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- 1. Contemporary Ideas on Ship Stability and Capsizing in Waves
- 2. FLUID MECHANICS AND ITS APPLICATIONS Series Editor: R. MOREAU MADYLAM Ecole Nationale Supérieure d’Hydraulique de Grenoble Boîte Postale 95 38402 Saint Martin d’Hères Cedex, France Aims and Scope of the Series fundamental role. As well as the more traditional applications of aeronautics, hydraulics, heat and mass transfer etc., books will be published dealing with topics which are currently in a state of rapid development, such as turbulence, suspensions and multiphase ﬂuids, super and hypersonic ﬂows and numerical modeling techniques. It is a widely held view that it is the interdisciplinary subjects that will receive intense scientiﬁc attention, bringing them to the forefront of technological advancement. Flu- ids have the ability to transport matter and its properties as well as to transmit force, therefore ﬂuid mechanics is a subject that is particularly open to cross fertilization with other sciences and disciplines of engineering. The subject of ﬂuid mechanics will be highly relevant in domains such as chemical, metallurgical, biological and ecological engineering. This series is particularly open to such new multidisciplinary domains. The median level of presentation is the ﬁrst year graduate student. Some texts are mono- graphs deﬁning the current state of a ﬁeld; others are accessible to ﬁnal year undergrad- uates; but essentially the emphasis is on readability and clarity. The purpose of this series is to focus on subjects in which ﬂuid mechanics plays a Volume 96 For further volumes: http://www.springer.com/series/5980
- 3. Editors Jean Otto de Kat • Kostas Spyrou • Naoya Umeda Marcelo Almeida Santos Neves • Vadim L. Belenky Contemporary Ideas on Ship Stability and Capsizing in Waves
- 4. No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com) means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose Springer Dordrecht Heidelberg London New York Library of Congress Control Number: 2011929340 © Springer Science+Business Media B.V. 2011 of being entered and executed on a computer system, for exclusive use by the purchaser of the work. ISSN 0926-5112 e-ISBN 978-94-007-1482-3ISBN 978-94-007-1481-6 Marcelo Almeida Santos Neves Department of Naval Federal University of Rio de Janeiro Parque Tecnologico, 21941-972 Rio de Janeiro Brazil Jean Otto de Kat Esplanaden 50 1098 Copenhagen Denmark Jean.Dekat@maersk.com masn@peno.coppe.ufrj.br Vadim L. Belenky Naval Surface Warfare Center Carderock D 20817-5700 West Bethesda Maryland USA vadim_belenky@hotmail.com Kostas Spyrou School of Naval Architecture National Technical University of Athens Iroon Polytechnion, 157 73 Athens Greece and Marine k.spyrou@central.ntua.gr Naoya Umeda Department of Naval Osaka University Graduate School of Eng Yamadaoka Suita 2-1 565-0971 Osaka Japan DOI 10.1007/978-94-007-1482-3 umeda@naoe.eng.osaka-u.ac.jp MacArthur Blvd 9500 Editors A. P. Moeller - Maersk A/S Cover design: eStudio Calamar S.L. Architecture and Oce Quadra 7 Architecture and Oce Zographou 9 Stability Criteria Evaluation and Performance Based Criteria Development for Damaged Naval Vessels, by Andrew James Peters © Copyright QinetiQ Limited 2010, by permission of QinetiQ Limited.
- 5. v Table of Contents Preface ....................................................................................................................xi 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 K.J. Spyrou Conceptualising Risk..............................................................................................47 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 Yury Nechaev SOLAS 2009 – Raising the Alarm.......................................................................103 Dracos Vassalos and Andrzej Jasionowski 2 Stability of the Intact Ship Effect of Initial Bias on the Roll Response and Stability of Ships in Beam Seas ........................................................................................................119 Historical Roots of the Theory of Hydrostatic Stability of Ships........................141 Horst Nowacki and Larrie D. Ferreiro The Effect of Coupled Heave/Heave Velocity or Sway/Sway Velocity Initial Conditions on Capsize Modeling..............................................................181 Leigh S. McCue and Armin W. Troesch A. Yucel Odabasi and Erdem Uçer
- 6. Table of Contentsvi The Use of Energy Build Up to Identify the Most Critical Heeling Axis Direction for Stability Calculations for Floating Offshore Structures.................193 Joost van Santen Some Remarks on Theoretical Modelling of Intact Stability...............................217 N. Umeda, Y. Ohkura, S. Urano, M. Hori and H. Hashimoto 3 Parametric Rolling An Investigation of Head-Sea Parametric Rolling for Two Fishing Vessels..................................................................................................................231 Marcelo A.S. Neves, Nelson A. Pérez, Osvaldo M. Lorca and Claudio Simple Analytical Criteria for Parametric Rolling...............................................253 K.J. Spyrou 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 4 Broaching-to Parallels of Yaw and Roll Dynamics of Ships in Astern Seas and the Effect of Nonlinear Surging ....................................................................363 K.J. Spyrou A. Rodríguez
- 7. Table of Contents vii Model Experiment on Heel-Induced Hydrodynamic Forces in Waves for Realising Quantitative Prediction of Broaching.............................379 H. Hashimoto, Naoya Umeda and Akihiko Matsuda Perceptions of Broaching-To: Discovering the Past ............................................399 K.J. Spyrou 5 Nonlinear Dynamics and Ship Capsizing Use of Lyapunov Exponents to Predict Chaotic Vessel Motions ........................415 Leigh S. McCue and Armin W. Troesch Applications of Finite-Time Lyapunov Exponents to the Study of Capsize in Beam Seas ......................................................................................433 Leigh S. McCue Nonlinear Dynamics on Parametric Rolling of Ships in Head Seas....................449 Marcelo A.S. Neves, Jerver E.M. Vivanco and Claudio A. Rodríguez 6 Roll Damping A Simple Prediction Formula of Roll Damping of Conventional Cargo Ships on the Basis of Ikeda’s Method and Its Limitation .........................465 Yuki Kawahara, Kazuya Maekawa and Yoshiho Ikeda A Study on the Characteristics of Roll Damping of Multi-Hull Vessels.............487 Toru Katayama, Masanori Kotaki and Yoshiho Ikeda 7 Probabilistic Assessment of Ship Capsize Capsize Probability Analysis for a Small Container Vessel................................501 E.F.G. van Daalen, J.J. Blok and H. Boonstra Efficient Probabilistic Assessment of Intact Stability..........................................515 N. Themelis and K.J. Spyrou Probability of Capsizing in Beam Seas with Piecewise Linear Stochastic GZ Curve ............................................................................................531 Vadim Belenky, Arthur M. Reed and Kenneth M. Weems Probabilistic Analysis of Roll Parametric Resonance in Head Seas....................555 Vadim L. Belenky, Kenneth M. Weems, Woei-Min Lin and J. Randolf Paulling
- 8. Table of Contentsviii 8 Environmental Modeling Sea Spectra Revisited ...........................................................................................573 Maciej Pawłowski On Self-Repeating Effect in Reconstruction of Irregular Waves.........................589 Vadim Belenky 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 Maciej Pawłowski 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 Flooding Simulation.............................................................................................689 Pekka Ruponen 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 Roll Motions.........................................................................................................735 Luis Pérez-Rojas, Elkin Botía-Vera, José Luis Cercos-Pita, Antonio Souto-Iglesias, Gabriele Bulian and Louis Delorme
- 9. Table of Contents ix 11 Design for Safety Design for Safety with Minimum Life-Cycle Cost..............................................753 Romanas Puisa, Dracos Vassalos, Luis Guarin and George Mermiris 12 Stability of Naval Vessels Stability Criteria Evaluation and Performance Based Criteria Development for Damaged Naval Vessels...........................................................773 Andrew J. Peters and David Wing A Naval Perspective on Ship Stability .................................................................793 Arthur M. Reed 13 Accident Investigations New Insights on the Sinking of MV Estonia........................................................827 Andrzej Jasionowski and Dracos Vassalos Experimental Investigation on Capsizing and Sinking of a Cruising Yacht in Wind ......................................................................................................841 Naoya Umeda, Masatoshi Hori, Kazunori Aoki, Toru Katayama and Yoshiho Ikeda Author Index.......................................................................................................855
- 10. Preface 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 Stability Workshop (Trieste, 2001), 6th Stability Workshop (Long Island, 2002), STAB 2003 Conference (Madrid), 7th Stability Workshop (Shanghai, 2004), 8th Stability Workshop (Istanbul, 2005), STAB 2006 Conference (Rio de Janeiro), 9th Stability Workshop (Hamburg, 2007), 10th 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) xi
- 11. List of Corresponding Authors 1 Stability Criteria Review of Available Methods for Application to Second Level Vulnerability Criteria Vadim Belenky vadim_belenky@hotmail.com A Basis for Developing a Rational Alternative to the Weather Criterion: Problems and Capabilities K.J. Spyrou spyrou@deslab.ntua.gr Conceptualising Risk Dracos Vassalos d.vassalos@strath.ac.uk Evaluation of the Weather Criterion by Experiments and its Effect to the Design of a RoPax Ferry Shigesuke Ishida ishida@nmri.go.jp Evolution of Analysis and Standardization of Ship Stability: Problems and Perspectives Yury Nechaev int@csa.ru SOLAS 2009 – Raising the Alarm Dracos Vassalos d.vassalos@strath.ac.uk 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 Horst Nowacki nowacki@naoe.tu-berlin.de xiii E. Uçer ucerer@itu.edu.tr
- 12. List of Corresponding Authorsxiv The Effect of Coupled Heave/Heave Velocity or Sway/Sway Velocity Initial Conditions on Capsize Modeling Leigh S. McCue mccue@vt.edu The Use of Energy Build Up to Identify the Most Critical Heeling Axis Direction for Stability Calculations for Floating Offshore Structures Joost van Santen joost.vansanten@gustomsc.com Some Remarks on Theoretical Modelling of Intact Stability Naoya Umeda umeda@naoe.eng.osaka-u.ac.jp 3 Parametric Rolling An Investigation of Head-Sea Parametric Rolling for Two Fishing Vessels Marcelo de Almeida Santos Neves masn@peno.coppe.ufrj.br Simple Analytical Criteria for Parametric Rolling K.J. Spyrou spyrou@deslab.ntua.gr Experimental Study on Parametric Roll of a Post-Panamax Containership in Short-Crested Irregular Waves Hirotada Hashimoto h_hashi@naoe.eng.osaka-u.ac.jp Model Experiment on Parametric Rolling of a Post-Panamax Containership in Head Waves Harukuni Taguchi taguchi@nmri.go.jp Numerical Procedures and Practical Experience of Assessment of Parametric Roll of Container Carriers Vadim Belenky vadim_belenky@hotmail.com Parametric Roll and Ship Design Riaan van’t Veer riaan.vantveer@sbmoffshore.com
- 13. List of Corresponding Authors xv Parametric Roll Resonance of a Large Passenger Ship in Dead Ship Condition in All Heading Angles Toru Katayama katayama@marine.osakafu-u.ac.jp Parametric Rolling of Ships – Then and Now J. Randolph Paulling RPaulling@aol.com 4 Broaching-to Parallels of Yaw and Roll Dynamics of Ships in Astern Seas and the Effect of Nonlinear Surging K.J. Spyrou spyrou@deslab.ntua.gr Model Experiment on Heel-Induced Hydrodynamic Forces in Waves for Realising Quantitative Prediction of Broaching Hirotada Hashimoto h_hashi@naoe.eng.osaka-u.ac.jp Perceptions of Broaching-To: Discovering the Past K.J. Spyrou spyrou@deslab.ntua.gr 5 Nonlinear Dynamics and Ship Capsizing Use of Lyapunov Exponents to Predict Chaotic Vessel Motions Leigh S. McCue mccue@vt.edu Applications of Finite-Time Lyapunov Exponents to the Study of Capsize in Beam Seas Leigh S. McCue mccue@vt.edu Nonlinear Dynamics on Parametric Rolling of Ships in Head Seas Marcelo A.S. Neves, masn@peno.coppe.ufrj.br 6 Roll Damping A Simple Prediction Formula of Roll Damping of Conventional Cargo Ships on the Basis of Ikeda’s Method and Its Limitation Yoshiho Ikeda ikeda@marine.osakafu-u.ac.jp
- 14. List of Corresponding Authorsxvi A Study on the Characteristics of Roll Damping of Multi-hull Vessels Toru Katayama katayama@marine.osakafu-u.ac.jp 7 Probabilistic Assessment of Ship Capsize Capsize Probability Analysis for a Small Container Vessel E.F.G. van Daalen E.F.G.van.Daalen@marin.nl Efficient Probabilistic Assessment of Intact Stability K.J. Spyrou spyrou@deslab.ntua.gr Probability of Capsizing in Beam Seas with Piecewise Linear Stochastic GZ Curve Vadim Belenky vadim_belenky@hotmail.com Probabilistic Analysis of Roll Parametric Resonance in Head Seas Vadim L. Belenky vadim_belenky@hotmail.com 8 Environmental Modeling Sea Spectra Revisited Maciej Pawłowski mpawlow@pg.gda.pl On Self-Repeating Effect in Reconstruction of Irregular Waves Vadim Belenky vadim_belenky@hotmail.com New Approach To Wave Weather Scenarios Modeling Alexander B. Degtyarev deg@csa.ru 9 Damaged Ship Stability Effect of Decks on Survivability of Ro–Ro Vessels Maciej Pawłowski mpawlow@pg.gda.pl
- 15. List of Corresponding Authors xvii Experimental and Numerical Studies on Roll Motion of a Damaged Large Passenger Ship in Intermediate Stages of Flooding Yoshiho Ikeda ikeda@marine.osakafu-u.ac.jp Exploring the Influence of Different Arrangements of Semi-Watertight Spaces on Survivability of a Damaged Large Passenger Ship Riaan van’t Veer riaan.vantveer@sbmoffshore.com Time-Based Survival Criteria for Passenger Ro-Ro Vessels Dracos Vassalos d.vassalos@strath.ac.uk Pressure-Correction Method and Its Applications for Time-Domain Flooding Simulation Pekka Ruponen pekka.ruponen@napa.fi 10 CFD Applications to Ship Stability Applications of 3D Parallel SPH for Sloshing and Flooding Dracos Vassalos d.vassalos@strath.ac.uk Simulation of Wave Effect on Ship Hydrodynamics by RANSE Dracos Vassalos d.vassalos@strath.ac.uk A Combined Experimental and SPH Approach to Sloshing and Ship Roll Motions Luis Pérez-Rojas luis.perezrojas@upm.es 11 Design for Safety Design for Safety with Minimum Life-Cycle Cost Dracos Vassalos d.vassalos@strath.ac.uk
- 16. List of Corresponding Authorsxviii 12 Stability of Naval Vessels Stability Criteria Evaluation and Performance Based Criteria Development for Damaged Naval Vessels Andrew J. Peters AJPETERS@qinetiq.com A Naval Perspective on Ship Stability Arthur M. Reed arthur.reed@navy.mil 13 Accident Investigations New Insights on the Sinking of MV Estonia Dracos Vassalos d.vassalos@strath.ac.uk Experimental Investigation on Capsizing and Sinking of a Cruising Yacht in Wind Naoya Umeda umeda@naoe.eng.osaka-u.ac.jp
- 17. 1 Stability Criteria
- 18. M.A.S. Neves et al. (eds.), Contemporary Ideas on Ship Stability and Capsizing in Waves, 3 Fluid Mechanics and Its Applications 96, DOI 10.1007/978-94-007-1482-3_1, © Springer Science+Business Media B.V. 2011 Review of Available Methods for Application to Second Level Vulnerability Criteria Christopher C. Bassler*, Vadim Belenky*, Gabriele Bulian**, Alberto Francescutto**, Kostas Spyrou***, Naoya Umeda**** *Naval Surface Warfare Center Carderock Division (NSWCCD) - David Taylor Model Basin, USA; **Trieste University; Italy; ***Technical University of Athens, Greece; ****Osaka University, Japan Abstract The International Maritime Organization (IMO) has begun work on the development of next generation intact stability criteria. These criteria are likely to consist of several levels: from simple to complex. The first levels are expected to contain vulnerability criteria and are generally intended to identify if a vessel is vulnerable to a particular mode of stability failure. These vulnerability criteria may consist of relatively simple formulations, which are expected to be quite conservative to compensate for their simplicity. This paper reviews methods which may be applicable to the second level of vulnerability assessment, when simple but physics-based approaches are used to assess the modes of stability failure, including pure-loss of stability, parametric roll, surf-riding, and dead-ship condition. 1 Introduction Current stability criteria have been used for several decades to determine the level of safety for both existing and novel ship designs. However, their current state is not representative of the level of our understanding about the mechanisms of dynamic stability failure of ships. Acknowledging this deficiency, the IMO has begun work on the development of next generation intact stability criteria to address problems related to dynamic phenomena and expand the applicability of criteria to current and future ship designs. The new generation criteria should include a range of fully dynamic aspects in their formulation. A thorough critical analysis of the existing situation at the beginning of the revision of the Intact Stability Code and a description of the present state- of-the-art can be found in a series of works by Francescutto (2004, 2007). The current framework of new generation intact stability criteria started to take shape at the 50th session of IMO Sub-committee on Stability and Load lines and
- 19. on Fishing Vessels Safety (SLF) with SLF-50/4/4 (2007), submitted by Japan, the Netherlands, and the United States and SLF-50/4/12 (2007), submitted by Italy. A significant contribution was also made in SLF-50.INF.2 (2007), submitted by Germany. These discussions resulted in a plan of development of new generation of criteria (see Annex 6 of SLF 50/WP.2). Following this plan, the intersessional Correspondence Group on Intact Stability developed the framework in a form of the working document that can be found in Annex 2 of SLF 51/4/1 (2008), submitted by Germany. Annex 3 of this paper contains the draft terminology agreed upon for the new criteria that was also developed by the Correspondence Group. The most current version of the framework document can be found in Annex 1 of the report of the Working Group on Intact Stability (SLF 51/WP.2). To facilitate these discussions, Belenky, et al. (2008) presented a broad review of issues related with development of new criteria, including the physical background, methodology, and available tools. It is the intention of the authors of this paper to provide a follow-up review, with a particular focus on methods for application to second level vulnerability criteria. The views and opinions expressed in this paper are solely and strictly those of the authors, mainly for facilitating wider and deeper discussion inside the research community. They do not necessarily reflect those of the delegations that involve the authors, the intersessional correspondence group on intact stability, or the working group on intact stability of the International Maritime Organization. The contents of this paper also do not necessarily indicate a consensus of opinion among the authors, but the authors agree on the need for further discussion of the contents. 2 Overview of New Generation of Intact Stability Criteria The major modes of stability failures were listed in section 1.2 of the 2008 IS Code, part A. They include: Phenomena related to righting lever variations, such as parametric roll and pure loss of stability; Resonant roll in dead ship condition defined by SOLAS regulation II-1/3 Broaching and manoeuvring-related problems in waves A multi-tier structure of new criteria has been identified from the ongoing discussion at IMO. Each of the four identified stability failure modes: pure-loss of stability, parametric roll, surf-riding and broaching, and dead ship conditions is evaluated using criteria with different levels of sophistication. The intention of the vulnerability criteria is to identify ships that may be vulnerable to dynamic stability failures, and are not sufficiently covered by existing regulation (SLF 51/WP.2, Annex 1). C.C. Bassler et al.4
- 20. Review of Available Methods for Application to Second Level Vulnerability Criteria 5 A first level of criteria (vulnerability criteria) could be followed by other stability assessment methods with increasing levels of sophistication. The first level of the stability assessment is expected to be relatively simple, preferably geometry based if possible. The function of this first level is to distinguish conventional ships, which are obviously not vulnerable to stability failure. The second tier is expected to be more sophisticated and assess vulnerability to dynamic stability with enough confidence that the application of direct calculation could be justified. The expected complexity of the second level vulnerability criteria should consist of a spreadsheet-type calculation. An outline of the process is shown in Fig. 1. The methods will most likely be different for each stability failure mode. 3 Pure Loss of Stability Differences in the change of stability in waves, in comparison with calm water, were known to naval architects since late 1800s (Pollard & Dudebout, 1892; Krylov 1958). However, it was uncommon until the 1960s that attempts to calculate the change of stability in waves (Paulling, 1961) and evaluate it with a series of model tests (Nechaev, 1978; available in English from Belenky & Sevastianov, 2007) were made. As a distinct mode of stability failure, pure-loss of stability was identified during the model experiments in San-Francisco Bay (Paulling, et al., 1972, 1975; summary available from Kobylinski & Kastner, 2003). Accurate calculation of the change of stability in waves presents certain challenges, especially at high speed, as the pressure around the hull and waterplane shape are influenced by the nonlinear interaction between encountered, reflected, and radiated waves (Nechaev, 1989). Stability Failure Mode Vulnerability Criteria Level 1 Ship Design Pass Fail Vulnerability Criteria Level 2 Pass Fail IMO IS Code Performance-Based CriteriaFor each mode Fig. 1. The proposed assessment process for next generation intact stability criteria IMO IS Code
- 21. The physical mechanism of pure-loss of stability is rather simple: if the stability is reduced for a sufficiently long time, a ship may capsize or attain large roll angle (see Fig, 2, taken from Belenky, et al., 2008). This phenomenon is presently addressed, although in a very short and qualitative form, by the MSC.1/Circ.1228 (2007). MSC.1/Circ.1228 does not provide any ship-dependent information concerning the considered failure mode. The difficulty of developing criteria for pure-loss of stability is not limited to the calculation of the GZ curve in waves. A probabilistic characterization of the associated stability changes is also difficult, due to nonlinear nature of stability changes in waves. Boroday (1967, 1968) developed a method for the assessment of statistical characteristics for restoring moments in waves, followed by an energy balance- based method for evaluation of the probability of capsizing (Boroday & Netsevatev, 1969). Using energy balance methods for changing stability in waves was also the focus of Kuo & Vassalos (1986). Dunwoody (1989a, 1989b), Palmquist (1994), and Bulian & Francescutto (2006a) considered statistical characteristics of GM and other elements of the righting moment as a stochastic process. The alternative approach of the “effective wave” was proposed by Grim (1961). In this method, the length of the wave is equal to ship length— the wave is unmovable. However, its amplitude is a random process (Fig. 3). Fig. 2. Capsizing due to pure-loss of stability Fig. 3. 0 50 100 150 50 50 100 150 200 t, s Time history of roll and capsizing Phase of encounter wavePhase of heeling moment 0 10 20 30 40 50 60 0.5 1 Wave trough Calm water Wave crest Magnitude of heeling moment -0.5 deg Righting / Heeling Arm, m C.C. Bassler et al.6 A concept of an effective wave
- 22. Review of Available Methods for Application to Second Level Vulnerability Criteria 7 The amplitude of the effective wave is calculated in order to minimize the difference between the effective and encountered waves. Umeda, et al. (1993) performed comparative calculations and suggested the effective wave to be of satisfactory accuracy for engineering calculations. The effective wave approach was used in a number of works; of special interest are those with a “regulatory perspective.” Helas (1982) discussed the reduction of the righting arm in waves. To consider the duration where stability is reduced, the amplitude of the effective wave was averaged over a time sufficient to roll to a large angle, about 60% of natural roll period. As a result, the average effective wave amplitude becomes a function of Froude number. Analyzing results of the calculation performed for different vessels in different sea conditions, recommendations were formulated for a stability check in waves. Small ships and ships with “reduced stability” were meant to be checked using the GZ curve— evaluated for the effective wave with averaged amplitude against conventional calm water stability criteria. Umeda and Yamakoshi, (1993) used the effective wave to evaluate the probability of capsizing due to pure-loss of stability in short-crested irregular stern-quartering waves with wind. The capsizing itself was associated with departure from the time-dependent safe basin. The effect of initial conditions was taken into account using statistical correlation between roll and the effective wave. Instead of the Grim effective wave, Vermeer (1990) assumed waves to be a narrow-banded stochastic process. Capsizing is associated with the appearance of negative stability during the time duration, and it is sufficient for a ship to reach a large roll angle under the action of a quasi-static wind load. The probability of capsizing is considered as a ratio of encountered wave cycles, where the wave amplitude is capable of a significant and prolonged deterioration of stability. Themelis and Spyrou (2007) demonstrated the use of the concept of a “critical wave” to evaluate probability of capsizing. Because pure-loss of stability is a one wave event, it is straightforward to separate the dynamical problem from the probabilistic problem. First, the parameters that are relevant to the waves capable of causing pure-loss of stability are searched. Then, the probability of encountering such waves is assessed. To complete the probabilistic problem, the initial conditions are treated in probabilistic sense. Bulian, et al. (2008) combined the effective wave approach with upcrossing theory, where the phenomenon of pure-loss of stability is associated with an up- crossing event for the amplitude of the effective wave. Such an approach allows the time of exposure to be considered directly and produces a time-dependent probability of stability failure. An additional advantage of upcrossing theory is the possibility to characterize the “time above the level”, which in this context is the duration of decreased stability. The mean value of this time can be evaluated by a simple formula, but evaluation of other characteristics is more complex (Kramer and Leadbetter, 1967). In summary, it seems that the second level vulnerability criteria for pure-loss of stability are likely to be probabilistic, due to the very stochastic nature of the
- 23. phenomenon. For criteria related to this mode of stability failure, the time duration while stability is reduced on a wave could be included. Methods useful for addressing pure-loss of stability may include, but are not limited to, the effective wave, the narrow-banded waves assumption, the critical wave approach, and upcrossing theory. As Monte-Carlo simulations may be too cumbersome for vulnerability criteria, at this time, simple analytical and semi-analytical models seem to be preferable. 4 Parametric Roll Parametric roll is the gradual amplification of roll amplitude caused by parametric resonance, due to periodic changes of stability in waves. These stability changes are essentially results of the same pressure and underwater geometry changes that were the cause in the pure-loss of stability mode. However, in contrast to pure-loss of stability, parametric roll is generated by a series of waves of certain frequency. Therefore, parametric roll cannot be considered as a one wave event (Figure 4). Research on parametric roll began in Germany in the 1930s (Paulling, 2007). In the 1950s, the study of parametric roll for a ship was continued by Paulling & Rosenberg (1959). Paulling, et al. (1972, 1975) observed parametric roll in following waves during model tests in San-Francisco Bay (summary available from Kobylinski & Kastner, 2003). Later it was discovered that this phenomenon could occur in head or near head seas as well (Burcher, 1990; France, et al., 2003). Fig. 4. Partial (a) and total (b) stability failure caused by parametric resonance 0 100 200 300 400 500 t, s -20 20 , deg 0 100 200 300 400 500 t, s -50 50 100 150 200 , degb) a) C.C. Bassler et al.8
- 24. Review of Available Methods for Application to Second Level Vulnerability Criteria 9 Parametrically excited roll motion is at present widely recognized as a dangerous phenomenon, where the cargo and the passengers and the crew may be at risk. Hull forms have evolved from those considered in the development of the 2008 IS Code, as MSC Res. 267(85)— basically the same ship types used 30-50 years ago for the definition of Res. A. 749(18) (2002). For certain ship types, present hull forms appear to have resulted in the occurrence of large amplitude rolling motions associated with parametric excitation. According to this evidence, there is a compelling need to incorporate appropriate stability requirements specific to parametric roll into the new generation intact stability criteria. The simplest model of parametric resonance is the Mathieu equation. It can describe the onset of instability in regular waves. Because the Mathieu equation has a linear stiffness term, it is incapable of predicting the amplitude of roll, and not applicable in irregular waves. Nevertheless, it was used by ABS (ABS, 2004; Shin, et al., 2004) to establish estimates for initial susceptibility to parametric excitation, which corresponds more with intended use for the first level vulnerability criteria. More complex 1(.5)-DOF models include nonlinearity in the time-dependent variable stiffness term. This nonlinearity leads not only to stabilization of roll motion, e.g. establishing finite roll amplitude, but also to a significant alteration of instability boundary in comparison of Ince-Strutt diagram— with unstable motions found outside of the Mathieu instability zone (Spyrou, 2005). This should cause concern about relying on the standard linear stability boundaries. Inclusion of additional degrees of freedom, primarily pitch and heave (Neves & Rodríguez, 2007) results in additional realism. These are especially important for parametric roll in head seas, where pitch and heave may be large. Spyrou (2000) included surging as well. These models, even with their low dimensional character, seem to give reasonable agreement with model tests (Bulian, 2006). Evaluation of the steady-state amplitude of parametric roll can be carried out using asymptotic methods, such as the method of multiple scales (Sanchez & Nayfeh, 1990). Approximate analytical expressions exist for the determination of the nonlinear roll response curve (e.g. ITTC, 2006). The potential of these methods is discussed in Spyrou et al. (2008). The introduction of irregular waves results in drastic changes in the way parametric roll develops. Large oscillations may not be persistent anymore: it may stay, but it may also be transitory (Figure 5). Such behavior has significant influence on the ergodicity of the process— the ability to obtain statistical characteristics from one realization if it is long-enough. Theoretically, the process remains ergodic, as the integral over its autocorrelation function still remains finite, but the time necessary for convergence may be very large. As a result, the term “practical non-ergodicity” is used (Bulian, et al., 2006; Hashimoto, et al., 2006).
- 25. Distribution of parametric roll may be different from Gaussian. Hashimoto, et al. (2006) obtained distributions with large excess of kurtosis from model tests in long- and short-crested irregular seas (Fig. 6). Roll damping has more influence on roll amplitude in irregular waves. The damping threshold defines if a wave group can contribute to parametric excitation and therefore, to results in an increase of the roll amplitude. Characteristics of wave groups, such as number, length, and the height of waves in the group, may also have an influence on how fast the roll amplitude increases. In a sense, a regular wave can be considered as an infinite wave group. Therefore, using regular waves could be too conservative for an effective vulnerability criterion, due to the transient nature of this phenomenon (SLF48/4/12). Focusing on the instability boundaries in irregular waves, Francescutto, et al., (2002) used the concept of “effective damping,” also mentioned in Dunwoody (1989a); that is meant to be subtracted from roll damping. Bulian, et al. (2004) used the Fokker-Planck equation and the method of multiple scales for the same purpose. They utilized Dunwoody’s spectral model of GM changes in waves (1989a, 1989b). Furthermore, Bulian (2006) made use of Grim’s effective wave Fig. 5. Sample records of simulated parametric roll in long-crested seas (Shin, et al., 2004) Fig. 6. Distribution of roll angles during parametric resonance in (a) long-crested and (b) short crested seas 0 200 400 600 800 1000 1200 1400 -40 -20 0 20 40 Time, s RollAngle,deg Time, s 0 200 400 600 800 1000 1200 1400 -40 -20 0 20 40 RollAngle,deg a) b) C.C. Bassler et al.10
- 26. Review of Available Methods for Application to Second Level Vulnerability Criteria 11 concept. Despite overestimating the amplitude of roll, the boundary of instability was well identified. Application of the Markov processes for parametric roll in irregular waves was considered by Roberts (1982). To facilitate Markovian properties, (dependence only on the previous step), white noise must be the only random input of the dynamical system. Therefore, the forming filter, usually a linear oscillator, must be part of the dynamical system to ensure irregular waves have a realistic-like spectrum. These complexities allow the distribution of time before the process reaches a given boundary to be obtained— the first passage method. This method is widely used in the field of general reliability. Application of this approach to parametric roll was considered by Vidic-Perunovic and Jensen, (2009). Time-domain Monte-Carlo simulations (Brunswig, et al., 2006) are increasingly used as computers become more powerful and are integral part of any naval architect’s office. At the same time, caution has to be exercised when using Monte- Carlo simulations, due to statistical uncertainty and other methodological challenges that may limit its use for second level vulnerability criteria. Nevertheless, having in mind the difficulties associated with prediction of amplitude for parametric resonance in irregular waves, Monte-Carlo method should not be dismissed. Themelis & Spyrou (2007) used explicit modelling of wave groups as a way to separate problems related to ship dynamics from the probabilistic problem. This method exploited the growth rate of parametric roll due to the encounter of a group, i.e. by solving the problem in the transient stage. Thereafter, the probability of encounter of a wave group can be calculated separately. In principle, this may allow enough simplification to make Monte-Carlo methods into feasible and robust tool for the second level vulnerability criteria for parametric roll. In summary, it seems that the second level vulnerability criteria for parametric roll may be related to the size of the instability area in irregular seas. The group wave approach seems to be promising, as a method to build a criterion based on amplitude. 5 Surf-Riding and Broaching Broaching is the loss of controllability of a ship in astern seas, which occurs despite maximum steering effort. Stability failure caused by broaching may be “partial” or “total”. It can appear as heeling during an uncontrollable, tight turn. This is believed to be intimately connected with the so called surf-riding behaviour. Moreover, it can manifest itself as the gradual, oscillatory build-up of yaw, and sometimes roll, amplitude. The dynamics of broaching is probably the most dynamically complex phenomenon of ship instability. Well known theoretical studies on broaching from the earlier period include Davidson (1948), Rydill (1959), Du Cane & Goodrich (1962), Wahab & Swaan (1964), Eda (1972); Renilson & Driscoll (1981), Motora, et al.(1981); and more recently, e.g. by Umeda &
- 27. Renilson (1992), Ananiev & Loseva (1994), and Spyrou (1996, 1997, 1999). Broaching has also been studied with experimental methods. Tests with scaled radio-controlled physical models have taken place in large square ship model basins; e.g. Nicholson (1974), Fuwa (1982), De Kat & Thomas (1998). Surf-riding is a peculiar type of behaviour where the ship is suddenly captured and then carried forward by a single wave. It often works as predecessor of broaching and its fundamental dynamical aspects have been studied by Grim (1951), Ananiev (1966), Makov (1969), Kan (1990), Umeda (1990), Thomas & Renilson (1990), and Spyrou (1996). Key aspects of the phenomenon are outlined next. In a following or quartering sea, the ship motion pattern may depart from the ordinary periodic response. Figure 7 illustrates diagrammatically the changes in the geometry of steady surge motion, under the gradual increase of the nominal Froude number (Spyrou 1996). An environment of steep and long waves has been assumed. For low Fn, there is only a harmonic periodic response (plane (a)) at the encounter frequency. However, as the speed approaches the wave celerity, the response becomes asymmetric (plane (c)). The ship stays longer in the crest region and passes quickly from the trough. In parallel, an alternative stationary behavior begins at some instance to coexist. This results from the fact that the resistance force which opposes the forward motion of the ship may be balanced by the sum of the thrust produced by the propeller and the wave force along the ship’s longitudinal axis. This is surf-riding and the main feature is that the ship is forced to advance with a constant speed equal to the wave celerity. It has been shown that surf-riding states appear in pairs: one nearer to wave crest and the other nearer to trough. The one nearer to the crest is unstable in the surge direction. Stable surf- riding can be realised only in the vicinity of the trough and, in fact, only if sufficient rudder control has been applied. Fig. 7. Changing of surging/surf-riding behavior with increasing nominal Froude number G x Fn Periodic surging Surf-riding equilibria At wave trough At wave crest Fncr1 Fncr2 cos(2xG/ a b c d C.C. Bassler et al.12
- 28. Review of Available Methods for Application to Second Level Vulnerability Criteria 13 Surf-riding is characterised by two speed thresholds: the first is where the balance of forces becomes possible (plane(b)). The second threshold indicates its complete dominance in phase-space, with sudden disappearance of the periodic motion. It has been pointed out that, this disappearance is result of a global bifurcation phenomenon, known in the nonlinear dynamics literature as homoclinic saddle connection (Spyrou, 1996). It occurs when a periodic orbit collides in state- space with an unstable equilibrium— the surf-riding state near to the wave crest. A dangerous transition towards some, possibly distant, undesirable state is the practical consequence. 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 10 10 20 G, m G, m/s . Surf-riding Surf-riding Surging Surging
- 29. From the above, one sees that, susceptibility to this type of broaching could be assessed by targeting three sequential events: the condition of attraction to surf- riding; the condition defining an inability to stay in stable surf-riding, thus creating an escape; and the condition of reaching dangerous heeling during this escape— manifested as tight turn. For the first condition, we already have deterministic (analytical and numerical) criteria based on Melnikov and other methods (Ananiev, 1966; Kan, 1990; Spyrou, 2006). For the second, criteria have been discussed, linking ship manoeuvring indices with control gains (Spyrou, 1996, 1998). The third condition can be relatively easily expressed as the balance of the roll restoring versus the moment of the centrifugal force that produces the turn. A proper probabilistic expression of these conditions is still wanting. However, the critical wave approach may provide a very practical way of dealing with this (Themelis & Spyrou, 2007). Efforts in similar spirit were reported by Umeda (1990) and Umeda, et al. (2007). In reviews of broaching incidents, those that had happened at low to moderate speeds (i.e. far below the wave celerity of long waves) are usually distinguished from the more classical high-speed events. These incidents could have some special importance, because they seem to be relevant also to vessels of larger size, which otherwise unlikely to broach the nature of such a broaching mechanism g. It has been shown that, a Further increase of the commanded heading was found to cause a rapid increase of the amplitude of yaw oscillation, leading shortly to a turn backwards of the steady yaw response curve and, inevitably, a sudden and dangerous jump to resonant yaw (Spyrou, 1997). The transient behaviour looks like a growing oscillatory yaw which corresponds closely to what has been described as cumulative-type broaching. Fig. 9. Phase plane with surf-riding only Fn>Fncr2 are through surf-riding. Let’s consider now , which we will call “direct broaching” since the ship is not required to go through surf-ridin fold bifurcation is possible to arise for a critical value of the commanded heading, This phenomenon can have a serious influence on the horizontal plane dynamics, creating a stable sub-harmonic yaw (Fig. 10). 300 200 100 0 100 10 10 20 G, m Surf-riding Surf-riding Surf-riding G, m/s . C.C. Bassler et al.14
- 30. Review of Available Methods for Application to Second Level Vulnerability Criteria 15 Instability could be an intrinsic feature of the yaw motion of a ship overtaken by very large waves. It has been pointed out that, starting from Nomoto’s equation, the equation of yaw in following waves becomes Mathieu-type (in Spyrou (1997) the derivation and discussion on this and its connection with the above scenario was presented). The amplitude of the parametric forcing term is the ratio of the wave yaw excitation to the hull’s own static gain— maneuvering index K. The damping ratio in this equation receives large values, unless there is no differential control. Due to the large damping, the internal forcing that is required in order to produce instability is much higher (steeper waves) than that at the near zero encounter frequency (i.e. for the mechanism of broaching through surf-riding). Simple criteria connecting the hull with rudder control have accrued from this formulation and could be directly in the vulnerability criteria. A probabilistic formulation has not yet been attempted. 6 Dead Ship Condition Both the term and the concept of dead ship conditions took its root during the steamer era. When a ship looses power, it becomes completely uncontrollable— a significant disadvantage in comparison with ship equipped with sails. The worst possible scenario is a turn into beam seas, where the ship is then subjected to a resonant roll combined with gusty wind. Traditional ship architecture, with a superstructure amidships, made this scenario quite possible. Significant changes in marine technology have led to more reliable engines and a variety of architectural Fig. 10. Direct broaching One period motion Stable sub- harmonic yaw One period motion Stable sub- harmonic yawFlip bifurcation Fold bifurcation One period motion Amplitudeofperiodicmotion C commanded heading Rudderangle Yaw angle
- 31. types. Nevertheless the dead ship condition— zero speed, beam seas, resonant roll, and wind action, maintains its importance. The present Intact Stability Code (MSC.267(85), 2008), as well as its predecessor (IMO Res. A749(18), 2002), contains the “Severe Wind and Rolling Criterion (Weather Criterion).” This criterion contains a simple mathematical model for ship motion under the action of beam wind and waves. However, some parameters of this model are based on empirical data. Therefore, it may not be completely adequate for unconventional vessels. Thus MSC.1/Circ.1200 (2006) and MSC.1/Circ. 1227 (2007) were implemented, to enable alternative methodologies to be used for the assessment of the Weather Criterion on experimental basis. The problem of capsizing represents the ultimate nonlinearity: the dynamical system transits to motion about another stable equilibrium. When the wave excitation amplitude approaches a critical value, a number of nonlinear phenomena take place. The boundary of the safe basin becomes fractal (Kan & Taguchi, 1991) as a result of the self-crossing branches of an invariant manifold (Falzarano, et al., 1992). Considering the deterioration of the safe basin (Figure 11), Rainey and Thompson (1991) proposed a transient capsize diagram. It uses the dramatic change of the measure of integrity of the safe basin as a boundary value. Another obvious way to improve criterion is to consider more realistic, e.g. stochastic, models for the environment. This does not necessarily means that the new criterion must be probabilistic. The improvement will be made by adjusting parameters of deterministic criterion to reach similar level of safety for all the vessels which meet it. Studies of probabilistic methods for these criteria were carried out starting in the 1950s (Kato, et al., 1957). As the conventional weather criterion uses an Fig. 11. ith increasing of wave amplitude 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 SMOOTH BOUNDARY FRACTAL BOUNDARY AreaofSAFEBASIN 1 FINAL CRISIS Normalized amplitude of excitation a b c d e f C.C. Bassler et al.16 Deterioration of safe basin w
- 32. Review of Available Methods for Application to Second Level Vulnerability Criteria 17 energy balance approach, an attempt to consider it from probabilistic point of view seems logical (Dudziak & Buczkowski 1978; a brief description is available from Belenky & Sevastianov, 2007). Similar ideas are behind the Blume criterion (Blume, 1979, 1987), which uses a measure of the remaining area under the GZ curve. Bulian & Francescutto (2006b) and Bulian, et al. (2008) developed a method where the righting arm is locally linearized at the equilibrium, due to the action of a steady beam wind. The probability of large roll is calculated using an “equivalent area” to account for the nonlinearities of the righting arm. The piecewise linear method (Belenky, 1993; Paroka & Umeda, 2006) separates probability of capsizing into two less challenging problems: upcrossings through maximum of the GZ curve and capsizing if such upcrossing has occurred (Figure 12). The probability of upcrossing is well defined in upcrossing theory. While the probability of capsizing after upcrossing can be found using the probability of roll rate at upcrossing that lead to capsizing; this is a linear problem in the simplest case. Statistical linearization can be used to evaluate parameters of the piecewise linear model. Simplicity and a direct relation with time of exposure are among the strengths of the piecewise linear method. In principle, a critical wave approach can be used for dead ship conditions as well. However, the outcome of capsizing vs. non-capsizing strongly depends on initial conditions. Therefore, a critical wave group approach similar to Themelis and Spyrou (2007) may be also considered as a candidate for vulnerability criteria. 7 The Choice of Environmental Conditions If vulnerability criterion is deterministic and based on a regular wave assumption, the wave is characterized with height and length. These characteristics must then be related to a corresponding sea state and further on with a safety level. This formulation is not new— this approach is used for evaluation of extreme loads. A typical scheme for calculation of extreme loads is based on long-term assumption, so a number of sea states are considered. An operational profile is Fig. 12. Separation principle in a piecewise linear method t Capsized equilibrium Threshold Upcrossing leading to capsizing Upcrossing not leading to capsizing
- 33. usually assumed based on existing experience; it includes the fraction of time that a ship is expected to spend in each sea state. Short-term probability of exceedance is calculated for each sea state; then the formula for the total probability is used to determine the life-time probability of exceedance of the given level. This level is typically associated with significant wave height and zero-crossing, or mean period. These data are then directly used to define a regular wave that is expected to cause extreme loads. Similar ideas underlie the reference regular wave in the Weather Criterion in the 2008 IS Code, which was determined from the significant wave steepness, based on the work of Sverdrup and Munk (1947). With all of the similarities, a direct transfer of methods used for extreme loads to stability applications may be difficult. Structural failure is associated with exceeding certain stress levels, actual physical failure is not considered in Naval Architecture. Stresses do not have their own inertia, they follow the load without time lag. The same can be observed about the loads, they occur simultaneously with a wave. They do not depend on initial conditions; even in a nonlinear formulation and do not have multiple responses for the given position of a ship on a wave. Nevertheless, these motion nonlinearities affect only short-term probability, while the scheme as a whole may be applicable. Determining the equivalence between regular and irregular waves represents a problem by itself. The use of significant wave height or steepness has been a conventional assumption, but it does not have a theoretical background behind it. The general problem of equivalency between regular and irregular roll motions was considered by Gerasimov (1979; a brief description in English is available from Belenky & Sevastianov, 2007) in the context of statistical linearization with energy conservation, resulting in an energy statistical linearization. The idea of a regular wave as some sort of equivalent for sea states is attractive because of its simplicity. However, the physics of some stability failures may be sufficiently different in regular and irregular waves. As it was mentioned in the section regarding parametric roll, a regular wave is an infinitely long wave train. Roll damping in this case has a very small influence on the response amplitude, while in irregular seas damping has a significant influence on the variance of response. Therefore, the regular equivalent of irregular waves cannot be applied universally. The application of different environmental models is possible for different stability modes. Surf-riding/broaching is an example of a one-wave event where the concept of a critical wave may fit well. If some of vulnerability criteria are made probabilistic, the next important choice is time scale: long-term or short term. Here, short-term refers to a time interval where an assumption of quasi-stationarity can be applied. Usually this is an interval of three to six hours, where changes of weather can be neglected. Long-term consideration covers a larger interval: a season, a year, or the life-time of a vessel. The short-term description of the environment is simpler and can be characterized with just with one sea state or spectrum. However, if chosen for vulnerability criteria, justification will be required as to why a particular sea must be used. This C.C. Bassler et al.18
- 34. Review of Available Methods for Application to Second Level Vulnerability Criteria 19 choice is important because sea states which are too severe may make the criterion too conservative and diminish its value. Therefore, special research is needed in order to choose the sea state “equivalent” or “representative” for a ship operational profile. This may naturally result in ship-specific sea-state to use for assessment. An alternative to the selection of a limited set of environmental conditions may be the use of long-term statistics considering all the combinations of weather parameters available from scatter diagrams, or appropriate analytical parametric models (see, e.g. Johannessen, et al., 2002). 8 Summary and Concluding Comments Vulnerability to pure-loss of stability is caused by prolonged exposure to decreased stability on a wave crest, and is likely to be characterized by probabilistic criteria. Methods for a first attempt for vulnerability criteria include the effective wave, a narrow-band or envelope presentation, the critical wave approach, and upcrossing theory. 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 be deterministic. 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. 9 Acknowledgments
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- 38. Review of Available Methods for Application to Second Level Vulnerability Criteria 23 Spyrou KJ (1996) Dynamic instability in quartering seas: The behaviour of a ship during broaching J. Ship Res 40 1 :46-59 Spyrou KJ (1997) Dynamic instability in quartering seas-Part III: Nonlinear effects on periodic motions. J Ship Res 41 3:210-223 Spyrou KJ (1999) Annals of Marie–Curie fellowships, edited by the European commission. http://www.mariecurie.org/annals/volume1/spyrou.pdf Spyrou KJ (2000) On the parametric rolling of ships in a following sea under simultaneous nonlinear periodic surging. Phil Trans R Soc London, A 358:1813-1834 Spyrou KJ (2005) Design criteria for parametric rolling. Ocean Eng Int, Canada 9 1:11-27 Spyrou KJ (2006) Asymmetric surging of ships in following seas and its repercussions for safety. Nonlinear Dyn, 43:149-172 Spyrou KJ, Tigkas I, Scanferla G, Themelis N (2008) Problems and capabilities in the assessment of parametric rolling. Proc 10th Int Ship Stab Workshop. Daejeon, Korea 47-55 Sverdrup HU, Munk WH (1947) Wind, sea and swell, theory of relations for forecasting. Hydrogr Off Publ 601 Themelis N, Spyrou KJ (2007) Probabilistic assessment of ship stability trans. SNAME, 115:181-206 Thomas G, Renilson M (1992) Surf-riding and loss of control of fishing vessels in severe following seas. Trans RINA 134:21-32 Umeda N (1990) Probabilistic study on surf-riding of a ship in irregular following seas. Proc of 4th Int Conf onn Stab of Ships and Ocean Veh. Naples 336-343 Umeda N, Renilson MR (1992) Broaching— A dynamic analysis of yaw behavior of a vessel in a following sea. In Wilson PA (ed) Maneuvering and Control of Mar Craft, Comput Mech Publ. Southampton 533-543 Umeda N, Yamakoshi Y (1993) Probability of ship capsizing due to pure loss of stability in quartering seas. Nav Archit and Ocean Eng: Sel Pap Soc of Naval Arch of Japan 30:73-85 Umeda N, Shuto M, Maki A (2007) Theoretical prediction of broaching probability for a ship in irregular astern seas. Proc 9th Int Ship Stab Workshop. Hamburg Umeda N, Hori M, Hashimoto H (2007) Theoretical prediction of broaching in the light of local and global bifurcation analysis. Int Shipbuild Prog 54 4:269-281 Vermeer H (1990) Loss of stability of ships in following waves in relation to their design characteristics. Proc of 4th Int Conf on Stab of Ships and Ocean Veh. Naples Vidic-Perunovic J, Jensen JJ (2009) Estimation of parametric rolling of ships – comparison of different probabilistic methods. Proc 2nd Int Conf on Mar Struct. Lisbon, Portugal Wahab R, Swaan WA (1964) Course-keeping and broaching of ships in following seas. J Ship Res 7 4
- 39. M.A.S. Neves et al. (eds.), Contemporary Ideas on Ship Stability and Capsizing in Waves, 25 Fluid Mechanics and Its Applications 96, DOI 10.1007/978-94-007-1482-3_2, © Springer Science+Business Media B.V. 2011 A Basis for Developing a Rational Alternative to the Weather Criterion: Problems and Capabilities K.J. Spyrou School of Naval Architecture and Marine Engineering, National Technical University of Athens, 9 Iroon Polytechneiou, Zographos, Athens 15780, Greece The feasibility of developing a practical ship dynamic stability criterion based on nonlinear dynamical systems’ theory is explored. The concept of “engineering integrity” and Melnikov’s method are the pillars of the current effort which can provide a rational connection between the critical for capsize wind/wave environment, the damping and the restoring characteristics of a ship. Some discussion about the dynamical basis of the weather criterion is given first before describing the basic theory of the new method. Fundamental studies are then carried out in order to ensure the validity, the potential and the practicality of the proposed approach for stability assessment of ships in a wider scale. 1 Introduction The weather criterion adopted as Resolution A.562 by IMO’s Assembly in 1985 was a leap beyond the “statistical approach” of the earlier Rahola-type general intact ship stability criteria of 1969 where the safety limits were based empirically on GZ characteristics of vessels lost even a 100 years ago (IMO 1993). Despite the criticism the weather criterion has a rational basis in the sense that there is some account of ship roll dynamics integrated within the stability assessment process. Nonetheless this analysis has a simplified character and in the light of recent research advances on the one hand and new trends in design on the other, it seems that the time has come for looking more seriously into the potential of alternative approaches. It is no surprise that at IMO the discussion about the development of new criteria has been re-opened. In the current paper a few steps are taken towards clarifying whether the concept of “engineering integrity” of dynamical systems could be used for setting up an improved and “general purpose” stability criterion (Thompson 1997). This corresponds basically to implementing a Abstract
- 40. 26 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 still unanswered. 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 K.J. Spyrou
- 41. A Basis for Developing a Rational Alternative to the Weather Criterion 27 Fig. 1 The IMO weather criterion Fig. 2 The structure of the weather criterion A number of extra comments pertain to the modeling of system dynamics: although the input of energy from the waves is taken into account for the calculation of the attained angle θ1 and the ship is like being released in still-water from the angle θ1 (Fig. 2). The only energy balance performed concerns potential energies in the initial and final positions. An advantage of the criterion is that the damping is somehow present in the calculation of the resonant roll angle θ1 both in terms of hull dimensions, form θ θ1 θ0 θ2 θc lw1 lw2 θ0 Restoring and heeling lever b a θ θ2 , it is not considered during the final half-cycle
- 42. 28 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 . θ θdownflooding 25 deg GZC c GZmax 0 Fig K.J. Spyrou
- 43. A Basis for Developing a Rational Alternative to the Weather Criterion 29 3 The Concept of “Engineering Integrity” Several articles have been published about the concept of “loss of engineering integrity” of dynamical systems. The reader who is not acquainted with the basic theory and terminology is referred for example to Thompson (1997). The basic idea is that the safety robustness of an engineering system can be represented by the integrity of its “safe basin” comprised by the set of initial conditions (in our case pairs of roll angle and velocity) which lead to bounded motion. It is known that the basin could suffer some serious reduction of area (“basin erosion”) once some critical level of excitation is exceeded given the system’s inertial damping and restoring characteristics. The reason for this is the complex intersection of “manifolds” i.e. those special surfaces which originate from, or end on, the unstable periodic orbits corresponding to the vanishing angles which become time- dependent when there is wave-forcing. Detailed analysis about these phenomena can be found in various texts see for example Guckenheimer & Holmes (1997). The initiation of basin erosion can be used as a rational criterion of system integrity which in our case and for a naval architectural context would be translated as sufficient dynamic stability in a global sense. The methods that exist for predicting the beginning of basin erosion are the following: a) Use of the so-called Melnikov analysis which provides a measure of the “closeness” of the critical pair of manifolds and hence can produce the condition under which this distance becomes zero. For capsize prediction the method is workable when the damping is relatively low (strictly speaking it is accurate for infinitesimal damping - however several studies have shown that the prediction is satisfactory for the usual range of ship roll damping values). Ideally the method is applied analytically; or it may be combined with a numerical part when the required integrations cannot be performed analytically. b) Direct numerical identification of the critical combination of excitation damping and restoring where the manifolds begin to intersect each other. This method is more accurate but requires the use of specialised software. c) Indirect identification of the critical condition by using repetitive safe basin plots until the initiation of basin erosion is shown. The same comment as in (b) applies here. 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:
- 44. 30 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) Where: 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): 20 0 ε xf τε τ O Μ d (2) d t( ) x f x( ) Hamiltonian K.J. Spyrou
- 45. A Basis for Developing a Rational Alternative to the Weather Criterion 31 Where 0τ is a phase angle in the range /2τ0 0 . Since the denominator is of the order of 1 the function 0τM (which is called Melnikov function) is to first order a good measure of the distance of manifolds: τττ,τxgτxfτ 00 dM (3) The wedge symbol means to take the cross product of the vectors f and g . 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 further). 5 Melnikov’s Method for Ship Rolling With a Wind-Induced Bias We have applied the method for the following family of restoring curves which have the bias as free parameter: 32 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: 20 3 1 2 112 21 βττΩsin1 xFaxxaxx xx (5) If the bias is relatively strong the unperturbed part is represented by the vector: 3 1 2 11 2 1 xf axxax x (6) Where
- 46. 32 2 1 x x x (7) While the perturbation is: 2β-Ωτsin 0 xF xg (8) After substitution into (3) the Melnikov function takes the form of the following integral: DE MMdxFxM 0 - 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: F a 0.055 Ω=0.8 Ω=0.9 Ω=1 0.940.920.90.880.860.840.820.8 0.035 0.04 0.045 0.05 K.J. Spyrou
- 47. A Basis for Developing a Rational Alternative to the Weather Criterion 33 sasaas d sd 112 τ 23 2 2 (10) With some manipulation the above can be written as: 22 qpshs d ds (11) The parameters that appear in (11) are defined as follows: a aa qand a a p a h 3 212 3 122 , 2 (12) The solution of (11) is: τ1cosh 1ττ 22 aqp qp xs ∓ (13) Differentiation of (13) yields: 2 22 2 τ1cosh τ1sinh1 aqp aqpaq x d dx d ds (14) 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: q pqp q p pqqpqph F arccoshμsin μπsinh 6ππ arccosh32 β 22 22222 (15) The sustainable wave slope is: 2 ν . Ων φF Ak crit (16)
- 48. 34 where νφ is the vanishing angle and xx x II I δ ν . 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 (Papagiannopoulos 2001). a) Higher-order GZ In the first instance we considered the family of fifth-order restoring polynomials 53 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 GZ K.J. Spyrou
- 49. A Basis for Developing a Rational Alternative to the Weather Criterion 35 sudden reduction of basin area beyond some critical wave forcing for all the members of this family i.e. that the concept is “robust” with respect to the characteristics of the restoring function. Fig. 6 Scaled “quintic” restoring curves In Fig. 7 are shown some typical results of this investigation for an initially hardening GZ at four different levels of damping. The points on the curves are identified with measurement of the basin area and scaling against the area of the corresponding unforced system. The curves confirm that for the fifth-order levers the concept of engineering integrity is applicable without any problem i.e. there is indeed a quick fall of the curve beyond some critical forcing. This critical forcing should be approximately equal to the forcing determined from the Melnikov method. An advantage however of the direct basin plot is that it allows to determine fractional integrities also like 90% 80% etc. as function of damping and excitation (90% integrity means that the remaining basin area is 90% of the initial). This is quite important for criteria development since it provides the necessary flexibility for setting the boundary line; for example one could base the maximum sustainable wave slope at 90% integrity rather than at 100% which might be too stringent.
- 50. 36 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 integrity 1 0.8 0.6 0.4 0.2 1 0.8 0.6 0.4 0.2 0 0.05 0.1 0.15 0.2 0.01 0.05 0.1 0.2 b 0.25 integrity Forcing amplitude F 0 0.05 0.1 0.15 0.2 0.25 Forcing amplitude F c=1, Ω=1 c=1, Ω=0.88 K.J. Spyrou
- 51. A Basis for Developing a Rational Alternative to the Weather Criterion 37 wave steepness considered in the weather criterion as function of natural period under the assumption of resonance condition. Fig. 8 Critical wave environment for an existing ship on the basis of the new concept 7 Use of the Concept of Engineering Integrity for Design Optimisation 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 nondimensionalised position l x 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 order polynomial
- 52. 38 4 4 3 3 2 21 2 xaxaxa B xX (19) 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 xn zzxZ 1, (20) where xtsxn (21) 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 K.J. Spyrou
- 53. A Basis for Developing a Rational Alternative to the Weather Criterion 39 Fig. 10 Roll damping for each examined hull Although originally the height z is non-dimensionalised with respect to the draught, in our case this is done with respect to the depth because the hull-shape above the design waterline is very important in stability calculations ( 01 z ). To narrow further on the free parameters of this investigation we have fixed the length, the beam and the depth; respectively to 150m, 27.2m and 13.5m. The displacement was also kept constant at 23710 t which means that only the block coefficient and the draught were variable. In addition we have considered only hulls fore-aft symmetric. KG was linked to the depth while the windage area was set to be 4.1 L x T. Fig. 11 Dynamic stability performance: Left bars are for waves only; while the right bars are for wind and waves. 0,08 0,07 0,06 0,05 0,04 0,03 0,02 0,01 0 1 3 5 7 9 11 S h i p N o 13 15 17 19 0,035 0,03 0,025 0,02 (H/λ)critical 0,015 0,01 0,005 0 1 3 5 7 9 11 S h i p N o 13 15 17 19
- 54. 40 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 1.5 1.0 0.5 0 0 50 Heel angle (degrees) GZ (m) Area (m-rad) 100 K.J. Spyrou
- 55. A Basis for Developing a Rational Alternative to the Weather Criterion 41 side or accumulation of water on the deck. Also, we have left out of the present context any discussion about stability in following seas, a matter which deserves a separate approach due to the different nature of the dynamics involved. Although the following-sea is the most dangerous environment of ship operation, it was not seriously addressed up to now in the IMO Regulations. As a matter of fact, ships are designed with no requirement for checking their stability at the most critical condition that they may encounter although the primary modes of ship capsize in a following sea environment are now well understood. Of course such a matter is not raised for the first time and in the academic world several approaches have been discussed in the past: some are based on the area under the time-varying GZ curve (butterfly diagram see e.g. Vassalos 1986) while more recently there is a trend for taking into account “more fully” system dynamics. For parametric instability already exist general criteria that have been developed in the field of mechanics. However it is doubted whether these are known to ship designers (for an attempt to summarize these criteria see for example Spyrou et al. 2000). Furthermore the clarification of the dynamics of broaching-to has opened up new possibilities for the development of a simple criterion for avoiding the occurrence of this phenomenon too. It is emphasized that the proposed methodology is a platform for developing also formulations targeting the following-sea environment. Furthermore since this approach is a global method and is not based on ordinary simulation it presents also distinctive advantages over the so-called “performance-based methods” a term which in an IMO-style vocabulary is assumed to mean the execution of a limited number of experimental or simulation runs. Our analysis up to this stage shows that although the proposed method can produce meaningful results there are a few practical problems whose solution is uncertain and would require further research. A deeper look into the Melnikov- based approach is presented in a companion paper (Spyrou et al 2002). Some questions that immediately come to mind are: • How to deal with down-flooding angles or more generally limits of stability that are lower than the vanishing angle? The problem arises from the fact that the method targets the phenomena affecting the basin boundary while the amplitude of rolling may be well below that level if there is flooding of non-watertight compartments. The formulation of Melnikov’s method as an energy balance might give the idea for a practical solution. • How to combine the deterministic and the stochastic analysis? This happened quite meaningfully in the weather criterion and some comparable yet not obvious in terms of formulation approach is required. • How to adjust the level of stringency of the criterion? This could be achieved perhaps by setting the acceptable level of integrity to a fractional value such as 90% rather than the 100% that is currently considered in research studies.
- 56. 42 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 values. • 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 environmental conditions. • 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. References Falzarano J Troesch AW and Shaw SW (1992) Application of global methods for analysing dynamical systems to ship rolling and capsizing. Int J of Bifurc and Chaos 2:101-116. Frey M and Simiu E (1993) Noise-induced chaos and phase-space flux Physica D 321-340. Guckenheimer J and Holmes P (1997) Nonlinear oscillations dynamical systems and bifurc of vector fields. Springer-Verlag N.Y. Himeno Y (1981) Prediction of roll damping – state of the art. Report No. 239. Dep of Naval Archit and Marine Eng The Univ of Michigan Ann Arbor MI. Hsieh SR Troesch AW and Shaw SW (1994) A nonlinear probabilistic method for predicting vessel capsizing in random beam seas. Proc of the Royal Soc A446 1-17. IMO (1993) Code on Intact Stability for All Types of Ships Covered by IMO Instruments. Resolut A.749(18) London. Min KS and Kang SH (1998) Systematic study on the hull form design and resistance prediction of displacement-type-super-high-speed ships. J of Marine Sci and Technol 3: 63-75. NES 109 (2000) Stability Standard for Surface Ships. U.K. Ministry of Defence Issue 1 Publication date 1st April 2000. 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 Athens Sept. 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. K.J. Spyrou
- 57. A Basis for Developing a Rational Alternative to the Weather Criterion 43 Spyrou KJ (2000) Designing against parametric instability in following seas. Ocean Eng, 27 653. 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. Vassalos D (1986) A critical look into the development of ship stability criteria based on work/energy balance. R.I.N.A. Transactions, 128, 217-234. Watanabe (1938) Some contribution to the theory of rolling I.N.A. Transactions, 80, 408-432. Wiggins S. (1992) Chaotic transport in dynamical systems Springer-Verlag N.Y. Appendix I: Melnikov’s Method as Energy Balance Consider again the roll equation with bias: τcos11β Faxxxxx (I1) The corresponding Hamiltonian system is: 011 axxxx (I2) with kinetic energy 2 2 1 KE x (I3) and potential energy Fig. I1 Energy balance around the homoclinic orbit 625- 50 Yamagata, M. (1959) Standard of stability adopted in Japan, RINA Transactions, 101, 417-443.
- 58. 44 432 43 1 2 1 11PE x a x a xcaxxx (I4) 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 (Fig. I1): τββDE 2 dxdxx (I5) 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. τττcosβWEDE 0 2 dxFdx (I7) 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 K.J. Spyrou
- 59. A Basis for Developing a Rational Alternative to the Weather Criterion 45 specific formulation and adaptation for the ship capsize problem is owed to Hsieh et al (1994). The key idea is that the probability of capsize in a certain wave environment is linked to the rate of phase-flux. The formulation is laid out as follows: 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): ( )2 ε ε OM H M PM H M pH A D s D D s D s + ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ −⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ +⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ Α = Φ σσ σ (II1) 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:
- 60. 46 p q pqqpqphM D arccosh32β 3 1 22222 (II2) 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: 00 0 22 2τ dSSdSME FxME (II3) The spectral density function of the wave forcing is linked to the wave spectrum SFSF 2 (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 - τ τ π2 1 dexS i x (II4) 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 ). , , K.J. Spyrou
- 61. M.A.S. Neves et al. (eds.), Contemporary Ideas on Ship Stability and Capsizing in Waves, 47 Fluid Mechanics and Its Applications 96, DOI 10.1007/978-94-007-1482-3_3, © Springer Science+Business Media B.V. 2011 Conceptualising Risk Andrzej Jasionowski, Dracos Vassalos SSRC, NAME, University of Strathclyde / Safety At Sea Ltd Abstract Ever present jargon and colloquial notion of risk, coupled with obscuring of the simplicity inherent in the concepts of risk by the sophistication of computer software applications of techniques such as event trees, fault trees, Bayesian networks, or others, seem to be major factors inhibiting comprehension and systematic proliferation of the process of safety provision on the bases of risk in routine ship design practice. This paper attempts to stimulate consideration of risk at the fundamental level of mathematical axioms, by proposing a prototype of a comprehensive yet plain model of risk posed by the activity of ship operation. A process of conceptualising of substantive elements of the proposed risk model pertinent to the hazards of collision and flooding is presented. Results of tentative sensitivity studies are presented and discussed. 1 Introduction It is well recognised that the engineering discipline entails constant decision making under conditions of uncertainty, that is decisions are made irrespective of the state of completeness and quality of supporting information, as such information is never exact, it must be inferred from analogous circumstances or derived through modelling, and thus be subject to various degrees of approximation, or indeed, the information often pertains to problems involving natural processes and phenomena that are inherently random. No better example of such decision making process in engineering can be invoked, than that of development of regulations for the provision of safety. The process of rules development is well recognised to rely on both, (a) qualified “weigh ing” of a multitude of solutions derived on the basis of scientific research, engineering judgement, experiential knowledge, etc, and (b) debates accommodating the ubiquitous political agendas at play and governing societal concerns. The result of such complex and predominantly subjective pressures is often a simplistic compromise normally in the form of deterministic rules which does not reflect the underlying uncertainty explicitly. t
- 62. A. Jasionowski and D. Vassalos48 Although such standards have formed the main frame of safety provision over a century or so, the acknowledgement of existence of these uncertainties, brought about by the occurrences of accidents with undesirable consequences on one hand, and realisation that economic benefit could be compromised by inbuilt but potentially irrelevant margins, on the other hand, instigated a search such as (SAFEDOR 2005) for better methodologies for dealing with the issues of safety. Notably, such search has been enhanced, or possibly even inspired, by the tremendous progress in computational technologies, which by sheer facilitation of fast calculations have allowed for development of numerical tools modelling various physical phenomena of relevance to engineering practice, at levels of principal laws of nature. Such tools present now the possibility to directly test physical behaviour of systems subject to any set of input parameters, and thus also allow derivation of an envelope of the inherent uncertainty. Existence of such methods, however, is not sufficient for progressing with the development of more efficient means of safety provision, without a robust framework for reasoning on the results of any such advanced analyses undertaken for safety verification processes. Such means evolve naturally from the principles of inductive logic; a field of mathematics dealing with reasoning in a state of uncertainty, (Jaynes 1995) (Cox 1961). In particular the branch comprising probability theory forms a suitable framework for such reasoning in the field of engineering. However, while the concept of probability is known to engineers very widely, it does not seem that the many unresolved to this day subtleties associated with either, its fundamental principles or relevant linguistics, are recognised by the profession. This is the situation especially when a special case of application of the probability theory, namely that of risk, is considered. The uptake of risk assessment process, or more recently strife to develop and implement risk-based design paradigms, (SAFEDOR 2005) (Konovessis 2001) (Guarin 2006), are examples of a new philosophy that could alleviate the aforementioned problems of safety provision, if they can become suitable for routine practice. This in turn, can only be achieved if fundamental obstacles such as lack of common understanding of the relevant concepts are overcome. Therefore, this paper sets to examine some lexical as well as conceptual subtleties inherent in risk, offers some suggestions on more intuitive interpretations of this concept, and puts forward a proposal for risk modelling process as a contributory attempt to bringing formalism in this development. 2 The Risk Lexicon As has been the case with the emergence of probability in the last 350 years or so, (Hacking 1975), the terminology on risk has not evolved into anything that can be regarded as intuitively plausible lexicon, as neither have the pertinent syntax nor semantics been universally endorsed.
- 63. Conceptualising Risk 49 Any of tautological phrases, such as: “level of safety standard”, “hazard threatening the safety”, semantically imprecise statements such as: “collision risk”, “risk for collision”, “risk from collision”, “risk of collision”, ”risk from hazard aspects of …”, simple misnomers such as “safety expressed as risk”, “safety risk”, or philosophically dubious slogans such as “zero risk1 ”, can be found in the many articles discussing or referring to the concepts of risk. A few other, easily misconceiveable terms used in this field can be mentioned: (a) likelihood, chance, probability (b) frequency, rate (c) uncertainty, doubt, randomness (d) risk, hazard, danger, threat (e) analysis, assessment, evaluation (f) risk control measures, risk control options (g) safety goals, safety objectives, safety functional statements, safety performance, relative safety 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 quality. 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. 2.1 Safety 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: 1 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 justification.
- 64. A. Jasionowski and D. Vassalos50 safety is the state of acceptable risk This definition is unambiguous and it is practical. It emphasizes explicitly that safety is a state. It thus cannot be calculated, it can only be verified by comparison of the actually established quantity of risk with the quantity of risk that is considered acceptable by relevant authoritative bodies; a criterion or standard, in other words. Thus, it follows that statements such as “higher safety” are incompatible with this definition, as the safety is either attained or it is not and no grades of safety can be distinguished. Appropriate statement would be “higher safety standard”, which directly implies lowered levels of risk as acceptable. Also, expressions such as “safety performance” become semantically imprecise unless, contrary to intuitive perception, the word “performance” implies a binary mode of compliance/no compliance with the set standard, rather than a scale, which in turn, relates to risk not safety. 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 sought after. The direct application of the concept of risk is one such method, especially since the above definition of safety implies the “presence of risks”. The key question now is what is risk?
- 65. Conceptualising Risk 51 2.2 Risk 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 any interpretation. For instance, the syntax of the above definition implies differentiation of the semantics of the phrases such as “hazard” and “consequence”, without any qualification how such differentiation is attained. Thus, the sinking of the vessel can be considered as a hazard, which can lead to the loss of human life, but it can also be considered as an ultimate consequence, or in other words as the loss. See here Fig. 1 and the subsequent discussion. While the merit of proposal of such definitions of risk could be its generic nature, overly lack of any specificity deprives it of practical significance. The following alternative definition is suggested, which is equally generic yet it is comprehendible more intuitively and, apparently, more pragmatic: risk is a chance of a loss The word “chance” is one of the most fundamental phrases referring to, or being synonymous with the natural randomness inherent in any of human activities and which, in this case, can bring about events that are specifically undesirable, the “loss”. The engineering aspects of the concept of chance are effectively addressed by the fields of probability and statistics, both of which offer a range of instruments to quantify it, such as e.g. expectation, and hence there is no need for introduction of unspecific terms such as “combination”. These well known mathematical tools will be discussed in section 3. This definition of risk resolves also, to some extent, the manner of its classification. Namely reference to risk should be made in terms of the loss rather than in terms of any of the intermediate hazards in the chain of events leading to the loss, a nuance just mentioned.
- 66. A. Jasionowski and D. Vassalos52 Fig. 1 The concept of a chain of events in a sample scenario leading to the loss. Each of the events is a hazard after it materialises. A scenario can be identified by a principal hazard, see Table 1. For instance, semantics of a statement “risk to life” or “risk of life loss” seem to indicate sufficiently precisely the chance that an event of “fatality” can take place, without much space for any other interpretations. On the other hand, a statement such as “risk of collision” can be interpreted as a chance of an event of two ships colliding; however, given a further set of events that are likely to follow, such as flooding propagation, heeling, capsizing, evacuation/abandoning and fatalities, see Fig. 2, it becomes unclear what is the “loss”, and thus what is the risk? Furthermore, the above definition supports lexically the recently more pronounced efforts to introduce the concept of holism in risk assessment. Namely, rather than imposing a series of criteria for acceptable risk levels in a reductionism manner, whereby separate criteria pertaining to different hazards are introduced, such as e.g. criteria on fire hazards or criteria on flooding hazards (e.g. probabilistic concept of subdivision, (SLF 47/17 2004) ), a unique criterion should be proposed that standardises the acceptable risk level of the ultimate loss, e.g. criteria for loss of life, environmental pollution, etc, and which risks account for each hazard that can result in the loss, a view already mentioned in (Skjong et al. 2005) or (Sames 2005). This concept underlines the risk model proposed in the following section 3. 3 Risk Modelling As mentioned above, the fields of probability and statistics provide all the essential instruments to quantify the chance of a loss. 3.1 Frequency Prototyping To demonstrate the process of application of these instruments for risk modelling, perhaps it will be useful to stress the not necessarily trivial difference between the concepts of probability and frequency. Namely, to assess a “probability of an event”, a relevant random experiment, its sample space and event in question on that sample space must all be well defined. The relative frequency of occurrence of this event, whereby relative implies relative to the sample space, is a measure of probability of that event, since such relative frequency will comply with the axioms of probability.
- 67. Conceptualising Risk 53 In the case of “an experiment” of operating a ship, however, whereby events involving fatalities can occur in the continuum of time, the pertinent sample space is customarily not adopted, and rather, the rate of occurrence of these events per unit time is used. The rate can be greater than unity, and obviously can no longer be referred to as probability. Thus, also use of the term “likelihood” would seem to be displaced here since it is synonymous with probability rather than frequency. Therefore, application of the term frequency per ship per year is customary, also since its quantification relies on the historical data for the relevant fleet. Lack of a well defined sample space, however, does not prevent one from using concepts of probability, such as e.g. distribution, expectation, etc, in analyses and analytical modelling. 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 ship operation (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 principal hazards2 , 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. 2 an event of a loss scenario is a hazard that materialised
- 68. A. Jasionowski and D. Vassalos54 Deriving from the above, it is hereby proposed that the frequency frN (N) of occurrence of exactly N fatalities per ship per year is derived based on Bayes theorem on total probability, as follows: (1) 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 2004) 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). (2) jNj n j hzN hzNprhzfrNfr hz 1 NcepwhzNpr kji n k kj i n j iN Hsflood ,, 3 1
- 69. Conceptualising Risk 55 Where the terms wi and pj are the probability mass functions of three, i=1 ..3, possible ship loading conditions and a j=1 … nflood flooding extents, respectively, assigned according to the harmonised probabilistic rules for ship subdivision, (SLF 47/17 2004), or (Pawlowski 2004). The term ek is the probability mass function derived from (15) for the sea state Hsk, where mHsk 40 , 1 4 Hsk nkHs 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). 30 ln)( 30 ,,,,,, Nt Nc fail Nt kjikjikji fail (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: ijcrity ijcritk kji sHs sHsHs )( 1,, (4) 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 Appendix 2. Fig. 3 The concept of “capsize band”, (Jasionowski et al. 2004), for critical sea states of 0.5 and 4.0m. The concept of a capsize band 6 5 4 3 2 1 0 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.5 1 1.5 2 2.5 Hs crititcal pdf(Hs) pcap=cdf(Hs) 3 3.5 4 4.5 5
- 70. A. Jasionowski and D. Vassalos56 1 NtNtNt failfailfail (6) tNNt failfail 1 (7) 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. 2001). 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 equation (9).
- 71. Conceptualising Risk 57 ifrNF N Ni NN max (9) 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). iFNERisk N i NPLL max 1 (10)
- 72. A. Jasionowski and D. Vassalos58 As it is known in the field of statistics, it is always desirable to find a statistic which is not only consistent and efficient, but ideally is sufficient, as then all relevant information is contained in one such number. In case of a statistic which is not sufficient, additional information is required to convey the characteristics of the underlying random process. Since the expectation is hardly ever a sufficient statistic, it is necessary to examine what additional information is required additionally to (10) to quantify risk sufficiently comprehensively. Namely, it is well known that the public tolerability for large accidents is disproportionally lesser than the tolerance for small accidents even if they happen at greater numbers. This tendency is not disclosed in the model (10), as simply the same expected number of fatalities can result from many small accidents or a few larger ones. It is for this reason, that model (10) is also referred to as a risk-neutral or accident-size-neutral, (Evans 2003). Thus, public aversion to large accidents cannot be built into any relevant criteria based on PLL in its current form and without recurs to some form of utility function. Although there are strong suggestions negating such a need and supporting the view that a risk-neutral approach is sufficient, e.g. (Skjong et al. 2005), it does not seem that the above simple fact of real life can be ignored. Indeed, recently proposed criteria for stability based on the so called “probabilistic concept of subdivision” defy this risk-neutral philosophy, as ships with more passengers are required higher index of damage stability. Therefore, a proposal by (Vrijling and Van Gelder 1997) shown as equation (11) seems worth serious consideration as an alternative to (10), as it allows for controlled accommodation of the aversion towards larger accidents through stand. dev. and a risk-aversion index k. (11) 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). NkNERisk
- 73. Conceptualising Risk 59 However, since it is known that collision and flooding is one of the major risk contributors, a sample study aiming to demonstrate the usefulness of the proposed risk model is undertaken here based on only this loss scenario. Consider a RoRo ship, carrying some 2200 passengers and crew, and complying with the new harmonised rules on damaged ship stability by meeting the required index of subdivision of 0.8 exactly, see Figure 6. Consider also that the evacuation arrangements allow for two different evacuation completion curves, as shown in Figure 7. The question is what is the effect of given evacuation curve on risk to life posed by this ship? The risk is calculated for nhz=1 in (1), using constant frhz(hz1)=2.48•10–3 per ship year

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