Contemporary Ideas on Ship Stability
and Capsizing in Waves
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
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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
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
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
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
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)
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
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
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
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
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
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
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
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
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
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
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
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).
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
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 &
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
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
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
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
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
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
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
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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C.C. Bassler et al.22
Review of Available Methods for Application to Second Level Vulnerability Criteria 23
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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
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 .
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( )
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:
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
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
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.
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)
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
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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.
A Basis for Developing a Rational Alternative to the Weather Criterion 43
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.
Thompson JMT (1997) Designing against capsize in beam seas: Recent advances and new
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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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
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 ).
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.
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
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?
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.
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
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
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).
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
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
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).
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).
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
year. Element (2) is estimated based on the information shown in Figure 6 and
Fig. 8 demonstrates the effect of the evacuation completion curve on the
conditional probability prN(N|hz1). It seems that the majority of this scenario
variants are either no fatalities at all or a very large number of fatalities. Note that
because of this trend, the function estimates at these limits determine the ultimate
risk, and that the impression of higher overall frequencies for 60-min evacuation
curve seen in Fig. 8, must be viewed with care.
Fig. 6 Elements of index A, sample case.
Fig. 7 Hypothetical evac. completion curve
0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02
time to evacuaute [minutes]
60 minutes 120 minutes
A. Jasionowski and D. Vassalos60
Fig. 8 Prob. prN(N|hz1) for sample case
The resultant FN curve derived on the basis of (9) can be seen in the Figure 9.
It seems that the distribution is not affected significantly by different evacuation
completion curves, though when comparing the risk of 0.881 and 0.939 fatalities
per ship year, estimated according to equation (10) for 60 minutes and 120
minutes evacuation completion curves, respectively, a difference of some 6% can
be seen as a considerable design/operation target worth achieving.
Fig. 9 FN plot, effect of evacuation completion characteristics on risk
Another interesting test was undertaken, whereby a hypothetical ship with only
two flooding extents possible was used together with the 60minutes evacuation
completion curve shown in Figure 7. Two cases were assumed for the characteristics
of flooding extents, such that in both cases the resultant index A remained the same.
0 500 1000 1500 2000
Number of fatalities
evacuation complete 120min
evacuation complete 60min
Namely in case 1, 1 1 1 1
1 1 2 20.1, 0.3000, 0.9, 0.8560,p s p s and for case 2,
2 21 2 2
1 1 2 22.1, 0.6000, 0.8, 0.8505,p s p s were assumed. The resultant F
curves are shown in Figure 10 below.
Conceptualising Risk 61
Fig. 10 FN plot, effect of subdivision on risk, 60-min evacuation completion curve
As can be seen, contrary to the commonly expressed view, two ships with
different subdivision but the same index A, seem unlikely to have the same level
5 Further Work
The process of conceptualising risk discussed in this article is one of the many needed
before the vision of Risk Based Design, a design paradigm utilising the concept of
risk for the purpose of safety verification, can become a reality. The fundamental
elements that need to be in place are (a) an explicit risk model capable of
accommodating a number of loss scenarios, sufficient to meaningfully quantify
risk, and (b) the criteria, which will reflect both, the societal risk tolerability, and as
importantly, the epistemic and the aleatory uncertainties of the proposed risk model.
This paper discussed the concepts of safety and risk. Many lexical issues, often
ignored in the communication on these concepts, have been pointed out. Some
suggestions on more intuitive definitions have been made. A process of risk
modelling has been explained to some extent, and a simple yet comprehensive
model has been presented. A tentative model for probability of exactly N fatalities
and conditional on the occurrence of a collision ∩ flooding loss scenario has been
presented. Results of calculations of contribution to risk from this scenario have
been presented. Preliminary results seem to indicate that the common notion that
two different ships with the same index of subdivision correspond to the same
level of risk is not justified.
A. Jasionowski and D. Vassalos62
The present research was funded by the EC 6th
Framework Programme (2002-
2006), project EC FP6 IP 516278 SAFEDOR SP2.1 . The
support received is hereby gratefully acknowledged.
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Vassalos D, Jasionowski A, Guarin L (2005) Passenger Ship Safety – Science Paving the Way.
Int Ship Stab Workshop, Istanbul
Vrijling JK, Van Gelder PHAJM (1997) Societal Risk And The Concept Of Risk Aversion. Dep
of Civil Eng, Delft Univ Of Technol, Delft
Conceptualising Risk 63
Appendix 1 A Note on the Expected Rate of Fatalities
It can be shown that the area under the FN curve represents the expected number
of fatalities per ship year, (Vrijling and Van Gelder 1997):
Appendix 2 A Note on the Interpretation of the Probability
The survival factor “si” in (SLF 47/17), represents the probability of an event that
a vessel survives after flooding of a set of compartments i due to a collision with
another vessel and subsequent flooding.
The question is what does “survive” mean? The following is the proposed
The relationship between the survival factors, considering only the final stage
of flooding, and the ship parameters are:
Ks finali (13)
This has been derived from (14), i.e. the relationship between the parameters
GZmax and Range and the critical sea state3
, Hscrit, established through physical
model experiments, (Tagg and Tuzcu 2002); and the cumulative distribution
function collisionHsF of the sea states recorded at the instant of collision and given by
equation (15), see also Fig. 11. Note, that for the purpose of derivation of (13), it
was implied that Hscrit ~ Hscollision, whereby the relationship between (14) and (15)
was obtained through regression analyses rendering directly formula (13).
4 max RangeGZ
a sea state causing the vessel capsizing during about half of the 30minutes scaled model tests, the
damage opening modelled was that known as SOLAS damage
A. Jasionowski and D. Vassalos64
It is worth now offering the form of hard numbers in explaining the meaning of
the probability (13), given the underlying relationships (14) and (15). One such
form could read as follows: should the vessel suffer 100 collision incidents, say in
her lifetime, always leading to flooding of the same spaces “i ” with a survival
factor of, say, si,final = 0.9, then 90 of these incidents will take place in a sea state
with significant wave height below Hs(si, final)=2m, see equation (16), with the
vessel remaining afloat for at least 30minutes at a relevant damage-state
equilibrium (in other words the vessel will “survive”). The remaining 10 of these
incidents will take place in a sea state with the significant wave height above
Hs(si, final)=2m, and the vessel will not be able to sustain this for the 30 minutes
after collision (the vessel will not “survive”), see the note on critical sea state3
Fig. 11 CDF of significant wave heights at instants of collisions, eq.(15), (Tagg and Tuzcu 2002).
This interpretation follows the “philosophy” behind the formulation for the
si,final, discussed in (Tagg and Tuzcu 2002), currently underlying the new
harmonised Chapter II Part B of the SOLAS Convention, (SLF 47/17 2004).
The relationship between the significant wave heights at the instant of collision,
here equal to the critical sea state3
, and the “ s ” factors is given by equation (16),
(Tagg and Tuzcu 2002), which is derived from equation (15).
The aforementioned time of 30 minutes derives from the re-scaled model test
duration assumed during the campaign of physical model experiments of the
HARDER project, (Tagg and Tuzcu 2002), which underlines the relationship
between ship parameters and the sea states3
leading to capsize.
recognized that a considerable accumulation of the experimental experience is
required to correctly evaluate the safety.
Some effects of this assessment were already discussed (Bulian et al., 2004,
Francescutto et al., 2004). However, they were not based on full experiments included
in the guidelines. With this background the authors conducted experiments with a
Ro-Pax ferry model following the guidelines and examined the above mentioned
items. The previous paper (Taguchi et al, 2005, hereafter just referred as “previous
paper”) reported the results except the wind tunnel tests. In this paper, the effects
of this experiment-supported assessment by full experiments are reported. In the
following chapters the items explained in the previous paper are mentioned concisely.
2 The Weather Criterion and its Alternative Assessment
The weather criterion evaluates the ability of a ship to withstand the combined
effects of beam wind and waves. The criterion requires that area “b” should be
equal to or greater than area “a” (see Fig. 1), where
lw1 : steady wind heeling lever at wind speed of 26 m/s
lw2 : gust wind heeling lever (lw2 = 1.5 lw1)
: roll amplitude in beam waves specified in the code
: downflooding angle or 50 degrees or angle of second intercept between lw2
and GZ curves, whichever is less.
Fig. 1 Weather criterion
In the revised code, lw1 and 1
can be evaluated by model experiments in the
following conditions and it is allowed to consider the heeling lever as dependent
on the heeling angle like the broken line in Fig. 1,
lw1 : to the satisfaction of the Administration
: when the parameters of the ship are out of the following limits or to the
satisfaction of the Administration;
Angle of heel
S. Ishida et al.
Evaluation of the Weather Criterion 67
- B/d smaller than 3.5
- (KG/d-1) between -0.3 and 0.5
- T smaller than 20 seconds
where B, d and KG are the breadth, draft and the height of CG above keel of the
ship respectively, and T is the natural rolling period.
3 The Subject Ship
The subject ship is a Japanese Ro-Pax ferry. Table 1, Figs. 2 and 3 show the principal
particulars, the general arrangement and GZ curve respectively. Compared to general
European Ro-Pax ferries this ship has finer shape. From Fig. 3 it is found that up to
about 40 degrees the GZ curve has a very small nonlinearity to heel angle.
Table 1 Principal particulars
Length between perpendiculars Lpp [m] 170.0
Breadth B [m] 25.0
Depth D [m] 14.8
Draft D [m] 6.6
Displacement W [ton] 14,983
Blockage coefficient Cb 0.521
Area of bilge keels Abk [m2
Vertical center of gravity KG [m] 10.63
Metacentric height G0M [m] 1.41
Flooding angle f
Natural rolling period Tr [sec] 17.90
Lateral projected area AL [m2
Height of center of AL above WL Hc [m] 9.71
Fig. 2 General arrangement
Fig. 3 GZ curve
4 Wind Heeling Lever lW1
The wind heeling lever is estimated from the heeling moment when the ship is
drifting laterally by beam wind. Therefore, wind tunnel tests and drifting tests are
4.1 Wind Tunnel Tests
The wind tunnel tests were conducted at “Pulsating wind tunnel with water channel”
of NMRI (National Maritime Research Institute), with wind section of 3m in
width and 2m in height. The test arrangement is shown in Fig. 4. The connection
between the model and load cell had a rotating device for testing the model in
heeled conditions. In heeled conditions, the height of the model was adjusted by
the adjusting plate to keep the displacement constant when floating freely. By
using the model of 1.5m in length, the blockage ratio was kept less than 5%,
which is requested by the guidelines. The gap between the model and the floor
plate was kept within approximately 3mm and covered by soft sheets for avoiding
the effect of downflow through the gap.
0 20 40 60 80
Angle of Heel [deg]
S. Ishida et al.
Evaluation of the Weather Criterion 69
Fig. 4 Arrangement for wind tunnel tests
The wind speed was varied from 5m/s to 15m/s in upright condition and confirmed
that the drag coefficient is almost constant in this speed range. For the full tests a
wind speed of 10m/s was used, corresponding to the Reynolds’ number of 1.52 105
as defined by the following equation:
is the uniform wind speed outside the boundary layer, B is the breadth
of the model and is the kinematic viscosity coefficient of air.
The horizontal force Fwind, the heeling moment M and the lift force L were
measured by the load cell. The heeling moment M was converted to the one with
respect to point O, defined as Mwind in the guidelines, by the following equation:
M M F l cos L l sin (2)
where l is the distance from the centre of the load cell to the point O, which is defined
as the cross point of the centre line of the ship and waterline in upright condition.
4.2 Results of Wind Tunnel Tests
The measured drag coefficient CD, lift coefficient CL and heeling moment coefficient
CM are shown in Fig. 5. They are nondimensionalized by the following equations:
Fig. 5 Results of wind tunnel tests
In the figure the angle of heel is defined as positive when the ship heels to lee
side as shown in Fig. 4. The broken line is the heeling moment coefficient calculated
from the current weather criterion (IMO, 2002, called standard weather criterion
It is characteristic in Fig. 5 that at all the quantities (CD, CL and CM) vary
significantly with heel angle. As for the heeling moment, it is smaller than the
standard criterion and further reduces when the ship heels, especially to lee side.
The lift force is not so small and close to the drag force when the heeling angle is -
5 degrees (weather side). However, the adjustment of the vertical position of the
model is not necessary since the lift force is 0.7% of the displacement of the ship
in the assumed wind speed of 26m/s.
The result is also shown in Fig. 6 as the height of the centre of wind force above
waterline , lwind, by the following equation. It can be observed that the centre of
wind force is also a function of heel angle.
/wind wind wind
l M F (5)
-35 -30 -25 -20 -15 -10 -5 0 5 10 15 20 25 30 35
Angle of heel (deg.)
CM (standard criterion)
S. Ishida et al.
Evaluation of the Weather Criterion 71
Fig. 6 Height of the centre of wind force above waterline (model scale)
Although it is not requested in the guidelines, the effect of encounter angle, ,
was investigated. Fig. 7 shows the wind heeling moment coefficient, CM. Here
<90 means following wind. The figure shows that the wind heeling moment is
almost at the maximum in beam wind condition ( =90 degrees). This fact supports
the assumption of existing regulations. However, for developing performance based,
physics based criteria, the information on the effect of encounter angle to heeling
moment might be necessary.
Fig. 7 Wind heeling moment coefficient for various encounter angles
-35 -30 -25 -20 -15 -10 -5 0 5 10 15 20 25 30 35
Angle of heel (deg.)
Centre of lateral
Angle of heel [deg.]
4.3 Drifting Tests
The detail of drifting tests was reported in the previous paper. Here, the height of the
centre of drift force above waterline, lwater, is shown in Fig. 8. lwater was calculated
in the same manner as equation (3). The angle of heel is positive when the ship heels
to the drift direction. The drift speed (towing speed) was decided, as requested in
the guidelines, to make the measured drift force equal to the wind force at the
wind speed of 26m/s in ship scale. Because drifting tests were conducted before
wind tests, the wind drag coefficient, CD, was assumed to be from 0.5 to 1.1. In
upright condition and CD=0.8 the drift speed was 0.195m/s (1.80m/s in ship scale).
Fig. 8 Height of the centre of drift force above waterline
Fig. 8 shows that the centre of drift force is above half draft (which is assumed
in the standard criterion) and is generally above the waterline for this ship. This
phenomenon was reported by Hishida and Tomi (1960), Ishida (1993), Ishida and
Fujiwara (2000) and referred in IMO/SLF (2003). This is due to the more
dominant effect of the bottom pressure distribution than the side pressure when
breadth/draft ratio is large. For the cross sections with this proportion, high
position of the centre of sway force can be easily found in hydrodynamic tables of
Lewis Form. This fact suggests that potential theory would explain this phenomenon.
However, the effect of separated flow, e.g. from bilges, was also pointed out (Ishida
and Fujiwara, 2000). It was confirmed experimentally in the previous paper that
lwater reduces when the draft is enlarged.
Angle of Heel (deg.)
S. Ishida et al.
Evaluation of the Weather Criterion 73
4.4 Determination of lw1
The heeling moments by wind, Mwind, and by drifting, Mwater, both around point O,
were divided by the displacement, , and the wind heeling lever, lw1, was calculated
as a function of heel angle (equation (6)). Figs. 9 and 10 show the results.
Fig. 9 Wind heeling lever, lw1, evaluated by the tests
Fig. 10 Wind heeling lever, lw1, compared with the GZ curve
Wind + Drift tests
Angle of Heel (deg.)
Wind + Drift tests
Angle of Heel [deg]
In Fig. 9, the heeling levers due to wind ( wind
M ) and drift motion ( water
are also included. In both figures, lw1 at angles greater than 30 degrees (tested
range) is assumed to keep the same value as at 30 degrees as prescribed in the
guidelines. Figs. 9 and 10 show that, in the considered case, the wind heeling lever
estimated by wind and drift tests is sensibly smaller than that required by the
standard weather criterion.
5 Roll Angle 1
The formula of roll angle 1
in the weather criterion implies the maximum amplitude
out of 20 to 50 roll cycles in beam irregular waves. And 1
is related to the resonant
roll amplitude, 1r
, in regular waves, whose height and period are equal to the
significant wave height and mean wave period of the assumed irregular waves
(IMO, 2006, Watanabe et al., 1956). The reduction factor is 0.7 (see equation (7))
and this alternative assessment estimates 1r
instead of 1
by model experiments.
5.1 Direct Measurement Procedure
In the guidelines, this procedure is called “Direct measurement procedure” because
the resonant roll angle, 1r
, is measured directly in waves with the steepness specified
in the IS Code and the period equal to the natural roll period.
The results of experiments were mentioned in the previous paper. Here, Fig.11
is shown again. In the figure, “s” is the wave steepness, which is tabled in the
Code as a function of the natural roll period. For this ship s=0.0383 (1/26.1), but
lower steepnesses were also used. Due to the linearity of the GZ curve the
amplitudes reach the maxima at the vicinity of the natural roll frequency in all
steepnesses. From this result, 1
was decided as 19.3 degrees (=0.7 1r
S. Ishida et al.
Evaluation of the Weather Criterion 75
Fig. 11 Roll amplitude in regular waves
5.2 Alternative Procedures
In the guidelines, alternatives procedures are included, i.e. “Three steps procedure”
and “Parameter identification technique (PIT)”. In the “Three steps procedure”,
the roll damping coefficient N is estimated by roll decay tests. And, the effective
wave slope coefficient r is estimated by roll motion tests in waves with smaller
value of s. Finally, 1r
(degrees) is calculated by the following equation:
This method was adopted when the standard weather criterion was developed
and is based on linear theory except roll damping. The previous paper showed that
the estimated value of 1
is 19.5 degrees when the resonant roll amplitude at
s=1/60 is used. This value of 1
is very close to that by the “Direct measurement
procedure” due to the linear feature of GZ curve.
The PIT is a methodology to determine the parameters included in the equation
of roll motion. Once all the parameters are decided by test data at small wave
steepness, the roll amplitude at prescribed s can be extrapolated. In the guidelines,
an equation with 9 parameters is presented, in which nonlinear features of roll
damping, GZ curve and wave exciting moment are included. PIT analysis is not
carried out in this paper. However, the difference of 1
by PIT from other 2
procedures is expected to be limited because of the linearity of GZ curve.
Wave frequency / Natural roll frequency
6 Alternative Assessment of the Weather Criterion
In the guidelines, simplified procedures on the wind heeling lever, lw1, are also
mentioned for making the assessment practically easier. For wind tunnel tests, the
lateral horizontal force Fwind and the heeling moment Mwind can be obtained for the
upright condition only and considered as constants (not depending on heeling
angle). Instead of drifting tests, the heeling moment Mwater due to drift can be
considered as given by a force equal and opposite direction to Fwind acting at a
depth of half draft in upright condition, as assumed in the standard criterion. And
the combinations of complete procedures and simplified procedures are to the
satisfaction of the Administration.
The comparison of the assessments of the weather criterion using experimental
results is summarized in Table 2. In the table all the possible combinations of the
wind tests and the drifting tests, complete procedures and simplified procedures
are included. As for 1
, the standard criterion and the result of “Direct measurement
procedure” are included. The results of “Three steps procedure” can be omitted here
since the estimated 1
was almost equal to the one of “Direct measurement
procedure” for this ship.
Table 2 Assessments of weather criterion by experiments
The last line of Table 2 shows the critical values of the vertical centre of gravity
(KG), in which the ratio of area b/a=1 (see Fig. 1). These last results are to be taken
with some caution, since the effects of changing the vertical centre of gravity on Tr
(natural roll period) and on the other quantities related to roll motion, including
, have been neglected.
Table 2 shows that the alternative assessment by model experiments can change
b/a significantly with respect to the standard criterion. For this ship, b/a’s of the
right side of the table are smaller than those of the left side. This tendency comes
from the increased 1
obtained by experiments and it was suggested in the previous
S. Ishida et al.
Evaluation of the Weather Criterion 77
paper that 1
was enlarged by the small damping coefficient (N=0.011 at 20
degrees). On the other hand, the lw1 evaluated through all the combinations of the
wind tests and drifting tests, complete procedures and simplified procedures, tends
to make b/a larger than the standard criterion. It has to be noted that the leading
cause of the fluctuations is the large variation in the vertical centre of hydrodynamic
pressure when evaluated through the drifting tests.
Fig. 12 shows the critical values of KG. From Table 2 and Fig. 12, it is recognized
that the changes in the critical value of the vertical centre of gravity are more
contained than b/a, but as much as 0.4m at the maximum from the standard criterion.
Fig. 12 Variation of critical KG
In this paper, the results of the alternative assessment of the weather criterion by
model experiments are reported. Almost full tests, included in the interim guidelines,
were conducted. Considerable fluctuations of the assessment and their reasons are
clarified. In order to make the assessment more uniform and to remove the word
of “interim” from the guidelines, more extensive confirmation from the experience
gained through the application of the guidelines is needed.
Some parts of this investigation were carried out as a research activity of the SPL
research panel of the Japan Ship Technology Research Association in the fiscal
year of 2005, funded by the Nippon Foundation. The authors express their sincere
gratitude to the both organizations.
Bulian, G, Francescutto, A, Serra, A and Umeda, N (2004) The Development of a Standardized
Experimental Approach to the Assessment of Ship Stability in the Frame of Weather
Criterion. Proc. 7th
Int Ship Stab Workshop 118-126
Francescutto, A, Umeda, N, Serra, A, Bulian, G and Paroka, D (2004) Experiment-Supported
Weather Criterion and its Design Impact on Large Passenger Ships. Proc. 2nd
Int Marit Conf
on Des for Saf 103-113
Hishida, T and Tomi, T (1960) Wind Moment Acting on a Ship among Regular Waves. J of the
Soc of Nav Archit of Japan 108:125-133 (in Japanese)
IMO (2002) Code on Intact Stability for All Types of Ships Covered by Instruments. Int Marit
IMO (2003) SLF46/6/14 Direct Estimation of Coefficients in the Weather Criterion: submitt by
IMO (2006) SLF49/5 Revised Intact Stability Code prepared by the Intersessional
Correspondence Group (Part of the Correspondence Group’s report): submitt by Germany
Ishida, S (1993) Model Experiment on the Mechanism of Capsizing of a Small Ship in Beam
Seas (Part 2 On the Nonlinearity of Sway Damping and its Lever). J of the Soc of Nav Archit
of Japan 174: 163170 (in Japanese)
Ishida, S and Fujiwara, T (2000) On the Capsizing Mechanism of Small Craft in Beam Breaking
Waves. Proc. 7th
Int Conf on Stab of Ships and Ocean Veh (STAB2000) B:868-877
Taguchi, H, Ishida, S and Sawada, H (2005) A trial experiment on the IMO draft guidelines for
alternative assessment of the weather criterion. Proc. 8th
Int Ship Stab Workshop
Watanabe, Y, Kato, H, Inoue, S et al (1956) A Proposed Standard of Stability for Passenger
Ships (Part III: Ocean-going and Coasting Ships). J of Soc of Nav Archit of Japanese 99:29-46
S. Ishida et al.
modeling-forecasting-decision-making using complex integrated intelligent systems
(IS). The analysis of “informational snapshots” of the investigated situations allows
one to formulate the basic principles determining “latent” information processes.
Currently new approaches for intelligent technologies, including non-algorithmic
control which is naturally parallel and non-deterministic, are under development.
The period of development of the studies about ship stability is marked by a
change of paradigms as shown in Fig. 1.
Theoretical-experimental paradigm marks the beginning of research on ship
stability in waves. At this stage, the basis of experimental and theoretical methods
occur within the framework А. N. Кrylov’s hypothesis- the ship does not affect
the wave (Lugovsky 1966). The main attention here is given to the development of
theoretical models for the description of the righting moment and estimation of
dynamic roll of a ship in waves (Boroday and Netsvetaev 1969, 1982). Physical
pictures of roll and capsizing ships from experimental research with radio-controlled
models of ships with natural waves were also used (Nechaev 1978, 1989). The
basic scientific result of the research was the determination of critical situations
for a ship in waves (complete loss of stability, low-frequency resonance, broaching)
and justification of the necessity for theoretical models of dynamics to account for
nonlinear wave interference components — for ships with small L/B ratios having
high Froude numbers. The generalization of the results using physical models
allowed the author to develop linear regression formula (submitted to IMO in 1965
(Nechaev 1965)), describing the function of the righting moment for a preliminary
estimation of stability in waves. The main works of Russian researchers on this
paradigm, were performed within 1960-1970. The theoretical-experimental paradigm
was implemented on the basis of digital and analog computers with serial processing.
Computer paradigm, where the majority of works on ship dynamics in waves is
currently carried out in Russia, is connected to the further development of stability
theory using methods and models of research of nonlinear dynamical systems (ex.
Fokker-Plank-Коlmogorov equation, method of Monte-Carlo, method of moments,
phase plane, catastrophe theory), and also using modern methods of data processing
of physical experimental data with fast algorithms.
Most of the principle advancements in ship dynamics in waves were achieved
within the frames of this paradigm (works performed during 1971-1995).
Development of software and analysis procedures for ship stability in waves was
the most important part of these works; with the particular emphasis on probabilistic
models and criteria (Boroday et al. 1989, Nechaev 1978, 1989). Availability of
high-performance computers allowed the author to carry out robust analysis of
experimental results and develop a nonlinear regression model describing the
spatial function of the righting moment on waves (Nechaev 1978, 1989). Different
types of bifurcations, including strange attractors and the phenomenon of
deterministic chaos were also studied within the framework of the computer
paradigm; leading to a better understanding of the behavior of nonlinear dynamical
systems describing large-amplitude motions of a ship in waves. Implementation of
Evolution of Analysis and Standardization of Ship Stability 81
the computer paradigm is related with processing of information using parallel and
Fig. 1 The approaches to paradigms for information processes for research of ship stability in
The phenomenon of artificial intelligence; its application in ship dynamics
marked a new paradigm. Artificial intelligence is a principle component of on-
board intelligent systems (IS) with intended applications for prediction and control
of ship motions in waves as well as monitoring of safe navigation. Development
of these systems involves significant difficulties related to formalization of
knowledge and very complex computation algorithms. This concept considers
stability in waves as a complex, open set of coordinated dynamical procedures that
are carried out for analysis of extreme situations with an objective of safe navigation.
Implementation of new technology for ship control and the prediction of stability
on waves requires extensive use of mathematical modeling and supercomputers
(Nechaev et al. 2002, Nechaev and Zavialova 2002).
The development of new generation IS with a dynamic knowledge base and a
system of adaptation requires reconsideration of general principles of organization
and formalization of knowledge of interaction between a ship and its environment,
as well as with hardware and software (Nechaev and Petrov 2005). The requirements
of self-training and self-organizing in a continuously changing and fuzzy environment
lead to re-examination of the contents of adaptive components of the knowledge
base. These requirements also have increased a role of mathematical modeling for
prediction of ship behavior in waves as a complex dynamic object (DO). Most of
the information here comes from sensors of a measuring system; analysis, prediction
and interpretation of extreme situations is carried out on the basis of this information.
As a result, problems of integration of information and an analysis of alternatives
were revealed. Complimenting the knowledge base with solvers for numerical,
logical, and combinatorics problems, enabled the author to achieve significant
progress in one of the most complex areas of knowledge engineering – development
of methods for processing of information in dynamic environment.
The development of new methods of control of information processing gave birth
to new generation onboard IS for motion and stability control. The associative
approach of this technology allows organizing the process of making decisions for
control of a dynamical object by integration of information and algorithms, taking
into account restrictions and under-determinate of models. This new paradigm of
the control and prediction of ship stability in waves with on-board IS is focused on
extreme situations while using a wide variety of mathematical models (Nechaev
et al. 2001, 2006). The model of an extreme situation is based on direct interaction
with the environment; in this context the application of parallel computing is both
necessary and natural. This new type of control is based on the measurements
from sensors and results of simulation; such an approach considerably changes
organization of data processing by the onboard IS, making it distributed and
independent of number of processors. A new conceptual approach is underway to
substitute non-random and consecutive computational processes; this process is
based on models of the current situation and associative self-organizing in parallel
Thus, the research and development in the area of ship stability in waves allows
consideration of a model of ship/environment interaction as a complex holistic
open system. One of the main focuses of the analysis is the development of
integrated on-board IS, using models based on fuzzy logic and neural networks. It
allows development of more flexible systems and adequate description of a
phenomenon of ship behavior as complex dynamic object in conditions of
uncertainty and incompleteness of the initial information.
3 Analysis and Criteria of Stability of Ship in Waves.
Background Theory and Validation
Dynamics of the interaction of a ship with environment as a problem of stability in
waves generally can be described as follows:
Evolution of Analysis and Standardization of Ship Stability 83
where Х is a n-dimensional vector of phase coordinates; Y is a m-dimensional
vector stochastic excitations; F(X, t) is the area of phase coordinates corresponding to
a safe operation; ),,( tYXf is a vector-valued function describing the dynamical
system and 00 )( Xtx are initial conditions.
The solution of a problem (1) results in allowable values of output, used for
formulation of stability criteria:
Here J ,,1 … are the area of possible values of criteria, taking into account
uncertainty and incompleteness of the initial information; CRKG — is the critical
elevation of a centre of gravity of a ship.
Using a concept of KGCR, it is possible to associate probability of capsizing
with a probability of a parameter exceeding some allowable value
kkCRG dCdCCCPZZP ,,,, 11 …… (3)
where is the area where variables kCC ,,1 … satisfy the following inequality
CRkG ZCCZ ),,( 1 … (4)
An alternative formulation of a problem is that the value of CRKG can be
exceeded with the given probability 0P :
0)( PZZR CRG (5)
Therefore, the problem formulation (1), (2) consists of developing an algorithm
for the analysis and judgment of stability in waves with an estimate of correctness
of formulation of criteria in the conditions of uncertainty and incompleteness of
the initial information.
The following theoretical principles are formulated for the research of ship
stability in waves and further development of stability standards. Consider an
information operator processing information on dynamics of interaction of a ship
with environment. A structure of this operator should reflect the following properties
of information flows (Nechaev 1995):
Relatively small uncertainty in input may be related with significant uncertainty of
the output. This uncertainty must be accounted correctly, as the output
represents vital stability information.
Formulation of stability criteria should take into account properties of a ship as
dynamical system under external excitation. A normal operating area is around
stable equilibria with a corresponding safe basin parameter space. The behavior
of system on and in vicinity of the boundaries depends on the physics of the
problem. Uncertainty leads to appearance of a “gray” area limited by “safe”
and “unsafe” boundaries. Small excursions through the “safe” boundary lead to
equally small changes in the state of the dynamical system, while similar
excursions through the “unsafe” boundary lead to a qualitatively different state
of the system that is incompatible with safe operation.
Practical implementation of these theoretical principles for setting boundary
values for criteria results in difficulties caused by the uncertainty and incompleteness
of the input data. The generalized model developed within the framework of this
approach allows formal consideration of ship stability in waves, including both
analysis and criteria.
Research of ship stability in waves includes consideration of direct and inverse
problems. The solution of a direct problem is the transition from a known structure
and inputs of a dynamical system to the characteristics of output and stability
criteria. The objective of the inverse problem is synthesis; it is transition from the
desirable characteristics and known criteria to unknown structure of the dynamical
system and characteristics of its components. These models characterize the
theoretical formalized core of stability regulations.
Formulation of the problems of the control and prediction of stability in waves
is inherently related with conditions of uncertainty and incompleteness of the
input data. Therefore, the problem of validity of the mathematical models becomes
the most important. Let ),,1(),( niaFi … be a criterion of validity defined from
the analysis of a mathematical model describing a certain type (or types) of ship
motions. This criterion is either a function of parameters ),,1(, kja j … , or a
function of the solution of differential equations. The coefficients of mathematical
model ja , satisfying given parametrical, functional and criteria restrictions, make
an allowable area aE in a space of criteria )( aEF . Setting the accuracy of
approximation of parameters ja as ),,1(, kjj … , and the accuracy of criteria
Evolution of Analysis and Standardization of Ship Stability 85
),,1(, nii … , it is possible to present criterion for the proximity of the data
from mathematical modeling to data from physical experiments, which is the
criterion of validity of the mathematical model:
FFFFF ,,11 … (6)
where index P stands for calculation and E for experiment.
The approximation of area aE is necessary for the evaluation of coefficients of
a mathematical model with the given accuracy. It is achieved by finding values
,,1),(),()(min … (7)
under conditions for determining allowable area aE
i FFF (8)
Here 1a parameters that are within the boundaries ***
jjj aaa , N – number
of experiments; **
iF - allowable criteria limits (level of validity), assigned with
account for accuracy of the experiment.
Attempt to make dependences (6)-(8) specific, leads to the following
formulation. Let's present mathematical model (1) as local discrete transformation:
),,( 1 iiimi VSWS , (9)
where iS is the modeled state of a ship in the i-th instant of time; while 1iS is the
state corresponding to the previous instant of time i – 1; iW and iV are variables
discriminating state of environment and internal state of the dynamical system in
the i-th instant of time.
The condition of validity is determined as
mCcSY , (10)
where S is a domain limiting meaningful behavior of model; mC is a set of
objectives for modeling. For example, if a model of ship motions is linear and its
domain of meaningful behavior is limited to relatively small-amplitude motions, it
is valid if the objective is a conventional seakeeping analysis.
Development of the model, indeed, includes a choice of the type of the model
and its domain. To quantify deviation from a penalty function is introduced:
The discussion at IMO during 1965-67 was focused on the problem of validity of
mathematical models of stability and the correct design of physical experiments.
Observed differences between the theoretical and experimental data may be
used as criteria of validity with the peak data of motion time history (deterministic or
probabilistic expressions for linear or angular motions, velocities and accelerations),
as well as the limitation of divergence between time histories of phase trajectories.
Alternatively, the applicability of theoretical distributions (Gaussian, Rayleigh,
Weibull, etc) can be tested against the empirical data.
The improvement of mathematical models of ship stability in waves can be
facilitated by improvement of the theoretical background and experimental
technology of hydro-aerodynamic research, the use of effective methods for
processing and analyzing experimental results. The validity of mathematical
models can be improved by the development of more reliable ways of evaluation
of components of these models by better accounting for distortions that a moving
vessel introduces into the wave field (interference and diffraction of waves, change
of a field of pressure), as well as a more complete description of the spatial and
temporary structure of wind air-flows and roll damping forces.
Mathematical models for stability in waves (Boroday and Netsvetaev 1969,
1982, Lugovsky 1966, 1971, 1980, Nechaev 1978, 1989) are usually developed
for extreme situations. The background of these models is a description of the ship
interaction with environment, so it is a nonlinear dynamic system with six degrees
of freedom (Кrylov , 1958). A specific type of mathematical model is determined
by the physics which are observable during experiments. For example, when
considering stability in following waves or stability in breaking waves it is enough
to use the differential equations of drift and roll. While considering broaching, it is
necessary to use a system of at least four differential equations, including the
surge, sway, roll and yaw. In more complex situations, it is necessary to formulate
hypotheses with all six degrees of freedom included. It is important to include a
term describing continually changing stability in waves in the equation of roll
motions. The nonlinear spatial function of the righting moment in waves is
represented by the following formula (Nechaev, 1978,1989):
where max),( l and min),( l are the magnitude of stability changes while
the ship is on a wave crest or a wave trough, at various course angles ï ; ),( l is
Evolution of Analysis and Standardization of Ship Stability 87
the righting arm determined by interpolation on and for the various moments
of time, is the phase of a ship relative to a wave.
Fig. 2 Nonlinear function, describe of righting component: 1 – initial function; 2 – transform
function; 3 – wave shape
The geometric interpretation of function ),,( tM is given in Fig. 2.
4 Methods for the Analysis of Nonlinear Dynamic Systems
in the Modeling of Ship Stability in Waves
The research on ship stability in waves as a complex nonlinear dynamic system
requires understanding of the physics governing the phenomena. These problems
not always can be resolved within the frames of existing concepts and approaches
and may require development of the new approaches, methods and models.
Nonlinearity is, probably, one of the most important properties of the system,
which defines the methods applicable for analysis, such as: method of Monte-
Carlo, method of the moments, method of action functional, equation of Fokker-
Plank-Кhоlmogorov, method of a phase plane and catastrophe theory. These methods
are helpful in both the qualitative and quantitative interpretation of complex physical
phenomena associated with stability of ship in waves.
The author applied the Monte-Carlo method for stability and roll of a ship in
waves in 1975; it is described in the monograph (Nechaev, 1978). This method
plays a role of a bridge between theory and experiment. Being quite a powerful
tool of research in statistical dynamics, the Monte-Carlo method is related with
handling rather extensive volumes of data. It requires high speed of computation
1 2 3
and a large volume of memory. For the solution of such problems, high-performance
computers with parallel data processing technologies are best.
The mathematical model for the application of the Monte-Carlo method is
transformed into a dynamical system containing stochastic nonlinear function
representing the righting moment. This function can be presented as
Parameter (t) and phase (t) are considered random variables with normal and
uniform distribution, respectively.
The hypothesis of Gaussian distribution of the process (t) is checked by the
analysis of the time history of process. The correlation function )(*
tR of the process
(t) is estimated using the data of real sea waves. Calculations show, that the
tR can be presented with the following approximation:
where () and () — parameters to be found by the calculations.
For modeling the stochastic process (t), the method of forming filters is used
in time domain. The parameters of the filter can be found from variance )(*
and autocorrelation function )(*
The filter is described by system of the differential equations:
Evolution of Analysis and Standardization of Ship Stability 89
Here )(1 tW and )(2 tW are white noise stochastic processes; () and () are
parameters of the autocorrelation function; (t) is the auxiliary stochastic process;
M and *
D are operators of mean and variance; *
is the delta function.
This mathematical model is a part of practical procedure of modeling the
system with the given initial conditions. An important point here is the necessity
of formation of the random initial conditions determined by expressions (17). As
shown by calculations, the Monte-Carlo method has a number of advantages over
other methods of solution (method statistical linearization, theory of Markov
processes) for problems of statistical dynamics of nonlinear systems. The main
advantages are compactness of the calculation scheme, stability of results in a case
of hardware failure, and a rather simple way to estimate the accuracy of results.
Another convenient feature is that there is no limitation on the structure of differential
equations as any type on nonlinear terms can be included.
A traditional approach of classical dynamics considers either deterministic
(regular) or stochastic (irregular) processes described by the appropriate nonlinear
differential equations of roll motions. However, the nonlinear deterministic dynamical
system can exhibit chaotic behavior, when phase trajectories that were initially
close, become divergent in the limited area of phase space. This phenomenon of
the nonlinear system is known as deterministic chaos.
Nonlinear dynamics makes a clear distinction between simple (usual) and
strange (chaotic) attractors. The simple attractors are encountered quite frequently
when describing nonlinear roll and capsizing of a ship on waves. The geometrical
interpretation of a simple attractor describing a dissipative system represents on a
phase plane either a focus or a limit cycle. For both these cases, all phase trajectories
have a shape of spirals that converge to a stable equilibrium or to steady-state
stable cycle. Strange attractors have more complex structure, as they contain both
stable (attractors) and unstable (repeller) trajectories. Essentially, they are saddle-
type trajectories that are being attractors in one dimension and repellers in another;
they form a set of layers connecting to each other in quite complex way .
Fig. 3 Formation of a limiting cycle caused by of group waves
The papers (Degtyarev and Boukhanovsky 1995, 1996) describe the development
of parametric resonant motions of a ship caused by passage of group of waves. It
is shown that complex structures of oscillatory roll modes are formed in this case.
These structures are sets of attractors looking like unstable limit cycles. The cycle-
attractor can lose stability through several scenarios. A passing group of large
waves stabilizes the limit cycle described due to the nonlinearity of large-amplitude
response (Fig. 3). This cycle appears when the wave height in a sequence of waves
in a group exceeds the critical value of creating enough parametric excitation to
reach practically constant amplitude of response. However, the subsequent gradual
reduction of wave heights breakes the conditions for parametric resonance and the
A more complex scenario is realized when the stable cycle collides with an
unstable cycle (Fig. 4). Such a situation is much rarer and it can be generated by a
consecutive passage of waves groups containing wave of significantly different
height. The first wave group with small heights results in a relatively small-
amplitude limit cycle, while the second group generates large-amplitude cycle.
Appearance and loss of stability of an oscillatory mode (“birth and death of a
cycle” using a terminology by А.А. Аndronov) occur due to limitedness of a rolling
resonant zone on a rather small time interval where large-amplitude motions are
generated by a passage of wave groups.
Fig. 4 Appearance (A) and loss of stability of a cycle (B) (C)
The problems and methods of controlled chaos is an area of intensive research
in recent decades. In the beginning, the concept of deterministic chaos was considered
as an exotic phenomenon that is of interest to mathematicians only; the very
possibility of occurrence of chaotic response in practical applications was
doubtful. However, later on, chaotic dynamics was found to be present in many
dynamical systems across disciplines: in mechanics, laser physics and radio-
physics, chemistry and biology, economy, and health science. Systems with chaos
demonstrate both good controllability and surprising flexibility: the system
quickly reacts to external excitations, while keeping a mode of motions. The
combination of controllability and flexibility in part is the reason why chaotic
dynamics is a characteristic type of behavior for many dynamical systems
describing ship motions in waves.
From the point of view of the synergetic approach, chaotic systems represent
possibilities for realization of self-organizing processes. One of the most typical
scenarios for the transition to chaos is through a sequence of period-doubling
bifurcations, which is observed for systems with viscous friction under action of
excitation forces. The research of chaotic behavior of a nonlinear dynamical
system describing roll motions was carried out by systematic variation of control
parameters. The result was shown on Poincare map, allowing to clearly see the
Evolution of Analysis and Standardization of Ship Stability 91
period-doubling and sub-harmonic bifurcation (Nechaev 1993). Another scenario is
the mechanism of transition to chaotic response, known as intermittency, where
intervals of deterministic chaos alternate with almost periodic motions (Fig. 5).
Fig. 5 Occurrence of chaos of an alternated type: А – Duffing model; IB – Mathieu model
at presence of an initial heel angle.
The Lyapunov exponent is an important tool for study of behavior of chaotic
systems. It can be demonstrated with the Duffing equation, which is one of the
interpretations of nonlinear rolling (Nechaev, 2007). Introduction of the third
dimension for the independent variable allows presentation of the Duffing equation as
a system of the third order:
This system is defined with two positive parameters: coefficient of linear damping
k and amplitude of excitation B. Negative divergence is determined by damping
The linearized system leads to the following matrix
The results of calculations for В = 10, k =0.1 are shown in a Fig. 6. These
parameters lead to the appearance of chaotic behavior of the system (18), which is
confirmed by time histories, phase trajectories and the Lyapunov exponent.
Theoretical development of chaotic dynamics has revealed a series of possible
practical applications, including ship motions and stability in waves, where the chaotic
responses may be encountered as a result of complex interaction of the ship and
environment. Moreover, there are possible practical applications, where the control
over nonlinear system is realized by changing a degree of its “chaoticness”. Methods
for the solution of similar problems have been developed recently, especially for
application of different control algorithms for chaotic systems.
Fig. 6 Attractor and Lyapunov exponents for the Duffing equation
5 Analysis of Extreme Situations and Control of Ship
Stability in Waves
The evaluation and prediction of stability changes for motion control of a ship in
waves are carried out using the information from acting sensors (Nechaev (ed.)
2001). In this case, the extreme situation caused by deterioration of stability can
be described in compact set with variable ),,1( mz … . The analysis produces
scenarios for the possible development of extreme situations, taking into account
physics of the interaction between a ship and environment. When the probabilities
of encounter of extreme situations are estimated, measures of statistical uncertainty
have to be evaluated as well. Let number of situations be N, while M(w) is an average
Evolution of Analysis and Standardization of Ship Stability 93
estimate, and D(w) is the variance estimate of a value representing possible
consequences of the situation.
mjNqN jj … (21)
N is the predicted total number of extreme situations for the considered
period of time; jq is the fraction of situations for the j-th class in the distribution
F(w), determined by the formula
)( maxmin jjj wwwPq (22)
The value jq is calculated with known distribution F(w) and its parameters
M(w) and D(w). The accuracy of the forecast under the formula (21) is
characterized by the standard deviation:
1 ][ jj qNNqN
random deviation 20
][Nqj and 20
][ jqN included in the formula (23) are
determined by statistical processing of results from the mathematical modeling.
The development of an algorithm for the extreme situation prediction is carried
out using the information from a measuring system. This information is pre-
processed and represents a time-series reconstructed on the basis of the Takens’
theorem. According to the Takens’ theorem, a phase portrait restored as:
is equivalent topologically to the attractor of initial dynamic system.
6 Method of Action Functional
The search of a universal principle that could help to describe behavior of a dynamic
system, has resulted in discovery of a principle “of least action”, similar to a
principle of “minimum potential energy” that determines the position of equilibria.
According to a principle of the least action, the variation problem is a search of a
stationary value of the action integral, defined on the interval ),( 21 tt . Responses of the
dynamical system corresponding to all possible initial conditions are compared.
If the interval of time )( 12 tt is small enough, the action integral has a minimal
value as well as the stationary value.
The implementation of the algorithm of functional action for ship dynamics in
waves is considered in (Nechaev and Dubovik 1995). The advantages of the
method can be summarized as follows:
use of the information for an external disturbance, received during real-time
taking into account significant nonlinearity of a problem for any kind of
function – possibility of the solution without use of a priori data on behavior of
system and its initial conditions.
Functional action is defined by an asymptotic expression (Nechaev (ed.) 2001,
Nechaev and Dubovik 1995):
It allows estimating probability xP that critical set D (capsizing) will be
achieved. This estimation is carried out using optimum control for the stability
criteria formulated above. The solution of such problem (25) is given by the main
term of logarithmic asymptotic expansion of the probability at 0 .
Fig. 7 Probability of capsizing as a function of time (А), dynamics of roll of a ship on waves (B)
and polar diagram (С)
This approach makes possible the development of new methods for analysis
and prediction of behavior of nonlinear dynamical systems in real-time, including
stability control in waves. The results of a method of functional action are presented
as a probability of capsizing as a function of time (Fig. 7A, В). Then interpretation
of the results for the given external conditions allows “drawing a picture” of
capsizing for a ship in a considered extreme situation. Additionally information on
heeling moment leading to capsizing can be made available on the screen. Using the
method of the functional action it is possible to evaluate combinations of dangerous
Evolution of Analysis and Standardization of Ship Stability 95
speeds and headings relative to waves and present those in a form of a polar
diagram (see Fig. 7C).
7 Study of Ship Stability Using Climatic Spectra
and Scenarios of Extreme Situations
Modelling the environment is one of the most complex problems that need to be
solved while developing procedures for onboard IS. Contemporary approaches to
the solution of these problems require new presentations of wind-wave fields in
the ocean. These include the conception of climatic officially accepted on 18th
Assembly of IMO in 1993 along with the concept “wave climate” (Boukhanovsky
et al. 2000). These concepts open the opportunities for more detailed description
of waves in different parts of the World Ocean.
The hydrodynamic model of waves in the spectral form is represented as the
equation of the balance of wave energy:
Here N is the spectral density of wave action; it is function of latitude ,
longitude , wave number k, and the angle between a wave direction and a parallel
, as well as frequency and time t. The equation (26) relates inflow of energy
from wind, dissipation and redistribution, as well as nonlinear interaction between
frequency components of the process. Often, the source function G is written as a
sum of three components dsnlin GGGG (incoming energy from wind to
waves, weak nonlinear interaction in a wave spectrum, and dissipation of wave
Examples of classification of wave spectra are shown in Fig. 8. There are six
classes of spectra: A – swell; B – wind waves; C – combined swell and wind
waves systems with prevalence of swell; D – combined swell and wind waves
systems with prevalence of wind waves; E – combined swell and wind waves
systems without clear division and with prevalence of swell; F – combined swell
and wind waves systems without clear division and with prevalence of wind
Fig. 8 The typical normalized spectra of sea waves; vertical axis is value spectral density
S()/Smax and horizontal axis is circular frequency , in sec–1
It is important to note that the standard calculation of dynamics of complex
objects in real sea waves is done with a one-peak-spectrum, which may result in a
type II error in comparison with climatic spectrum. Therefore, climatic spectra
must be used for development of onboard IS, as type II errors may result in
8 Paradoxes of Stability Standards
The concepts of complexity and randomness are closely related. Simple criteria are
often preferred over the complex ones. However, while gaining better understanding
of physics of ship/environment interaction, it becomes clear that the problem of a
choice of criteria is a very complex one. That’s why sometimes there are unexpected
results and paradoxes (Nechaev, 1997).
Paradox of “ideal” norms. It is known that the guarantee (defined as a probability
that failure will not occur if an object satisfies the norms, see also (Belenky and
Sevastianov 2007, Belenky et al. 2008)) for any, even most primitive, norms can
be made very close to unity (ideal norms), if
where KG, CRKG are the actual and critical position of center of gravity.
А B C
D E F
Evolution of Analysis and Standardization of Ship Stability 97
It can only be achieved by making norms stricter. Simple calculations show
that considerably more physically valid stability criteria may result in further
deviations from the ideal, compared to simplified IMO criteria. The paradox is
that the result is achieved by the extremely uneconomical way, at the expense of
making a norm stricter than the average norms, even when it is not really
Paradox of zero probability. The paradox of zero-probability is directly related
to probabilistic stability criteria. Practically, the probability of capsizing equals to
zero (based on casualty statistics, probability of capsizing is about 10–4
corresponds to a range of risk (1-10) 10–4
similar to landing helicopters, horse
races with obstacles and sports car races), but this event is not impossible. There is
a paradox, whether it is possible to compare “chances” of events having zero-
probability. Also whether it is real when the conjunction of events having zero-
probability can result in finite-value probability, i.e. the addition of many “nothings”
results in “something”. Probabilistic analysis of stability failures leads to a concept
of rare events coming from “unusual occurrences” (combination of external conditions
for assumed situations, encounter with extreme waves, etc.). The small probability
of such events may make a false impression that simultaneous combination of many
adverse factors is practically impossible, like conditions that lead to large-amplitude
rolling (synchronous and parametrical resonance). This paradox is also closely
related to the problem of rarity.
Paradoxes of the distributions. Many paradoxes are related with the application
of theoretical distributions. One such peculiarity is caused by asymptotic
properties of theoretical distributions and results in insensitivity of the probability
of stability failure to the large changes of the centre of gravity in the vicinity
10 P . Other problems are related to the change of distributions by a nonlinear
dynamical system, depending on the level of excitation.
It is known that for the normal distribution, the average Ni /*
random variable is an unbiased estimate 0)( *
E and converges
P at the large values N. However if the distribution is not
known a priori, the estimate *
may end up being biased with the minimum
variance; in case of multi-dimensional distributions, such an estimate may simply
not exist for quadratic loss function 2*
)( . If such a check was not
performed during the statistical analysis the resulting criteria may be not valid.
Paradox of a choice of boundaries for a criterion. The paradox of rational
choice boundaries for a stability criterion is very important. Use of fuzzy boundaries
for stability criteria makes sense due to the random nature and uncertainty of the
input data that is used as a basis for calculation scheme of evaluation of stability
(Fig. 9) (Nechaev, 1997).
The errors of an inclining experiment, limited accuracy of data on loading
conditions of a ship, as well as the uncertainty of other factors, may result in the
true value of the parameter Х actually belonging to the interval ];[ 00 XX .
Then according to the requirements when 0
XX , the norm is observed exactly,
while in reality 0X . The area of acceptance of a hypothesis 0XX , i.e. the
interval ( XX ,1 ) equals ( - significance value). Then the probability of a
type II error for deviation from hypothetical value qX equals ; the value 1 –
characterizes “power of the criterion”. Reducing leads to a reduction in the
probability of a type I error (the zero hypothesis is rejected, when it is correct).
However, the probability is increased of the type II error and the “power” of
criterion is reduced.
Fig. 9 Errors of practical use of stability criteria: 0X – parameter; ),( minmax XX - area of
actual change Х; A – area zero-hypothesis (1) and alternative hypotheses (2); B – areas
corresponding to a type II error
Paradoxes of computer implementation. Many paradoxes are related to computer
implementation of stability problems. Any statistical solution, which can be
implemented on the computer, now becomes available. As a result, “stable” and
multi-dimensional methods, requiring huge number of operations, have entered
engineering practice without a sufficient theoretical justification. Meanwhile, it is
possible to justify many empirical collisions, using robust statistical methods.
Unfortunately this is a common practice in stability assessment and regulation.
Evolution of Analysis and Standardization of Ship Stability 99
9 Future Directions
The further developments of the analysis and criteria of stability can be
achieved by advancements in the following strategic directions:
1. Development of hydrodynamic model of the interaction of a ship with external
environment while under action of extreme waves.
2. Study of physical response of roll and capsizing of a ship in extreme situations
using climatic spectra.
3. Development of a hydrodynamic model for nonlinear interaction of ship waves
with incident waves in storm conditions.
4. Further development of criteria for ship stability in waves using new
approaches for description of uncertainty in complex dynamic environments.
Development of effective means of the control of ship stability in waves using
algorithms, which take into account the change of ship dynamics in time.
The considered problems of the analysis and criteria of ship stability in waves reflects
only a small portion of the research applications in which the ideas developed by
the author, his students and colleagues have found a place. The foundation for these
works combines stability and the theory of oscillators and offers a methodology
for better understanding how both these concepts are applied for modeling. Thus
the author emphasizes applications that attract both scientific and practical interest
while researching complex behavior of ships in various extreme situations. Making
decisions for the control of these situations may benefit from the new approaches
based on formulations of uncertainty and construction of system of knowledge
using artificial intelligence and fuzzy mathematics.
Advancements in cybernetics, information technology and the general theory
of systems led to a new scientific picture of the world. This picture has found its
reflection in new approaches to the problem of stability of a ship in waves. The leading
role belongs to nonlinear dynamics and is related to the concept of synergy that has
changed the understanding of the relation between chaos and order, entropy and
information. The theory of synergy came from works on the theory of bifurcation
of dynamical systems and was further developed by new generations of researchers.
Complexity of structures and processes plays a key role for modeling of the
stability of a ship in waves. Their instability, randomness, and transitive nature led
to new scientific paradigm. Changes of the structure of a dynamical system are studied
within this paradigm, rather than a behavior of a system with unchangeable structure.
The problem of analysis and development of stability criteria is one of important
avenues for ensuring safety of operation. Uncertainty and incompleteness of the input
information are inherent for the complex problem of ship stability. Information
technology for monitoring ship dynamics in waves being developed by the author
and his colleagues may be implemented into on-board intelligent systems. Such
technology should not be reflected in the final form of criteria, as a mathematical
formulation allows more flexible analysis of stability in complex situations.
The author is grateful to his advisor prof. N.B. Sevastianov, who formulated the
general approach to study the very difficult problem of ship stability in waves, and
also to professors S.N. Blagoveshchensky, I.К. Boroday, N.N. Rakhmanin, V.V.
Semenov-Тyan-Shansky and V.G. Sizov – for their support and advice.
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D. Vassalos and A. Jasionowski104
and accurately is paramount. Unfortunately, a close scrutiny of the work that led
to the current formulation of the s-factor revealed that it is simply the result of a
series of unjustified compromises, inadvertently creeping in during the rule-making
process. In the form, due to be adopted in 17 months time, the s-factor derives
from a regression analysis of only a filtered set of old cargo ships and as such it
will be unsuitable even for this category of ships, considering the evolutionary
changes in most cargo ship types. More importantly, recent research results indicate
that, on average, the formulation to be adopted seriously underestimates the
inherent survivability of cruise ships whilst it drastically undermines the survivability
of Ro-Ro passenger ships. What is of crucial significance, there is little consistency
between the actual survivability of both these vessel types (the most safety-critical
of ships) and that postulated by the currently proposed formulation. The problem
does not end here: since this formulation was used to calculate Index-A for a
representative sample of various vessel types so as to evaluate the Required Index
of Subdivision (in principle, the safety level to be adopted), these results will also
be in disarray. In short, SOLAS 2009 is in need of major introspection!
This paper attempts to lay the bare facts and in so doing to raise the alarm,
hoping that in the time remaining no effort will be spared in ensuring that the best
ever regulatory achievement in our industry will not fall flat on its face before it
2 The Probabilistic Concept of Ship Subdivision
The first probabilistic damage stability rules for passenger vessels, deriving from
the work of Kurt Wendel on “Subdivision of Ships”, (Wendel 1968) were introduced
in the late sixties as an alternative to the deterministic requirements of SOLAS
‘60. Subsequently and at about the same time as the 1974 SOLAS Convention was
held, the International Maritime Organisation (IMO), published Resolution A.265
(VIII). The next major step in the development of damage stability standards came
in 1992 with the introduction of SOLAS part B-1 (Chapter II-1), containing a
probabilistic standard for cargo vessels, using the same principles embodied in the
1974 regulations. This, in turn, necessitated launching at IMO the regulatory
development of “Harmonisation of Damage Stability Provisions in SOLAS, based
on the Probabilistic Concept of Survival” in the belief that this represented a more
rational approach to addressing damage stability.
However, the slowly changing world of IMO engulfed this process and it almost
stagnated. In this state of affairs, the compelling need to understand the impact of
introduction of probabilistic damage stability regulations in the design of cargo
and passenger ships and the growing appreciation of deeply embedded problems
in both the rules and the harmonisation process itself, provided the motivation for
the adoption of an in-depth evaluation and re-engineering of the whole probabilistic
framework through the EC-funded €4.5M, 3-year project HARDER (HARDER
SOLAS 2009 – Raising the Alarm 105
2003) The overriding goal of this project was to develop a rational procedure for
probabilistic damaged ship stability assessment, addressing from first principles
all relevant aspects and underlying physical phenomena for all types of ships and
damage scenarios. In this respect, HARDER became an IMO vehicle carrying a
major load of the rule-development process and fostering international collaboration
at its best – a major factor contributing to the eventual success in achieving
harmonisation and in proposing a workable framework for damage stability
calculations in IMO SLF 47.
A stage has now been reached where the draft text of the major revision to the
subdivision and damage stability sections of SOLAS Chapter II-1 based on a
probabilistic approach has been completed, following final amendments in January
2005 to Regulation 7-1 involving calculation of the “p” factor. The revised
regulations were adopted in May 2005 by the IMO MSC and will be entering into
force for new vessels with keels laid on or after 1 January 2009.
One of the fundamental assumptions of the probabilistic concept of ship
subdivision in the proposed regulations is that the ship under consideration is
damaged, that is the ship hull is assumed breached and there is (large scale)
flooding. This implies that the cause of the breach, the collision event with all the
circumstances leading to its occurrence, are disregarded, and hence the interest
focuses on the conditional probability of the loss of stability.
In other words, risk to life is assumed to be irrelevant on the likelihood of
occurrence of a collision event that ends in hull breaching and flooding. For this
reason, the regulations imply the same level of “safety” irrespective of the mode
of operation that can e.g. take place in area of varying density of shipping (congestion
of traffic), or indeed can be so different depending on ship type, or can involve
vastly different consequences, etc, all of which might imply considerably different
levels of actual risk. This said, all risk-related factors (e.g. size of ship, number of
persons on board, life saving appliances arrangement, and so on) are meant to
indirectly be accounted for by the Required Index of Subdivision, R. Summarizing,
the probability of ship surviving collision damage is given by the Attained Index
of Subdivision, A, and is required not to be lesser than the Required Index of
Subdivision, R, as given by expression (1):
A/R Attained/Required Index of Subdivision
j loading condition (draught) under consideration
J number of loading conditions considered in the calculation of A (normally 3
i represents each compartment or group of compartments under consideration
for each j
Draught and GM
D. Vassalos and A. Jasionowski106
I set of all feasible flooding scenarios comprising single compartments or
groups of adjacent compartments for each j
wj probability mass function of the loading conditions (draughts)
pi probability mass function of the extent of flooding (that the compartments
under consideration are flooded)
sij probability of surviving the flooding of the compartment(s) under
consideration, given loading (draught) conditions j occurred
The index A can thus be considered as the expected value of the “s-factor”,
with “p- and w-factors” being the respective likelihoods, reflected in worldwide
ship collision statistics:
Consequently, (1-A) can be considered as the marginal probability of (sinking/
capsize) in these scenarios, and as such it can be used for deriving the collision-
related risk contribution, (Vassalos 2004).
The required Index of subdivision, R (derived principally from regression on
A-values of representative samples of existing ships) represents the “level of
safety” associated with collision∩flooding events that is deemed to be acceptable
by society, in the sense that it is derived using ships that society considers fit for
purpose, since they are in daily operation.
The new regulations represent a step change away from the current deterministic
methods of assessing subdivision and damage stability. Old concepts such as
floodable length, criterion numeral, margin line, 1 and 2 compartment standards
and the B/5 line will be disappearing.
With this in mind, there appears to be a gap in that, whilst development of the
probabilistic regulations included extensive calculations on existing ships which
had been designed to meet the current SOLAS regulations, little or no effort has
been expended into designing new ships from scratch using the proposed regulations.
However, attempting to fill this gap through research and though participation in
a number of new building projects revealed a more serious problem that affects
consistency and validity of the derived results and hence of the whole concept.
These findings constitute the kernel of this paper and are described next, following
a brief reminder on the derivation in project HARDER and the eventual adoption
at IMO of the survival factor “s”.
It has to be pointed out that despite the wide-raging investigation undertaken in
HARDER, the survival factor “s” adopted in the new harmonised probabilistic
rules designates simply the probability of a damaged vessel surviving the dynamic
effects of waves once the vessel has reached final equilibrium, post-damage. All other
3 Probability of Survival – “S-Factor”
SOLAS 2009 – Raising the Alarm 107
pertinent factors and modes/stages of flooding are not accounted for. Moreover,
whilst considering 7 ship models in the test programme (HARDER 2001),
representing a range of different types, sizes and forms, namely: 3 Passenger ships
(2 Ro-Ro’s and one cruise liner) and 3 dry cargo ships (Ro-Ro Cargo Ship,
Containership, and Bulk Carrier - a Panamax Bulk Carrier was added later), the
formulation for the s-factor focused on Ro-Ros and non-Ro-Ros. This categorisation
derives primarily from the capsize mechanisms pertinent to these two categories.
Therefore, and without labouring the detail of the formulation (for which SSRC
was responsible), the following two approaches were followed:
3.1 s-Factor for Low Freeboard Ro-Ro Ships
It was always the intention to base the survival factor for Ro-Ro ships on the SEM
methodology, (Vassalos et al. 1996). In brief, the method statically develops the
volume of water that will reduce the damage GZ curve to exactly zero and at the
ensuing critical heel angle, θ, two parameters: h – the dynamic water head and f –
the freeboard are calculated as shown in Figure 1.
The statistical relationship between parameters (h) and (f) and the significant
wave height of the critical survivable sea state (HS) is then established.
Fig. 1 The SEM Methodology
D. Vassalos and A. Jasionowski108
Accounting finally for the likelihood of occurrence of this sea state at the time
of casualty led to the following relationship (Figure 2):
Fig. 2 SEM-Based s-Factor for Ro-Ro Ships (25 Cases – 4 Ship Models)
3.2 s-Factor for non-Ro-Ro Ships
Having applied the SEM methodology to non-Ro-Ro ships and achieved what was
considered unsatisfactory results, the focus shifted towards the more traditional
damage stability criteria employing the use of properties of the GZ curve, such as
GZ max, GZ Range, and GZ Area. In simple terms, the correlation of these three
traditional parameters with the observed survival sea states from the non-Ro-Ro
model tests was first established, shown here in Figures 3 to 5.
Fig. 3 Regression of GZ range
s = exp(-exp (f –8h +1.3)s = exp(-exp (f –8h +1.3)
Range vs. Hs
0 5 10 15 20 25 30 35 40 45
SOLAS 2009 – Raising the Alarm 109
Fig. 4 Regression of GZ area
Fig. 5 Regression of GZ max
A review of these results led to discarding the GZ area parameter due to lack of
correlation as well as most points not falling naturally on the observed trends.
Adopting finally an existing SLF format and with Hs limited to 4m, similar to Ro-
Ro vessels, yielded the following final formulation (Figure 6):
Area vs. Hs
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
GZ Area (m-rad)
GZmax vs. Hs
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
D. Vassalos and A. Jasionowski110
Fig. 6 GZ-Based “s”-Factor for Conventional Ships (25 Cases – 3 Ship Models)
It may be noticed that 25 damage cases were used in the formulations described
in the Figure 5 and another 25 damage cases in Figure 6. Thus, the whole data set
used in project HARDER, and shown here in Figure 7, comprises 51 data points
(the one point not accounted for relates to the cruise ship).
Fig. 7 Data set of 51 measured points of survivability in the HARDER project
Based on the aforementioned developments the following points are noteworthy:
The formulation for the survival factor in the new harmonised probabilistic
regulations for damage stability, due to be adopted in January 2009, is based
solely on conventional cargo ships, Figure 6. OOOPS!!!
The SEM methodology was never adopted; hence there is no formulation for
survival factor in SOLAS 2009 that could be applied to anything other than
s = [ (GZmax / 0.12) (Range / 16) ]1/4s = [ (GZmax / 0.12) (Range / 16) ]1/4
0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 0.26 0.28 0.3 0.32 0.34 0.36 0.38 0.4 0.42 0.44
SOLAS 2009 – Raising the Alarm 111
Examining also Figure 7, leads to some additional and even more disturbing
with marginal static stability characteristics the vessel survives sea states
with Hs in excess of 4m. Several tests since then clearly demonstrate that
the current formulation underestimates the survivability for cruise ships;
as a general trend the bigger the vessel the larger the error!
On the other end of the spectrum, the formulation due to be adopted could
overestimate the survivability of Ro-Ro ships. In other words, survivability
of Ro-Ro ships can be lower than what the formulation implies.
Having made these sobering observations, there is yet another blunder to
consider: the development of the Required Index, R, for passenger ships. The
original results, shown in Figure 8, represent the A-values calculated on the basis
of the new formulation for a representative sample of ships.
Considering the arguments made above, it may justifiably be proposed that the
observed trends simply reflect the gross oversight of the fact that the s-factors
considered in the new harmonised rules had nothing to do with passenger ships.
Fig. 8 Rnew for all passenger ships
Not realising this fact, rather than questioning the method of calculation, it was
attempted instead to make sense of the results at hand, using an approach
questionable at best, which was surprisingly adopted at IMO; the tidal wave of the
new probabilistic rules managed to push an excellent framework through the door
but it seems that a lot of rumble went through with it.
All PASS ships
0 25 50 75 100 125 150 175 200 225 250 275 300
Linear (Rnew (SLF))
Linear (Rnew (HARDER))
The single result on the cruise ship, which was never used, shows that even
D. Vassalos and A. Jasionowski112
4 Emerging Evidence for Raising the Alarm on Solas 2009
4.1 SAFENVSHIP Project
In the proposed formulation for the s-factor, described above, the still water GZ
curve characteristics are calculated by assuming that the lowest-most spaces
within the damaged zone flood instantly and that static balance between vessel
mass, displaced water and floodwater describe fully the vessel stability.
To gain better understanding of the meaning of the above approximation and,
in general, of the stability of modern passenger ships, a study was undertaken,
(Jasionowski, 2005), including probabilistic damage stability calculations, numerical
time-domain simulations and scaled-model tank testing of three cruise ship designs,
considered representative of modern passenger ships. Figure 9 shows a sample
result of the study. Altogether 33 different damage cases of the three vessels were
investigated, chosen to provide a range of s-factors and distribution of damages
along the whole length of the vessel; damage openings were chosen with a level of
“high severity” in mind. Deriving from this study the following points are noteworthy:
Of the 33 cases considered, 16 were found to lead to vessel capsizing within
two hours, sometimes rapidly. Of the 16 “capsizing” cases, 10 had an s-factor
between 0 and 1, with some having s=1.0. Of the remaining 17 “surviving”
cases, 7 had an s-factor equal to zero!
The study demonstrates that traditional GZ-curve characteristics cannot adequately
describe the behaviour (and hence the destiny) of a damaged ship with the
complexity in watertight subdivision of internal spaces found on a typical cruise
vessel, and consequently, that the formulation for the s-factor in its present
form cannot meaningfully represent the average resistance of such ships to
capsize when subjected to flooding, following collision damage. Another reason
for this deficiency derives from explicitly neglecting the presence of multiple
free surfaces (MF phenomenon specific to ships with complex
watertight subdivision, which substantially weakens or indeed leads to complete
erosion of stability at any of the stages of flooding (this is a finding post-
S? effect), a
SOLAS 2009 – Raising the Alarm 113
Fig. 9 A map of “s” factors (upper plot) and the expression p.(1-s) for all damage cases
considered as a function of the damage locations. The red boxes in the lowest-most plot indicate
cases where the ship capsized within 2 hours simulation time, whereas green boxes indicate
4.2 Newbuilding Design Experience
With the introduction of the new probabilistic rules, the dilemma of prescriptive
SOLAS-minded designers must be overcome. Using an approach, known as platform
optimisation, SSRC/SaS developed a fully automated optimisation process leading
to ship designs with an acceptable level of flooding-related “risk” (i.e., compliance
with statutory requirements) that can be optimised along with cost-effectiveness,
functionality and performance objectives. Alternatively, high survivability internal
ship layouts can be developed, without deviating much from the current SOLAS
practice, thus making it easier for ship designers to relate to the proposed procedure.
This procedure has been extended to address in full the damage survivability of
new building cruise and RoPax vessels from first principles in a systematic and all
embracing way, as explained in (Vassalos 2007).
Moreover, a direct link between Index-A and flooding risk has been established,
thus facilitating a better understanding of the meaning of all the factors involved
and of their significance in affecting safety. This also led to new evidence
concerning the inconsistency and inaccuracy in the details of the new probabilistic
rules, as summarised in Figure 10.
0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 210 220 230 240 250 260
C1 - STD Opened
0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 210 220 230 240 250 260
C1 - STD Opened
D. Vassalos and A. Jasionowski114
Fig. 10 Cumulative probability distributions for time to capsize for a traditional Ro-Ro vessel
and a large new building cruise ship. Deepest draft loading conditions.
The cumulative distributions for time to capsize, for the ships shown in Figure 10,
are derived using two approaches: (a) an analytical expression based on the
formulation of the new probabilistic rules and (b) time-domain numerical simulations
the values of these distributions tend
to 1-A within a reasonable time period of several hours. Therefore, based on these
results which seem to be typical for these groups of vessels, the following
observations can be made:
For the Ro-Ro vessel, (Jasionowski 2007), approximately one in three collision
∩flooding events would lead to capsize within an hour as inferred from either
analytical solution or numerical simulation (this is not a good standard!)
Application of the new rules indicates that the cruise ship will follow a similar
fate in approximately one every 10 events whilst using first-principles time-
domain simulation tools the rate becomes approximately 1:100. This is a big
These and other similar results have alerted the industry to the extent that a new
investigation is being launched in an attempt to sort out the details in the new rules
before they are adopted in January 2009, in the belief that the probabilistic rules
constitute one of the most significant developments in the rule-making history of
the maritime profession.
Based on the results and arguments presented in this paper the following conclusions
can be drawn:
of 500 damage cases derived my means of Monte Carlo sampling that represent the
statistics of damage characteristics used in these rules. Both ships comply with the new
rules. Moreover, as explained in (Vassalos 2007),
SOLAS 2009 – Raising the Alarm 115
The index A of new probabilistic rules on ship subdivision can be interpreted as
a value that reflects the average survivability of a vessel following collision
damage and flooding in a seaway. As such, an accurate calculation of the survival
probability in these rules is of paramount importance.
This being the case, there is new evidence mounting that indicates gross errors
in the derivation of survival factors, demanding swift action by the profession
to avert “embarrassment” in global scale.
Such action is already being planned but it needs a wider participation to ensure
that results will be available on time to re-engineer a correct formulation before
the new rules are enforced.
More importantly, in the knowledge that the probabilistic rules are inconsistent
and inaccurate, due care is needed in designing new ships; in particular, the use
of time-domain simulation tools or physical model tests must be fully exploited.
The financial support of the UK Maritime and Coastguard Agency and of the
European Commission in the research undertaken is gratefully acknowledged.
Special thanks are due to Royal Caribbean International and to Carnival for
providing the inspiration needed to delve beyond trodden paths, even when it is
not called for.
HARDER (2001) Task 3.1 – Main Report of Model Tests. Report Number 3-31-D-2001-13-1
HARDER (2003) Harmonisation of Rules and Design Rationale. U Contact No. GDRB-CT-
1998-00028. Final Technical Report
Jasionowski A, (2005) Survival Criteria For Large Passenger Ships. SAFENVSHIP Project,
Final Report, Safety at Sea Ltd
Jasionowski A, York A, (2005) Investigation Into The Safety Of Ro-Ro Passenger Ships Fitted
With Long Lower Hold. UK MCA RP564, Draft Report MCRP04-RE-001-AJ, Safety at Sea
Vassalos D (2004) A risk-based approach to probabilistic damage stability. 7th
the Stab and Oper Safety of Ships, Shanghai, China
Vassalos D (2007) Risk-Based Design: Passenger Ships. SAFEDOR Mid-Term Conf, the Royal
Inst of Naval Archit, Brussels
Vassalos D, Pawlowski M, Turan O (1996) A Theoretical Investigation on the Capsizal
Resistance of Passenger/Ro-Ro Vessels and Proposal of Survival Criteria. Joint Northwest
Wendel K (1968) Subdivision of Ships. Diamond Jubilee Int Meeting, N.Y.
A.Y. Odabasi and E. U er120
method, harmonic balance method and the method of averaging to solve roll
equation with or without bias.
Another important experimental study was made by Grochowalski (1989). He
presented experimental study on the mechanism of ship capsizing in quartering
and beam waves. He used a 1:14 scaled GRP model which was fitted with
bulwarks, freeing ports, superstructure and stern ramp and also had a large
centreline skeg and a single four blade propeller. He made experiments for two
load conditions (light and full load condition). He used free model runs to give an
insight into the kinematics of ship rolling and also captive tests to identify the
composition of wave exciting hydrodynamic forces and moments. In his study, he
indicated the danger created by bulwark submergence and accumulation of water
on deck and also stability reduction on a wave crest. In his experiments he also
found that the quartering waves could be as dangerous as beam waves.
Cardo, Francescutto and Nabergoj (1981) used Bogoliubov-Krylov-Mitropolsky
asymptotic method to determine first order steady state solution of roll equation
without bias in ultra harmonic and sub harmonic resonance regions. (Nayfeh and
Khdeir 1986) used method of multiple scales to determine a second order
approximate solution in the main resonance region for the nonlinear harmonic
response of biased ships in regular beam waves.
Nayfeh (1979) obtained the first and second order solutions of Equation (1) in
main, sub harmonic and super harmonic resonance regions with the method of
multiple scales. This equation is similar to the equation used in this article.
However, the solutions obtained in this article are different from Nayfeh’s
solutions due to transformations made in Equation 1.
Jiang et al. (1996) investigated the influence of bias on capsizing both in terms
of the reduction of the safe basin and in terms of phase space flux. Their
conclusions were based on Melnikov analyses and also supported by extensive
Macmaster and Thompson (1994) examined capsize of ships excited by the
naturally propagating wavefront created by a laboratory wave maker that is
suddenly switched on. They used transient capsize diagrams and showed the
importance of bias. Cotton et al. (2000) used the same equation shown below to
examine the sensitivity of capsize to a symmetry breaking bias in beam seas and
stated that the case α=0 to be a sensible model of ship roll motion.
Spyrou et al. (1996) used Melnikov method to obtain analytical formulas which
can characterize capsizability of ships taking into account the presence of some
roll bias. They have tested these formulas against numerical simulations targeting
Effect of Initial Bias on the Roll Response and Stability of Ships in Beam Seas 121
the process of safe basin erosion. Their mathematical model for rolling motion
was also Equation (2).
The aim in the first part of this study is to present available methods in the
solution of roll equation (method of multiple scales, Bogoliubov-Mitropolsky
asymptotic method and numerical methods), compare the obtained results and also
show the sensitivity of solution to the effect of equation parameters. It was also
observed from the solutions that neglecting the effect of cubic coefficient of
restoring term is not sensible.
In the second part of this study, stable solution bounds of the biased roll equation
were obtained for different equation parameters and the effect of these parameters
on the size and symmetry of the bounds were also examined.
In the concluding part of the paper it is stated that boundedness of solution, in
the sense of Lyapunov, constitute a sound basis for the intact stability research.
2 Methodology of the Investigation
2.1 Equation of Motion and Preliminary Considerations
In this paper, while the rolling motion of the ship was being modelled, interactions
between rolling and other modes of motion have been ignored and the ship was
considered to have a rigid body and sea water was ideal and incompressible.
Under these assumptions, rolling motion of a ship was written as in Equation (3).
windR MtEMDI , (3)
θ : Rolling angle with respect to calm sea surface
: Roll angular velocity
I : Mass moment of inertia including the added mass moment of inertia.
MR(θ) : Restoring moment. Its coefficients can be obtained either from a
computed righting arm curve by curve fitting or from the characteristics of the
righting arm curve, such as GM, GZmax, area under the righting arm curve, by
constrained curve fitting.
D(θ, ) : Damping moment. In this study, the linear damping moment
associated with wave produced by the hull was only taken care of.
E(t) : Roll exciting moment, due to external force. When the wave elevation is
induced by regular transversal waves, the exciting term may be written
as tCosEtE w where Ew is the amplitude and Ω the angular frequency
Mwind : Wind Moment
A.Y. Odabasi and E. Uçer122
In this study, for the sake of simplicity, the following approximations have
been used 3
21)( kkM R and sD ),( where Δ is the displacement
and s is the linear damping coefficient
Under the above assumptions Equation (3) can be written as
windw MtCosEkksI 3
Dividing both sides of the equation by the mass moment of inertia, the
expression given in Equation (5) is obtained.
This type of roll equation was also examined from other author’s earlier
If θ = x + θs transformation is made, Equation (5) can be re-written as
0 c3c , 2s1 c3d , 22 cd ,
In the following parts of this paper Equation (6) were used as simplified rolling
2.2 Main, Sub and Super Harmonic Resonance Investigation
In this study, wave excitation was assumed to be approximately half, twice and
equal to the natural frequency to make an investigation in sub, super and main
harmonic resonance regions. Then biased roll equation was solved in these regions
by using the method of multiple scales, Bogoliubov-Mitropolsky asymptotic
method and numerical methods. The effects of parametric resonance have not
Finally, the stable solution bounds of roll motion were obtained for different
bias angles, linear damping coefficients, and amplitudes and frequencies of wave
excitation in the sub, super and main harmonic resonance regions by using
Mathematica NDSolve routine and the Fortran program written by the authors.
The necessary data used in numerical computation were shown in Table 1.
Effect of Initial Bias on the Roll Response and Stability of Ships in Beam Seas 123
Table 1 The data used in numerical calculations.
θs (rad.) ω0 d1 d2 ˆ F
0.005 0.610257 0.001975
0.100 0.607020 0.039499
0.200 0.597179 0.078998
0.131664 0.04,0.07,0.10,0.13 0.08,0.10,0.12
2.3 Solution Methods
2.3.1 Method of Multiple Scales
Method of multiple scales assumes that the expansion representing the response to
be a function of multiple independent variables, or scales instead of a single
variable. In this process, first the new independent variables are introduced.
n For n=0,1,2…… where ε is a small parameter
Time derivatives then can be re-written as
To illustrate the implementation of the method first order approximate solution
of Equation (6) in the main resonance region is provided below.
In the main resonance region where the excitation frequency is near the natural
frequency of the ship ( 0 ), the linear damping coefficient, quadratic
and cubic restoring moment coefficients and finally amplitude of the excitation
frequency were considered to have order of є. Thus, Equation (6) was written in as;
If the expressions shown in Equation (7a) and (7b) are substituted into
Equation (8), Equation (9) is obtained.
A.Y. Odabasi and E. Uçer124
The solution of above equation can be represented by an expansion as
If the expansion shown in Equation (10) is substituted into Equation (9) and
coefficients of like powers of ε are equated, we obtain the following equations.
The solution of Equation (11a) is
, A and A are the
only functions of T1 as we are searching for first approximate solution.
If x0 solution shown in Equation (12) is substituted into Equation (11b), a new
differential equation is obtained. The right hand side of this equation includes
secular terms. These secular terms are eliminated by equating the coefficients of
Effect of Initial Bias on the Roll Response and Stability of Ships in Beam Seas 125
If A and A are substituted into the above equations, after some transformations
which will not be shown here due to limited space, Equation (14a) and (14b) are
where (΄) symbol represents derivative taken with respect to T1.
Defining 1T and , the above equation becomes
The above equations can also be written in the form of the equations below.
The first order approximate solution of biased roll equation is written as:
where a and γ can be found from Equation (16a) and (16b).
This method had been used to obtain an approximate solution of roll equation
from many scientist and researchers (Nayfeh 1979, Nayfeh and Khdeir 1986).
A.Y. Odabasi and E. Uçer126
Thus, second order approximate solution and also solution in sub and super
harmonic resonance regions weren’t explained here.
Due to decrement in linear damping coefficient or increment in initial bias
angle and amplitude of wave excitation, the magnitude of coefficient “a” in the
steady state solution increases in main, sub and super harmonic resonance regions.
As can be seen from Fig. 1 and 2, when the wave excitation frequency is
approximately half of the natural frequency, coefficient “a” takes its value in
negative small values of σ (deviation between excitation frequency and natural
frequency). The jump phenomenon was also observed when linear damping
coefficient µ was equal to 0.04 and amplitude of the wave excitation (F) taken
value higher or equal to 0.12.
Fig. 1 a-σ graph The effect of initial bias angle
Fig. 2 a-σ graph The effect of amplitude of excitation
As can be seen from Fig. 3, if the wave excitation frequency is approximately
twice of the natural frequency, coefficient a in the first and second approximate
solution of rolling equation goes to directly to zero without any oscillation when
time goes to infinity. If the linear damping coefficient is increased, coefficient “a”
goes to zero faster.
m=0.04 F=0.12 d2
-0.15 -0.12 -0.09 -0.06 -0.03 0
0.03 0.06 0.09 0.12 0.15
w0=0.597179 d1=0.078998 d2=0.131664 m=0.04
-0.15 -0.12 -0.09 -0.06 -0.03 0
0.03 0.06 0.09 0.12 0.15
Effect of Initial Bias on the Roll Response and Stability of Ships in Beam Seas 127
Fig. 3 a-t graph (µ=0.04, F=0.12, σ =-0.05, 02 , θs=0.2 rad.)
As can be seen from the Fig. 4 and Fig. 5, while the other parameters of the
equation are fixed if we take σ =-0.1 instead of σ =0.1 in the main resonance
region, the coefficient “a” gets much larger values (nearly twice of the former).
Fig. 4 a-t graph (µ=0.04, F=0.12, σ =0.1, 0 , θs=0.2 rad.)
Fig. 5 a-t graph (µ=0.04, F=0.12, σ =-0.1, 0 , θs=0.2 rad.)
2.3.2 Bogoliubov-Mitropolsky Asymptotic Method
While examining the main resonance region by Bogoliubov-Mitropolsky asymptotic
+εΔ, linear damping coefficient, quadratic and cubic term of restoring
moment and finally amplitude of excitation force have been assumed to be order
of ε. Under these assumptions biased roll equation were written as
The above equations can be written in more efficient form as
where ψ = Ω t+β
The solution is assumed to have the form (Bogoliubov and Mitropolsky 1961),
...........,,1 tauCosax (20)
and a and β in the above equation must satisfy the following expressions
To obtain first order approximate solution of biased roll equation, only first
terms were taken in the right hand side of the Equation (20), (21a) and (21b).
If the first term of expression shown in Equation (20) is put into the left hand
side of the Equation (19), Equation (22) is obtained. If it is put into right hand side
of the Equation (19), Equation (23) is obtained.
A.Y. Odabasi and E. U erç
Effect of Initial Bias on the Roll Response and Stability of Ships in Beam Seas 129
If the right hand sides of the Equation (22) to Equation (23) are equated and the
terms which have Cosψ and Sinψ are brought together, the following expressions
are obtained, and with the help of other terms, u1 solution can be found as:
SinfaA 12 (24a)
If A1 and B1 are substituted into the Equation (21a) and Equation (21b), the
following expressions are obtained:
The first order approximate solution of biased roll equation is the given as:
where a and β coefficients are determined from Equation (26a) and (26b).
In sub and super harmonic resonance regions instead of using Equation (18),
Equation (28) is used. Equation (28) is obtained by making tCos
transformation in the Equation (6).
2.3.3 Numerical Methods
Biased roll equation was solved by Mathematica 5-NDSolve with 50000 steps in
sub, super and main harmonic resonance regions for different initial conditions and
parameters of the equation. The results also controlled with Fortran IVPAG routine.
Stable bounds for different equation parameters were obtained by Mathematica
5-NDSolve and the program (“Rk.f90”) written in Fortran 90 language by the
authors. “Rk.f90” uses Runge Kutta Fourth order method with time step 0.001 second
to solve biased roll equation and finds out the initial conditions cause to solution
goes to infinity when time goes to infinity. Solutions of “Rk.f90” program are the
same as the solutions obtained by Mathematica 5-NDSolve.
3 Compliance of Perturbation & Asymptotic Results
with Numerical Solution
The solutions obtained by using method of multiple scales (MSM) shown in Fig.
6-11 in and Bogoliubov-Mitropolsky asymptotic method solutions shown in
solutions obtained by using MSM and asymptotic method in sub, super and main
harmonic resonance regions, have better compliance with numerical solution rather
than the first order approximate solutions. Second order approximate solutions
only have difference with numerical solutions in transient region. However, the
first order approximate solutions also have differences in steady state region.
Although this good compliance of perturbation and asymptotic results with
numerical solution in stable bounds, perturbation and asymptotic methods could
not explain unstable cases.
Fig. 6 Comparison of second order approximate MSM and numeric solution when 0
ω=0.597179 d=0.078998 σ=0.1 F=0.12
100 150 200 250 300 350 400
Fig. 12-17. As can be seen from the figures, the second order approximate
A.Y. Odabasi and E. U erç
Effect of Initial Bias on the Roll Response and Stability of Ships in Beam Seas 131
Fig. 7 Comparison of second order approximate MSM and numeric solution when 02
Fig. 8 Comparison of second order approximate MSM and numeric solution when 20
Fig. 9 Comparison of first order approximate MSM and numeric solution when 0
MSM O(ε^2)x(0)=0 dx/dt=0.56 ω=0.59718 Ω=1.29436 d=0.078998
50 100 150 200
250 300 350 400
0 50 100 150 200 250 300
x(0)=0.08045 dx/dt=0.21482 ω=0.59718 Ω=0.28359 d=0.078998
x(0)=0 dx/dt=0.459 ω=0.59718 Ω=0.69718 d=0.078998
Fig. 10 Comparison of first order approximate MSM and numeric solution when 02
Fig. 11 Comparison of first order approximate MSM and numeric solution when 20
Fig. 12 Comparison of second order approximate asymptotic and numeric solution when 0
0 50 100 150
x(0)=0 dx/dt=0.56 ω=0.59718 Ω=1.29436 d=0.078998
300 350 400
x(0)=0.1 dx/dt=0.2 ω=0.59718 Ω=0.2835 d=0.078998
0 50 100 150 200 250 300 350
0 50 100 150 200 250 300 350 400
μ=0.1,F=0.12, Ω=0.697179, ω=0.597179 x(0)=0 dx/dt=0.535
A.Y. Odabasi and E. U erç
Effect of Initial Bias on the Roll Response and Stability of Ships in Beam Seas 133
Fig. 13 Comparison of second order approximate asymptotic and numeric solution when 02
Fig. 14 Comparison of second order approximate asymptotic and numeric
solution when 20
Fig. 15 Comparison of first order approximate asymptotic and numeric solution when 0
0 50 100
μ=0.04,F=0.12, Ω=1.294358,ω=0.597179 x(0)=0 dx/dt=0.56 Bias=0.2 rad.
150 200 250 300 350
0 50 100
Ω=0.285755 F=0.12 x(0)=0.25 y(0)=0.268 Bias=0.2 rad.
150 200 250 300 350 400
0 50 100 150
μ=0.1,F=0.12, Ω=0.697179,ω=0.597179 x(0)=0 dx/dt=0.535
200 250 300 350 400
Fig. 16 Comparison of first order approximate asymptotic and numeric
solution when 02
Fig. 17 Comparison of first order approximate asymptotic and numeric solution when 20
4 Existence of Bounds for Initial Conditions and its Dependence
As can be seen from Table 2, in the main resonance region, when σ = -0.1 (σ:
Deviation of excitation frequency from natural frequency) and |F| ≥ 0.1 (F:
Amplitude of wave excitation), there is no stable bound. If amplitude of wave
excitation is decreased to 0.08, there are stable bounds for very large linear
damping coefficients. The effect of sign and magnitude of σ can easily be seen
from Table 3 When σ = 0.1, there are stable bounds even for |F|=0.12.
The increment of the amplitude and initial bias angle or decrement of linear
damping coefficient causes the narrowing of stable solution bounds.
0 50 100 150 200 250 300 350 400
x(0)=0 y(0)=0.56 ω=0.597179 Ω=1.29436 Bias=0.2 rad.
0 50 100
150 200 250 300
ω=0.597179 Ω=0.285755 Bias=0.2 rad.
A.Y. Odabasi and E. U erç
Effect of Initial Bias on the Roll Response and Stability of Ships in Beam Seas 135
Table 2 Stable bounds in main resonance region for σ = -0.1
Initial Bias Angle
Initial Bias Angle
Initial Bias Angle
|F|=0.12 - - -
|F|=0.10 - - -
for µ ≥0.13
for µ ≥0.1
Table 3 Stable bounds in main resonance region for σ = 0.1
Initial Bias Angle
Initial Bias Angle
Initial Bias Angle
For µ > 0.10
For µ ≥ 0.07
For µ ≥ 0.04
For µ > 0.07
For µ > 0.04
For µ ≥ 0.04
For µ > 0.04
for µ ≥0.04
for µ ≥0.04
There is another factor which effects size of stable bounds. That is the phase
angle which will be given to excitation force.
The stable bound obtained for μ=0.1, d1=0.039499, d2=0.131664, ω0=0.60702,
F=±0.12 and σ = 0.1 shown in Table 4.
Table 4 Boundaries of stable bound for μ=0.1 |F|=0.12 θs=0.1 rad. σ =0.1
F=0.12 µ=0.1 F=-0.12 µ=0.1
x (rad.) x (rad/s) x (rad.) x (rad/s)
0.000 0.634 0.000 0.844
0.200 0.605 -0.200 0.854
0.400 0.558 -0.400 0.842
0.600 0.491 -0.600 0.806
0.800 0.404 -0.800 0.742
1.000 0.298 -1.000 0.636
1.200 0.170 -1.065 0.587
1.427 0.000 -1.065 0.000
Let’s take a point which is inside the stable bound shown in Table 4. For example,
0)0(x and 634.0)0(x . If this point is taken as the initial condition, some
points on the solution orbit in x- x phase space are out of the stable bound (see
Table 5 and Fig. 18).
Table 5 Some points out of the stable bound
t (sn.) x (rad.) x (rad/s)
19.2 -0.033582 0.764827
19.3 0.042842 0.763051
19.4 0.118912 0.757785
19.5 0.194286 0.749140
19.6 0.268632 0.737270
Fig. 18 Solution orbit in x- x phase space for initial condition x(0)=0 x (0)=0.634
As it can be seen from Fig. 19, if x(0)=0.194286 x (0)=0.749140 are taken as
initial condition and solve the biased roll equation, the solution is unstable (when
(when time goes to infinity, solution also goes to infinity).
Fig. 19 Solution orbit in x- x phase space for initial condition x(0)=0.194286 x (0)=0.749140
The solution is stable for x(19.5)=0.194286 and y(19.5)=0.74914 initial condition,
1 2 3 4
A.Y. Odabasi and E. U erç
Effect of Initial Bias on the Roll Response and Stability of Ships in Beam Seas 137
Fig. 20 Solution orbit in x- x phase space for initial condition x(19.5)=0.19428 x (19.5)=0.74914
The solution was showed unstable for x(0)=0.194286 x (0)=0.74914 initial
condition. If δ=1.220519 rad. phase angle is put into the excitation force and the
biased roll equation is solved for x(0)=0.194286 x (0)=0.749140 initial condition
again, the solution is stable (see Fig. 21). Hence, the effect of phase angle which
will be given the excitation force were proved.
Fig. 21 Solution orbit in x- x phase space for initial condition x(0)=0.194286
x (0)=0.74914 δ=1.220519 rad.
In sub and super harmonic resonance regions, the increment of the amplitude
and initial bias angle or decrement of linear damping coefficient causes to narrow of
stable solution bounds and also the phase angle which will be given to excitation
force effects size of stable bounds like main resonance region.
Some of the stable bounds obtained when the excitation frequency is
approximately half of the natural frequency are shown in Fig. 22-24. As can be seen
from these figures, the symmetry of the stable bounds corrupts after the initial bias
angle increases. Fig. 25 shows sample of orbit for frequency equal to half the natural
Fig. 22 Stability bound for µ=0.04 θs=0.20 rad. |F|=0.12 σ =-0.078
Fig. 23 Stability bound for µ=0.04 θs=0.10 rad. |F|=0.12 σ =-0.04
Fig. 24 Stability bound for µ=0.04 θs=0.005 rad. |F|=0.12 σ =-0.03
A.Y. Odabasi and E. U erç
Effect of Initial Bias on the Roll Response and Stability of Ships in Beam Seas 139
Fig. 25 Sample solution orbit in x- x phase space for 20
5 Concluding Remarks
Due to decrement in linear damping coefficient or increment in initial bias angle
and amplitude of wave excitation, the magnitude of coefficient a in the steady
state solution increases in main, sub and super harmonic resonance regions.
When the wave excitation frequency is approximately half of the natural
frequency, coefficient “a” takes its maximum value in negative small values of σ
(Deviation between excitation and natural frequency) and also there is jump
phenomenon when linear damping coefficient is equal to 0.04 and amplitude of
the excitation is higher or equal to 0.12.
If the wave excitation frequency is approximately twice of the natural
frequency, coefficient “a” in the first and second approximate solution of rolling
equation goes to directly to zero without any oscillation when time goes to
infinity. If the linear damping coefficient, is increased coefficient “a” goes to zero
While the other parameters of the equation are fixed if σ =-0.1 is taken instead
of σ =0.1 in the main resonance region, the coefficient “a” gets much bigger
values (nearly twice of the former).
The second order approximate solutions of asymptotic method and the method
of multiple scales have better compliance with numerical solutions in main, sub
and super harmonic resonance rather than first order approximate solution of the
asymptotic method and the method of multiple scales. The second order approximate
solutions only have differences with numerical solutions in transient part of the
solutions. However, the first approximate solutions also have differences with
numerical solutions in steady state part of the solutions. Although this good
compliance of perturbation and asymptotic results with numerical solution in
stable bounds, perturbation and asymptotic methods could not explain unstable
The magnitude of linear damping coefficient, initial bias angle, amplitude and
frequency of wave excitation and the phase angle which are given to excitation
force effects the size of stable solution bounds in main, sub and super harmonic
resonance regions. The increment of the amplitude and initial bias angle or
decrement of linear damping coefficient causes the narrowing of stable solution
bounds. The phase angle of the excitation force can make an unstable initial condition
stable initial condition. Thus, importance of this parameter shouldn’t be neglected.
The initial bias angle is also affected on the symmetry of the stable solution
bounds. The increment of initial bias angle causes to increase asymmetries.
Bogoliubov N N, Mitropolsky Y A (1961) Asymptotic methods in the theory of nonlinear
oscillations. Intersciense NY
Cardo A, Francescutto A et al. (1981) Ultra harmonics and subharmonics in the rolling motion of
a ship: steady state solution. Int Shipbuild Prog 28:234-251
Cotton B, Bishop S R et al. (2000) Sensitivity of capsize to a symmetry breaking bias. In:
Vassalos D, Hamamato M et al. (eds). Contemp Ideas on Ship Stab. Elsevier Oxford
Grochowalski S (1989) Investigation into the physics of ship capsizing by combined captive and
free running model tests. Trans of Soc of Nav Archit and Mar Eng 97:169-212
Jiang C, Troesch A W et al. (1996) Highly nonlinear rolling motion of biased ships in random
beam seas. J of Ship Res 40 2:125-135
Macmaster A G, Thompson J M T (1994) Wave tank testing and the capsizability of hulls. Proc
of the R Soc London A 446:217-232
Nayfeh A H (1979) Nonlinear oscillations. John Wiley NY
Nayfeh A H, Khdeir A A (1986) Nonlinear rolling biased ships in regular beam waves. Int
Shipbuild Prog 33:84-93
Spyrou K J, Cotton B et al. (1996) Analytical expressions of capsize boundary for a ship with
roll bias in beam waves. J of Ship Res 46 3:125-135
Wright J H G, Marshfield W B (1980) Ship roll response and capsize behavior in beam seas.
Trans of RINA 122:129-149
A.Y. Odabasi and E. U erç
H. Nowacki and L.D. Ferreiro142
theory, who quite independently and almost simultaneously arrived at their
landmark results for hydrostatic stability criteria. BOUGUER developed the theory
and introduced the terminology of the metacenter and the metacentric curve.
EULER defined the criterion of the initial restoring moment, which for hydrostatic
stability amounts to an equivalent concept. The full implementation of
computational methods for evaluating these criteria and their acceptance by
practitioners took even several decades longer. Paradoxically this chain of events
raises two nearly contradictory questions, which this paper will address:
Why did it take so long for these formalized, quantitative criteria for the
stability of ships of arbitrary shape to be pronounced?
When at last the discoveries were made, why then did two independent, but
equivalent solutions suddenly appear almost at the same time?
The knowledge required to evaluate the stability of a ship rests on many
concepts and requires many autonomous discoveries to be made and to be brought
into concerted application. These include:
The idea of conceptual experiments in dealing with mechanical systems.
The abstraction of thinking in terms of resulting forces and moments of weight
and buoyancy (“lumped effects”) substituted for distributed effects of gravity
The axiom of force equilibrium, here between weight and buoyancy force
(Principle of ARCHIMEDES).
The axiom of moment equilibrium in a system at rest.
A definition and a test for system stability.
A method for finding the combined weight and center of gravity (CG) for
several weight components.
A principle for finding the resultant buoyancy force and its line of action
(through the center of buoyancy, CB).
A method for calculating volumes and their centroids, first for simple solids,
then for arbitrary ship shapes.
An analytical formulation of a stability criterion (for infinitesimal and for finite
angles of inclination).
Although the physical principles of hydrostatic stability were already established
by ARCHIMEDES, it took a long time before the analytical formulation of
stability criteria could be pronounced for the general case of ships, essentially by
means of calculus, and before numerical evaluations became feasible. BOUGUER
and EULER were the first to find an analytical criterion for initial stability,
BOUGUER in terms of the metacenter. EULER in terms of the initial restoring
moment. BOUGUER went beyond this in several practical aspects. The intriguing
question remains: How did two scientific minds work independently to come to
rather equivalent results? What were their sources, their background, their
approach, their logic, their justification and verification? Which were their unique
original thoughts and how did they differ?
Historical Roots of the Theory of Hydrostatic Stability of Ships 143
To answer these questions it is not sufficient to look only at their final results
and conclusions, but it is necessary to examine more closely the methodical
approaches taken to the ship stability issue and to compare the trains of thought by
which the individual authors arrived at their results. In this article we will review
the developments that led to this historical stage when modern hydrostatic stability
theory was founded.
ARCHIMEDES of Syracuse (ca. 287-212 B.C.), the eminent mathematician,
mechanicist and engineering scientist in antiquity, is also the founder of ship
hydrostatics and hydrostatic stability, which he established as scientific subjects
on an axiomatic basis. ARCHIMEDES was brought up in the early Hellenistic era
in the tradition of Greek philosophy, logical rigor and fundamental geometric
thought. ARCHIMEDES was well educated in these subjects in Syracuse and very
probably also spent an extended study period in Alexandria, the evolving Hellenistic
center of science, at the Mouseion (founded in 286 B.C.). There he met many
leading contemporary scientists, e.g., DOSITHEOS, ARISTARCHOS and
ERATOSTHENES, with whom he maintained lifelong friendships and scientific
correspondence. Steeped in this tradition of Greek geometry and mechanics,
ARCHIMEDES learned how to excel in the art of deductive proofs from first
premises usually based on conceptual models, i.e., models of thought, by which
physical reality was idealized. But ARCHIMEDES was also unique, as many
legends on his engineering accomplishments tell us, in applying practical observation
to test his scientific hypotheses and to develop engineering applications, although
these achievements are not mentioned in his own written work.
We are fortunate that many, though not all, of ARCHIMEDES’ treatises are
preserved, all derived from a few handwritten copies made in the Byzantine
Empire during the 9th
centuries and transmitted to posterity on circuitous
routes to resurface essentially through the ARCHIMEDES revival during the
Renaissance. Luckily, one Greek manuscript, which was later lost, had been
translated into Latin by the Dominican monk Willem van MOERBEKE. This
translation, which was published in 1269 and was later named Codex B, also
contained ARCHIMEDES’ treatise “On Floating Bodies”, Books I and II. It
became the only accessible reference to ARCHIMEDES’ work on hydrostatics for
many centuries until in 1906 most surprisingly an old 10th
century palimpsest was
rediscovered in a Greek monastery in Constantinople by J. HEIBERG (Heiberg
1906-07). This palimpsest, called Codex C by HEIBERG, under the writing of a
century monk of a Greek prayerbook had retained significant traces of the
H. Nowacki and L.D. Ferreiro144
rinsed off text of a Greek manuscript from ARCHIMEDES including the Greek
version of “On Floating Bodies”. ARCHIMEDES’ texts were soon reconstructed
from this source, transcribed and translated into modern languages (Archimedes
2002). Meanwhile this palimpsest, which had disappeared in private possession in
the aftermath of the Greek-Turkish war in 1920-22, has turned up again recently
and is under new scientific evaluation at the Walters Art Museum in Baltimore
(Netz and Noel 2007). It is on these sources primarily that we can base a reliable
evaluation of ARCHIMEDES’ contributions to ship hydrostatics today.
ARCHIMEDES preceded his work on hydrostatics by a number of other
treatises establishing certain axioms of mechanics:
2.1.1 The Law of the Lever:
In his treatise “The Equilibrium of Planes” ARCHIMEDES first deals with the
equilibrium of moments about a fulcrum in a lever system. Although he claims to
deduce this principle from geometric reasoning alone, it is actually understood
today that the law of the lever is equivalent to the axiom of moment equilibrium in
mechanics. Second, he introduces the concept of “centroids” of quantities (areas,
volumes, weights) into which the quantities can be “lumped” as concentrated effects
so that moment equilibrium is retained. Third, he proposes a method for finding
the “compound centroid” of a system of components, e.g., a center of gravity. Finally,
he proves the critical “centroid shift theorem” i.e., a rule for the shift of the system
centroid when some quantity is added to, removed from or shifted within the
system. All of these concepts and results are essential physical principles as
prerequisites for his work on hydrostatics.
In his treatise “The Quadrature of the Parabola” ARCHIMEDES illustrates by the
example of the parabola how the Greeks determined areas and volumes of
elementary shapes without the availability of calculus. Here he uses a method,
well-known since EUDOXUS (Boyer 1939, 1949), based on an inscribed polygonal
approximant which is continually refined by interval halving until under the given
premises it converges to the given curve within a specified error tolerance. This
type of deduction was later called “method of exhaustion”. Thus the quadrature
problem is reduced to that of evaluating the sum of a finite, truncated or sometimes
even infinite series of approximations. This method of quadrature generally is not
equivalent to calculus for lack of a limiting process to infinitesimal step width, but
has nonetheless inspired many future developments until this day.
Historical Roots of the Theory of Hydrostatic Stability of Ships 145
2.1.3 The Method of Mechanical Theorems:
This famous treatise, which was actually also discovered in the palimpsest of 1906,
explains how ARCHIMEDES used a reasoning based on principles of mechanics
(like moment equilibrium of volume quantities) in deriving geometrical results (like
volume centroids). He regarded such findings as propositions for later rigorous
deduction by strictly geometrical proofs.
2.1.4 On Floating Bodies:
In this treatise ARCHIMEDES makes use of all these prior results and proceeds to
lay the foundations for ship hydrostatics and stability in the following steps
(Archimedes 2002, Czwallina-Allenstein 1996):
Book I of this treatise begins with Postulate 1 describing the properties of a
fluid at rest axiomatically.
“Let it be supposed that the fluid is of such character that, its parts lying
evenly (N.B.: at the same level) and being continuous (N.B.: coherent), that part
which is thrust the less is driven along by that which is thrust the more and
that each of its parts is thrust by the fluid which is above it in a perpendicular
direction, unless the fluid is constrained by a vessel or anything else”.
Although ARCHIMEDES does not use the word “pressure” and the Greeks did
not know that concept in antiquity, he does infer that parts under more pressure
would drive parts under less pressure so that a fluid cannot be at rest unless the
pressure is uniform at a given depth, while the weight of a vertical column of fluid
rests on the parts below it. From these very simple axiomatic premises, which do not
permit evaluating the local pressure anywhere in the fluid, ARCHIMEDES is able
to derive the principles of hydrostatic equilibrium and stability of floating bodies.
This is achieved by considering the equilibrium of the resultant buoyancy and
gravity forces and of their moments.
ARCHIMEDES’ famous Principle of Hydrostatics is stated in Book I,
“Any solid lighter than a fluid will, if placed in the fluid, be so far
immersed that the weight of the solid will be equal to the weight of the fluid
displaced” (Archimedes 2002).
The proof of this law, usually pronounced today as Δ = γ V, is brilliantly brief
and conclusive. In all brevity it rests on the argument that in equilibrium the solid
is at rest in a fluid at rest, thus if the body is removed from the fluid and the cavity
left by its underwater volume is filled with fluid matter, then the fluid can only
remain at rest if the replacing fluid volume weighs as much as the solid, else the
fluid would not remain in equilibrium and hence at rest.
H. Nowacki and L.D. Ferreiro146
In Book II, mainly Proposition 2, ARCHIMEDES deals with the stability of
hydrostatic equilibrium by treating the special case of a solid of simple shape, viz.,
a segment of a paraboloid of revolution of homogeneous material whose specific
gravity is less than that of the fluid on whose top it floats. In equilibrium it floats
in an upright condition. The stability is tested by inclining the solid by a finite
angle to the vertical, but so that the base of the segment is not immersed. The
equilibrium is defined as stable if the solid in the inclined position has a restoring
moment tending to restore it to the upright condition.
For the homogeneous solid this stability criterion is readily evaluated
geometrically by examining the lever arm between the buoyancy and the gravity
force resultants (Fig. 1). The buoyancy force acts through the centroid of the
underwater volume B, which ARCHIMEDES finds for the inclined paraboloid from
theorems proven earlier. The gravity force or weight acts through the Center of
Gravity R of the homogeneous solid. Our conventional righting arm, the projection of
BR on the horizontal, is positive. Instead of using this stability measure,
ARCHIMEDES takes a shortcut for this homogeneous solid by splitting off and
removing the weight of the submerged part of the solid Δ1 and the corresponding
equal share of the buoyancy force, which have no moment about B because they
both act through B. Thus only the weight of the abovewater section of the solid Δ2,
acting through C, and the equal and opposite buoyancy force increment, acting
through B, are taken into account. The centroid C is found from B and R by
applying the centroid shift theorem when removing the underwater part from the
system. This yields a positive “incremental righting arm” for the force couple of
Δ2, acting through B and C respectively. The restoring moment is thus positive
and the solid will return to the upright position.
This application of the hydrostatic stability criterion by ARCHIMEDES is
limited to the special case of a homogeneous solid of simple parabolical shape. It
demonstrates the physical principles of the hydrostatic stability problem for a
finite angle of inclination. It does not extend to floating bodies of arbitrary shape
and of non-homogeneous weight distribution, hence actual ships. Still the
foundations were laid to enable others much later to treat the generalized case of
the ship on the same fundamental grounds.
A more detailed appraisal of ARCHIMEDES’ contributions to the hydrostatic
stability of floating systems and a thorough historical account is given by
NOWACKI (Nowacki 2002).
Historical Roots of the Theory of Hydrostatic Stability of Ships 147
Fig. 1 Restoring moments, righting arms for inclined homogeneous paraboloid, based on
ARCHIMEDES’ “On Floating Bodies”, Book II (Archimedes 2002) (figure from (Nowacki
2.2 Stevin and Pascal
In the beginning of the modern era of science the manuscripts of ARCHIMEDES
had been rediscovered by several humanists during the Renaissance and were
made accessible in print after 1500 by several editions in Greek, Latin and later by
translations into modern languages (Archimedes 2002, Czwallina-Allenstein 1996,
Simon STEVIN (born 1548 in Bruges, died 1620 in Leyden), the celebrated
Flemish/Dutch mechanicist and hydraulic engineer, who knew and admired
ARCHIMEDES’ works, was probably the first modern scientist who resumed and
resurrected the study of hydrostatics and applied it to hydraulics, but also to ships.
In his important work “De Beghinselen des Waterwichts” (Stevin 1955), first
published in Dutch in 1586 and translated into Latin by Willibrord Snellius in
1605, he deals with the principles of hydrostatics and hydraulics. He adopts the
idea of “specific gravity” of a fluid from ARCHIMEDES and introduces the concept
of “a hydrostatic pressure distribution”, as we would call it today, proportional to
the weight of a water prism reaching down to the depth in question. This enables
him to calculate water loads on walls in a fluid, an important foundation in
hydraulic engineering. He also rederives ARCHIMEDES’ Principle of Hydrostatics.
However when he proceeds to examine the hydrostatic stability of a ship in his
later note “Van de Vlietende Topswaerheyt” (Stevin 1586), he correctly reconfirms
that for equilibrium buoyancy and weight force resultants must act in the same
H. Nowacki and L.D. Ferreiro148
vertical line through the centers of buoyancy (CB) and gravity (CG). But he
erroneously concludes that for stability the CB must always lie above the CG. This
error occurred because – unlike ARCHIMEDES - he neglected the centroid shifts
resulting from the volume displacement from the emerging to the immersing side in
heel. Despite this flaw in an application STEVIN deserves high recognition for
founding modern hydrostatics on the concept of hydrostatic pressure.
Blaise PASCAL (born 1623 in Clermont, died 1662 in Paris) is also counted
among the founders of modern hydrostatics. He was familiar with ARCHIMEDES’
and probably with STEVIN’s work. In his “Traité de l’ équilibre des liqueurs”,
Paris (1663), he arrives at similar conclusions and justifications regarding the
fundamentals of hydrostatics in a fluid as STEVIN did, though he did not deal with
ships. The assertion and experimental verification of hydrostatic laws also being
applicable to air belong to his original contributions to this subject.
Christiaan HUYGENS (born 1629 in The Hague, died 1695 in The Hague), the
eminent Dutch physicist, at the youthful age of 21 made an excursion into
hydrostatic stability, which is not well known. In 1650 he wrote a three volume
treatise “De iis quae liquido supernatant” (Huygens 1967a), in which he applies
the methodology of ARCHIMEDES to the stability of floating homogeneous
solids of simple shape, reconfirming ARCHIMEDES’ results and extending the
applications to floating cones, parallelepipeds, cylinders etc., at the same time
studying the stability of these solids through a full circle of rotation. He never
published this work in his lifetime because he felt it was incomplete or “of small
usefulness if any” or in any case not sufficiently original in comparison to
ARCHIMEDES. He wanted the manuscript to be burnt, but it was found in his
legacy and at last published in 1908. HUYGENS must be admired for his deep
insights into ARCHIMEDES’ fundamentals of hydrostatics and for his own creative
extensions. He recognized e.g. that for homogeneous, prismatic solids their specific
weight and their aspect ratio are the essential parameters of hydrostatic stability.
In his derivations he used a formulation based on virtual work as a principle of
These examples of famous physicists between 1500 and 1700 working in
hydrostatics and on hydrostatic stability of floating solids and ships demonstrate
that the foundations inherited from ARCHIMEDES were understood by certain
specialists, but had not yet been extended or applied to the design and stability
evaluation of actual ships. Although the physical principles of stability were
understood, the practical evaluation of volumes, volume centroids, weights and
centers of gravity for floating bodies of arbitrary shape and non-homogeneous
weight distribution de facto still posed substantial difficulties. These were not
overcome before the advent and application of calculus during the 18th
Historical Roots of the Theory of Hydrostatic Stability of Ships 149
The French mathematics professor Paul HOSTE, s.j., (1652-1700) was the first to
attempt to quantify the problem of ship stability. He did so in his 1697 treatise of
naval architecture, Théorie de la construction des vaisseaux, which was appended
to his book on naval tactics, written at the behest of Admiral TOURVILLE (Hoste
1697). However, he did not apply calculus to the problem, which was not yet
HOSTE assumes, without citing STEVIN, that the CG and CB were in a vertical
line; but he also allows without proof that the CG could be above the CB.
However, to explain how this could be possible without the ship tipping over, he
also assumes that the buoyancy force was equally divided between the two halves
of the ship, forming the base of a triangle. He continues this error by stating:
“If the center of gravity of the ship is known, the force with which it has to carry
sail is easily known, which is no other thing than the product of the weight of the
ship by the distance between these centers (of weight and displacement)” (Hoste
In modern terms, the righting moment is equal to Δ x ( KG – KB ). In other
words, the higher the center of gravity, the more stable the ship.
Although he does not provide a theoretical means for determining this “power
to carry sail”, HOSTE does describe a procedure which could empirically
demonstrate this: The inclining experiment. HOSTE asserts that, by measuring the
angle of inclination due to suspending a weight from a boom at a certain height, that
the “force to carry sail” can be determined (Hoste 1697, p. 56). Although HOSTE’s
argument contains several fundamental errors, it was the first attempt to express
the stability of a ship in mathematical terms, and his book remained the only
published inquiry into stability for almost half a century.
2.5 La Croix
César Marie de LA CROIX (1690-1747) was the head of administration and finance
for the Rochefort dockyard, and maintained the records for the galley fleet. He
was not a scientist or engineer. Yet he developed some fundamental concepts on
ship motions and hydrostatic stability, derived for a floating parallelepiped
(La Croix 1736), so certainly before BOUGUER’s and EULER’s work appeared.
LA CROIX was interested in the motions, but also in the hydrostatic restoring
moments for this parallelepiped when heeled. He correctly understood the role of
weight and buoyancy forces acting in the same vertical line in equilibrium. He
also followed ARCHIMEDES in his stability criterion by examining the heeled
body and requiring positive righting arms, accounting for the wedge volume
shifts. But he falsely determined the influence of the wedge shift moments and
H. Nowacki and L.D. Ferreiro150
hence the righting arms. Besides, since he lacked integration methods based on
calculus, he was not able to generalize his results for arbitrary section and
waterplane shapes, hence to make them applicable to ships.
Yet although LA CROIX’s work was flawed, it may have triggered EULER’s
renewed interest in ship hydrostatics. In 1735 EULER was asked by the Russian
Imperial Academy of Sciences to review the treatise (La Croix 1736), which LA
CROIX had submitted there prior to publication. EULER quickly responded and
appraised the merits of LA CROIX’s problem formulation, but also pointed out the
weaknesses of the solution. He then put on record his own correct solution for the
initial restoring moment, hence stability criterion, for the parallelepiped (or any
other body of constant cross sectional shape), which he claimed to have found
earlier (Euler 1972). This dates EULER’s earliest written mention of the criterion
for this special case to 1735. In 1736 in reply to LA CROIX ’s rebuttal EULER
also provided the complete explicit derivation of this result.
2.6 Sailing Vessel Propulsion and Maneuvering
Questions of ship propulsion, ship motions and maneuvering, but also of ship
stability have always aroused an acute interest, not only among seafarers and
shipbuilders, but also in the scientific community. The propulsion of sailing
vessels by the forces of wind, e.g., was investigated by ARISTOTLE in antiquity
and later is associated with such celebrated names of scientists as Francis BACON,
Thomas HOBBES and Robert HOOKE, among others.
Toward the end of the 17th
century the subject of the propulsion of sailing
vessels had gained new scientific interest, also under the influence of the navies in
applying scientific principles to ship design and operation. Treatises and
monographs on the theory of maneuvering and on the equilibrium of aerodynamic
propelling forces and hydromechanic response had appeared, notably by Ignace-
Gaston PARDIES (1673), Bernard RENAU d’ELISSAGARAY (1689), Christiaan
HUYGENS (1693), Jakob BERNOULLI (1695) and Johann (I) BERNOULLI (1714).
This had led to lively controversies, but also to an increasing depth of understanding
of the mechanics of the sailing vessel. In fact through the various treatises the
principles for applying hydrodynamic underwater and aerodynamic sail forces to
the vessel in order to establish its equilibrium position had become well understood.
At the same time the importance of allowing for trim and heel (as well as yaw)
under these forces and moments had been recognized and dealt with, although the
quantitative assessment of the hydrostatic restoring effects remained largely an
open issue. Both BOUGUER and EULER would begin their investigations into
the nature of hydrostatic stability with the study of propulsion and maneuvering of
Historical Roots of the Theory of Hydrostatic Stability of Ships 151
2.7 Displacement Estimates Before Calculus
The infinitesimal calculus was important to the development of stability for several
reasons, including the practical evaluation of underwater volume and volume
centroids. Yet a number of methods to calculate underwater volumes (and therefore
displacements) were developed in the 1500s and 1600s, well before any theoretical
framework for stability was available to make use of it. Why would shipbuilders
go through the trouble of making such calculations? It appears that there were two
separate reasons for this: The development of the gunport, and the measurement of
The gunport was introduced in the early 1500s, and came into wide use by the
middle of the century. This greatly increased the firepower of naval ships, but
brought two problems; ships got much heavier, and with large holes in the side of
the ship, the available freeboard got dramatically smaller. Thus, the margin for
error in estimating the waterline was considerably reduced. To handle this problem,
shipbuilders like Anthony DEANE (1638-1721) developed methods to calculate
how much armament, ballast and stores should be loaded on a ship to bring it to
the correct waterline below the gunports. DEANE and others made use of ship’s
plans, which were just beginning to be employed by shipbuilders as a construction
guide. In his manuscript “Doctrine of Naval Architecture” (Deane 1981), never
published but widely circulated, DEANE demonstrates two methods to calculate
the area underneath waterlines at each “bend” or frame of the hull; using either (1)
an approximation for the area of a quarter-circle or (2) by dividing the area into
rectangles and triangles. DEANE then sums the areas for each frame, multiplies by
the frame spacing and multiplies the volume by the specific weight of water to
obtain the displacement. He does this for several different waterlines, including
the desired waterline below the gunports. When a ship is launched, he can
immediately determine the light displacement, and then calculate how much
weight should be added to arrive at the design waterline.
A second reason for introducing displacement calculations was to more
accurately measure the cargo capacity of a ship. For example, from 1646 to 1669
the Dutch and Danish governments carried on a series of negotiations on cargo
measurement. In 1652 a Dutch mathematician Johannes HUDDE identified the
basic issue of measuring cargo deadweight by determining the difference between
the weight of the ship empty and fully-laden, using a difference-in-drafts method.
He suggested to the Dutch authorities that they measure the waterplane areas at
each draft of an actual ship in the water (not from plans), by taking measurements
to the hull from a line extended at the side of the ship parallel to the centerline.
The space between hull and an overall rectangle formed by the length and beam
was then divided into trapezoids and triangles, the areas calculated and summed,
multiplied by the difference in drafts and multiplied by seawater density, to obtain
cargo tonnage. Although the suggestion was never used, HUDDE’s cousin
H. Nowacki and L.D. Ferreiro152
Nicolas WITSEN reported it in his 1671 shipbuilding manuscript “Aeloude en
hedendaegsche scheepsbouw en bestier” (Witsen 1671; Cerulus 2001).
3 Development of Stability Theory
3.1.1 Biographical Sketch
Pierre BOUGUER (Fig. 2) was born on 10 February 1698 in the French coastal
town of Le Croisic, near Saint Nazaire at the mouth of the Loire. Educated in the
Jesuit school in Vannes, he quickly showed himself a child prodigy. When his
father Jean BOUGUER, a royal hydrographer and mathematician, died when
Pierre was only 15, he applied for his father’s position. After initial hesitation by
the authorities, he passed the rigorous exam and was given the post of Royal
Professor of Hydrography.
BOUGUER won several Royal Academy of Sciences prizes for masting,
navigation and astronomy before he was 30. He moved to the port of Le Havre in
1731, about the same time he became a member of the Academy. His work caught
the attention of the French Minister of the Navy MAUREPAS (1701-1781), who, like
COLBERT before him, was convinced of the strategic benefit of ship theory as a
way of compensating by quality against the quantitative superiority of the British
navy. MAUREPAS supported BOUGUER’s research and sponsored his publications.
In 1734, BOUGUER became involved in a controversy over the Earth’s shape;
those who believed in DESCARTES’ vortex theories of physics held that the Earth
was elongated at the poles, while those who accepted NEWTON’s theories of
gravitational attraction argued that the Earth was wider at the equator due to
centrifugal force. MAUREPAS also held the view that a full understanding of the
Earth’s shape was essential to navigation, so he sent BOUGUER, along with
several other members of the Academy of Sciences (accompanied by two Spanish
naval officers) on a Geodesic Mission to Peru to determine the length of a meridian
arc for a degree of latitude at the equator. One of his companions on the Mission
was the young Spanish lieutenant Jorge JUAN Y SANTACILIA, who would later
become a prominent naval constructor and author of a highly recognized treatise on
BOUGUER spent ten years away (1735-1744), during which time he not only
surveyed and calculated a meridian arc length of three degrees of latitude, he also
performed various experiments on gravity and astronomy. It was during this Mission,
in the peaks and valleys of the Andes far from the ocean, that he wrote much of
his monumental work “Traité du navire”, the first comprehensive synthesis of
Historical Roots of the Theory of Hydrostatic Stability of Ships 153
Fig. 2 Portrait of Pierre BOUGUER.
“Traité du navire” was published in 1746, shortly after BOUGUER’s return. He
remained unmarried, lived in Paris and devoted himself to revising his geodesic work,
and carrying out further studies of naval architecture, astronomy, optics,
photometry and navigation. BOUGUER died in Paris on 15 August 1758, aged 60.
Further details on BOUGUER’s biography and scientific work are presented by
FERREIRO (Ferreiro 2007) and DHOMBRES (Dhombres 1999).
3.1.2 Early Work on Ship Theory
In 1721 BOUGUER was asked by the Academy of Sciences in Paris to compare
the accuracy of two methods of calculating cargo tonnage being proposed to the
Council of the Navy for port fees. Although the mathematician Pierre VARIGNON
proposed estimating the volume of a ship as a semi-ellipsoid, BOUGUER found
that the best results came from a proposal of Jean-Hyacinthe HOCQUART of
H. Nowacki and L.D. Ferreiro154
Toulon, who used a difference-of-waterplanes method that employed equal-width
trapezoids to estimate waterplane areas. This was similar to HUDDE’s approach but
allowed a direct calculation of areas from ship’s plans. BOUGUER would later adopt
and refine this “method of trapezoids” for his own stability work (Mairan 1724).
In 1727 the Academy offered a Prize for the best treatise on masting, which
BOUGUER’s entry won. He postulated that a “point vélique”, the intersection of
the sail force and the resistance of water against the bow, should be directly above
the center of gravity to minimize trimming by the bow. In his treatise he relied on
HOSTE’s theories to explain stability, although where HOSTE implied that the
advantage of doubling is through an increase in the center of gravity, BOUGUER
invoked ARCHIMEDES to point out that the buoyancy of the added portion of
the ship moves the center of buoyancy laterally when heeling, thus increasing the
righting moment. Still, this treatise did not yet show any insights into the
evaluation of trim or heel angles or restoring moments that he would develop five
years later (Bouguer 1727).
3.1.3 The Metacenter
BOUGUER probably began formulating his theory of stability around 1732, after
he had moved to Le Havre, for he tested it using the little 18-gun frigate Gazelle,
laid down in that dockyard in May 1732 and delivered in January 1734 (Demerliac
1995). However, no letters or manuscripts from that period survive to confirm
this, and his derivation of the metacenter would not appear in print until 14 years
later in “Traité du navire”. The following steps illustrate BOUGUER’s derivation
of the metacenter in his “Traité du navire” (Bouguer 1746, pp. 199-324).
STEP 1: Premises and axioms
BOUGUER implicitly defines the hydrostatic properties of fluids, based on the
principle of hydrostatics that weight and buoyancy are equal, opposite and act in
the same vertical line. He does so without proof, without the use of equations and
without mentioning ARCHIMEDES by name, although he shows general familiarity
with his work. He also implies through geometrical arguments that the pressure of
the fluid follows a hydrostatic distribution with depth and is everywhere normal to
the surface of the submerged body.
Historical Roots of the Theory of Hydrostatic Stability of Ships 155
Fig. 3 BOUGUER’s diagram of the metacenter (Bouguer 1746, fig. 54).
p = submerged volume
Γ = upright CB
γ = inclined CB
AB = upright waterline
ab = inclined waterline
I = CG of an unstable ship
G = CG of a stable ship
g = metacenter
1 = centroid of immersed wedge
2 = centroid of emerged wedge
3 = centroid of body without wedges
STEP 2: Magnitude of buoyancy force
BOUGUER resolves the submerged surface into very small elements (though
without using any calculus notations at this point) and equates the vertical
components of the hydrostatic pressure forces to the weight of the water column
resting on top of the element in the interior of the submerged volume conceptually.
Hence, the total pressure resultant is equal to the total weight of water filling the
submerged volume. He thus reconfirms the law of ARCHIMEDES by pressure
STEP 3: Measurement of volume and centroids
BOUGUER first suggests two methods for calculating the volume of the ship as
a regular solid. The first is to model the ship as an ellipsoid, as originally proposed by
VARIGNON, and the second is to divide it into prisms. The set of quadrilateral
prisms then forms a polyhedral approximant of the hull surface for volume
summation. BOUGUER sees clearly that, by analogy, the same principle of
H. Nowacki and L.D. Ferreiro156
polygonal approximation also holds for evaluating the area of planar curves. He
thus quickly homes in on a quadrature rule for curves based on equally wide
trapezoids, his favorite rule, also known as trapezoidal rule. He uses it to measure
areas of waterline “slices” and then combining those two-dimensional slices to
obtain a three-dimensional volume. Although he borrowed the idea from
HOCQUART’s 1717 proposal, BOUGUER refined the method, first by dividing
each waterplane into many sections (HOCQUART only proposed three sections),
and second by taking the areas of several waterlines to develop the entire volume
of the hull (HOCQUART took only one “slice”).
BOUGUER then arrives at the conclusion that his quadrature rule is suitable for
evaluating the integral of any continuous function of one variable or, if applied
recursively, for continuous functions of any number of independent variables.
Interestingly he thus interprets the analytical formulation of stability criteria,
which he introduces later, as being equivalent to the discretized quadrature rules
presented here earlier. The ancestry of his concepts from both practical shipbuilding
traditions and modern calculus is still leaving some visible traces.
BOUGUER then digresses for many pages into using the trapezoidal rule to
calculate incremental waterlines for estimating a ship’s payload capacity, and
gives various rules for tunnage admeasurement.
For finding the centroid of the underwater volume, the center of buoyancy CB,
BOUGUER then begins by explaining, in very simplistic terms, how to use the
sum-of-moments method to determine the centroid of an object. He then derives
the area centroid of a planar figure (2D case), then the volume centroid of a solid
(3D case), which for a ship he calls the “center of gravity of the hull”. He then
discusses by example how to evaluate these expressions numerically by the
method of trapezoids.
STEP 4: Stability criterion
The center of buoyancy having been determined, BOUGUER next explains why he
chooses the metacenter as the initial stability criterion, using the geometrical
argument shown in Fig. 3.
The ship’s center of gravity g is always in the same vertical line as the center of
buoyancy Γ, but this geometry is not constant due to the ship’s movement. If the
ship has a very high center of gravity I, and moves even a little from the upright
position (waterline A-B) to another position (waterline a-b), it is no longer stati-
cally stable; the center of buoyancy moves from Γ to γ, i.e., away from the vertical of
the center of gravity, and the vertical force of buoyancy shifts from Γ-Z to γ-z. The
ship’s weight, centered at I and on the opposite side of the inclination from the
new center of buoyancy, tends to push the ship even further over, rendering it
unstable. However, if the center of gravity of the ship is at G, below the
intersection g of the upright and inclined vertical forces of buoyancy, then the
center of gravity is on the same side as the new center of buoyancy and the
resulting force always tends to restore the ship to the horizontal.
BOUGUER states in his definition of the metacenter (Bouguer 1746, pp. 256-
Historical Roots of the Theory of Hydrostatic Stability of Ships 157
“Thus one sees how important it is to know the point of intersection g,
which at the same time it serves to give a limit to the height which one can give
the center of gravity G, it [also] determines the case where the ship maintains
its horizontal situation from that where it overturns even in the harbor without
being able to sustain itself a single instant. The point g, which one can justly
title the metacenter [BOUGUER’s italics] is the term that the height of the
center of gravity cannot pass, nor even attain; for if the center of gravity G is
at g, the ship will not assume a horizontal position rather than the inclined
one; the two positions are then equally indifferent to it: and it will
consequently be incapable of righting itself, whenever some outside cause
makes it heel over.”
BOUGUER does not use the terms “stable” or “unstable”, but rather states that
G is either lower or higher than g.
Step 5: Evaluation of the criterion
The determination of this point of intersection, BOUGUER says, reduces to the
question of the distance between the centers of buoyancy Γ and γ of the
submerged body upright and just slightly inclined. BOUGUER employed the all-
important shifting of equal immersed and emerged wedges to determine this
distance, by drawing a triangle between points 1, 2 and 3. Since the centers of
buoyancy Γ and γ lie on the legs of that triangle, the distance Γ – γ must be
proportional to the distance between points 1 and 2, and the distance between Γ
and point 3 must be proportional to the ratio of the volume of the underwater hull
and the wedges. This geometrical explanation sets the stage for the second half of
the explanation, the mathematical analysis of the mechanics of stability.
Step 6: Results
In the next section, BOUGUER uses calculus to determine the three unknowns:
The distance from point 1 to point 2; the volume of the wedges; and the volume of
the hull. BOUGUER imagines that Fig. 3 represents the largest section of the ship,
which actually extends through the plane of the page in the longitudinal direction
x, with the immersed and emerged wedges actually an infinite sequence of
triangles of width y (the largest being b = F-B, or the half-breadth of the hull at the
waterline) and height e ( = H-B) going through the length of the hull at a distance
dx from each other; integrating, the volume of the wedge is
The second unknown is the distance between the centers of the wedges, points
1 to 2; but since the center of a triangle is two-thirds the height from apex to base,
it is straightforward to obtain this distance as
H. Nowacki and L.D. Ferreiro158
Distance 1- 2 =
The third unknown, the volume of the hull p, is derived using the trapezoidal
rule. Putting the three together, the distance is:
Finally, observing that the triangle Γgγ (i.e., between the centers of buoyancy
and the intersection of their vertical lines of force) is similar to the triangles
formed by the immersed and emerged wedges, the height of the metacenter g
above the center of buoyancy Γ is found via Euclid to be:
This is the now-famous equation for the height of the metacenter.
3.1.4 Implications of the Metacenter
BOUGUER develops in thorough detail both the theoretical and practical aspects
of the metacenter. He derives the metacenter for various solids (ellipsoid,
parallelepiped, prismatic body) and presents the procedures for its practical,
numerical calculation for ships. These explanations were detailed enough for
practical applications and became the foundation for later textbooks.
But BOUGUER also charged ahead beyond the initial metacenter for infinitesimal
angles of heel when he introduced the concept of the “métacentrique”, i.e., the
metacentric curve for finite angles of heel. First, he clearly recognized that the
same physical principles and stability criteria apply to an inclined position of
the ship as they do for the upright case. Second, he recognized that the metacentric
curve for finite angles is in fact the locus of the centers of curvature of the curve
of the centers of buoyancy. This was a brilliant, original insight. The locus of the
centers of curvature of a curve was known since Christiaan HUYGENS’s work on
the pendulum clock (Huygens 1967b) under the name of “développée” (or evolute).
Third, BOUGUER also knew how to construct the evolute of a given curve. For a
wall-sided ship (or parallelepiped) the metacentric curve is a cusp shaped curve
composed of two hyperbolas lying above the metacenter, as BOUGUER showed.
His demonstration examples for the “métacentrique” do document that he understood
Historical Roots of the Theory of Hydrostatic Stability of Ships 159
how to approach stability for finite angles of heel, though he never used a
“righting arm” criterion.
On practical aspects of initial stability he recommends that the widest point of
the ship be no lower than where the maximum heel would be before it begins to
tumble-home, or even that ship sides be straight or flared throughout. In other
words, BOUGUER was advising to avoid tumble-home altogether, although in
somewhat oblique terms.
BOUGUER next provides a numerically-worked example of the application of
the metacenter to a real ship, underlining the importance of Gg, the distance
between the center of gravity and the metacenter. Using the Gazelle as his model,
BOUGUER explains how to account for the weight of each part of the ship,
including the frames, planking, nails, etc., how to measure the center of gravity of
each piece using the keel as the reference point, and how to sum the moments to
obtain the overall center of gravity G. He then calculates displacement and center
of buoyancy of the ship using the trapezoidal rule, which enables him to find the
metacenter g. His calculations confirmed that, once properly ballasted, the Gazelle
would be stable.
BOUGUER continues for almost 50 pages to outline the practical implications
of the metacenter on hull design and outfitting. In many cases the implications are
re-statements of what constructors already knew; but BOUGUER provided for the
first time a rigorous analysis of why they were true. Some of the main points he
brings out are:
The greatest advantage to stability lies in increasing the beam. BOUGUER
claims that the “stability” varies as the cube of the beam, by which he must
have been referring to the transverse moment of inertia I T, while the metacentric
height requires closer scrutiny. But he wanted to emphasize the rapid increase of
stability with beam. This also explains for the first time why doubling a ship
improves stability, although BOUGUER does not explicitly state so.
Stability is improved by diminishing the weight of the topsides; though this
was well-understood, BOUGUER details how to accurately assess the effects.
BOUGUER also correctly describes for the first time how to evaluate the inclining
experiment, for whose basic idea he credits HOSTE, in order to determine Gg, the
distance between the center of gravity and the metacenter. In Fig. 4, he demonstrates
how to use a known weight suspended from a mast to incline the ship and measure the
angle of heel. Using the law of similar triangles, he shows that the ratio of the
ship’s displacement and the suspended weight is proportional to the ratio of Gg to the
angle of heel and distance the weight moves; on the assumption of small angles of
heel this allows Gg (i.e., the reserve of stability) to be calculated directly without
knowing the exact position of the metacenter.
BOUGUER demonstrates how to use his metacenter by detailing the calculations
of weights and centers of gravity for Gazelle while it was still being framed out. It
is therefore curious that he did not verify this by performing an inclining
experiment on Gazelle. It would appear that, while BOUGUER was completing
his initial stability work in 1734, he was caught up in the events that led to the
H. Nowacki and L.D. Ferreiro160
Geodesic Mission, and would not return to the subject until he was in the
mountains of Peru.
Fig. 4 BOUGUER’s diagram of an inclining experiment, (Bouguer 1746, fig. 55).
3.2.1 Biographical Sketch
Leonhard EULER (Fig. 5) was born in Basel, Switzerland on 15 April 1707. He
was the son of Paulus EULER, parson of the Reformed Church, and his wife
Margaretha née BRUCKER. He went to a Latin grammar school in Basel and, as
his father recognized his talent early, took private lessons in mathematics. In 1720
he enrolled at the University of Basel as a student and later as a young scientist,
received a Magister’s degree, then in
theology. But he soon turned his main interest to mathematics and mechanics,
which he studied under Johann (I) BERNOULLI, who was recognized as a leading
mathematician of that era. In fact Johann (I) BERNOULLI, who was 40 years
EULER’s senior, saw EULER’s maturing genius and invited him to join the
Saturday afternoon “privatissimum” in mathematics, a private circle held in the
BERNOULLI’s home, where he also met and made friends with younger members of
the BERNOULLI family, notably Niklaus (II), Daniel (I) and Johann (II)
BERNOULLI. In this period Johann (I) BERNOULLI laid the foundations from
which EULER’s later mathematical fame developed.
In 1727 EULER, who had been looking for an academic position, received an
offer from the Russian Imperial Academy of Sciences in St. Petersburg, which
Tsar Peter had initiated and which had been opened in 1725, where Niklaus (II)
and Daniel (I) BERNOULLI held appointments as professors from this beginning.
EULER accepted this prestigious offer, starting as an Adjunct (élève) for a modest
in the first three years in philosophy where he
Historical Roots of the Theory of Hydrostatic Stability of Ships 161
salary, and arrived there – after a three month voyage by coach and by ship – in
June 1727. EULER began his scientific career in St. Petersburg very successfully,
advanced to a professorship in physics and full membership in the Academy in
1731, and spent his “First St. Petersburg Period” (1727-1741) very productively,
working on a wide range of scientific subjects and publishing more than 50
treatises and books.
In 1741, amidst political transitions and uncertainties in Russia, EULER received
an attractive offer from King FREDERICK II of Prussia to come to Berlin and to
work for the Royal Academy of Sciences, which was to be founded. EULER
moved there in 1741, continued his illustrious scientific work on an increasing
variety of subjects, and when the Academy at last opened in 1746 soon became
recognized as one of its most eminent members and as the leader of the mathematical
class. He remained immensely productive and published some of his most famous
books and treatises during his period in Berlin from 1741 to 1766.
Despite EULER’s scientific fame and the considerable merits he earned in the
Berlin Academy during its formative first two decades he never developed a
relationship with King FREDERICK that made him feel recognized, appreciated
and understood. This was one contributing factor why after 25 years in Prussian
service he decided to return to St. Petersburg. There he was welcomed and honored at
all levels, and spent his “Second St. Petersburg Period (1766-1783)” in relative
comfort, peace and recognition despite his weakening health and fading eyesight.
His scientific creativity never ceased. A large share of his more than 800 treatises
and books also stems from this second period in Russia. His work encompasses
the full gamut of scientific topics in his era, not only in mathematics and mechanics,
but also in other branches of physics, astronomy, ballistics, music theory,
philosophy and theology. He died in St. Petersburg in 1783.
More detailed accounts of EULER’s life and scientific vita are given by
BURCKHARDT et al. (Burckhardt et al 1983), FELLMANN (Fellmann 1995) and
CALINGER (Calinger 1996).
3.2.2 Early Work in Ship Theory
In 1726 the Académie des Sciences in Paris invited contributions to a prize
competition on the optimum placement, number and height of masts in a sail
propelled vessel. EULER as a student and protégé of Johann (I) BERNOULLI
was of course familiar with the earlier treatises and disputes that had arisen about
the forces acting on a sail propelled maneuvering vessel between RENAU, HUY-
GENS, Johann (I) BERNOULLI a.o. So he felt well enough prepared, also
encouraged by Johann (I) BERNOULLI, to submit a treatise (“De Problemate
Nautico…”) (Euler 1974) in 1727, which gained honorable mention, an “accessit”
equivalent to a second prize. Pierre BOUGUER was awarded the first prize. In this
treatise EULER, who still treads in the foot-steps of Johann (I) BERNOULLI, does
not yet show deeper insights into ship hydrostatics although he does draw
H. Nowacki and L.D. Ferreiro162
attention to the requirement that for a ship before the wind the propelling sail force
is limited by the acceptable forward trim angle. But he has no reliable physical
basis for estimating that angle. Yet this early experience in his life at the age of 20
gave EULER a first thorough acquaintance with the mechanics of ships,
familiarized him with nautical and ship construction terminology and created in
him a lasting interest in ships as topics to be studied by methods of mathematical
analysis and engineering mechanics.
Fig. 5 Portrait of Leonhard EULER.
In St. Petersburg EULER soon had ample opportunities to return to these
subjects. It is not clear when he first occupied himself more deeply with ship
hydrostatics. But evidently the appearance of LA CROIX’s treatise on the hydrostatic
stability of a parallelepiped in 1735 found EULER well prepared, when the
Academy asked for his review, to detect certain flaws in LA CROIX’s derivations
and to present the correct result for this simple shape, which first established
EULER’s initial restoring moment criterion on hydrostatic stability. In his second
review in 1736, in response to LA CROIX’s rebuttal, EULER went beyond his first
results, dealt with some other prismatic shapes (trapezoidal and triangular prisms) and
Historical Roots of the Theory of Hydrostatic Stability of Ships 163
alluded to his general approach to stability (Euler1972). This evidence is sufficient to
date EULER’s earliest written mention of the hydrostatic stability criterion.
There is also some indirect evidence of EULER’s earliest results on hydrostatic
stability in his correspondence with the Danish naval constructor and naval attaché
in London, Friderich WEGERSLØFF, which he maintained between 1735 and
1740. In a letter of 14 September 1736 WEGERSLØFF acknowledges receipt of a
solution for the hydrostatic stability problem, on 22 May 1738 EULER replies and
gives a derivation of his stability results and also mentions some experimental
verification (Euler 1963). It is not clear from the context what sort of tests he had
By 1737 EULER’s interests in ship theory were well known at the Imperial
Academy. Thus - probably not without his own prior knowledge or suggestion - he
was commissioned by the Academy to write a book on this subject, resulting in his
monumental two-volume “Scientia Navalis”. This work encompasses a
presentation of the complete scope of ship theory according to the state of the art
at that time and includes many new findings and derivations by EULER. The
chapters in the two parts deal with ship hydrostatics, ship resistance and propulsion,
maneuvering and ship motions. Many results have been of lasting value and have
served as foundations for the growing scientific body of ship hydromechanics.
According to EULER’s own report in the Preface of “Scientia Navalis”, he had
worked on this manuscript from 1737 to 1740 (Euler 1967, pp. 15/16). When he
left St. Petersburg in 1741 he had completed the first part, which contains all four
chapters on hydrostatic equilibrium and stability, and half of the second part. The
remaining sections of Part 2 were finished by 1741 in Berlin. Unfortunately, he
had much difficulty finding a publisher for this voluminous opus, which thus was
not published until 1749 by the Academy in St. Petersburg.
Comprehensive appraisals of EULER’s overall contributions to ship theory and
related matters are given by HABICHT (Habicht 1974), MIKHAILOV (Mikhailov
1983) and TRUESDELL (Truesdell 1983).
3.2.3 The Initial Stability Criterion
To understand the history of EULER’s involvement in ship theory it is important
to read the Preface to his “Scientia Navalis”. Here he explains that his work will
go beyond the established disciplines of hydrography or nautical science and will
concentrate on a physical and analytical investigation of the mechanics of ships, at
rest and in motion, for which fundamental theoretical works were still missing at
that time. In hydrostatics EULER departs from the Principle of ARCHIMEDES, to
whom in the preface of “Scientia Navalis” he gives credit and praise. But he adds
that the hydrostatic stability of ships must be newly approached and quantified in
order to be able to distinguish between stable and unstable equilibrium of ships at
the design stage. The experience of naval architects (“architecti navales”) alone, long
H. Nowacki and L.D. Ferreiro164
established as it may be, will not be sufficient to prevent unexpected stability
EULER nowledges the motivation he received from reviewing LA
CROIX’s treatise in 1735-1736, which prompted him to investigate more
profoundly the transverse and longitudinal stability of ships. EULER also gives
very favorable credit to BOUGUER’s “Traité du Navire”, which had appeared 3
years before “Scientia Navalis”, but he takes much care also to avert any suspicion
of plagiarism by calling on the Imperial Academy as witnesses that he had written
“Scientia Navalis” essentially between 1737 and 1741 during which period there
was no communication with BOUGUER who was in Peru. These statements were
never disputed between the two authors, who corresponded in a respectful and
amicable fashion on other subjects later.
EULER’s derivation of his stability criterion in “Scientia Navalis” proceeded in
the following steps (Euler 1967, pp. 3-166):
STEP 1: Premises and axioms
In the first chapter of Book I, EULER deals with the equilibrium of bodies
floating in water at rest. In essence he rederives the Hydrostatic Principle of
ARCHIMEDES from the modern viewpoint of infinitesimal calculus by integration of
the hydrostatic pressure distribution prevailing in a fluid over the surface of the
body. The properties of the fluid at rest and the use of calculus for this purpose
both were new at the time when EULER wrote these passages. As for the pressure
distribution he states axiomatically in the opening paragraph of his book:
“Lemma: The pressure which the water exerts on the individual points of a
submerged body is normal to the body surface; and the force which any surface
element sustains is equal to the weight of a straight cylinder of water whose base
is equal to the same surface element and whose height is equal to the depth of the
element under the water surface”.
These brief axioms, viz. the normality of pressure to a surface and the inferred
equality of the pressure at a point in a given depth in all directions, were the first
analytical formulation for the properties of the fluid and are regarded as the
necessary and sufficient conditions for the foundation of hydrostatics (Truesdell
STEP 2: Magnitude of buoyancy force
From these premises EULER proceeds to define by integral calculus the buoyancy
force as the pressure resultant and the center of buoyancy (EULER calls the CB:
“centrum magnitudinis”), through which it acts, as the volume centroid of the
submerged part. He reconfirms also that for equilibrium the buoyancy and weight
forces must act in the same vertical line and must be equal in magnitude and
opposite in direction. He then illustrates these principles by examples of simple
shapes like parallelepipeds and prisms of triangular and trapezoidal cross section.
For each of these solids he finds the possible equilibrium conditions over a full
circle of rotation as a function of the specific weight of these homogeneous solids,
not unlike HUYGENS in his unpublished treatise of 1650.
Historical Roots of the Theory of Hydrostatic Stability of Ships 165
In Chapter II, briefly digressing from hydrostatics, EULER discusses the
resulting motions of a floating body if it is temporarily displaced from its
“upright” equilibrium position. Here he explains the “lumping” of masses in their
center of gravity (CG), introduces the definition of the principal axes of inertia for
ships, underscores the significance of the CG as a reference point, e.g., for
decomposing a resulting motion into the translation of the CG and the rotation
about it. EULER will adhere to the CG as his system reference point, also when
returning to ship motions later.
STEP 3: Measurement of volumes and volume centroids
In stark contrast to BOUGUER, EULER confines himself to analytical
definitions of stability criteria, volumes, centroids, areas, moments of inertia etc.
He does not address their numerical evaluation at all. He is taking it for granted
that once the shape of the ship or body is defined by some function the
integrations can be readily performed. In his examples he usually deals with
simple shapes where the integrations can be performed in closed form.
STEP 4: Stability criterion
In Chapter III, the main chapter on hydrostatic stability, EULER defines his
stability criterion right away (Proposition 19) by:
“The stability which a body floating in water in an equilibrium position
maintains, shall be assessed by the restoring moment if the body is inclined from
equilibrium by an infinitesimally small angle”.
EULER illustrates this principle by discussing cases of unstable, neutral and
stable equilibrium of ships and adds that it is necessary to quantify stability in
terms of the restoring moment because even a stable ship may be in danger by
external heeling moments and may require righting moments of greater
magnitude. (The issue is not only whether the ship is initially stable or not, but
also how much “stability capacity” it has).
STEP 5: Evaluation of the criterion
EULER then enters into the determination of the restoring moments by first
examining a planar cross section of arbitrary shape (or thin disk) floating upright
(Fig. 6). The Figure AMFNB = “AFB” is inclined by a small angle dw so that the
new floating condition has the waterline ab. Since the center of gravity G remains
his reference point, also for later purposes, he draws the parallel lines MN and mn
to the waterlines before and after inclination, also through G. The center of
buoyancy of the cross section is designated by O. A normal VOo to the inclined
waterline is drawn through O. The equality of the immersed and emerging wedges
requires that ab intersect AB at the center point C so that AC = BC.
H. Nowacki and L.D. Ferreiro166
Fig. 6 EULER’s figure for centroid shift in inclined cross section, 2D case, (Euler 1967, fig. 39).
Let the displacement per unit length and the equal buoyancy force of the cross
section before inclination be denoted by M = γ (AFB). For the inclined position
the restoring moment is composed of three contributions:
1. The effect of the original submerged volume forming a positive restoring
couple of forces through G and V:
M GV = M GO dw (5)
2. The effect of the submerged wedge CBb whose cross section area is:
and whose restoring moment about G hence is:
(qo + GV) (7)
where qo =
3. Likewise for the emerging wedge ACa the restoring moment is
(po - GV) (8)
where po =
For all moments combined, replacing γ by M/ (AFB):
Historical Roots of the Theory of Hydrostatic Stability of Ships 167
M REST = M GO dw +
The expression in square brackets has the dimension of a length and is the now
well known result, corresponding to, in BOUGUER’s later terminology, the
metacentric height of the cross section. This is how EULER’s restoring moment
and BOUGUER’s metacentric height are connected. Note that the GO term
reverses sign if G lies above the center of buoyancy O, as is common in cargo
ships. EULER discusses this result at some length and for several simple shapes.
In Proposition 29 EULER then arrives at the general three-dimensional case of
a floating body of arbitrary, in particular asymmetrical shape, whose waterplane is
drawn in Fig. 7. He denotes:
M = displacement or weight of body
V = submerged volume
GO = distance between CG and CB, positive for G above B
CD = reference axis of waterplane through centroid of waterplane area,
parallel to axis through G
CX = x = abscissa from origin C
XY = y = ordinate in upper part of waterplane
XZ = z = ordinate in lower part of waterplane
p and q = area centroids of upper and lower parts of waterplane
P and Q = equivalent pendulum lengths of waterplane area parts w.r.t. axis CD
Mdw [GO +
H. Nowacki and L.D. Ferreiro168
Fig. 7 EULER’s figure for the derivation of the stability criterion for a body of arbitrary shape,
3D case (Euler 1967, fig. 48).
EULER derives, since the axis CD runs through the waterplane centroid:
dx = ∫ z2
dx, and hence
PR + QS =
( )y z dx
from which in analogy to the planar case the restoring moment, divided by the
angle dw, for the ship becomes:
M (GO +
In the special case of port/starboard symmetry (y = z), with
dx; and TI
In our familiar notation (GO= GB , OM= BM ) the restoring moment simplifies
M ( GB + BM ) = M GM (12)
Historical Roots of the Theory of Hydrostatic Stability of Ships 169
STEP 6: Results
This summarizes the course of steps EULER took to derive the initial restoring
moment. EULER never used the word metacenter. EULER’s result thus is the
initial restoring moment, divided by (M dw), which he uses as his stability criterion.
Again EULER illustrates this result by many examples for simple shapes of
solids, even by an analytical formulation for a shiplike body with parabolic section
shapes. But he does not present any numerical calculations for an actual ship
although he discusses many practical implications of his results.
3.2.4 Implications of the Stability Criterion
EULER interprets his results for practical applications to ships, mainly in Part II of
“Scientia Navalis”. For EULER ships are “floating objects, carrying a cargo or
payload, symmetrical starboard and port, propelled by rowing and/or sails”.
Among the major conclusions he draws, we quote:
Stability is judged by the restoring moment divided by the angle dw.
Transverse and longitudinal stability are clearly distinguished and both
addressed. EULER also derives an expression for combined heel and trim
under oblique sail force moments, deriving the oblique restoring effects by
rotation of the reference axis from the principal axes.
To assess stability you need to know the ship’s displacement, the centroids CB
and CG, the waterplane area and shape, which infers the restoring moment
(without using the metacenter).
To improve stability, lower the CG, raise the CB and/or widen the beam.
The addition, removal and internal shifts of weight as well as the role of
ballast for stability are discussed.
The use of doubling (“soufflage”) with its pros and cons is mentioned.
In Chapter IV of Part 1 EULER addresses practical problems of inclining ships
under external moments or by internal weight shifts, though again only analytically,
He also addresses the issue of external wind loads and required stability,
especially for small ships vs. big ships, cargo ships vs. war ships.
EULER finally also makes the interesting “philosophical” remark that sailors
knew about measures of stability all along. When the sailors say, “This ship can
sustain a strong wind in its sails”, then EULER claims they mean the same thing as
he expresses by his restoring moments.
3.3 Comparison of Approaches
Based on the facts presented above it is now possible and in fact intriguing to
compare the approaches taken by BOUGUER and EULER in their creative work
on hydrostatic stability. This will bring out certain commonalities, but also
underscore the differences. We have examined several aspects.
H. Nowacki and L.D. Ferreiro170
BOUGUER worked on the subject of ship hydrostatic stability from about 1732 to
at least 1740, and his treatise was published in 1746. EULER was engaged with
this topic between about 1735 to at least 1740, “Scientia Navalis” appeared in
1749. During these formative periods of their new concepts there existed no
contacts and no communications between them. Thus their work originated
independently. But after their first scientific acquaintance through their participation
in the 1727 Paris Academy award competition it is likely that they both underwent a
common fermentation period on related subjects. Probably they were both looking for
the missing pieces in the mosaic of sailing ship maneuvering, among which the
hydrostatic response was foremost. It appears that their deeper familiarity with
ARCHIMEDES originated during the period between 1727 and 1735.
BOUGUER writes that he was familiar with the work of PARDIES, HOSTE,
RENAU, most likely also with relevant publications by HUYGENS and the
BERNOULLIs. He shows close acquaintance with ARCHIMEDES’ principles,
although he does not mention his name. On calculus he quotes the well-known
treatise by NEWTON and Roger COTES, but very likely he also thoroughly read
de l’HÔPITAL, whose notation in the LEIBNIZ style he preferably uses. His
knowledge of calculus was mostly self-taught, based on occasional tutorials and
recommendations from mathematics professor Charles René REYNEAU, who
wrote the basic textbook on the subject, “Analyse démontrée” in 1708.
EULER also knew ARCHIMEDES, HOSTE, RENAU, whom he mentions. He
learnt calculus first-hand from the BERNOULLIs, who stood firmly in the
LEIBNIZ tradition. EULER’s book on mechanics, “Mechanica” in 2 volumes,
where he displays exquisite knowledge of the principles of equilibrium and
motions of mechanical systems, had appeared in 1736.
Although both authors are not very generous in giving references on their
sources it is possible to a certain extent to trace their intellectual ancestry by
looking at their scientific and technical terminologies, even if they wrote in
different languages, French and Latin. We have analyzed their vocabularies in
their main treatises on hydrostatic stability and identified commonalities and
distinct differences. They shared much common ground by using such established
words as equilibrium, stability, weight, buoyancy, specific weight, centers of gravity
and buoyancy, inclination (derived from ARCHIMEDES) and pressure, hydrostatics
(from STEVIN and PASCAL). BOUGUER is unique in inventing the metacenter
and using the evolute (from HUYGENS), but never the word “restoring moment”,
while EULER’s vocabulary was vice versa. This indicates where their approaches
Historical Roots of the Theory of Hydrostatic Stability of Ships 171
3.3.3 Stability Criteria:
Sections 3.1 and 3.2 have shown that BOUGUER and EULER derived equivalent
stability criteria, but used different approaches. To sum up the comparison:
Premises and axioms: BOUGUER’s work is founded on the premises of
ARCHIMEDES, though augmented by a hydrostatic pressure law (STEVIN). EULER
defines the axioms of hydrostatics a bit more strictly by postulating the direction
independence of pressure and its normality to a surface.
Magnitude of buoyancy force: Both agree and reconfirm the law of
ARCHIMEDES by pressure integration and calculus.
Measurement of volumes and centroids: BOUGUER takes the route of
establishing numerical procedures, based on quadrature rules, for measuring volumes,
centroids, areas etc. before giving analytical definitions of the stability criterion. In
passing he skillfully adapts the “method of trapezoids” to shipbuilding applications.
EULER skips this step entirely and drives directly at analytical formulations for
his stability criterion.
Stability criterion: BOUGUER’s invention and choice for a measure of stability is
“metacentric height”, a geometric quantity representative of initial stability capacity.
EULER chooses the “initial restoring moment”, a physical quantity, also measuring
the ship’s initial stability capacity. Although the two ideas are closely connected,
their meanings are conceptually different. BOUGUER never refers to “restoring
moments”, EULER never uses “metacenter”. Neither author recurs to ARCHIMEDES’
idea of a “righting arm”.
Evaluation of the criterion: BOUGUER derives the metacenter mainly from
geometric arguments and, allowing for the shift of wedge volumes and hence CB,
determines the metacenter as the point of intersection of the upright and inclined
buoyancy directions. EULER uses physical arguments of moment equilibrium and
shift of volumes and finds the restoring moment simply by a summation of moments.
Results: BOUGUER’s result is metacentric height, a distance. EULER’s result is a
restoring moment. The two results are equivalent in practice, but not equal in
3.3.4 Audience, Style and Language:
BOUGUER’s “Traité du navire”, the first modern synthesis of theoretical naval
architecture, is written in French for a readership of scientists and constructors
who are to be introduced to the use of theory in practical ship design and
shipbuilding. BOUGUER benefited from both his experience as a hydrographer
and his collaboration with ship constructors. His style is clear and logical,
explaining many practical details, almost as in a textbook. He is concerned about
how this methodology can be implemented. This style does not cause any
sacrifices in rigor. His language is lucid, his own new ideas come across crisply.
H. Nowacki and L.D. Ferreiro172
EULER in his Latin text of “Scientia Navalis” writes as an applied mechanicist/
physicist and addresses mainly the scientific community of his era. Among those
who were educated to read scientific treatises of this sort in Latin, which was still
a lingua franca in the academic world, EULER’s brilliance of style, his inimitable
logic and clarity were highly praised. These attributes also apply to his “Scientia
Navalis”. His work on hydrostatic stability is still valid today. But at the same
time he did not have any experience with practical shipbuilding applications. Thus
he left many details for implementation by later successors, which delayed the
spreading of his methods, certainly in comparison to BOUGUER.
Yet in the final balance the scientific and engineering community must be
grateful to both BOUGUER and EULER.
Table 1 gives a synopsis of important milestones and ideas in the development of
the theory of hydrostatic stability by indicating where certain elements of this
theoretical knowledge first occurred. The importance of ARCHIMEDES’s physical
insights and the relevance of BOUGUER’s and EULER’s parallel discoveries
described by calculus become clearly visible in this condensed tabular summary of
Table 1. Historical development of stability concepts
Historical Roots of the Theory of Hydrostatic Stability of Ships 173
4 Further Work
4.1 Practical Applications
Stability theory quickly found direct applications in the day-to-day practice of ship
design. This occurred in two ways: first, by the increasing sophistication of
calculations for weights, centers of gravity and the metacenter within the design
process; and second, by the use of inclining experiments to validate stability.
These applications were surprisingly widespread; stability theory was quickly
incorporated in navies where there was already a strong institutional development
of scientific naval architecture, notably in France, Denmark, Sweden and Spain.
Those countries also created schools of naval architecture during the 18th
where students were weaned on stability theory and naturally used it when they
became constructors at the dockyards. This was not the case for the British or
Dutch navies, which had provided little direct support for scientists working on
ship theory, and which did not establish permanent schools of naval architecture
until the 19th
4.1.1 Stability Calculations During Design
The most famous of these schools of naval architecture was the Ecole du Génie
Maritime, established in 1741 in France by the scientist Henri Louis DUHAMEL
DU MONCEAU (1700-1782), whom MAUREPAS had appointed to Inspector-
General of the Navy.
DUHAMEL worked with BOUGUER to create the first comprehensive textbook
for the students, based on “Traité du navire”. DUHAMEL’s genius was to take
BOUGUER’s complex mathematics and render them into step-by-step instructions
on how to calculate the metacenter. The textbook, “Elémens de l’architecture
navale” (Duhamel 1752) became the standard reference for both students and
DUHAMEL DU MONCEAU was also involved in developing the Ordinance of
1765 under Navy minister Etienne François, Duke of CHOISEUL, which created
the Corps du Génie Maritime and formalized the data to be included on ship’s
plans: “centers of gravity and resistance, and height of the metacenter” as well as
accompanying calculations and a tabulation of hull materials. Later, the
standardized plans introduced in 1786 by DUHAMEL’s successor as Inspector-
General, Jean-Charles de BORDA, listed the specific immersion of the hull at full
and light load in tonnes-per-cm equivalent, which allowed a rapid estimate of the
effect of loading weights on the ship.
H. Nowacki and L.D. Ferreiro174
4.1.2 Inclining Experiments
Another facet of rapid adoption of the metacenter was that the first recorded inclining
took place in 1748, just two years after BOUGUER published his work. Francois-
Guillaume CLAIRIN-DESLAURIERS, then a junior constructor at Brest, performed
the experiment on the newly-built 74-gun ship Intrépide, apparently out of curiosity to
test the new theory. He hung two 24-pound cannons (each weighing over 2
tonnes) from a buttress built on the side of the ship (Fig. 8), and although the buttress
broke, he was able to take enough measurements to ascertain that the ship’s GM
was 1.8m (Clairin-Deslauriers 1748).
Fig. 8 CLAIRIN-DESLAURIERS’ drawing of the inclining of Intrépide
4.1.3 CHAPMAN and SIMPSON’s Rule
Another important contribution to the promulgation of hydrostatic stability
calculations came from the Swedish naval constructor and innovative ship
designer Fredrik Henrik CHAPMAN. CHAPMAN (1721-1808), the son of the
superintendent of the Göteborg dockyard, was endowed by this background with
much practical shipbuilding knowledge. He visited Britain in 1750 to improve his
knowledge in mathematics and other fundamentals. He took classes from Thomas
SIMPSON (1710-1761), a gifted mathematician trained in NEWTON’s tradition
and an instructor at the Royal Military Academy in Woolwich near London.
SIMPSON reportedly taught CHAPMAN a quadrature rule for planar curves,
which yields a numerical approximation of the integral of a continuous function
using at least one equally spaced double interval, hence at least three offset points
per double interval, or any desired number of further double intervals of the same
spacing. This quadrature rule later became known as “SIMPSON’s Rule”,
especially in shipbuilding, although the rule has several historical precursors and
SIMPSON never claimed to have invented it. CHAPMAN, after further visits to
France and Holland and having picked up much valuable background including
Historical Roots of the Theory of Hydrostatic Stability of Ships 175
the recent findings by BOUGUER on the metacenter, returned in 1757 to Sweden
to continue his career as an important ship constructor. In this process he soon
made it his habit to calculate ship displacements (1767), also as a function of draft,
and assessing the metacenter, applying SIMPSON’s Rule for numerical
integration. His engineering methods applied in ship design became well known
internationally soon after the publication of his book Tractat om Skepps-Byggeriet
(Chapman 1775), where he explained his calculations in much detail (Harris
1989). Thus CHAPMAN helped to promulgate the use of the metacenter and to
make its calculation a routine matter among naval architects. At the same time
SIMPSON’s Rule, which offers certain efficiency advantages over the trapezoidal
rule for an equal number of intervals, was popularized and is still much favored
4.2 Extensions of Stability Theory
4.2.1 Hydrostatic Stability About Other Axes
In their work on the masting of sailing ships in 1727 it must have occurred to both
BOUGUER and EULER that in general the external wind moments would act in
an oblique direction to the ship’s centerplane so that heeling and trimming would
result simultaneously. EULER remembered this open question and in “Scientia
Navalis”, Part 2, returned to this issue. He derived an expression for the restoring
moment about an oblique axis between the axes of heel and trim. The moments of
inertia about this axis could be constructed, when the moments of inertia IT and IL
for transverse and longitudinal stability were known, by using an ellipse of inertia
moment construction. Thus this method was able to predict at least the initial
hydrostatic response (small angles) to a wind moment in an oblique plane in terms
of simultaneous heel and trim angles. EULER mentioned on the side, too, that
longitudinal stability was always much greater than transverse stability.
It was some time later that the Spanish naval constructor and engineering
scientist Jorge JUAN Y SANTACILIA (1713-1773), who knew BOUGUER and his
work very well from the Peruvian expedition and also was in correspondence with
several European Academies, concerned himself not only with ship stability, but
also with ship oscillations, especially with pitching motions. In this context he
extended BOUGUER’s concepts of the metacenter to longitudinal inclinations and
first introduced the definition of the longitudinal metacentric height GML. He
pointed out that it by far exceeds the transverse metacentric height. His book
“Examen Maritimo”, which appeared in 1771, combined the theoretical insights of
his days with much practical design experience and became a widely used
reference and textbook (Juan y Santacilia 1771).
H. Nowacki and L.D. Ferreiro176
4.2.2 Large Angles of Heel
A broader extension of stability theory occurred not on the European continent but
in Britain. In two papers presented to the Royal Society of London, the British
mathematician George ATWOOD examined the inclination of ships at large angles
of heel. The first paper (Atwood 1796) based on a thorough knowledge of
ARCHIMEDES, BOUGUER and EULER, reviews the fundamental physical
principles of hydrostatic stability, applied to finite angles of heel. Here ATWOOD
investigates the stability properties of homogeneous solids of simple shape
(parallelepiped, cylinder, paraboloid) through 360 degrees of rotation as a function of
body draft at rest (or specific weight). He finds numerous equilibrium positions (8
or 16), only some of them stable. He develops a shifted wedge volume method for
finite heeling angles and reviews the numerical quadrature rules, settling for
STIRLING’s 3 interval rule. This paper already drew full attention to the fact that
stability must be judged over a range of finite, practical heel angles; initial
stability alone is inadequate as a stability measure.
In the second paper (Atwood and Vial du Clairbois 1798), which ATWOOD
coauthored with the French constructor Honoré-Sébastien VIAL DU CLAIRBOIS,
the investigation was extended to actual ships, and for the first time a numerical
analysis of the “righting moments” of ships over a large range of heeling angles
was performed. ATWOOD and VIAL DU CLAIRBOIS introduced the term GZ for
the “righting arm”, which was again numerically evaluated by a wedge volume
shift method. This successfully demonstrated the feasibility of numerical stability
analysis for actual ships over a range of finite heeling angles. The necessity of
performing such calculations rather than just relying on initial stability measures
was again underscored. Yet in shipbuilding practice it still took several more
decades before the initial difficulties in numerical integration could be overcome
by more robust methods and instruments. Even half a century later such stability
evaluation had not yet become a routine matter.
It was not before the early 19th
century that the knowledge on the metacenter
progressed one further step by the work of French constructor and mathematician
Charles DUPIN from metacentric curves to metacentric surfaces (Dupin 1813).
DUPIN stated that the ship for finite angles of inclination in whatever direction
possesses a surface of centers of buoyancy. This surface at every point has two
principal curvatures, thus one can construct two sheets of metacentric surfaces,
i.e., evolute surfaces of the buoyancy surface. The two metacenters for the upright
condition, MT and ML, each is a special point on one of the surfaces.
It is probably fair to say that at this level of perception the issue of hydrostatic
stability had been fully resolved in terms of the relationships between ship geometry,
physical responses to inclinations, and differential geometry of metacentric surfaces
for finite angles of heel.
Historical Roots of the Theory of Hydrostatic Stability of Ships 177
ARCHIMEDES laid the foundations for the stability of floating systems, introduced a
stability measure similar to the righting arm and presented an approach for assessing
the ability of a floating inclined solid to right itself. But his applications were limited
to simple geometrical shapes. Fortunately his manuscript “On Floating Bodies”
survived in a few copies and became more accessible again by a Latin translation
in the 13th
c. and also in print after 1500. Yet it took a few more centuries before
modern hydrostatic stability was established and could be applied to actual ships,
in particular also at the design stage.
The interest in scientific solutions for ship stability gained new momentum for
practical and scientific reasons by about 1700:
The leading navies were getting more concerned about stability risks with
increasing ship sizes, gunports that were close to the water and increasingly heavier
In the beginnings of the Age of Enlightenment, new expectations were raised
with regard to the capabilities of science to predict physical phenomena and the
performance of technical systems.
Mathematical breakthroughs occurred in infinitesimal calculus, analytical and
The modern age of engineering science made rapid progress in mechanics and
hydromechanics, including the study of equilibrium and stability of mechanical
All of this contributed to an atmosphere in the scientific community in the early
c. that was open to new challenges, also in application to ships. Both
BOUGUER and EULER were actively involved in these general developments and
certainly exposed to this unique zeitgeist. BOUGUER responded to the recognized
problem of hydrostatic stability more as an engineering scientist, EULER reacted
rather more like an applied mechanicist and mathematician. Both were able to
reformulate and solve this problem in their own unique and original ways.
As our comparisons in this paper have reconfirmed, BOUGUER’s and EULER’s
nearly simultaneous work was not only performed quite independently, which was
never doubted, but was also distinctly different in approach and justification. Both
investigated the ability of the ship to right itself after an infinitesimal heeling
displacement. BOUGUER reasoned mainly geometrically and for the inclined ship
derived a length measure, the metacentric height, as the decisive geometric
measure of initial stability. EULER argued mainly on mechanical grounds and
deduced the initial restoring moment as his measure of hydrostatic stability. Despite
these contrasting styles of justification, as we appreciate today of course, both
stability measures are in fact equivalent and can be converted into each other. We
may still benefit from both lines of thought and should be grateful for the insights
we owe to these two congenial pioneers.
H. Nowacki and L.D. Ferreiro178
We have explained why it took several more decades before these landmark
discoveries in ship stability were fully accepted and widely applied in shipbuilding
practice. But the foundations laid by BOUGUER and EULER have remained of lasting
value as secure cornerstones in our knowledge for designing safe and stable ships.
The authors are pleased to acknowledge the support and encouragement they
received during the preparation of this work from many colleagues, in particular
from Jean DHOMBRES, Emil A. FELLMANN, Jobst LESSENICH and Gleb
MIKHAILOV. The first author also expresses his gratitude to the Max Planck
Institute for the History of Science Berlin, where he is a Visiting Scientist and
where his work was much promoted by personal communications and the excellent
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L.S. McCue and A.W. Troesch182
thoroughly investigate the one degree of freedom problem, Vassalos and Spyrou
(1990) and Thompson and de Souza (1996) make clear the importance of studying
Murashige and Aihara (1998) investigate the case of a forced flooded ship
experimentally and mathematically. While they improve upon the traditional one-
dimensional model through inclusion of the effects of water on deck, their model
is based upon assumptions that cause one to neglect sway and heave degrees of
The work presented in this paper seeks to expand upon the one degree of
freedom models previously discussed, and the computationally intensive models
developed for fully nonlinear simulation and/or water on deck models which are
currently being developed and validated (see for example Huang et al, 1999 and
Belenky et al, 2002). This paper discusses results from a quasi-nonlinear time
domain model developed by Lee (2001) which is far less computationally
intensive than its fully nonlinear counterparts allowing for investigation of far
greater quantities of data. Additionally, the numerical model is validated by three
degree of freedom experimental results conducted at the University of Michigan
Marine Hydrodynamics Laboratory (Obar et al, 2001; Lee, 2001, Lee et al, 2006).
Over one hundred eighty separate experiments were used in the study to validate
the numerical model (Obar et al, 2001; Lee, 2001, Lee et al, 2006). For each test
case a simple box barge was excited at a frequency of 6.8 rad/s. The waves acting
upon the model had a wave height to wave length ratio of 0.02. Each release,
however, featured different initial conditions in roll, roll velocity, sway, sway
velocity, heave, and heave velocity. These values were measured in 1/30th
second increments for the duration of the experimental run. Approximately 37%
of the test runs resulted in capsize suggesting that initial conditions were a
significant contributor to subsequent dynamics. While roll initial conditions (i.e.
initial roll angle and roll velocity) are well known to influence capsize (e.g.
Thompson, 1997), little has been said about heave initial conditions. A reduced-
model nonlinear time-domain computer program is used in this work to provide
guidance in interpreting the experimental results.
Qualitative behavior of the capsize dynamics is presented in a number of formats.
The roll phase space is characterized by “safe/unsafe basins” (e.g. Thompson,
1997). These figures show capsize boundaries based upon roll initial conditions.
The evolution or change of these boundaries is investigated for varying sets of
heave and sway initial conditions. From similar analyses, “integrity curves” are
generated for systematic variations in the initial heave displacement and heave
velocity as well as sway displacement and velocity. Each point on an “integrity
curve” represents the ratio of safe (i.e. non-capsize) basin area for a given wave
amplitude to a similar safe basin area derived from zero wave amplitude. In this
way, the integrity curves show the relative influence of incident wave excitation
on capsize relative to vessel safety in the absence of incident waves.
Results from the simulation are presented here with discussions of accuracy and
time of computation. Qualitative guidelines are suggested and recommendations are
Sway/Sway Velocity Initial Conditions on Capsize Modeling 183
made for determining whether experimental programs may be sensitive to test
2 Numeral Analysis
2.1 Definition of Numerical Model
The works of Lee (2001) and Lee et al (2006) develop a quasi-nonlinear three
degree of freedom ‘blended’ hydrodynamic model to simulate highly nonlinear
roll motion of a box barge. The model uses an effective gravitational field to
account for the centrifugal forces due to the circular water particle motion in
addition to the earth’s gravitational field. This results in a local time dependent
gravitational field normal to the local water surface. Additionally the model
assumes long waves. Equation 1 below gives the equations of motion for the
numerical model in global coordinates where a and b are added mass and damping
coefficients, m and I are the mass and moment of inertia of the body, ge is the time
dependent effective gravitation, fD
are diffraction forces, is the time dependent
displaced volume of water, and GZ is the ship’s time dependent roll righting arm.
The global and body fixed coordinate systems for the model are defined in
L.S. McCue and A.W. Troesch184
Fig. 1 Coordinate system definition, Model scale definitions: 18.25 cm draft, 1.12 cm freeboard,
66 cm length, and 30.48 cm beam.
While the numerical model cannot account for the dynamics of water on deck,
such as sloshing, it does account for hydrostatic effects due to deck immergence.
The forcing due to the waves is modeled as a cosine wave. To simulate laboratory
transients, the wave “ramps” from zero until steady state. An experimentally
determined envelope curve was generated to create a ramped cosine wave that
closely resembles the waves generated in the tank (Obar et al, 2001). All release
times for the model are defined relative to the maximum transient wave crest as
shown in Figure 2 below and indicated by the notation “to”.
Fig. 2 Sample wave profile (top) with enlarged region showing definition of points relative to to
Sway/Sway Velocity Initial Conditions on Capsize Modeling 185
While this ‘blended’ model has admitted weaknesses due to the simplifications
employed, it has the distinct advantage of computational efficiency. Other, multi-
degree of freedom models that simulate various dynamics of water on deck run
significantly slower than real time (Belenky et al, 2002). However, this model,
whose results are qualitatively experimentally validated (Lee et al, 2006), runs in a
fraction of the actual run time. The work that makes up this numerical study
represents years worth of real time data; using this quasi-nonlinear model such
data can be collected in a matter of days.
2.2 Numerical results
As discussed in the work of Soliman and Thompson (1991), one can consider a set
of model releases in which all initial conditions, save roll and roll velocity, are
held constant. A sweep is made of roll and roll velocity initial condition pairs with
conditions resulting in capsize marked in black and non capsize left white yielding
a ‘safe basin’. Figure 3 shows a sample safe basin generated with this numerical
model (note e denotes excitation frequency, n represents vessel natural frequency in
roll, similarly te and tn represent excitation and natural periods respectively).
Fig. 3 Safe basin for release time of to-2 with sway, sway velocity, heave, and heave velocity
initial conditions set equal to zero at release. en≈3, h/=0.019.
L.S. McCue and A.W. Troesch186
Fig. 4 Safe basin for release times starting from to-2 in 30 degree increments with en≈3,
h/=0.019, roll angle from -30 to 30 deg and roll velocity from -30 to 30 deg/s; i.e. the left hand
column represents safe basins at to-2, to-1.5, to-1, to-.5, to, to+.5, to+1, and to+1.5. All heave and sway
displacements and velocities initially set equal to zero. x and y axes represent roll and roll
Sway/Sway Velocity Initial Conditions on Capsize Modeling 187
One can then consider a set of safe basins with some third parameter varied.
For example, Figure 4 shows a series of safe basins for different release times. The
figure shows safe basins for 48 different release times starting from to-2 until to+2.
Thus a shift between two safe basins represents a 30 degree shift in phase location
along the wave profile (i.e. 1/12 ). This figure illustrates how crucial accurate
accounting of time, and implicitly sway location, is on predicting vessel stability.
Additionally, this plot demonstrates the impact transients in the incident wave
profile have upon the safe basins. The safe basins before steady state, such as from
to-2 to to-1, exhibit no periodicity (i.e. to-2 and to-1’s safe basins are quite different),
while basins from to to to+2 produce a periodicity in that every 360 degrees of
phase (or one wave length) the safe basin is virtually identical (i.e. to+.5 and to+1.5
are nearly the same).
It should be noted that a change in sway displacement can be explicitly written
as a shift in time. For example, upon reaching steady state, a sway displacement
equal to one wave length at time tx could instead correspond to a sway displacement
of zero at a time tx+1, that is time moved forward through one wave period. Thus a
shift in time corresponds to a change in sway displacement if time is referred to by
a particular location in the wave profile, e.g. the maximum wave crest, to (Figure 2).
As described by Thompson (1997), one can then introduce the idea of integrity
values. Integrity is a ratio of safe area for given initial conditions to safe area for
some reference case. For a typical integrity curve, wave frequency and all non-roll
initial conditions would be held constant while wave amplitude is varied. As an
example, Figure 5 presents an integrity curve for the box barge numerical model
described in the previous section. In Figure 5 sway, sway velocity, heave, and
heave velocity are all set to zero at the instant of release. A safe basin is generated
for varying roll and roll velocity initial conditions (between -20 to 20 degrees and
–15 to 15 degrees/sec respectively). This is done for multiple wave amplitudes
with integrity values being calculated as the ratio of safe area at a particular wave
height to safe area for a wave height of zero. Thus the curve begins at one. The
resulting curve is an integrity curve as defined by Thompson (1997) in which one
notes the crucial characteristic ‘Dover cliff’ (1997) at which point there is a
distinct, rapid loss of stability. One can note that in Figure 5, integrity values are
greater than one for wave amplitude to wave length ratios of 0 to approximately
0.007. Physically, integrity values greater than 1.0 indicate that the presence of
waves at certain wave amplitudes can have a stabilizing effect upon the vessel
relative to the zero wave condition.
L.S. McCue and A.W. Troesch188
0 0.002 0.004 0.006 0.008 0.01 0.012 0.014
Wave Amplitude per Wave Length (o/)
Fig. 5 Integrity Curve. Sway, sway velocity, heave, and heave velocity 0 at time of release.
In Lee et al (2006) a detailed study is conducted considering the ways in which
these curves shift for varying initial conditions of sway, sway velocity, heave, and
heave velocity. This results in series of curves such as those found in Figure 6. In
this case one sees the variations in the location of the cliff; however, direct
comparison is difficult as the normalization factor for each curve varies. This is
analysed in detail for the 6 state variables in Lee et al (2006).
Fig. 6 Heave Integrity Curves. Sway, sway velocity, and heave velocity 0 at time of release.
Te/Tn~3. Note significant change with respect to heave in location of cliff on cur