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INTERNATIONAL MARITIME ORGANIZATION

IMO

E

SUB-COMMITTEE ON STABILITY AND LOAD LINES AND ON FISHING VESSELS SAFETY 50th session Agenda item 4

SLF 50/INF.2 26 January 2007 ENGLISH ONLY

REVISION OF THE INTACT STABILITY CODE

Proposal on additional intact stability regulations

Submitted by Germany

SUMMARY Executive summary:

This document outlines an approach with the aim to derive additional intact stability criteria and describes tools and methods that can be used

Action to be taken:

Paragraph 6

Related documents:

MSC 78/INF.5; SLF 48/4/7, SLF 49/5/2 and SLF 50/4/1

Background 1 One of the major reasons for this initiative of a revision of the IS Code (resolution A.749(18), as amended) was the increase of numbers of ships at risk observing severe rolling which mainly resulted in damages to the ships and its cargo in recent years and, fortunately, in few occasions only injuries to persons. It is recognized that ship hull forms of certain ship types have undergone rapid developments. It was mainly governed by the demand of an ever increasing carrying capacity (passengers and cargoes) and other economic demands such as higher speeds at optimized fuel consumptions. This vivid change in characteristics was not matched by a due consideration of the dynamic behaviour in seemingly moderate sea states (e.g., pure loss or parametric roll). 2 Germany had undertaken a Formal Safety Assessment (FSA) proving compelling need for making the presently recommendatory IMO intact stability criteria mandatory and including dynamic stability criteria (MSC 78/INF.5). The results of this FSA prove the feasibility and the cost-effectiveness of this legislative measure and the task was included into the IMO work programme. 3 In order to back up the longer term work tasks, Germany (SLF 48/4/7) supported the implementation of at least some preliminary description of these phenomena and the need for them to be addressed as “dynamic stability criteria” within part A, chapter 2 of the actual IS Code presented in the Correspondence Group report. This was supported by a proposal for probabilistic intact stability criteria for parametric rolling and pure loss phenomena (SLF 49/5/2).

SLF 50/INF.2 - 2 -

I:\SLF\50\INF-2.doc

4 The purpose of this proposal on additional intact stability regulations is to provide a further step towards the development dynamic stability criteria. 5 A general methodology and a possible implementation for consideration are attached to this document, as set out in the annex. Action requested of the Sub-Committee 6 The Sub-Committee is invited to note the information provided in this report and take action as appropriate.

***

2

ANNEX

Proposal

on Additional Intact Stability Regulations

Submitted by Germany

This document outlines a general methodology proposed to derive additional intact

stability criteria and describes a particular implementation including the tools and

methods that will be used. This proposal was prepared in collaboration between

Germanischer Lloyd AG and Prof. Dr.-Ing. H. Söding.

Contact:

Germanischer Lloyd AG, Department Fluid Dynamics,

Vorsetzen 32/35, 20459 Hamburg, Germany

Hamburg, 28 January 2007

3

Contents

Chapter 1. Motivation, Scope and Formulation ............................................................................................................................4 1.1. Why additional regulations on intact stability? ..................................................................................................................4 1.2. Scope: Vessels Addressed by the New Regulations ........................................................................................................6 1.3. Scenarios of Heel Casualties Addressed..........................................................................................................................6 1.4. Proposed Form of the New Stability Criteria .....................................................................................................................8

Chapter 2. General Methodology for Intact Stability Criteria ........................................................................................................9 2.1. Three-Step Methodology of Deriving New Stability Criteria: From Performance To ‘Prescriptive’ Criteria .......................9 2.2. Evaluation of the Long-Term Stability Characteristics ......................................................................................................9

2.2.1. Total Probability Formulation .....................................................................................................................................9 2.2.2. Assumptions about distributions ..............................................................................................................................10 2.2.3. Evaluation of the Short-Term Stability Characteristics ............................................................................................10

2.3. Selection of the Appropriate Safety Level .......................................................................................................................10 2.4. Simple ‘Prescriptive’ Criteria for ‘Everyday’ Use .............................................................................................................11

Chapter 3. A Proposal for Implementation .................................................................................................................................12 3.1. Selection of the Probabilistic Measure of Stability ..........................................................................................................12 3.2. Suggested Form of the New Stability Criteria .................................................................................................................12 3.3. Calculation of the Long-Term Probabilistic Stability Characteristics ...............................................................................12 3.4. Calculation of the Short-Term Probabilistic Stability Characteristics...............................................................................13 3.5. Necessary Number of Simulations for Short-Term Statistics ..........................................................................................13 3.6. Acceleration of Calculations: Extrapolation of rS over H1/3 ..............................................................................................14 3.7. Assumptions about the Distribution p(µ,v | H1/3,T1).........................................................................................................15 3.8. Assumptions about the seaway ......................................................................................................................................16 3.9. Evaluation of the integrals in (3.2)...................................................................................................................................17 3.10. Considerations to the Selection of the Appropriate Safety Level ..................................................................................17 3.11. Influence of parameters T1, µ, v, KG on roll motion......................................................................................................18 3.12. Method to Derive Criteria: Linear Discriminant Analysis ...............................................................................................19 3.13. Validation of the prescriptive criteria .............................................................................................................................20

Bibliography................................................................................................................................................................................20

Appendices.................................................................................................................................................................................22

Appendix A. Simulation Methods and Validation ...................................................................................................................22 A.1. Method ROLLS (roll simulation) .................................................................................................................................22 A.2. Method GL SIMBEL ...................................................................................................................................................26 A.3. Modeling the seaway..................................................................................................................................................29 A.3. Review of the validation of the simulation methods ROLLS and SIMBEL..................................................................29

Appendix B. Estimation of the average rate of casualties......................................................................................................32

Appendix C. Discriminant Analysis ........................................................................................................................................32 References to the Discriminant Analysis ...........................................................................................................................39

Attachments ...............................................................................................................................................................................42 Attachment1. Global Seaway Statistics .................................................................................................................................42 Attachment 2. Validation of ROLLS and GL SIMBEL in regular waves .................................................................................42 Attachment 3. Validation of ROLLS in irregular waves ..........................................................................................................42

4

Chapter 1. Motivation, Scope and Formulation

1.1. Why additional regulations on intact stability?

The rules of Chapter 3.1 of IMO Res. A 749(18) (IS-Code) have been developed to safeguard against

excessive heel and capsizing due to effects of the seaway. Other causes of excessive heel like

transverse wind, water ingress and cargo shift are handled by other regulations. To estimate whether

these rules are adequate only the effects of the seaway on ships will be considered. Both events,

excessive heel (say, beyond ±40 degrees) and capsizing will be called heel casualty in the following.

The general intact stability regulations of Chapter 3.1 specify parameters of the intact, still-water

righting arm curve which have the dimension of a length. They were originally derived for ships of

length ≤100m; their basis were capsizing events of ships most of which had a length <50m. The

question arises whether it is appropriate to prescribe minimum values measured in a length unit

without any scaling to ships of sizes ranging in length from 24m to about 360m. If one considers the

smaller ships as models of the larger ones, one would rather apply Froude scaling. That would mean

that righting arms of a larger ship should be larger in proportion to ship length to generate a

corresponding safety.

Froude scaling is the generally accepted basis of seakeeping model tests. However, for estimating

ship safety in the real seaway, Froude scaling would require that also the length and height of waves

which are responsible for heel casualties obey to Froude scaling. For natural seaways described by

characteristic wave period T1 (=2π/ω1, where ω1 is the circular frequency of the centre of gravity of the

area under the wave spectrum) and significant wave height H1/3, a non-dimensional ‘significant

steepness’ of the seaway may be defined by

1/3

2

1

2 Hs

gT

π= , (1.1)

(g is the gravity acceleration), because 2

1 / 2gT π may be considered as the characteristic wavelength

of the seaway. Froude scaling would require that s is independent from the characteristic period T1 for

those seaways which are of interest here, i.e. which involve the danger of a heel casualty.

Alternatively, the peak (modal) period Tp or the zero-upcrossing period Tz could also be used

instead of T1. That would change the s values, but it would not influence the general conclusions

derived from them.

To indicate which s values occur, Table 1.1 from Söding (2001) is reproduced here. It is a ‘scatter

table’ for the North Atlantic, which corresponds closely to the table assumed by the ISSC for

determining long-term structural loads. The table shows that, for fixed characteristic period T1, the

probability of occurrence of different H1/3 values drops to vanishingly small values (in the table

indicated as 0) at a certain limiting significant height. To establish that limiting height, Fig. 1.1 shows

the s value for the rightmost nonzero entry in each line of Table 1.1 as markers. (Values H1/3 and T1

for the midpoint of the intervals are used.) The line shown in Fig. 1.1 approximates the upper limit of

the markers; it corresponds to the formula

1

max 10.082 0.0023s s T−= − ⋅ . (1.2)

From (1.1) and (1.2) follows the maximum significant wave height as a function of characteristic

period T1 as

( )

1 2

1/ 3 max 1 10.013 0.00036H s T gT

−= − ⋅ . (1.3)

5

Fig. 1.1. Maximum wave steepness according to Table 1 (°) and formula (1.2) (line)

Fig. 1.1 shows that simple Froude scaling is not adequate because the maximum wave steepness

decreases with characteristic period. Expression (1.2) can be used to estimate approximately the ratio

between the required righting arm values for a ship of 360m length to those of a small ship of 36m

length which is considered as a model of the large ship. To derive this ratio, the following

assumptions will be made, which are only rough approximations to reality, but appear sufficient to get

an impression of the required scaling ratio of righting arms:

• The steepest seaways of a given period are most relevant to heel casualties.

• Most relevant for heel casualties are seaways in which the characteristic wave length 2

1 / 2gT π is equal to ship length. (In simulations sometimes smaller, sometimes larger

characteristic wave lengths are found to have maximum effect.)

• If the significant wave height is increased for constant T1, than equal safety requires an

increase of the righting arm curve which is assumed here as proportional to H1/3.

Using these approximations, the ratio α of required righting arms of the large and the small ship can be

determined from the scale factor λ =10 as

( )( )

( )( )

max 1max, 360m max

max, 36m maxmax 1

2 360m / 15.2s 0.04710 10 10 6.6

4.80s 0.0712 36m /

L

L

s T gs s

s ss T g

πα λ

π

=

=

= ⋅= = = = ≈

= ⋅. (1.4)

This shows that the minimum admissible values of the righting arm curve (levers or areas both of

which have the dimension of a length) should increase substantially with ship length, although less

than proportional. But that is not accounted for in Chapter 3.1 of the IS-Code. That is a clear

indication that the IS Code should either be changed or supplemented.

Table 1.1. Relative frequency of occurrence⋅106 of seaways in the North Atlantic having parameters H1/3, T1 as

indicated. For example, the value 0.5 for significant wave height designates the interval between 0 and 1m.

6

The finding that large ships need larger maximum righting levers than small ships corresponds to

practical experience. Many captains of large ships stated that maximum righting levers of 20cm are

much too small according to their experience.

Note that in most cases such small righting arms are not admissible anyway because they would not

comply with the damage stability criteria or with the weather criterion. However, new intact stability

criteria are required because it has not been demonstrated that the damage stability rules and the

weather criterion imply sufficient safety from heel casualties for intact ships in extreme seaways

(conditions for which these criteria are not intended). Besides, these criteria can be satisfied by means

which do not help to reduce heel casualties in intact condition (like a narrow interior subdivision).

Numerous model experiments and motion simulations have shown that large ships may experience

excessive roll motions even if they have much larger righting lever curves than those required by

Chapter 3.1 of the IS-Code; in many cases also ships which comply with all current stability

requirements (including damage stability and weather criterion) were found to be not safe from heel

casualties. Roll angles leading to passenger injuries were reported from passenger ships, and roll

angles exceeding 40 degrees, with immersion of the bridge wing and loss of deck containers, have

been observed on several large container vessels.

Why are so few capsizing casualties reported for large intact ships? Large tankers and bulk carriers

are normally not in danger because they have very large metacentric height and righting levers which

exceed those of Chapter 3.1 of the IS-Code typically by factors in the range of 10. Furthermore, their

righting arms vary typically much less between wave crest and trough than those for ships with smaller

block coefficient, and their low speed decreases the frequency of wave riding and thus of broaching-to.

For these ships no additional regulations would be necessary. On the other hand, large container

ships, passenger vessels and many roro ships and ferries have large freeboard which effectively

reduces the danger of capsizing; but the large freeboard does not exclude the occurrence of heel angles

beyond 40 degrees which may cause severe injuries to passengers and crew, failure of the machinery

and damage or loss of substantial parts of the cargo. For these types of ship the IS-Code has to be

changed or supplemented to avoid the severe consequences of excessive roll motions.

These dynamic stability phenomena have been investigated by several national research programs

in Germany since the late 60’s. Based on this knowledge and a large amount of simulations, a design

criterion for ships suffering from large righting arm variations in seaway has been developed and

proposed at SLF 49 meeting (see SLF 49/5/2(2006)). Notwithstanding the proposed practical

criterion, this paper should generalise the views emerging in Germany on the development of intact

stability criteria, reflect ongoing research in the development of particular tools and methods, and

present respective background information.

1.2. Scope: Vessels Addressed by the New Regulations

If the new rules are based on a reasonable model of the physics of extreme roll motions, they will have

influence on all vessels prone to the dynamic stability problems. Predominantly these are large vessels

(length more than 250m) and vessels of intermediate size (more than 100m). Thus it is expected that

the new stability regulations will increase the safety for medium-sized and large ships, for which the

dynamic stability problems are most important. It seems appropriate that the regulations should aim to

reduce the average rate of capsizing of large intact ships to vanishingly low values, whereas excessive

roll angles (e.g. beyond 40 degrees) may be tolerated to occur with low but not vanishing average rate.

1.3. Scenarios of Heel Casualties Addressed

As stated previously, only effects of the seaway on intact ships are dealt with here. The ships are

assumed to be sufficiently watertight as prescribed in the existing regulations.

The following scenarios may involve heel casualties under these conditions:

(1) Pure loss of stability on a wave crest

(2) Parametric rolling

(3) Resonant rolling in beam sea

7

(4) Shift of cargo or other items due to excessive accelerations in a transverse direction parallel

to the decks

(5) Broaching-to

Various combinations of such scenarios may occur in practice. In spite of that it appears

reasonable to base the planned additional stability regulations on these scenarios, as was also proposed

in Krüger et al.(2006).

Items (1) and (2) are both caused by changes of the righting arm curve in the seaway. Item (1) may

be regarded as an extreme case of item (2). It seems thus plausible that both scenarios can and should

be dealt with by the same regulations which specify a maximum height of the ship’s mass centre of

gravity depending on the ship’s draft and, if necessary, trim.

Regarding item (3), the seaway conditions under which resonant rolling occurs depend on the

righting arm curve, but for arbitrary righting arms resonance conditions may occur. Thus ships have to

be safe also in resonance conditions. Here the roll damping is of main importance. Roll damping is

larger if the ship has substantial forward speed; thus the most critical case may be that of no forward

speed in beam seas, for example due to failure of the propulsion machinery. When assessing the heel

casualty rate of the example ships the case of machine failure has to be included. Further, at least in

some example ships the roll damping has to be varied. When establishing the stability criteria the

effect of roll damping has to be taken into account, for example by minimum requirements for the

bilge keels.

Regarding item (4), safety against shifting of cargo or other heavy masses should be established by

comparing

(a) the transverse acceleration atol parallel to the ship’s decks which is tolerated by the lashings

and other means against shift of heavy masses, with

(b) the acceleration aocc which may occur in heavy seaways. aocc must include the component

of the gravity acceleration parallel to the deck, sing ϕ , due to the heel angle ϕ.

A standard value for atol, for example

2

tol sin 40 6.3m/sa g= =� , (1.5)

may be applicable in most cases.

If in a simulation atol is smaller than aocc, a heel casualty is assumed. To limit its rate of occurrence,

minimum values of KG (corresponding to maximum values of metacentric height GM) will be

required. Correspondingly, in the stability criteria upper limits for GM or righting arms will be

necessary.

Broaching-to, item (5) above, is regarded sometimes as a manoeuvring problem to be solved by

manoeuvring, not stability regulations. However, it seems that in many cases broaching-to cannot be

avoided by reasonable manoeuvring standards. Broaching-to occurs in steep aft and quartering

seaways, especially if the Froude number exceeds 0.20 or 0.25. In principle, broaching-to could be

avoided by sailing with a lower speed. However, due to wind forces and drift forces of the seaway,

speeds corresponding to a Froude number less than 0.2 requires low or vanishing propeller load, and in

this condition the ship may be unable to keep its course. Thus for combinations of quartering, steep

seaways and relatively high speed broaching-to may be unavoidable for small to medium-sized vessels.

To simulate its effect, probably more involved methods are required than for the other conditions

discussed above. In any case, it must be established that the simulation method which is used for cases

of high speed in quartering waves can handle broaching correctly. When this has been demonstrated,

broaching does not require any special treatment for computing the heel casualty rate. Regarding the

stability criteria, to limit the heel casualty rate requires sufficiently large righting arms also in cases of

broaching. Whether the necessary righting arms must depend on the parameters such as rudder size,

overshoot angle in zigzag test etc., must be established by varying the manoeuvring qualities of some

example ships. It may turn out that the yaw moments caused by the waves are so large that all the

details which are important for manoeuvring in still water are negligible for broaching in steep

seaways.

8

1.4. Proposed Form of the New Stability Criteria

Following the above considerations, a representative sample of vessels should be selected. To ensure

representativeness of the sample used, international collaboration under the coordination of IMO

would be helpful.

The aim is to establish the limits for the height of the centre of gravity of any given ship as a

function of its draft and, if necessary, of its trim angle. It appears unnecessary to specify the loading

case explicitly because the influence of different but reasonable inertial radii on roll for fixed draft,

trim and GM is small.

The limits of the height of the centre of gravity should ensure sufficient safety against heel

casualties, which is quantified by an appropriate probabilistic stability measure (for example,

‘capsizing index’ in SLF 49/5/2 (2006) or the rate of heel casualties proposed here in Chapter 3.

The new criteria specifying the minimum acceptable stability limits will look like upper limits for

the height of the centre of gravity for different drafts, although the following extensions will also have

to be done sooner or later:

• lower limits of the height of the centre of gravity to avoid excessive accelerations leading

to a shift of heavy masses on board, people discomfort etc.;

• if roll damping proves important for the dynamic stability problems addressed, its lower

limits should be also determined.

Variation of these parameters will lead to the size of the sample being equal to the number of

vessels considered times number of drafts times number of the centre of gravity height variations etc.

9

Chapter 2. General Methodology for Intact Stability Criteria

This chapter outlines the framework and the general methodology proposed here as a common basis

for deriving new intact stability criteria. Concrete implementation of this methodology may however

vary, see for example SLF 49/5/2 (2006) and Chapter 3 of this proposal.

2.1. Three-Step Methodology of Deriving New Stability Criteria: From Performance To ‘Prescriptive’ Criteria

As proposed in SLF 49/5/2 (2006) and also supported here (see Chapter 3 for the details of

implementation), new intact stability criteria should be derived in the following three major steps

given the sample of vessels:

• First, long-term stability characteristics of all vessels in the sample are evaluated using a

probabilistic ‘performance-based’ procedure. Examples of such characteristics are the

‘capsizing index’ proposed in SLF 49/5/2 (2006) and the rate of heel casualties proposed

here and described in Chapter 3, but other reasonable choices are indeed possible.

• Second, an appropriate limit of the acceptable stability performance needs to be selected,

and the vessels in the sample are divided into ‘safe’ and ‘unsafe’ accordingly. Various

concepts can be applied to identify such a limit: Experience from the past (statistics and

accidents study) can be used for this purpose, or analysis of the current safety level, as it is

proposed here in Chapter 3, in the spirit of ‘objective-driven rules’. In any case, such a

limit may need to be re-adjusted later.

• Although any vessel can be assessed following the same probabilistic performance-based

procedure as the one used for the vessels in the sample, this would be impractical for

everyday use and will be done only in particular cases. Therefore, simplified ‘prescriptive’

stability criteria need to be derived from the above analysis, such as the ‘simplified

capsizing index’ in SLF 49/5/2 (2006).

Such criteria represent simple mathematical expressions not requiring complex

calculations. For example, they may relate the required still-water righting levers to the

righting-lever alterations between wave crest and trough.

2.2. Evaluation of the Long-Term Stability Characteristics

2.2.1. Total Probability Formulation

For a given draft, trim and height of the centre of gravity, a long-term probabilistic stability

characteristic needs to take into account how often different values of the other main parameters

influencing ship safety will occur, namely seaway parameters (especially the significant wave height

and the characteristic wave period), and operational parameters (such as course angle relative to the

seaway and ship speed).

It is proposed to calculate the long-term probabilistic stability characteristics using a total

probability formulation, integrating over a multi-dimensional probability density function (of the

seaway and operational characteristics, see 2.2.2) multiplied with the probabilistic short-term stability

performance measure evaluated for a given seastate and operational characteristics (see 2.2.3).

Evaluation of such long-term performance measure thus requires explicit assumptions about the

distributions of the seaway and operational parameters, and a procedure for the evaluation of the short-

term probabilistic stability measures.

10

2.2.2. Assumptions about distributions

Many feasible choices exist for the distributions used in the calculation of the long-term probabilistic

stability measures. Moreover, this area attracts significant research interest and certainly with

increasing knowledge and thus increasing reliability of such estimations more accurate models will be

applied.

Presently, for the safety regulations regarding a minimum stability standard for ships without

restrictions of service area, the following assumptions seem appropriate:

• the annual average scatter table for the North Atlantic should be used for the seaway

distribution, neglecting the fact that a vessel spends part of its operational time in less

severe areas or in port;

• the course angle and ship speed distributions should not take into account operational

aspects, such as weather routing or the actions of the crew aimed at improving safety in

heavy seaways. This should be also reflected in the selected safety level.

2.2.3. Evaluation of the Short-Term Stability Characteristics

To evaluate the short-term probabilistic stability characteristics of a vessel in a given seaway for given

operational conditions, it is proposed here to use numerical simulations of ship motions in a seaway.

Simulation method Rolls was proposed in SLF 49/5/2 (2006) and a combination of Rolls and

GL SIMBEL in Chapter 3 of this proposal (see Appendix 4.1 for the description and validation of

Rolls and GL SIMBEL), although the choice is certainly not limited to these methods. In principle,

any validated simulation code can be used for these calculations if the dynamic stability phenomena of

interest are modeled sufficiently accurate. Moreover, wide collaboration in this area, with cross-

validation and benchmarking of the methodology using different simulation tools for a shared

selection of vessels would improve the quality of and confidence in the new stability criteria.

Simulations have to be performed for a large number of vessels, and for each vessel, for additional

variations of draft, height of the centre of mass, different operating conditions and seaway parameters.

Besides, at the limit between safe and unsafe which is of interest in deriving minimum stability

requirements, the simulations will have to be performed over long simulation time to ensure the

necessary confidence of the estimates. Therefore, means of decreasing the computational effort to a

feasible amount without unacceptable decrease in accuracy are needed.

The previous proposal SLF 49/5/2 (2006) used for this purpose the method by Blume (1987), while

the present proposal (see Chapter 3) makes use of the idea of Tonguć and Söding (1986), extrapolating

the rate of heel casualties over the significant wave height. Indeed, the choice is not limited to these

two methods; comparison with other approaches, for example Söding (2005) and Jensen and Pedersen

(2006) would be useful.

2.3. Selection of the Appropriate Safety Level

The most comprehensive approach to the selection of the acceptable minimum stability performance

would be a cost-benefit analysis. The aspired degree of safety due to the new regulations should be

such that additional costs of reducing losses from heel casualties has approximately the same effect as

it would have when the same resources are used to decrease other losses, for example due to fire or

collision. (If this condition were not satisfied, it would be worthwhile to decrease the safety of the

casualties which are expensive to avoid, and use the released resources to improve the safety against

the ‘cheap to avoid’ casualties.)

At present, this principle seems premature to quantify the degree of safety which the rules should

provide, and a more feasible way appears to start from an estimate of the contemporary level of safety,

for example using numerical simulations, statistical data (see Chapter 3), and accident data. In any

case, the figure reflecting the minimum acceptable safety is not an absolute constant and will

presumably change, maybe even during the work on the new stability criteria, for example for the

following reasons:

11

• The level of safety aspired initially appears too low or too high (for example, too expensive

to achieve).

• The generally accepted risk level changes.

• The methods applied to assess the stability performance of ships contain a systematic error

which can be estimated quantitatively.

2.4. Simple ‘Prescriptive’ Criteria for ‘Everyday’ Use

The aim so far has been to establish the limits of the height of the centre of gravity (and possibly, other

characteristics such as roll damping) depending on draft (and possibly, trim), at which the ship

experiences, roughly, the minimum acceptable stability performance (for example, the limit rate of

heel casualties as proposed in Chapter 3). These results are obtained using numerical simulations of

ship motions in a series of seaways.

The same procedure cannot be used routinely for all new ships, therefore, these results, obtained

for a sufficiently large sample of vessels, are now used to derive ‘prescriptive’ criteria, which are

much easier to check, for example, minimum requirements for the still-water righting arm curve, or

righting-arm alterations in waves like in SLF 49/5/2 (2006) etc. Such ‘prescriptive’ criteria should be

derived by a rational method aimed to ensure that, if the criteria are satisfied but not over-satisfied, the

ship will have stability characteristics in waves which are as close as possible to the minimum

acceptable performance.

Indeed, many methods can be used to derive such criteria, for example the linear regression

analysis proposed in SLF 49/5/2 (2006) or linear discriminant analysis proposed here in Chapter 3.

Additionally, validation of the new ‘prescriptive’ criteria would be necessary on a possibly large

basis, for example, testing the criteria on vessels not used in their derivation, comparison with existing

or concurrently emerging criteria, testing on vessels which experienced casualties.

Finally, for cases in which the ‘prescriptive’ criteria appear not applicable, and perhaps also for

other cases where clear benefits of a ‘performance-based’ evaluation exist, the regulations should also

accept the direct assessment of the stability performance through numerical simulations. Naturally,

this would require that such numerical methods are acknowledged.

12

Chapter 3. A Proposal for Implementation

This chapter describes a particular implementation of the general methodology outlined in Chapter 2.

Note that different building blocks may be used following the same general methodology; see for

example SLF 49/5/2 (2006) for another implementation.

3.1. Selection of the Probabilistic Measure of Stability

Using the the average rate of losses (i.e. the average loss per time) as a probabilistic measure of safety

eliminates the need to fix a reference time. It may not be always appropriate to relate the losses to the

expected lifetime of the ship, because both the benefit rate of applying the new rules and the cost rate

of satisfying them may not depend substantially on the ship’s lifetime. For the same reason it appears

appropriate to limit the expected rate of heel casualties rather than the probability of such casualties

during any fixed time. Nevertheless, the latter measure can be easily calculated from the known rate

of heel casualties if necessary.

3.2. Suggested Form of the New Stability Criteria

The aim is to establish limits for the height KG of the centre of gravity of any given ship depending on

its actual draft d and, if necessary, on its trim angle θ. The KG limits should ensure sufficient safety

against heel casualties, or, more specifically, a small tolerable rate of heel casualties. Thus, as an

additional intact stability criterion, upper limits of KG have to be established to avoid heel casualties.

Possibly, also other criteria will be derived, for example for for the lower limits of KG (i.e. upper

limits of GM) to avoid excessive accelerations leading to a shift of heavy masses on board.

3.3. Calculation of the Long-Term Probabilistic Stability Characteristics

If the ship’s draft and trim are fixed, there remain the following main parameters influencing the

casualty rate in a certain seaway and operating condition, i.e. the short-term rate of heel casualties:

• significant wave height H1/3

• characteristic wave period, for example the mean period T1

• course angle µ relative to the main direction of wave propagation

• ship speed v

• height KG of the centre of gravity

The variable KG is treated differently from the other variables because the aim here is to establish

limiting values of KG.

To estimate the safety of a certain KG the approach has to take into account how often different

values of the other parameters will occur. This leads to the long-term heel casualty rate

( )max

2

1/ 3 1 1/3 1 1/ 3 1

0 0 0 0

( , , , ) ( , , , , ) d d d d

v

L Sr KG p H T v r H T v KG H T v

π

µ µ µ∞ ∞

= ⋅∫ ∫ ∫ ∫ , (3.1)

where p(H1/3,T1,µ,v) is the four-dimensional probability density for the seaway and operation state

characterized by these four parameters, and rS(H1/3,T1,µ,v,KG) is the short-term heel casualty rate.

To enable using scatter diagrams of seaway, (3.1) is written in a slightly different form:

( )max

2

1/ 3 1 1/3 1 1/3 1 1/ 3 1

0 0 0 0

( , ) ( , | , ) ( , , , , ) d d d d

v

L Sr KG p H T p v H T r H T v KG H T v

π

µ µ µ∞ ∞

= ⋅ ⋅∫ ∫ ∫ ∫ . (3.2)

13

Here, the two-dimensional probability density p(H1/3,T1) of the seaway data follows from a scatter

table1 by dividing the entries of such a table by the corresponding intervals ∆H1/3 and ∆T1. The

resulting values represent p(H1/3,T1) at the centre of each tabulated interval. If values at different

parameter pairs H1/3,T1 are required, the logarithm of the probability density will be interpolated

linearly over both parameters. p(µ,v | H1/3,T1) is the two-dimensional probability density of wave

direction and speed for a given pair of seaway data H1/3,T1.

3.4. Calculation of the Short-Term Probabilistic Stability Characteristics

Application of formula (3.2) requires to determine rS(H1/3,T1,µ,v,KG) for a large number of parameter

combinations, including different draft and possibly trim conditions. Each rS has to be found from a

numerical simulation, possibly over a long time range, ensuring the necessary confidence level of the

statistical estimates. Thus means of decreasing the computational effort without unacceptable

decrease in accuracy are required. Further, the ways to handle the probability density p(µ,v| H1/3,T1)

must be discussed.

To estimate the short-term heel casualty rate, the simulation methods ROLLS and GL SIMBEL will

be used. The number of heel casualties (for example, the number of times that 40 degrees roll is

exceeded) will be counted and divided by the time for which the simulation was performed. If a heel

casualty occurs, the simulation is continued with a small roll angle and zero roll velocity as new initial

values. Whether the simulation is done in one long run or in several shorter ones is without influence

as demonstrated in Appendix A.3 except for cases of very high casualty rates which are irrelevant for

establishing the limit of admissible KG values.

3.5. Necessary Number of Simulations for Short-Term Statistics

The more heel casualties is encountered during a simulation, the better is the approximation for the

average heel casualty rate r, but the higher is also the numerical effort. Fig. 3.1 shows the probability

of finding various numbers of casualties during time T; the rate r of casualties varies between 5/T and

20/T. The reasoning underlying this figure is given in Appendix B.

Fig. 3.1. Probability of finding various numbers of casualties during time T if the average

casualty rate times T is 5, 10 or 20

1 There are several sources of long-term wave statistics: Söding (1990), IACS (2001), ISSC (1994),

NATO (1983) and (1987). It is worthwhile to study the influence of the choice of the long-term scatter

diagram on the result. Should this influence be important, this choice must be further discussed.

14

From Fig. 3.1 follows that, if the average number of casualties during a simulation time T is 20, in a

simulation a number between 15 and 25 casualties will be encountered with a probability of 78.3%.

Thus, with this probability the average casualty rate is approximated with an accuracy of 25%. In

view of the fact that the casualty rates are very sensitive to small changes of KG, and that this short-

term rate is averaged over a large number of seaways to the long-term rate, it seems possible that a

smaller number of casualties, e.g. 10, is sufficient to estimate the short-term casualty rate.

3.6. Acceleration of Calculations: Extrapolation of rS over H1/3

Tonguć and Söding (1986) argue that, for small values of rS, the dependence of rS on H1/3 can be

estimated from the probability distribution of maxima in a Gauß process, which are, approximately,

Rayleigh-distributed. They do not assume that the maxima of roll motion are Rayleigh-distributed;

they are certainly not. Instead, they assume that large roll amplitudes occur with a certain (unknown)

probability if a certain (also unknown) number of successive wave amplitudes exceed a certain

(unknown) limit value. The following relation follows from this general assumption:

2

1/3ln /Sr A B H− = + . (3.3)

This relation has been tested in a large number of simulations. Fig. 3.2 shows an example; it

demonstrates also the typical error range of this formula. The case with ϕmax=60 degrees (angle at

which a casualty is assumed) corresponds quite well to the linear relation between ln Sr− and 2

1/ 3H−

,

while for ϕmax=40 degrees there are deviations from the linear relation for smaller values of H1/3. This

was found also for various other cases (see for example Fig. 3.3).

0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01 0.011 0.012 0.0130

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

6

6.5

7

7.5

8

8.5

9

1/ H1/3

2

10 m30 20 1115 14 13 12

ln(c

ap

siz

era

tep

er

roll

pe

rio

d)

φ = 40maxφ = 60max

1/3H =

−

Fig. 3.2. Dependence of the logarithm of the average short-term heel casualty rate rS on H1/3. rS in 1/average roll

cycle duration; estimated from 20 casualties.

This deviation from (3.3) may be understood: Using the same reasoning as in Tonguć and Söding

(1986), the following relation between rS and H1/3 is found:

2

1/3ln /Sr A nC H− = + , (3.4)

where n is the number of successive wave amplitudes which all exceed the (unknown) limit value. In

Tonguć and Söding (1986) it was assumed that n is independent from H1/3; this results in (3.3).

However, in a lower seastate the average roll amplitude will be lower, and thus more successive high

waves may be necessary to increase the roll amplitude from its average value to the limit value.

Perhaps an approximate relation between n and H1/3 can be established by combining theoretical

15

reasoning with simulations. If not, the linear relation can be used and the fact had to be tolerated that

the estimated casualty rate may be slightly higher than that which would follow from simulations in

lower significant wave height.

Formulae like (3.3) and (3.4) can be used not only to substitute the variation of H1/3; they also allow

eliminating the need for very long simulations which would be required to determine the low heel

casualty rates at the limit between safe and unsafe. Suitable values of H1/3 to be used in the

simulations are those for which the rate of a heel casualty is between 0.02 and 0.002 per average roll

period.

3.7. Assumptions about the Distribution p(µµµµ,v | H1/3,T1)

In steep head waves, the maximum attainable ship speed is usually much lower than the maximum

speed in still water. Thus it seems necessary to determine the maximum continuous ship speed vmax in

head seas depending on µ, H1/3 and T1. Krüger et al. (2006), par. 5.4.2 state that they developed an

estimation formula for this quantity which depends on the ship’s block coefficient. If that estimation

formula is documented in detail and found applicable, it is recommended for use also here. Otherwise

an own formula of that kind should be established. A very good reference for a method to do so is

Schenzle et al. (1974).

Additionally, the minimum ship speed vmin should be determined depending on the propulsion plant.

For example, in ships with a controllable pitch propeller the minimum speed may be assumed as zero,

whereas in ships with a fixed-pitch propeller driven by a diesel engine, 15 to 25 percent of the design

speed seems appropriate, provided that this is more than vmax. If not, this course angle µ relative to the

wave direction appears unfeasible in the actual seaway and should be excluded from the integral in

(3.2), whereas the probability density in the feasible range must be increased correspondingly.

Sailing with high speed is certainly more frequent than with low speed. However, to assume

always maximum speed would neglect the (often much higher) probability of heel casualties for lower

ship speed. Therefore, it is proposed here to use a probability density of speed which, for each

combination of µ, H1/3 and T1, is proportional to v (but not minv v− ) between the limits minv and maxv .

The proportionality constant follows from the total probability of the parameter combination µ, H1/3,

T1.

Failures of the propulsion machinery occur too often to be neglected. (An example of a capsizing

during failure of the main engine is the containership E.L.M.A. Tres, which capsized in a hurricane in

Fig. 3.3. The same as in Fig. 3.2 for different mean wave directions and significant wave periods

16

1981.) Thus it is proposed to add a separate case of failed machinery. This leads to a) a particularly

calculated conditional heel casualty rate, b) the need to establish a corresponding safety level, which

should take into account the time fraction (probability) of the occurrence of such a condition, and c) an

additional ‘prescriptive’ criterion.

The heel casuality rate for this particular case can be calculated using the following modification of

formula (3.2):

( ) e 1/ 3 1 1/ 3 1 0 1/3 1

0 0

( , ) ( , , ,0, )d dL Sr KG p H T r H T KG H Tµ∞ ∞

= ⋅∫ ∫ . (3.5)

Here, µ0 is the angle between the ship’s longitudinal axis and the wind speed vector which will

occur under the action of the wind on the ship without forward speed. The formula assumes that the

direction of the wind vector (opposite to the usual designation of the wind direction) and the seaway

direction are the same, and it neglects the time necessary for the ship’s course angle to adjust to that

direction.

If the longitudinal centre of gravity of the above-water lateral area of a ship is located near to that

of its under-water lateral area, the ship will turn transverse to the wind (µ0=90 degrees) and the

seaway. On the other hand, a feeder ship without deck containers will turn its bow into the wind due

to the wind yaw moments produced by its high deck house. Thus it is recommended here to estimate

the angle µ0 and to use only this one angle as µ for the condition of failed propulsion machinery. The

stationary wind angle may be determined using wind force data by Blendermann (1993) in

Brix (1993).

Although several authors attempted to estimate the frequency of occurrence of different wave

encounter angles µ, see for example van Daalen et al. (2006), such estimates do not appear to be

applicable in general. Facts influencing the µ distribution are:

• Weather routing tends to increase the frequency of stern waves.

• The slower speed which can be attained in head waves tends to increase the frequency of

head waves because the ships need more time to cross the area of head seaway.

In view of the uncertainties associated with an attempt to establish a correct distribution, and of the

moderate influence which reasonable deviations from a uniform distribution would have, a uniform

distribution of µ values is applied here except for the unfeasible directions mentioned before.

3.8. Assumptions about the seaway

To derive wave spectra from the values T1, H1/3 and µ, the JONSWAP spectrum will be used. It

requires to specify a peak enhancement factor γ, which may vary, roughly, between 1 and 8 for wind

sea. It is planned to use only a fixed value γ=3.3, which was found as average value in the original

JONSWAP measurements. Because of the moderate influence of γ on rS, and in view of the many

simulations required to establish a single casualty rate, that simplification seems both necessary and

tolerable.

Contrary to wind seas, in swell there occur widely varying spectra. Especially spectra in which the

wave energy is concentrated in a very narrow frequency range may occur in swell and may be relevant

for heel casualties, for example if this frequency range corresponds to a (direct or parametric)

resonance condition. On the other hand, the probability that, in a narrow spectrum, resonance

conditions are met, is smaller, and smaller changes of speed and/or course angle may suffice to avoid

resonance conditions in such a case. Thus swell may be most important in case of failed propulsion

machinery. It is planned to investigate whether swell should be taken into account explicitly, for

instance by selecting also larger peak enhancement factors for H1/3-T1 pairs which are typical for swell.

Another parameter governs the distribution of wave energy over the direction α of wave

propagation in a single short-term seaway having main direction µ. Usually this distribution is

assumed proportional to the function

( )cos if 90 , otherwise 0n α µ α µ− − < �. (3.6)

17

For the exponent n, values of 2 and 4 are applied most often. Presumably, the effect of variable n

values is negligible, but this has still to be investigated.

Sometimes it is speculated that ‘cross seas’ where wind see and swell have different directions and

characteristic periods may be more dangerous than wind sea alone. However, it has never been

demonstrated nor made probable that cross seas cause higher rates of heel casualties than wind sea

alone. Therefore, most probably cross seas need not be included; also this has to be tested.

3.9. Evaluation of the integrals in (3.2)

To evaluate the 4-fold integral in (3.2) the subroutine Gint (Söding (1990)) appears suitable. The

routine aims to use as few evaluations of the integrand as possible to obtain a specified accuracy. In

this method, the quadruples H1/3,T1,µ,v for which simulations are performed are densely concentrated

where the integrand, i.e. the probability density of the seaway times probability density of the

operating condition times heel casualty rate, depends sensitively on these four parameters, while

ranges where at least one of the three factors is small are covered only by a coarse grid of evaluation

points. To establish limiting values for KG depending on draft, the fourfold integral has to be

evaluated for different values of KG and draft. The computational effort can be reduced further by

specifying lower accuracy if the long-term rate of heel casualties turns out to be far below or far above

the limit rate.

3.10. Considerations to the Selection of the Appropriate Safety Level

According to various statistics published in the journal “Shipping Statistics and Market Review”, for

the world’s merchant fleet of ships exceeding 500 BRZ the number of ship losses per year due to

adverse weather conditions is about 20. Because the world’s merchant fleet comprises about 37 000

ships >500 BRZ, the current total loss rate due to bad weather is, for a single ship, about 5⋅10-4 per

year. Not all these losses are heel casualties; on the other hand, there are ship losses from unknown

reasons; part of these losses may be due to capsizing. And the majority of heel casualties do not lead

to total loss of the ship. Thus, the current rate of heel casualties in larger ships is probably in the range

of 10-2 to 10-3 per year. To reduce this rate, the selected ‘limit’ (accepted) rate should be smaller; thus

a rate below 10-3

per year would probably constitute an increase in safety.

When fixing a limit rate for casualties, the following cases should be distinguished: a) cases where

an increase in safety over the current value requires large additional effort, and b) cases where even a

small effort can improve the safety substantially. Stability rules as planned here belong to type b).

This is illustrated in Fig. 3.4, showing a dependence of the annual heel casualty rate calculated using

(3.2) as a function of the height above the keel of the centre of gravity KG for a container vessel. Note

that a decrease of KG by about 0.3m decreases the heel casualty rate by one order of magnitude. This

enables selection of a low limit rate of heel casualties because it costs only slightly more than a higher

rate.

As discussed in Chapter 2, an important question in the selection of the minimum acceptable safety

level is whether the new rules should guarantee safety only under the condition that the crew acts at

improving the safety in severe seaways, or whether the ship should be safe to a prescribed minimal

degree in arbitrary operating conditions which are physically possible. It seems appropriate that

minimum stability requirements for ships without restrictions of service area should not take into

account operational aspects influencing the safety as, at least presently

• it cannot be guaranteed that the necessary operational measures in heavy seaways are specified

in sufficient detail for every ship and always applied by the master,

• there is no reliable means allowing the master to estimate the seaway conditions, so that he

can follow the recommendations,

• the minimum safety standards aimed at here are attainable and not too restrictive even without

additional operational considerations.

Therefore, in calculations of the heel casualty rate:

• Weather routing as well as course and speed changes due to heavy seaways are not considered.

18

• The seaway distribution for the North Atlantic Ocean is assumed; it is neglected that the ship

spends part of its time in less severe sea areas or in port.

• It is assumed that the actual KG value is always at the allowed limit. (Whether this occurs at a

constant draft or in variable loading conditions is without influence because the same tolerable

rate of heel casualties is assumed for all load cases.)

These three simplifications increase the computed heel casualty rate compared to the correct rate.

To account for this, a relatively high limit rate is proposed here: 10-3 per year.

This figure should not be considered as an absolute standard; it might be changed even during the

work on the new stability criteria as discussed in Chapter 2.

3.11. Influence of parameters T1, µµµµ, v, KG on roll motion

To obtain some feeling for the influence of these parameters on the roll motion, Fig. 3.5 taken from

Brunswig et al. (2006) shows the roll angles of a large containership L⋅B⋅T=317m⋅43.2m·14.4m;

GM=1.26m) in steady-state response to regular head (right side) and stern (left side of Fig. 3.5) waves

of 10m height. The responses were simulated using the program ROLLS. Wave length and Froude

number were varied. The figure shows parametrically excited roll motions. The small blue ‘island’ to

the left is due to parametric ‘resonance’ for the frequency ratio c=1. Much more important is the

‘resonance’ near to the frequency ratio c=0.5. Maximum responses occur, however, at c≈0.4 because

the frequency ratio was defined using the roll natural frequency for small-amplitude motions, whereas

actual roll motions have large amplitude and thus higher frequency, because the righting arm curve

exceeds the linear function GM⋅ϕ for the most relevant heel angles in this type of ship. The figure

shows that a wide range of ship speeds may lead to severe roll motions.

The relatively sharp resonance peak over wave length (but at different wave lengths for different

Froude number) is an artefact of using regular waves. In natural seaways a broad range of T1 values

(corresponding to widely varying characteristic wave lengths gT12/2π) may cause severe roll motions.

This is shown in Fig. 3.6. For larger (smaller) metacentric height the resonance region of large roll

motions would shift parallel to the curves of constant c to the lower right (upper left).

Fig. 3.4. Long-term heel casualty rate as a function of KG

19

Fig. 3.6. Maximum roll angle in natural head and stern wind seas during 0.5 hours for the same ship as Fig. 3.5

3.12. Method to Derive Criteria: Linear Discriminant Analysis

In the previous sections it was described how numerical simulations of ship motions in seaways can be

used for a sufficiently large number of vessels to establish the limits of KG (and, possibly, also other

parameters such as roll damping) as a function of draft and perhaps trim, for which the ship

experiences the maximum allowed rate of heel casualties.

These results shall be used to derive ‘prescriptive’ criteria which are much easier to check (i.e.

without carrying out simulations of ship motions in seaways), like for example minimum requirements

for the still-water righting-arm curve.

The ‘prescriptive’ criteria will be derived by the linear discriminant analysis. They are defined in

such a way that, if the criteria are satisfied (but not over-satisfied), the ship will experience heel

casualty rates which approximate the limit rate as closely as possible.

Suppose that heel casualty rates have been determined for a number of examples called cases in the

following. Each case refers to a certain ship variant and loading condition. Variants may vary with

respect to draft, trim, KG, maximum speed in still water, bilge keel area, rudder size or whatever else

may influence the casualty rate. After performing numerical simulations to determine the casualty rate

according to (3.2), or similar formulae like (3.5) for the case of failed machinery, the cases will be

grouped into two classes: ‘unsafe’ cases, for which the computed casualty rate is larger than the limit

rate, and ‘safe’ cases, for which the opposite is true.

For a new case, for example a new ship at a certain draft and KG value, the decision must be made

whether it is safe or unsafe, but without performing the cumbersome simulations necessary to calculate

the casualty rate. Instead, the aim is to use parameters which appear characteristic for the degree of

safety of the case and which can be determined with small effort.

Fig. 3.5. Roll amplitudes in head and stern waves of 10m height. From Brunswig et al. (2006).

20

Krappinger and Sharma (1974) describe a rational method to do so (see Appendix C). As a

measure of the ‘discriminating quality’ of a parameter (‘criterion’) x they define the ratio

( )2

safe unsafe

safe unsafe

2x x

v vα

−=

+, (3.7)

where safex is the mean value of x for the safe cases, and safev is the variance of x for the safe cases;

correspondingly for unsafex and unsafev . α should be determined for various parameters x and select that

which has the largest α, i.e. which allows best to decide whether a new case is safe or unsafe. Further,

the value of x has to be selected that distinguishes between safe and unsafe cases. One may, for

instance, select the value which minimises the number of incorrect decisions but other choices (for

example, allowing fewer ‘unsafe’ errors than ‘safe’ ones) are also possible.

An important extension of this principle is the use of an expression containing several x values,

combined into a complex criterion, for example

1 1 1 1 2 2 3 3 or ln

n nc x c x c x c x c x+ ⋅⋅⋅+ + + . (3.8)

The constants ci are determined from an optimisation problem, using the method of least squares, to

maximise the discriminating quality α of the expression. Krappinger and Sharma (1974) show that

such expressions can have higher discriminating quality than the best single parameter xi.

In the application under consideration, one could, for instance, test various parameters of the still-

water righting lever curve (for example, GM, h(40°), hmax, area under the curve) and of righting lever

curves in assumed waves of different length, height and phase angle, and then combine these

parameters to an expression which contains only the most relevant parameters and has the maximum

discriminating quality. If other data, such as dimensions of the bilge keel or maximum ship speed in

still water, turn out to have substantial influence on the casualty rate, these should also be included.

3.13. Validation of the prescriptive criteria

One or, more likely, several prescriptive criteria (for the different casualty scenarios) must be

validated for cases not used in their derivation. In particular, it is planned to determine the expected

long-term heel casualty rates for ships which are at the limit of satisfying

• only the rules of Chapter 3.1 of A 749(18),

• all existing stability regulations applicable to the actual ship,

• the newly proposed prescriptive criteria,

• alternative variants (for example, simpler, more strict, or less strict) of the new criteria.

Further it appears worthwhile to test the criteria for ships which experienced a heel casualty; the

new criteria must specify all such cases as unsafe.

Here, collaboration, benchmarking studies and discussions on national and international level

would be most helpful.

Bibliography

Blendermann, W. (1986) Die Windkräfte am Schiff. Report No.467, Institut für Schiffbau, Hamburg

Blendermann, W. (1993) Wind loads, in: Brix, ed., Manoeuvring Technical Manual

Blendermann, W. (2002) Short-term and long-term statistics of the wind loads on ships, Ship Technology

Research 49

Blume, P. (1976) Zur Frage der erregenden Längskraft in von achtern kommenden regelmäßigen Wellen. Report

No.334, Institut für Schiffbau, Hamburg

Blume, P. (1979) Experimentelle Bestimmung von Koeffizienten der wirksamen Rolldämpfung und ihre

Anwendung zur Abschätzung extremer Rollwinkel, Schiffstechnik 25(1) 3-29

Blume, P. and Hattendorff, H.G. (1984) Stabilität und Kentersicherheit moderner Handelsschiffe. Report

No.184/84, Hamburg Model Basin (HSVA)

21

Blume, P. (1987) Development of new stability criteria for modern dry cargo vessels, Proceedings, PRADS 1987

Böttcher, H. (1986) Ship motion simulation in a seaway using detailed hydrodynamic force coefficients,

Proceedings, 3rd Int. Conf. on Stability of Ships and Ocean Vehicles STAB’86, Gdansk

Brix, J. (ed.) (1993) Manoeuvring Technical Manual. Seehafen Verlag, ISBN 3-87743-902-0

Brunswig, J., Pereira, R. and Kim, D.-W. (2006) Validation of parametric roll motion predictions for a modern

containership design, Proc., 9th Int. Conf. on Stability of Ships and Ocean Vehicles STAB’2006

Chang, B.-C. (1999) On the survivability of damaged ro-ro vessels using a simulation method. PhD thesis,

Institut für Schiffbau, Hamburg

van Daalen, E.F.G., Boonstra, H. and Blok, J.J. (2006) Capsize probability analysis for a small container vessel,

International Maritime Organisation, Sub-Committee on Stability and Load Lines and on Fishing Vessels

Safety, SLF 49/INF.7, 19 May

Grim, O. (1961) Beitrag zu dem Problem der Sicherheit des Schiffes im Seegang, Schiff und Hafen 490

Hennig, J., Billerbeck, H., Clauss, G., Testa, D., Brink, K.-E. and Kühnlein, W.L. (2006) Qualitative and

quantitative validation of a numerical code for the realistic simulation of various ship motion scenarios,

Proceedings, 25th Int. Conf. on Offshore Mechanics and Arctic Engineering OMAE 2006, Hamburg,

Germany, June 4-9, Paper OMAE2006-92245

IACS Recommendation No.34 (2001) Standard Wave Data

ISSC (1994) Int. Ship and Offshore Structures Congress, Jeffery, N.E. and Kendrick, A.M. (ed.), St.John’s

Jensen, J.J. and Pedersen, P.T. (2006) Critical wave episodes for assessment of roll, Proceedings, 9th Int. Marine

Design Conf. IMDC’06, Ann Arbor, Michigan, 399-411

Kröger, P. (1987) Roll Simulation von Schiffen im Seegang, Schiffstechnik 33(4) 187-216

Krappinger, O. and Sharma, S.D. (1974) Sicherheit in der Schiffstechnik (in German). Jahrbuch

Schiffbautechnische Gesellschaft, 329-355

Krüger, S. and Cramer, H. (2001) Numerical capsizing simulations and consequences for ship design. Proc.

Schiffbautechnische Gesellschaft

Krüger, S., Hinrichs, R., Kluwe, F. and Billerbek, H. (2006) Towards the development of dynamic stability

criteria, Hansa 143(6) 204-214

Allied Naval Engineering Publication ANEP11 (1983) Standardized Wave and Wind Environments for NATO

Operational Areas, April

Allied Naval Engineering Publication ANEP14 (1987) Seasonal Climatology of the North Sea (16-Year-

Statistics), October

Pereira, R. (1988) Simulation nichtlinearer Seegangslasten, Schiffstechnik 35(4) 173-193

Pereira, R. (1989) Ermittlung der Belastungen von Schiffen in steilem Seegang durch Simulation, Proc.

Schiffbautechnischen Gesellschaft 83, Summer Meeting, Berlin, 145-158

Pereira, R. (2003) Numerical simulation of capsizing in severe seas, Proceedings, 8th Int. Conf. on the Stability

of Ships and Ocean Vehicles STAB’2003, Madrid

Pereira, R. and Puntigliano, F. (1993) Simulation of loads and motions of ships in severe seas and during

manoeuvres, Proceedings, 3rd Pan American Congress of Applied Mechanics PACAM III, São Paolo,

Januar

Petey, F. (1986) Forces and moments due to fluid motions in tanks and damaged compartments, Proceedings, 3rd

Int. Conf. on Stability of Ships and Ocean Vehicles STAB’86, Gdansk

Schenzle, P., Boese, P. and Blume, P. (1974) Ein Programm-System zur Berechnung der Schiffsgeschwindigkeit

unter Dienstbedingungen. Report No.303, Institut für Schiffbau Hamburg

Schlachter, G. (1989) Belastung von Schiffen unter Berücksichtigung nichtlinearer Einflüsse. Report No.488,

Institut für Schiffbau Hamburg p.70

SLF 49/5/2 (2006) Proposal of a probabilistic intact stability criterion. Submitted by Germany. International

Maritime Organisation, Sub-Committee on Stability and Load Lines and on Fishing Vessels Safety, 49th

session, 13 April

Söding, H. (1982) Leckstabilität im Seegang. Report No.429, Institut für Schiffbau

Söding, H. (1990) Multi-dimensional integration, Ship Techn. Res. 37(4) 199-200

Söding, H. (2001) Global seaway statistics, Ship Techn. Res. 48 147-153

Söding, H. (2005) Design waves for maximum ship loads, Ship Technology Research 52(3) 148

Tonguć, E. and Söding, H. (1986) Computing capsizing frequencies of ships in a seaway, Proceedings, 3rd Int.

Conf. on Stability of Ships and Ocean Vehicles STAB’86, Gdansk

22

Appendices

Appendices contain details of the numerical tools proposed in Chapter 3.

Appendix A. Simulation Methods and Validation

A.1. Method ROLLS (roll simulation)

Introduction

The method ROLLS (ROLL Simulation) was established by Söding (1982) and further developed by

Kröger (1987), Petey (1986). It was distributed to different universities and companies in the 90th and

undergoes continuous development since then (see for example Krüger and Cramer (2001)). Thus

different versions exist which are based on the same basic theory and origin.

If no tanks or flooded compartments are included, the simulation runs very fast so that it can be

used routinely even for studies involving large number of seaways and ship operation options.

The method was developed to simulate accurately the roll motion whereas other motions are treated

primarily to take account of their influence on the roll motions, as methods not including the influence

of the five other degrees of freedom on the roll motion cannot predict correctly roll motions of ships in

steep waves.

Computation of nonlinear hydrodynamic forces during simulation requires high computational

effort, therefore the degrees of freedom are separated in ROLLS into two groups:

• those degrees of freedom for which hydrodynamic effects are small but forces and

moments depend strongly nonlinearly on wave and motion amplitudes (roll and surge

motions) and

• those degrees of freedom for which hydrodynamics is important but nonlinear effects are

less important (heave, pitch, sway and yaw motions).

For these different groups of motions different computational methods are applied. Heave, pitch,

sway and yaw motions are computed in the frequency domain using a linear strip method. From the

motion transfer functions found in this way, the motion history in an irregular seaway is found by the

superposition of the reactions in regular waves. For the surge and roll motions, on the other hand, the

relatively small hydrodynamic effects are only roughly approximated, while the nonlinear hydrostatic

and Froude-Krylov forces are taken into account carefully.

The method can also handle damaged ships by including the sloshing as well as the in- and outflow

of water in ship compartments. This feature has been used also for intact ships to account for the

effect of water on deck between the bulwark and hatches Krüger et al. (2006).

Current developments of the method try to take into account more accurately the hydrodynamics

for roll and surge, and the nonlinearities for the other degrees of freedom. Especially the linear

treatment of the yaw motion in the present program version has the effect that broaching-to in

following waves cannot be simulated, in spite of its strong coupling to the danger of capsize. On the

other hand, other causes of capsizing, such as the dangerous decrease of the righting moments on a

wave crest, parametric roll excitation, or water on deck, are accounted for accurately enough by the

ROLLS method.

Roll Motion Equation

The fundamental equation of roll motion is

23

d

d

HM

t=

��

, (A.1)

where M�

is the moment exercised by external forces on the ship, and H�

is the angular momentum of

the ship. Both quantities are described in a coordinate system ξ,η,ζ, the orientation of which is

defined relative the horizon (ζ points downward, ξ into the average forward direction of the ship),

whereas the origin of this coordinate system is fixed to the centre of gravity G of the ship’s mass.

Further, also use a coordinate system x,y,z is used which is fixed to the ship and participates in its

rotations; the axes point forward, to starboard and down, respectively. Its origin is assumed in the

midship section and midship plane at the height of the keel.

The column of coordinates [ξ,η,ζ]T on the first coordinate system of the vector ξ

� from the origin

of this system G to a point, and the column of the coordinates [x,y,z]T expressed in the second

coordinate system of the vector x�

from the origin K of the second system to the same point are related

by the equation

K K K[ , , ] [ , , ] [ , , ]T T Tx y zξ η ζ ξ η ζ= ⋅ +T . (A.2)

Here, T is a 3×3 matrix, which depends on the heel angle ϕ, trim angle θ and yaw angle ψ.

Consistently with the approximations used in the ROLLS method, T is linearised with respect to θ

and ψ but not ϕ; then T becomes:

1 sin cos cos sin

cos sin

sin cos

θ ϕ ψ ϕ θ ϕ ψ ϕ

ψ ϕ ϕ

θ ϕ ϕ

− + = − −

T . (A.3)

The angular momentum is defined as

all masses

dd

dH m

t

ξξ= ×∫

���

. (A.4)

Combining equations (A.1) to (A.4) gives, after some algebraic transformations and linearisation

with respect to θ and ψ, for the first component Mξ of M�

in the system ξ,η,ζ (the heeling moment), in

the case of a symmetrical mass distribution, the following heel moment equation:

( ) ( ) ( )2 2sin cos sin cosGx Gxz GxzM I I Iξ ψ ϕ θ ϕ ϕ ϕ θ θϕ ϕ ψ ψϕ = − + + + − +

���� � �� � , (A.5)

where IG... are the mass moments of inertia referred to the centre of gravity G:

( ) ( ) ( )( )2 2

d , dGx G G Gxz G GI y y z z m I x x z z m = − + − = − − ∫ ∫ . (A.6)

If the mixed moment of inertia IGxz is zero, i.e. if the ship’s masses are arranged, in the average, in

equal heights in the forebody and in the afterbody, a simpler equation can be used

GxM Iξ ϕ= �� . (A.7)

Practically, however, in most ships the centre of masses is higher in the afterbody than in the

forebody, giving positive IGxz. In this case, the remaining terms in (A.5) constitute an inertial coupling

between heel, pitch and yaw motions. Other coupling terms between the different degrees of freedom

are contained in Mξ.

The heeling moment Mξ is approximated as a sum of the following contributions:

Moment due to Roll Acceleration: the added moment of inertia Ix´´ due to the acceleration of the

surrounding fluid gives a contribution

''I xM I ϕ= − �� . (A.8)

Here, Ix´´ (referred to the longitudinal axis through G like the moments and motions) can be

determined by potential flow calculations (for example, by a panel method), but except for large B/T

ratios, MI is relatively small and can be approximated by simple formulae.

24

Damping Moment: this is assumed to consist of a linear and a quadratic term:

D L QM d dϕ ϕ ϕ= − −� � � . (A.9)

The coefficient dL contains a small speed-independent part caused by wave generation of the rolling

ship, and another, in most cases larger part which is proportional to the forward speed of the ship and

is caused by lift forces generated by the hull, propeller and rudder. To approximate it, either the

model data (see for example Blume (1979)) or approximations based on slender-body or strip theory

can be used. The coefficient dQ takes account of the bilge keels. An additional model is necessary if

the ship is equipped with active (steered) roll damping fins, whereas the effect of (seldom used) fixed

fins is treated similarly to a ship rudder.

Weight: because the moment reference point is G, the weight does not contribute to Mξ.

Froude-Krylov Pressure: the pressure in still water or in a wave, not modified by the ship. This is a

very important contribution. It is related to the righting-arm curve, including its changes in waves.The

program uses stored tables of righting arms calculated hydrostatically in a wave-shaped water surface.

This procedure uses the concept of Grim’s equivalent wave (see Grim (1961)): in a natural seaway, the

irregular water surface which would occur along the midship plane if the ship were not present is

approximated, within the length range of the ship, by a suitable standard shape, which produces,

approximately, the same righting moment curve over heel as the real, irregular water surface.

As such a standard shape, the following expression for ζ is used (it is slightly generalised compared

to Grim’s original approach):

( ) ( ) ( ) ( ) ( ), cos 2 /x t a t b t x c t xζ π λ= + + . (A.10)

The formula contains a length-averaged increase or decrease a(t) of the water surface height with

respect to the average water level, a length-averaged lengthwise slope of the water surface b(t) and a

wavy shape with amplitude c(t). The wave crest or trough is assumed always at the midship section,

and the wavelength λ is selected constant for a given ship, in most cases equal to the ship length or

slightly larger, because waves of such a length have the most significant influence on the righting

levers.

The time functions a(t), b(t) and c(t) are computed efficiently from the transfer functions between

these quantities and the regular waves constituting the seaway. Before starting the simulation, these

transfer functions are determined using the least-square method (see Kröger (1987)). A reduction

factor e-kz

may be applied to take account of the smaller pressure variations along the ship’s bottom

than at the water surface, the so-called Smith effect.

Further, before starting the simulations, a table of righting arms is determined depending on draft d,

trim angle θ, heel ϕ and wave amplitude c. For each time instant within the simulation, the heave

motion ζK and the pitch motion θ are determined, using transfer functions which are computed by the

strip method. For interpolating the righting arm h from the table described above, actual values of

draft and trim are determined as

average ship draft Kd a ζ= − + , (A.11)

average trim bτ θ= − + . (A.12)

These give the following moment due to heel in a wave:

AM g hρ= − ∇ , (A.13)

where ∇ is the actual displacement corresponding to the instantaneous draft and trim of the ship.

Instead of h, this value MA could be tabulated; however, in the view of the following considerations it

seems better to tabulate only the righting arm h.

Calling m33 the added mass for heave motion, approximately (neglecting coupling with the other

motions)

( )33 Km m g mgζ ρ+ = − ∇ +�� , (A.14)

where m is the ship’s mass. The value ∇ following from this equation is inserted into (A.13) giving

25

( ) 33A K KM m g h m hζ ζ= − − +�� �� . (A.15)

Vertical Ship Acceleration: the vertical acceleration of the ship changes the water pressure. The

moment caused by this effect depends on the “centre of gravity of the added mass” m33. As an

approximation, the centre of gravity B of the displacement volume is used. This gives a moment

33V KM m hζ= − �� , (A.16)

which cancels with the second term in (A.15), giving

( )A V KM M m g hζ+ = − − �� . (A.17)

Together with the dependence of h on the wave height c the term Km h� may contribute to the

parametric excitation of roll motions.

Moment MSY due to sway and yaw motions: this contribution is determined by a linear strip method.

Before starting a simulation, the transfer function between this moment and the waves is determined.

Moment ME due to the wave excitation: in computing the moment MA+MV, a horizontal water

surface is assumed in the transverse direction of the ship. The transverse inclination of the wavy water

surface, however, generates another contribution to the external moment. In the ROLLS method, this

moment is accounted for also by a transfer function, computed using the strip theory.

Moment MW due to transverse or oblique wind: contrary to many published or rule-based

assumptions, this moment is assumed constant over time in ROLLS because ships have usually much

larger length than the height. This leads to largely independent random fluctuations of the wind force

acting on different cross-sections of the ship, which reduces substantially the fluctuations of the wind

speed averaged over the ship side area as compared to the point-wise wind fluctuations; see for

example Blendermann (2002).

For a given wind speed and direction the wind heeling moments are determined from the data by

Blendermann (1993); an additional counterforce resulting from the wind-generated drift motion on the

underwater body may be added according to Blendermann (1986).

Adding the above-listed contributions to the external moment gives, together with (A.5) and (A.7), the

following ordinary differential equation for the roll angle ϕ:

( ) ( ) ( ) ( ) ( )

( )

2 2cos sin , , ,

'' sin cos

Gxz L Q SY E W

Gx x Gxz

I d d m g h d c M M M

I I I

ψ ψϕ ϕ θ θϕ ϕ ϕ ϕ ζ τ ϕϕ

ψ ϕ θ ϕ

+ − + − + − − + + + =+ − +

�� ���� � � � �

�� . (A.18)

(The corresponding roll equation given in Brunswig et al. (2006) as (4) is incorrect; the simulations in

the paper were done with the program using the correct version of roll equation (A.18).)

Surge Motion

The surge motion is simulated using the following motion equation:

( )

11

1G

T t R X

m mξ

− − +=

+�� . (A.19)

Here, T(1-t) is the propeller thrust less thrust deduction, calculated depending on the instantaneous

speed and (if variable) the propeller rate of evolution, R is the ship resistance, which is taken from

model tests or assumed proportional to the instantaneous ship speed squared, X is the longitudinal

force due to waves, and m11 is the added mass for the longitudinal motion.

It has been shown by Blume (1976) that the Froude-Krylov force, i.e. the force due to the wave

pressure undisturbed by the ship, is nearly alone responsible for the surge motion; the longitudinal

diffraction forces are negligible for such slender bodies as ships. The Froude-Krylov force can be

determined using a simple assumption that, for a given ship section, the buoyancy force is directed

normal to the wave contour (with a reduction factor e-kz

) and thus contains a longitudinal component.

26

Integration of Motion Equations

The four ordinary first-order differential equations

/ right-hand-side of (4.19)Gd dtξ =� , (A.20)

/G Gd dtξ ξ= � , (A.21)

/ right-hand-side of (4.18)d dtϕ =� , (A.22)

/d dtϕ ϕ= � (A.23)

are integrated numerically using the fourth-order Runge-Kutta method with a fixed time step size.

Determination of the Linear Wave Responses

The time dependencies of draft d, trim θ, wave height c and the ship motions sway, heave, pitch and

yaw influence the right-hand side of (A.18) and thus have to be determined. They are assumed

linearly depending on wave amplitudes and are computed using complex transfer functions ( )ˆ ,Y ω µ ,

where ω is the circular wave frequency (not encounter frequency) and µ is the wave direction with

respect to the ξ axis. The natural seaway is modelled as a linear superposition of regular harmonic

waves; nonlinear effects such as sharper wave crests than wave troughs are neglected. The linear

responses r(t) are added from the contributions corresponding to the regular waves making up the

seaway as

( ) ( ) ( )w

cos

1

ˆ ˆRe , n n G n

Ni t k

n n n

n

r t Y eω ξ µζ ω µ −

=

= ∑ . (A.24)

Here, n is the index of a regular wave characterised by ωn and µn, kn is the wave number, and ˆnζ is

the complex amplitude of wave n, determined from the wave spectrum S(ω,µ) as

( )ˆ 2 ,ni

n n n n ne Sεζ ω µ ω µ= ∆ ∆ , (A.25)

where εn is a random phase angle distributed uniformly between 0 and 2π.

To avoid self-repetition of the generated seaway, the range of frequencies and wave encounter

angles is subdivided into a sufficient number of rectangles of equal area (i.e. all regular components

have equal amplitudes) and besides, within each rectangle the frequency ωn and encounter angle µn are

selected as random variables. Especially in cases with small roll damping (for instance, for ships with

small forward speed) more wave components are expected to give more accurate results. Further, each

test is repeated for several sets of random phases εn frequencies ωn and wave headings µn. In all these

runs, different wave tracks result; thus sufficient confidence can be expected in a statistical sense, for

example with respect to the frequency distribution of roll angles or the average rate of capsizes.

A.2. Method GL SIMBEL

Because ROLLS cannot model broaching-to, also a more involved method is required which can

handle nonlinear manoeuvring motions in a seaway and especially their coupling with the roll motion.

Work on this more involved method GL SIMBEL was also started in Hamburg between 1980 and 1990

and continues until today. It is based primarily on work by Söding (1982), Böttcher (1986) and

Pereira (1988).

In GL SIMBEL all 6 degrees of rigid-body ship motions are simulated using nonlinear motion

equations. Transformations between the inertial and the ship-fixed coordinate systems and the rigid-

body motion equations (i.e. the relations between force and moment on the one hand and translational

and rotational accelerations on the other hand) are exact for arbitrarily large rotation angles. However,

to attain reasonable computing times simplifications have to be made also in GL SIMBEL in

determining the external forces (and moments, which are always included without mentioning). These

forces comprise:

27

• ship weight

• Froude-Krylov forces due to the pressure field of the waves unchanged by the ship

• radiation and diffraction forces, i.e. forces due to the influence of the ship on the pressure

field

• rudder force

• propeller force

• wind force (not used here)

Froude-Krylov force

The waves are treated linearly according to the Airy theory, assuming that nonlinear effects due to the

ship outweigh those due to the exciting seaway. The force results form integrating the pressure field

over the wetted ship surface. The pressure field in the assumed natural seaway is added up from the

Airy pressure fields of all component waves.

Rudder force

The rudder force is determined by either a lifting-line or a panel method as in manoeuvring

simulations. Naturally, the propeller slipstream and the maximum rudder lift (stall angle) are of main

importance. For simulations in a seaway the time-varying emergence of the upper part of the rudder

and the orbital fluid motions of the undisturbed wave are accounted for. For determining the rudder

angle the data of a simple autopilot are specified.

Propeller force

The longitudinal propeller thrust can be determined from polynomial regressions for KT for series

propellers. These data may be modified to meet the actual propeller at design conditions. The

longitudinal component of the orbital velocity and the time-dependent ship speed is accounted for in

determining the advance coefficient, and the thrust is reduced in case of propeller emergence.

In oblique flow due to yaw and drift motion and due to orbital motion in a wave, a propeller exerts

also a substantial transverse force. Various versions of GL SIMBEL use different methods to model the

transverse propeller forces which are important for the manoeuvring behaviour of the ship.

Radiation and diffraction forces

In GL SIMBEL, radiation and diffraction force and moment are determined by a nonlinear strip method.

The force and moment exerted by the water on the ship sections are determined depending on the

section immersion and the waterline inclination, together with the time derivatives of the relative

motions between the section and the wave-induced motion of the surrounding water. Also velocities,

accelerations and acceleration derivatives up to a sufficiently large order are taken into account to

model the previous history of relative motion or, what is essentially the same, the frequency

dependence of the added mass, damping and wave excitation forces. Radiation forces are

hydrodynamic forces generated by the ship motions, whereas diffraction forces are the difference

between the forces on the non-moving ship in waves and the Froude-Krylov forces. This difference is

due to the change of the wave pressure field by the presence of the non-moving ship.

Due to nonlinearity the radiation and diffraction force cannot be subdivided into separate

contributions from radiation and diffraction. For instance, if a part of the ship hull emerges from the

water due to ship motions, the diffraction force on that hull part must be zero and thus cannot be

computed for the non-moving ship. Therefore the sum of radiation and diffraction force is determined

using the relative-motion principle: The force is assumed to depend on the motion of the ship minus

the motion of the water at the considered ship cross section. Thus the wave motion averaged over a

ship cross-section is assumed to determine the fluid flow and the force on that cross section. And the

total force is added from the contributions of ship sections, taking into account the longitudinal

interactions due to the forward speed of the ship in a simplified manner as in the linear strip method.

28

For small motion amplitudes the complex added mass matrix a (containing masses in the real part

and damping constants in the imaginary part) of a ship section is the proportionality constant between

(relative) motion acceleration u�� and force per length:

f au= �� . (A.26)

For ship sections 3 degrees of freedom have to be considered: transverse and vertical translations

and the roll motion. Thus the added mass matrix of a ship section is a complex 3 by 3 matrix, while f

and u�� are 3-component column vectors.

To avoid CFD calculations (for instance using a panel method) for each section at each time instant

of the simulation because that needs much CPU time, the added mass matrices for a number (for

example, 25) of ship sections are computed before starting the simulation. The matrices are

determined for a sufficient number of (encounter) frequencies, immersion drafts and inclination angles

of the waterline, including slightly submerged sections and immersion of the deck. During the

simulation values for the actual waterline inclination and immersion are interpolated.

A special difficulty for simulating ship motions in a natural seaway is the frequency dependence of

these matrices. Especially the hydrodynamic damping due to waves generated by the relative motions

between waves and ship depends extremely on frequency: it is zero for zero frequency as well as for

infinite frequency, and it attains a maximum somewhere in between. The added masses vary often by

a factor of 3 or more between different frequencies. However, in a natural seaway a multitude of

frequencies are present at the same time, but due to the nonlinearity contributions from different

frequencies cannot be simply added up. One solution of this problem is to change from the frequency

dependence of the matrices to a relation between force and acceleration which involves not only the

present acceleration, but also the accelerations during previous times (for example for the last 30

seconds) by using impulse-response functions. It appears that a better (especially faster) solution is

what is now called the state space model: Instead of a proportionality between force and acceleration,

a relation is established between acceleration and a few of its time derivatives on the one hand, and of

the force and some of its time derivatives on the other hand:

2

0 1 2 0 1 22A u A u A u B f B f B f

t t

∂ ∂+ + + ⋅⋅⋅ = + + + ⋅⋅⋅

∂ ∂� ���� �� �� . (A.27)

Here, the 3×3 matrices Ai, Bi are independent on frequency but still dependent on section shape,

submergence and heel; thus they depend implicitly on time. Because (A.27) is a homogeneous

equation, for one of the matrices Ai, Bi a unit matrix can be selected. The other matrices are

determined from the frequency-dependent complex added mass matrices a such that (A.26) and (A.27)

give approximately the same relation between force and moment for sinusoidal relative motions of

small amplitude and arbitrary frequencies in the least-squares sense.

For ships with forward speed, in analogy to the strip method ‘‘substantial’’ derivatives are used

vt x

∂ ∂−

∂ ∂, (A.28)

where v is the instantaneous speed ahead, instead of most of the partial time derivatives in (A.27).

Further the matrices A1 etc. are included into the derivative to take account of the dependence of the

matrices on longitudinal coordinate x and (implicitly) on time. The latter is especially important to

obtain correct slamming forces on sections during immersion of the bottom or of a flaring upper part.

Further details of the state-space formulation have been selected partly by theoretical considerations to

obtain coincidence with known solutions in special cases, and partly by comparing different

alternatives with experiments performed in the model tank of Institut für Schiffbau for bodies of

different shape performing forced oscillations with small and also with large amplitudes (compared to

the average body draft) and various frequencies Schlachter (1989). Especially bodies emerging from

the water require modifications of the method.

The modified version of (A.27) is integrated ‘‘substantially’’ over time for all terms and all

sections to determine the section force f per length due to diffraction and radiation. It was not easy to

obtain a stable and accurate integration; if the damping force resulting from the modified equation

(A.27) is negative for any combination of motions within a small range of frequencies (even if these

are uninterestingly low of high), the force will show self-excited oscillations. Special care is required

29

for sections with an immersed deck, and when sections enter the water after having been completely

emersed.

To determine the actual submergence of ship sections the steady wave field due to the forward

speed of the ship, and for higher speed also the squat of the ship has to be taken into account. Also the

steady ship resistance can be regarded as part of the radiation and diffraction force. Numerous

methods are employed to determine it as a function of speed.

Because the ship may perform rudder-induced and wave-induced yaw and sway motions, hull

forces due to drift and yaw motions have to be included in the simulations. Here the methods used for

manoeuvring simulations of ships in still water, including empirical relations for viscous forces due to

flow separation, are included (Pereira and Puntigliano (1993)).

A.3. Modeling the seaway

Both ROLLS and GL SIMBEL model the seaway as a linear superposition of sinusoidal waves.

Typically 15 to 20 wave frequencies and 8 to 15 wave angles constitute a Cartesian grid in the

frequency – wave angle plane. In each rectangle of the grid a certain combination of frequency and

angle is selected, independent from the other rectangles, by a random procedure using a constant

probability density within the rectangle. The amplitude of the wave is chosen as the square root of the

integral of the seaway spectrum over the rectangle. The integrals over rectangles are calculated to a

prescribed (high) accuracy using adaptive refinement. These 100 to 200 regular waves are

superimposed to model the seaway. Sometimes other procedures for selecting the data of the regular

waves are applied; however, in every case it is a common practice to vary the number of regular waves

for each particular case before doing the actual simulations to make sure that the seaway discretisation

is appropriate.

On the one hand, this procedure neglects the nonlinearity of the seaway, for namely that wave

crests are, in the average, sharper and higher than wave troughs. As experience shows, this is tolerable

for the applications considered because the wave nonlinearities are far less important than the

nonlinearities of the ship responses, and it must be tolerated because nonlinear models of the seaway

would increase the computing time too much.

However, this method of modeling the seaway is criticized for another reason: In the real seaway

one finds that the auto-correlation function R(τ) of the surface height ζ(t) at a fixed position drops to

nearly zero for τ values exceeding, say, 1 minute. The superposition of sine waves, on the other hand,

results in autocorrelation functions which vary between positive and negative values, roughly in the

size range (square root of number of frequencies used) × (variance of ζ). This difference in

autocorrelation function indicates that also the time functions ζ(t) of the real seaway and the model

seaway are different. It is easy to verify that such differences must occur if the difference in

neighbouring frequencies used for the superposition exceeds 1/T where T is the simulation time.

That would be important if the aim were to model ship responses in a given wave train. However,

here only statistical results are of interest such as frequency distribution of maxima or the average rate

of exceeding limit values. Fig. A.1 shows that these distributions – here for the function ζ itself

instead of responses to ζ – can be modelled accurately with even less than 100 regular waves in the

superposition; the remaining differences of a few exceedances are random errors.

Fig. A.2 shows that the result is also the same, except for random errors, if a) 3 simulations of

duration 2000s are averaged, or b) a single simulation is extended over 6000 seconds.

A.3. Review of the validation of the simulation methods ROLLS and SIMBEL

Regular waves

Brunswig et al. (2006) (Attachment 2) compared the roll motions found in model experiments with

those simulated by the methods ROLLS and GL SIMBEL for a large containership

(L⋅B⋅T=317m⋅43.2m⋅14.4m). They used regular head and stern waves deviating by 2 to 5 degrees from

being exactly longitudinal to initiate roll motions which may be increased by parametric excitation.

30

The ratio between wave encounter frequency and roll natural frequency was selected near to the values

2 and 1, i.e. ratios where parametric roll may occur.

Of practical importance for parametric rolling in regular longitudinal waves is nearly alone the

frequency ratio 2. For this ratio parametrically excited roll motions occur already in waves of 2m

height, whereas for the frequency ratio 1 up to wave heights of 8m no parametric roll motions could be

found, neither in experiments nor in simulations. For 10m wave height ROLLS showed moderate

parametric rolling, whereas GL SIMBEL failed to do so, and the model experiments showed parametric

number of exceedances

limit values

Fig. A.1. Number of times that various limit values (abscissa) are exceeded by a wave track

superimposed from 148 (continuous lines), 74 (broken lines) or 37 frequencies (dotted lines)

within 6000s simulation time. 2 random realizations plotted for each case.

number of exceedances

limit values

Fig. A.2. Number of times that various limit values (abscissa) are exceeded by a wave track

superimposed from 74 frequencies. Full lines: Average of 3 simulations each of 2000s

duration; broken lines: 1 simulation of 6000s. 4 random realizations plotted for each case.

31

rolling in one of 4 cases. Thus, for this case 10m appears to be the limiting wave height above which

ship motions without substantial roll become unstable.

Concentrating on the important frequency ratio 2 and its neighbourhood, Figure A.3 shows the

maximum roll angle according to the model experiments and to the simulation methods ROLLS and GL

SIMBEL for 10m wave height. It demonstrates that both ROLLS and GL SIMBEL give results which

are suitable to predict a) whether parametric roll occurs, and b) which roll amplitudes occur in case of

parametric resonance. There is only one case where ROLLS shows parametric rolling whereas the

experiment and GL SIMBEL do not. Naturally the limits of stability of the non-rolling condition can

only be predicted with a certain accuracy; thus a few cases must be expected where the simulation

shows parametric roll whereas the experiment does not, or vice versa. Further, near to the limit of

parametric roll there must exist conditions in which both a non-rolling and a parametrically excited

roll motion are stable; which of these conditions is actually realised depends on the initial conditions.

The reason for this is the dependence of the roll natural frequency on the roll amplitude: If the ship has

already a substantial roll amplitude, the frequency of encounter may match the resonance ratio 2

accurately enough to increase the rolling further, whereas a ship having a smaller initial roll motion

and thus, typically, a lower roll frequency, may miss the resonance condition for parametric excitation.

One should note, however, that this sensitivity to the initial conditions is a specialty of regular waves;

in natural seaways there is no such sensitivity.

Fig. A.3. Maximum roll angle occurring in regular waves of 10m height in the neighbourhood of a frequency

ratio 2 according to model experiments and two simulation methods GL SIMBEL and ROLLS

Irregular waves

Results of the simulation method ROLLS and of model experiments in long-crested irregular but

deterministic wave trains were compared for a roro ship (Lpp⋅B⋅T=190m⋅26.5m⋅7.35m) by Hennig et al.

(2006) (Attachment 3). Fig. A.4 shows an example. In the simulation the wave train was represented

by superimposing 91 regular waves. As explained before, this can represent the actual wave train only

during a time range which is much shorter than the duration of the experiment. (Times on the abscissa

32

are full-scale values.) Thereafter differences between actual and simulated waves occur, which,

however, do not change the frequency distribution of surface height, wave crest height etc. except for

small random errors. Correspondingly the simulated and measured roll motion is different, but the

probability distribution of its maxima and minima are nearly identical. Because the aim here is

obtaining the probability distribution of the roll maxima and minima, not the actual time track of roll

motions, Fig. A.4 demonstrates also that the method ROLLS seems appropriate for this purpose.

There is another reason for differences between the simulated and measured time track of roll angle

for later times: There were course changes of the model (the first one was performed at about t=400s)

which were neglected in the simulation.

Fig. A.4. Taken from Hennig et al. (2006): Measured (blue) and simulated (red) roll motion of a roro model in

an irregular long-crested wave train

Further verifications were made for GL SIMBEL and ROLLS; see for example Chang (1999) and

literature cited by Hennig et al. (2006) and Brunswig et al. (2006).

Appendix B. Estimation of the average rate of casualties

Consider experiments, where the probability is p that the outcome of the experiment is A (e.g.

capsising), while the outcome not A has probability 1–p. Find the probability pm,n that in n

independent experiments the number of outcomes A is exactly m. Here, only the limit of small

probability p is considered. Then Poisson’s formula is applicable:

,

( )

!

mnp

m n

npp e

m

−= . (A.29)

To apply this to the case under consideration, the simulation time T is subdivided into n intervals

each of length T/n. n is assumed so large that only zero or one, but not more casualties occur in each

interval; however the duration of intervals is sufficient to treat them as independent from each other

with respect to the outcome ‘casualty’ or ‘no casualty’. That is possible if the average rate of

casualties is so small that the initial conditions with which the simulation is continued after one

casualty are without influence on the motion at the instant of the next casualty.

Under this condition (A.29) can be applied. The casualty probability p within any interval is

/p rT n= . (A.30)

Inserting this into (A.29) gives an expression which does not depend on n:

,

( )= =

!

mrT

m n m

rTp p e

m

−. (A.31)

This formula gives the results plotted in Fig. 3.1.

Appendix C. Discriminant Analysis1

1. The discriminant analysis can be regarded as one of the typical instruments to solve safety

problems. It is explained in detail below on a classic example from the area of ship safety, namely the

problem of capsize safety. It is well known that the physics of capsizing in real environment is very

complex. A pure theoretical (i.e. analytic or computational) prediction of the physical features, which

1 translated from Krappinger, O. and Sharma, S.D. (1974) Sicherheit in der Schiffstechnik (in German). Jahrbuch

Schiffbautechnische Gesellschaft, 329-355

33

a ship must possess to be safe from capsizing, appears almost hopeless1. To do this, first of all a

comprehensive nonlinear three-dimensional theory of seaway and seakeeping must be developed. The

method of model testing, which is always used in ship hydrodynamics if the methods of rational

mechanics are not sufficient, was indeed also applied to the studies of capsize safety, but it is very

time-consuming and expensive. The number of variable parameters is so large, that systematic and

comprehensive model tests for identification of general capsize criteria are hardly accomplishable. So

far, such studies only concern particular cases. Therefore, practice makes do mostly with simple

empiric criteria. One of the most known is based on an idea by Rahola (1939). According to it, the

thresholds of stability parameters are deduced through the comparison of the parameters of ships

involved in stability accidents with those that proved safe in operation, see for example Seefisch

(1965). We will show how this approach can be extended and rationalised by using the discriminant

analysis.

2. Our example is based on the data taken from a working document2 of IMCO (Inter-Governmental

Maritime Consultative Organisation, London). In Table 1 ten specific parameters (from GM to e) of

the righting levers are given for 19 ‘unsafe’ ships, which have undergone stability accidents. The

figures do not apply however to the state during the accident, but to the full load arrival state. In Table

2, the corresponding particulars for 61 ‘safe’ ship are gathered, that have not experienced (at least until

the survey) stability accidents3. For completeness’ sake it must be mentioned that the following cases

were removed from the original IMCO list: first, all accidents that are attributed not to insufficient

stability, but for example to cargo shift due to improper stowage; second, all ships with timber cargo,

because these ships require a special treatment; and third, ships with insufficient information. Further,

all remaining data were inspected, and some evidently erroneous figures were corrected through

lofting the righting lever curves as good as possible. Additional particulars of the mentioned ships can

be taken from the specified IMCO document or from the earlier documents referred to there, using the

original numbers of cases accepted here.

Table 1. Stability data of 19 ‘unsafe’ ships

3. The task now is to identify a stability criterion for ‘safe’ ships through comparison of the stability

parameters in Tables 1 and 2. Following Rahola (1939), it stands to reason to compare first the

frequency distributions (approximated through the cumulative frequencies in the samples) of the

1 From the translator: note that the paper was written in 1974

2 Document Nr.\ IS VI/3 dated 7 June 1966

3 Because of space limitations, there is no elaboration on the justification of the selection of exactly these samples

for the purpose of the study. Note however, that for a study of capsizing safety choice of other samples can also

be advisable.

34

available parameters in the both groups, see Figures 4a-i. It can be seen immediately that some

parameters (for example, h40 or e) separate the groups indeed more distinct than others (for example,

GM or ϕm), although complete separation is not achieved in any case. There are at least two different

interpretations for the overlapping of the curves F1 and 1–F2 in Figures 4a-i: one can assume that some

of the ships in Table 2 are actually not at all safe, but have had luck so far not to be exposed to any

real danger, or, alternatively infer that none of the parameters at hand suffices as a stability criterion if

taken alone.

For practical purposes, a separation should be undertaken despite the overlapping. So, one could

consider for example a particular extremal value from group 1 as a boundary for the necessary stability

(long-dashed lines in Figure 4). However, it is not quite satisfactory, as the minimum value for GM

would than be 0.85m, and only 15% of the ships from the ‘safe’ group 2 would achieve it. Another

possibility would be drawing the threshold through the intersection of the curves F1 and 1–F2. In this

case, the percentage of wrong decisions would be equal for the both groups (short-dashed lines in

Figure 4). This was proposed for the first time from the Polish delegation in the above mentioned

Table 2: Stability data of 61 ‘safe’ ships

35

IMCO document, and leads after all to a reasonable separation of the groups in about 80% of the cases

if one of the parameters h30, h40, hm or e is used. However, even this approach is not entirely

satisfactory as the threshold and the ‘success rate’ depend too much on the random (because of the

small sample size) position of the curves F1 and 1–F2 in the neighbourhood of the intersection.

A relatively insensitive threshold of a criterion x is the average

( ) ( )1 21

2x x x = +

of both the average values

Fig. 4a-e

36

( ) ( )

1

1, 1,2,

knk k

i

ik

x x kn =

= =∑

where n1 and n2 mean the number of the elements in the group 1 and 2, respectively. Each of the values ( )1

x , x and ( )2

x are shown with three vertical lines in Figures 4a-i.

Completely independent from the selection of the threshold, there is a question about the

discriminating quality of one criterion in comparison with the others. As a statistically relatively

reliable and descriptive measure for this appears the ratio

( ) ( )

( ) ( )

22 1

2 11

2x x

x x

v v

α − = +

,

where ( )1

xv and ( )2

xv mean the variances of the criteria within the groups:

Fig. 4f-i

Fig. 4a-i. Frequency distributions of the stability parameters studied in the

considered samples of ‘unsafe’ and ‘safe’ ships

37

( ) ( ) ( ) 2

1

1, 1,2

1

knk k k

x i

ik

v x x kn =

= − = −∑ .

The larger the value of α, the better is the differentiation, see Figure 4. Sorted according to this

measure α, the nine stability parameters studied follow the ranking

40 m 30 20 m v 10, , , , , , , ,h h h e h h GMφ φ ,

which agrees completely with our intuitive perception of their importance for the ship stability.

4. It seems worthwhile to try to improve the ‘success rate’ of the differentiation between the groups

through combination of more criteria. A naïve possibility to do this would be assembling several

criteria side by side. This would lead, for example, to the requirement that the righting lever curve of

a ‘safe’ ship must exceed everywhere the corresponding threshold of group 2, see Figure 5. It is then

immediately clear that this would be an unrealistic, or at least excessive requirement. It appears better

to combine m different criteria xj (where j=1,...,m) into one single ‘super criterion’, for instance using a

linear combination

0

1

m

j j

j

z c c x=

= +∑ .

Fig. 5. Average values of the stability parameters

This leads to the question how the coefficients cj should be selected. To answer, let us introduce

the already applied above separation measure α also for z. With the average values and variances for z

( ) ( )

1

, 1,2m

k k

j j

j

z c x k=

= =∑ ,

( ) ( )' '

1 ' 1

m mk k

z j j jj

j j

v c c Q= =

=∑∑ ,

with

( ) ( ) ( ) ( ) ( )' ' '

1

1

1

knk k k k k

jj ji j j i j

ik

Q x x x xn =

= − − −∑ ,

one obtains

( ) ( )

( ) ( )

22 1

2 11

2z z

z z

v v

α − = +

.

An optimal (in a certain sense) separation of the groups can be achieved if the cj are selected so that

α is maximal. The well known conditions for this are

38

/ 0, 1,...,jc j mα∂ ∂ = = .

From this system of equations, cj can be defined up to a multiplicative constant. From here not

relevant reasons, this coefficient as well as the constant c0 are selected so that

( ) ( )2 1 1

2z z α= − = .

Then one obtains (see for example Anderson (1958), Schmetterer (1966))

( ) ( ) ( )1 2 1

' ' '

' 1

m

j jj j j

j

c Q x x−

=

= − ∑ ,

with

( ) ( )1 2

' ' '

1

2jj jj jjQ Q Q = +

and

( ) ( ) ( ) ( ) ( )1 2 1 2 1

0 ' ' '

1 ' 1

1

2

m m

jj j j j j

j j

c Q x x x x−

= =

= − + − ∑∑ ,

where ( )1

'jjQ

− means the inverse matrix of the matrix 'jjQ .

Linear combinations of the criteria built with these coefficients are referred to as discriminant

function following Fisher (1936).

Now one can calculate the composite criterion for all elements z and define (as it is done above for

single criteria) the frequency distributions in the groups and an appropriate threshold zG. To allocate a

new element, for which only the criteria xj are known, into one of the groups, the calculated z-value

needs to be compared with the threshold zG; namely

z > zG: group 1

z < zG: group 2

5. For the example considered above, we have determined all 502 possible 2- to 9-term linear

discriminant functions from the nine available stability parameters using a computer program.

Because of space reasons only some selected examples are reproduced here, namely all 2- and 3-term

discriminant functions (Figures 6 and 7, respectively). It can be seen that higher α-values are attained

here than with the simple criteria (Figure 4). For the 36 possible criteria pairs, the highest values of α

are achieved in cases (a) and (b) shown in Figure 6. Especially interesting is case (c) in Figure 6, as it

combines the two most common stability parameters, namely GM and h40. At first sight it may

surprise to see in Figures 6a-c that the parameters h10, e and GM have negative coefficients. The

explanation for this is the interaction of different parameters. For example, it can be seen in Figure 8a,

that an increase of GM with constant h40 changes the righting arm curve in such a way, that appears

intuitively as a reduction of stability1. Exactly this is expressed by the negative coefficient of GM (as

for example in the case shown in Figure 6c). Therefore, the results of the discriminant analysis appear

to confirm the hypothesis that larger GM-values should be compensated through larger levers to

achieve equivalent safety. Similar considerations apply to the interdependence between h40 and e (see

Figures 6b and 8b).

One can interpret the linear discriminant function as a set of hyperplanes z = const in the m-

dimensional space of the criteria xj. The separation of the groups (or samples) derived through z = zG

can be illustrated particularly easy for m = 2. This is shown in Figures 9a-c for the same three cases as

used in Figures 6a-c.

In conclusion, it may be noted that one can extend the linear discriminant analysis in several

directions. For example, one can study right away nonlinear approaches to the criteria xj, according to

the above algorithm, as long as they are linear with respect to the unknown coefficients cj. We have

tested this on the example of GM and h40 with the following result:

1 For the righting lever curve with the larger GM, there is stronger roll excitation and a smaller range

39

z α

401.26 10.48 2.22GM h− + − 2.59

( )2

400.10 10.01 2.67GM h− + − 2.50

402.55 10.75 1.18GM h− + − 2.71

One can see that GM represents a better pair together with h40 than GM or even (GM)2. One can

also say that the influence of the increase in GM (with constant h40) grows weaker than linear. This

‘nonlinear’ discriminant function is shown in Figure 9c.

References to the Discriminant Analysis

Anderson, T.W. (1958) An introduction to multivariate statistical analysis. John Wiley & Sons, New York

Fisher, R.A. (1936) The use of multiple measurements in taxonomic problems, Annals of Eugenics 7 179-188

Rahola, J. (1939) The judging of stability of ships and the determination of the minimum amount of stability,

Helsinki

Schmetterer, L. (1966) Einführung in die mathematische Statistik (2. Aufl.) Springer-Verlag, Wien--New York

Seefisch, F. (1965) Stabilitätsbeurteilung in der Praxis, Jahrbuch STG 59 578-593

Fig. 6. Frequency distributions of some two-parameter-criteria

40

Fig. 7. Frequency distributions of some three-parameter-criteria

Fig. 8. Scheme to explain the negative

coefficients in the discriminant functions. In

case (a) ship 2 has a smaller GM than ship 1

with the same h40 and appears safer. In case

(b) ship 2 has a smaller e than ship 1 with the

same h40 and appears safer.

41

Fig. 9a. Graphic representation of

the ships in groups 1 (symbol +)

and 2 (symbol °) as well as the

differentiating line

40 3024.03 17.30 2.07 0h h− + − =

obtained from the discriminant

analysis

Fig. 9b. Graphic representation of

the ships in groups 1 (symbol +)

and 2 (symbol °) as well as the

differentiating line

4018.75 26.53 2.36 0h e− − =

obtained from the discriminant

analysis

42

Fig. 9c. Graphic representation of

the ships in groups 1 (symbol +)

and 2 (symbol °) as well as the

differentiating line

401.26 10.48 2.22 0GM h− + − =

obtained from the linear

discriminant analysis (solid line)

and from the nonlinear

discriminant analysis

402.55 10.75 1.18 0GM h− + − =

(dashed line)

Attachments

Attachments contain three papers referred in the proposal, which might be useful for highlighting the

details of the methods used in Chapter 3.

Attachment1. Global Seaway Statistics

See the attachment attachment1.pdf containing the paper Söding, H. (2001) Global seaway statistics,

Ship Techn. Res. 48 147-153

Attachment 2. Validation of ROLLS and GL SIMBEL in regular waves

See the attachment attachment2.pdf containing the paper Brunswig, J., Pereira, R. and Kim, D.-W.

(2006) Validation of parametric roll motion predictions for a modern containership design, Proc., 9th

Int. Conf. on Stability of Ships and Ocean Vehicles STAB’2006

Attachment 3. Validation of ROLLS in irregular waves

See the attachment attachment3.pdf containing the paper Hennig, J., Billerbeck, H., Clauss, G., Testa,

D., Brink, K.-E. and Kühnlein, W.L. (2006) Qualitative and quantitative validation of a numerical code

for the realistic simulation of various ship motion scenarios, Proceedings, 25th Int. Conf. on Offshore

Mechanics and Arctic Engineering OMAE 2006, Hamburg, Germany, June 4-9, Paper OMAE2006-

92245

Global Seaway Statistics

Heinrich S�oding� TU Hamburg�Harburg�

� Purpose of the work

Young and Holland ������ published a global wave statistic based on radar measurements of theGEOSAT satellite� Their �Atlas of the oceans� gives probability distributions of the signi�cant waveheight H��� �Fig�� shows an example at �� points� Fig�� distributed regularly between �� � latitudein the world�s oceans� For many purposes this information is not enough� Data about the signi�cantperiod of the seaways are missing� the measuring duration of � years is not su�cient to give reliabledata for seldom occurring extreme seaways� and the accuracy of the radar measurements is not withoutdoubt� Fig���

ISSC ������ published another wave statistic� Table I� which is probably the most�used �wavescatter table� within naval architecture and ocean engineering� It gives information about both waveheight and wave period� but it covers only the North Atlantic� i�e� the region hatched in Fig�� TheISSC table is based on hindcasted wave spectra determined from the wind �eld during �� years� ANEP������� For the North Atlantic this table is estimated to give more reliable wave data for extendedtime periods than the data by Young and Holland �������

This work combines both sources to produce tables of the relative frequency with which signi�cantwave height and wave period are both within certain intervals at the �� points indicated in Fig��

Table I� Relative frequency of occurrence ���� of seaways in the North Atlantic having parametersH���� T� as indicated� The value ��� for signi�cant wave height H��� designates the intervalbetween � and �m height� According to ISSC �������

T��s� H����m�from to FCUM ��� ��� ��� ��� �� ��� �� ��� ��� �� ���� ���� ���� ���� � �� ���� ���

�� ��� ��� ��� � � � � � � � � � � � � � � � ���� �� �� ��� ��� � � � � � � � � � � � � � � ��� ��� ��� ���� ���� �� � � � � � � � � � � � � � ���� �� ��� ����� ��� ���� ��� � � � � � � � � � � � � ��� ��� �� � �� �� ����� � �� � � � � � � � � � � � � ���� �� �� ���� � �� ����� ��� � � �� � � � � � � � � � � ��� �� ���� �� ���� ���� �� ��� � ��� � � � � � � � � � � �� ���� ��� �� ����� ��� ���� � �� ��� ����� ��� ��� � � � � � � � ����� ���� ��� ��� ��� �� � � � ���� ���� ���� ��� �� �� � � � � � � ����� ���� ��� ��� � �� ��� ���� �� ���� �� �� ���� ��� � � � � � ����� ���� ��� �� ���� ��� � � � � � �� ��� ��� ���� ��� � � � � ����� �� ��� �� ���� ���� ���� ��� ��� ���� ��� ��� ���� ��� �� �� � � � ��� �� �� � �� �� � �� ��� ���� ��� ���� ���� �� ���� ��� � � ��� ��� �� ��� �� � � �� � �� �� �� �� ��� ���� �� � � � � � ���� ���� ����� �� �� �� � �� �� ��� ��� �� �� ��� �� �� ��� � �� �

FCUM �� ���� ��� ��� ���� ��� �� ��� �� ��� �� �� � ��� ��� ��� ���

� De�nitions

The signi�cant wave height H��� is the average height� measured from wave trough to wave crest�of the largest ��� of all waves which occur in a space and time interval within which the seaway canbe considered as� approximately� stationary� To characterize the signi�cant period �time durationbetween one wave and the next of the seaway several di�erent quantities are used� Here we use the�center of gravity period� T�� i�e� the period which corresponds to the frequency �� � ��T� wherethe area of the spectrum S�� has its center of gravity� thus ��T� is the energy�weighed average ofthe frequency of the regular wave components �Fourier terms comprising the total seaway�

�L�ammersieth �� D ����� Hamburg� h�soeding�tu�harburg�de

Schi�stechnik Bd� � � ����Ship Technology Research Vol� � � ��� � �

Fig��� Example �Pt��� � �� �� ��of signi�cant wave heightdistribution published byYoung and Holland ������

Fig��� Comparison between measurementsof H��� obtained from GEOSATdata and from a wave buoy�Dobson et al �����

Fig�� Position of points � to �� of Young and Holland �������The ISSC wave statistic refers to the area designated as NA�

� � Schi�stechnik Bd� � � ����Ship Technology Research Vol� � � ���

Other characteristic periods used for describing the seaway are� e�g�� the �peak period� Tp� i�e�the period at which the wave energy density per frequency is maximum� and the period T� � �����where �� is the mean zero�upcrossing circular frequency of the seaway� The correct interpretation ofthe period T� used here is essential for the statistics� An approximate relation is

T� � ���� � Tp ��

� Procedure used

As indicated by Fig�� the ISSC seaway table refers� approximately� to the point �� of Young and

Holland ������� Fig� compares the cumulative distributions of H��� according to the ISSC table andaccording to Young and Holland ������� For the North Atlantic� we trust the ISSC distribution more�but for other areas we have no ISSC data� Therefore we map the distribution for the North Atlantic�index NA to other locations of the oceans using the following relation�

fNA�H����NA� T��NA�H����NA�T��NA � f�H���� T��H����T� �

It says� Seaways in the North Atlantic which have characteristic data H����NA� T��NA �with an accuracy

of ��

��H����NA and ��

��T��NA where the � values are supposed to be small are even likely as seaways

at any other location which have characteristic data H���� T�� if there is a proper relationship betweenH����NA� T��NA on the one hand and H���� T� on the other hand�

Fig� � Probability Pr that H��� exceedsthe abscissa value acc� Young and

Holland ������ �Pt� ��� full lineand acc� to the ISSC scatter table�Table I� broken line

Fig��� Relation between signi�cant waveheight �in m at point � �abszissaand point �� �ordinate� thick lineused to transform the probabilitydistribution of wave data at point�� to that at point �

For H��� this relation is established by the cumulative frequency distributions at point ��� on theone hand� and at any point P of the �� points for which data are given� on the other hand�

H����NA � H���

H�������

H����P��

Schi�stechnik Bd� � � ����Ship Technology Research Vol� � � ��� � �

The fraction on the right�hand side of �� should be chosen such that the cumulative frequency ofH����P at point P� and of H������� at point ��� are alike� Thus the fraction depends on H����P itself�To avoid undue complexity� this condition is exactly satis�ed only at two points of the cumulativeprobability functions� for a probability of not exceeding H��� of �� and ���� Between these two valuesthe ratio H��������H����P is linearly interpolated� whereas for very small and very large values of H����i�e� outside of this interval� the ratio is kept constant to avoid inaccuracies by excessive extrapolation�The resulting relation between H������� and H����P is indicated for point P � � in Fig���

An important parameter of the seaway is its steepness� i�e� the relation between the characteristicvalues of wave height and wave length� For instance� a fully developed seaway has� for arbitrary windspeed� the same steepness� Because wave length is proportional to T �

�� wave steepness is proportional

to H����T��� As shown in Table I� beyond a certain value of H����T

��the probability of occurrence

drops sharply to zero� This characteristic of scatter tables based on hindcasts of seaways like TableI is reproduced only if we correlate T �

�of di�erent points P and �� just as we correlate H���� That

means�

T����NA � T���

sH�������

H����P�

Scatter tables which are based on visual observation instead of measurements do not showthis sharp drop of the frequency of occurrence beyond a certain wave steepness� This is a strongindication that such tables� and results based on them� are misleading and should not be used�at least not for predicting extreme values �e�g� extreme loads� It is not correct to argue thatthe large number of observations decreases the e�ect of the inaccuracy of each observation totolerable levels� Positive and negative errors cancel only for the most probable seastates� not for ex�treme ones� because extreme observed values result if the correct value and the error are both extremes�

Fig��� Natural logarithm of the probability density function fNA�H���� T� in m��s�� of signi�cantwave height �abszissa and period T� �curve parameter for the North Atlantic

To determine the scatter tables of arbitrary points P � we have to determine also the two�dimensionalprobability density function of H���� T� for the North Atlantic such that it corresponds to Table I�Because the probability density changes by many orders of magnitude� its logarithm was interpolatedinstead of the density itself� Fig�� shows the natural logarithm of the interpolated probability density

��� Schi�stechnik Bd� � � ����Ship Technology Research Vol� � � ���

function� Using the interpolated logarithm� the density was computed and integrated numericallywithin all the rectangles corresponding to the entries in Table I� Results were compared with TableI� and the density function description was modi�ed until a reasonable correspondence was attained�An exact correspondence would require a highly irregular probability density function� which appearsunrealistic� Therefore deviations from Table I were accepted to improve the smoothness of the densityfunction�

Using these techniques� the scatter table of any point P was determined as follows�

�� The ranges of H��� and T� values were subdivided into the intervals corresponding to the rowsand columns of the table� Compared to the ISSC table� a �ner and more regular subdivisionwas used� All T� intervals have the same breadth of �s� and the H��� intervals change from ���over � to m� Especially for the milder seaways in tropical regions a �ner subdivision seemedappropriate for small H��� values�

� The values of H��� which� at point P � correspond to a probability of not being exceeded of ��and ��� were read in� The values were measured from the �gures in Young and Holland �������

�� For each of the rectangular intervals of H��� and T� of the scatter table� the values H����NA andT��NA were determined which� according to the relations �� and � � correspond to the upperand lower limiting values of the rectangle�

� The probability p that� at a time selected at random� the seaway characteristics are within therectangle� follows from integrating � over the area of the rectangle�Z

fNA�H����NA� T��NAdH����NAdT��NA � p ��

The integral was determined numerically using ��� points in a Cartesian mesh within the rect�angle� For each of these points� the density fNA was determined interpolating the logarithm ofthe density�

�� The probability of the seaway belonging to any of the rectangular domains of the scatter table issummed up� The result was nearly but not exactly � because� on the one hand� the probabilitydensity function is not exactly scaled� and� on the other hand� there is a �nite probability thatthe seaway characteristics are outside of the range covered by the scatter table� e�g� the waveperiod can be � �s� As a correction� the integrated probabilities p for each rectangle weredivided by the total probability� to attain ���� probability that seaway belongs to any of thetabulated intervals�

�� �� The output was complemented by the cumulative probabilities of exceeding the selectedlimiting values of H��� and T�� The result was printed in Latex format as shown in Table II forpoint no� � as an example� The full set of tables for points � to �� is printed in S�oding �� ��

and can be loaded from the www�hsva�de�services�seak�

Fig�� compares the cumulative probability distribution of wave height at point � given by Young

and Holland ������ with that of Table II� There are substantial di�erences� It was not aimed tominimize these di�erences because they are supposed to be caused by systematic di�erences betweenthe results of the satellite measurements and the hindcasted values� Instead� the procedure used herewas aimed to reproduce approximately� for all �� points� the di�erences found in the wave heightdistribution between the ISSC table and point �� of Young and Holland ������ as depicted in Fig� �This goal is achieved largely but not exactly� An exact correspondence between Figs� and � couldhave been attained by

�� avoiding the smoothing of the probability density function fNA�

Schi�stechnik Bd� � � ����Ship Technology Research Vol� � � ��� ���

� adapting the ratio H��������H����P in �� to the published probability distributions at all insteadof at two probabilities�

Both changes would lead to not smooth probability distributions which may correspond better to theexisting data but appear physically unreasonable� Thus I think my derived tables are not far fromthe optimum estimate which can be derived from the presently available data�

Fig��� Probability Pr that the signi�cant wave height exceeds the abscissa value according to Young

and Holland ������ �Point �� full line and according to the ISSC scatter table �broken line

References

ANEP II�I ������� Allied Naval Engineering Publication Standardized wave and wind environments for NATO

operational areas

DOBSON� E� MONALDO� F� GOLDHIRSH� J ������� Validation of Geosat altimeter�derived wind speeds

and signi�cant wave heights using buoy data� J Geophys Res � � pp�����������

ISSC ������� Int Ship and O�shore Structures Congress� Je�ery� NE and Kendrick� AM �ed�� StJohn�s

S�ODING� H � ����� Global seaway statistics� Schriftenreihe Schi�bau Report ���� TU Hamburg�Harburg

YOUNG� IR� HOLLAND� GJ ������� Atlas of the oceans� Wind and wave climate� Pergamon� Oxford

�� Schi�stechnik Bd� � � ����Ship Technology Research Vol� � � ���

Table II� Relative frequency of occurrence ���� of seaways at point � of Fig� having parametersH���� T� as indicated

T��s� H����m�from to FCUM ���� ���� ���� ���� ���� ���� ���� ���� ��� ���� ��� ���� ���� ��� ����� ����� �����

��� ��� ��� � � � � � � � � � � � � � � � ���� ��� ��� �� �� � � � � � � � � � � � � � � ���� �� ��� � �� ��� � � � � � � � � � � � � � ��� ��� ��� ��� ��� ��� ��� � � � � � � � � � � � � ���� �� �� ��� ���� ���� � �� ����� ��� � � � � � � � � � � ��� ��� � �� �� ��� ���� ��� ���� ���� ��� � � � � � � � � ���� ��� � �� ��� ��� ���� ��� ����� � ��� �� ����� � �� �� � � � � � � ���� �� ���� ���� ��� � � ���� ��� ���� ���� ���� ��� �� � � � � � � �� ���� ��� ���� ���� ���� �� ���� ����� ��� ���� ��� � ���� ��� � � � � ����� ���� �� ���� �� � � ��� �� � ���� ����� �� � � ��� �� ��� � � ����� ���� ��� � ���� � �� �� �� � ��� �� ����� ��� ��� �� ���� � ��� � ����� ���� ��� �� �� ��� ��� �� � ���� � ��� ��� ��� � ���� ��� �� �� �� ����� ��� �� � �� �� �� ��� �� ���� � � � � �� �� ���� ���� �� � �� �� ���� ���� �� ��� ��� ��� �� ��� �� ��� ��� ���� �� ���� ��� ���� ��� ��� �� ������ ��� �� � �� ��� �� �� � � �� � � � ��� ��� � � � ����� ���� � � �� ��� �� ��� �� ��� �� ��� ��� ��� ��� �� �� ��� ��� ����� ���� �� �� � � � ��� ��� ��� �� ��� ��� ��� ��� �� ��� ��� ������ � �� � � �� �� � �� �� � �� � �� �� � � �� � ��� ��� �� ���� ����� �� �� � �� �� �� �� �� � �� �� �� �� � �� �� ������ ���� ����� � �� �� �� �� �� �� �� �� � �� �� � �� �

FCUM ��� ��� ��� ���� ���� �� � �� ���� �� ��� �� ��� �� � �� � �����

Point �� Latitude �� �� Longitude �

T��s� H����m�from to FCUM ����� � ��� ����� ����� �����

���� ��� �� � � � � ���� ���� � � � � ����� ���� �� �� � � � ����� � �� � � � � � �� �� ���� ����� �� � � � ����� ���� ����� �� � � �

FCUM ����� ����� ����� ����� �����

The tables give ��� times the probability of encountering a seaway �for the upper left entry used asan example having a signi�cant period T� between � and s� and a signi�cant height H��� between �and ���m�

Cumulative probabilities FCUM of not exceeding T� values for any H���� or H��� values for any T��are given in percent�

Schi�stechnik Bd� � � ����Ship Technology Research Vol� � � ��� ���

Proceedings of the 9th International Conference on Stability of Ships and Ocean

Vehicles 1

Validation of Parametric Roll Motion Predictions

for a Modern Containership Design

Jorg Brunswig, Germanischer LloydRicardo Pereira, Germanischer Lloyd

Daewoong Kim, Daewoo Shipbuilding & Marine Engineering Co., Ltd.

ABSTRACT

This paper describes recent efforts to validate two nonlinear time domain programs forsimulation of ship motions. The results of the methods ROLLSS and GL SIMBEL werecompared with model test measurements of a modern post-Panmax containership modelcarried out at Hamburg Ship Model Basin (HSVA). This model was designed by DaewooShipbuilding & Marine Engineering Co., Ltd. (DSME). Furthermore, to obtain reliableroll damping coefficients, roll damping tests were performed prior to the calculations. Fordifferent load cases the occurrence of parametric roll in regular waves was investigatedfor a range of speeds, wave lengths and wave heights. The computed and measured rollmotions revealed a significant nonlinear behaviour with respect to wave height.

Keywords: Parametric Roll, Simulation, Roll Damping, Validation

1 Introduction

The results of an internal research projectDYNAS - Dynamic Stability - carried outat Germanischer Lloyd (GL) since 2003 arepresented. The first phase of this projectjust finished. A systematic applicationof the nonlinear sea keeping methods GLSIMBEL (Pereira, 2003) and ROLLSS(Petey, 1988) to predict the motion be-haviour of modern Panmax-, Post-Panmaxcontainerships in severe sea ways was per-formed. The focus of the first phase wasto validate the methods ROLLSS and GLSIMBEL with emphasis on parametric roll.The second phase of DYNAS aims to es-tablish GL classification rules to avoidparametric roll, pure loss of stability, andbroaching-to phenomena.

2 Parametric Roll

Large modern containerships are suscep-tible to what is known as parametricrolling (SNAME 2003). Dangerous para-metric roll motions with large amplitudesin waves are induced by the variation oftransverse stability between the positionon the wave crest and the position in thewave trough. Parametric roll primarily oc-curs under the following conditions:

• Slender hull

• The primary wave system’s wave-length varies between half and twicethe ship’s length

• The wave height exceeds a thresholdlevel

• Almost ahead or astern wave head-ing

Proceedings of the 9th International Conference on Stability of Ships and Ocean

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• Low roll damping

• The natural roll frequency ωr of theship is about half the encounter fre-quency ωe or almost equal to the en-counter frequency.

For frequency ratios of ωr

ωe= 1

2in head

waves or following waves, the stabilityvaries with the encounter frequency ωe

which is approximately twice the roll fre-quency ωr of the ship. The stability at-tains a minimum and maximum twice dur-ing each roll motion. The ship reaches themaximum roll angle in the wave trough,where the up-righting moment is large dueto the increased stability. On the wavecrest with low stability, the roll motioncrosses zero. During one encounter periodthe ship gains energy twice and shows largeamplitudes of symmetric rolling.For frequency ratios of 1:1 in followingwaves, the stability attains a minimum anda maximum once during each roll motion.This situation is characterised by asym-metric rolling, i.e. the amplitude withthe wave crest amidships is greater thanthe amplitude on the opposite side. Inhigher waves, parametric roll tends to oc-cur within a bandwidth of frequency ratiosbetween 0.9 and 1.1, see Figure 7.

3 Simulation Methods

Because of two restrictions, linear seakeeping methods like GL PANEL (Pa-panikolaou/Schellin, 1991) or GL STRIP(Hachmann, 1991) are not suited to pre-dict parametric roll. They do not accountfor stability changes caused by a passingwave, because the pressure forces are onlyintegrated up to the undeformed water sur-face. In addition, these methods are re-stricted to small amplitude ship motions.Therefore, they are incapable of predictinghighly nonlinear phenomena such as para-metric roll, which often leads to large rollangles.

The methods to be validated here remedythese problems by simulating in the timedomain and treating the motions nonlin-early. We approach parametric roll inves-tigations by the following two-step process:

• ROLLSS (2 nonlinear degrees of free-dom -surge and roll, very fast) is usedto perform a large number of simu-lations to quickly identify regions ofparametric roll occurrence.

• GL SIMBEL (6 nonlinear degreesof freedom, slower) is used to yieldmore accurate results in these re-gions of interest.

3.1 ROLLSS

The method was first established bySoding (1982) and further developed byKroger (1986) and Petey (1988). Thistime domain method uses response am-plitude operators (RAO) computed withGL STRIP to determine the sway, heave,pitch, and yaw motions and simulates thesurge and roll motions nonlinearly. Therighting lever arm is calculated at eachtime step, using the concept of the equiva-lent wave (Soding 1982). The wave eleva-tion at location x and time t is calculatedby a superposition of all wave componentsin the chosen seaway spectrum:

ζ(x, t) =

nω∑

n=1

ℜ[

ζn · ei(ωnt−knx cos µn)]

(1)

where ωn is the wave frequency, kn thewave number and µn the wave direction.The equivalent wave is given by

ζe(x, t) =

nω∑

n=1

ℜ[(

an + bnx +

+ cn cos2πx

λ

)

eiωnt

]

(2)

where the coefficients an, bn and cn are de-termined using the following minimisation

Proceedings of the 9th International Conference on Stability of Ships and Ocean

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problem:

∫ L2

−L2

(ζ(x, t) − ζe(x, t))2dx = min! (3)

The wavelength of the equivalent wave isλ = Lpp, which is expected to yield thelargest parametric roll excitation. The rollequation reads:

ϕ =M−dLϕ−dQϕ|ϕ|−(g−z)mh(ϕ,T,ϑ,t)−ϑΘxz sin ϕ

Θxx

(4)where M is the wave excitation momentobtained from GL STRIP, and dL and dQ

are linear and quadratic damping coeffi-cients, respectively. Gravity and ship massare denoted by g and m, and z and ϑ arethe heave and pitch accelerations calcu-lated with GL STRIP. The righting leverarm h was precomputed as a function ofroll angle ϕ, draught T and pitch angleϑ using a hydrostatic method (user-codedNAPA Macros).

3.2 GL SIMBEL

The development of this method datesback to the eighties. It is primarily basedon work of Soding (1982), Bottcher (1986)and Pereira (1988, 1989, 2003). Themethod simulates large amplitude rigidbody motions of mono- and multi hullvessels in six degrees of freedom, andshear forces and bending moments are de-termined. In determining the externalforces, several assumptions and simplifica-tions were made. These forces comprise:

• forces and moments due to weight,

• Froude-Krylov wave pressures undis-turbed by the ship,

• radiation and diffraction pressure,i.e. forces due to the influence of theship on the pressure field,

• speed effects (resistance and ma-noeuvring forces),

• propeller and rudder forces (in-cluding a proportional-integral-differential -PID- heading controlleror a track-keeping controller)

• forces due to fins and bilge keel ac-tions,

• wind forces,

• as well as forces due to fluid mo-tion in tanks and damaged compart-ments.

Due to the nonlinearity of large shipmotions, radiation and diffraction forcescannot be calculated separately. Radia-tion forces are generated by the ship mo-tions. Radiation forces represent the dif-ference between the forces of the non-moving ship in waves and the Froude-Krylov forces (undisturbed wave forces).If the ship partly emerges from the water,the diffraction component of the emergedpart must be zero. Therefore, radiationand diffraction forces are determined us-ing the relative-motion hypothesis. Theforce is assumed to depend on the motionof the ship minus the motion of the wa-ter at the different cross-sections. The or-bital velocity of the wave components ofthe seaway is averaged over a ship crosssection. The total force is obtained by in-tegrating the contribution of the ship sec-tions. Longitudinal interactions due to for-ward speed effects are treated in the sameway as for linear strip methods. The pres-sure distribution not only depends on theinstantaneous acceleration of the ship, butalso on the preceding accelerations (mem-ory effects). For linear computations inregular waves, these memory effects resultin the frequency dependence of the com-plex added mass matrix, which containsthe proportionality constant between therelative motion acceleration of a ship sec-tion and the force per length :

f = Au (5)

Proceedings of the 9th International Conference on Stability of Ships and Ocean

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where f and u are 3-component columnvectors, while A is a complex 3 by 3 ma-trix. For simulations of motions in a nat-ural seaway, the frequency dependency ofthese matrices constitutes a challenge, be-cause many frequencies occur at the sametime. A solution to this problem is the useof convolution integrals (impulse-responsefunctions), which account for the depen-dency of these forces on the accelerationsat different time steps. The state spacemodel is a faster solution. It uses a relationbetween acceleration and force derivatives:

A0u+A1∂

∂tu+A2

∂2

∂t2u... = B0+B1f+B2f+...

(6)

The 3 × 3 matrices Ak, Bk are frequencyindependent but depend on the actual sub-mergence and roll motion. Implicitly, theyare time dependent and are computed fromthe frequency dependent coefficients by re-gression analysis. During the simulation,the actual waterline inclination and im-mersion are interpolated.

4 Roll Damping

The damping moment model comprises alinear and a quadratic term:

dD = −bL ϕ − bQ ϕ |ϕ| . (7)

The coefficient bL contains a small speed-independent part caused by wave gener-ation of the rolling ship and another (inmost cases larger) part that is proportionalto the forward speed of the ship and iscaused by lift forces generated by hull, pro-peller, and rudder. The coefficients aretaken from Blume (1979). The coefficientbQ accounts for bilge keel effects. Blumepresents his model test results in diagramsfor different breadth/depth ratios and dif-ferent block coefficients. In his plots, thenon-dimensional roll damping coefficientφStat/φRes is used, which denotes the ratio

between the heel angle due to a static mo-ment and the roll amplitude for the equiva-lent resonant roll moment. The ratio is de-pendent on Froude number. To check theaccuracy of Blume’s coefficients for a mod-ern containership, model tests were car-ried out. The experiments have been donewith forced roll motion induced by rotat-ing masses. The linear damping constantbL was determined from forced roll motionmodel tests, using the following equation:

bL =m g GM0

ωr

(

φStat

φRes

)

5◦, (8)

whereωr - roll resonance frequencyGM0 - metacentric height.

The nonlinear roll damping coefficient bQ

was also determined from forced roll mo-tion tests. For the resonance angle φRes =20◦ an effective linear roll damping coeffi-cient beff was calculated:

beff =mgGM0

ωr

(

φStat

φRes

)

20◦. (9)

For a roll oscillation with frequency ωr andamplitude ϕA = 20◦, it was assumed thatthe linear roll damping coefficient beff isequivalent to the quadratic damping con-stant bQ (without bilge keels).To estimate bQ, the linear component bL

was subtracted from beff :

bQ =3π

8 ω0 ϕA

(beff − bL) (10)

The results are shown in Figure 2 and com-pared with Blume’s coefficients derivedfrom experiments carried out in the seven-ties. The figure clearly shows that the re-sults for zero speed are reasonable, but de-viate significantly for higher Froude num-bers. For the highest Froude numbers,twice the values compared to Blume’s weredetermined. In all subsequent calculations,the coefficients derived from our experi-ments were used.

Proceedings of the 9th International Conference on Stability of Ships and Ocean

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A comparison with other approaches, likeIkeda et al. (1978), is recommended if noexperimental data are available. Ikeda’sroll damping approach includes friction,eddy, lift and wave damping for naked hull,normal force damping of bilge keels, hull-pressure damping due to bilge keels andwave damping of bilge keels. Blume’s co-efficients are of experimental nature andincludes with exception of bilge keels alleffects related in Ikeda. For the bilge keeleffects we introduce an equivalent momentacc. to Gadd (1964).

5 Preparative Calcula-

tions

A large number of simulations in regularwaves was carried out prior to the modeltests to determine which situations regard-ing wave length and ship speed are relevantfor validation purposes. The selected hullform from DSME is shown in Figure 1.The main particulars and investigated loadcases of the ship were:

Length over all 332 [m]

Length b. perp. 317 [m]

Breadth 43.2 [m]

Design draught 14.4 [m]

rxx/B 0.384 [-]

ryy/Lpp 0.255 [-]

rzz/Lpp 0.254 [-]

Load case 1

GM0 1.26 [m]

Mass 140283 [t]

Draught aft 14.647 [m]

Draught fore 14.238 [m]

Load case 2

GM0 3.8 [m]

Mass 122908 [t]

Draught aft 12.949 [m]

Draught fore 12.728 [m]

The first load case was chosen as the onewith the smallest realistic GM0 in the sta-bility booklet. Load case 2 was chosen torepresent the upper range of realistic trans-verse stability values. The purpose of thisload case was to show the effect of a largerGM0 on the frequency ratios and the max-imum roll angles in parametric roll situa-tions.

The range of wave lengths was set from70 to 650 m, and Froude numbers fromzero speed to Fn=0.25 were investigatedfor wave heights ranging from 2 to 10 m.ROLLSS calculations were performed for59 wave lengths, 26 speeds, and 5 waveheights. To reduce the computational ef-fort, the number of wave lengths was de-creased to 16 for the GL SIMBEL calcula-tions.

Figures 3 to 9 show the results for load case1. The left hand side of the figures depictsthe results in following waves, whereasthe right hand side shows the results inhead waves. ROLLSS and GL SIMBELyielded similar areas of parametric roll oc-currence with small differences in the pre-dicted maximum roll angles. As expected,these areas developed in the vicinity of thefrequency ratio 0.5. They become largerfor higher wave amplitudes. The bound-ary towards smaller wavelengths shows asudden transition from quiescence to largeroll amplitudes. The decrease of roll anglestowards longer waves is smoother. OnlyROLLSS predicted parametric roll withthe frequency ratio 1:1 at a wave heightof 10 m, see Figure 7.

Figure 12 shows the results for load case 2.The curves of constant frequency ratio areshifted towards lower speeds in followingwaves and towards higher speeds in headwaves. The area of occurrence of para-metric roll is narrower than predicted forthe smaller GM0, and maximum roll anglesreach only 10◦ to 15◦.

Proceedings of the 9th International Conference on Stability of Ships and Ocean

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6 Model Tests

In the German research project ROLL-S,a new sophisticated test procedure for afully automated motion measurement ofa free running ship model was developed(Kuehnlein et al. 2003). The ship’s coursewas controlled by the master computer us-ing telemetry. Ship motions in six degreesof freedom were accurately registered bycomputer controlled guidance of both tow-ing and horizontal carriage. The speed ofthe propeller had to be fixed before start-ing the test. This resulted in a strong de-crease of the model speed as soon as para-metric rolling developed, making it diffi-cult to hold a constant velocity during atest run or to reach a given Froude num-ber.

The model tests were carried out in 10 mhigh regular waves for load case 1 only.The model scale was 1:53. The angle ofencounter was close to 0◦ (following waves)or 180◦ (head waves). Because of the lim-ited number of test runs, it was impossi-ble to generate result plots as detailed asfor the preparative simulations. Tables 1and 2 summarise the obtained roll angles ofthe model tests and the simulations. Fig-ures 10 and 11 show the same data as barcharts.

In following waves for the frequency ra-tio 1:1, GL SIMBEL simulations did notpredict parametric roll for λ/Lpp = 0.9 ,compare Fig. 7 and 9. We should men-tion that due to time constraints not somany simulations habe been carried outwith GL SIMBEL as with ROLLSS. Thearea where this kind of parametric roll oc-curs is very small and small differences onthe approaches may predict the paramet-ric roll in closed areas but not exact atthe same. For this frequency ratio (testruns no. 38, 39, 40, 52), it was difficultto determine the maximum roll angle inthe experiments, because the roll motionstill increased when reaching the end of the

model basin. Therefore, the test runs forthis condition were repeated several times,and the results did not show a clear trendfor the maximum roll angles, see Table 2.

7 Conclusions

A large number of simulations were car-ried out using GL SIMBEL and ROLLSS.Model tests in regular waves were per-formed and the results were compared tothe simulations. The results of simulationsand experiments compared favourably formost test runs. The head sea experiments32, 33b and 34a for situations close to thesteep transition from quiescence to largeroll angles showed bad correlation with thesimulations.For the frequency ratio 1:1 in followingwaves (test runs no. 38, 39, 40, 52) itwas difficult to determine the maximumroll angle in the experiments because theroll motion still increased when reachingthe end of the model basin.Both methods demonstrated their abilityto predict the occurrence of parametricroll in head and following seas. AlthoughROLLSS is based on a simpler mathemat-ical model than GL SIMBEL, both meth-ods predicted similar roll. The main factorthat causes the parametric roll, changes inthe wetted form of the ship, is capturedin both methods. ROLLSS take this intoaccount with pre-calculated heeling armcurves and GL SIMBEL calculates the hy-drostatic pressure at the actual position ofthe ship at each time instant. For otherheadings the accuracy for GL SIMBEL isexpected to be better due to the nonlinearcoupling off all ship motions in GL SIM-BEL. This has not been experimentallyvalidated.The importance to reevaluate the rolldamping coefficients for modern ship de-signs was demonstrated. The next phaseof DYNAS focuses on the validation of thenumerical codes in irregular sea ways.

Proceedings of the 9th International Conference on Stability of Ships and Ocean

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References

[1] Blume, P. (1979): ExperimentelleBestimmung der Koeffizienten derwirksamen Rolldampfung und ihreAnwendung zur Abschatzung ex-tremer Rollwinkel, Schiffstechnik,Vol. 26

[2] Bottcher, H. (1986): Ship motion sim-ulation in a seaway using detailed hy-drodynamic force coefficients, STAB1986, 3rd International Conference onStability of Ships and Ocean Vehicles,Gdansk

[3] Gadd, G.E. (1964): Bilge kiels andbilge vanes, Report Nr. 64, NationalPhysical Laboratory - Ship Division

[4] Hachmann, D. (1991):Calculation ofPressures on a Ship’s Hull in Waves,Ship Technology Research, Vol. 38 No.3

[5] Ikeda, Y.; Himeno, Y.; Tanaka, N.(1978): A Prediction Method forShip Roll Damping, Department onNaval Architectural, University of Os-aka Prefecture, Report Nr. 00405,

[6] Kroger, H.P. (1986): Rollsimulationvon Schiffen im Seegang, Schiffstech-nik, Vol. 33

[7] Kuehnlein, W.L.; Brink, Kay-Enno;Hennig, J. (2003): Innovative deter-ministic seakeeping test procedures,STAB 2003, 8th International Con-ference on the Stability of Ships andOcean Vehicles, Madrid

[8] Papanikolaou, A.D.; Schellin, T.E(1991): A Three-Dimensional PanelMethod for Motions and Loads ofShips with Forward Speed, Ship Tech-nology Research, Vol. 39 No. 4

[9] Pereira, R. (1988): Simulationnichtlinearer Seegangslasten, Schiffs-technik, Vol. 35, 4:173-193

[10] Pereira, R. (1989): Ermittlungder Belastungen von Schiffen insteilem Seegang durch Simulation,Jahrbuch der SchiffbautechnischenGesellschaft, Vol. 83:145-158, Som-mertagung, Berlin

[11] Pereira, R. (2003): Numerical Sim-ulation of Capsizing in Severe Seas,STAB 2003, 8th International Con-ference on the Stability of Ships andOcean Vehicles, Madrid

[12] Petey, F. (1988): Ermittlung der Ken-tersicherheit lecker Schiffe im Seegangaus Bewegungssimulationen, ReportNr. 487, Institut fur Schiffbau derUniversitat Hamburg

[13] SNAME AD HOC PANEL #13(2003): Investigation of head-seaparametric rolling and resulting ves-sel and cargo securing loads, MarineTechnology (41)

[14] Soding, H. (1982): Leckstabilitat imSeegang, Report Nr. 429, Institut furSchiffbau der Universitat Hamburg

Proceedings of the 9th International Conference on Stability of Ships and Ocean

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Figure 1: Containership Hull Form

Proceedings of the 9th International Conference on Stability of Ships and Ocean

Vehicles 9

0

0.05

0.1

0.15

0.2

0.25

0.3

0 0.05 0.1 0.15 0.2

ϕ sta

t/ϕre

s

Froude number

Blume - linear part with bilge keelBlume - quadratic part with bilge keel

Experiments with bilge keel - linear partExperiments with bilge keel - quadratic part

Figure 2: Comparison of the non-dimensional roll damping coefficient φStat/φRes acc. toBlume with model tests

c=0.3

=0.4

c=0.5c=

0.6

c=0.7

c=0.7

c=0.7

c=0.8

c=0.8

c=0.9

c=0.9

c=1

c=1

c=1

c=1.1

c=1.1

c=1.

c=1.2

c=1.2

c=1.2

c=1.3

c=1.3

c=1.4 c=0.3

c=0.4

c=0.4

c=0.5

c=0.6

Froude number

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0.25 0.2 0.15 0.1 0.05 0 0.05 0.1 0.15 0.2 0.25

0.5 0.5

1 1

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2 2

Max roll angle45403530252015105

Figure 3: ROLLSS Results for Wave Height 2m

Proceedings of the 9th International Conference on Stability of Ships and Ocean

Vehicles 10

c=0.4

c=0.5

c=0.6

c=0.6

c=0.7

c=0.7

c=0.7

c=0.8

c=0.8

c=0.8

c=0.9

c=0.9

c=0.9

c=1

c=1

c=1

c=1.1

c=1.1

c=1.2

c=1.2

c=1.3

c=1.3

c=1.4c=0.2

c=0.3

c=0.4

c=0.5

c=0.6

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0.25 0.2 0.15 0.1 0.05 0 0.05 0.1 0.15 0.2 0.25

0.5 0.5

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Max roll angle45403530252015105

Figure 4: ROLLSS Results for Wave Height 4m

c=0.4

c=0.5

c=0.5

c=0.6

c=0.6

c=0.7

c=0.7

c=0.7

c=0.8

c=0.8

c=0.8

c=0.9

c=0.9

c=1

c=1

c=1

c=1.1c=1.1

c=1.1

c=1.2

c=1.2

c=1.2

c=1.3c=1.4c=0.2

c=0.3

c=0.4

c=0.4

c=0.5

c=0.5

c=0.6

Froude number

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0.25 0.2 0.15 0.1 0.05 0 0.05 0.1 0.15 0.2 0.25

0.5 0.5

1 1

1.5 1.5

2 2

Max roll angle45403530252015105

Figure 5: ROLLSS Results for Wave Height 6m

c=0.4

=0.5

c=0.5

c=0.6

c=0.6

c=0.7

c=0.7

c=0.7

c=0.8

c=0.8

c=0.8

c=0.9

c=0.9

c=1

c=1

c=1

c=1.1

c=1.1

c=1.1

c=1.2

c=1.2

c=1.2

c=1.3

c=1.4

c=0.2

c=0.3

c=0.4

c=0.5

c=0.5

c=0.6

Froude number

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0.25 0.2 0.15 0.1 0.05 0 0.05 0.1 0.15 0.2 0.25

0.5 0.5

1 1

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2 2

Max roll angle45403530252015105

Figure 6: ROLLSS Results for Wave Height 8m

Proceedings of the 9th International Conference on Stability of Ships and Ocean

Vehicles 11

c=0.4

c=0.5

c=0.6

c=0.6

c=0.7

c=0.7

c=0.7

c=0.8

c=0.8

c=0.9

c=0.9

c=0.9

c=1

c=1

c=1

c=1.1

c=1.1

c=1.1

c=1.2

c=1.2

c=1.2

c=1.3c=1.4

c=0.2

c=0.3

c=0.4

c=0.4

c=0.5

c=0.5

c=0.6

Froude number

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ng

th/L

pp

0.25 0.2 0.15 0.1 0.05 0 0.05 0.1 0.15 0.2 0.25

0.5 0.5

1 1

1.5 1.5

2 2

Max roll angle45403530252015105

Figure 7: ROLLSS Results for Wave Height 10m

c=0.2

c=0.3c=0.4

c=0.5

c=0.6

c=0.3c=0.4

c=0.5

c=0.5

c=0.6

c=0.

7

c=0.7

c=0.8

c=0.

8

c=0.

8

c=0.

9

c=0.

9

c=1c=

1

c=1

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h/L

pp

0.25 0.2 0.15 0.1 0.05 0 0.05 0.1 0.15 0.2 0.25

0.5 0.5

1 1

1.5 1.5

2 2

Max roll angle35302520151050

Figure 8: GL SIMBEL Results for Wave Height 8m

c=0.3

c=0.4c=0.5

c=0.5c=0.6

c=0.4

c=0.5

c=0.

6

c=0.6

c=0.

7

c=0.7

c=0.8

c=0.

8

c=0.8

c=0.9

c=0.

9

c=1

c=1

Froude number

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h/L

pp

0.25 0.2 0.15 0.1 0.05 0 0.05 0.1 0.15 0.2 0.25

0.5 0.5

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2 2

Max roll angle35302520151050

Figure 9: GL SIMBEL Results for Wave Height 10m

Proceedings of the 9th International Conference on Stability of Ships and Ocean

Vehicles 12

Table 1: Comparison of Maximum Roll Angles for Head Waves (Wave Height 10m)

Test Run µ [◦] λ/Lpp Fn [-] ϕmax [◦]

No. Experiments GL SIMBEL ROLLSS

10 178 0.6 0.059 0.0 0.1 0.018 178 1.0 0.040 30.0 32.1 31.019 178 0.9 0.022 31.0 33.3 32.0

21 b 175 0.9 0.065 36.0 37.2 38.022 175 0.9 0.030 37.0 32.2 33.023 177 1.1 0.052 29.0 31.9 28.024 177 1.3 0.076 22.0 28.6 24.026 177 1.5 0.068 18.0 21.8 17.027 177 1.7 0.061 17.0 15.0 7.528 177 2.0 0.085 0.3 0.9 0.031 177 1.0 0.052 31.0 33.6 32.032 177 1.0 0.108 1.4 37.7 39.0

33 a 177 1.0 0.055 31.0 34.0 32.033 b 177 1.0 0.131 22.0 33.2 41.034 a 177 1.0 0.099 26.0 38.4 38.034 b 177 1.0 0.146 0.0 0.0 0.0

Table 2: Comparison of Maximum Roll Angles for Following Waves (Wave Height 10m)

Test Run µ [◦] λ/Lpp Fn [-] ϕmax [◦]

No. Experiments GL SIMBEL ROLLSS

38 3 0.9 0.197 2.5 0.0 15.039 3 0.9 0.204 4.0 0.2 15.040 5 0.9 0.204 10.0 0.2 16.052 5 0.9 0.206 15.0 0.2 16.054 2 0.7 0.095 0.7 0.1 15.055 2 0.8 0.063 18.5 15.2 22.056 0 0.8 0.063 18.0 15.2 22.057 0 1.0 0.060 17.0 10.7 17.0

Proceedings of the 9th International Conference on Stability of Ships and Ocean

Vehicles 13

Figure 10: Comparison of Simulations and Tests in 10m Head Waves

Figure 11: Comparison of Simulations and Tests in 10m Following Waves

Proceedings of the 9th International Conference on Stability of Ships and Ocean

Vehicles 14

c=0.6

c=0.8

c=0.

c=0.9

c=1

c=1

c=1.1

c=1.1

c=1.1

c=1.2

c=1.2

c=1.2

c=1.3

c=1.3

c=1.4

c=1.4

c=1.5

c=1.5

c=1.6

c=1.6

c=1.6

c=1.7

c=1.7

c=1.7

c=1.8

c=1.8

c=1.8

c=1.9

c=1.9

c=1.9

c=2

c=2

c=2

c=2.1

c=2.1

c=2.1

c=2.2

c=2.3

c=2.4

c=0.3

c=0.4c=0.5

c=0.6

c=0.7

c=0.7

c=0.8c=

0.9c=

1

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0.25 0.2 0.15 0.1 0.05 0 0.05 0.1 0.15 0.2 0.25

0.5 0.5

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2 2

Max roll angle45403530252015105

Figure 12: ROLLSS results for GM0=3.8m, Wave Height 10m

March 27, 2006 11:0

Proceedings of OMAE 200625th International Conference on Offshore Mechanics and Arctic Engineering

Hamburg, Germany, June 4-9, 2006

OMAE2006-92245

QUALITATIVE AND QUANTITATIVE VALIDATION OF A NUMERICAL CODE FORTHE REALISTIC SIMULATION OF VARIOUS SHIP MOTION SCENARIOS

Janou Hennig∗MARIN

NetherlandsEmail: j.hennig@marin.nl

Heike BillerbeckFlensburg Shipyard

GermanyEmail: heike@fsg-ship.de

Gunther F. ClaussOcean Engineering SectionTechnical University Berlin

GermanyEmail: clauss@naoe.tu-berlin.de

Daniel TestaOcean Engineering SectionTechnical University Berlin

GermanyEmail: testa@naoe.tu-berlin.de

Kay-Enno BrinkNaval Architect

GermanyEmail: kay.brink@freenet.de

Walter L. KuhnleinHamburg Ship Model Basin

GermanyEmail: kuehnlein@hsva.de

ABSTRACTThere is an ongoing discussion on safety guidelines to be

considering more recent developments in ship design. Numeri-cal simulations of ship motions are considered as powerful toolfor the safety evaluation of a given design. However, the conse-quent use of numerical codes calls for their thorough validationwhich has to be performed both qualitatively and quantitatively.This paper focuses on a code used and further developed by theFlensburg Shipyard. For its validation, the capsizing scenarioin steep wave sequences is realized in the wave tank first. Thededicated computer controlled experimental technique ensuresthe exact phase correlation of wave excitation and resultant shipmotions. Thus, the registered wave and the track of the shipmodel in the model test serve as input to the numerical simu-lation which results in the specific motion time traces. These arenow directly compared to the motion registrations from the modeltests. First results of the validation by direct comparison of timeseries have been presented in earlier publications, still with therestriction that only a few cases have been investigated. In thispaper, the promising method is applied to another scenario in along-crested sea state including steep wave combinations. Dif-

∗Address all correspondence to this author.

ferent aspects are discussed which results in the conclusion thatthe method is feasible for free running ships in stern and sternquartering seas.

1 INTRODUCTIONSimulation programs have improved significantly during the

last years and are considered routinely in the design process atshipyards and for the basic investigation of ship safety (Cramerand Tellkamp (2003)). First results from test calculations for ex-isting ships are presented by Kruger (2002), and the conclusionsfrom numerical motion simulations regarding seakeeping capa-bilities of different ships correlate well with operational experi-ence. Many examples show, that simulation tools are well suitedto investigate accidents, see e. g. France et al. (2001), McTag-gart and de Kat (2000), Soding (1987a). With respect to designoptimization it can furthermore be demonstrated that the conse-quent application of numerical investigations in the early designphase allows to efficiently improve the design with respect to in-tact safety. Numerical motion simulations are a valuable tool forship design, investigation of accidents and operational guidanceif they are used appropriately.

1 Copyright c© 2006 by ASME

In the framework of the ongoing discussion on safety guide-lines accounting for recent developments in ship design, numer-ical simulations of ship motions are considered as powerful toolfor the safety evaluation of a given design. Numerical assess-ments are faster and less expensive than experiments but requirea thorough validation by model tests as the required quality ofnumerical simulation tools is not agreed upon by rules and regu-lations.

Validation denotes the process of testing the ability to accu-rately model a physical phenomenon (whereas verification refersto the process of testing the accuracy of numerical implementa-tions of a mathematical model). Validation of numerical methodsby model tests commonly comprises the following approaches:

• Comparison of amplitudes and phases in regular waves• Analysis of statistical results in irregular waves• Investigation of basic ability to model the phenomenon qual-

itatively

A validation by direct comparison of time series is frequentlyused for regular waves but not for irregular seas as the exact cor-relation of time and position for wave excitation and ship motionis not available in standard test set-ups. Additionally, a directcomparison of measured and simulated time series is extremelycomplicated if all six degrees of freedom are considered: Smalldeviations in the ship reaction might add up over time. Thus,a small deviation in the surge motion changes the wave trainsencountered in the moving reference frame of the vessel.

In this paper, a state-of-the-art numerical method for thesimulation of ship motions in realistic seas is validated by cap-sizing tests with deterministic wave sequences. The new type ofvalidation allows for comparison of time series from model testsand numerical simulations in all degrees of freedom directly inthe moving reference frame of the ship. The validation is basedon the exact modelling of deterministic wave sequences both innumerical and experimental simulation.

Based on deterministic wave sequences and fully automatedcapsizing tests the wave elevation is known at all tank positionsand instances – i. e. in the moving reference frame of the ship.Thus, it is possible to extent the validation to more details.

In Hennig (2005), the method of validation is described indetail and a first promising case is presented whereas this paperaddresses some important details to be accounted for. Further-more, another test case reveals the conclusion that the procedureof validation as well as the numerical method itself are feasibletools for the evaluation of a given ship design.

2 NUMERICAL SIMULATION TOOLVarious numerical motion simulation programs exist world-

wide – from rather simple linearized tools to sophisticated non-linear codes. All simulation methods are based on assumptions

and simplifications in their mathematical model in order to re-duce computing time. Still some of the highly non-linear meth-ods have longer computing times than simulated time, whichstrongly reduces their practical applicability for safety assess-ments.

The numerical motion simulation tool Rolls is based on amethodology developed by Kroger (1987) and Petey (1988) andfurther developed by Soding (1987b), and Cramer and Kruger(2001). Rolls simulates the motion of intact and damaged shipsin time domain in all six degrees of freedom in regular wavesand irregular long or short crested seas. For heave, pitch, swayand yaw, response amplitude operators (RAO) are used – cal-culated linearly by means of strip theory. The surge motion issimulated assuming a hydrostatic pressure distribution under thewater surface for the determination of the surge-inducing waveforces. The roll motion is simulated non-linearly with the right-ing arm in waves being determined for every time step usingGrim’s effective wave as modified by Soding (1987a). The un-derlying theoretical model (Cramer and Kruger (2001)) resultsin very short simulation times and is basically able to predict thefollowing mechanisms of large roll motions and capsizing verywell (Clauss et al. (2005)):

• Resonance excitation• Parametric excitation• Loss of stability phenomena• Combinations of the above.

3 DETERMINISTIC CAPSIZING TESTSA deterministic wave generation and transformation tech-

nique as described previously in Clauss and Hennig (2003) andClauss et al. (2004) is applied to a deterministic capsizing testprocedure at the Hamburg Ship Model Basin. The fully com-puter controlled test procedure allows for highly accurate and re-producible investigations of various intact stability phenomena.Deterministic wave sequences are generated with respect to anexact time-space correlation of ship encounter (Hennig (2005)).

Numerical investigations are used to identify critical condi-tions with respect to resonance phenomena or pure loss of stabil-ity. Corresponding wave sequences are generated – in particularwith respect to timing between waves and vessel – and realizedin the model basin. This shows that the numerical tool is ableto model the basic mechanism of large roll motions (qualitativevalidation).

In addition, the measured scenario is re-modelled in the nu-merical simulation to allow the direct – quantitative – compari-son of model test and numerical motion simulation.

An example of reproducible test results for parametric rolldue to a high deterministic wave sequence from astern (signifi-cant wave height Hs = 9.36 m, peak period TP = 11.66 s) is givenin Fig. 1. Apart from large container ships, also other modern

2 Copyright c© 2006 by ASME

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Figure 1. Comparison of two capsizing test runs in deterministic wavesequences from astern inducing parametric roll (v = 10.4 kn, µ = 3◦):The encounter of ship and wave train demonstrates the very good repro-ducibility of capsizing tests with deterministic wave sequences: The wavetrain is encountered under identical conditions. However, the sensitivityof capsizing processes is obvious – although the conditions and resultantmotions are almost identical, the RO-RO vessel (GM = 1.27 m) cap-sizes during the second test run only (light green graph, bottom).

ship designs are susceptible to parametric excitation such as RO-RO, RoPax, ferries and cruise vessels. The test conditions forthe RO-RO vessel (metacentric height GM = 1.27 m, ship speedv = 10.4 kn, ship heading µ = 3◦ – waves from astern) are sim-ulated prior to the test. In both test runs, the ship encountersthe wave train at almost identical conditions (top) and shows asimilar roll response. However, in the first test (dark graph) theship roll motions exceed 50◦ but the capsize occurs only in thesecond run (light graph). This is also an example for both thehigh reproducibility of capsizing tests with deterministic wavesequences and the sensitivity of the mechanism leading to cap-sizing. It should be noted that in reality, a capsize at a roll angleof more than 50◦ might have occurred though – eventually dueto shift of cargo, vehicles – or engines.

4 DESCRIPTION OF TEST CASEThe capsizing case used here to validate the simulation tool

against tests and to examine some important factors of influencewith respect to the accuracy of the simulation is a combinationof parametric roll and loss of stability on the wave crest. As hasbeen shown previously, the simulation tool is able to predict suchmodes of roll reliably.

Parametric excitation occurs when the roll angle increases

Figure 2. RO-RO vessel – model scale 1:34 – in a high wave sequencefrom astern which finally leads to capsizing – Lpp = 190 m, B = 26.5 m,T = 7.35 m.

dangerously due to non-linear processes (disturbances) whichcause roll motions at half the frequency of the exciting wave.The ship rolls with twice the period of the pitch motion (whichis approximately the encountered wave period). The followingfactors contribute to parametric roll:

• Encounter period close to one-half of the natural roll period• Large waves• Wave length comparable to ship length• Wide, flat sterns in combination with pronounced bow flare,

as this increases the stability variations when the wavepasses along the hull.

Loss of stability at the wave crest refers to the quasi-staticloss of transverse stability (associated with an excessive rightingarm reduction) for the ship in the wave crest condition. Thismode occurs typically in following to stern quartering waveswhen the frequency of encounter becomes low. The ship can cap-size when it experiences temporarily a critically reduced (possi-bly negative) righting arm for a sufficient period of time, whilethe wave crest overtakes the ship slowly and the ship is surgingor surf-riding periodically. In irregular seas, only one encoun-tered wave of critical length and steepness is sufficient to causethe sudden catastrophic event.

In realistic conditions, these two phenomena usually com-bine. Ships with low stability in a wave crest condition expe-rience a shift in roll period towards longer periods in followingseas, while the large righting moment in the wave trough con-dition yields much roll energy into the ship motion. Thus, theremaining stability in the wave crest condition might not be suf-

3 Copyright c© 2006 by ASME

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mesured (blue) and simulated (red) roll motion

Figure 3. Comparison of time series from numerical simulations (red)and deterministic test results (blue) of capsizing due to parametric excita-tion (TP = 9.9 s and Hs = 7.1 m) – first simulation interval.

ficient to prevent a capsize.The sea state realized in the model basin is a JONSWAP

spectrum with γ = 3.3 which refers to a possible standard tradein the North Sea for the investigated ship (Fig. 2). The peak pe-riod is TP = 9.9 s and the significant wave height is Hs = 7.1 m.As in the last year’s paper (Clauss et al. (2005)), a defined se-quence of this sea state has been realized in the basin accordingto previous simulations to identify the most critical wave condi-tions. A registration of this wave sequence relatively close to thewave maker (x = 6 m) is given in Fig. 3 (top). The ship sailsin almost stern seas (µ = 3◦) at an average speed of v = 8.8 kn(full scale) through the basin. The associated track is shown inthe next graph. Thus, the ship encounters the wave train given inthe following graph. The methods to calculate the moving refer-ence frame wave train are explained in Cramer et al. (2004). Asa consequence, a roll motion of more than 40◦ is measured. As aresult of the increasing roll angle due to parametric roll in com-bination with a higher wave group, the ship capsizes after 1000 s(see Fig. 4).

5 REPRESENTATION OF DETERMINISTIC WAVE SE-QUENCESA very important aspect of the validation is the correct rep-

resentation of the wave sequence according to the Shannon sam-pling theorem: A signal is represented correctly if the samplingfrequency is at least twice the signal bandwidth. It means thatthe discrete samples are a complete representation of the signalif the bandwidth is less than half the sampling rate, which is re-ferred to as the Nyquist frequency. Thus, it can be reproducedat full length and without loss of information. This seems to bea well-known basic and therefore almost not worth to be stated.However, it is so easy to disregard it if caring for a reduction ofcomputational time which is often achieved by a reduction of testdata. An example of possible problems is given in Fig. 5.

The stationary wave train as has been measured at x = 6 mis fed into the numerical simulation in order to consider the waveelevation at every instant at the ship sailing at a defined velocity.To reduce the amount of data the wave train is represented by 91frequency values. The reduction of data is performed as follows:The full time trace is Fourier transformed. The frequency stepswhich are not in the range of the significant spectrum are filteredout. From the remaining values not all given time steps are cho-sen. If a too short time interval is considered only this intervalcan be reconstructed later on. For a longer simulation time thereis no agreement between the time series due to the reduced dataset and the original one. This is illustrated in Fig. 5: As a conse-quence of the wrong wave representation, also the measured rollmotion is not in agreement with the simulated one.

6 DISCUSSION OF RESULTSFig. 3 shows a comparison of experimental and numerical

simulation for the first 400 s: The above graph is the station-ary wave registration at the position close to the wave maker(x = 6 m). This wave train is calculated at the measured movingposition of the keel point of the ship (second graph). This resultsin the wave train as given in the third graph from above. Both thespectral representation of the stationary wave train as well as theactual position of the ship serve as input to the numerical simu-lation (where the first time step considered is defined to be zero).Also the x-position of the ship has to be given as input in order tobe able to compare measurement and calculation. As even smalldisturbances in the surge motion and position of the ship result inlarge differences in the encountered wave train, the ship is guidedin x-direction according to the measured x-series. Consequently,the wave train encountered in the simulation corresponds wellwith the wave train encountered in the model test and the resultsfor roll are comparable. However, all other degrees of freedomare free during the simulation. The graph below gives the resul-tant roll motion as response to the moving reference wave trainfrom both numerical and experimental simulation. The agree-ment is good in the first wave group inducing high roll angles.

4 Copyright c© 2006 by ASME

Fig. 6 shows the roll motion at the entire simulation period.In the beginning of the simulation the agreement of calculatedand measured time traces is good. However, after approximately400 s, the phase is shifting considerably.

This phenomenon can be explained by the turning manoeu-vre which has to be performed during the experiment but is notconsidered in the numerical simulation where the width of themodel basin is no restriction. The change of course happens dur-ing the first interval where no significant roll motion is identified.The course change in the turning manoeuvre (Fig. 6) implies aninitial roll angle to the ship which now increases again as theencountered wave group induces parametric excitation again. Inthe numerical simulation, no external roll angle excitation is in-troduced to the system, i. e. the initial roll angle results fromthe small wave excitation. As can be seen in the time traces forroll, these initial conditions are already shifted against each otherwhich is maintained during the ongoing simulation. Thereforethe phenomenon of parametric roll remains the same but appearsupside down. For this reason, in Fig. 4, the roll motion is in-verted.

A phase shift of the simulated time trace by 180◦ (Fig. 6,second picture) results in good agreement at the interval from205 s on. Thus, the time traces are compared in two parts - start-ing from the beginning until the course change due to the turn-ing manoeuvre in the model test, and starting from that coursechange up to the end of the registration where the simulation hasbeen inverted (Fig. 3 and 4). In any case, the ship capsizes in thefatal second wave group.

Another issue can be observed in the comparison of numeri-cal and experimental simulation: a slight time shift in the results.In general, good agreement between measured and calculateddata is observed. However, there is a slight time shift occurringbetween both which is not clear as related to the good agree-ment between simulated and measured wave data at the ship.Therefore, additionally to the so far described time traces, alsothe measured relative wave probe data have been analyzed. Dur-ing the model tests, an ultrasonic wave probe is registering thedisturbed wave train at the riding towing carriage. As the ship issailing between wave generator and carriage, disturbances by theship model are to be expected. Thus, the wave height and shapeis not correct. However, a comparison of a registration by thiswave probe and the calculated wave train at the position of theprobe (non-constant speed!) might give an indication of possibleerrors in the calculation procedure.

Fig. 7 shows a comparison between wave probe registra-tion and calculation from the stationary wave train close to thewave maker. For the time sequences of interest, a slight devia-tion in time can be observed which means that this deviation (asreflected in the roll motion) is already induced by the wave anal-ysis. As we deal with a time - and not a phase - shift only, this isnot of great influence to the main results.

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mesured (blue) and inverted simulated (red) roll motion

Figure 4. Comparison of time series from numerical simulations (red)and deterministic test results (blue) of capsizing due to parametric exci-tation (TP = 9.9 s and Hs = 7.1 m) – second interval with inverted rollmotion.

7 CONCLUSIONS AND PERSPECTIVESAs demonstrated in previous publications, the presented

method of validation is an excellent tool to validate numericalmethods related to the prediction of large roll motions by a directcomparison of time series from model tests and simulations. Thevalidation – applied to a different test case – shows once againthat the numerical tool is feasible for free running ships in irreg-ular seas. In particular, roll motions due to defined wave groupsare well predicted.

In this paper, examples are given to illustrate what is im-portant to perform a proper validation. Thus, the course anglemeasured in the model test has to be considered as it might intro-duce an ”artificial” roll angle due to the turning manoeuvre whichis not identical in the numerical simulation. Also the numeri-cal representation of the wave train is of greatest significance.These findings demonstrate that the detailed knowledge of theused tools is crucial to evaluate the safety of a given ship designproperly.

The cases of head and head quartering seas as well as variousship speeds and encounter angles still have to be investigated.

ACKNOWLEDGMENTThe authors are indebted to the German federal Ministry of

Research and Education (BMBF) for funding the projects SIN-SEE and LASSE.

5 Copyright c© 2006 by ASME

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Figure 5. Comparison of wave representation at position of ship by 91 values (red) and original sampling rate (blue). If a too low number of samples ischosen the wave train cannot be applied over the entire simulation interval. Calculated roll motion due to wave representation with 91 values (bottom). Theroll motion deviated considerably in the second half of the simulation interval since the wave is not represented by a sufficient number of samples.

REFERENCESG. F. Clauss, J. Hennig, and C. E. Schmittner. Modelling Ex-

treme Wave Sequences for the Hydrodynamic Analysis ofShips and Offshore Structures. In PRADS 2004 - 9th Inter-national Symposium on Practical Design of Ships and OtherFloating Structures, Lubeck-Travemunde, Germany, 2004.

G. F. Clauss, J. Hennig, H. Cramer, and K.-E. Brink. Vali-dation of Numerical Motion Simulations by Direct Compar-ison with Time Series from Ship Model Tests in Determin-istic Wave Sequences. In OMAE 2005 - 24th InternationalConference on Offshore Mechanics and Arctic Engineering,Halkidiki, Greece, 2005. OMAE2005-67123.

G.F. Clauss and J. Hennig. Deterministic Analysis of ExtremeRoll Motions and Subsequent Evaluation of Capsizing Risk.In STAB 2003 - 8th International Conference on the Stabilityof Ships and Ocean Vehicles, Madrid, Spain, 2003.

H. Cramer and S. Kruger. Numerical Capsizing Simulationsand Consequences for Ship Design. In STG Summermeeting,Gdansk, Poland, 2001.

H. Cramer and J. Tellkamp. Towards safety as a performancecriteria in ship design. In RINA International Conference onPassenger Ship Safety, London, UK, 2003.

H. Cramer, K. Reichert, K. Hessner, J. Hennig, and G. F. Clauss.Seakeeping Simulations and Seaway Models and ParametersSupporting Ship Design and Operation. In PRADS 2004 - 9thInternational Symposium on Practical Design of Ships and

Other Floating Structures, Lubeck-Travemunde, Germany,2004.

W. N. France, M. Levadou, T. W. Treakle, J. R. Paulling, K.Michel, and C. Moore. An Investigation of Head-Sea Paramet-ric Rolling and its Influence on Container Lashing Systems. InSNAME Annual Meeting, 2001.

J. Hennig. Generation and Analysis of Harsh Wave Environ-ments. Dissertation, Technische Universitat Berlin (D 83),2005.

P. Kroger. Simulation der Rollbewegungen von Schiffen im See-gang. PhD thesis, Universitat Hamburg, 1987.

S. Kruger. Dynamic Stability of Ro-Ro Ships in Waves. InConference on Design and Safety of Ro-Ro Passenger Ships,Copenhagen, Denmark, 2002.

K. McTaggart and J. O. de Kat. Capsize risk of intact frigatesin irregular seas. In SNAME transactions, volume 108, pages147–177, 2000.

F. Petey. Abschlussbericht zur Erweiterung des Vorhabens Leck-stabilitat im Seegang. Technical report, Institut fur Schiffbauder Universitat Hamburg, 1988.

H. Soding. Ermittlung der Kentergefahr aus Bewegungssimula-tionen. Schiffstechnik, 34, 1987a.

H. Soding. Simulation der Bewegungen intakter und leckerSchiffe. 23. Fortbildungskurs, 1987b.

6 Copyright c© 2006 by ASME

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Figure 6. Measured roll motion (blue) in comparison with numerical simulation (red): The first sequence – up to 400 s – shows a good agreement betweennumerical and experimental results. Then, due to a change of course in the model test within a relatively calm period of time, the roll motions show a phaseshift of 180◦ between each other. This is highlighted by the inversion of the second part in the picture below. Now the agreement appears to be reasonableagain.

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Figure 7. Time shift due to shift in ultrasonic wave probe data – top: measured data in blue, calculated from measured stationary wave data in red,bottom: wave at ship.

7 Copyright c© 2006 by ASME