Hydrodynamic Coefficients for Calculation Of

ELSEVIER

Marine Structures 9 (1996) 745-758 © 1996 Elsevier Science Limited

Printed in Great Britain. All rights reserved 0951-8339/96/$15.00

0 9 5 1 - 8 3 3 9 ( 9 5 ) 0 0 0 2 3 - 2

Hydrodynamic Coefficients for Calculation of Hydrodynamic Loads on Offshore Truss Structures

Ove T. G u d m e s t a d a & Geir M o e b

aStatoil A/S, Forus, N-4035, Stavanger, Norway bNTH, Trondheim, Norway

(Received 14 December 1994; accepted 24 June 1995)

ABSTRACT

The ctwrent American Petroleum Institute's recipe [API RP 2A WSD, Recom- mended practice for planning, designing and constructing fixed offshore platforms, working stress design. AP1, USA, 1993.]for calculation of hydrodynamic loads on offshore truss structures is compared with the corresponding North Sea Design Practice, as given by the rules of Det Norske Veritas. Most emphasis is put on the hydrodynamic coefficients and the estimation of design current as these issues are identified to be particularly critical

Use of the updated AP1 (1993) recommendations in which the drag coefficient for roughened cylinders is increased from a minimum of O.6 (AP11991) to 1.05 (AP! 1993) and where current is included, could lead to a general increase in the estimated load level on slender offshore structures [Petrauskas, C., Heideman, J.C. & Berek, E.P., Extreme wave force calculation procedure for the 20th edition of AP1 RP 2A. OTC paper 7153, In Proc. OTC 1993, Houston, Texas, 1993, pp. 201'-211].

The main emphasis with regard to the impact of the new API recommendations, howew,r, is that a consistent approach is provided to the calculation of lO0-yr directhmal loads. This includes taking into account the effect of marine growth on force coefficients, modifying the wave kinematics for directional spreading, and considering current blockage effects, conductor shielding effects, and joint occur- rence of wave height and current (i.e., using the associated current as being repre- sentative of the current that would lead to the lO0-yr load).

It is concluded that a consistent approach, such as that underlying the new API RP 2A (1993) re&'pe, is preferable to the current North Sea Design Practice [Det Norske Veritas, Environmental conditions and environmental loads. D N V classification notes 30.5,1991. ] in this field, and thus that t he North Sea Design Practice should.be updated. This relates in particular to selection of hydrodynamic coeffi-

745

746 O. T. Gudmestad, G. Moe

cients. Measurement programmes to obtain full scale global force data simultaneously with wave and current data are furthermore recommended. © 1996 Elsevier Science Limited.

Key words: wave forces, hydrodynamic coefficients, offshore truss structures, API Recommended practice.

1 INTRODUCTION

The International Organisation for Standardisation (ISO) is developing a new world-wide code for offshore structures, supported by USA (API), UK (Health and Safety Executive, HSE), Norway (Norwegian Petroleum Direc- torate, NPD) as well as other countries, w Part 1 (General principles) of this code has already been completed. Currently extensive work on steel jacket structures is underway. In connection with this effort, it is necessary to select the load recipe to use for the calculation of environmental loads.

The state-of-the-art for estimating hydrodynamic loads on truss structures (jackets), assuming quasi-static behaviour, is to use a regular design wave and Morison's equation, 13 involving the following features:

- - w a v e height and corresponding period, and a current velocity; - - w a v e particle kinematics and current velocity profile; - -values of hydrodynamic coefficients.

Since the first API RP 2A appeared in 1969, and the first NPD Regulations and DNV Rules were issued a few years later, design guidelines have undergone changes. However, the state-of-the-art, sea load calculation recipe for jacket structures has remained practically the same over the last 10-15 years. The API approach introduced in 19771 and the so-called standard North Sea Practice for jackets 6 were relatively close until API introduced its 20th ed., 3 with substantial changes in the recommended design practice. In view of this, it is seen prudent to summarise the API changes, to discuss the background for these changes, and to point to the need for harmonisation of the API RP 2A recommendations and North Sea Design Practice, see Table 1.

2 THE API CHOICE OF HYDRODYNAMIC COEFFICIENTS

The present API procedure for the calculation of hydrodynamic loads on slender offshore structures is described in API RP 2A. 3'4'17 The main differences between the present and the previous API procedures 2 are given in Table 2.17 It should be noted that the previous versions of API RP 2A 2 refer to DNV 6 for determination of hydrodynamic coefficients for non- circular cross-sections and for vertical forces on conductor guide frames

Calculation of hydrodynamic loads 747

TABLE 1 Parameters for Calculation of Deterministic Hydrodynamic Loading

1tern AP1 (1993) 3,4 North Sea Design Practice z

Wave kinematics

Wave kinematics factor

Shielding filctor

Current velocity

Hydrodyn. coefficients

Wave hei#tt

Stokes/Stream function

Include, if directional spreading

To be calculated

100 year wave + current load

High drag coeff, value Lower inertia coeff, value

About 1 m lower than for previous API recommendations 2

As API

Normally not included

To be calculated

100 year wave in comb with 10 year current

Low drag coeff, value Higher inertia coeff, value

TABLE 2 20th vs 19th Edition of API RP 2A Wave Force Procedures 2' 3, 4 and Gulf of Mexico Meto-

cean Criteria 17

Consideration 19th ed. 2 20th ed. 3' ~

Wave force procedure

Current

Current blockage factor

Wave period

Wave kinematics factor

Force coefficients

Conductor shielding factor

Marine growth

Gulf of Mexico metocean criteria

Not included

Not applicable

Unmodified

Not applied

Ca = 0"6--1-2 Cm = 1.3-2.0

Not applied (Factor = 1.0)

Mentioned

Omnidirectional wave

Included

Factor of 0.7-1.0 applied to current. Value depends on number of legs and wave direction

Doppler effect used

Factor is 0.88 for hurricanes and 0.95-1.0 for extratropical storms

For drag dominated forces Ca (smooth) = 0.65 Ca (rough) = 1.05 Cm (smooth) = 1.6 Cm (rough) = 1.2

Function of spacing/diameter ratio

Value of 1-5in used to 150ft depth for Gulf of Mexico

Directional with current, function of longitude and water depth

748 O. T. Gudraestad, G. Moe

(Sections 2.3-16.2 and 4, respectively). For global horizontal forces it was required that a platform in the Gulf of Mexico or other US waters was designed for a certain minimum force level, the reference force level (Section 2.3.4g2). This force level was to be achieved by using the reference level wave height, Morison equation with Ca = 0.6 and Cm = 1-5 (for 6 ft outer diameter, and larger Cm for larger outer diameter), zero storm current, and appropriate wave theory. Consequently, if one believed that a lower wave height was appropriate, then it was still required that the same force level be achieved by increasing Ca, using a storm current, etc.

The updated API Recommended Practice is based on a consistent treatment of all variables involved in calculating hydrodynamic load. For a review of selection of wave kinematics models, see Ref. 9. A considerable increase in hydrodynamic loads results from the use of updated hydrodynamic coefficients and inclusion of current, especially if load reducing factors, such as shielding, blockage, etc. 17 are not considered. It should be noted that API 3'4 now assumes that the wave and current loading are based on a joint probability assessment aiming at obtaining the 10 -2 per year environmental loading. 1° Since real ocean waves are directional and irregular, the selection of the regular wave to use in such analyses may still, however, be a matter of some controversy.

Quantification of load differences found by utilizing the 1993 vs the 1991 edition of API RP 2A (i.e., 20th vs. 19th) has been discussed by the API Task Group on Wave Force Commentary. 18 Whether or not the new API recommendations lead to higher or lower forces depends on previous practice and the wave direction. Impact on steel weight for Gulf of Mexico jackets depends on the amount of optimization one might perform with respect to directional criteria. Also, for every one % increase in force, the increase in steel weight is about 0.25-0-20%. For the Gulf of Mexico, the Task Group 3'4 also came up with revised wave heights which were on the order of 1 m lower (for depths > 150 ft) than those determined in accordance with the 19th ed. 2

The net effect on forces was a significant increase in the principal wave direction in deep water, but in some cases broadside forces were lower than found from the 19th ed. For shallow water, the effect is dependent on structure location because the current is spatially variable. For the North Sea, sensitivity studies indicate that the 100-yr force can be significantly lower depending on the choice of wave kinematics factor and associated current than that determined using current UK practice utilizing the 50-yr wave height and 50-yr current criteria. 18

For static regular wave analysis, global platform wave forces can, according to API, 3'4 be calculated by the Morison's formula with particle kinematics taken from a so-called design wave, in combination with prescribed force coefficients. The design wave is usually determined in two steps. In the


first step (called long term statistics) the highest significant wave height and its associated period are predicted from field data, usually based on averages of 20 rain periods, registered at 3 hour intervals. The use of individual storms may be a better strategy, however, since these may be assumed to be statistically independent events. 15'21 In the second step (short term statistics), the expected amplitude al of the highest wave for such an extreme seastate is estimated, assuming linearity, so that the higher peaks will be Rayleigh distributed. For predictions of surface geometry, second order terms may be added, increasing the peaks and decreasing the depths of the troughs, relative to the linear estimate, but the particle velocities seem to be better predicted on basis of the linear amplitude al. An adjustment factor to account for wave directionality is a useful concept and is included in the API recipe which, however, also recommends the use of a regular nonlinear design wave, e.g., a Stokes 5th order wave. An approach that may prove more advantageous is to predict the wave form and the particle kinematics from the statistically based new-wave. 2°' 12

API recommends the following drag and inertia values for unshielded circular cylinders:

Smoot]~ cylinders: Ca = 0.65, C,, = 1.6 Rough cylinders: Ca = 1.05, C,n = 1.2.

These values are said to be appropriate for

- - the case of a steady current with negligible waves;

o r

- - t h e case of large waves with UmoTapp/D > 30

where

Umo = maximum horizontal particle velocity at storm mean water level under the wave crest from a two-dimensional wave kinematics theory 9

Tapp = apparent wave period D = platform leg diameter at storm mean water level.

For wave dominant cases with UmoTapp/D < 30, the hydrodynamic coefficients for nearly vertical members are modified by 'wake encounter'. Such situations may arise with large diameter caissons in extreme seas or ordinary platform members in lower seastates (typically considered in fatigue analysis).

Further details as to selection of Ca and Cm in accordance with the recent API recommended practices are given in Refs 3 and 4, comm. C3.2.7.

For dynamic analysis, the API procedure 3'4 recommends time history methods based on simulated random waves. Frequency domain methods

750 o. T. Gudmestad, G. Moe

may be used for the global dynamic analysis, provided the linearization of the drag force can be justified.

The hydrodynamic coefficients developed for use with individual deterministic waves can, according to API, also be used for random wave analysis (either time or frequency domain) of fixed platforms by using:

--significant wave height

and

- -spectra l peak period

to calculate K, the Keulegan-Carpenter Number. 3'4

3 H Y D R O D Y N A M I C COEFFICIENTS R E C O M M E N D E D BY DNV

The DNV rules 6' 7 represent North Sea Design Practice for the calculation of hydrodynamic loads on offshore truss structures. These suggest tentative values of Cm for different cross-sectional shapes. For circular cylinders the value amounts to 2-0. It is in particular noted that DNV call for use of the selected Cm value in 'Conjunction with the acceleration of water particles as calculated using an appropriate wave theory' (see also Ref. 23).

The selection of Ca values should, according to Ref. 6, take into account the variation of Ca as a function of:

- - Reynold's number, Re; --Keulegan-Carpenter number, Kc; - - roughness number kr/D where kr is the effective roughness height

and D the diameter of member, kr/D = 10 -2 in the absence of more reliable data for marine growth);

- -var ia t ion of cross-sectional geometry.

Tentative values for the drag coefficient for a circular cylinder of varying roughness in steady flow are shown in Fig. 1, while tentative values in the supercritical regime in steady flow for some in-service marine roughnesses are given in Fig. 2. Note that the following values of surface roughness k, could be used in the determination of the drag coefficient:

k, (metres) Steel, new uncoated 5 x 10 -5 Steel, painted 5 x 10 -6 Steel, highly rusted 3 x 10 -3 Concrete 3 x 10 -3 Marine growth 5 x 10-3-5 x 10 -2


1.4

1.2

1.0

o = 0.s

~ 0.6

~ 0.4

0.2

[kdDxl0-3f 3.1 V ~ " ¢ , ~ " ° - " ° " L I 2/" ,( L z . 1.4 , , .°. ."

~ I I I I I I1 I I , , SIm~ltil(kr/D"~';) 1 I I I 10 5 10 6

Reynold's number (Re)

Fig. 1. The drag of sand-roughened cylinders in steady uniform flow.

DNV 6 finally state that 'hydrodynamic drag coefficients for a rough cylinder in oscillating flow are subject to approval in each case'. For a smooth cylinder in oscillating flow, Ref. 6 states that the drag coefficient should not be less than 0.7.

It should be noted that the marine growth should be included in the estimate of the member diameter. In order to reduce the outer diameter and to use the drag coefficient applicable for smooth members, Statoil decided to use antimarine growth coating on their Veslefrikk jacket which was installed in the Northern North Sea. 5

DNV updated their relevant document in 1991. 7 The updated document

1.4

1"2 t 1.0

v

0.8

o 0.6

g04 i 0.2

0 10 -4

: : . ,"

Szechenyi ~ u ~, • :-', ,

:::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ....:.: :+.................. . . . . .-...... : - . : : : : : ~ : ~ : f f ~ . , ~ . : ................ ::::::::!: :iiiii?i: '

. . . . . . . . . Range of 'in service' marine 1, roughness ,.]

106<Re<6x106 [q 5xl0s<Re< I 5x104<Re<6x106 0.j 6x106 :[.~

Achenbach ! t Sand NMI

Simulated marine roughness I I I I I I II1 I I I I I I l l [ I i i i i n iJ

10-3 10-2 i0 -I Relative roughness (kr/D)

Fig. 2. Drag of rough cylinders at high Reynolds numbers in steady uniform flow.


takes into account new information as follows (utilizing member diameters which include growth thickness):

- - it allows the use of Cm values other than 2.0 provided that the chosen values can be justified. Cm values for circular cylinders with in-service marine roughness should, however, normally not be less than 1-8.

It can thus be seen that DNV in general prescribe higher C,~ values than the new API recommended practices. 3'4

For selection of Ca values, it is stated that 'Hydrodynamic drag coefficients for circular cylinders in oscillatory flow with in-service marine roughness should normally not be less than 1-1. The drag coefficient for a smooth circular cylinder in oscillatory flow should not be less than 0-7'.

DNV's tentative values for the drag coefficient as a function of Keulegan- Carpenter number Kc for smooth and marine growth covered circular cylinders for supercritical Reynold's numbers are given in Fig. 3. The figure is valid for a free flow field without any influence of a fixed boundary.

The drag coefficient for steady current is equal to the asymptotic value for oscillatory flow for Kc equal to infinity. For combined wave and current action, the increase of Kc due to the current may be taken into account.

If a deterministic wave analysis based on the 100-year design wave is used to calculate global loads, a reduction in the drag coefficients may, however, be appropriate. In such cases the drag coefficient for circular cylinders is not to be taken as less than Ca = 0.6, where no, or moderate marine growth is considered and Ca = 0.7, where marine growth is considered.

1.5

.~ 1.0

U

0.5

1/20 , ~

1/100 . . . . , N ~

k/D< 1 / 1 0 0 0 0 ~ " ' ° " , , . . . . . . . . . . . . . . . . . . (smooth) ~

Diagram based on logarithmic interpolation between k/D = 1/20 and kiD = 1 / 10000

I I I 0 I 0 20 30 40

K¢

Fig. 3. Drag coefficient ca as function of Kc for cylinders in waves. Re > 5.10-5. 7


Note that this reduction should not apply to the design of individual members.

Reference 7, furthermore, allows for the introduction of group effects (shielding and blockage etc.). It should also be noted that the effect of marine growth on appurtenances such as anodes, etc. should be considered when selecting effective diameters and drag coefficients. Standard industry practice :is to increase the drag coefficient by 7-10% to account for anodes.

From the above it can be concluded that the drag coefficients recommended by DNV in general are consistent with the API recommended practices, except for the acceptance of using a Cd value of 0.7 for rough members in global deterministic wave analysis. If this value is used together with group effects (blockage and shielding) and the best estimate of wave and current kinematics based on joint probability calculations, the global force for a slender offshore structure may be substantially less than found by using the recent API recipe. 3'4 Note, however, that it is not North Sea Design Practice 1:o consider joint probability of waves and currents.

A brief comparative analysis of the loading on a riser platform located in 82 m water depth in the North Sea has been carried out using the input values given in ".Fable 3. The results of this analysis are presented in Table 4.

4 NPD's A P P R O A C H TO SELECTION OF H Y D R O D Y N A M I C COEFFICIENTS

The Norwegian Petroleum Directorate 16 in a commentary to Section 26 state that 'Simplified deterministic analysis based on commonly used hydrodynamic coefficients, may be used for structures, water depths, environ-

TABLE 3 Input Values for Comparison Analysis

Structural system Jackets riser platform

Water depfla Wave height Wave period 10 year return current 0 m-32.5 m elev.

32.5 m-52.5 m elev. 52.5 m-wavecrest

Joint current with 100 yr wave Marine growth + 2 m to 40 below MWL

40 m below MWL to bottom North Sea kinem.factor Current blockage factor

81.7 m (lowest astronomical tide) 27.0 m 14.5 s mean, 12.5-16.5 s (90% interval) 0.55~/s 0"55m/s-0.65m/s (linear extrapolation) 0-65m/s-l.0m/s (linear extrapolation) 0.25m/s 100mm 50 mm

0.95 0.85


TABLE 4 Sensitivity Analysis for Calculation of Loads on a North Sea Jacket Riser Platform

Base shear Overturning load moment

North Sea standard practice 16-5 s wave period 100% 100% API hydrodyn, coeff. 123% 118% +API wave period 14.8 s 113% 115% +API current 0-25m/s 92% 96% +Kinematics factor 0.95 83% 87% API recipe results API, 19933,4 83% 87%

mental conditions etc. if extensive experience shows that the method is applicable'.

In the Guidelines concerning loads and load effects, Ref. 16 in paragraph 4.3.3.1 b states that:

- - If Kc is greater than 2, the wave load can be calculated by means of the Morison formula, with Ca and Cm given as functions of the Reynold figure Re, the Keulegan-Carpenter figure Kc and relative roughness.

According to present North Sea Design Practice, the 'commonly used hydrodynamic coefficients' means the coefficients recommended by DNV 6' 7 (and previously referenced by API2). A clarification of the NPD rules with respect to those updated coefficients presently recommended by API 3'4 is urgently recommended, taking into account that:

- - t h e r e has been no damage to offshore structures in the North Sea resulting from possible inadequate calculation of hydrodynamic loading

- - the new API recommendation practices 3'4 contain a consistent procedure for calculation of hydrodynamic loads utilizing consistent values of wave kinematics s and shielding/blockage etc. together with the most appropriate hydrodynamic coefficients

- - the new API recommended practice assumes that the wave plus current load is found by use of joint probability assessments for wave and current, while NPD require use of a combination of a wave with annual probability of exceedance of 10 -2 with a current having an annual probability of exceedance of 10 -1 for an ultimate limit design code check

- - a n y change in North Sea Design Practise be calibrated with today's inherent safety level in mind, to estimate the safety level for present and new structures.


5 ASSF, SSMENT OF THE PROVISIONS OF THE MOST RELEVANT CODES

The preferred codes are those which use consistent values for all parameters in the load recipe. To use conservative estimates of wave plus current kinematics in combination with low drag factors for deterministic analysis is not considered consistent. In order to assess the safety margins that result when using the latter approach, a full calibration is necessary for any new concept and any new application (waterdepth, etc.). API, on the other hand, suggest an approach which is considered to be consistent. This facilitates application to new concepts and new conditions and also makes code modification easier and is believed to be a good argument for selecting the new API RP 2A recommendations 3 as the most relevant code for estimating hydrodynamic loads on truss structures.

6 ASSESSMENT OF UNCERTAINTIES IN ESTIMATE OR HYDRODYNAMIC COEFFICIENTS

In view of the very large difference between the API recommended hydrodynamic drag coefficient for rough members (Cd = 1-05) and the standard North Sea deterministic global design practice (Cd=0 .7 for rough members), key attention should be focused upon resolving the major differences between the input to the load calculation recipes.

In probabilistic analyses using state-of-the-art values for all parameters together with best estimate of parameter uncertainties, the uncertainty in the API reconamended hydrodynamic coefficients could be established by revi- siting data from relevant model tests. For the drag coefficient a variation of less than 10% is expected.

7 LATEST RESEARCH RESULTS ON USE OF MORISON'S EQUATION AND SELECTION OF HYDRODYNAMIC

COEFFICIENTS

A review of the use of Morison's equation with discussion of the selection of hydrodynamic coefficients was presented by Moe & Overvik. 14 Recent research 22 on marine roughened cylinders also supports the use of the API recommended drag factor. The use of Morison's equation is in general supported even though some reluctance exists concerning some details such as:

- - the automated selection of the 100 year wave and associated period; 11

756 o. T. Gudmestad, G. Moe

- - the uncompromizing adherence to a vector form of Morison's formula, based on instantaneous (free field) wave particle velocities and accel- erates and current particle velocities. (Note that wake formation under unidirectional wave and current is unclear);

- - no special treatment of horizontal members parallel to the wave front.

8 CONCLUSIONS AND RECOMMENDATIONS

It is concluded that the new API recipe 3' 4 for the calculation of hydrodynamic loads on truss structures uses state-of-the-art values for hydrodynamic coefficients as well as for other parameters influencing the load recipe. North Sea Design Practice for global deterministic analysis 7 combines a low drag factor with conservative estimates of wave kinematics and current.

It is strongly recommended that the most relevant values for all the parameters influencing the estimation of loads on offshore truss structures are used. Following this recommendation it will be necessary to carry out careful full scale measurements of global force data simultaneously with wave and current measurements to identify the design (10 -2 per year) loading as well as the wave plus current wave kinematics level which should be used in conjunction with proper hydrodynamic coefficients. It is recommended that North Sea practice be changed to incorporate such measurements and most relevant hydrodynamic coefficients. Dramatic increases in design loads are not expected, although the total loads at certain locations may increase above the design values in current use.

Any attempt to utilise the low drag coefficient normally used in the North Sea Design Practice in combination with the other parameters of the API recipe, 3 i.e., wave kinematics factor, low current value, current blockage and shielding, i.e., parameters from different recipes, may result in an under- estimation of design loads potentially leading to unsafe structures, and this must by all means be avoided.

A C K N O W L E D G E M E N T S

The authors would like to express thanks to Professor T. Moan for his interest in the ongoing work and for his initiative within the ISO organisation to harmonise API recommended practice and Nor th Sea Design Practice for calculation of hydrodynamic loads on truss structures. Furthermore, the authors express thanks to C. Petrauskas for valuable comments and to J. I. Dalane who carried out the riser platform comparison analysis.


R E F E R E N C E S

1. API RP 2A, Recommended Practice for Planning, Designing and Constructing Fixed Offshore Platforms, 9th Edition, API, USA, 1977.

2. API RP 2A, Recommended Practice for Planning, Designing and Constructing Fixed Offshore Platforms, 19th Edition, API, USA, 1991.

3. API RP 2A WSD, Recommended Practice for Planning, Designing and Constructing Fixed Offshore Platforms, Working Stress Design, 20th Edition, API, USA, 1993.

4. API RP 2A LRFD, Recommended Practice for Planning, Designing and Constructing Fixed Offshore Platforms, Load and Resistance Factor Design, First Edition, API, USA, 1993

5. Baerheim, M. & Fossan, T.I., Weight optimization of the Veslefrikk jacket. OTC paper 6189, In Proc. OTC 1989, Houston, Texas, 1989, pp. 689-700.

6. Det Norske Veritas, Rules for the Design, Construction and Inspection of Offshore Structures, Appendix B, Loads, DNV, Norway, 1977.

7. Det Norske Veritas, Environmental Conditions and Environmental Loads, DNV Classification notes no. 30.5, DNV, Norway, 1991.

8. Gudmestad, O.T., Measured and predicted deepwater wave kinematics in regular ancl irregular seas. Marine Struct., 6 (1993) 1-73.

9. Gudmestad, O.T. & Karunakaran, D., Wave kinematics models for calculation of wave loads on truss structures. OTC paper 7421. In Proc. OTC 1994, Hous- ton, Texas, 1994, pp. 413-424.

10. Heideman, J.C., Hagen, O., Cooper, C. & Dahl, F.E., Joint probability of extreme waves and currents on Norwegian shelf. J. Waterway, Port, Coastal & Ocean Engng, 115 (1989) 534--546.

11. Haver, S., On the joint distribution of heights and periods of waves, Ocean Engng, 14 (1987) 359-376.

12. Jonathan, P., Taylor, P.H. & Tromans, P.S., Storm waves in the Northern Sea. In Proc. Boss '94 conference, MIT, 2 (1994) pp. 481-494.

13. Morison, J.R., O'Brian, M.P., Johnson, J.W. & Schaaf, S.A., The forces exerted by surface waves on piles. Petroleum Trans., AIME, 189 (1950) 149- 157.

14. Moe, G. & Overvik, T., The use of the Morison equation, a review of field measurements. In Proc. of E&P Forum Workshop, Wave and Current Kine- matics and Loading, FIP, Paris (1989) pp. 305-332.

15. Nolte, K.J., Statistical methods for determining extreme seastates. In POAC 1973, pp. 705-742.

16. Norwegian Petroleum Directorate, Regulations Concerning Loadbearing Struc- tures in the Petroleum Activities with Guidelines Concerning Loads and Load Effects, NPD, Stavanger, Norway, 1994.

17. Petrauskas, C., Heideman, J.C. & Berek, E.P., Extreme wave force calculation procedure for the 20th edition of API RP 2A. OTC paper 7153. In Proc. OTC 1993, Houston, Texas, 1993, pp. 201-211.

18. Petrau,;kas, C., 19th vs 20th ed. forces. Private communication, Oct. 1994. 19. Thomas, G.A.N. & Thorp, G., The upstream oil and gas industry's initiative in

the development of international standards. SPE paper 23325. In Proc. Asia Pacific Oil & Gas Conf., Singapore, February 1991, pp. 127-136.

20. Tromans, P.S., Anaturk, A. & Hagemeijer, P., A new model for the kinematics


of large ocean waves--application as a design wave. In Proc. ISOPE Conf., Edinburgh, 1991, pp. 64-71.

21. Tromans, P.S., Efthymiou, M., Van de Graaf, J.W. & Taylor, P.H., Extreme storm loading on f ixed offshore platforms. Publication 1102, Shell Research, Rijswijk, The Netherlands, 1992.

22. Wolfram, J., Jusoh, I. & Sell, D., Uncertainty in the estimation of fluid loading due to the effects of marine growth. In Proc. OMAE, Vol. II, 1993, pp. 219-228.

23. Zelt, J., Gudmestad, O.T. & Skjelbreia, J.E., Fluid accelerations under irregular waves. Appl. Ocean Res., 17 (1995) 43-54.

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Hydrodynamic Coefficients for Calculation Of