An introduction to the mathematical modelling of the behaviour of floating offshore structures in marine environment

NATIONAL RESEARCH COUÑCIL CANADA CONSEIL NATIONAL DE RECHERCHES CANADA

ARCTIC VESSEL AND MARINE RESEARCH INSTITUTE

L'INSTITUT DE RECHERCHÉ MARITIME ET SUR LES NAVIRES ARCTIQUES

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AN INTRODUCTION TO THE MATHEMATICAL MODELLING OF THE BEHAVIOUR OF FLOATING OFFSHORE STRUCTURES IN MARINE ENVIRONMENT

29 August, 1983

Author:

r.ing. Jacek, S. Pawówski

Approved:

a.

D.C. Murdey

1. Introduction

i i Typical Offshore Structures

TABLE OF CONTENTS

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Page

i

1.2 Some Problems of the Design of

Semisubmersibles 6

Mathematical Modelling of the Behaviour of

Floating Offshore Structures in Waves 12

2.1 Fundamentals of the Mathematical

Modelling of Physical Objects 12

2.2 Existing Structural Models òf Floating

Offshore Structures 14

2.3 Core. Hydrodynarnic Programs 20

Conclusion 34

1 INTRODUCTION

1.1 Typical Offshore Structures

During the last decade, the exploration of oil and gas resources from the seabed has experienced a very rapid development. The number of. mobile drilling units

(MODUs) has increased from about 200 in 1970 to 450 in 1980 and above 600 in 1982; and on an average, the rate of growth amounts to 9.8%. The fleet of MODUs comprises four major types of units: jack-ups (about 430 in 1982), semisubmersibles (about 120 in 1982), drillships and drill barges (about 80 in 1982), and submersibles (about 20 in 1982),. [2]

Typical. operations performed by MODUs as vessels are transit and station keeping The main difference bet-ween the various types of MODUS listed above is that

semisubmersibles and driliships remain afloat during station keeping whereas drill barges, jack-ups and submersibles are grounded in order to perform the. same operation Drilling barges have flat bottoms, they are towed to the location and then ballasted o the Seabed (in iater depths up to approx. 9 m). Jack-ups consist of a floating deck structure. which is equipped with legs and can be raised by them to a sufficient height above sea level. at the location. Submersibles are principally operated in the Gulf of Mexico, they consist of a fixed height deck structure supported by several columns which are attached to a barge or pon-toon construction. On location, the submersible is ballasted down to the seabed. Drillships can be con-rn sidered as developed from drill barges and they

provide better mobility and increased water depth cap-ability. The semisubmersible drilling units (SSDUs) evolved from submersible designs.. They consist of a large deck structure supported by a number of columns attached to underwater displacement hulls On loca-tion, these hulls are positioned at approximately 16 m below. the water surface in order to minimise, the

interaction of waves with the structure. Driliships and SSDUs maintain position by means of adequate'

1

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moöring and/or dynamic positioning systems

Good water depth capabilities of SSDU5 ccmbined with their good motion characteristics in waves, which pro-vide a wide weather window for drilling operations,

[2], Ç3], and their good mobility explain the follow-ing rates of annual growth of the numbers of various units between 1970 and 1982, E2],

Submersible +1 .3%

Jack-up +12.3%

Drillshjp & +3.8%

nil barge

Semisubmersible +14. 0%

(thrusters) The water depth capabilities of various MODUs are shown below for guidance (they depend on environmental conditions), t21: Submersible 3 to 46 m Jack-up 3 to 107 in Dril.lship (moored) 30 to 460 m Drillship (d.p.). 122 to 1830 in Semisubmersible (moored) 66 to 610 in Semisubmersible (d.p.) 122 to 1830 m

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3

During 1983 the number of SSDUs will increase

approximately by 30%, [2] These statistics reflect

the advantages that semisubmersibles display in harsh environmental conditions and deeper waters, as the exploration of seabed resources of jdrocarbons

spreads to such aréas as North Sea and North Atlantic, and. the waters of South America and South East Asia, although their costs per Unit are high (about $1OO million or more per unit in comparison wlth a jack-up of the cost around $40 million, _{E3 ]).}

The exploitation öf hydrocarbon. resources from the seabed is typically devided into three phases, _[3] Initially a geophysical survey and exploration are carried out, often by converted trawlers, which are followed by exploratory or wild-cat drilling either by drilling ships or SSDUs. Due to their large deôk load capabilities, drillships are suitable for operations

in deep waters and remote areas, whereas an SSDU requires the support of two or three handling/tug/ supply boats and a continuous logistic suppört. When the exploratory drilling has confirmed a conmercial potential of the resource, a whole system of the exploitation of the resource is developed. Such systems are most often based on periitnent drilling units of fixed or compliant kind The fixed struc-tures are steel-template jackets or concrete gravity-base platforms, these can achieve very considerable dimensions and water depths (weights öf 41,000 and 898,000 tons respectively) at the depths of the order of 150 m, [4]. First permanent drilling platforms of

the compliant type are the Hutthn tension leg platform (TLP) for the North Sea (depth 148 m) and the Block 280 guyed tower for the Gulf of t4exicö (depth 300 m). The total costs of biggest permanent structures varies

from about $1 billion to above $2.5 billion, _[4]. Of these kinds of structures, only TLPs are floating structures. _{A TLP consists of the floating hull which}

is much larger than that of a semisubmersible but of very similar geometry. _{The hull is pulléd down at} each corner by an array of highly tensioned vertical tethers, made of high-tensile-strength steel the other ends of which are' fixed tò the seabed. The

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tethers are pre-tensioned so far that they never go slack éven, in the maximum waves to be encountered in every loo years Due to this construction, a TLP

moves in waves thereby diminishing wave forces exerted upon the floating hull (in the instance of the Hutton platform the departures from the vertical are designed

to reach maximum välues of. 24 ñ., [4])

A system of offshOre structures for the exploration and/or permanent exploitation of a seabed resource is supported by a flotilla of service support vesssels,

[3], such as supply, small crane, diving support and rescue vessels Besides, storage and hotel vessels of semisubmersible type are employed. The cônstruction of a permanent exploitation system requires the

employment of large transport barges and crane vessels (in order to transport and position jackets, top-side units, etc ) The exploitation system often includes

loading terminals for tankers, which usually tnvolve a single buoy or an articulated buoy construction. ànd a storage tanker, thereby representing a systth of

mechanically connected floating bodies.

Oñ the basis Of the foregoing brief discussion of typical offshore structures and systems, the hydrody-namic problems inherent in their design, construction and operation can be classified as the ones of the interaction of water waves and urrents, in the pre-sence of effects of other environmental phenomena such as winds and ice, with:

fixed structures of template-jacket type., fixed structires ôf gravity-base type,

C) floating structûres f seinisubmersible and TLP type,

ships and barges at zero añd small fôrward speeds, systems of mechánically connected floating bodies. The prob1em a) and b) both cöncern fixed structures, whose displacements with respéct to the reference

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configutation of static equilibrium are of structural elastic origin. The difference between the two pro-blems is geñeraily due to the different ratio öf the characteristic dimensions (in the direction of flow induced by the waves) of structural members, with res-pect to orbital displacements of water partiçles in waves. These ratios are usually small for the type a) of struc.tures and therefore hydrodynamic loads ön the structure can well be predicted by Morison's formulai see e.g. _[5]. För the structures of the type b), the ratios are usually large which. makes the diffraction aspect of the interaction of the structure with waves primarily important and.invalidates the application of Morison's formula, see e.g. [6] and [7].

For the structures of the type e) and e) both sthall and large ratios described above occur. The small ones normally apply to such structural members of semisub-mersibleS as brases and ttusses whéreas large values correspond to the flow around displacement hulls. The same situation occurs for various structural members of TLPs and structures of the type e) which can be coni-idered as instances of the structures c) and d)

combined.

Floating and compliant structures change their confi-gurations in waves not only due to structural

elasticity but also in the rigid body modes öf motion which are not suppressed by thecönStr'aints. _These

latter displacements with respect tö the mean

configu-ration are as a rule anorder of magnitude larger than

those resulting from elastic distortions and, owing to their relatively small frequencies (frequencies of encöunter which are equal to the frequencies of the oncoming waves for a structure of zero mean speed), they give rise to so called radiation effects

(spread-ing öf gravity waves in water due to the motions of _the structure) The radiation and diffraction phenomema are of particular importance if strudtural members of considerable displacement are positLoned in the

proxi-mity of or at the free surface _{Significant waterplane}

areäs, if they occur, produce largehydrostatic forces

as a result of rigid body motions in the vertical plane. In the instance of structures of the type e) nonlinear, with respect to the motions and oncoming wave elevation, mean and slowly varying forces become important

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The problems of the interaction of waves and currents with the structures of the type d) are closest to

classic seakeeping problems of naval architecture; viscous effects are important for the roll mode of motion and can be critical for stability in that

mode. The interaction with waves of semisubmersibles in transit condition falls within the same òlass of problems.

The above rudimentary review of the hydrodynamic prob-lems related to the design, construction and operation of typical offshore structures indicates that most of

these problems must be dea.t with in connection with the modelling of the behaviour in waves of SSDUs. Besides, the quoted above statistics of the growth of

the offshore fleet show the practical importance of this type öf offshore structures. Therefore a

compilation and development of. a hydrodynamic model and corresponding software applicable to the

investigation of the behaviour in waves of

semisubmersibles seems to be most appropriate, bóth from the scientific and practical.standpoint, as a starting point for the mathematical modelling of hydrodynamic properties of offshore structures.

1.2 Some Problems of the Design of Semisubmersibles The purposes of mathematical modelling in technical sciences are. manifold and will be discussed to some extent in the next section. One of the most important of them is to assist the designer in the decision

makin9 process of designing a new physical object, see e.g. _L8J. Hence, in the context of the present report it is wörthwhile to review some of the current pro-blems of the design of semisubmersibles.

Semisubmersibles represent.a relatively well developed type of MODU, the greatest advantage of which is its generally good performänce in waves, in the sense of the induced motions (in particular in the heave mode),

for waves of periods T 20 sec., [9]. This charac-teristic is of consïderable importance for operations in harsh environmental conditions, e.g. a compärative study indicates, [3], that the number of workdays per year in Northern North Sea (so called weather window, within which operating is possible owing to

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designs of a crane vessel, a semisubmersible, ship and flat bottom barge, are respectively 320, 110 and 80. A typical strudtural donfiguratioh of aSSD(J, [3], consists of the deck structure supported by 2, 3, or 4 pairs of columns. At their lower ends, on port and starboard, the columns are connected by twinparalle.i pontoon hulls of the length of 8.0 to 120 m. he

breadth to depth ratios of the pontoons vary between

1 1 to 2 6 Columns are usually circular but

dia-meters may differ between corner columns and those arouñd, amidships. Vessels are usual1 desiqned to Operate at three principal draughts, L3]:

a transit draught where the decks of the pontoon hulls are just clear of the water with a freeboard of betweeh 0.5 to i m (transit speeds may be of the

order of 12 knots),

a drillingdraught, usually of 18 to 24 m, when the

vessel is fully stored with drilling materials and consumables and with ballast water in the pontoons,

C) a survival draught which is about 4 m less than the operating draught. and is designed to keep the deck platform above storm waves.

Variable deck load (VDL) cönstitutes the principal parameter on which the design of an SSDU is based, usually it is defined as "the weight which can be car-ried at a point located a small distance above the

main storaqe area (typically 5 ft. abovethe pipe

racks)t1, _[2]. VDL varies between 2,00.0 to 4,000

tonnes, depending on designs, _{[2], [3]} A semisubmer-sible can also be described by the, value of variable payload

which

_{includes consumable items stored below}

pipe racks, such as fuel oil, bulk materials, potable

water, etc.and therefore represents a greater figure

then VDL, _[2]. VDL and variable payload figures determine how often the vessel must be supplied ánd. hence they decide about'.the amount of logistic support

required by the vessel However, VDL values are limited by the stability of vessel. Besides it must be possible tO achieve the survival draught rapidly if a storm approaches and it is a common practice to dis-charge overböard sufficient quantities of ballast and consumables in order to obtain the necessary change of

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draught. Therefore procedures are èlàborated which dictate how these operations should be performed with-out violating stability criteria, [3]. The necessity of discharging consumables overboard has the obvious adverse effect of increasing the demand for logistic support.

Taking into account the description of MODUs presented in the preceding section and the above discussion of the VDL and/or variable payload factor,. the opera-tional capabilities of an SSDU, apart from mobility, can be specified by the following principal characte-ristics (which are subjected to appropriate additional requirements for storage capacities and equipmen.t

sys-tems):.

weather window (dependent on enviromental

conditions and motion characteristics), variable deck load and/or variablé payload

(dependent on. the stability of the vessel),

C) water depth (dependent on mooring .and/òr dp system capbiiity).

The majorcompromise in the design of an SSDU must be

made between the weather windöw and variable deck load characteristics This results from the very principle on which the. concept of a semisubmersible is based

i.e. that of positioning the voluminous members of the displacement part of the hull as deep under the

waterplane as possible thereby improving the motion characteristics. However such a solutiôn produces

small waterplane areas and is strongly disadvantageous for stability The importance of this compromise is greatly amplified by the recent tendency to increase

considerably the VDL values (up to about 10,000 tonnes, however, primarily for semisubmersible

floating production facilities, [2]) and to eliminate the di.fference between drilling and survival draughts without a reduction of payload, e.g. [io], [ii], in

order to increase endurancé without supply. On the other hand, it is recognized, e.g. [2], [12], that a

number of form factors, such as: shape of pontoon hulls, spacing between pontoon hulls, number of

columns, spacing between columns, column diameters and form, have a significant effect, on the characteristics of motion in waves and besides must be 'decided upon in

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preliminary stages of design, e.g _[io]. It follows that the availability of appropriate hydrodynamic models is of primary impörtance for the designing of the form of an SSDU.

The geometry and dimensions of a sexsubmersib1e, in

conjunction with its mass distribution, also determine the structural loads on the vessel for given environ-mental conditions. Pârt of these loads appear direct-ly as a result of fluid exerted stresses on the sur-face of the vessel and others follow from induced motions in the form of inertial and time dependent weight loads, see e.g. _[12]. Therefore., essentially a hydrodynamic model, leads to the evaluation of the

loads and a structural analysis model is employed to determine the resulting stresses. The stresses are examined against two main criteria, i.e. the extreme stresses and fatigue effects, see e.g. [i], _[12],

[13], [14 ], _[15]. Apart from its safety aspects, the

proper prediction of fatigue effects influences costly inspection, maintenance and repair activities.

The significance of elastic properties of the vessel structure for the prediction of loads at higher fre-quencies of excitation (which correspond to elastic distortion modes of the vessel) can generally improve the evaluation of stresses and fatigue phenonena,

[16]. Therefore, in their full versatility, hydrody-namic models of the behaviour of floating offshore structures in waves should evolve in the direction of hydroelastic, see [17], models. Such a development is also full.y justified by the logic of sciencè which. requires the unification of at first separate models of physical phenomena.

Safety aspects of the design of major technical con-structions are of paramount moral (including the pro-fessional integrity of the designer _{and researcher),} legal and economic significance. Although there appear arguments, see... [9], to the effect that SSDUs are relatively safe in comparison with other offshore structures and ships, the fact remains that what is sometimes called rather euphemistcally a mishap, [3], experienced by an SSDtJ or other semisubmersible often amounts to a major disaster, as the recent accidents. of the Alexander Kielland and the Ocean Ranger

exemplify, see eg. [9]. _{Owing to the numbér of}

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(e.g. in severe storm conditions and icy watersof the North Altantic), the character öf the operations

(drilling, often in remote areas) and sizé and öost of the vessel, any signifi.cant.accident concerning an SSDU creates a serious threat of the loss of human lives and property, brings the risk of an ecological

disaster andimplies the necessity of dangerous and

cöstiy salvage operations and/or, if the worst comes to the worst, disposing of the wreck which may create a navigational obstacle (as e.g in the instance of the Ocean Ranger).

Over the last 15 years, rules and regulations applying to the design and construction of, MODUs have been

developed with the purpose of assuring a fundamental standard of safety. These rules and regulations are formulated on three levels, in the manner similar to the formulation of rules and regulations concerning the design and construction of ships. Firstly, the rules of the Classificatiòn Societies (such as

American Bureau of Shipping, Bureau Ventas, Det Norske Ventas, Lloyds, etc.) which are of primary

concern to the owner of a newly built véssel and are related to the insurance. Secondly, the régulations of the Governmental Authorities which apply to vessels operating. in the areaé within a country's

jurisdic-tiori. Usually, the owner will stipulate approval by

one Classification Society and at leaSt tw different governmental regulations in order to maintain opera-tional flexibility. The highest level of Safety regulations for ships are International Conventions

(e.g. SOLAS for ships), prepared by 1MO (an agency of

UNO) or Codes, formulated by the same örganizatio,

which, although they do not have the legal power öf Conventions, greatly influence design and construction practices thröughöut the world, as they give good

gu.idélïnés and potentially can be included in part or as a whole in future Convetitiòns. At present there

are no International Conventions applying directly to the design and construction of offshote structures, there. is however 1140's MODU Code and the formulation of an International Convention öf that kind seems

inevitable in the future.

For any floating structue the fundamental prerequi-site of safety is the capacity to remain afloa.t and maintain a configuration with respect to the free

surface of water, which prevents the loss of lives and/or of the vessel, even if the usual operátional

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characteristics of the vessel have been impaired owing to severe environmental cönditiöns, damage or malfunc-t:ioning of vessel's sytems (such as e.g. the ballast-ing system). This principle has long been recognized in naval architecture and recently, partly at least as a result of bitter experiences, is being forcefully introduced into the regulations concerning SSDUs, see e g _[15] Typically the principle is expressed in terms of the requirements to fulfil intact stability, damage stability and reserve buoyancy criteria, see

e.g. _{[2], [15].} All these criteria are interrelated

and can be fully considered only by taking into

account the interaction of the vessel with waves and currents, Sand therefore their proper formulation

requires the employment öf hydrodynamic models of the behaviour of semisubmersibles in waves. Taking into account the character of the criteria, the models should reach beyond the approximation of small

harmonic motions in order to be useful for the inves-tigation of evolution of the vessel configuration

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- 12

2. MATHEMATICAL MODELLING OF THE BEHAVIOUR OF. FLOATING

OFFSHORE 'STRUCTURES IN WAVES

2.1 Fundamentals of the Mathematical Modelling of Physical Objects

In general the. process öf mathematical modelling òf a physical object can be considered, see [8], as

composed of the following consecutive stages:

a) the definition òf the object of the process, 'b) the creatior of the empirical relation

system,

the creation of a formal, mathematical system which corresponds to the empirical relätion system.

The formal mathêmatical System, expressed in a closed or aÏgorithmic formE constïtutes the mathematical model of the physical object, if it is understood in terms of the correspondence to the empirical relation system.

I should be pointed out that the physical öbject

cannot be identified by the time and place in which it exists. Instead it represents the class of' things and phenomena of which it is an abstraction and comprises only those features of their affinity that concern the

investigator, [8]

It follows that the mathematical models which can be considered as describing the same aspect of physical reality may vary according to the kind of empirical relation and formal systems of which they are com-posed In particular the class of models that is

founded on the empirical and formal systems of physical sciences is distinguishable The models belonging to this class have been given the name of

structural models, (8] , because, in the above

ex-plained sense, they are created within the empirical and formal structure o.f physical sciences.

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13

-It is the characteristic feature of structural, models that they are constructed on the basis of the particular physical laws (which can be considered as elementary mathematical models) that are applicable to the relevant phenomena. As a result öf the local naturê of these laws*) with respect to time nd spacé variables (i.e. the fact that they express relations between quantities at a given point in Space and a ¿pêòified time), the application of structural models requires the knowledge of the

corresponding detailed descriptions of their objects It follows that in technical problems of designing new

physical entities, structural models can be émplöyed

efficiently in order to analyse the properties of advanced designs which have already been specified in sufficient detail.

This "analytic" property of advanced structural modéls has the adverse effect of impairing their usefulness for the decision making pròcess of desïgn, especiálly at its

preliminary Stages, (8] . Owing to that, the class of

rionstructural models acquires significance in relation to design applications A nonstructural model does not

involve elementary models of physical laws but instead its formal system directly quantifies relations between the global parameters describing the properties of the object Thereby the required dèscription of the object is reduced very considerably in comparison with what would be

necessary for the application of a corresponding structural model. Besides, theré exists a possibility of selecting the kind of the formal system (i.e. the mathematical f°rm of the model) to be employed, according to the needs of the prospective application of the nonstructural model, [8]. Therefore purposefully identified

nonstructural

models constitute convenient tools of

scrutinizing

the impli-cations of the decisions to be taken in the process of design, and can effectively assist the designer in the synthetic task of creating a new physical entity.

*) They preserve this character _{even in variational or wèak} formulations since then they still express relations between space-time tensor

fields.-MTB-141 14

-As a result they are also suitable as normative parts of rules and regulations.

It follows from the. above discussion that structural models are necessarily compatible, in a natural manner,. with physical structures of their objects. Special attention and techniques, which are still being developed, are required to assure similar

compatibility for nonstructural models, see [8] , [181

[19] and [20]. Generally it appears that well established structural models provide the proper information upon which the identification of the

corresponding high quality nonstructural models should be founded, (8], (18], in particular in connection with the choice of the arguments and responses of a nonstructural model Besides the results obtained within catastrophe theory, see e.g. (21] , indicate

that in the vicinity of the bifurcation set of the object quite strong qualitative links exist betwee.n the physical structure of the object and the form öf .a corresponding nonstructural model, at least for the objects classified as autonomus or, more specifically, gradient systems Taking into account that the points in the bifurcation set (in the space of so-called

control parameter.s of the object) are the. points of the transition of the object from one steady-state into another, it appear..s that results of catastrophe theory should be included in the methodology of the identification of nonstructural models (based on the employment of either numetical or physical

experiments) concerning the stability of floating bodies (understood in the sense which has been

discussed at the end of the section 1.2 in connection with safety prerequisites for such bodies).

22 Existing Structural Models of Floating Offshore

Structures

It has been indicated in the. preceding sub-sect.ion that at the beginning of the process of mathematical modelling the object of the process must be specified and that the object is specified in terms of the

common features of the class of .things and phenomena which are to be represented by the model.. In

technical sciences this class of things and phenomena is shaped by cLrrent design and industrial practice, although, as a result. of the generali.zing charàcter of

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-the creation of an object and its ma-thematical _model, the product of the process of mathematical modelling in i.ts most fruitfúl applications should enable the designer to reach beyond such a practice.

Therefore it is worthwhile to review existinq struc-turai models of the behaviour of floating

off-shore structures in waves with a view of specifying the object of a future mathematical _{model Here it} has been chosen to focus attention _{on the relevant} program packages which are at present employed by _some Classification Societies, recognized consultant firms and shipyard design offices, _{as the most} represent-ative of the. current design and industrial practice and of relatively well documented _{range of} appli-cations. _{InfOrmation about these packages is partlÏ} contained in the references _{[12], [13], [15], [22]}

-[38].

The typical scope Of techniäal _{devices to which the} packages apply (as far _{as floating structures are} con-cerned) -is illustrated in Fig1 1 which is reproduced

fröm [33].. _{In -thè Inträduction it has been pointed} out that semisubmersibles _{constitute a class of}

float-ing offshore structures of perhaps most _{general range} of problems required to be dealt with in design, con-struction and operation _{This is exemplified by Fig}

1 which shows the ranqe of the application of a well

developed program package On the basis of a typical semisubmersib].e design. The _{same pattern is followed} in [25] , _{[27] , [28] , [29] , [32] , [34] and [35]}

. The

other floating structures to which _{the packages apply} are production and multibody loading and _mooring

systems, see e.g. [22], _{[26], [291, [30], [31],}

[33],

[36] and [37] . - -

-Fig. i also illustrates how a hydrodynamic

program package (a part of AQWA) is imbedded in a system of program packages which ultimately analyses _the

strength (ASAS, see [38]) and dynamic _{behaviour (OMS,} VORTOS) of the structure or its components. _The

arrangement of one of the most extensive _package

systems SESAM'80, is shown in. Fig. 2, _{reproduced from} [23]. _{In the schema of Fig. 2 the. place occupied by} the program WAJAC (,whiçh is _{a hydrodynamic load}

program for steel-template structures) _corresponds also to the. position of floating structures programs WADIF(NV14.58)and WÄLMOS(NV1461), _{[23], [22]. Similar}

arrangement is shown in greater detail -in _{Fig. 3,} reproduced from [25].

-STRUCTURAL MON ITOR ING FATIGUE ANALYSIS VORTEX ShEDDiNG (ASAS,ADEPT,VOIUI'OS) I)YNAHIC AJJALY;IS AND MOItONS PRIiI)ECTION 01: CRANES (AQWA, OMS) STRUCTURAL I)ES IGN ( AQ WA, ASAS) MOOR ING ANALYSIS (AQWA , OÑS) ENVIRONMENTAL MONITORING DATA ANALYSIS (OtIS.ADEPT) PILEI) FOIJNI)ATION ANALYSIS

ATKINS

RESEARCH AND DEVELOPMENT SPECIAUST SERVICES

STRUCTURAL DYNAMICS (ASAS) DYNAMIC ANALYSIS AND MOTION PREDICTION OF VESSEL (AQWA, MS) ASSESSMENT AN D CONTRO L OF MACI I ENE VI IIRATION(ADEPT)

i H

o-'

WIND FLOW OVER DECKS (CAFE)

i

STRESS I)1STR1'BUTION ANALYSES (ASAS) STAB IL ITY ANALYSIS (AQWA

, Oh.ISÌ

FINITE ELEI.IENT STRUCTURAL ANALYSIS OF NODI

(ASAS)

RISER RESPONSE I)ESEGN ANI) ANALYSIS (AQWA,VCIUFOS)

SESAM'80

L&DS WAVES WINOS TEPIPERATURE S LAUNCH ¿ POSTFEN PROCESscçs t Art PRE -PROCESSORS GEOMETRY LOAD :ATA 't&TEP!AL PPErES

r

-rL

r'

-il

II-

_hl

CAD

III

_. ----a. I INPUT FILE FRIS STRA STP(JCTURAL ANALYSIS STATIC tXAPIIC

LI(A?

NONLINEAR

-

17

-LLJTION FILE POSTAME' CODE CHECKS AW DEOMETWr MATERIAL OPERTIES I

L

j

StilLS - PILES STATIC OYNANIC NONLINEAR

Fig. 2

The SESAM'80 Program Architecture

TAT

STEO

ETENED

CALCULATIONS

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PLOY VERITAS _FAIGUE

PRINT _{ALSC ¡API EARTHQII.XE}

CDM

PRDDRAMME

LI

ARY

LOADIÑO CONDITIO$

H

STATIC FORCL3ON EAcEI ELEMENT FORCES ON J EAcH ELEiENT GT 1302 GT 1303 SYRCUTURAL ANALYSIS 'T' SIX MOTION TRANSFER FUNCTIONS ..M: 21AV5 NSYIGN PWATIC PROPERTIES Sr R EIS TRANSFER FUNCTIONS CAIfrSSWWIMIIp 2 T ¿j O SIREIS CONCENTRATION FACIORK UIVO..M.aI GT1300 GT13O1

STAT iST ICAL CALCULAT ION

FIGURE 3 TRES$ RESPONSE FUNCTION ANO PROBASILISrIC CURVE Il N O L.0 S TIME BEFORE

I FAILURE FOR A GIVEN

I___ [MISSBLE K VALUE FOR A GIVEN DURATIOÑ GEOMETRIC DATA REGULAR WAVES

FATIGUE CALCULAI ION (MINER LAW) TRIM HEEL ANGLE MOTION RESPONSE FUNCTION ANO PROBABILISTIC CURVE ,Q-S io' 2øt1,ty DISPLACEMENT DRAUGHT 18 MTB-141

MTB-141 19

-On the basis of. these typical examples it can be determthed that a program package representing the model of hydro-dynamic properties of floating structures should be com-patible at least with packages ana]y5ing:

a). the mass distribution and hydrostatic

equi-librium of the structure,

the dynamics öf mechanical çonstraints imposed upon the. structure (such as catenary, hawser or any other kind of mooring system) ànd/or of dynamic positioPing systems *),

structural strength of the structure,

the dynamics of attachable devices (such as the riser, cranes, etc ) which have negligible

influence upon the motionS of the ètructure. These conclusions are valid also if the floating structure

is understood as a multi-body system rather than a single floating body.

The compatibility of the hydrodynamic program ckage with those listed under a) and b) is indispensible for the eva-luation of motions of the structure ( and therefore as well

for the evaluation of corresponding hydrodynamic loads), whereas the other two kinds of packages assure the proper

range of applicability for the hydrodynamic package Hence, the packages a), b) and the hydrodynamic package can be considered as a sub-unit of the. package system. However, it should be pointed out that thé possibility of consider-ing program packages c) separately from the hydrodynamic program package follows from the assumption that the interaction of elastic properties of the structure with hydrodynamic loads is negligible Such an assumption cannot be maintained if fully hydroelastic approach is

attempted.. .

*) _{Here, the inclusion Of dynataic pOSitiOning systems}

does not follow directly from the examples referred to

above, but is justified by. the alternative or complementary role these systems have in relatjon to mooring systems in station keeping, see the Introduction and Figure 6.

MTB-141

20

In Fig. 4. the scope. of applicability of. a typical developed hydrodynamic program package is illustrated, reproduced. from [33.]. The illustration shows that it is possible to use the package for the analysis of the behaviour in waves of a complicated multibody mooring-loading system. The.

behavjour in waves of semisubmersibles or floating

permanent structures (such as e.g. TLP's) constitute then particular instances of the analysis to which the package can be applied. In Fig. 5 the block schema of the same program package is shown, and Fig 6 shows the block schema of another similar program package (MOSSI, [34]

These schemata exemplify the underlying Organization of hydrodynamic packages, which consists of the. core

hydrodynamic program (e g AQWA-LINE in Fig 5) and a set of complementary dynamic programs (e g AQWA-FER,

AQWA-NAUT, AQWA-DRIFT in Fig 5) The core program

provides values of the first order, average drift and low frequendy hydrodynamic forces exerted upon the structure by waves and currents, whereas the complementary programs

analyse the motions and resulting loads on the structure occurring under the influence of mechanical constraints

and/or dynamic positioning system (see Fig 6) and hydrodynamic and wind forces.

The results obtained from the complementary programs are

initially in either spectral response operators (or man

force coefficients) or time, series form (e.g. this is the main difference between AQWA-FER, AQWA-NAUT and AQWA-DRIFT programs, as shown ïn Fig. 5, and such programs as TDS of

(26]) ánd require further processing for the purpose of the. evaluation of statistical properties of the motions and loads. This is performed by standard programs or routines as it is partly shown in Fig. 3.

2.3 Core. Hydrodynamic PrograíPs

It has been indicated a.bove that in the organization of hydrodynamic program packages core hydrodynamic programs can be distinguished, by means of which values of f.irst order, average drift and low frequency hydrodynamic forces are computed, It follows that the core hydrodynamic

prögram constitute.s a'n indispensible and central part of any package of this kind and that its qualities decide

DYNANIC ANALYSIS AND MUTtON PkEDICTION O CRANES

HAWSER TENSIONS AND

PERFOMANCE

ARTI CULATIONS (RESTRICTED FREEDOMS)

SLOW DRIFT FORCES AND MOTIOÑS (FISHTAILIÑG)

LARGE ¡DESIGN WAVE TIME HISTORY ANALYSIS

RISER RESPONSE 'DESIGN AND ANALYSIS

STABILITY DURING LOADING AND DEBALLASTING

Figure 4

VESSEL MOTÏON PREDICtION

DERIVATION OF .kDDED MASS AND DAMPIÑd COEFFICIENTS

TIME HISTORY MOTIÖN DATA

WIND, tffiRENT AND SEASTATE. CONDITIONS MODELLING

EXTERNAL Si'IFFNESS INPUTS

FINITE WATER ÓEP

EFFECT OF RUSTERS MULTIPLE BODY FACI LITY. MTB-141

AQWA:THE CAPABILITIES

SIGNIFICANT MdTIONS AND 'FORCES AT SPECIFIED POINTS

AQW4-LWH 111M

AQWA SUITE STRUCTURE

AOWA-FER

ELEMENTAL FLOATING SYSTEM: DESCRI PTI ON

AO WA-LINE

DATA BASE

IIYDRODyI4AFIIC COEFFICIENTS FR014 LINEAR DIFPRACTION/

RADIATION PROGRAM FREQUENCY TIME T IXYIAIN IXY4AIN RECULAR NAME g IRREGULAR WAVE I il

Figure 5

AOWA-NAUT AO WA-DRIF T I RISER DESCRIPTION AOWA -RISER DATA BASE OF

MEAN AND SIGNIFICANT

EQUILIORILEI AND

STABILITY

!ROPERTIES

Slow DRIFT AHI) WAVE FREQUENCY NOTIONS

TIllE HISTORY OF

MOTIONS AND LOADS

TIME HISTORY 0F

MOTIONS AND LOADS

MEAN AND SIGNIFICANT RISER MOTIONS AND

LOADS OUTPUT OUTPUT OUTPUT OUTPUT OIITPUT L EQUIUBRIUH ANALYSIS NOTIONS ANALYSIS RISER ANALYSIS

t--q---,-. Dynamic cable analyses Low fre mo

-

23 -t t H m

LTt111oI

i Figure 6 of the package.

Existing hydrodynamic programs vary considerably with

respect to the ranqe ofphysical _{phenomena they take into}

account and the level of analytical _{sophistication. The} fundamental phenomena which must be modelled by

corresponding structura], models _{are waves and currents (in} principle also the interaction between _{the two should he} allowed for) and their interaction with free or constrained floating bodies of various sizes _{and shapes. The bodies} can be considered as riqid or elastic and the influence of other boundaries of the fluid domain, such as the bottom and banks should be reckoned with _{if necessary.}

All of these problems have extensive _{literature which is} rapidly growing owing to the economic _{impact of offshore} exploitations of seabed resources. _{Recent comprehensive} reviews of the problems and their _{existing solutions can he} found in _{[5] , [7]}

, [39J and [40]

The major difficulties _{encountered in attempts to model the} relevant phenomena are those _{connected with viscous effects}

in the fluid flow, _{nonhinearities and so-called memory} effects (see e.g. [41), [42], [43)) and radiation

conditions. _{The latter two of which are pertinent to the} presence of the free surface of the fluid.

Water wave theories which are normally applied to the

investigation of the behaviour of _{floating bodies in waves,} neglect the viscosity of water and _{consider waves}

independently from currents. _{Nonlinear wave theories are} employed, as a rule, when the modelling of the interaction

MOSSI

Linked to or available for MOSSI Linked to MOSSI ICABDYN or

LINDYN STOCCAI MTB-14 i Thtus Veseel r---I Dynamic posi-tiOn system Dritt torce coeftcents

First order motion

transfer Unctions L.

MTB-141 24

-with the floating body is foUnded on so-called design wave concept, see. e.g. _{[5],, [131, [391 and the application of} Morison's equation to the evaluation of wave induced

forces. So far attempts to apply nonlinear wave

representation to the evaluation of diffraction forces have met only with limited success, see [7] . Linear (Airy)

wave theory gives realistic ptedictions of sea waves on the basis of spectral representation, [7], and is generally used for the evaluation of diffraction and radiation

effects, but is often also applied when hydrodynamic fotces are determined from Morison's equation.

Fluid exerted loads resulting from viscous effeçts in the flow are evaluated by means of semi-empirical formulae Their effects become important when the ratio ¡D, of the relative displacement tS of water particles with respéöt to the characteristic dimension D of the body (measured in the direction of the relative displacement), is close to i or

larger (the instance of so-called Small bodies, _[39]). In the opposite instance the flöw can be considered to be very close to that of the ideal fluid, although several

limitations, which are d-iscussed jn e.g. _{[7], should be} imposed on such a general statement.

The range of flows for which wave diffraction phenomena can be neglected is characterized by another simple parameter D/X, with À denoting the wave length, and _{corresponds to}

the condition D/X<O.2. _{If this condition is fulfilled,}

.the in-line (i.e. In the direction of the instantanèous horizontal fluid velocity) wave induced force per unit length, on a fixed vertical circular column of diameter D (small with respect to its length and À), can be expressed by Semi-empirical Morisön's formula:

dF = C p _lTD2 u° + _{i C PD u/U/, (1)}

m

_r

with p denoting water density, _{u and. u° signifying the} horizontal components of the instantaneous fluid velocity and acceleration in the undisturbed wave, and. Cm and Cd denoting inertia and drag coefficients _{réspectively. Under} the abOve mentioned cöndition the application of Morison's formula is usually extended to floating _{bodies in waves by} considering relative velocity and acceleration, e g in

[37] it is used in the form:

= - f p

ds _{+ CM + CDV}

/

/

₍₂₎ n

for cylindrical structural members, with _{p denoting water} pressure due to undisturbed flow in the wave, _{n denoting}

MTB-141 25

-the outward surface unit _{normal vector, ds signifying} the element of peripheral arc length, CM denoting added mass coefficient, CD denoting quadratic drag coefficient, _{an axd}

signifying components of relative fluid acceleration _and velocity respectively in the plane perpendicular to the member centerline and estimated on the centerline. _The integra. on the rhs of _{(2) can be recognized as the} Froude-Krilov force per unit length. _Further general-izations of Morison's formula, _{which cover also small} bodies of noncylindrical shape, _{are presented e.g. in}

[39]

and [44]. _{If, in addition to D/XO.2,}

the condition, /Di

is satisfied, the drag component can be dropped out of Morison's formula and the added mass or inertia

coefficients can be determined on the basis of the

assumption of ideal fluid flow without _{the presence of free} surface. _{This latter approach is subject to the same}

limitations as the application df _{the /D criterion.} Owing to complexities of visous _{flow phenomena, see e.g.}

[39] , [7] , [46] + [49] , _{the inertïa and drag coefficients}

(Cm and Cd) ot Morison's formula, depend, in _a

complicated way on three nondimensionaj. _{parameters of the} flow: _{Keulegan-Carpenter numbêr K}

Um T/D, Reynolds number Re _Um

D/v,

_{and relative roughness number}

kID, where u _{denotes the maximum water particle speed during} the wave cycle of the period T, V _{is the kinematic}

coefficient of water viscosity and k is _{the average}

roughness height. _{In particular it is becoming recognized} that roughness can have very strong influence upon drag forces, [49] , (50] Attempts are also being made to rationalize the approach to the evaluation of viscous effects, _[51]

Apart from viscous effects, significant _{local loads often} occur on cylindrical members of offshore _{structures as a} result of slamming, [7] , [39] . However, they do not. considerably influence global fluid flow phenomena and motions of the structure. _{.An extensive discussion of} slamming phenomena and their mathematical modelling is presented in [51]

It is generally accepted that for b/A _{0.2 the diffraction} (and radiation) phenomena of the interaction between a moving structure and, waves cannot be neglected. However,

it should also be noticed that

_{for d/A0.5, with d}

MTB-141 26

-and X signifying the length of oncoming or induced wave the influence of free surface is negligible Taking into account the deep water steepness limit for breaking waves (H/X <1/7 with H denoting the wave height) i't is realized that for D/X.o.2 the ratio 6/D must be small since 6<H. It follows that diffractiön and radiation effects can be modelled with the assumption of the ideal fluid flow. Viscous effects can still be induced by small protruding

elements or sharp edges, and remain important in the range of resonant motions. However, they do not dominate the

flow and can again be considered on semi-empirical basis. Generally applicable diffraction-radiation mödels are

foundéd on linear wave theory and are lihearized with respect to the amplitudes of thé motions of the structure

Tij, j=1, 2 , n, with n denoting the number of degrees of

freedom, and the wave amplitude a. It follows that all related phenomena depend harmonically upon time and thereforé they are usually represented as products of complex amplitudes (such as

nj)

and thé factor exp

(iwt),

with i= /-i, w representing the circular frequency of oscillation and t denoting the time variable The wave amplitude is made.real by the appropriate choice of the origin of time variable

The generalized fluid exerted forces F acting upon the structure, are determined by thé formulae:

F. = - exp(-iwt) _f pL. 6? dS, (3)

J

_s0

for

j=

1, 2,....n where S. denotes the. wetted surface in the reference configuration in calm water, p is the fluid pressure (equal to zero on the free surface),

L

_signifiés the normal to S0 (directed into the fluid domain), is the

instantaneous radius versor; p, and 6 are linearized as

Tj.

explained above. In particular:

,p = p(iw +

ge 3.r),

(4)

with _{representing the potential of the fluid velocity, g} signifying the. accerleratiori of gravity and e being the versor directed vertically upwards.

MTB-14 i

27

-For a rigid member of the structure (3) becOmes:

iwP f

o A e).dS + (5)

So

+ linear hydrostatic terms, for j 1, 2,..., 6. The potential i sought för in the form:

= ₊

J

exp(iwt), (6)

where

,

(j=1, 2,.., n.) dénote respectively the

complex amplitude incoming wave, diffraction- and radiation. potentials The incoming wave potential is known (Airy wave potential) whereas the other potentials must satisfy the Laplace equation, linear impermeability conditions

(resulting from the kinematics of the incoming wave train for _cD and the kinematics of bOdy displacements in the j-th mode for ), linear free surface

condition

and

radiation condition at infinity or a matching conditiOn ön a control surface in the fluid domain, see e g [52], [53]

After inserting (6) intO (5) arid. taking into account mass inertia terms, the equilibrium of inertia, hydrodynamic and hydrostatic forces is expressed in thi form:

[w2 (M+A) + jw

+ C] _{= FK} (7)

wherè the following notation has been adopted:

M -

the inertia matrix A - the added mass matrix

B the hydrodynainic damping matrix C - hydrostatic restoring forces matrix

- the vector of transfer functions,

with the components of the real matrices A and B defined by:

-

iwBjk

P

¿k

°' 0A ciS, (8)

FK denoting the vectOr of Froudê-Krjlov forces:

MTB-141

-

28

arid

D

signifying the diffraction torces:

f

. j wp

f

₍ , A jÇ ₎ dS

Dj

.

A discussion of the. symmetry relations fulfilled by the

added mass and daitiping matrices:

Ajk. = Akj

Bjk = Bkj

and Haskind .rlations leading to:

= -iwp f

.

I dS

Dj

can be found in [52] .

This latter expression allows to

cömpute the diffraction forces without solving the

boundary value problem

or

_D

The computational effort invOlved in the evaluation of

hydrodynamic forces is very considerably reduced if the

assumption of slenderness is fulfilled for an essentially

horizontal structural member

The generalized hydrodynamic

fOrces per unit length of a vertical strip of thé structure

are then determined by the formulae:

dF.

[p f

'

(io,

o A

e) .dCj

_,

(13)

C

[54], with

'

denoting the velocity potential analogous to

'1

(see the equation (6)) but with

defined by

appropriate two-dimensional boundary value problems, and C

representing the contour of the strip. The operator.6

in (13) is defined as:

I

(14)

where U signifies the forward speed of the vessel in. the

positive direction of the löngitudinàl. coordinate

x.

By

means of that operator forward speed is introduced in a

very expedient manner (forward speed should be taken into

account in transit condition)

The strip-method has

acquired rich literature, e g

[55] ,

[56] , (57] ,

[58] and

references.

(10)

G=

¡x-x'!

+ G',

MTB-141 29

-Numerical approaches to the evaluation of the diffraction and radiation potentials discussed above are reviewed e g in (531, [591, [60] and [61]. They can be divided intò four fundamental groups, i.e. methods based upon conformal mappings, the method of integral equations, hybrid element method and boundary element method. Conformal mapping approaches (multipole expansions) by their very nature have the scope of applications limited to plane flows, see [62] and references. They preceded other methods in development and therefore strip-method algorithms used to be based

upon. them. The other methods are not limited by the dimensionality of the problem.

In integral equations methöds (the names singularity or Green's function methods are also in use), the sought for velocity potential is expressed in terms of a singularity distribution (sources and dipoles can be employed, see

e.g. [28]) on the reference wetted surface of the body

S0.

Following [63], it is possible to write the potential in the form:

(i)

1/(4.ir'J

f (i') G (ï") dS,

(15)

So

with denoting the radius vector of a point, in the fluid

dothain, ' representing the radius vector of a point in So,

f (i') signifying the unknown density of source

distribution, and G representing the Green's function *)

(16)

withG' (the regular part of G) satisfying the Laplace

equation in the fluid domain, and choseñ in süch a way that G fulfills the radiation condition, free surface and all

impermeability conditions with the exception of that. on

S0.

It follows that , () as expressed by the formulae (15) also satisfies these conditions. The impermeability

condition on So imposed upon () then yields the integral

equation with respéct to thé uñknown density f (x):

-f. () + 1/21T)

_f

f (')

G (i; ') dS= 2v

₍₎

(17)

o n

on

S0,

where v denotes the velocity in the direction of

*) A review of Green's functions for radiation and

diffraction problems in regular water waves is presented in

MTB-141 30

-the normal to So, as specified by -the condition. The

equation (17) can be solved for f() numerically in a

discretized form, and thus an approximation to (X) can be found from (15).

Integral equations methods are conceptionally simple but

have some diadvantages which impair their applications.

The expressions for the regular part of Green's functions are complicated and cumbersome to handle numerically. Besides, at present, they are basically limited to. linear boundary value problems .in the frequency domain, with the exception of two-dimensional problems, see e.g. [66] and

[67.].. Another feature of these methods Which may cause

problems is the occurrence of so-called irregular

frequencies corresponding to the natural frequencies of slosh modes inside the body.

The hybrid element method is an. extension of the finite element method to problems of flow in infinite fluid domain. Usual finite element approaches, see e.g. [68],

[69] and [70] , are applied to formulate the. numerical problem för the ápproximation öf the. potential q, by means of so-called shape functions in a finite fluid domain

contained between the free. surface, impermeable boundaries and a control surface. The infinite fluid domain bounded by the control surface from the inside is considered to be the superelement and hence the name of the method follows. The main difficulty of the method .is the imposition of

proper boundary conditions on the cóntrol surface in order to make the solution for q, in the finite domain unique

This corresponds to the imposition of the radiation condition in the integral equat.ions methods. Two

approaches have been eláborated to solve the problems of boundary condition on the control surface, [53]. In both

the finite element solution is matched on the control

surface with the general form of the potential valid inside the superelement. This fôrm can be assumed in terms of a source distribution of the type (15) on the control surface or in terms of an eigenfunction. expansion (both known

representations are for linear harmonic waves).

The main disadvantage of thehybrid element method consists

in the necessity to elaborate a discretization of

three-dimensional fluid region. Besides, the scope of its applications is limited by available general solutions valid in the superelement.

In the boundary element method distributions of fundamental singularities (sources and/or dipoles in infinite domain) on the boundaries of the fluid domain are employed in order to represent the sought for potential q,, see e.g. _(71],

[72], (53]. FOr a source distribution this results in a representation of q, analogous to (15) with the exception that G' is deleted from the formula (16) and in (15) the

MTB-141 - 31

-integral is taken over the whole boundary of a finite fluid domain (this domain is defined essentially in the same way as in the hybrid element method). By imposing boundary conditions upon the representation of the potential an integral equation is formulated which, when solved in a discretized form, provides an approximate solution for .

The problem of finding the proper boundary condition

on the öontrol surface is fundamentally the. same as in the hybrid element method.

It follows from the above discussion that the boundary element method avoids the. use of numerically difficult Green's functions and considerably reduced the effort of descritization and volume of computations in comparison with the hybrid element method by diminishing the

dimension of the discretized domain by one. Although the difficulty with the formulation of the boundary condition on the control surface. rèmains. However the làtter problem may appear to be easier to solve within thè boundary

element method since the reduction of the dimension of discretized domain means that the control surface can be placed further away from the body than in the hybrid element method for the same required cömputational capacity. It follows that approximate forms of the

radiation condition can be imposed on the control surface (73], or that the boundary condition on the control surface can be abandoned altogether for transient flows of

sufficiently short duration., [74]

Returning to the equations (7) it should be nóticed that they are applicable to the. investigation of the behaviour. in waves of structures for which diffraction-radiation flow phenomena are totally dominant. In the more general

instance Mori,sonts formula must be utilized for evaluating fluid exerted forces on small.members*). .Then the drag term must be linearized ïn order to ensure the linearity of the resulting equations. This_can be achieved by

minimizing the error function E:

= C

// - C

,

(18)

in the mean square sense over the cycle of motion, and Leads to:

CDLCD8

V a

with Va denoting the amplitude of v, see e.g. [30] and

[75]. .

*) Often it appears convenient to apply Morison's formula (in the form of the formula (2)) to the whole structure as a first appröximation.

MTB-141 32

-If the stiffness of constraints imposed upon the structure (such as mooring) is taken into account in a linear form, the equations (7) are tr.asformed into:

[-(M+A+A') + iwB+B') +C+C'

= FK (20)

with the matrices A', B' and he excit.iñg. force

resulting from the application of the linearized Morisön's formula and C' representing the stiffness matrix of the constraints. Thé equations (20) are of quasilinear

character owing to the dependence of the matrix B' and the exciting förce

M upon amplitudes of the motions of the structure, and hence their solution requires an iterative approach.

The vector having been determined from the equations (20), it is possible to evaluate mean and slowly varying forces exertéd on the structure by

the

waves. 'he methods to compute these forces are. still being developed and can

be classified into two groups, as"far-field" and

"near-field" approaches, a recent review of these methods is presented in [76] and [77]. The far-field methods are less demanding computationally but give only the value of horizontal mean force, whereas the ñear-field approach allows to evaluate the vertical mean force and low

frequency forces induced by the waves, besides the latter approaçh gives more. insight intö thé mechanism of the generation of these forces..

Thenear-field method begins with the general expression

for hydrodynamic forces:

F.= - fp

(iì,

A).dS,

j= 1,2,.,6,

(21)

J

s J

(compare with the formulae (5)), where S, and denote the instantaneous wetted surface, normal versor and radius vector respectively, and p is given by .the full expression:

p

= -

p ₍ tS .

/

V /2 +

g ea),

rather than by the equation (4),

_and

the potential

includes also higher order potentials than those comprised in the formula (6), see. e.g. [781*). The method continues with extracting from the expressions (21) and (22) .the

lowest order mean and slowly varying terms, _{[78] , [80].} Mean forces can also result from drag effects and are then determïned by means öf Morison's formula, _[7].

(22)

*) _{An extensive discussion of the two-dimensional}

case in the contex.t of strip-theory can be found in _[79].

MTB-1 4.1

33

-Among the program packages dicussed in sub-section 2.2 ,i is interesting to observe the tendency which presents

itself clearly from [12] , (13] , [15] and [25] , [26] , [27] ,

(28) This is the tendency of development from older core

hydrodynamic programs based upon the app.ication Qf strip-theory and Morison's formula towards three-dimensional formulations by means of an integral equation method and Morison's formula At the same time far-field methods for the estimation of drift forces are being replaced by the near-field approach. The core. hydrodyniic program of [33], is also founded òn än Integral equatiofis method.

It appears that för the analysis. of strongly nonlinear

behaviour of the structure such as a Tr.P, Morison's formula formulation may be preferred for the whole structure,

combined wiht the application of a predictor-corrector method for the integration of the equations in the time domain, [361,. [37]. However, noñe Of the core programs is developed on the basis of hybrid element or boundary

MTB-141 34

-3. CONCLUSION

The foregoing discussion of fundamental problems involved in the mathematical modelling of the behaviour of floating offshore structures in marine environment can be summarized in broad terms as follows:

the object Of the modelling consists of the phenomena of the interaction of rigid and/or elastic impermeable bodies (single or 'multiple) with sea waves, currents, winds and ice,

displacements of the bodies should be considered as subject to constraints (such as resulting from the action of moorings or mechanical connections between multiple bOdies) and/or influenced by dynamic positioning systems, or as referred to a configuration moving with a small horizontal speed.

The purpose Of the modelling is twofold:

a). to develop a sophïsticated structural model (or

models) which would advance the understanding of the modelled phenomena and be applicable to the analysis of developed design alternativesÉ with the purpose of enhancing better solutions of the basic design problems of safety (defined by

intact and damage stability, reserve buoyancy and structural integrity) and efficiency (e.g. the compromise between requirements concerning weather-window, stability and variable deck load for SSDUs)..

b) on thebasis of ade.quat structural mödels,

verified by model tests, to identify

corresponding nonstructural models applicable at preliminary design stages and/or as normative parts of rules and regulations.

From the state of the art it can be inferred that the core hydrodynamic model should provide a three-dimensional.

solution of the linear diffraction-radiation problem in the frequency domain (harmonic waves) for large rigid bodies and include the application of Morison's formula for small rigid bodies Besides the model should allow to evaluate méan ánd 10w frequency hydrodynamic forces. However, with a view to better satisfying the purposes of the modelling

(as described above) the model should also constitute a good initial stage for the development of nonlinear, time domain and hydroelastic models It seems that, at present, models based on hybrid element or boundary element methods can give the best prospeçt of fulfilling all of these.

MTB-141

- 35

-In order to ensure proper applicability of th Cçre hydrodynamic program (which corresponds to the core

hydrodynamic model) the program should be understood as a part of a basic program package related to the analysis of motions and loads in waves of such offshore structures as semisubmersibles and tension-leg platförms. It föllows that two compi-ementary programs, covering:

the analysis of mass distribution and hydrostatic equilibrium of the structure, the analysis of the dynamic properties of the.

tooring system,

should be developed together with the cOre program. Besides the core program should be made compatible with standard programs for spectral and statistical analysis of time processes and for the analysis of structural strength.

MTB-141 36

-4. REFEREÑCES

Ronen, E.MQ.,"Offshore Exploration and Srvice".

Mobile Units, their Market Design and Safety. Ships en Werf, No. 7, 1982, pp99, 106.

Rodnight, T.V., "Development of: the Modern

Semi-Submersible Drilling Unit". Proceedings of the International Symposium, Semi-Submersjbles: the New Generations. London March 17 and 18, 1983.

Penney, P.W., "Offshore Vehicle Design: A Review of the Craft, their Tasks and Equipment", Trans!. North East Coast Engineers and Shipbuilders vol 98, August 1982, pp 109, 118.

Ellers, F.S., "Advanced OffshöreOLl Platforms",

Scientific American Vol 246, No. 4, April 1982, pp 39 49

Hallam, M.G., Heaf, N.J., Wootton, L.R., "Dynamics

of Marine Structures"1 CIRA Underwater Engïneering

Group, London .1978.

[61 Rules and Regulations for the Construction and

Classificat,iÒnof Offshore Platforms, Bureau Veritas 1975.

[7] Standing, R.G., "Wave Loading on Offshore

Structures: A Review", NMIR102, February 1981.

[8], Pawlowski, J.S., "On the Application of

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