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Proceedings of OMAE 2001 Conference The 20th International Conference on Offshore Mechanics and Arctic Engineering

Rio de Janeiro, Brazil, 3-8 June, 2001

OMAE2001/OFT-1280

PREDICTION OF EXTREME MOTIONS AND CAPSIZING OF SHIPS AND OFFSHORE MARINE VEHICLES

Jan O. de Kat MARIN

Wageningen, the Netherlands

J. Randolph Paulling Professor Emeritus

University of California, Berkeley

ABSTRACT This paper presents an overview of developments related to the dynamic behavior of intact and damaged ships and offshore platforms in extreme weather conditions. Capsize mechanisms are discussed, including flooding through damage openings and the behavior of flood water. The paper gives a brief analysis of the major accidents of three semisubmersible platforms, where the loss of stability played a role in their foundering. The development of regulations accounting for dynamic stability of MODUs is reviewed. The paper treats in some detail the aspects associated with the numerical modeling of intact and damaged vessels and moored offshore platforms in heavy seas. INTRODUCTION Research using scale models in realistic wave conditions combined with analytical work helped considerably in enhancing understanding of the nature of the ship capsize process, thus allowing for predictions of a vessel capsize using either physical or numerical methods. However, unlike the more established predictions of ship performance in calm water and in waves (seakeeping), there are no accepted standards for capsize predictions. Addressing this need, the 21st ITTC set up a Specialist Committee on Ship Stability. The main tasks were to examine techniques for carrying out model tests to investigate capsize of intact and damaged vessels and provide guidelines for such tests, and to assess the methods available for numerical simulations of capsize of intact and damaged vessels. In its second term, the Committee is coordinating a comparative study of the available mathematical models for the prediction of dynamic intact and damage stability in waves and of finalizing

guidelines for experimental testing of intact and damage stability. Based partly on the ITTC work, this paper presents state-of-the-art developments concerning the physics and modeling of extreme ship motions and capsizing in heavy seas. Regarding offshore applications, the paper describes three major accidents where (damage) stability played a key role in the foundering of the moored platforms. The subsequent development of dynamic stability criteria for MODUs is reviewed. The paper highlights similarities and differences relevant to the numerical modeling of extreme motions of ships and moored platforms.

CAPSIZE MECHANISMS FOR INTACT SHIPS Extreme rolling and - ultimately - capsizing in critical wave and operational conditions may occur according to the following mechanisms: ! Static loss of stability ! Dynamic loss of stability ! Broaching ! Combined modes with additional factors The following sections describe the physics of the above capsize modes. The 22nd ITTC Proceedings [1] provide a comprehensive literature review with regard to numerical and physical modeling of capsizing of intact and damaged ships; the relevant references have been omitted from this paper for reasons of brevity. Furthermore [2] contains a collection of state-of-the-art research papers on intact and damaged stability.

1 Copyright © 2001 by ASME

Static Loss of Stability This refers to the quasi-static loss of transverse stability (associated with an excessive righting arm reduction) in the wave crest or simply due to negative GM. This mode occurs typically in following to stern quartering waves with low encounter frequencies. The ship can capsize when it experiences temporarily a critically reduced (possibly negative) righting arm for a sufficient amount of time, while the wave crest overtakes the ship. One encountered wave of critical length and steepness may be sufficient to cause the sudden catastrophic event.

Dynamic Loss of Stability A vessel can lose stability dynamically in conjunction with extreme rolling motions and lack of righting energy under a variety of conditions. This capsize mode may be associated with the phenomena described below. • Dynamic Rolling with 6 coupled degrees of freedom: This

mode of motion can occur at forward speed in stern quartering seas. Here all six degrees of freedom are coupled, where in addition to roll, surge, sway and yaw can exhibit large amplitude fluctuations. The motion is characterised by asymmetric rolling: the ship rolls heavily to the leeward side in phase with the wave crest approximately amidships (position of reduced stability) and rolls back to the windward side in the wave trough, albeit with a shorter half-period and smaller amplitude. Due to the associated surging behaviour, the ship spends more time in the wave crest than in the trough. The actual roll period may exceed the natural roll period significantly. In the case of a capsize, the roll motion typically builds up over a number of wave encounters to a critical level.

• Parametric Excitation: Parametric excitation results from the time-varying (hydrostatic) roll restoring characteristics of a ship typically found in longitudinal waves. The periodic changes in static righting arm during the repeated passage of a wave crest followed by the trough can cause large amplitude roll motions, which occur at approximately the natural roll period and simultaneously at twice the wave encounter period. The wavelength must be of the order of the ship length. Parametric rolling - also referred to as low cycle resonance - is sensitive to roll damping and can result in capsizing. When a ship travels at the mean group speed in following seas, parametric excitation can occur during the passage (in a regular fashion) of a wave group with a sufficient number of encountered waves of critical height and length. Ships with a wide transom stern and great bow flare may be more prone to parametric rolling. It has been observed in head seas at low speeds for some container ships and cruise ships having the aforementioned hull form

features. This mode of motion can be relevant to moored FPSOs with tanker hull form.

• Resonant Excitation: In principle large amplitude roll

motions can result when a ship is excited at or close to its natural roll frequency. Roll resonance conditions are determined by the combination of GZ curve characteristics, weight distribution, roll damping, heading angle (e.g., beam seas), ship speed, wavelength and height.

• Impact Excitation: Steep, breaking waves can cause severe

roll motions and may overwhelm a vessel. The impact due to a breaking wave that hits a vessel from the side will affect the ship dynamics and may cause extreme rolling and capsizing. Possible damage to deck structures and subsequent water ingress may cause additional detrimental effects. This capsize mode is relevant especially to smaller vessels in steep breaking seas, such as are encountered on shoals, bars and river entrances.

Broaching Broaching is related to the loss of course keeping of a steered ship in waves. A variety of broaching modes exist: • Successive overtaking waves (low speed); • Low frequency, large amplitude yaw motions; • Broaching caused by a single wave. The first mode may occur in steep following seas at low ship speed, where the ship is gradually forced to a beam sea condition during the passage of several steep waves. The other modes occur at higher speed, typically at a Froude number Fn > 0.3. The third mode is usually characterised by quasi-steady surfriding at wave phase speed and steadily increasing yaw angle; this broaching mode has been also been observed for ships in combination with bow submergence during surfriding. Loss of rudder effectiveness due to pitching and orbital wave motion can play a role.

Combined Modes and Additional Factors Some of the modes of motion and capsizing described above can occur sequentially or in a combined fashion, ultimately leading to capsize. For instance, in astern seas a ship can start to surfride at high speed, broach and subsequently lose stability in the wave crest. In addition, water on deck can occur in conjunction with (and hence influence) the capsize modes discussed above. Large amplitude relative motions and breaking waves can result in the temporary flooding of the deck, where free surface effects and sloshing can lead to reduced stability. Furthermore, deck edge submergence results in loss of waterplane area and righting arm. If a bulwark is present, its submergence will influence the forces acting on the vessel.


Wind may have an important influence on extreme roll motions in beam sea conditions, especially. Wind dynamic effects are routinely included in mooring assessments of offshore drilling vessels and can easily be included in stability assessment as well (see discussion on unsteady wind further below). Cargo shift as a consequence of large amplitude rolling and high accelerations can induce capsizing.

FLOODING AND CAPSIZE MECHANISMS OF DAMAGED SHIPS A significant amount of research during the last decade has been aimed at understanding and improving the survivability of damaged ro-ro ferries and naval ships. Transient flooding Transient flooding following sudden damage occurrence may result in large (transient) roll angles, or - in the extreme case - in capsizing, depending on the damage size, initial stability characteristics and internal geometry of the ship. It is common for e.g. ro-ro ferries and certain naval ships to have cross flooding arrangements, which are effective in transferring floodwater and thereby reducing any asymmetry in the final damage condition.

Figure 1. Modern ro-ro ferry with two-compartment damage A ferry with a typical cross flooding arrangement is shown in figure 1 [3]; the cross duct in the double bottom has a width of three frames. To assess the effectiveness of the cross duct during transient flooding, the occurrence of damage was simulated by a sliding door on the port side. This was done in calm water and in waves. Figure 2 shows the transient response in calm water for one loading condition. This figure shows the measured roll motion and the water elevation at the following locations of the

damaged compartments underneath the main ro-ro deck (measured close to the cross duct openings): aft port and starboard side compartments (REL4) and forward port and starboard side compartments (REL6).

Figure 2. Transient roll response and water levels in damaged PS and intact SB compartments of ro-ro ferry following slow damage occurrence Model tests and simulations suggest that during the first stages of water ingress cross ducts are not very effective in reducing maximum transient roll peaks, which can be significantly larger than the static equilibrium heel angle. Increasing the cross-sectional area of the ducts will reduce the cross-flooding time, but it is not possible to achieve complete equalisation within one roll period. Damaged ro-ro ferry behavior in waves A damaged ro-ro ferry subjected to wind and waves may be vulnerable to the gradual accumulation of water on the car deck. Under these conditions a capsize represents almost a quasi-static process: the mean roll angle may increase gradually while the wave-frequency roll motions are small, until a critical amount of water has amassed on the deck. At that stage, the passage of one critical wave group is sufficient to force in flood water beyond the critical level, after which the ship capsizes rather fast due to static loss of stability. It is noted that this capsize mode is typical only for ro-ro ferries, where the accumulation of water is dominated by the occurrence of critical wave groups, vertical relative motions and associated freeboard exceedance. Figure 3 illustrates the capsize process of the ferry discussed above in waves with Hs = 4 m and low GM [3]. A damaged, compartimented ship without continuous deck will exhibit different capsize mechanisms than a ro-ro vessel.


Figure 3. Capsize of ro-ro ferry drifting in beam seas

SAFETY OF MOORED OFFSHORE PLATFORMS

Three Stability Related Disasters The early 1980s saw three major catastrophes involving floating vessels involved in offshore oil drilling activities. They were the loss of the semisubmersible accommodations platform Alexander Kielland in the Norwegian sector of the North Sea in 1980, the loss of the semisubmersible drilling platform Ocean Ranger in the Atlantic Ocean off eastern Canada, and the loss of the exploratory drilling ship Glomar Java Sea in a South China sea typhoon. All three casualties involved great loss of life. While all three casualties occurred during periods of very severe weather and sea conditions, there were essential differences in each case in the initial casualty mechanism. Directly or indirectly, loss of stability was a major contributing factor to these accidents. The Alexander Kielland was a pentagon-type semisubmersible designed originally as a drilling semisubmersible but utilized from the time of delivery as an accommodation platform. The platform was outfitted to accommodate 348 persons and, at the time of loss, 27 March 1980, was moored at the Ekofisk Field in the Norwegian sector of the North Sea. Of the 212 persons on board at the time, 123 were lost. Wind and sea conditions were about 16-20 meters per second and 6-8 meters significant height. The casualty was initiated by a fatigue fracture of a diagonal member bracing one of the five main vertical tubular columns supporting the upper deck structure. Failure of other members followed in rapid succession, resulting in loss of the main column itself. The platform then heeled to about 30-45 degrees, submerging a portion of the upper deck volume, causing flooding of this nominally watertight portion of the

structure. Downflooding of other legs and pontoons then took place and the platform capsized in about 20 minutes. See [4]. The Ocean Ranger was a twin-hull semisubmersible drilling platform conducting exploratory drilling at a location approximately 170 miles east of St. John�s, Newfoundland. Weather forecasts indicated that a severe storm with winds of 70 knots and maximum wave heights of 7 meters would pass between the drilling site and St. John�s at about noon on Sunday February 14, 1982. Conditions at the site continued to deteriorate during Sunday and Monday forcing Ocean Ranger to cease drilling operations on Monday afternoon. Later that evening, Ocean Ranger reported winds of 90-100 knots and 15 meter seas. Still later, there were reports of a broken portlight in the ballast control room, sea water getting into the ballast control panel, and a list of up to 15 degrees. At approximately 0100 Ocean Ranger issued a Mayday call and reported that the list was increasing followed by a second Mayday at 0130. At 0338, the standby vessel Nordentor reported that Ocean Ranger had disappeared from its radar. There were no survivors. Post accident investigations were carried out on the wreck in 75 meters water depth by sonar, an ROV and divers who were able to enter the ballast control room to recover numerous pieces of equipment and documents. It was concluded that the loss was the result of a chain of events involving a coincidence of severe storm conditions, design inadequacies and lack of knowledgeable human intervention.. The failed portlight probably allowed seawater to enter the ballast control console, causing an electrical malfunction of the controls which allowed on board ballast water to gravitate to the forward ballast tanks and/or allowed sea water to enter those tanks. The increase in draft forward allowed boarding seas to flood the forward chain lockers and increase the trim. Either lack of detailed knowledge or failure of system controls prevented the crew from correcting the situation by pumping ballast. See [5] and [6]. The Glomar Java Sea was a 120 m long ship built in 1975 expressly for exploratory drilling at sea. In October 1983 the ship was anchored on location south of Hainan Island conducting exploratory drilling operations. At 1339 on October 25, Typhoon Lex was about 120 miles to the east of the vessel and was expected to pass about 30 miles to the north. Winds were forecast to be 60 knots gusting to 75, with seas of 11 to 12 meters from northwest and 9 meter northeast swells. At 2341 hours Glomar Java Sea reported a list of 15 degrees but the crew had been unable to determine the cause. At 2346, communication with the ship was lost and not reestablished. The sunken vessel was found by side scan sonar and videotaped extensively by ROVs. These revealed major structural damage, especially to the starboard side shell plating amidships. The US Coast Guard concluded that the most probable cause of the loss was capsizing due to severe sea conditions aggravated by typhoon-strength winds. The starboard list probably


contributed to the casualty and the structural damage probably occurred after the ship sank and struck the seafloor. The cause of the list could not be definitely determined but may have been loss of the deck load of drill pipe and/or flooding of interior spaces through incompletely closed deck openings. See [7] and [8]. In the decade following these casualties, there was a surge of research activity aimed at understanding more fully the behavior of ships and platforms in extreme ocean environments and to improving stability standards, both intact and damaged. Traditional criteria had been based almost exclusively on stability computations performed under the assumption of behavior in calm water under the action of a stepwise wind heeling moment. There was an obvious need for new criteria that incorporated dynamic effects of wind, wave and vessel motion.

The Development of Dynamic Stability Criteria for MODUs In 1986, the American Bureau of Shipping began a development project with industry support and participation as reported in [9]. The objectives were threefold: # Parametric analysis of semisubmersibles� behavior in

realistic environmental conditions to determine geometric and operating parameters affecting their dynamic motion.

# Calibration of analytical techniques and quantification of their limitations through correlation with model tests in extreme wind and sea conditions.

# Assessment of the adequacy of safety margins afforded by the existing criteria in terms of downflooding and capsize considering the motion responses of a parametric series of semisubmersibles to a realistic extreme environment.

In order to meet these goals, a comprehensive survey of existing four, six and eight column twin hull MODUs was first carried out. The geometrical parameters, internal subdivision, wind sail area and other properties were classified and a series of generic units were designed to be used as a basis for computational and model studies. The computations consisted of intact and damaged stability according to rules then in effect and dynamic studies of wave- and wind-induced behavior. Model tests were performed on one representative design in order to determine (1) the limits of the computational methods, (2) effects of second-order low-frequency motions, and (3) downflooding (run-up) occurrence for intact and damaged conditions. Computations of platform motions utilized both linear frequency-domain techniques and nonlinear time-domain simulations. Agreement between predictions and measurements was good in waves up 30 meters in height, giving confidence to the use of computed responses for further rule development and in demonstrating rule compliance for an actual unit.

Essentially, the new rule that was developed under this study specifies stability margins in terms of maximum predicted dynamic responses. The dynamic response, to be determined by an appropriate time domain simulation technique, is to include the effects of random waves, random wind, moorings and current. Relative motions between wave and platform are to be a part of the computations in order to establish the occurrence of, and margins against downflooding. The work is described in [9], [10] and [11], and in the American Bureau of Shipping Rules [12].

NUMERICAL MODELING

Extreme motions of intact vessels Work on time domain capsize prediction started in the mid 1970s, notably at the University of California, Berkeley. Several techniques are available to predict extreme ship motions in waves [2]. Time domain simulations offer the advantage of the ability to account for nonlinearities in the ship system and external forces in a comprehensive fashion. This section presents a brief description of a time domain method that has been applied in [13]; its principles are similar to other models, given e.g. in [14]. The model consists of a non-linear strip theory approach, where linear and non-linear potential flow forces are combined with maneuvering and viscous drag forces. The non-potential force contributions are of a nonlinear nature and based on (semi)empirical models. The basics of this approach apply also to offshore platforms. The derivation of the equations of motions is based on the conservation of linear and angular momentum. These are given in principle in the inertial (earth-fixed) reference system. Euler�s method is applied for deriving the equations of motion in terms of a rotating, ship-fixed coordinate system. The equations of motion are given by:

( )

)pqI xx,0-I yy,0(

)prI zz,0-I xx,0(

)qrI yy,0-I zz,0(

q)uG-pvG.(m0

p)wG-ruG.(m0

r)vG-qwG.(m0

-

M z

M y

M x

F z

F y

F x

= G.+

∑

∑

∑

∑

∑

∑

∞ xaM0 &&)]([][

(1)

[M0] is the generalized (6x6) mass matrix of the intact ship, [a] is the added mass matrix, and x&& is the acceleration vector at the center of gravity; p, q and r represent the rotational velocities for roll, pitch and yaw, respectively. The summation signs in the right hand side represent the sum of all force and moment contributions, which result from: • Froude-Krylov force (nonlinear) • Wave radiation (linear)


• Diffraction (linear) • Viscous and maneuvering forces (nonlinear) • Propeller thrust and hull resistance (nonlinear) • Appendages -- rudders, skeg, active fins (nonlinear) • Wind (nonlinear) • Internal fluid (nonlinear) • Mooring system (nonlinear) Large angles must be retained in the matrices for transformation between the ship-fixed and the earth-fixed coordinate system in relation to acceleration vectors and rotational velocity vectors. One of the most important force contributions that should be treated "exactly" is the integrated hydrostatic and dynamic wave pressure. This represents the Froude-Krylov force, which should be obtained by pressure integration over the instantaneous wetted surface of the hull at each time step. This will account for a large part of the nonlinearities that affect the ship response. Linear wave theory is used to describe the sea surface and wave kinematics. In the case of irregular waves, the model makes use of linear superposition of sinusoidal components with random phasing. Linear, 3D, transfer functions can be used in the determination of the diffraction forces. The wave radiation forces are based on linear retardation functions and convolution integrals (including forward speed terms):

τττ dkxtt

jkKtkxjkCtkxjkBtkxjkAtjkF )()(0

)()()()( &&&& −∫−−∞−∞−=

(2) The retardation functions can be obtained as follows in terms of the 3D hydrodynamic damping coefficients:

ωτωωπ

τ djk

Bjk

Bjk

K )cos(0

)(2

)( ∫∞ ∞−=

(3)

Viscous effects include roll damping due to hull and bilge keels, wave-induced drag due to wave orbital velocities, and non-linear maneuvering forces with empirically determined coefficients. The quasi-steady hull forces resulting from the motions in the horizontal plane consist of a linear and non-linear part; for instance, the sway force is:

F2,H(t) = FH,L(u,v,r,θ;t) + FH,NL(v(x,t), T(x,t), CDx(u(t),x), u,v,r,ηt) (4)

where v(x,t) is the local transverse velocity at (sectional) location x, T(x,t) is the local draft, CDx is the local cross-flow drag coefficient, and ηt is the local wave orbital velocity in

transverse direction. The roll moment resulting from lift and hull drag forces has the following nature:

Mx,H(t) = FH,L.z(t) + FH,NL(t).z(x,t) + Kur.U(t).r + Kup.U(t).p +

Kpp|p|p (6)

The propeller thrust depends on the propeller characteristics and instantaneous inflow conditions. The hull resistance is a function of instantaneous speed and draft. Appendage forces are estimated by using wing theory in the case of a rudder or active fin, and by using pressure drag in the case of a skeg. Interaction between rudder and hull is accounted for. The equations of motion are solved in the time domain using e.g. a 4th order Runge-Kutta scheme.

Extreme motions of damaged ship In analogy with the intact ship method above, the derivation of the equations of motions for a ship or platform subjected to flooding through one or more damage openings is based on the conservation of linear and angular momentum for six coupled degrees of freedom. Here the fluid inside the ship is considered in a dynamics sense as a free particle with concentrated mass. With this assumption, classical rigid body dynamics can be used to derive the equations of motion, see e.g. [15] and [16]. Time-varying mass Many of the numerical integration schemes used in time-domain simulations are designed to deal with systems of first-order equations of the form

( , , )x vv f v x t

==

&

& (7)

In our numerical simulations we may have to deal with terms corresponding to time-varying mass. There are at least two sources of such terms: (1) added mass that varies with time due to the changing underwater geometry, and (2) body mass that varies with time due to the accumulation of flooding water. The effect of time varying mass is handled as follows. First we note that the equations were originally derived by taking the time derivative of momentum to yield

( )d dv dmF mv m vdt dt dt

= = + (8)

The second term on the RHS is merely the velocity times the rate of change of mass. This term may be moved to the LHS and combined with the force vector:

( ')dm dvF v m mdt dt

− = + (9)


The term m� is the mass that varies with time, while m is the actual physical mass of the body. After rearranging, this equation can now be written,

'

dmF vdv dtvdt m m

−= =

+& (10)

which is of the form required for numerical integration. Using the above approach and generalized mass matrices yields the following equations of motion for a damaged vessel in the ship-fixed coordinate system:

( )[ ] [ ( )] [ ] &&0 fM a M x+ + . =

F

F

F

M

M

M

-

( m + m ).( w q - v r)

( m + m ).( u r - w p)

( m + m ).( v p - u q)

( I - I )qr

( I - I )pr

( I - I )pq

G

x

y

z

x

y

z

0 f G G

0 f G G

0 f G G

zz,0 yy,0

xx,0 zz,0

yy,0 xx,0

∞

∑

∑

∑

∑

∑

∑

- +

+ additional

terms

[M ].xf G& &

(11) The matrix [M0] is the generalized mass matrix of the intact ship, [a] is the added mass matrix that is part of the linear radiation forces (the convolution integrals are part of the terms in the RHS). [Mf] is the 6x6 matrix containing all ship-acceleration related, time-dependent inertia terms associated with the flood water, including non-zero off-diagonal terms. The summation signs in the RHS represent the sum of all external force contributions, as for the intact case (including the presence of damage fluid). The �additional terms� in the RHS of the equations of motion stem from cross products, which appear when expressing the conservation of momentum in a ship-fixed coordinate system, and from the motion of the fluid relative to the ship. For example, the conservation of linear momentum in the ship-fixed system results in the following equation: F v v r v r v

v v r = ( m + m ).( + ) + m .( + ( )+ 2. )

+ m .( + + )0 f G G f f f

f G f

& & &

& & &

ω ω ω ω ωω

⊗ ⊗ + ⊗ ⊗ ⊗⊗

(12) where the vectors vG = (uG, vG, wG) and ω = (p, q, r) represent the linear and angular velocity vectors in the ship-fixed coordinate system, respectively; vf is the velocity vector of the center of gravity of the flood water with time-dependent mass mf expressed in the ship-fixed reference system. All terms resulting from eq. (9) are retained. The conservation of angular momentum can be derived in a similar fashion.

Water ingress and fluid loading Hydraulic flow To estimate the flow rates of water entering a compartment, the flooding model is typically based on the Bernoulli equation [15]. This analysis is applied to each damage opening or holes between two compartments. It assumes stationary flow conditions and no loss of energy due to friction or increased turbulence. Based on the difference in pressure head, the velocity through a damage opening can be calculated. Figure 4 presents a sketch for the flow through an orifice, where the discharge velocity is given by:

2 1 2v 2g(H H )= − (13)

2H equals zero for the free discharging orifice. The height 1H is considered as the height from the free surface plane to the center of the hole. The flow over a weir (which for example applies when the damage opening is partly above the waterline) is calculated using equation (13) as well, with 2H equal to zero.

Figure 4. Flow through a free discharging orifice (left) and through a fully submerged orifice (right) To obtain the total discharge through an opening, the following empirical formulation is used:

d 2Q C v A= (14) where A is the area of the damage and dC is the discharge coefficient. This coefficient accounts for a combination of several effects (such as friction losses) and can be found in hydraulics textbooks. For a sharp-edged orifice a typical value is around 0.60, which is valid for high Reynolds numbers. If the (damage) opening is relatively large, equation (11) has to be integrated over the height of the opening since the average velocity value obtained from equation (10) is not sufficiently


accurate. Furthermore, using thermodynamic gas laws the flow through openings and compression of air can be modeled. Quasi-dynamic fluid loading Based on the computed inflow and outflow of fluid through all openings, the fluid mass inside a compartment is known at each time step. A simple yet practical approach is to assume that the water level of the flood water inside any compartment remains horizontal (earth-fixed) at all times. This implies that the damage fluid causes a vertical force (due to gravity) to act on the ship and that any sloshing effects are neglected. The associated ship-fixed force and moment components can be determined through the appropriate transformations; these are then added to the RHS of eq. (11). This approach gives adequate results for engineering purposes, so long as sloshing is not dominant; this applies to forces and moments acting on a ship during transient flooding in a variety of compartments and during forced oscillations [15].

VOF method There are accurate ways to describe dynamic fluid motion and associated forces. As has been shown in [17], it is possible to simulate complex flow situations accurately by means of a Volume of Fluid (VOF) method. The elements or cells must cover the whole volume of the domain and must be limited in size to obtain accurate flow details. This results in a large number of unknowns to be solved for relatively simple configurations; the method is computationally very intensive. To demonstrate the application of 3D VOF calculations, a partially filled engine room of 15 m width, similar to the one shown in figure 5, was modeled using a mesh of 54 by 45 by 33 cells (cell size was 0.33 m). The objects representing the main engines and gearbox were modeled as impermeable blocks.

Figure 5. Model of floodable engine room compartment

2

2

2Water height - location H3 [m]

2

2Water height - location H6 [m]

time (s)

0 20 40 60 80 100 120 140 160

2

2

CalculatedMeasured

Water height - location H9 [m]

time (s)

Figure 6. Time histories of computed (VOF) and measured water levels at three locations inside engine room An example of the comparison between computed and measured forced roll tests in terms of water elevations inside the compartment is shown in figure 6. here the water level is 3 m, the period of oscillation 8 s and the roll amplitude 10 degrees -- sloshing occurs in these conditions. The associated sway force and roll moments show excellent agreement with the measured data. Current research aims at coupling a VOF model for description of the fluid dynamics inside a damaged ship to a time domain simulation model of the overall vessel motions.

Unsteady Wind Effects Time varying wind effects have been included in a number of the commonly used stability criteria for many years. The heeling work-righting energy type of criterion used by many regulatory bodies hypothesizes a heeling moment generated by a wind of specified strength acting on the ship�s profile area. The stability requirement is then expressed as a minimum required ratio of the area under the righting moment curve to the area under the heeling moment curve starting from zero heel angle. This heeling moment area is just the work done by a heeling moment that is initially zero when the ship is upright, then rises instantly to its maximum value before the ship develops any appreciable heel. Thus it corresponds to an increase of wind strength that is a stepwise function of time. The maximum dynamic heel angle is reached when the potential (righting) and kinetic (heeling) energy are equal. The stepwise gust is a highly idealized model of the real wind behavior since the wind blowing over land or sea is not steady in velocity or direction but varies randomly with time. This wind turbulence is now often taken into consideration by civil


engineers in the design of high rise buildings, bridges and other lofty structures, and also by naval architects, especially those engaged in offshore structure design. The intensity of the turbulence in the boundary layer within few hundred feet of the surface of the ground or sea is strongly dependent on the roughness of the terrain. A sufficient number of wind measurements have been made over the open sea to reveal some characteristic differences in turbulence over waves compared to turbulence over terrestrial roughness. As any open sea small boat sailor knows, when sailing in large swells, the wind strength experienced by his boat when on the crest of a swell is appreciably greater than the wind strength when in a deep trough. This suggests that there should be some relation between the time-dependent nature of winds over the sea and the height of the waves. If the frequency content of the wind induced heeling moments is correlated with the wave-induced moments, there exists the possibility of reinforcement between the two in inducing rolling motion of the ship. In common with waves, the turbulent component of the wind velocity is usually represented in the form of a spectral density function where this function is found to depend on mean wind strength, height and roughness of the terrain. [18] and [19] contain several wind spectral formulations together with a discussion of the characteristic differences between wind turbulence over land and sea. Of greatest importance, the wind spectra for use over the sea have higher components at the low frequency end than land based spectra, whereas the high frequency tails are very similar. Figure 7 contains plots of the two of the most commonly used wind spectra: one, developed by Davenport [20], has been used as a basis for land structures, and the second, by the American Petroleum Institute [21], is intended for use in the design of floating offshore structures.

Figure 7. Davenport and API Wind Spectra for Vmean = 30 m/s

The formula for the API spectrum is:

2 /( )( ) 5/3[1 1.5 / ]

f fz pS ff f f p

σ=

+ (15)

where S(f) = spectral density of time varying part of the wind strength at elevation z, f = frequency in Hz, fpz/Vz = 0.01 to 0.1 with a nominal value of 0.025, Vz = one hour mean wind speed at elevation z, σ(z) = standard deviation of the time dependent

part of the wind speed, σ(z)/Vz = I(z) = turbulence intensity

0.125( ) 0.15( / ) , ,0.275( ) 0.15( / ) , ,

I z z z z zo x

I z z z z zo x

−= ≤−= >

where zx is the thickness of the surface layer = 20 meters. The mean wind speed at a height z is given by a power law,

( )0.125/V V z zz o o= (16)

where Vo = wind speed at reference height zo (zo is usually taken as 10 meters). The Davenport spectrum is given, in SI units, by

2(1220 / )2( ) 4 2 4/3[1 (1220 / ) ]

V fzS f KVzf Vz

=+

(17)

Here, K = surface roughness coefficient, 0.005 < K < 0.05, depending on the roughness of the terrain. Davenport�s expressions for the reference wind and wind velocity profile differ somewhat from those of the API, but for comparative purposes they may be as defined above. The crossover point between the two spectra occurs in the vicinity of 0.02 Hz or a period of 50 seconds. Thus the difference in the two spectra is expected to be of importance for long period phenomena such as the horizontal plane motions of moored vessels or the rotational motions of semisubmersible platforms. [18] and [19] give examples of other spectral formulations in which this crossover is observed at higher frequencies of 0.025 to 0.03 Hz. For this reason, it is recommended to use an ocean-based wind spectrum when investigating floating vessel motion phenomena, especially those characterized by natural periods of greater than about 30 seconds.


As in the case of spectral representation of random waves, a random wind field may be approximated as the sum of a number of sinusoidal components, each having random phase and an amplitude related to the spectral density function. The random wind field will have a nonzero constant term corresponding to the mean wind velocity, and, in general both the mean wind and the turbulence intensity will vary with altitude. Typically, if S(f) is the mean spectral density over a frequency interval δ f centered on frequency fi and Uo is the mean wind speed at height z, an approximation to the random wind field is given by

∑=

−=N

iiiiz tfffStU

1

)2cos()(2)( επδ (18)

where the εi are random and uniformly distributed in (0,2π). Using the appropriate spectral formulation, which in most cases includes expressions for wind velocity variations with elevation, equation (18) enables one to simulate the wind time history. This can then be combined with drag and area coefficients for the structure in question, e.g., the exposed profile area of the ship, and an approximation to the time-dependent wind heeling moment computed. This is most effectively used when embedded in a time-domain simulation of the vessel or platform response. Such a procedure is suggested to be used in conjunction with the American Bureau of Shipping Dynamic Response Based Intact Stability Criterion. MOORING SYSTEMS The nonlinear dynamic behavior of a moored ship or platform contains certain features related to the mooring system itself. The fundamental purpose of a mooring is to maintain the floating vessel in its mean position in order for it to perform its intended function. In emergency situations, moorings are relied on to prevent a vessel that has, for example, lost power from going aground. Other than dynamically-positioned units, vessels engaged in offshore drilling and production operations are usually maintained in position by some kind of mooring arrangement. Ideally, the mooring system is intended to maintain the floating vessel in a constant unvarying mean position at the sea surface but, as a result of dynamic effects, this ideal can only be approximately attained. The mooring lines, which may be chains, wire rope or a combination of the two, have properties of elasticity, mass and hydrodynamic drag that can affect the dynamic response of the vessel to waves and wind. As perhaps the most obvious affect, the mooring provides a spring or restoring term to the equations of horizontal motion of surge, sway and yaw, making these motions subject to resonance in their response to wave excitation. At the same time they can

change the resonance characteristics in the other three motions, and the roll especially is affected. The spring effect of the mooring lines is usually computed by means of a procedure that takes into consideration the elastic stretch, the weight, the length of line, including the amount lying on the sea floor, the presence of spring buoys or clump weights and other features. The result can be thought of as a graph of the tension and the horizontal and vertical forces at the fairlead end of the line versus the horizontal and vertical offset of that end point from its initial position These quantities are normally nonlinear functions of position. For a small offset from the initial point, the forces are dominated by weight effects. At large offset and high tension the line elasticity dominates. Under certain circumstances, there may be significant effects due to line dynamics. If we consider the rolling motion of a ship moored by a spread array of anchors, rolling of the ship will be accompanied by appreciable motion of the uppermost part of the mooring lines. If the line tension is high, the motion of an individual line will be in a direction nearly normal to the line and it will experience fluid drag. Since a typical ship is very lightly damped in roll, the cumulative drag of a taut spread mooring system may have a significant effect on the wave-induced roll motion. This is discussed in [22]. In considering stability against capsize, moorings may have a number of effects that should be taken into consideration. They provide the resistance to oppose the mean disturbance of wind or current and to maintain the vessel in its desired position. The height of the attachment points (fairleads) must be taken into consideration in computing the wind overturning moment. Here most rules specify that mooring are be considered in the most unfavorable light, i.e., they are to be neglected unless their inclusion worsens the overturning moment or other response effects. Finally, it may be necessary to consider the effect on vessel behavior of damage to the mooring such as the loss of one or more lines, since this is most likely to occur in severe environmental conditions. LOW FREQUENCY PHENOMENA There are a number of phenomena to which the above term may be applied. Two examples of the phenomena are rolling in beam seas having a �grouping� characteristic, and the surge-sway-yaw of a moored vessel. Figure 8 is a simple example illustrating the rolling motion of a twin hull semisubmersible platform in beam seas consisting of the superposition of two trains of sinusoidal waves of different frequency.


Figure 8. Rolling motion of a twin hull semisubmersible in bichromatic beam seas In the simulation shown here, the two wave frequencies were chosen such that the resultant wave consists of a relatively high frequency term modulated by another sine wave of much lower frequency. This is a simplified example of wave grouping that might be experienced at sea in swells propagating from a distant storm center. The wave-induced roll moment in this case contains a component that is of the same low frequency as the wave envelope in addition to the wave frequency component. This low frequency component is of viscous drag origin and is a result of the time-varying immersed lengths of the columns with passage of successive wave troughs and crests. In the example shown, the envelope frequency was adjusted to be near the natural frequency of roll of the platform, consequently near resonant roll motion is excited. The motion excited by the moment at the higher wave frequency is also visible but is of much lower amplitude. Another slowly varying wave force or moment occurs as a result of wave diffraction and reflection about the floating vessel. This is a potential flow effect that is independent of viscosity. For a moored ship or platform in a current, there can be low frequency surge-sway-yaw motions generated as a result of large-scale vorticity in the wake. This is not unlike the vortex-induced oscillation of cylindrical bodies such as pilings, cables or pipes in a current. Wave groups are a normal occurrence in the real oceans where a group is simply a sequence of several successive waves of greater than average height. Their occurrence may be explained by imagining the real random sea to be composed of a large number of regular waves of different amplitudes, frequencies and random phasing. When a number of the larger components, that is, those grouped around the peak of the spectrum, occur nearly in phase with one another, their amplitudes will reinforce and a group of higher waves, having an average frequency near that of the spectral peak will result. As a result of the relation between wave length and celerity, the different components will, after a time, disperse so that phase coincidence no longer holds. The group is seen therefore, to be of an evanescent nature, continually forming and dissipating. A moored platform will be subject to low-frequency (slow drift) oscillations related to second order potential flow effects. Wave

groupiness plays an important role in this respect. The low-frequency oscillations (typically surge, sway and yaw) can influence the roll motions and possibly flooding of a damaged platform. Slow drift motion amplitudes are sensitive to damping resulting from a variety of sources, including viscous drag on the platform, second order potential (wave drift) damping effects, presence of current and wind, and mooring system. Wave groups may have an importance in selecting conditions for model testing or numerical simulations in which the objective is to find extremes of behavior in very long time periods. Ongoing research includes ways to perform deterministic testing (and simulations) whereby the most likely extremes will be encountered within a short time period.

CONCLUSIONS This overview paper starts with a description of possible capsize mechanisms for intact and damaged ships. It proceeds with the description of the foundering of two semisubmersible platforms and one offshore drill ship; stability played a major role in these accidents. In the aftermath, research in the 1980s culminated in the development of dynamic stability criteria for MODUs. The numerical modeling of extreme motions and capsizing has been addressed for intact and damaged ships and offshore platforms. While the physics of offshore platform dynamics allow in principle a similar modeling approach as for ships, offshore platforms require special attention to different aspects. These include the effects of unsteady wind, mooring system, low-frequency roll oscillations, and slow drift motions.

REFERENCES [1] 22nd International Towing Tank Conference, Proceedings, Volume 2: Final Report of the Specialist Committee on Stability, Seoul and Shanghai, Sept. 1999, pp. 399-431 [2] D. Vassalos, M. Hamamoto, A. Papanikolaou and D. Molyneux, Eds., Contemporary Ideas on Ship Stability, Elsevier, 2000 [3] J.O. de Kat, M. Kanerva, R. van 't Veer and I. Mikkonen, "Damage Survivability of a New Ro-Ro Ferry", Proceedings of the 7th International Conference on Stability for Ships and Ocean Vehicles, STAB 2000, Launceston, Tasmania, Feb. 2000 [4] Norges Offentlige Utredninger, �Alexander L. Kielland Ulykken�, NOU 1981: 11 [5] U. S. Coast Guard Marine Casualty Report, �Mobile Offshore Drilling Unit (MODU) Ocean Ranger, O. N. 615641, Capsizing and Sinking in the Atlantic Ocean on 15 February 1982 with Multiple Loss of Life�, Report No. USCG 16732/0001 HQS 82, May 1983


[6] Royal Commission on the Ocean Ranger Marine Disaster, �Report One: The Loss of the Semisubmersible Drill Rig Ocean Ranger and its Crew�, St John�s, Newfoundland, 1984 [7] U. S. Coast Guard Marine Board of Investigation on the Capsizing and Sinking of the Drillship Glomar Java Sea, O. N. 56182 in the South China Sea on 25 October 1983 with loss of life�, 16732/GLOMAR JAVA SEA, Mar 5, 1986. [8] U. S. National Transportation Safety Board, Marine Accident Report �Capsizing and Sinking of the United States Drillship Glomar Java Sea in the South China Sea October 25, 1983, NTSB/MAR-87/02 (Supersedes: NTSB/MAR-84/08) [9] S. G. Stiansen, Y. S. Shin, G. Shark, �Development of a New Stability Criteria for Mobile Offshore Drilling Units", Proceedings, Offshore Technology Conference, Paper No. OTC 5802, 1988, pp 505-514. [10] J. I. Collins, T. W. Grove, �Model Tests of a Generic Semisubmersible Related to a Study Assessing Stability Criteria�, Proceedings, Offshore Technology Conference, Paper No. OTC 5801, 1988, pp 498-503 [11] G. Shark, Y. S. Shin, J. S. Spencer, �Dynamic-Response Based Intact and Residual Damage Stability Criteria for Semisubmersible Units�, Transactions, SNAME, vol. 97, 1989, pp 213-242. [12] American Bureau of Shipping, �Rules for Building and Classing Mobile Offshore Drilling Units�, Part 3 Hull Construction and Equipment, 1994. [13] K. McTaggart and J.O. de Kat, "Capsize Risk of Intact Frigates in Irregular Seas", Proceedings SNAME Annual Meeting, Vancouver, Oct. 2000 [14] Umeda, N., Munif, A. and Hashimoto, H., "Numerical Prediction of Extreme Motions and Capsizing for Intact Ships in Following/Quartering Seas", Proc. Fourth Osaka Colloquium on Seakeeping Performance of Ships, OC 2000, Osaka, Oct. 2000, pp. 368-373 [15] R. van 't Veer and J.O de Kat, "Experimental and Numerical Investigation on Progressive Flooding and Sloshing in Complex Compartment Geometries", Proceedings of the 7th International Conference on Stability for Ships and Ocean Vehicles, STAB 2000, Vol. A, Launceston, Tasmania, Feb. 2000, pp. 305-321 [16] L. Letizia and D. Vassalos, "Formulation of a Non-Linear Mathematical Model for a Damaged Ship with Progressive Flooding", Int. Symposium on Ship Safetyin a Seaway, Kaliningrad, May 1995. [17] E.F.G. van Daalen, et al. "Anti-Roll Tank Simulations With a Volume of Fluid (VOF) Based Navier-Stokes Solver", Proceedings ONR Symposium on Naval Hydrodynamics, Val de Reuil, Sept. 2000 [18] M. K. Ochi, Y. S. Shin, �Wind Turbulent Spectra for Design Consideration of Offshore Structures�, Proceedings, Offshore Technology Conference, Paper OTC 5736, 1988, pp 461-467.

[19] G. Z. Forristall, �Wind Spectra and Gust Factors Over Water�, Proceedings, Offshore Technology Conference, Paper OTC 5735, 1988, pp 448-460. [20] A. G. Davenport �Spectrum of Horizontal Gustiness Near the Ground in Strong Winds�, Quart. J. Royal Met. Soc., vol. 87, April 1961. [21] American Petroleum Institute Recommended Practice for Planning, Designing and Constructing Tension Leg Platforms, RP2T, 1992 [22] W. C. Webster, �Mooring-Induced Damping�, Ocean Engineering, Vol 82, No. 6, 1995, pp571-591.


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