1. Contemporary Ideas on Ship Stability
and Capsizing in Waves
2. FLUID MECHANICS AND ITS APPLICATIONS
Series Editor: R. MOREAU
Ecole Nationale Supérieure d’Hydraulique de Grenoble
Boîte Postale 95
38402 Saint Martin d’Hères Cedex, France
Aims and Scope of the Series
As well as the more traditional applications of aeronautics, hydraulics, heat and mass
transfer etc., books will be published dealing with topics which are currently in a state
of rapid development, such as turbulence, suspensions and multiphase ﬂuids, super and
hypersonic ﬂows and numerical modeling techniques.
It is a widely held view that it is the interdisciplinary subjects that will receive intense
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ids have the ability to transport matter and its properties as well as to transmit force,
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highly relevant in domains such as chemical, metallurgical, biological and ecological
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The purpose of this series is to focus on subjects in which ﬂuid mechanics plays a
For further volumes:
Jean Otto de Kat • Kostas Spyrou • Naoya Umeda
Marcelo Almeida Santos Neves • Vadim L. Belenky
Contemporary Ideas on Ship
Stability and Capsizing
Table of Contents
List of Corresponding Authors ..........................................................................xiii
1 Stability Criteria
Review of Available Methods for Application to Second Level
Vulnerability Criteria ...............................................................................................3
Christopher C. Bassler, Vadim Belenky, Gabriele Bulian, Alberto
Francescutto, Kostas Spyrou and Naoya Umeda
A Basis for Developing a Rational Alternative to the Weather Criterion:
Problems and Capabilities......................................................................................25
Andrzej Jasionowski and Dracos Vassalos
Evaluation of the Weather Criterion by Experiments and its Effect
to the Design of a RoPax Ferry ..............................................................................65
Shigesuke Ishida, Harukuni Taguchi and Hiroshi Sawada
Evolution of Analysis and Standardization of Ship Stability: Problems
and Perspectives .....................................................................................................79
SOLAS 2009 – Raising the Alarm.......................................................................103
Dracos Vassalos and Andrzej Jasionowski
2 Stability of the Intact Ship
Effect of Initial Bias on the Roll Response and Stability of Ships
in Beam Seas ........................................................................................................119
Historical Roots of the Theory of Hydrostatic Stability of Ships........................141
Horst Nowacki and Larrie D. Ferreiro
The Effect of Coupled Heave/Heave Velocity or Sway/Sway Velocity
Initial Conditions on Capsize Modeling..............................................................181
Leigh S. McCue and Armin W. Troesch
A. Yucel Odabasi and Erdem Uçer
6. Table of Contentsvi
The Use of Energy Build Up to Identify the Most Critical Heeling Axis
Direction for Stability Calculations for Floating Offshore Structures.................193
Joost van Santen
Some Remarks on Theoretical Modelling of Intact Stability...............................217
N. Umeda, Y. Ohkura, S. Urano, M. Hori and H. Hashimoto
3 Parametric Rolling
An Investigation of Head-Sea Parametric Rolling for Two Fishing
Marcelo A.S. Neves, Nelson A. Pérez, Osvaldo M. Lorca and Claudio
Simple Analytical Criteria for Parametric Rolling...............................................253
Experimental Study on Parametric Roll of a Post-Panamax
Containership in Short-Crested Irregular Waves .................................................267
Hirotada Hashimoto, Naoya Umeda and Akihiko Matsuda
Model Experiment on Parametric Rolling of a Post-Panamax
Containership in Head Waves..............................................................................277
Harukuni Taguchi, Shigesuke Ishida, Hiroshi Sawada and Makiko Minami
Numerical Procedures and Practical Experience of Assessment
of Parametric Roll of Container Carriers .............................................................295
Vadim Belenky, Han-Chang Yu and Kenneth Weems
Parametric Roll and Ship Design .........................................................................307
Marc Levadou and Riaan van’t Veer
Parametric Roll Resonance of a Large Passenger Ship in Dead Ship
Condition in All Heading Angles.........................................................................331
Abdul Munif, Yoshiho Ikeda, Tomo Fujiwara and Toru Katayama
Parametric Rolling of Ships – Then and Now......................................................347
J. Randolph Paulling
Parallels of Yaw and Roll Dynamics of Ships in Astern Seas
and the Effect of Nonlinear Surging ....................................................................363
7. Table of Contents vii
Model Experiment on Heel-Induced Hydrodynamic Forces
in Waves for Realising Quantitative Prediction of Broaching.............................379
H. Hashimoto, Naoya Umeda and Akihiko Matsuda
Perceptions of Broaching-To: Discovering the Past ............................................399
5 Nonlinear Dynamics and Ship Capsizing
Use of Lyapunov Exponents to Predict Chaotic Vessel Motions ........................415
Leigh S. McCue and Armin W. Troesch
Applications of Finite-Time Lyapunov Exponents to the Study
of Capsize in Beam Seas ......................................................................................433
Leigh S. McCue
Nonlinear Dynamics on Parametric Rolling of Ships in Head Seas....................449
Marcelo A.S. Neves, Jerver E.M. Vivanco and Claudio A. Rodríguez
6 Roll Damping
A Simple Prediction Formula of Roll Damping of Conventional
Cargo Ships on the Basis of Ikeda’s Method and Its Limitation .........................465
Yuki Kawahara, Kazuya Maekawa and Yoshiho Ikeda
A Study on the Characteristics of Roll Damping of Multi-Hull Vessels.............487
Toru Katayama, Masanori Kotaki and Yoshiho Ikeda
7 Probabilistic Assessment of Ship Capsize
Capsize Probability Analysis for a Small Container Vessel................................501
E.F.G. van Daalen, J.J. Blok and H. Boonstra
Efficient Probabilistic Assessment of Intact Stability..........................................515
N. Themelis and K.J. Spyrou
Probability of Capsizing in Beam Seas with Piecewise Linear
Stochastic GZ Curve ............................................................................................531
Vadim Belenky, Arthur M. Reed and Kenneth M. Weems
Probabilistic Analysis of Roll Parametric Resonance in Head Seas....................555
Vadim L. Belenky, Kenneth M. Weems, Woei-Min Lin
and J. Randolf Paulling
8. Table of Contentsviii
8 Environmental Modeling
Sea Spectra Revisited ...........................................................................................573
On Self-Repeating Effect in Reconstruction of Irregular Waves.........................589
New Approach To Wave Weather Scenarios Modeling......................................599
Alexander B. Degtyarev
9 Damaged Ship Stability
Effect of Decks on Survivability of Ro–Ro Vessels............................................621
Experimental and Numerical Studies on Roll Motion of a Damaged
Large Passenger Ship in Intermediate Stages of Flooding...................................633
Yoshiho Ikeda, Shigesuke Ishida, Toru Katayama
and Yuji Takeuchi
Exploring the Influence of Different Arrangements of Semi-Watertight
Spaces on Survivability of a Damaged Large Passenger Ship.............................643
Riaan van’t Veer, William Peters, Anna-Lea Rimpela
and Jan de Kat
Time-Based Survival Criteria for Passenger Ro-Ro Vessels...............................663
Andrzej Jasionowski, Dracos Vassalos and Luis Guarin
Pressure-Correction Method and Its Applications for Time-Domain
10 CFD Applications to Ship Stability
Applications of 3D Parallel SPH for Sloshing and Flooding...............................709
Liang Shen and Dracos Vassalos
Simulation of Wave Effect on Ship Hydrodynamics by RANSE........................723
Qiuxin Gao and Dracos Vassalos
A Combined Experimental and SPH Approach to Sloshing and Ship
Luis Pérez-Rojas, Elkin Botía-Vera, José Luis Cercos-Pita,
Antonio Souto-Iglesias, Gabriele Bulian and Louis Delorme
9. Table of Contents ix
11 Design for Safety
Design for Safety with Minimum Life-Cycle Cost..............................................753
Romanas Puisa, Dracos Vassalos, Luis Guarin and George Mermiris
12 Stability of Naval Vessels
Stability Criteria Evaluation and Performance Based Criteria
Development for Damaged Naval Vessels...........................................................773
Andrew J. Peters and David Wing
A Naval Perspective on Ship Stability .................................................................793
Arthur M. Reed
13 Accident Investigations
New Insights on the Sinking of MV Estonia........................................................827
Andrzej Jasionowski and Dracos Vassalos
Experimental Investigation on Capsizing and Sinking of a Cruising
Yacht in Wind ......................................................................................................841
Naoya Umeda, Masatoshi Hori, Kazunori Aoki, Toru Katayama
and Yoshiho Ikeda
During the last decade significant progress has been made in the field of ship stability:
the understanding of ship dynamics and capsize mechanisms has increased through
improved simulation tools and dedicated model tests, design regulations have been
adapted based on better insights into the physics of intact and damaged vessels,
and ships can be operated with better knowledge of the dynamic behaviour in a
seaway. Yet in spite of the progress made, numerous scientific and practical
challenges still exist with regard to the accurate prediction of extreme motion and
capsize dynamics for intact and damaged vessels, the probabilistic nature of
extreme events, criteria that properly reflect the physics and operational safety of
an intact or damaged vessel, and ways to provide relevant information on safe ship
handling to ship operators.
This book provides a comprehensive review of the above issues through the
selection of representative papers presented at the unique series of international
workshops and conferences on ship stability held between 2000 and 2009. The
first conference on the stability of ships and ocean vehicles was held in 1975 in
Glasgow (STAB ‘75) and the first international workshop on ship stability was
held in 1995 in Glasgow. The aim of the workshop was to address issues related to
ship capsize and identify ways to accelerate relevant developments. Since then the
workshops have been held each year in succession, whereby each third year a
STAB conference has been held as a cap of the preceding workshops and to
present the latest developments.
The editorial committee has selected papers for this book from the following
events: STAB 2000 Conference (Launceston, Tasmania), 5th
(Trieste, 2001), 6th
Stability Workshop (Long Island, 2002), STAB 2003 Conference
Stability Workshop (Shanghai, 2004), 8th
Stability Workshop (Istanbul,
2005), STAB 2006 Conference (Rio de Janeiro), 9th
Stability Workshop (Hamburg,
Stability Workshop (Daejeon, 2008), and STAB 2009 Conference (St.
Petersburg). The papers have been clustered around the following themes: Stability
Criteria, Stability of the Intact Ship, Parametric Rolling, Broaching, Nonlinear
Dynamics, Roll Damping, Probabilistic Assessment of Ship Capsize, Environmental
Modelling, Damaged Ship Stability, CFD Applications, Design for Safety, Naval
Vessels, and Accident Investigations.
We would like to express our sincere gratitude to all of the authors who have
presented papers at the conferences and workshops; to the organizers and all of the
participants of these events; to the members of the International Standing Committee
for the Stability of Ships and Ocean Vehicles; and to Professor Marcelo Neves for
his role as main editor of this book.
Dr. Jan Otto de Kat
Chairman of the International Standing Committee for the Stability of Ships
and Ocean Vehicles (on behalf of the editorial committee)
11. List of Corresponding Authors
1 Stability Criteria
Review of Available Methods for Application to Second
Level Vulnerability Criteria
A Basis for Developing a Rational Alternative
to the Weather Criterion: Problems and Capabilities
Evaluation of the Weather Criterion by Experiments
and its Effect to the Design of a RoPax Ferry
Evolution of Analysis and Standardization of Ship
Stability: Problems and Perspectives
SOLAS 2009 – Raising the Alarm
2 Stability of the Intact Ship
Effect of Initial Bias on the Roll Response and Stability
of Ships in Beam Seas
Historical Roots of the Theory of Hydrostatic Stability of Ships
12. List of Corresponding Authorsxiv
The Effect of Coupled Heave/Heave Velocity or Sway/Sway
Velocity Initial Conditions on Capsize Modeling
Leigh S. McCue
The Use of Energy Build Up to Identify the Most Critical
Heeling Axis Direction for Stability Calculations
for Floating Offshore Structures
Joost van Santen
Some Remarks on Theoretical Modelling of Intact Stability
3 Parametric Rolling
An Investigation of Head-Sea Parametric Rolling for Two
Marcelo de Almeida Santos Neves
Simple Analytical Criteria for Parametric Rolling
Experimental Study on Parametric Roll of a Post-Panamax
Containership in Short-Crested Irregular Waves
Model Experiment on Parametric Rolling of a Post-Panamax
Containership in Head Waves
Numerical Procedures and Practical Experience of Assessment
of Parametric Roll of Container Carriers
Parametric Roll and Ship Design
Riaan van’t Veer
13. List of Corresponding Authors xv
Parametric Roll Resonance of a Large Passenger Ship in Dead
Ship Condition in All Heading Angles
Parametric Rolling of Ships – Then and Now
J. Randolph Paulling
Parallels of Yaw and Roll Dynamics of Ships in Astern Seas
and the Effect of Nonlinear Surging
Model Experiment on Heel-Induced Hydrodynamic Forces in Waves
for Realising Quantitative Prediction of Broaching
Perceptions of Broaching-To: Discovering the Past
5 Nonlinear Dynamics and Ship Capsizing
Use of Lyapunov Exponents to Predict Chaotic Vessel Motions
Leigh S. McCue
Applications of Finite-Time Lyapunov Exponents to the Study
of Capsize in Beam Seas
Leigh S. McCue
Nonlinear Dynamics on Parametric Rolling of Ships in Head Seas
Marcelo A.S. Neves,
6 Roll Damping
A Simple Prediction Formula of Roll Damping of Conventional
Cargo Ships on the Basis of Ikeda’s Method and Its Limitation
14. List of Corresponding Authorsxvi
A Study on the Characteristics of Roll Damping of Multi-hull Vessels
7 Probabilistic Assessment of Ship Capsize
Capsize Probability Analysis for a Small Container Vessel
E.F.G. van Daalen
Efficient Probabilistic Assessment of Intact Stability
Probability of Capsizing in Beam Seas with Piecewise
Linear Stochastic GZ Curve
Probabilistic Analysis of Roll Parametric Resonance
in Head Seas
Vadim L. Belenky
8 Environmental Modeling
Sea Spectra Revisited
On Self-Repeating Effect in Reconstruction of Irregular
New Approach To Wave Weather Scenarios Modeling
Alexander B. Degtyarev
9 Damaged Ship Stability
Effect of Decks on Survivability of Ro–Ro Vessels
15. List of Corresponding Authors xvii
Experimental and Numerical Studies on Roll Motion
of a Damaged Large Passenger Ship in Intermediate
Stages of Flooding
Exploring the Influence of Different Arrangements
of Semi-Watertight Spaces on Survivability of a Damaged
Large Passenger Ship
Riaan van’t Veer
Time-Based Survival Criteria for Passenger Ro-Ro Vessels
Pressure-Correction Method and Its Applications
for Time-Domain Flooding Simulation
10 CFD Applications to Ship Stability
Applications of 3D Parallel SPH for Sloshing and Flooding
Simulation of Wave Effect on Ship Hydrodynamics
A Combined Experimental and SPH Approach to Sloshing
and Ship Roll Motions
11 Design for Safety
Design for Safety with Minimum Life-Cycle Cost
16. List of Corresponding Authorsxviii
12 Stability of Naval Vessels
Stability Criteria Evaluation and Performance Based
Criteria Development for Damaged Naval Vessels
Andrew J. Peters
A Naval Perspective on Ship Stability
Arthur M. Reed
13 Accident Investigations
New Insights on the Sinking of MV Estonia
Experimental Investigation on Capsizing and Sinking
of a Cruising Yacht in Wind
19. on Fishing Vessels Safety (SLF) with SLF-50/4/4 (2007), submitted by Japan, the
Netherlands, and the United States and SLF-50/4/12 (2007), submitted by Italy. A
significant contribution was also made in SLF-50.INF.2 (2007), submitted by
Germany. These discussions resulted in a plan of development of new generation
of criteria (see Annex 6 of SLF 50/WP.2). Following this plan, the intersessional
Correspondence Group on Intact Stability developed the framework in a form of
the working document that can be found in Annex 2 of SLF 51/4/1 (2008),
submitted by Germany. Annex 3 of this paper contains the draft terminology
agreed upon for the new criteria that was also developed by the Correspondence
Group. The most current version of the framework document can be found in
Annex 1 of the report of the Working Group on Intact Stability (SLF 51/WP.2).
To facilitate these discussions, Belenky, et al. (2008) presented a broad review of
issues related with development of new criteria, including the physical
background, methodology, and available tools. It is the intention of the authors of
this paper to provide a follow-up review, with a particular focus on methods for
application to second level vulnerability criteria.
The views and opinions expressed in this paper are solely and strictly those of
the authors, mainly for facilitating wider and deeper discussion inside the research
community. They do not necessarily reflect those of the delegations that involve
the authors, the intersessional correspondence group on intact stability, or the
working group on intact stability of the International Maritime Organization. The
contents of this paper also do not necessarily indicate a consensus of opinion
among the authors, but the authors agree on the need for further discussion of the
2 Overview of New Generation of Intact Stability Criteria
The major modes of stability failures were listed in section 1.2 of the 2008 IS
Code, part A. They include:
Phenomena related to righting lever variations, such as parametric roll and pure
loss of stability;
Resonant roll in dead ship condition defined by SOLAS regulation II-1/3
Broaching and manoeuvring-related problems in waves
A multi-tier structure of new criteria has been identified from the ongoing
discussion at IMO. Each of the four identified stability failure modes: pure-loss of
stability, parametric roll, surf-riding and broaching, and dead ship conditions is
evaluated using criteria with different levels of sophistication.
The intention of the vulnerability criteria is to identify ships that may be vulnerable
to dynamic stability failures, and are not sufficiently covered by existing regulation
(SLF 51/WP.2, Annex 1).
C.C. Bassler et al.4
20. Review of Available Methods for Application to Second Level Vulnerability Criteria 5
A first level of criteria (vulnerability criteria) could be followed by other
stability assessment methods with increasing levels of sophistication. The first
level of the stability assessment is expected to be relatively simple, preferably
geometry based if possible. The function of this first level is to distinguish
conventional ships, which are obviously not vulnerable to stability failure. The
second tier is expected to be more sophisticated and assess vulnerability to
dynamic stability with enough confidence that the application of direct calculation
could be justified. The expected complexity of the second level vulnerability
criteria should consist of a spreadsheet-type calculation.
An outline of the process is shown in Fig. 1. The methods will most likely be
different for each stability failure mode.
3 Pure Loss of Stability
Differences in the change of stability in waves, in comparison with calm water,
were known to naval architects since late 1800s (Pollard & Dudebout, 1892; Krylov
1958). However, it was uncommon until the 1960s that attempts to calculate the
change of stability in waves (Paulling, 1961) and evaluate it with a series of model
tests (Nechaev, 1978; available in English from Belenky & Sevastianov, 2007)
were made. As a distinct mode of stability failure, pure-loss of stability was
identified during the model experiments in San-Francisco Bay (Paulling, et al.,
1972, 1975; summary available from Kobylinski & Kastner, 2003).
Accurate calculation of the change of stability in waves presents certain
challenges, especially at high speed, as the pressure around the hull and waterplane
shape are influenced by the nonlinear interaction between encountered, reflected,
and radiated waves (Nechaev, 1989).
Criteria Level 1
Criteria Level 2
IMO IS Code
Performance-Based CriteriaFor each mode
Fig. 1. The proposed assessment process for next generation intact stability criteria
IMO IS Code
21. The physical mechanism of pure-loss of stability is rather simple: if the stability
is reduced for a sufficiently long time, a ship may capsize or attain large roll angle
(see Fig, 2, taken from Belenky, et al., 2008).
This phenomenon is presently addressed, although in a very short and qualitative
form, by the MSC.1/Circ.1228 (2007). MSC.1/Circ.1228 does not provide any
ship-dependent information concerning the considered failure mode.
The difficulty of developing criteria for pure-loss of stability is not limited to
the calculation of the GZ curve in waves. A probabilistic characterization of the
associated stability changes is also difficult, due to nonlinear nature of stability
changes in waves.
Boroday (1967, 1968) developed a method for the assessment of statistical
characteristics for restoring moments in waves, followed by an energy balance-
based method for evaluation of the probability of capsizing (Boroday & Netsevatev,
1969). Using energy balance methods for changing stability in waves was also the
focus of Kuo & Vassalos (1986). Dunwoody (1989a, 1989b), Palmquist (1994),
and Bulian & Francescutto (2006a) considered statistical characteristics of GM
and other elements of the righting moment as a stochastic process.
The alternative approach of the “effective wave” was proposed by Grim
(1961). In this method, the length of the wave is equal to ship length— the wave is
unmovable. However, its amplitude is a random process (Fig. 3).
Fig. 2. Capsizing due to pure-loss of stability
0 50 100 150
Time history of roll
Phase of encounter wavePhase of heeling moment
0 10 20 30 40 50 60
Righting / Heeling Arm, m
C.C. Bassler et al.6
A concept of an effective wave
22. Review of Available Methods for Application to Second Level Vulnerability Criteria 7
The amplitude of the effective wave is calculated in order to minimize the
difference between the effective and encountered waves. Umeda, et al. (1993)
performed comparative calculations and suggested the effective wave to be of
satisfactory accuracy for engineering calculations.
The effective wave approach was used in a number of works; of special interest
are those with a “regulatory perspective.” Helas (1982) discussed the reduction of
the righting arm in waves. To consider the duration where stability is reduced, the
amplitude of the effective wave was averaged over a time sufficient to roll to a
large angle, about 60% of natural roll period. As a result, the average effective
wave amplitude becomes a function of Froude number. Analyzing results of
the calculation performed for different vessels in different sea conditions,
recommendations were formulated for a stability check in waves. Small ships and
ships with “reduced stability” were meant to be checked using the GZ curve—
evaluated for the effective wave with averaged amplitude against conventional
calm water stability criteria.
Umeda and Yamakoshi, (1993) used the effective wave to evaluate the probability
of capsizing due to pure-loss of stability in short-crested irregular stern-quartering
waves with wind. The capsizing itself was associated with departure from the
time-dependent safe basin. The effect of initial conditions was taken into account
using statistical correlation between roll and the effective wave.
Instead of the Grim effective wave, Vermeer (1990) assumed waves to be a
narrow-banded stochastic process. Capsizing is associated with the appearance of
negative stability during the time duration, and it is sufficient for a ship to reach a
large roll angle under the action of a quasi-static wind load. The probability of
capsizing is considered as a ratio of encountered wave cycles, where the wave
amplitude is capable of a significant and prolonged deterioration of stability.
Themelis and Spyrou (2007) demonstrated the use of the concept of a “critical
wave” to evaluate probability of capsizing. Because pure-loss of stability is a one
wave event, it is straightforward to separate the dynamical problem from the
probabilistic problem. First, the parameters that are relevant to the waves capable
of causing pure-loss of stability are searched. Then, the probability of encountering
such waves is assessed. To complete the probabilistic problem, the initial conditions
are treated in probabilistic sense.
Bulian, et al. (2008) combined the effective wave approach with upcrossing
theory, where the phenomenon of pure-loss of stability is associated with an up-
crossing event for the amplitude of the effective wave. Such an approach allows
the time of exposure to be considered directly and produces a time-dependent
probability of stability failure.
An additional advantage of upcrossing theory is the possibility to characterize
the “time above the level”, which in this context is the duration of decreased
stability. The mean value of this time can be evaluated by a simple formula, but
evaluation of other characteristics is more complex (Kramer and Leadbetter, 1967).
In summary, it seems that the second level vulnerability criteria for pure-loss of
stability are likely to be probabilistic, due to the very stochastic nature of the
23. phenomenon. For criteria related to this mode of stability failure, the time duration
while stability is reduced on a wave could be included. Methods useful for
addressing pure-loss of stability may include, but are not limited to, the effective
wave, the narrow-banded waves assumption, the critical wave approach, and
upcrossing theory. As Monte-Carlo simulations may be too cumbersome for
vulnerability criteria, at this time, simple analytical and semi-analytical models
seem to be preferable.
4 Parametric Roll
Parametric roll is the gradual amplification of roll amplitude caused by parametric
resonance, due to periodic changes of stability in waves. These stability changes
are essentially results of the same pressure and underwater geometry changes that
were the cause in the pure-loss of stability mode. However, in contrast to pure-loss
of stability, parametric roll is generated by a series of waves of certain frequency.
Therefore, parametric roll cannot be considered as a one wave event (Figure 4).
Research on parametric roll began in Germany in the 1930s (Paulling, 2007). In
the 1950s, the study of parametric roll for a ship was continued by Paulling &
Rosenberg (1959). Paulling, et al. (1972, 1975) observed parametric roll in
following waves during model tests in San-Francisco Bay (summary available
from Kobylinski & Kastner, 2003). Later it was discovered that this phenomenon
could occur in head or near head seas as well (Burcher, 1990; France, et al., 2003).
Fig. 4. Partial (a) and total (b) stability failure caused by parametric resonance
0 100 200 300 400 500 t, s
0 100 200 300 400 500
C.C. Bassler et al.8
24. Review of Available Methods for Application to Second Level Vulnerability Criteria 9
Parametrically excited roll motion is at present widely recognized as a
dangerous phenomenon, where the cargo and the passengers and the crew may be
at risk. Hull forms have evolved from those considered in the development of the
2008 IS Code, as MSC Res. 267(85)— basically the same ship types used 30-50
years ago for the definition of Res. A. 749(18) (2002). For certain ship types,
present hull forms appear to have resulted in the occurrence of large amplitude
rolling motions associated with parametric excitation. According to this evidence,
there is a compelling need to incorporate appropriate stability requirements
specific to parametric roll into the new generation intact stability criteria.
The simplest model of parametric resonance is the Mathieu equation. It can
describe the onset of instability in regular waves. Because the Mathieu equation
has a linear stiffness term, it is incapable of predicting the amplitude of roll, and
not applicable in irregular waves. Nevertheless, it was used by ABS (ABS, 2004;
Shin, et al., 2004) to establish estimates for initial susceptibility to parametric
excitation, which corresponds more with intended use for the first level vulnerability
More complex 1(.5)-DOF models include nonlinearity in the time-dependent
variable stiffness term. This nonlinearity leads not only to stabilization of roll
motion, e.g. establishing finite roll amplitude, but also to a significant alteration of
instability boundary in comparison of Ince-Strutt diagram— with unstable
motions found outside of the Mathieu instability zone (Spyrou, 2005). This should
cause concern about relying on the standard linear stability boundaries.
Inclusion of additional degrees of freedom, primarily pitch and heave (Neves &
Rodríguez, 2007) results in additional realism. These are especially important for
parametric roll in head seas, where pitch and heave may be large. Spyrou (2000)
included surging as well. These models, even with their low dimensional character,
seem to give reasonable agreement with model tests (Bulian, 2006).
Evaluation of the steady-state amplitude of parametric roll can be carried out
using asymptotic methods, such as the method of multiple scales (Sanchez &
Nayfeh, 1990). Approximate analytical expressions exist for the determination of
the nonlinear roll response curve (e.g. ITTC, 2006). The potential of these methods is
discussed in Spyrou et al. (2008).
The introduction of irregular waves results in drastic changes in the way
parametric roll develops. Large oscillations may not be persistent anymore: it may
stay, but it may also be transitory (Figure 5).
Such behavior has significant influence on the ergodicity of the process— the
ability to obtain statistical characteristics from one realization if it is long-enough.
Theoretically, the process remains ergodic, as the integral over its autocorrelation
function still remains finite, but the time necessary for convergence may be very
large. As a result, the term “practical non-ergodicity” is used (Bulian, et al., 2006;
Hashimoto, et al., 2006).
25. Distribution of parametric roll may be different from Gaussian. Hashimoto, et al.
(2006) obtained distributions with large excess of kurtosis from model tests in
long- and short-crested irregular seas (Fig. 6).
Roll damping has more influence on roll amplitude in irregular waves. The
damping threshold defines if a wave group can contribute to parametric excitation
and therefore, to results in an increase of the roll amplitude. Characteristics of
wave groups, such as number, length, and the height of waves in the group, may
also have an influence on how fast the roll amplitude increases. In a sense, a
regular wave can be considered as an infinite wave group. Therefore, using regular
waves could be too conservative for an effective vulnerability criterion, due to the
transient nature of this phenomenon (SLF48/4/12).
Focusing on the instability boundaries in irregular waves, Francescutto, et al.,
(2002) used the concept of “effective damping,” also mentioned in Dunwoody
(1989a); that is meant to be subtracted from roll damping. Bulian, et al. (2004)
used the Fokker-Planck equation and the method of multiple scales for the same
purpose. They utilized Dunwoody’s spectral model of GM changes in waves
(1989a, 1989b). Furthermore, Bulian (2006) made use of Grim’s effective wave
Fig. 5. Sample records of simulated parametric roll in long-crested seas (Shin, et al., 2004)
Fig. 6. Distribution of roll angles during parametric resonance in (a) long-crested and (b)
short crested seas
0 200 400 600 800 1000 1200 1400
0 200 400 600 800 1000 1200 1400
C.C. Bassler et al.10
26. Review of Available Methods for Application to Second Level Vulnerability Criteria 11
concept. Despite overestimating the amplitude of roll, the boundary of instability
was well identified.
Application of the Markov processes for parametric roll in irregular waves was
considered by Roberts (1982). To facilitate Markovian properties, (dependence
only on the previous step), white noise must be the only random input of the
dynamical system. Therefore, the forming filter, usually a linear oscillator, must
be part of the dynamical system to ensure irregular waves have a realistic-like
spectrum. These complexities allow the distribution of time before the process
reaches a given boundary to be obtained— the first passage method. This method
is widely used in the field of general reliability. Application of this approach to
parametric roll was considered by Vidic-Perunovic and Jensen, (2009).
Time-domain Monte-Carlo simulations (Brunswig, et al., 2006) are increasingly
used as computers become more powerful and are integral part of any naval
architect’s office. At the same time, caution has to be exercised when using Monte-
Carlo simulations, due to statistical uncertainty and other methodological challenges
that may limit its use for second level vulnerability criteria. Nevertheless, having
in mind the difficulties associated with prediction of amplitude for parametric
resonance in irregular waves, Monte-Carlo method should not be dismissed.
Themelis & Spyrou (2007) used explicit modelling of wave groups as a way to
separate problems related to ship dynamics from the probabilistic problem. This
method exploited the growth rate of parametric roll due to the encounter of a
group, i.e. by solving the problem in the transient stage. Thereafter, the probability
of encounter of a wave group can be calculated separately. In principle, this may
allow enough simplification to make Monte-Carlo methods into feasible and
robust tool for the second level vulnerability criteria for parametric roll.
In summary, it seems that the second level vulnerability criteria for parametric
roll may be related to the size of the instability area in irregular seas. The group
wave approach seems to be promising, as a method to build a criterion based on
5 Surf-Riding and Broaching
Broaching is the loss of controllability of a ship in astern seas, which occurs
despite maximum steering effort. Stability failure caused by broaching may be
“partial” or “total”. It can appear as heeling during an uncontrollable, tight turn.
This is believed to be intimately connected with the so called surf-riding behaviour.
Moreover, it can manifest itself as the gradual, oscillatory build-up of yaw, and
sometimes roll, amplitude. The dynamics of broaching is probably the most
dynamically complex phenomenon of ship instability. Well known theoretical
studies on broaching from the earlier period include Davidson (1948), Rydill
(1959), Du Cane & Goodrich (1962), Wahab & Swaan (1964), Eda (1972);
Renilson & Driscoll (1981), Motora, et al.(1981); and more recently, e.g. by Umeda &
27. Renilson (1992), Ananiev & Loseva (1994), and Spyrou (1996, 1997, 1999).
Broaching has also been studied with experimental methods. Tests with scaled
radio-controlled physical models have taken place in large square ship model
basins; e.g. Nicholson (1974), Fuwa (1982), De Kat & Thomas (1998).
Surf-riding is a peculiar type of behaviour where the ship is suddenly captured
and then carried forward by a single wave. It often works as predecessor of broaching
and its fundamental dynamical aspects have been studied by Grim (1951),
Ananiev (1966), Makov (1969), Kan (1990), Umeda (1990), Thomas & Renilson
(1990), and Spyrou (1996). Key aspects of the phenomenon are outlined next. In a
following or quartering sea, the ship motion pattern may depart from the ordinary
periodic response. Figure 7 illustrates diagrammatically the changes in the geometry
of steady surge motion, under the gradual increase of the nominal Froude number
(Spyrou 1996). An environment of steep and long waves has been assumed.
For low Fn, there is only a harmonic periodic response (plane (a)) at the encounter
frequency. However, as the speed approaches the wave celerity, the response
becomes asymmetric (plane (c)). The ship stays longer in the crest region and
passes quickly from the trough. In parallel, an alternative stationary behavior
begins at some instance to coexist. This results from the fact that the resistance
force which opposes the forward motion of the ship may be balanced by the sum
of the thrust produced by the propeller and the wave force along the ship’s
longitudinal axis. This is surf-riding and the main feature is that the ship is forced
to advance with a constant speed equal to the wave celerity. It has been shown that
surf-riding states appear in pairs: one nearer to wave crest and the other nearer to
trough. The one nearer to the crest is unstable in the surge direction. Stable surf-
riding can be realised only in the vicinity of the trough and, in fact, only if
sufficient rudder control has been applied.
Fig. 7. Changing of surging/surf-riding behavior with increasing nominal Froude number
Periodic surging Surf-riding equilibria
a b c d
C.C. Bassler et al.12
28. Review of Available Methods for Application to Second Level Vulnerability Criteria 13
Surf-riding is characterised by two speed thresholds: the first is where the
balance of forces becomes possible (plane(b)). The second threshold indicates its
complete dominance in phase-space, with sudden disappearance of the periodic
motion. It has been pointed out that, this disappearance is result of a global
bifurcation phenomenon, known in the nonlinear dynamics literature as homoclinic
saddle connection (Spyrou, 1996). It occurs when a periodic orbit collides in state-
space with an unstable equilibrium— the surf-riding state near to the wave crest.
A dangerous transition towards some, possibly distant, undesirable state is the
To be able to show the steady motion of a ship overtaken by waves as a closed
orbit, one should opt to plot orbits on a cylindrical phase-plane. On the ordinary
flat phase-plane the bifurcation would appear as a connection of different saddles
(“heteroclinic connection”, see Figures 8 and 9 based on Makov). This is
characterised by the tautochronous touching of the “overtaking wave” orbits,
running from infinity to minus infinity, with the series of unstable equilibria at
crests. This has been confirmed numerically by Umeda (1999). Therefore, no
matter what terminology is used, there is consensus and clear understanding,
rooted in dynamics, about the nature of phenomena.
The initial attraction into surf-riding, caused by the unstable surf-riding
equilibrium near the crest, should be followed by capture of the ship at the stable
surf-riding equilibrium near the trough. However, this is a violent transition and, if
the rudder control is insufficient, it will end in broaching. The effectiveness of
rudder control in relation to the inception of this behaviour, and also characteristic
simulations, can be found in the literature (Spyrou, 1996, 1999).
Fig. 8. Phase plane with surging and surf-riding, Fncr1<Fn<Fncr2
300 200 100 0 100
29. From the above, one sees that, susceptibility to this type of broaching could be
assessed by targeting three sequential events: the condition of attraction to surf-
riding; the condition defining an inability to stay in stable surf-riding, thus
creating an escape; and the condition of reaching dangerous heeling during this
escape— manifested as tight turn. For the first condition, we already have
deterministic (analytical and numerical) criteria based on Melnikov and other
methods (Ananiev, 1966; Kan, 1990; Spyrou, 2006). For the second, criteria have
been discussed, linking ship manoeuvring indices with control gains (Spyrou,
1996, 1998). The third condition can be relatively easily expressed as the balance
of the roll restoring versus the moment of the centrifugal force that produces the
turn. A proper probabilistic expression of these conditions is still wanting.
However, the critical wave approach may provide a very practical way of dealing
with this (Themelis & Spyrou, 2007). Efforts in similar spirit were reported by
Umeda (1990) and Umeda, et al. (2007).
In reviews of broaching incidents, those that had happened at low to moderate
speeds (i.e. far below the wave celerity of long waves) are usually distinguished
from the more classical high-speed events. These incidents could have some
special importance, because they seem to be relevant also to vessels of larger size,
which otherwise unlikely to broach
the nature of such a broaching mechanism
g. It has been shown that, a
Further increase of the commanded heading was found to cause a rapid increase of
the amplitude of yaw oscillation, leading shortly to a turn backwards of the steady
yaw response curve and, inevitably, a sudden and dangerous jump to resonant yaw
(Spyrou, 1997). The transient behaviour looks like a growing oscillatory yaw
which corresponds closely to what has been described as cumulative-type broaching.
Fig. 9. Phase plane with surf-riding only Fn>Fncr2
are through surf-riding. Let’s consider now
, which we will call “direct broaching”
since the ship is not required to go through surf-ridin
fold bifurcation is possible to arise for a critical value of the commanded heading, This
phenomenon can have a serious influence on the horizontal plane dynamics, creating
a stable sub-harmonic yaw (Fig. 10).
300 200 100 0 100
C.C. Bassler et al.14
30. Review of Available Methods for Application to Second Level Vulnerability Criteria 15
Instability could be an intrinsic feature of the yaw motion of a ship overtaken
by very large waves. It has been pointed out that, starting from Nomoto’s equation,
the equation of yaw in following waves becomes Mathieu-type (in Spyrou (1997)
the derivation and discussion on this and its connection with the above scenario
was presented). The amplitude of the parametric forcing term is the ratio of the
wave yaw excitation to the hull’s own static gain— maneuvering index K. The
damping ratio in this equation receives large values, unless there is no differential
control. Due to the large damping, the internal forcing that is required in order to
produce instability is much higher (steeper waves) than that at the near zero
encounter frequency (i.e. for the mechanism of broaching through surf-riding).
Simple criteria connecting the hull with rudder control have accrued from this
formulation and could be directly in the vulnerability criteria. A probabilistic
formulation has not yet been attempted.
6 Dead Ship Condition
Both the term and the concept of dead ship conditions took its root during the
steamer era. When a ship looses power, it becomes completely uncontrollable— a
significant disadvantage in comparison with ship equipped with sails. The worst
possible scenario is a turn into beam seas, where the ship is then subjected to a
resonant roll combined with gusty wind. Traditional ship architecture, with a
superstructure amidships, made this scenario quite possible. Significant changes in
marine technology have led to more reliable engines and a variety of architectural
Fig. 10. Direct broaching
C commanded heading
31. types. Nevertheless the dead ship condition— zero speed, beam seas, resonant
roll, and wind action, maintains its importance.
The present Intact Stability Code (MSC.267(85), 2008), as well as its predecessor
(IMO Res. A749(18), 2002), contains the “Severe Wind and Rolling Criterion
(Weather Criterion).” This criterion contains a simple mathematical model for
ship motion under the action of beam wind and waves. However, some parameters
of this model are based on empirical data. Therefore, it may not be completely
adequate for unconventional vessels. Thus MSC.1/Circ.1200 (2006) and MSC.1/Circ.
1227 (2007) were implemented, to enable alternative methodologies to be used for
the assessment of the Weather Criterion on experimental basis.
The problem of capsizing represents the ultimate nonlinearity: the dynamical
system transits to motion about another stable equilibrium. When the wave excitation
amplitude approaches a critical value, a number of nonlinear phenomena take
place. The boundary of the safe basin becomes fractal (Kan & Taguchi, 1991) as a
result of the self-crossing branches of an invariant manifold (Falzarano, et al.,
1992). Considering the deterioration of the safe basin (Figure 11), Rainey and
Thompson (1991) proposed a transient capsize diagram. It uses the dramatic
change of the measure of integrity of the safe basin as a boundary value.
Another obvious way to improve criterion is to consider more realistic, e.g.
stochastic, models for the environment. This does not necessarily means that the
new criterion must be probabilistic. The improvement will be made by adjusting
parameters of deterministic criterion to reach similar level of safety for all the
vessels which meet it.
Studies of probabilistic methods for these criteria were carried out starting in
the 1950s (Kato, et al., 1957). As the conventional weather criterion uses an
Fig. 11. ith increasing of wave amplitude
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Normalized amplitude of excitation
C.C. Bassler et al.16
Deterioration of safe basin w
32. Review of Available Methods for Application to Second Level Vulnerability Criteria 17
energy balance approach, an attempt to consider it from probabilistic point of view
seems logical (Dudziak & Buczkowski 1978; a brief description is available from
Belenky & Sevastianov, 2007). Similar ideas are behind the Blume criterion
(Blume, 1979, 1987), which uses a measure of the remaining area under the GZ
curve. Bulian & Francescutto (2006b) and Bulian, et al. (2008) developed a method
where the righting arm is locally linearized at the equilibrium, due to the action of
a steady beam wind. The probability of large roll is calculated using an “equivalent
area” to account for the nonlinearities of the righting arm.
The piecewise linear method (Belenky, 1993; Paroka & Umeda, 2006) separates
probability of capsizing into two less challenging problems: upcrossings through
maximum of the GZ curve and capsizing if such upcrossing has occurred (Figure 12).
The probability of upcrossing is well defined in upcrossing theory. While the
probability of capsizing after upcrossing can be found using the probability of roll
rate at upcrossing that lead to capsizing; this is a linear problem in the simplest
case. Statistical linearization can be used to evaluate parameters of the piecewise
linear model. Simplicity and a direct relation with time of exposure are among the
strengths of the piecewise linear method.
In principle, a critical wave approach can be used for dead ship conditions as
well. However, the outcome of capsizing vs. non-capsizing strongly depends on
initial conditions. Therefore, a critical wave group approach similar to Themelis
and Spyrou (2007) may be also considered as a candidate for vulnerability criteria.
7 The Choice of Environmental Conditions
If vulnerability criterion is deterministic and based on a regular wave assumption,
the wave is characterized with height and length. These characteristics must then
be related to a corresponding sea state and further on with a safety level.
This formulation is not new— this approach is used for evaluation of extreme
loads. A typical scheme for calculation of extreme loads is based on long-term
assumption, so a number of sea states are considered. An operational profile is
Fig. 12. Separation principle in a piecewise linear method
Upcrossing leading to capsizing
Upcrossing not leading to capsizing
33. usually assumed based on existing experience; it includes the fraction of time that
a ship is expected to spend in each sea state. Short-term probability of exceedance
is calculated for each sea state; then the formula for the total probability is used to
determine the life-time probability of exceedance of the given level. This level is
typically associated with significant wave height and zero-crossing, or mean
period. These data are then directly used to define a regular wave that is expected
to cause extreme loads.
Similar ideas underlie the reference regular wave in the Weather Criterion in
the 2008 IS Code, which was determined from the significant wave steepness,
based on the work of Sverdrup and Munk (1947).
With all of the similarities, a direct transfer of methods used for extreme loads
to stability applications may be difficult. Structural failure is associated with
exceeding certain stress levels, actual physical failure is not considered in Naval
Architecture. Stresses do not have their own inertia, they follow the load without
time lag. The same can be observed about the loads, they occur simultaneously
with a wave. They do not depend on initial conditions; even in a nonlinear
formulation and do not have multiple responses for the given position of a ship on
a wave. Nevertheless, these motion nonlinearities affect only short-term
probability, while the scheme as a whole may be applicable.
Determining the equivalence between regular and irregular waves represents a
problem by itself. The use of significant wave height or steepness has been a
conventional assumption, but it does not have a theoretical background behind it.
The general problem of equivalency between regular and irregular roll motions
was considered by Gerasimov (1979; a brief description in English is available
from Belenky & Sevastianov, 2007) in the context of statistical linearization with
energy conservation, resulting in an energy statistical linearization.
The idea of a regular wave as some sort of equivalent for sea states is attractive
because of its simplicity. However, the physics of some stability failures may be
sufficiently different in regular and irregular waves. As it was mentioned in the
section regarding parametric roll, a regular wave is an infinitely long wave train.
Roll damping in this case has a very small influence on the response amplitude,
while in irregular seas damping has a significant influence on the variance of
response. Therefore, the regular equivalent of irregular waves cannot be applied
universally. The application of different environmental models is possible for
different stability modes. Surf-riding/broaching is an example of a one-wave event
where the concept of a critical wave may fit well.
If some of vulnerability criteria are made probabilistic, the next important choice
is time scale: long-term or short term. Here, short-term refers to a time interval
where an assumption of quasi-stationarity can be applied. Usually this is an interval
of three to six hours, where changes of weather can be neglected. Long-term
consideration covers a larger interval: a season, a year, or the life-time of a vessel.
The short-term description of the environment is simpler and can be characterized
with just with one sea state or spectrum. However, if chosen for vulnerability
criteria, justification will be required as to why a particular sea must be used. This
C.C. Bassler et al.18
34. Review of Available Methods for Application to Second Level Vulnerability Criteria 19
choice is important because sea states which are too severe may make the criterion
too conservative and diminish its value. Therefore, special research is needed in
order to choose the sea state “equivalent” or “representative” for a ship operational
profile. This may naturally result in ship-specific sea-state to use for assessment.
An alternative to the selection of a limited set of environmental conditions may
be the use of long-term statistics considering all the combinations of weather
parameters available from scatter diagrams, or appropriate analytical parametric
models (see, e.g. Johannessen, et al., 2002).
8 Summary and Concluding Comments
Vulnerability to pure-loss of stability is caused by prolonged exposure to decreased
stability on a wave crest, and is likely to be characterized by probabilistic criteria.
Methods for a first attempt for vulnerability criteria include the effective wave, a
narrow-band or envelope presentation, the critical wave approach, and upcrossing
Vulnerability criteria for parametric roll would be more useful if defined in
irregular seas. This criterion may be based on size of instability area, while the
wave group approach seems to be a good candidate for amplitude-based criterion.
Vulnerability to broaching through surf-riding may be judged by the vulnerability
to surf-riding. The critical wave approach seems to be the best fit; however, other
methods are not excluded. Vulnerability criterion for direct broaching is likely to
Vulnerability to stability failure in dead ship conditions may be judged using a
variety of methods, as it was the main area of application for both probabilistic
and deterministic methods. The modified weather criterion, piecewise linear method,
and critical wave or wave group approach may be among the first to be considered.
A rational choice of environmental conditions for vulnerability criteria is at
least as important as the criteria. An unrealistic environmental condition may lead
to incorrect results, even if the criteria are technically correct. However, several
possibilities do exist: a regular wave equivalent for life-time risk, a short-term sea
state deemed “representative” for a specific ship operational profile, and a long-
term approach using a scatter diagram for a representative part of the World Ocean.
The authors are grateful for the funding support from Mr. James Webster (NAVSEA)
for this work. The authors are grateful for the guidance and encouragement they
received from Mr. William Peters (USCG). The authors also appreciate Dr. Arthur
Reed and Mr. Terrence Applebee (NSWCCD) for their help and support.
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geometrical approach addressing global behavior and going beyond the ordinary
simulation (see for example Falzarano et al 1992). The principal aim is to
characterise system robustness to critical environmental excitations. Whilst these
ideas have attracted the interest of researchers in the field of ship stability in most
cases they have relied on simple generic forms of roll restoring. Hence the question
whether such an approach could be used for practical stability assessment remains
2 Basics of the Weather Criterion
The IMO version of the weather criterion follows closely Yamagata’s “Standard
of stability adopted in Japan” which was enacted as far back as 1957 (Yamagata
1959) while the basic idea had appeared in Watanabe (1938) or perhaps even
earlier. The ship is assumed to obtain a stationary angle of heel θ0 due to side wind
loading represented by a lever lw1 which is not dependent on the heel angle and is
the result of a 26 m/s wind. “Around” this angle the ship is assumed to perform
due to side wave action resonant rolling motion as a result of which it reaches
momentarily on the weather side a maximum angle θ1 (Fig. 1). As at this position
the ship is most vulnerable to excitations from the weather-side it is further
assumed that it is acted upon by a gust wind represented by a lever lw2 = 1.5 lw1.
This is translated into a 2247.15.1 increase of the experienced wind velocity
assumed to affect the ship for a short period of time but at least equal with half
natural period under the assumption of resonance. The choice of wind corresponds
to extreme storm situation. In fact the criterion corresponds to a kind of average
between the centre of a typhoon where the wind is very strong but the rolling is
assumed that it is not so violent due to the highly irregular nature of the sea; and
the ensuing situation which prevails right after the centre of the typhoon has
moved away where the wind velocity is lower but the rolling becomes more
intense as the waves obtain a more regular form. As pointed out by Rachmanin in
his discussion of Vassalos (1986) the criterion is based on a roll amplitude with
2% probability of exceedance which perhaps corresponds to Yamagata’s choice of
roll amplitude θ1 as 70% of the resonant roll amplitude in regular waves.
The requirement for stability is formulated as follows: should the ship roll
freely from the off-equilibrium position with zero angular velocity the limiting
angle θ2 to the lee-side should not be exceeded during the ensuing half-cycle. This
limiting angle is either the one where significant openings are down-flooded the
vanishing angle θν or the angle of 50
work done by the wind excitation as the ship rolls from the wind-side to the lee-
side should not exceed the potential energy at the limiting angle θ2.
degrees which can be assumed as an explicit
safety limit whichever of the three is the lowest. Expressed as an energy balance: the
41. A Basis for Developing a Rational Alternative to the Weather Criterion 27
Fig. 1 The IMO weather criterion
Fig. 2 The structure of the weather criterion
A number of extra comments pertain to the modeling of system dynamics:
although the input of energy from the waves is taken into account for the
calculation of the attained angle θ1
and the ship is like being released in still-water from the angle θ1 (Fig. 2). The
only energy balance performed concerns potential energies in the initial and final
positions. An advantage of the criterion is that the damping is somehow present in
the calculation of the resonant roll angle θ1 both in terms of hull dimensions, form
, it is not considered during the final half-cycle
and fittings. Also, the fact that the energy dissipated through damping during the
half-roll is not accounted is not so important because it does not reduce seriously
the maximum attained angle and in any case the result is conservative. However,
the damping function is taken as a pure quadratic which is today an unnecessary
simplification. Finally, as in most critical cases the nonlinear part of the restoring
curve “participates” in the dynamics, it appears unreasonable this effect to be left
unaccounted in the calculation of θ1 (the only nonlinearity considered concerns
the damping function).
Fig. 3 The naval version of the weather criterion
In 1962 Sarchin & Goldberg presented the “naval version” of the weather criterion
which whilst more stringent than the above it adheres to the same principle with
only few minor differences. This work is the basis for the stability standards
applied from most western Navies nowadays as evidenced by documents such as
N.E.S 109 (2000) of the British Navy, DTS 079-1 of the U.S. Navy, NAV-04-A013
of the Italian etc. For ocean-going naval vessels the wind speed is assumed to be
90 knots and it is varying with the square of the heel angle (seemingly an
improvement over the straight-line shown in . 1; yet perhaps an equally crude
approximation of reality, see the discussion of Prohaska in Yamagata 1959). The
amplitude of resonant roll due to beam waves is prescribed to 25 degrees which
means that the important connection with the roll damping characteristics of the
ship that is found in IMO’s and Yamagata’s criterion is lost. Here however is
required that the equilibrium angle (point c in Fig.3) does not exceed 20 degrees
and also that the GZ at that point is less than 60% of the maximum GZ. The energy
requirement of the criterion prescribes that a substantial margin of potential
energy should be available at the limiting angle position in excess of the overturning
energy. This is expressed through the well-known relationship 12 4.1 AA .
43. A Basis for Developing a Rational Alternative to the Weather Criterion 29
3 The Concept of “Engineering Integrity”
Several articles have been published about the concept of “loss of engineering
integrity” of dynamical systems. The reader who is not acquainted with the basic
theory and terminology is referred for example to Thompson (1997). The basic
idea is that the safety robustness of an engineering system can be represented by
the integrity of its “safe basin” comprised by the set of initial conditions (in our
case pairs of roll angle and velocity) which lead to bounded motion. It is known
that the basin could suffer some serious reduction of area (“basin erosion”) once
some critical level of excitation is exceeded given the system’s inertial damping
and restoring characteristics. The reason for this is the complex intersection of
“manifolds” i.e. those special surfaces which originate from, or end on, the unstable
periodic orbits corresponding to the vanishing angles which become time-
dependent when there is wave-forcing. Detailed analysis about these phenomena
can be found in various texts see for example Guckenheimer & Holmes (1997).
The initiation of basin erosion can be used as a rational criterion of system integrity
which in our case and for a naval architectural context would be translated as
sufficient dynamic stability in a global sense. The methods that exist for predicting
the beginning of basin erosion are the following:
a) Use of the so-called Melnikov analysis which provides a measure of the
“closeness” of the critical pair of manifolds and hence can produce the condition
under which this distance becomes zero. For capsize prediction the method is
workable when the damping is relatively low (strictly speaking it is accurate for
infinitesimal damping - however several studies have shown that the prediction is
satisfactory for the usual range of ship roll damping values). Ideally the method is
applied analytically; or it may be combined with a numerical part when the
required integrations cannot be performed analytically.
b) Direct numerical identification of the critical combination of excitation
damping and restoring where the manifolds begin to intersect each other. This
method is more accurate but requires the use of specialised software.
c) Indirect identification of the critical condition by using repetitive safe basin
plots until the initiation of basin erosion is shown. The same comment as in (b)
In the next Section is provided an outline of Melnikov’s method and application
for a family of restoring curves parameterised with respect to bias intensity (e.g.
produced by the wind excitation). The critical for capsize wave slope as a function
of the wind-induced heel and the damping is determined. In the remaining Sections is
targeted the concept of engineering integrity itself. More specifically the following
two topics are addressed:
The value of the concept for higher order restoring curves: the practically
interesting point here is to find whether it is workable for arbitrary types of
restoring. Towards this the family of 5th
order restoring functions is analysed. In
addition, the concept is applied for the assessment of an existing ship thus giving an
idea of how the method would work for “real-world” stability assessment.
Possible use of the concept in a framework of design optimisation: a simple
parameterised family of ship-like hull forms has been considered with main
objective to determine whether the method can discriminate meaningfully
between good and poor designs.
4 Outline of Melnikov’s Method
Assume the dynamical system
τxεxfτx ,g (1)
x is in our case the vector
ddxx /, of roll displacement (normalised with
respect to the vanishing angle) and roll velocity
Fig. 4 The safe basin and the distance of manifolds
xf represents the “unperturbed” Hamiltonian part of the dynamical system;
and ,xg is the “perturbation” which includes the damping and the explicitly
time-dependent wave-forcing. is a parameter that represents the smallness of
,xg . It can be shown that the closeness of manifolds is expressed as (Fig. 4):
d t( )
f x( )
45. A Basis for Developing a Rational Alternative to the Weather Criterion 31
Where 0τ is a phase angle in the range /2τ0 0 . Since the denominator
is of the order of 1 the function 0τM (which is called Melnikov function) is to
first order a good measure of the distance of manifolds:
τττ,τxgτxfτ 00 dM
The wedge symbol means to take the cross product of the vectors f and
The main intention when this method is applied is to identify the critical
combinations of design/operational parameters where the Melnikov function
admits real zeros which means that the manifolds begin to touch each other (it is
essential the crossing of manifolds to be transversal but we do not discuss this here
5 Melnikov’s Method for Ship Rolling With a Wind-Induced
We have applied the method for the following family of restoring curves which
have the bias as free parameter:
11(1 axxaxaxxxxR (4)
The parameter a indicates the strength of the bias assumed here to be due to
beam wind loading. Letting 1xx and 2xx the roll motion could be described
by the following pair:
If the bias is relatively strong the unperturbed part is represented by the vector:
While the perturbation is:
After substitution into (3) the Melnikov function takes the form of the following
2020 ττβττsinΩ (9)
where 0τEM is the part due to wave forcing that is explicitly time-dependent;
and DM is the part due to the damping.
Fig. 5 Critical F as function of the bias. To be noted the initial “sharp” fall of the critical F as the
bias a departs from 1 which corresponds to a symmetric system
For the calculation of (9) we need to have available an explicit expression in
terms of time of the roll velocity 2x corresponding to the unperturbed system. This
is determined as follows: the homoclinic orbit goes through the corresponding
saddle point at x = 1. With the coordinate change 1 xs the equation of the
unperturbed system becomes:
47. A Basis for Developing a Rational Alternative to the Weather Criterion 33
With some manipulation the above can be written as:
The parameters that appear in (11) are defined as follows:
The solution of (11) is:
Differentiation of (13) yields:
With substitution of (14) into (9) we can calculate the integral (9) with the
method of residues. After some algebra, one obtains that the Melnikov function
becomes zero when:
The sustainable wave slope is:
Ak crit (16)
where νφ is the vanishing angle and
The critical wave slope obtained with application of the above method is
shown in Fig. 5 as a function of the frequency ratio for different values of the bias.
Notably Melnikov’s analysis produces an identical result with energy balance
where the energy influx due to the forcing is equated with the energy dissipated
through damping around the remotest orbit of bounded roll of the corresponding
“unperturbed” system. This formulation based on energy balance is intuitively
quite appealing. It will be discussed in more detail in Appendix I. Melnikov’s
method has been applied also for irregular wave excitation (e.g. Hsieh et al. 1994).
In Appendix II we outline the main issues involved in the formulation of Melnikov’s
method for a stochastic wave environment.
6 Concept Evaluation for Realistic
Here our objective is twofold: firstly to clarify whether the engineering integrity
concept is meaningful when applied to higher-order and thus more practical GZ
curves. Secondly to apply the new method for an existing ship which satisfies the
weather criterion (this work was part of a diploma thesis at NTUA
a) Higher-order GZ
In the first instance we considered the family of fifth-order restoring polynomials
1 xccxxxR (17)
Characteristic GZ shapes produced by (17) as c is varied are shown in Fig. 6. A
feature of this family is that it covers hardening ( 0c ) as well as softening
( 0c ) type levers. It must be noted that the curves shown in Fig. 5 are scaled
which means that there is no need the increase of c to lead to a higher value of the
real maxGZ . The main purpose of the study was to confirm that there is indeed
49. A Basis for Developing a Rational Alternative to the Weather Criterion 35
sudden reduction of basin area beyond some critical wave forcing for all the members
of this family i.e. that the concept is “robust” with respect to the characteristics of
the restoring function.
Fig. 6 Scaled “quintic” restoring curves
In Fig. 7 are shown some typical results of this investigation for an initially
hardening GZ at four different levels of damping. The points on the curves are
identified with measurement of the basin area and scaling against the area of the
corresponding unforced system. The curves confirm that for the fifth-order levers
the concept of engineering integrity is applicable without any problem i.e. there is
indeed a quick fall of the curve beyond some critical forcing. This critical forcing
should be approximately equal to the forcing determined from the Melnikov
method. An advantage however of the direct basin plot is that it allows to determine
fractional integrities also like 90% 80% etc. as function of damping and excitation
(90% integrity means that the remaining basin area is 90% of the initial). This is
quite important for criteria development since it provides the necessary flexibility
for setting the boundary line; for example one could base the maximum sustainable
wave slope at 90% integrity rather than at 100% which might be too stringent.
Fig. 7 Integrity diagrams
b) Application for an existing ship
A ferry was selected which had operated in Greek waters with 153BPL m
B = 22.8m T = 6.4m bC = 0.548 and 004.10KG m. The GZ curve was known
while the damping as a function of frequency was determined using the well-
known method of Himeno (1981). For the wave-making part we used the panel
code Newdrift (1989) which is available at the Ship Design Laboratory of NTUA. In Fig. 8
is shown the sustainable wave slope for a range of frequencies around resonance
on the basis of repetitive basin plotting. From this diagram it is possible to
determine the range where survivability in a beam sea condition is ensured. This
graph could be contrasted against Fig. 11 of Yamagata (1959) which gives the
0 0.05 0.1 0.15 0.2
Forcing amplitude F
0 0.05 0.1 0.15 0.2 0.25
Forcing amplitude F
51. A Basis for Developing a Rational Alternative to the Weather Criterion 37
wave steepness considered in the weather criterion as function of natural period
under the assumption of resonance condition.
Fig. 8 Critical wave environment for an existing ship on the basis of the new concept
7 Use of the Concept of Engineering Integrity for Design
The final study reported here was based on another NTUA diploma Thesis
(Sakkas 2001). It includes application of the method for a family of simplified
hull forms whose offsets are given by the following simple equation:
zxZxXzxfy ,, (18)
The same family was used in a paper by Min & Kang (1998) of the Hyundai
Heavy Industries for optimising the resistance of high-speed craft. The shape of
waterlines is described by the function xX which depends only on the
x where x is the longitudinal position measured
from the middle of the ship and l is the entrance length with 11 x . The
function xX is expressed as the 4th
Fig. 9 Characteristic hull-shape of the considered family
The transverse hull sections depend on both longitudinal position and height.
They are expressed by the following simple function
zzxZ 1, (20)
with s, t free parameters.
After a few trials we found that for somewhat realistic hull shapes the
parameters 2a 3 and 4a should take values between –1 and 1 while the
parameters s and t should lie between 0 and 0.4. As s and t increase the hull
becomes more slender. A characteristic hull-shape can be seen in Fig. 9.
53. A Basis for Developing a Rational Alternative to the Weather Criterion 39
Fig. 10 Roll damping for each examined hull
Although originally the height z is non-dimensionalised with respect to the
draught, in our case this is done with respect to the depth because the hull-shape
above the design waterline is very important in stability calculations ( 01 z ).
To narrow further on the free parameters of this investigation we have fixed the
length, the beam and the depth; respectively to 150m, 27.2m and 13.5m. The
displacement was also kept constant at 23710 t which means that only the block
coefficient and the draught were variable. In addition we have considered only
hulls fore-aft symmetric. KG was linked to the depth while the windage area was
set to be 4.1 L x T.
Fig. 11 Dynamic stability performance: Left bars are for waves only; while the right bars are for
wind and waves.
1 3 5 7 9 11
S h i p N o
13 15 17 19
1 3 5 7 9 11
S h i p N o
13 15 17 19
The free parameters were then varied in a systematic way so that
1432 aaa and ts . In total 20 ships were collected for further stability
investigation. For each one of these ships the corresponding GZ curve was
determined by using the commercial programme Autoship while the damping was
calculated as described in the previous Section. The roll radius of gyration was
calculated with a well-established empirical formula. The nondimensional
damping value β at roll resonance for each ship is shown in Fig. 10.
For all these simplified ships we applied the weather criterion and we verified
that it is fulfilled. Thereafter knowing the restoring damping and inertial
characteristics for each hull we run the programme Dynamics of Nusse & Yorke
(1998) in order to determine the critical wave slope Ak at which basin erosion is
initiated. The wind loading was calculated as prescribed in the weather criterion.
The performance of each ship as determined from this procedure is shown in
the bar chart of Fig. 11. In the same figure is shown the stability performance of
the ships when they were subjected to beam wind loading of the same type and
intensity as the considered in the IMO weather criterion. The best hull determined
is shown in Fig. 12.
Fig. 12 Best hull and the corresponding restoring curve
8 Concluding Remarks
An investigation on the concept of engineering integrity and whether it can be
used for developing a practical ship stability criterion has been presented. In the
spirit of the weather criterion we have confined ourselves to a study of system
robustness to combined wind and wave excitations coming from abeam and under
the assumption that there are no phenomena such as wave breaking on the ship
Heel angle (degrees)
55. A Basis for Developing a Rational Alternative to the Weather Criterion 41
side or accumulation of water on the deck. Also, we have left out of the present
context any discussion about stability in following seas, a matter which deserves a
separate approach due to the different nature of the dynamics involved. Although
the following-sea is the most dangerous environment of ship operation, it was not
seriously addressed up to now in the IMO Regulations. As a matter of fact, ships
are designed with no requirement for checking their stability at the most critical
condition that they may encounter although the primary modes of ship capsize in a
following sea environment are now well understood.
Of course such a matter is not raised for the first time and in the academic
world several approaches have been discussed in the past: some are based on the
area under the time-varying GZ curve (butterfly diagram see e.g. Vassalos 1986)
while more recently there is a trend for taking into account “more fully” system
dynamics. For parametric instability already exist general criteria that have been
developed in the field of mechanics. However it is doubted whether these are
known to ship designers (for an attempt to summarize these criteria see for example
Spyrou et al. 2000). Furthermore the clarification of the dynamics of broaching-to has
opened up new possibilities for the development of a simple criterion for avoiding
the occurrence of this phenomenon too.
It is emphasized that the proposed methodology is a platform for developing
also formulations targeting the following-sea environment. Furthermore since this
approach is a global method and is not based on ordinary simulation it presents
also distinctive advantages over the so-called “performance-based methods” a
term which in an IMO-style vocabulary is assumed to mean the execution of a
limited number of experimental or simulation runs.
Our analysis up to this stage shows that although the proposed method can
produce meaningful results there are a few practical problems whose solution is
uncertain and would require further research. A deeper look into the Melnikov-
based approach is presented in a companion paper (Spyrou et al 2002). Some
questions that immediately come to mind are:
• How to deal with down-flooding angles or more generally limits of stability
that are lower than the vanishing angle? The problem arises from the fact that the
method targets the phenomena affecting the basin boundary while the amplitude
of rolling may be well below that level if there is flooding of non-watertight
compartments. The formulation of Melnikov’s method as an energy balance might
give the idea for a practical solution.
• How to combine the deterministic and the stochastic analysis? This happened
quite meaningfully in the weather criterion and some comparable yet not obvious
in terms of formulation approach is required.
• How to adjust the level of stringency of the criterion? This could be achieved
perhaps by setting the acceptable level of integrity to a fractional value such as
90% rather than the 100% that is currently considered in research studies.
On the other hand, the method seems that it can easily fulfil the prerequisites of
a successful stability criterion which in our own opinion are the following:
• It should have an unambiguous direction of stability improvement i.e. should
be formulated in such a way that the designer can maximise the stability margin of
a ship under consideration (not in isolation of course but in parallel with other
performance and safety matters) and not simply check for conformance to limiting
• The criterion should be representative of stability in a global sense i.e. should
not be based on a prescribed and narrowly defined condition where stability
should be ensured.
• It should always make clear to the designer the connection with the limiting
• The method should take into account the transient nature of the ship capsize
process and it should account with sufficient accuracy for system dynamics.
• It should be flexible in order to accommodate changes in design trends i.e.
should not be overly dependent on existing ship characteristics which means that
the degree “empiricism” intrinsic to the method should be minimal.
Falzarano J Troesch AW and Shaw SW (1992) Application of global methods for analysing
dynamical systems to ship rolling and capsizing. Int J of Bifurc and Chaos 2:101-116.
Frey M and Simiu E (1993) Noise-induced chaos and phase-space flux Physica D 321-340.
Guckenheimer J and Holmes P (1997) Nonlinear oscillations dynamical systems and bifurc of
vector fields. Springer-Verlag N.Y.
Himeno Y (1981) Prediction of roll damping – state of the art. Report No. 239. Dep of Naval
Archit and Marine Eng The Univ of Michigan Ann Arbor MI.
Hsieh SR Troesch AW and Shaw SW (1994) A nonlinear probabilistic method for predicting
vessel capsizing in random beam seas. Proc of the Royal Soc A446 1-17.
IMO (1993) Code on Intact Stability for All Types of Ships Covered by IMO Instruments.
Resolut A.749(18) London.
Min KS and Kang SH (1998) Systematic study on the hull form design and resistance prediction
of displacement-type-super-high-speed ships. J of Marine Sci and Technol 3: 63-75.
NES 109 (2000) Stability Standard for Surface Ships. U.K. Ministry of Defence Issue 1
Publication date 1st
Nusse HE and Yorke JA (1998) Dynamics: Numerical explorations Spinger-Verlag N.Y.
Papagiannopoulos S (2001) Investigation of ship capsize in beam seas based on nonlinear
dynamics theory. Diploma Thesis, Dep of Naval Archit and Marine Eng, Nat Technical Univ of
Newdrift Vers. 6 (1989) User’s Manual. Ship Design Laboratory, Nat Technical Univ of Athens.
Sakkas D (2001) Design optimisation for maximising dynamic stability in beam seas. Diploma
Thesis Dep of Naval Archit and Marine Eng, Nat Technical Univ of Athens Sept.
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Spyrou KJ (2000) Designing against parametric instability in following seas. Ocean Eng, 27
Spyrou KJ, Cotton B and Gurd B (2002) Analytical expressions of capsize boundary for a ship
with roll bias in beam waves. J of Ship Res 46 3 167-174.
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insights. Appl Mech Rev. 5 307-325.
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work/energy balance. R.I.N.A. Transactions, 128, 217-234.
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Wiggins S. (1992) Chaotic transport in dynamical systems Springer-Verlag N.Y.
Appendix I: Melnikov’s Method as Energy Balance
Consider again the roll equation with bias:
τcos11β Faxxxxx (I1)
The corresponding Hamiltonian system is:
011 axxxx (I2)
with kinetic energy
KE x (I3)
and potential energy
Fig. I1 Energy balance around the homoclinic orbit
Yamagata, M. (1959) Standard of stability adopted in Japan, RINA Transactions, 101, 417-443.
It is obvious that if we take as initial point the saddle (x=1 0x ) and as final
the same point after going over the homoclinic orbit (“saddle loop”) the difference
in potential energy will be zero. Zero will be also the difference in kinetic energy
(at the saddle point the roll velocity is obviously zero).
The energy lost due to the damping “around” the homoclinic orbit is represented
by the area A inside the loop times the damping coefficient which is expressed as
The input of energy on the other hand due to the wave excitation is the integral
(in terms of the roll angle) of the external roll moment taken around the
homoclinic orbit. Mathematically this is expressed as
τττsinττsinWE 00 dxFdxF (I6)
The time constant 0τ allows to vary the phase between forcing and velocity x .
The balance of energies requires therefore that i.e.
which is identical with (9) when set equal to zero i.e. we have obtained a
condition identical to the condition 0τ0 M described by (3).
Despite the clarity of the above viewpoint a note of caution is essential: the above
energy balance “works” also for other nonlinear global bifurcation phenomena (e.g. a
saddle connection) and application should be attempted only when it is already
ensured that the underlying phenomenon is an incipient transverse intersection of
manifolds which indeed generates basin erosion.
Appendix II: Formulation for Irregular Seas
In the “stochastic” version of Melnikov’s method the objective is to determine the
time-average of the rate of phase-space flux that leaves the safe basin. The basic
theory can be found in Wiggins (1990) and Frey & Simiu (1993). However the
59. A Basis for Developing a Rational Alternative to the Weather Criterion 45
specific formulation and adaptation for the ship capsize problem is owed to Hsieh
et al (1994). The key idea is that the probability of capsize in a certain wave
environment is linked to the rate of phase-flux. The formulation is laid out as
Let’s consider once more the roll equation, this time however with a stochastic
wave forcing ( )τF at the right-hand-side. It is noted that the equation should be
written in relation to the absolute roll angle because the presentation based on
angle relatively to the wave in this case is not practical. It can be shown that the
flux rate i.e. the rate at which the dynamical system loses safe basin area scaled by
the area A of the basin of the unperturbed system is given by the following
expression (Hsieh et al 1994):
where sH is the significant wave height σ is the RMS value of the Melnikov
function ( ) ( ) DE MMM −= 0ττ for unity sH and DM is the damping part of the
Melnikov function which is commonly taken as non explicitly time-dependent.
For wave forcing with zero mean the mean value of the wave part of the Melnikov
function is zero thus the mean ( )[ ]0τME is DM− . Also p, P are respectively the
standard Gaussian probability density and distribution functions.
The assumption can be made, supported however by simulation studies (Hsieh
et al 1994 Jiang et al. 2000), that the flux rate reflects reliably the capsize
probability. Conceptually this appears as a direct extension for a probabilistic
environment of the connection between basin area and capsizability of a ship
under regular wave excitation. The “behaviour” of equation (II1) is that the flux
rate increases quickly once some critical significant wave height is exceeded
gradually approaching an asymptote as sH tends to infinity. Therefore, someone
could set a desirable environment of ship operation in terms of a certain wave
spectrum, significant wave height etc. and then through proper selection of the
damping and restoring characteristics he could try ensure that his design “survives”
in this environment. This is translated in keeping the flux rate as determined by
(II1) below a certain limit. Hsieh et al (1994) have suggested to use the point where
the asymptote intersects the sH axis as the critical one. The various quantities that
appear in (II1) are calculated as follows:
For the damping term DM of the Melnikov function it is entailed to determine
the integral of some power of the time expression of roll velocity at the homoclinic
orbit. In some cases it is possible this to be done analytically. For example for the
system described by equations (5) it becomes after calculation:
pqqpqphM D arccosh32β
where h p q are function of the bias parameter a and they are defined in Section 5.
connects with the spectrum of wave elevation as follows:
2τ dSSdSME FxME (II3)
The spectral density function of the wave forcing is linked to the wave spectrum
(this is perhaps the main weakness of the method as this
relationship is valid only for a linear process).
The final quantity that needs to be calculated is the power spectrum of the
scaled roll velocity
For the homoclinic loop shown in Fig. I1 this is easily calculated numerically
although it can be proven that it receives also an exact analytical solution in terms
of hypergeometric functions.
With the above we can plot the significant wave height versus, for example, the
characteristic wave period which gives a straightforward platform for setting a
level of acceptability (e.g. survival up to a certain sH ).
62. A. Jasionowski and D. Vassalos48
Although such standards have formed the main frame of safety provision over a
century or so, the acknowledgement of existence of these uncertainties, brought
about by the occurrences of accidents with undesirable consequences on one hand,
and realisation that economic benefit could be compromised by inbuilt but potentially
irrelevant margins, on the other hand, instigated a search such as (SAFEDOR 2005)
for better methodologies for dealing with the issues of safety.
Notably, such search has been enhanced, or possibly even inspired, by the
tremendous progress in computational technologies, which by sheer facilitation of
fast calculations have allowed for development of numerical tools modelling
various physical phenomena of relevance to engineering practice, at levels of
principal laws of nature. Such tools present now the possibility to directly test
physical behaviour of systems subject to any set of input parameters, and thus also
allow derivation of an envelope of the inherent uncertainty.
Existence of such methods, however, is not sufficient for progressing with the
development of more efficient means of safety provision, without a robust framework
for reasoning on the results of any such advanced analyses undertaken for safety
Such means evolve naturally from the principles of inductive logic; a field of
mathematics dealing with reasoning in a state of uncertainty, (Jaynes 1995) (Cox
1961). In particular the branch comprising probability theory forms a suitable
framework for such reasoning in the field of engineering.
However, while the concept of probability is known to engineers very widely, it
does not seem that the many unresolved to this day subtleties associated with either,
its fundamental principles or relevant linguistics, are recognised by the profession.
This is the situation especially when a special case of application of the probability
theory, namely that of risk, is considered.
The uptake of risk assessment process, or more recently strife to develop and
implement risk-based design paradigms, (SAFEDOR 2005) (Konovessis 2001)
(Guarin 2006), are examples of a new philosophy that could alleviate the
aforementioned problems of safety provision, if they can become suitable for
routine practice. This in turn, can only be achieved if fundamental obstacles such
as lack of common understanding of the relevant concepts are overcome.
Therefore, this paper sets to examine some lexical as well as conceptual
subtleties inherent in risk, offers some suggestions on more intuitive interpretations
of this concept, and puts forward a proposal for risk modelling process as a
contributory attempt to bringing formalism in this development.
2 The Risk Lexicon
As has been the case with the emergence of probability in the last 350 years or so,
(Hacking 1975), the terminology on risk has not evolved into anything that can be
regarded as intuitively plausible lexicon, as neither have the pertinent syntax nor
semantics been universally endorsed.
63. Conceptualising Risk 49
Any of tautological phrases, such as: “level of safety standard”, “hazard
threatening the safety”, semantically imprecise statements such as: “collision
risk”, “risk for collision”, “risk from collision”, “risk of collision”, ”risk from
hazard aspects of …”, simple misnomers such as “safety expressed as risk”,
“safety risk”, or philosophically dubious slogans such as “zero risk1
”, can be
found in the many articles discussing or referring to the concepts of risk.
A few other, easily misconceiveable terms used in this field can be mentioned:
(a) likelihood, chance, probability
(b) frequency, rate
(c) uncertainty, doubt, randomness
(d) risk, hazard, danger, threat
(e) analysis, assessment, evaluation
(f) risk control measures, risk control options
(g) safety goals, safety objectives, safety functional statements, safety performance,
Although some preamble definitions of these and other terms are offered from
article to article, lack of generic coherence in discussions of the risk concepts
among the profession is not helpful in promulgating this philosophy as of routinely
This article makes no pretence of having resolved all these subtleties. It merely
attempts to adopt a scheme of thinking, which appears to allow for systematic
understanding of the concepts of safety and risk.
By way of introduction of this scheme, the term underlying all this discussion,
namely safety, shall be presented to formally set the terms of reference.
Without going into intricacies of how complex the definition of the term
“safety” can be, it is hereby emphasised that for all engineering purposes and
currently emerging wider acceptance of risk concepts, the following definition of
safety, (Vassalos 1999), should be adopted widely:
The Cournot’s lemma stating that “an event of small probability will not happen” has been
initially endorsed by many mathematicians, even acclaimed as the physical “logic of probable” by
some, however, the concept has been dismissed for the latter part of the last century on the grounds of
Bayesian interpretations of probability, i.e. that “zero” is subject to “personal” interpretation. Hence a
statement of “zero risk” remains a philosophical conjecture with little, or indeed, no physical
64. A. Jasionowski and D. Vassalos50
safety is the state of acceptable risk
This definition is unambiguous and it is practical. It emphasizes explicitly that
safety is a state. It thus cannot be calculated, it can only be verified by comparison
of the actually established quantity of risk with the quantity of risk that is considered
acceptable by relevant authoritative bodies; a criterion or standard, in other words.
Thus, it follows that statements such as “higher safety” are incompatible with
this definition, as the safety is either attained or it is not and no grades of safety
can be distinguished. Appropriate statement would be “higher safety standard”,
which directly implies lowered levels of risk as acceptable. Also, expressions such
as “safety performance” become semantically imprecise unless, contrary to
intuitive perception, the word “performance” implies a binary mode of compliance/no
compliance with the set standard, rather than a scale, which in turn, relates to risk
This definition also establishes clear distinction between risk and safety. Risk is
the benchmark vehicle to demonstrate a state of safety or lack of it. Thus, safety
cannot be “expressed as risk”.
What the definition does not resolve is the consistency of the relevant syntax
which has come to be used in the every day communication on safety.
For instance, expressions such as “fire safety” and “ship safety” imply
fundamentally different meanings despite use of the same syntax. While the
interpretation analogous to that of the colloquial speech examples such as “orange
juice” and “baby juice”, could be accepted or taken for granted, as it is done
today, such subtleties should be addressed and resolved for avoidance of any
ambiguities in engineering applications, especially when many new concepts such
as safety goals or others are being introduced. This, however, is beyond the scope
of the discussion of this article.
Another worthwhile note should be made here, that this definition is also fully
compatible with the universally accepted in today’s world mechanism of safety
provision through compliance with regulations. Irrespective of the deterministic or
more complex nature of the regulation, safety is said to have been attained, when
an acceptable criterion has been met. Although the inherent risk is not disclosed, it
is said to have been brought to an acceptable level. Very often expressions, such
as “relative safety” or “conditional safety” are used to describe the limited scope
of such regulations and undisclosed nature of the risk, though neither of the words
“relative” nor “conditional” has any quantifiable explanation.
But, as mentioned earlier, since simplistic and disparate regulations could
compromise commercial gain or undermine societal approval of the standards
proposed for safety provision, when critical accidents happen, new methods are
The direct application of the concept of risk is one such method, especially
since the above definition of safety implies the “presence of risks”.
The key question now is what is risk?
65. Conceptualising Risk 51
An ISO 8402:1995 / BS 4778 standard offers the following definition of risk:
“Risk is a combination of the probability, or frequency, of occurrence of a defined
hazard and the magnitude of the consequences of the occurrence”
A similar definition is put forward at IMO, MSC Circ 1023/MEPC Circ 392:
“Risk is a combination of the frequency and the severity of the consequence”.
Although rather widely endorsed, these definitions are so ambiguous that they
hardly qualify as useful for any engineering purposes. The ambiguity derives from
the following: (a) the word “combination” is open to any interpretation whatsoever,
both in terms of relevant mathematics as well as underlying physical interpretations
(b) the words “probability” and “frequency” have, in common engineering practice,
different physical interpretations, which itself can be arbitrary, (c) the words
“magnitude” or “severity” are unspecific, (d) the “consequence” is again open to
For instance, the syntax of the above definition implies differentiation of the
semantics of the phrases such as “hazard” and “consequence”, without any
qualification how such differentiation is attained. Thus, the sinking of the vessel
can be considered as a hazard, which can lead to the loss of human life, but it can
also be considered as an ultimate consequence, or in other words as the loss. See
here Fig. 1 and the subsequent discussion.
While the merit of proposal of such definitions of risk could be its generic
nature, overly lack of any specificity deprives it of practical significance.
The following alternative definition is suggested, which is equally generic yet it
is comprehendible more intuitively and, apparently, more pragmatic:
risk is a chance of a loss
The word “chance” is one of the most fundamental phrases referring to, or being
synonymous with the natural randomness inherent in any of human activities and
which, in this case, can bring about events that are specifically undesirable, the
“loss”. The engineering aspects of the concept of chance are effectively addressed
by the fields of probability and statistics, both of which offer a range of instruments
to quantify it, such as e.g. expectation, and hence there is no need for introduction
of unspecific terms such as “combination”. These well known mathematical tools
will be discussed in section 3.
This definition of risk resolves also, to some extent, the manner of its
classification. Namely reference to risk should be made in terms of the loss rather
than in terms of any of the intermediate hazards in the chain of events leading to
the loss, a nuance just mentioned.
66. A. Jasionowski and D. Vassalos52
Fig. 1 The concept of a chain of events in a sample scenario leading to the loss. Each of the events is a
hazard after it materialises. A scenario can be identified by a principal hazard, see Table 1.
For instance, semantics of a statement “risk to life” or “risk of life loss” seem to
indicate sufficiently precisely the chance that an event of “fatality” can take place,
without much space for any other interpretations. On the other hand, a statement
such as “risk of collision” can be interpreted as a chance of an event of two ships
colliding; however, given a further set of events that are likely to follow, such as
flooding propagation, heeling, capsizing, evacuation/abandoning and fatalities, see
Fig. 2, it becomes unclear what is the “loss”, and thus what is the risk?
Furthermore, the above definition supports lexically the recently more pronounced
efforts to introduce the concept of holism in risk assessment. Namely, rather than
imposing a series of criteria for acceptable risk levels in a reductionism manner,
whereby separate criteria pertaining to different hazards are introduced, such as e.g.
criteria on fire hazards or criteria on flooding hazards (e.g. probabilistic concept of
subdivision, (SLF 47/17 2004) ), a unique criterion should be proposed that
standardises the acceptable risk level of the ultimate loss, e.g. criteria for loss of
life, environmental pollution, etc, and which risks account for each hazard that can
result in the loss, a view already mentioned in (Skjong et al. 2005) or (Sames 2005).
This concept underlines the risk model proposed in the following section 3.
3 Risk Modelling
As mentioned above, the fields of probability and statistics provide all the essential
instruments to quantify the chance of a loss.
3.1 Frequency Prototyping
To demonstrate the process of application of these instruments for risk modelling,
perhaps it will be useful to stress the not necessarily trivial difference between the
concepts of probability and frequency. Namely, to assess a “probability of an event”, a
relevant random experiment, its sample space and event in question on that sample
space must all be well defined. The relative frequency of occurrence of this event,
whereby relative implies relative to the sample space, is a measure of probability
of that event, since such relative frequency will comply with the axioms of
67. Conceptualising Risk 53
In the case of “an experiment” of operating a ship, however, whereby events
involving fatalities can occur in the continuum of time, the pertinent sample space
is customarily not adopted, and rather, the rate of occurrence of these events per
unit time is used. The rate can be greater than unity, and obviously can no longer
be referred to as probability. Thus, also use of the term “likelihood” would seem
to be displaced here since it is synonymous with probability rather than frequency.
Therefore, application of the term frequency per ship per year is customary,
also since its quantification relies on the historical data for the relevant fleet.
Lack of a well defined sample space, however, does not prevent one from using
concepts of probability, such as e.g. distribution, expectation, etc, in analyses and
Thus, following from this preamble, it is proposed here that for the purpose of
risk modelling the following assumptions are made:
(a) The loss is assumed to be measured in terms of an integer number of potential
fatalities among passengers that can occur as a result of activity relating to
(b) The loss is a result of occurrence of a set of scenarios which can lead to this loss
(c) Scenarios are intersections of a set of events (all must occur) and are identified by
, e.g. fire
(d) The scenarios are disjoint (if one scenario occurs then the other does not)
These assumptions are shown graphically in Fig 2 below.
Fig. 2 Illustration of the concept of a compound event referred to as “fatality” by Venn’s
diagram, (Jasionowski 2005), which is a union of mutually exclusive scenarios, whereby each
scenario is an intersection of a set of relevant events and is identified by principal hazards, such
as collision, fire, etc.
an event of a loss scenario is a hazard that materialised
68. A. Jasionowski and D. Vassalos54
Deriving from the above, it is hereby proposed that the frequency frN (N) of
occurrence of exactly N fatalities per ship per year is derived based on Bayes
theorem on total probability, as follows:
Where hzn is the number of loss scenarios considered, and hzj represents an
event of the occurrence of a chain of events, (a loss scenario), identifiable by any
of the following principal hazards:
Table 1. Principle hazards
Principal hazards, hzj Average historical frequency
of its occurance, frhz(hzj)
1 Collision and flooding 2.48e-3, (Vanem and Skjong
2 Fire -
3 Intact Stability Loss -
4 Systems failure -
…, etc. -
Furthermore, frhz(hzj) is the frequency of occurrence of a scenario hzj per ship
per year, and prN(N|hzj) is the probability of occurrence of exactly N fatalities, given
loss scenario hzj occurred.
In principle, there is nothing new to this proposal, except, perhaps, for the
emphasis of the need to estimate the probability for occurrence of exactly N fatalities,
prN(N|hzj), conditional on the occurrence of any of the principal scenarios j, an
essential element, conceived during (Jasionowski et al. 2005), and often unaccounted
for in risk concept proposals, such as e.g. (Sames 2005).
Modelling of either of the elements of the equation (1) requires an in-depth
analysis of historical data as well as thorough analytical considerations of all real-
life processes affecting them, practical attainment of which remains a challenge,
requiring inter-disciplinary approach.
For a demonstration purposes, a prototype model of the probability of an event
of occurrence of exactly N fatalities conditional on the occurrence of a scenario of
collision ∩ flooding, that is scenario corresponding to j=1 in Table 1, will be
discussed. Namely, it can be shown that prN(N|hzj=1) can be expressed as (2),
(Jasionowski et al. 2005).
69. Conceptualising Risk 55
Where the terms wi and pj are the probability mass functions of three, i=1 ..3,
possible ship loading conditions and a j=1 … nflood flooding extents, respectively,
assigned according to the harmonised probabilistic rules for ship subdivision,
(SLF 47/17 2004), or (Pawlowski 2004). The term ek is the probability mass function
derived from (15) for the sea state Hsk, where mHsk 40 ,
and nHs is the number of sea states considered. The term ci,j,k(N) is the probability
mass function of the event of capsizing in a time within which exactly N number
of passengers fail to evacuate, conditional on events i,j and k occurring, and can be
tentatively assigned with model (3).
The term i,,j,k (with y) represents the phenomenon of the capsize band shown
in Fig. 3 that is the spread of sea states where the vessel might capsize. These can
be estimated as follows:
049.0039.0 critcrity HsHs (5)
Where is the cumulative standard normal distribution. The Hscrit(s) is
calculated from equation (16) of Appendix 2. The sij is the probability of survival,
calculated according to (SLF 47/17 2004). Its interpretation is discussed in
Fig. 3 The concept of “capsize band”, (Jasionowski et al. 2004), for critical sea states of 0.5 and
The concept of a capsize band
0 0.5 1 1.5 2 2.5
3 3.5 4 4.5 5
70. A. Jasionowski and D. Vassalos56
1 NtNtNt failfailfail (6)
tNNtN evacfail max (8)
Finally, the term Nevac(t) is the number of passengers evacuated within time t ,
and is referred to as an “evacuation completion curve”, see Fig. 4. Such a curve
can effectively be estimated on the basis of numerical simulations, (Vassalos et al.
Fig. 4 Evacuation completion curve.
It is worthwhile to now offer discussion of some preliminary results from a
sample case studies demonstrating risk sensitivity to some key input parameters.
The study is discussed below, after introducing the final elements of the risk
model in the next section section 3.2.
3.2 Risk as a Summary Statistic
The frequency of occurrence of exactly N fatalities per ship year, that is the
information provided by the model (1), once all its elements are calculable, allows
for direct application of fundamental concepts of statistics for describing the
statistical properties of the random events in question, in this case fatalities.
The most obvious one is use of a graphical plot of a relationship of this frequency
with the number of fatalities. For example, it is very common to graphically plot
the cumulative frequency of N or more fatalities, so referred to FN plot, given by
71. Conceptualising Risk 57
Where Nmax is the total number of persons considered (e.g. number of crew, or
number of passengers, or both).
An example of such relationship derived on the basis of historical data, rather
than analytical/numerical modelling, is shown in Figure 5 below.
Fig. 5 Example of an FN plot derived from historical data, (Lawson 2004).
However, while conceptually useful and, indeed, accepted widely as an expression
of risk especially when plotted together with related criteria lines, (Skjong et al.
2005), it has been well known that for the purposes of any consistent decision making,
(Evans 2003), some form of aggregate information, derived on the basis of such
distributions, is required.
Commonly used summary statistics, such as expected value, are examples of
such aggregate information.
It is hereby emphasised that this form of information be used to quantify the
“chance” of a loss, or the risk. Unsurprisingly, the expected number of fatalities,
E(N), given by (10) and often referred to as the potential loss of life, PLL, has
been used among the pertinent profession routinely. See Appendix 1 on the form
of equation (10).
72. A. Jasionowski and D. Vassalos58
As it is known in the field of statistics, it is always desirable to find a statistic
which is not only consistent and efficient, but ideally is sufficient, as then all
relevant information is contained in one such number. In case of a statistic which
is not sufficient, additional information is required to convey the characteristics of
the underlying random process.
Since the expectation is hardly ever a sufficient statistic, it is necessary to
examine what additional information is required additionally to (10) to quantify
risk sufficiently comprehensively.
Namely, it is well known that the public tolerability for large accidents is
disproportionally lesser than the tolerance for small accidents even if they happen
at greater numbers. This tendency is not disclosed in the model (10), as simply the
same expected number of fatalities can result from many small accidents or a few
larger ones. It is for this reason, that model (10) is also referred to as a risk-neutral
or accident-size-neutral, (Evans 2003). Thus, public aversion to large accidents
cannot be built into any relevant criteria based on PLL in its current form and
without recurs to some form of utility function.
Although there are strong suggestions negating such a need and supporting the
view that a risk-neutral approach is sufficient, e.g. (Skjong et al. 2005), it does not
seem that the above simple fact of real life can be ignored. Indeed, recently proposed
criteria for stability based on the so called “probabilistic concept of subdivision” defy
this risk-neutral philosophy, as ships with more passengers are required higher index
of damage stability.
Therefore, a proposal by (Vrijling and Van Gelder 1997) shown as equation (11)
seems worth serious consideration as an alternative to (10), as it allows for controlled
accommodation of the aversion towards larger accidents through stand. dev.
and a risk-aversion index k.
Which form of a statistic is to be used to quantify risk should be the important
next step in research in this field.
To summarise this chapter, as one can see, risk can be shown to be a statistic of
the loss, which can be systematically modelled and used for better informed
decision making during any design process.
To demonstrate potential benefits of some elements of the concept of a holistic
approach to risk modelling and contained in model (1) and (10), simple case
studies are discussed next.
4 Sensitivity Studies
As has been mentioned earlier, there are no readily available elements related to
even a handful of loss scenarios listed in Table 1 and necessary to complete model (1).
73. Conceptualising Risk 59
However, since it is known that collision and flooding is one of the major risk
contributors, a sample study aiming to demonstrate the usefulness of the proposed
risk model is undertaken here based on only this loss scenario.
Consider a RoRo ship, carrying some 2200 passengers and crew, and complying
with the new harmonised rules on damaged ship stability by meeting the required
index of subdivision of 0.8 exactly, see Figure 6. Consider also that the evacuation
arrangements allow for two different evacuation completion curves, as shown in
Figure 7. The question is what is the effect of given evacuation curve on risk to
life posed by this ship?
The risk is calculated for nhz=1 in (1), using constant frhz(hz1)=2.48•10–3