300 words. lllustratidns may not be copied. The absttact should contain conspicuous acknowledgment of where and by whom the paper is presented.

ABSTRACT

The mean and low-frequency horizon- tal wave drift forces .on 2 types of semi-submersible structures in regu- lar and in irregular waves are de- termined from model tests and calcu- lations. For the measurement of the low-frequency drift forces in irre- gular waves use is made of a special dynamic system of restraint. Comparison of measured and computed drift forces in irregular waves show increasing divergence between predictions based on 3-dimensional potential theory and results of ex- periments with increasing severity of the irregular sea conditions. Comparison between computed and measured mean drift forces in regular waves show increasing divergence at lower wave frequencies. A simple model for approximating viscous contributions to the drift forces in irregular waves is applied to some test results and it is shown that the correlatioq between measurements and predictions is improved. In order to gain more detailed insight in the mechanisms of the viscous contribution to the drift force tests were carried out with single fixed vertical cylinder in

regular waves. The results of tests confirm that in conditions of waves without current the major part of che viscous contribution to the drift force is confined to the splash zone of the cylinder.

INTRODUCTION

The motions and mooring forces of Semi-Submersibles and TLP' S moored in exposed locations are often dominated by wave effects. These may be sub- divided in first order wave frequency forces with frequencies corresponding to the individual waves and mean and low-frequency second order wave drift forces related to wave groups. From the point of view of the design of mooring systems both first and second order wave loads and the mo- tion and mooring load response need to be taken into account. At the design stage predictions of these quantities for a particular design can be based on computational methods, model tests or a rational combination of both, Computational methods for wave frequency loads and motion response for semi-submersibles have been in development since the early 70's. See reference [l] . Such methods are

based on linear hydrodynamic theory and have proved their worth an many occasions. Non- linear, mean wave drift forces on semi-submersible type structures can be computed based on the application of linear, 3-dimensional diffraction theory computational methods combined with either a far-field method or a near-field method for the evaluation of the second order wave loads on the structure. In case a far-field method is applied, generally only the mean second order horizontal drift forces can be calculated. See reference [2] and reference [3] , If a near-field or pressure integration method is applied the mean and low-frequency components of the drift forces can be computed for 6 degrees of freedom. See reference [ 4 ] . This type of computational method assumes the flow to be inviscid thus excluding any effects which might arise from separated flow around the structure. In the past efforts have been made to verify the computational methods for the mean and low-frequency or slowly varying wave drift forces on semi- submersible type structures. See reference 151. It has been surmised that the drift forces on semi- submersible type structures, which consist of relatively slender surface-piercing columns and sub- merged floaters, are in some cases significantly affected by viscous effects in the flow around the structural elements. Very little ex- perimental data is available which can give insight in such effects. See reference [10]. In order to increase insight in these effects MARLN, in co-operation with a number of off shore operators, design- ers and manufactures, carried out extensive model 'test programs and computations among others of the mean and slowly varying wave forces on slender and full semi-submersibles. In this paper a number of aspects of this research including the model test programs and the correlation between model test results and

results of computations are discussed. The findings of these studies have confirmed that significant viscous effects can be present in the low- frequency wave forces on such structures. As a result, a research program has been initiated by the Delft University of Technology into determining such effects on structu- ral elements of semi-subrnersibles such as the columns and the pon- toons. The purpose of this research is to determine for which of these elements the viscous effects play an important role and, if possible, to develop a rational computational procedure for taking such effects into account when determining the mean and slowly varying drift forces on the complete structure. In this paper some results of recent model tests carried out on a fixed verti- cal cylinder in waves are presented,

SECOND ORDER WAVE DRIFT FORCES ON A S m I - S U M R I B L E

The subjects of this investigation were a slender 8-column Semi-Submer- sible I with circular columns and a displacement of 23,270 tonnes and a full 6-column Semi-Submersible I1 with square columns and a displace- ment of 56,300 tonnes. The body plans of the semi-submersibles are given in Figure 1 and Figure 2. In the follow- ing all results of measurements and computations will be given for rhe full scale structures. The aims of the study were as Eollows :

-To increase insight in the mean and low-frequencyhorizontal wave excit- ing forces and motion responses of large semi-submersibles -To check the validity of computa- tional methods for the prediction of first order wave frequency mo- tions and low-frequency wave drift forces based on 3-dimensional potential theory.

The total scope of the research does not allow all aspects to be treated here. In this paper the results of the following investigations are presented:

-Results of model tests in regular waves to determine the mean hori- zontal wave drift force response. -Results of tests in irregular waves to determine the mean and low-fre- quency wave drift force records.

The model tests were carried out at a scale of 1: 40 in the Seakeeping Basin of MARIN. This basin measures 100 m X

24 m x 2.5 m.

MODEL TEST SET-UP IN THE BASIN

Measurements of the mean horizontal wave drift forces on a model in re- gular waves can be carried out using a soft-spring restraining or mooring system which consists of horizontal wires incorporating soft linear springs which are connected to force transducers mounted on the model, The mooring wires are connected at deck level. The set-up for tests in regular waves is shown in Figure 3 . In order to measure the mean and slowly varying horizontal wave drift forces in irregular waves, ideally the model should be moored in such a way that all low frequency motion response is suppressed while leaving the model completely free to carry out the motions at wave frequencies. The first requirement ensures that the measured force is not affected by dynamic magnification effects. The second requirement can be deduced from theoretical analysis of the second order wave drift forces which show that part of the total second order excitation forces are directly dependent on the structural motions at wave frequencies. See reference /41. As a consequence, the model re- S training sys tem must possess the characteristics of an ideal Dynamic Positioning system. For the model

tests a system consisting of hori- zontal restraining wires connected to controllable tension winches was selected. See Figure 4. The winches were operated based on an active control system with a feed-back loop supplemented by a feed-forward control loop. See Figure 5. The feed-back loop acted on the hori- zontal position error and the time derivative of the error (Proportio- nal-Differential control). The feed- forward control loop was based on the real-time measurement of the re- lative wave elevation on the up-wave columns of the semi - submersible. 1 t has been shown that a major part of the mean and slowly varying second order wave drift forces as predicted by potential theory, is due to terms related to the square of the instan- taneous relative wave elevation around the waterline of a floating structure. This has been demon- strated, among others, from model tests on a tanker. See reference [G]. Application of feed-back and feed- forward control still does not result in full suppression of low-frequency motions however. This due to the fact that the feed-forward loop is supply- ing an imperfect estimate for the instantaneous low frequency horizon- tal force. As a result, the total restraining force is not equal and opposite to the low frequency wave exciting force thus resulting in residual low frequency motions. See Figure 6. In order to obtain a besr estimate of the total low frequency wave force on the model, the measured restraining force is corrected for the residual horizontal motions of the vessel. This is carried out off- line after a test has been carried out. The basic assumption behind this process is that the instantaneous discrepancy between the true wave force and the measured restraining force results in horizontal motion accelerations which are described by the following relationship:

in which m represents the virtual mass of the vessel and x(t) the mo- tion accelleration. Assuming that the virtual mass is constant , the accele - ration force can be determined in the time domain by passing a double- differentiating filter over the time record of the low frequency horizon- tal motions. The best estimate of the time record of the horizontal drift force then follows from:

An example of time records of measured restraining force, residual surge motion, correction force and total drift force are shown in Figure 7. The results apply to Semi-Submer- sible 11.

In order to verify the accuracy of this procedure model tests were repeated using different settings of the dynamic restraining system. An'example of the results found for the spectral density of the slowly varying wave drift force on the Semi- Submersible I1 in irregular head seas is shown in Figure 8. The results apply to the system of restraint being adjusted to repre- senting a spring system (Proportio- nal control), a spring and damper system (Proportional-Differential control) and a P-D control including Feed-forward based on the relative wave elevation measurements. The results shown in the figure indicate that the spectral density of the drift force obtained from tests with significantly different characteris- tics of the restraining system are reasonably consistent.

TESTS IN REGULAR WAVES

Tests in regular waves were carried out for both semi-submersibles for a range of wave frequencies, wave amplitudes and wave directions. For the slender Semi-Submersible I tests in head seas were carried out with- out and with bracings. The results

are given in Figure 9 through Figure 13 for both structures in head waves and in beam waves as mean drift force transfer functions. In the figures the theoretical values found on the basis of 3 -dimensional potential theory computations ex- cluding the contribution from the bracings are also given.

Comparison between the results of mo- del tests and computations show rhat in the lower wave frequency range the mean drift forces tend to be consi- stently underestimated by the compu- tations. The effect of the bracings on the mean drift forces on the slender Semi-Submersible I in head seas is to increase slightly the mean drift forces as can be seen from the comparison between the measured results shown in Figure 9 and in Figure 11.

TESTS I N IRREGULAR WAVES

Tests were again carried out for both semi-submersibles. The slender Semi- Submersible I was tested with-out bracings .

Results of tests in irregular waves are given in the form of time traces of the measured low frequency drift forces compared with time traces of the corresponding predicted low fre- quency force based on 3-dimensional potential theory. In some cases the spectral density of the computed and measuxed forces are compared. For the time domain predictions, use was made of second order impuls response functions combined with the measured time trace of the undisturbed ir- regular wave record in the basin. See reference 151 . The time domain second order impuls response functions for the drift forces are obtained from the complete second order quadratic transfer functions computed in the frequency domain. The quadratic transfer functions were computed based on the pressure integration method. See reference [ 4 ] .

Tests were carried out in different irregular sea conditions in order to determine the dependency of the cor - relation between computed and mea- sured forces on the sea condition.

Time traces of the measured and com- puted drift force records for irre- gular haed seas are given in Figure 14 through Figure 17. The spectral densities of the computed and mea- sured surge drift forces on Semi- Submersible I1 are compared in Figure 18 through Figure 20. Comparison between measured and computed data show that the correlation is good in low sea conditions with relatively short mean periods. In higher sea conditions combined with corres - pondingly longer mean wave periods the correlation worsens. Beside significant differences in the force peak values, the phase shift between peaks in the measured and computed records increases in higher sea con- ditions. The trend is more or less the same for both types of semi- submersibles.

DISCUSSION OF RESULTS FROK TESTS IN REGULAR AND IRREGULAR WAVES

The results found from tests in regular and irregular waves with respect to the drift force seem to support each other in that in both cases the correlation between measurement and computations worsen with an increase in the wave period. The reduction in the correlation seems to be accompanied by an in- creasing phase shift between measured and computed forces. The peaks in the computed drift forces tend to shift relative to the wave groups. See Figure 17.This is related to the fact that in longer waves diffraction effects which make up the major part of the drift forces in shorter waves are reduced. In longer waves the low-frequency drift forces are to a larger extent dominated by the compo- nents related to the second order

set-down waves present in the irregu- lar wave field. The troughs of these waves, which have periods comparable to the wave group periods, are in phase with the peaks of the wave groups. See Figure 21. The low- frequency wave force components due to these waves are in phase with the horizontal acceleration of the fluid which is largest when the slope of the set-down waves is greatest. This occurs after the peak in the wave group passes the structure. Due to this effect the peaks in the computed wave drift forces in irregular waves with longer mean periods tend to lag behind the peaks in the wave groups. The measured records, however, still shown that the peak forces CO-incide with the peaks in the wave groups. A possible explanation is that in A

longer waves, viscous forces, which are dominated by velocity related effects, are becoming relatively more important. Since the fluid velocities are highest near the peaks in the wave group, peaks in viscous contri- butions to the drift forces will also tend to CO-incide with the peaks in the wave groups. In order to investigate this effect, a simple model for the viscous con- tributions in the horizontal drift forces has been investigated and some further comparisons between the measured and computed drift forces, including viscous contributions, on the slender semi-submersible carried out.

VISCOUS COMPONENTS OF THE DRIFT FORCES IN IRREGULAR WAVES

The model used to describe the vis- cous contribution to the drift forces is based on the assumption that Morison's equation for the drag force on a vertical cylinder in waves can be applied to the surface-piercing parts of the columns of a semi-sub- mersible. See reference [6] through reference [10]. For the case of waves without current it can be shown that, as a first approximation, the viscous

drag force contribution to the drift forces is confined to the splash zone on a column. The viscous drag term is determined from the following equation:

S ( t -1 ' ~ d ( ~ ) = %P Cd '(t) - l v ( t ) l .D dz

0

in which :

v(t) - relative horizontal velocity between the fluid and the column

D - column diameter ((t) - relative wave elevation Cd - drag coefficient

This contribution to the drift force could be evaluated for each column in the time-domain based on the un- disturbed wave elevation record, the frequency domain motion characteris- tics of the semi-submersible and an assumption regarding the drag coef- ficient in the above equation. The summation of the drag force on each column results in the estimated vis- cous drag force contribution to the drift force. The total drift force is found by adding the viscous and potential contributions.

The results of thes computations are shown in Figure 22 for the slender Semi-Submersible I in irregular head seas. In this figure the wave eleva- tion record, the potential part of the drift force and the viscous part of the drift force are shown in the top three traces. The lower trace shows the sum of the viscous force and the potential force compared with the total measured force. It is clear that the result of adding the viscous contribution is a clearly improved correlation with the measured force. In order to show the overall effect of adding a viscous contribution to the potential contri- bution spectra of the low-frequency

surge force in irregular head seas and sway force in beam seas on Semi- Submersible I are given in Figure 23 for three different sea conditions. Each figure shows the drag coeffi- cient Cd used for the computations of the viscous force contribution. The Cd values used for the computations of the viscous contribution in some cases had to be adjusted in order to achieve a reasonable fit with the measured data. This clearly is an unsatisfactory aspect of the simpli- fied model for the viscous effect which will need to be refined in the future. An important effect not accounted for is for instance, the shielding effects due to the proximity of the columns. However, the above results tend to confirm that there is a significant viscous effect in the drift forces on semi- submersible type structures which, in irregular waves without current seems to be concentrated in the splash zone of the columns. The analysis has been based on a rather slmple model for the viscous contribution which has not been verified to any greae extent. In the next section some results of ongoing detailed research carried out at the Delft University of Technology into such effects is described.

VISCOUS EFFECTS I N DRIFT FORCES ON A FIXED VERTICAL C Y L I D E R

In the previous section it was indi- cated that the most significant vis- cous contribution to the horizontal drift force on a semi-submersible seems to originate from the splash zone of the columns. In order to gain more insight in such effects, model tests have been carried out to deter- mine the distribution along the ver- tical of the mean horizontal drift force on a single vertical cylinder in regular waves. The work is part of an on-going Ph.D. project. See reference [Ill. The model tests were carried out in No. 2 towing tank of the Ship Hydro-

mechanics Department. This facility measures 80 m X 2.75 m X 1.25 m and is equiped with a single flap hydraulically operated wave-maker capable of generating regular and irregular waves. The basin is fitted out with a towing carriage with a special low speed carriage control for the simulation of current effects by towing. The model cylinder which had a dia- meter of 0.075 m is shown in Figure 24. At scale 1:100 this could be re- presentative of a column with a 7.5 m diameter. The splash zone and the sub-surface part are independently attached to a central core through force transducers measuring the horizontal force on each of the two sections. Model tests were carried out in regu- lar waves with and without current. For each test the vertical position of the cylinder was adjusted so that the through of the wave passing the cylinder passed just above the sepa- ration between the splash zone part of the cylinder and the sub-surface part of the cylinder. This ensured that the sub-surface part of the cylinder was fully submerged at all times. Results of measurements in regular waves without current of the mean horizontal drift force on the splash zone and the sub-surface zone are compared with results of calcula- tions of the relevant contributions to the drift forces based on 3-dimen- sional potential theory and the ap- plication of the pressure integration or near-field method in Figure 25 and Figure 26 respectively. According to the near-field theory for drift forces, the splash zone contribution is dependent on the square of the relative wave elevation around the cylinder while the drift force on the subsurface element is due to the non-linear pressure con- tribution in the Bernoulli pressure equation. For this reason the results of mean force measurements have been divided by the square of the undis- turbed wave amplitude. Results are

given for the model scale. The model tests were carried out for a range of wave frequencies corres- ponding to the longer waves for a semi-submersible. At scale 1:100 the wave frequencies tested in the model correspond to 0.3 r/s to 0.8 r/s at full scale. This is a range of frequencies relevant for extreme sea conditions. The results shown in Figure 25 and Figure 26 confirm that the greatest discrepancies between the poten~ial computations and the measurements of the mean forces are found for the splash zone of the cylinder. The measured mean forces are consistently significantly larger than the com- puted values. For the sub-surface part of the cylinder, differences also occur between measurements and computations. In a relative sense they appear to be of the same order as for the splash zone part. However, the absolute value of the forces is considerable lower and the differen- ces between measurements and compu- tations are less consistent. It can be concluded that these model tests poit to the splash zone contri- bution to the viscous part of the mean drift force as being the most important one.

FINAL -KS

In this paper we have shown some results of an extensive series of model tests on two semi-submersibles which confirm differences between computed and measured mean and low- frequency horizontal wave drift forces in regular and irregular waves . Application of a simple model for the viscous contribution to the drift forces indicated that irregular waves without current the major source of the viscous contribution was to be found at the splash zone part of the columns of a semi-submersible. Model test in regular waves with a fixed vertical cylinder representing a single column of a semi-submersible

or a TLF confirm that the largest discrepancies between computed and measured drift forces are indeed to be found in the splash zone. Further experimental investigations are required in order to be able to formulate a more detailed model for the viscous effects which can also take into account such aspects as- the interaction effects due to the proxi- mity of the columns of a semi-submer- sible .

REFERENCES

[l] Hooft, J.P.: 'Hydrodynamic Aspects of Semi-Submersible Platforms', Publication No. 400, Netherlands Ship Model Basin, 1972

[2] Newrnan, J.N.: 'The Drift Force and Moment on Ships in Waves', Journal of Ship Research, 1966

[3] Faltinsen, O.M. and Michelsen, F.C.: 'Motions of Large Struc- tures in Waves at Zero Froude Number', Symposium on Marine Vehicles, London, 1974

[4] Pinkster, J.A.: 'Low-Frequency Second Order Wave Exciting Forces on Floating Spructu- rest, Publication No. 650, Netherlands Ship Model Basin, Wageningen, 1980

[5] Pinkster, $ . A . and Huijsmans, R.H.M.: 'The Low Frequency Mo- tions of a semi-submersible in Waves', Boss182, Boston, 1982

[7] Huse, E.: 'Wave induced Mean Force on Platforms in Direc- tion Opposite to Wave Propa- gation', Norwegian Maritime Research, Vo1.5, No.1, 1977

[8] Standing, R.G., Brendling, W.J. and Jackson, G.E.: 'Full- scale Measured and Predicted Low-Frequency Motions of the Semi-Submersible Support Ves- sel 'Uncle John", First In- ternational Offshore and Polar Engineering Conference, Edinburgh, 1991

[g] Ferretti, C. and Berta, ET,: 'Viscous Effect Contribution to the Drift Forces on Float- ing Structures', International Symposium on Ocean Engineering Ship Handling, Gothenburg, '80

[l01 Chakrabarti, S .K. : 'Steady Drift Force on Vertical Cylin- der - Viscous vs. Potential', Applied Ocean Research, Vo1.6, No.2, 1984

Dev, A.K.: 'Experimental In- vestigations of Viscous Mean Drift Forces on a Fixed Verti- cal Circular Cylinder in Waves and Currents Part I t , Report No. 928-M, Ship Hydrodynamics Department, Delft University of Technology, 1992

161 Pijfers, J.G.L. and Brink, A.W.: 'Calculated Drift Forces of Two Semi-Submersible Plat- form Types in Regular and Ir- regular Waves' , Paper No. OTC 2977, Offshore Technology Conference, Houston, 1977

Flg. l-General arrangement of Semisubmerslble I.

Dimensions are given in metres

Fig, 3-Test setup for tests in regular waves.

Fig. P-General arrangement of Sembubmersible 11.

Fig. 4-Test satup for tests in irregular waves. Fig. S-Block diagram of control syatem for tests in irregufar waves.

V.ss.1

- RorLrontd aerionr c +

r t l l t i v e wave e levat ion

Feed-forwrrd antral Relative

system wave c lev . L

4- = 3.1 m S T1 - 7 . 0 . dir - lao*

Feed-back

Fw (Wave d r i f t force)

Fig. 6-Block diagram of forces acting on the structure.

Vessel: Mass m d damping characteristics

r ~lasiduai surge motions o'?.==--*.../ -- P , r-. -

-' L

control system

Motions *

10. I

LODMl

tf ,[ k y e d restraining force a

i! H ~ V I dirmctlon 1.50 d.qel.s U'-

1 ! 1,

W.". charact*rlstlcs: twL,, - 5 . 0 m \! I

f, . 11.30 ' 1OO.W ' Correction force b - m . f

System OZ ~.stzmint baled upon: - springs V \ , j " . *- -----m- sprlnqa + motion

i' \p " '-

-- rprlnqs + mtlan + relative wave

Horizontal mccLon

F ( F o r a from control

Fig. I-Spectra of drift forces obtained for different restraining system characteristics for the Fig. 7-Example of (a) measured restraining form, (b) cormction force for mottons, s m a aer conditions. and (a + b) total drift force record.

Fig. 9 - Mean Surge drift force on Semi-Submersible I in regular head waves

Fig. 10 - Mean sway drift force on Semi-Submersible I in regular beam waves

F i g . 11 - Mean surge drift force on Semi-submersible I in regular head waves including effect of brrcingr

10000 I - Measured -- Calculated

Flg. 18-Spectral density on surge drift force of Fig. 16 on Semisubmersible II.

Flg. 20-Spectral denrity of surge drlft force of Fig. 17 on Semi13ubmerblible 11.

Fig, 19-Spectral density of surge drln force of Fig. l6 on Semisubmeraible 11.

loo00

Fig. 21-Wave set.down in Irregular wave..

25.00 Force Calculated (potential parr) 1

t f 0 1 L

! N

t - 3 - h!

V)

0 0.25 0.50

I - Measured --.- Calculattd

-25.00 J

1 Force

-

- Musurd

Calculated (viscous ~ a r t f b

-

Force ,Total calculated a + b ..;&P--

-25.00

Fig. 22-Low-frequency surge drift force on Semiaubmersibia I In irregular heed rear,

H c u u r d durinq model test

-- calculated, pot*ntial coneributlon only

A---- caJ.cular.d, VIS-I .ff.Ct 1nclud.d

Hsad seas bean reas

w in rad/s w i n rrd/s

Flg. 23-Spectte of low-frequency drift forcbs on Semlsubmmlble I.

U. m e on-

,- W M .

Fig. 24-Arrangement and model netup of fixed verticel cyllnder.

0 0 1 2 3 4 5 6 7 %

OMEQA lR/S1

MEAN DRIFT FORCE INIM'21

a MEA8URED(I) MEABURED(H1

- POTENTIAL THEORY MIASURPO(L)

$00

80

B0

40

Fig. 25-Mean drift forcer on the splash zone part at the cylinder.

. .... . . .. IHJ . HIOHEIT *ET OF w r AMP. 111 - INTERMCOIATE SET O r W€ IUI ILJ LOWEST SET O r W E AMP.

-

- I : i o b 6 ;

! -

!

MEAN DRIFT FORCE INIM"21 -- IHJ . HIPHEST SET OV W E AMP. '

111 INTERMEOIATE SET or WE *UP. : ILJ . Lowest ac t of nur Aur.

0 1 2 3 4 6 8 7 8 OMEQA IR191

0 MEASURED(I) MEASURED(H1

POTGNTIAL THEORY MEASURED(L)

Fig. 26-Mean drifl forces on the subsurface part ot the cylinder.

Fig. 12 - Mean surge drift force on Semi-Submersible I1 in Rgular head waves.

Flg. 13-Mean away drift f o f ~ e on S~mlsubmenlble II In nguiar beam waves.

5.00 7 wave I / , , I ) I , ! . , ,

0

-5.00

Fig. 15-Low.frequency aurge drift force on Semlsubmersible II in imguirr head seas-Ha-3.1 m, T, -7.1 a.

5.00 7 Wave 4 m - 3-09; - 7.12

m 'U'

0 50 LOO t in seconds

Fig. 14-Low-frequency surge drlfi force on Semiaubmenlbi* I In irregular head seas.

10.3 m - M c u u d -- calculated bCr = 180.

100.00 ; Fbrce : '.

Fig. 17-Low-frequency surge drift force on Semisubmersible II In Irregular head seaaHa- 10.3 m, T, = 14.5 a.

tf 1 Force

-,00,00 ,pi - - T-J-.,;

Fig. 18-Low-frequency surge drift force on Semirubmersible II in irregular head war-Hs m 6.5 m, T, X 11.3 S.

614