structural reliability framework
DESCRIPTION
structureTRANSCRIPT
HSEHealth & Safety
Executive
Structural Reliability Framework for FPSOs/FSUs
Prepared by Universities of Glasgow and Strathclydefor the Health and Safety Executive 2004
RESEARCH REPORT 261
HSEHealth & Safety
Executive
Structural Reliability Framework for FPSOs/FSUs
Moatsos Ioannis MEng(Hons), GMRINA, MSNAME,SMIStructE, SMIMarEst
ASRANet Ship Group Championand
Purnendu K. Das B.E, M.E, Ph.D, MIStructE, FRINA, FIEProfessor of Marine Structures
Dept. of Naval Architecture and Marine EngineeringUniversities of Glasgow and Strathclyde
Henry Dyer Building100 Montrose Street
GlasgowScotland UK
Structural Reliability Analysis (SRA) is a tool for calculating failure probabilities of a structure that takesinto account all random variables involved in ship structural design and provides a better andalternative to deterministic approaches. With a number of different regulations based on structuralreliability theory already existent for quite some time now in the areas of Civil and Aeronauticalengineering and the Nuclear Power industry it has been only a matter of time before the ship/offshorefield implemented such procedure into its regulatory regime. Classification Societies have publishedsince the early 1990s suggestions (Det Norske Veritas Class Note 30.6 1992) of how reliability analysistechniques can be used to analyse structures and recently some have incorporated and acceptedthose (Bureau Veritas 2000) as an equivalent means of analyzing structures to supplement theirexisting regulations.
This research examines the structural reliability of FPSO/Tanker structures already in operation in theNorth Sea/West of Shetland area, using a component reliability approach that takes into account theultimate strength of the vessel’s structure. Corrosion effects on the structure are modelled using asimple mathematical model based on actual measurements. Emphasis is placed on the modelling ofloads for use in reliability analysis. Load combination methodology is used for the combination of wavebending, still water, slamming and thermal loads using different approaches such as the Ferry BorgesCastanheta Method, Turkstra’s Approach and the SRSS Rule. A number of FPSO/Tanker vessels areanalysed using First and Second Order Reliability Method (FORM/SORM) component reliabilitytechniques to obtain Partial Safety Factors and probabilities of failure.
This report and the work it describes were funded by the Health and Safety Executive (HSE). Itscontents, including any opinions and/or conclusions expressed, are those of the authors alone and donot necessarily reflect HSE policy.
HSE BOOKS
ii
© Crown copyright 2004
First published 2004
ISBN 0 7176 2890 6
All rights reserved. No part of this publication may bereproduced, stored in a retrieval system, or transmitted inany form or by any means (electronic, mechanical,photocopying, recording or otherwise) without the priorwritten permission of the copyright owner.
Applications for reproduction should be made in writing to:Licensing Division, Her Majesty's Stationery Office, St Clements House, 2-16 Colegate, Norwich NR3 1BQ or by e-mail to [email protected]
iii
ACKNOWLEDGEMENTS
The authors would like to express their sincerest gratitude to Emeritus Professor Douglas Faulkner for his guidance and useful comments and Mr. Michel Huther of Bureau Veritas, France for providing the authors with a copy of the latest Bureau Veritas Rules and Regulations for the Classification of Ships and also for his useful comments and remarks on this research. Also to Dr. Conrad De Souza, HM Principal Inspector in the Health and Safety Executive and Mr. Terry Rhodes, Head of Offshore Structures, SHELL EP Europe for all the FPSO structural and operational data provided.
The authors would also like to express their sincerest gratitude to the Alexander S Onassis Public Benefit Foundation for providing a Doctoral Scholarship during which this research was performed and completed.
iv
v
CONTENTS
ACKNOWLEDGEMENTS ........................................................................................................ iiiCONTENTS................................................................................................................................ vEXECUTIVE SUMMARY......................................................................................................... v i i1. INTRODUCTION......................................................................................................................1
1.1 Background ………………………………………………………………………………….. 11.2 Objectives of report ...................................................................................................................21.3 scope of the review....................................................................................................................2
2. LITERATURE REVIEW.......................................................................................................... 32.1 Introduction ...............................................................................................................................32.2 Rationally-Based Structural Design ..........................................................................................32.3 ultimate strength ........................................................................................................................42.4 Still Water Loads.......................................................................................................................62.5 Load combination......................................................................................................................72.6 Reliability Analysis ...................................................................................................................82.7 thermal Stress ............................................................................................................................9
3. LOADING............................................................................................................................... 113.1 Introduction .............................................................................................................................113.2 Modelling Still Water Bending Moment (SWBM) .................................................................123.3 Modelling Vertical Wave Bending Moment (VWBM)...........................................................143.4 Short and Long Term Response ..............................................................................................173.5 Stochastic Combination of Hull Girder Bending Moments.....................................................183.6 Ferry Borges-Castanheta Method............................................................................................18
4. ULTIMATE STRENGTH....................................................................................................... 214.1 Introduction .............................................................................................................................214.2 Approximate formulas for linear first yield and full plastic bending moments of ship hulls. .21
5. CORROSION MODELLING ................................................................................................. 255.1 Corrosion Wastage Model.......................................................................................................25
6. THERMAL STRESS MODELLING ON SHIP STRUCTURES........................................... 276.1 Stress Distribution in a composite box girder due to an arbitrary two-dimensional change of temperature. ...........................................................................................................................................27
7. RELIABILITY ANALYSIS ................................................................................................... 317.1 Fundamental concepts .............................................................................................................31
8. APPLICATION AND RESULTS........................................................................................... 338.1 ultimate strength and thermal stress analysis ..........................................................................338.2 Probability distribution of long-term wave induced load effects.............................................368.3 Application of reliability concepts to marine structures..........................................................378.4 Application to FPSO/Tanker Structures ..................................................................................378.5 Structural Reliability results ....................................................................................................38
9. SUMMARY AND CONCLUSIONS...................................................................................... 43REFERENCES............................................................................................................................ 45BIBLIOGRAPHY ....................................................................................................................... 53
vi
vii
EXECUTIVE SUMMARY
This report on FPSO reliability has been prepared for the Health and Safety Executive (HSE) by Mr. Ioannis Moatsos and Prof. Purnendu K. Das of the Department of Naval Architecture and Marine Engineering of the Universities of Glasgow and Strathclyde.
This study starts with an extensive review of published work in the areas of ultimate strength of ship structures; load modelling and load combination, thermal stresses on ship structures and reliability analysis.
The relevant theory for each of the sections is then described in detail and commented upon in order for the relevant framework and a suitable procedure for the analysis of marine structures and especially FPSOs/FSUs to be established.
A corrosion wastage model based on published research is used to account for the ageing of the vessels.
2 FPSO structures currently in operation in the North Sea are analyzed. Ultimate strength analysis, load modelling and load combination provide a reliability model accounting for ageing ships with the effects of member corrosion, cracking and structural renewal by considering a time varying corrosion rate and temperature stresses. All results obtained have been integrated to carry out structural reliability analysis assessing the structural integrity of FPSOs/FSUs under extreme loading. Probabilities of Failure of the hull girder and Reliability Indices for varying temperatures and years of corrosion are calculated.
viii
1
1. INTRODUCTION
1.1 BACKGROUND
After nearly 20 years of development, in 1992 the International Association of Classification Societies (IACS) published their Unified Rule UR S11 [1] as a standard for longitudinal strength. Faulkner [2] clearly demonstrated that the standard for wave-induced bending moments in hog and sag had changed very little over the previous 30 years or so, and were actually 1% and 8% lower respectively than those given by Murray’s (1955) equations. However, of greater importance was that:
�� although some allowance is incorporated for non-linear hull form effects and for slamming effects in faster ships, these substantially underestimate the increase in sagging wave-induced bending moments (Mws) when compared with numerous model tests and with some non-linear theoretical responses
�� although an assumed consensus decided that the maximum moments occur during the life of a ship (taken as 20 years) with a probability of exceedance (pe) in the order of 10-
8, in practice “identical” calculations by 8 classification societies for a 158.5 m container ship showed for this pe a variation in Mws between 1.0 and 1.8; or, for a given Mws , pe varied by 4 orders of magnitude.
This hardly conveys confidence, especially as such discrepancies were pointed out to ship CSs in 1988 from a Department of Energy study for ships operating as offshore installations; moreover, the sagging moment is usually the most critical. These potentially important weaknesses are corroborated from four other sources:
�� following a US Navy Workshop in 1975, Buckley [3] and others recognised that current standards for longitudinal strength were quite unable to cater for really extreme sea conditions
�� a comprehensive time domain International Ship Structure Committee (ISSC) study 1988-91 [4] showed that a 3 hour simulation, including some steep elevated waves, give Mw values which were 79% greater in sag and 40% greater in hog than the UR S11 values; a more recent ISSC’94 study found similar results
�� a very limited time series study for the DERBYSHIRE [5]for a single steep elevated wave with a crest height of 0.65 H gave values almost identical with the ISSC study for sag
Norwegian Petroleum Directorate (NPD) funded studies appear to have shown significant increases in sagging moments above UR S11 in FPSO and FSU installations.
In terms of the ultimate strength, there are a few strength programs or formulations [6], [7], [8] now available for calculating the ultimate strength of a hull girder. But there is a need to systematically review their accuracy and suitability for their use in reliability analyses. There is also a need to develop a reliability model to account for ageing ships [6] with the effects of member corrosion, cracking and structural renewal. A time varying corrosion rate is to be considered and recommended for this development. In dealing with ultimate strength, the effect of temperature stresses may also play an important role, which needs to be addressed for special situations. This may have a considerable effect, especially in fracture mechanics based assessments for the fatigue limit state.
2
The need also exists to carry out structural reliability analysis of FPSOs/FSUs and thus to assess the structural integrity under extreme loading. Detailed sensitivity studies will be carried out to establish the effect of various design variables contributing to the total failure probability. Recommendations for suitable partial safety factors for use in design will be made for a range of lifetime safety indices or return periods, together with a recommended design code format.
1.2 OBJECTIVES OF REPORT
The principal objectives of this report are to:
�� Review all published literature on the areas of Ultimate Strength, Load Modelling and Load Combination, Corrosion Modelling, Thermal Stress and Structural Reliability in the context of it’s relevance to FPSOs/FSUs.
�� Identify the most applicable theory for establishing a framework for the analysis of FPSO/FSU structures.
�� Describe the procedure that would enable the determination of the structural reliability of FPSOs/FSUs.
�� Present some of the results obtained using the aforementioned theory.
1.3 SCOPE OF THE REVIEW
The focus of this study is on the ultimate strength and load modelling of FPSO/FSU structures as a means of modelling the resistance of the structure and the loads occurring during it’s operational life for use in reliability analysis while also taking into account corrosion. The report is subdivided into 9 chapters.
Chapter 2 provides an extensive literature review on all the relevant areas that will be discussed.
Chapter 3 discusses the modelling of loads imposed on an FPSO during it’s operational life and the relevant theory that is applicable.
Chapter 4 discusses the concept of ultimate strength by using advanced closed form formulation to model FPSO structures.
Chapter 5 provides a brief description of the corrosion wastage model used.
Chapter 6 discusses the concept of thermal stresses imposed in marine structures and how these can be modelled on FPSO structures.
Chapter 7 discusses the concept of structural reliability analysis and the fundamental concepts of its theory.
Chapter 8 provides a description of the procedures followed and some of the results calculated
Chapter 9 finally provides a brief summary of the results and some conclusions.
3
2. LITERATURE REVIEW
2.1 INTRODUCTION
One of the most familiar and fundamental concepts in engineering is that of a system, which may be anything from a simple device to a vast multilevel complex of subsystems. A ship is an obvious example of a relatively large and complex engineering system, and in most cases the vessel itself is a part of an even larger system which influences with its behaviour, shape and the economics involved in all the processes involved in building and maintaining such a complex system. The ship consists of several subsystems, each essential to the whole system such as the propulsion subsystem, and the cargo handling subsystem. The structure of the ship can be regarded as a subsystem providing physical means whereby other subsystems are integrated into the whole and given adequate protection and suitable foundation for their operation. In general terms the design of an engineering system may be defined as:
“The formulation of an accurate model of the system in order to analyze its response-internal and external-to its environment, and the use of an optimization method to determine the system characteristics that will best achieve a specified objective, while also fulfilling certain prescribed constraints on the system characteristics and the system response.”
2.2 RATIONALLY-BASED STRUCTURAL DESIGN
The ever increasing demand for more efficient marine transport has lead engineers in that field to consider a number of significant changes in ship sizes, types and production methods over the last 40 years. Different types of vessels have appeared attempting to meet the demands of the shipping industry. The growing number of factors which give rise to this process of change The need for protection against pollution, new trade patterns that emerge, new types of cargos and the need to safely transport any type that might be considered dangerous, increasing numbers in production lines of standard ship designs and their consequent development to achieve a higher degree of efficiency and the development of structures in vehicles for the extraction of ocean resources, require scientific, powerful and versatile methods for structural design. One can say that we are at present in the midst of a progressive and gradual but profound change in the philosophy and practice of ship structural design. Based on individual ship designer and shipyard experience and ship performance, in the past ship structural design had been mostly empirical based on the structural designers’ accumulated experience This approach lead eventually to the publication of structural design codes, or “rules” as they are referred to in the industry, published by various ship classification societies. These “manuals” of ship structural design provided a simplified and care-free method for the determination of a ship’s structural dimensions. This procedure provided a time and cost effective method for design offices and simplified classification and approval process at the same time. The method unfortunately has several disadvantages including the inability to handle the large number of complex modes of structural failure, the possibility of unsuitable results in regard to the specific goals of the ship owner or the particular purpose or economic environment that the ship is required to operate in and the inability to distinguish between structural adequacy and over adequacy leading to increased cost and steel weight in a structure. Most important though is the large number of simplifying assumptions included which bound such design process within certain limits. For these reasons there is a trend within the shipbuilding and ship design industry and the academic institutions involved in research related to these fields toward “rationally-based” structural design, which may be defined according to Hughes [9] as:
4
“Design which is directly and entirely based on structural theory and computer-based methods of structural analysis and optimization, and which achieves and optimum structure on the basis of a designer-selected measure of merit.”
Thus rationally-based design involves a thorough and accurate analysis of all the factors affecting the safety and performance of the structure throughout its life, and a synthesis of this information, together with the goal or objective which the structure is intended to achieve, to produce that design which best achieves the objective and which provides adequate safety. This process involves far more calculation, but that can be more than offset by using computers. For this reason rationally-based structural design is necessarily a computer-based, semi-automated design process.
2.3 ULTIMATE STRENGTH
The longitudinal strength of a vessel, as is otherwise known the ability of a ship to withstand longitudinal bending under operational and extreme loads without suffering failure, is one of the most fundamental aspects of the strength of a ship and of primary importance for Naval Architects. Assessment of the ability of the ship hull girder to carry such loads involves the evaluation of the capacity of the hull girder under longitudinal bending and also the estimation of the maximum bending moment which may act on it. From the initial work of pioneers in the area of ship structural design such as the likes of Thomas Young and Sir Isambard K. Brunel and Timoshenko the fundamental idea to assess longitudinal strength of a ship’s hull was first presented by John [10]. By calculating the bending moment assuming the wave whose length is equal to the ship length he proposed an approximate formula to evaluate the bending moment at a midship section. He also calculated the deck maximum stress and by comparing that to the material breaking strength he managed to determine the panel optimum thickness. Although John’s theory remains in use until today, subsequent methods of stress analysis and wave loading have improved substantially with criteria that help to determine optimal thickness changing from breaking strength to yield strength. One of the most recent developments in that area is that of taking into account, when assessing the hull girder strength, the ultimate strength of the hull girder. With two methods being more dominant over others that are used to evaluate the ultimate hull girder strength of a vessel under longitudinal bending a number of different opinions exist on which leads to more accurate results. One calculates the ultimate hull girder strength directly and the other performs a progressive collapse analysis on a hull girder. Work by Caldwell [11] originally attempted theoretically to evaluate the ultimate hull girder strength of a vessel. He introduced “Plastic Design”, as it is known today, by considering the influence of buckling and yielding of structural members composing a ship’s hull. By introducing a stress reduction factor at the compression side of bending he managed to calculate the bending moment produced by the reduced stress which he considered as the ultimate hull girder strength. By not taking into account the reduction in the capacity of structural members beyond their ultimate strength his method overestimates the hull girder’s ultimate strength in general and since exact values of reduction factors for structural members were not known, the “real” value could not be calculated, only an approximation. Since then it has been improved by work carried out by Maestro and Marino [12] and Nishihara [13] who extended the formulation to include bi-axial bending, modified it to such extend as to be able to estimate the influence of damage due to grounding and to improve the accuracy of the strength reduction factor. By proposing their own formulae Edo et al. (1988) and Mansour et al. [14] performed simple calculations that lead to the development of further methods as the one proposed by Paik and Mansour [15]. By applying these Paik et al. [16], [17] performed reliability analysis considering corrosion damage and similar work that includes corrosion effects has also been published by Wei-Biao [18]. Although these methods do not explicitly take into account of the strength reduction in the members, the evaluated ultimate strength showed good correlation with
5
measured/calculated results from other cases. Paik and Mansour [15] compared the predicted results with those by experiments and Idealized Structural Unit Method (ISUM) analysis and the differences between the two were found to be between -1.9% and +9.1%. By taking into account the strength reduction (load shedding) of structural members when the collapse behaviour of a ship’s hull is simulated, a number of methods were developed which can be grouped under the overall term “Progressive Collapse Analysis” whose fundamental part could be the application of the Finite Element Method (FEM) considering both geometrical and material nonlinearities. Unfortunately the large amount of computer resources required to perform such type of analysis and to reassure the reliability of the subsequent results has lead to the development of simplified methods such as the one proposed by Smith (1977). This is done by taking into account the strength reduction of structural members after their ultimate strength as well as the time lag in collapse of individual members and Smith was also the first to demonstrate that the cross-section cannot sustain fully plastic bending moment. The accuracy of the derived results depends largely on the accuracy of the average stress strain relationships of the elements. Problems in the use of the method occur from modelling of initial imperfections (deflection and welding residual stresses) and the boundary conditions (multi-span model, interaction between adjacent elements). To improve this recent research is focusing on the development of more reliable stress-strain curves and it can be seen in work by Gordo & Guedes Soares [19] and Paik [20]. Smith also performed a series of elasto-plastic large deflection analysis by FEM to derive the average stress-strain relationships of elements and analytical methods have been proposed such as the one by Ostapenko (1981) which includes combined applied in-plane bending and shear as well as combined thrust and hydraulic pressure. Rutherford and Caldwell [21] proposed an analytical method combining the ultimate strength formulae and solution of the rigid-plastic mechanism analysis. In both methods, the strength reduction after the ultimate strength is considered. Yao [22] also proposed an analytical method to derive average stress-strain relationship for the element composed of a stiffener and attached plating which derives average stress-strain relationships for the panel by combining the elastic large deflection analysis and the rigid-plastic mechanism analysis in analytical forms from work performed by Yao and Nikolov [23] &[24]. Then by taking into account the equilibrium condition of forces and bending moments acting on the element the relationships are derived. When the stiffener part is elastic, a sinusoidal deflection mode is assumed, whereas after the yielding has started, a plastic deflection component is introduced which gives constant curvature at the yielded mid-span region. A number of practical applications of the methods mentioned here have been published with one of the most significant being the investigation of the causes of the loss of the Energy Conservation VLCC by Rutherford and Caldwell [21] by performing progressive collapse analysis. With simplified methods for progressive collapse analysis, the applications of ordinary FEM are very few due to the influences of both material and geometrical nonlinearities which have to be considered when applying such an incremental procedure. With a ship’s hull girder being, perhaps, too large for such kind of analysis to get rational results easily, a number of some significant works has, nevertheless, been published. Chen et al. [25] and Kutt et al. [26] performed static and dynamic FEM analyses modelling a part of a ship hull with plate and beam-column elements and orthotropic plate elements representing stiffened plate and discussed the sensitivities of the ultimate hull girder strength with respect to yield stress, plate thickness and initial imperfection based on the calculated results. Valsgaard et al. [27] analysed the progressive collapse behaviour of the girder models tested by Mansour et al. [15] and energy concentration but unfortunately the results of FEM analysis to evaluate ultimate hull girder strength are not so many at the moment because the number of elements and nodal points become very huge if rational results are required. Apart from Smith’s method, the Idealized Structural Unit Method (ISUM) is another simple procedure that takes into account a larger structural unit considered as one element thus reducing the computational time required to achieve results. The essential point of this method is to develop effective and simple element (dynamical model) considering the influences of both
6
buckling and yielding. Ueda et al. [28] developed plate and stiffened plate elements that accurately simulate buckling/plastic collapse behaviour under combined bi-axial compression/tension and shear loads. Paik improved this work and performed different progressive collapse analyses as published in Paik et al. [29] and [30], [31]. Ueda and Rashed (1991) improved their results with Paik [32] following with an attempt to introduce the influence of tensile behaviour of elements in ISUM. Finally Bai et al. [33] developed beam element, plate element and shear element based on the Plastic Node Method, as originally published by Ueda and Yao (1982) and managed to achieve progressive collapse analysis. While in Smith’s method accurate results are obtained when only the bending moment is acting, the ISUM can be applicable for the case with any combination of compression/tension, bending, shear and torsion loadings but sophisticated elements are required to get accurate results and to improve this ISUM elements are still under development. More recently the latest development is the area of ultimate strength related research have been published and commented in the 2000 ISSC [34], [35].Significant work has also been published by…
2.4 STILL WATER LOADS
The issue of statistical analysis of SWBM has been addressed since the 1970’s and quite a significant volume of published work exists on the subject. The idea that any ship has probability distributions of SWBM was first supported by Lewis et al. (1973) [36] based on limited data of several cargo ships and one bulk carrier. Ivanov and Madjarov (1975) [37] fitted the normalized maximum SWBM by a normal distribution according to full or partial cargo load conditions from eight cargo ships for periods of two to seven years. Mano et al. (1977) [38] studied the nature of still water conditions by surveying log-books of 10 container ships and 13 tankers, and concluded that their distribution is approximately normal, as shown in Fig.1
Figure 1 Distribution of SWBM of containerships after Mano
Dalzell et al. (1979) [39] examined the service and full scale stress data of a large, fast containership, a VLCC, and a bulk carrier. They concluded that the still water bending stress variations appear to be random and subject to the control of their extremes by the operators. In the early 1980’s some still water bending stresses of different ships were reported by Akita (1982) [40] as a result of work carried out in Japan. This information was presented separately for a group of 10 containerships as well as for a group of 8 tankers. Based on this, Kaplan et al. (1984) [41] found that the coefficient of variation (COV) of SWBM for containerships is 0.29 and for tankers it is 0.99 for ballast condition and 0.52 for full load condition. More recently, Soares and Moan (1988) [42] analyzed SWBM resulting from about 2000 voyages for 100 ships belonging to 39 ship-owners in 14 countries. Still water load effects were assumed to vary from voyage to voyage for a particular ship, from one ship to another in a particular class of hips and also from one class of ships to another. The analysis was centred on
7
the mean values and standard deviations of the loads in the critical midship region. The still water loads were treated as ordinary random variables and in the results the authors reported a very large variability for tankers.
2.5 LOAD COMBINATION
It is not an easy task to combine all the relevant extreme loads at the same time as and hence we have to focus our interest in the most important combination problem, the combination of SWBM and VWBM in the long term, i.e., in a ship’s design lifetime. For this to be achieved it would be interesting to also examine, as it has been discussed previously in all relevant elements of this thesis, the volume of published work on the subject. In the existing ship rules, such as DnV (2004)[43] and IACS Requirements S11 (2003) [1] and S7 (1989) [44] these two loads are simply added together following the peak coincidence method which assumes that the maximum values of the two loads occur at the same instant during a ship’s design lifetime. In addition IACS also specifies that the maximum SWBM and VWBM should not exceed their respective allowable values, even if one of the moments is negligible, based on Nitta (1994) [45]. Söding (1979) [46] combined SWBM and VWBM by modelling them as random variables. Although this approach is more rational than the peak coincidence method, it is simplistic to model SWBM and VWBM as random variables rather than physically correct random processes. Besides, Söding’s solution was conditioned upon the assumption that the SWBM is a normal variable and the VWBM is an exponential variable. Moan and Jiao (1988) [47] considered both SWBM and VWBM as stochastic processes, and, based on a particular solution by Larrabee (1978) [48] they introduced a load combination factor to reduce the conservatism inherent in the peak coincidence method as adopted by the existing ship rules. While this approach has been the most rational, the accuracy in applying Larrabee’s solution remains to be verified against other acceptable load combination methods. Furthermore Moan and Jiao (1988) [47] considered the sagging condition only. Extension is then needed with regard to the hogging condition. In terms of the problem of load combination (vertical and horizontal bending moments and shear forces), a number of authors have proposed interaction equations and methodology that leads to the prediction of ultimate strength associated with each load supposed to act separately. With the stress profile in real ship structures being quite complicated the load bearing capacity of stiffened panels is a direct result of the actual stress distribution and occurs from a number of different load actions. In order for one to achieve a detailed stress distribution in parts of a vessel’s structure, large scale numerical analysis using FEM has to be used and is certain cases such analysis might even have to be non-linear due to the local non-linear response rising from initial deformations, residual stresses and fabrication tolerances. It therefore becomes of great importance that the effects on the ultimate load bearing capacity from combined load actions to be taken into account. Different loads should be treated separately in order to asses the effect on the ultimate stiffened panel capacity. A number of different combinations of load control can be used to obtain the ultimate load of stiffened panels under combined loads, such as load-displacement control or load-load control. Fujikubo et al (1997) [49] and Yao et al. (1997) [50] showed is their published work that one might say that the influence of the actual loading method on the ultimate collapse load is negligible. In terms of the effect of lateral pressure, this increases the local buckling strength of a continuous stiffened plate under thrust and as proposed by Yao et al. (1997) [51] empirical formulae can be sued to calculate the buckling strength of a continuous stiffened plate field including the effect of lateral pressure which shows good correlation with numerical FEM analyses. Another important aspect influencing the level of redistribution of loads in large plated structures is that of the in-plane stiffness of the panel and in work by Steen (1995) [52] it can be seen that the initial stiffness under the action of transverse loads could be reduced by the
8
order of 40% and that the transverse buckling capacity is less sensitive to lateral pressures than the longitudinal buckling capacity. With respect to the effect of transverse thrust work by Yao et al. (1997) [51] provided relationship between average stresses in the longitudinal and transverse direction of a stiffened plate field by elasto-plastic large deflection FEM analyses of a stiffened plate subjected to bi-axial thrust. Available results on the effect of shear loads on the ultimate capacity of stiffened panels are scarce. Some results are reported by Steen and Balling Engelsen (1997) [53] for un-stiffened plates using the non-linear FEM with the general observation that the ultimate strength is close to the shear yield strength for a number of plates with varying slenderness and aspect ratios. As for the effect of horizontal bending, all vessels are subjected to both horizontal and vertical bending moments, especially in rough seas with significant roll motions. Under this case work by Yao et al. (1994) [54], Mansour et al (1995) [55] and Paik et al. (1996) [56], Gordo and Guedes Soares (1997) [57] has looked into formulations for different types of ships under combined bending. For vertical shear forces on ship hulls, this is sustained by the side shell plating and additional longitudinal bulkheads and it can be accounted for in a simplified manner through a procedure described in by the Japanese Ministry of Trasport (JMT) (1997) [58]. For merchant ships with closed cross-section, the influence of the shear force is negligible for normal operational loads but for other ship types data are not enough to make any significant conclusions on the effect of shear force on the ultimate hull girder strength, although work by Paik (1995) [15] & (1996) [56] has reported some results. The effect of corrosion damage can be assessed indirectly by a sensitivity study on the influence of plate and stiffener thickness on ultimate hull girder capacity as published in work by Jensen et al. (1994). [59] Yao et al. (1994) [54] reported that 30% overall reduction in the plate thickness leads to a strength reduction of 34% and 38% in the sagging and hogging condition respectively.
2.6 RELIABILITY ANALYSIS
Reliability theory of engineering structural systems has three significant parts. The identification of all possible dominant failure modes, the calculation of the failure probability and sensitivity with respect to the obtained dominant limit states and the determination of the upper and lower bounds of the overall structural system according to the correlation between the dominant failure modes and their failure probability. The identification of the dominant failure mode is performed by either traditional mechanics or a mathematical programming approach. Although all proposed methods in this field show a significant amount of effectiveness, the computational effort is still quite demanding for complex structural systems. No close form of the limit states can be obtained since numerical methods calculate the response. As an alternative, work has also focused on the analysis of structural components or the local behaviour of a structural system. The determination of the failure probability is the most researched part of all parts comprising the theory of reliability. Work by Cornell (1969), Hasoferd and Lind (1974) and Shinozhka et al.(1983) has produced the first order and second order moment theories (FOSM and SOSM) which are well established and have found an ever-increasing us in a significant amount of engineering fields. In this type of theory the integration of the joint probability density function of the design variables is circumvented by transformation of the actual problem into a least distance problem in standard normal space. Orthogonal transform is used to uncouple the correlated design variables which essential shows that the problem is in its core an optimization procedure. A number of insurmountable problems in numerical integration of highly dimensional JPDFs in normal space lead research into the conclusion that the failure probability of the structural system has to be given in a “weak form”. Instead of calculating the failure probability itself, an interval is given to bind the exact value. Two methods have been proposed to help achieve this,
9
the Wide Bound Method as proposed by Cornell (1967) [60] and the Narrow Bound Method as proposed by Ditlevsen (1979) [61]. They are first order and second order approximations respectively. However with the increase of failure modes and their correlation, the bounds will become too loose. In this case, formulation of higher order approximations can be developed or different point evaluation techniques can be used such as the ones proposed by Ang et al. (1981) [62] and Song (1992) [63].A modified bound method can also be found in Cornell’s (1967) [60] original work. Monte Carlo Simulation (MSC) also plays a very important role in different levels of reliability analysis. The high accuracy that the method produces is only dependent on the sampling number and is not affected by the distribution type and the number of basic variables. The method can be used in even those cases where the limit state function is not known implicitly and it is the only approach to highly non-linear problems. A number of variation reduction techniques have been proposed such as the Importance Sampling Method but as always the computation cost in large complex structural systems is still significantly high. The Response Surface Method (RSM) is one of the latest developments in the field of structural reliability analysis. It is very suitable in cases where the limit state function is known only point-wisely by such numerical methods as the FEM rather than in closed form. In essence, RSM is a system identification procedure, in which a transfer function relates the input parameters (loading and system conditions) to the output (response in terms of displacements or stresses). The observations required for the identification of the most suitable way to relate those two are usually taken from systematic numerical experiments with the full mechanical model and the transfer function obtained approximately defined as the response surface (RS). The basis of the RSM can be tracked back to the 50s in experiment field, but only recently, it has been introduced into the field of reliability analysis. It combines deterministic structural analysis software and the basic reliability ideas aforementioned. In addition to this, even for those problems that other approximate methods seem to be susceptible to, the RSM is shown to be superior in both accuracy and efficiency with its only drawbacks being the experiment design and the identification of unknown parameters in the RS which influence the whole algorithm. Work by Bucher (1990) [64] and Rajashekhar (1993) [65] have lead the ways of future research. Advanced algorithms based on that work can be found in work published by Kim et al. (1997) [66], Zheng & Das (2000) [67], [68] and Yu, Das & Zheng (2001) [69].
2.7 THERMAL STRESS
The fact that ship structures do encounter temperature conditions that induce thermal stresses and deformation of practical importance has been recognized for some time. Hechtman [70] lists a paper by Suychiro and Inokuty (published in the 1916 volume of the Japanese Society of Naval Architects Journal) as one of the first to be devoted to this topic. Following that papers by Hurst (1943) [71]; Corlett (1950) [72]; Jasper (1955) [73]; Hechtman (1956) [70] and Meriam, Lyman, Steidel and Brown (1958) [74] have been published and all relevant research has been well documented by the Ship Structures Committee (SSC) and summarize to a certain extent the pertinent investigations devoted specifically to the effect of environmental temperatures on ship structures. By way of comparison some 200 papers had appeared in the literature between the late 1940s and the 1960s that were written with the specific intent of contributing to the better understanding of temperature problems associated with aircraft structures. This number does not include some 400 additional investigations devoted to the creep behaviour of aircraft structures and materials and a textbook on thermal stresses with applications to aircraft and missiles written by Gatewood [75]. The ‘boom’ of the space race in the 1950s and the interest of the defence industries worldwide from the 50s until today has more that quadrupled the numbers of relevant published research since then [75]. One might say that the environmental temperature conditions that aircraft and missile structures vary significantly in magnitude and nature from those that ship structures encounter. That is true. The temperature conditions encountered by
10
supersonic aircraft and missiles are much more severe than those encountered by ships, and for this reason it is reasonable to expect that the aircraft structures’ thermal problems would receive considerably more attention. On the other hand, one cannot rightly say that the thermal problems in ship structures are of so much less importance as to justify the comparatively little attention they have received and are still receiving since the 1960s. Obviously, any single service condition that induces stresses in the 8,000 to 12,000 psi (55,000-83,000KN/m2) range cannot be considered negligible. That such thermal stresses can arise in ship structures from perfectly commonplace temperature conditions has been well documented [74].
11
3. LOADING
3.1 INTRODUCTION
The accurate prediction of loads and load effects is a fundamental task in the design of ship structures which in general consists of three major aspects:
�� The prediction of still water resultant loads and load effects �� The prediction of wave resultant loads and load effects �� The prediction of combined loads and load effects
The occurrence of still water loads exists from forces that result from the action of the ship self-weight, the cargo or deadweight and the buoyancy. They include bending moment, shear force and lateral pressure and remain constant under specific loading conditions. However, still water loads may vary from one load condition to another as a result of a variation of the aforementioned parameters and need to be considered as stochastic processes in the long term. Typically, the most important still water load component is the still water bending moment (SWBM). Wave loads are forces resulting from wave action and include vertical, horizontal bending and Torsional moments, shear forces, hydrodynamic pressure and transient loads such as springing and slamming. Due to the random nature of the ocean, wave loads are stochastic processes both in the short-term and in the long term. The most typical and important component of wave loading is the vertical wave bending moment (VWBM). Since still water loads and wave loads are of quite different random natures, their respective extreme value has been though that they would not occur at the same time in most cases. For the appropriate determination of rational values of the extreme combined loads, a load combination theory needs to be employed that considers each random load as a stochastic process in the time domain and combines two or more loads as scalar or vector processes. Mansour and Thayamballi (1993) [75] have addressed the issue of different load combination problems for ships and the primary types are:
�� Hull girder loads, i.e., vertical bending, horizontal bending and Torsional loads �� Hull girder loads and local pressure loads �� Hull girder loads and transient loads, i.e., springing, slamming or whipping moments.
The major hull girder loads of design interest are the vertical and horizontal bending moments and shear forces and sometimes the Torsional moments depending on the ship type such as in the case of multi-hulls or containerships which have low Torsional rigidity and therefore Torsional loads are of primary importance. In most, merchant vessels that do not have wide hatches, such as tankers with closed sections, Torsional stresses are nevertheless small. Horizontal bending moment and shear force excited by low frequency waves are of less concern, due to the fact that they do not result in significant stresses in ocean-going ship structures. For production marine structures designed for being always headed on to waves in operation, although the actual heading may be subjected to uncertainty when wind, current and waves are not collinear, the horizontal bending and shear effects are negligible. In cases when shear forces need to be accounted for, the focus may be on the stern quarter body which is likely to have more critical shear forces than the forward one. When examining hull girder loads the following are the typical extreme loads and load combinations of interest.
12
�� Extreme vertical bending loads, with coexisting values of vertical and horizontal bending and Torsional loads. Vertical bending moments are important in the midship region.
�� Extreme Torsional loads, with coexisting values of vertical and horizontal bending loads. This needs to be addressed in the forward quarter body of ship structures with low Torsional rigidity.
3.2 MODELLING STILL WATER BENDING MOMENT (SWBM)
It has been widely accepted [76] that the SWBM can be efficiently modelled as a Poisson rectangular pulse process in the time domain as illustrated in (Fig.2).
Figure 2 Modelling of SWBM by a Poisson rectangular pulse process
The reason behind such a conclusion lies in the fact that the SWBM at a given section of the ship is constant under a specific load condition but can easily vary from one load condition to another. The duration of each load condition can also be a random variable. The space variation of still waters mostly depends on the amount of cargo a vessel is carrying and the cargo’s distribution along the length of the ship. A number of measures are often taken to ensure that the maximum specified still water loads are not exceeded during ship operation. Captains are often encouraged by Classification Societies, Local Government Maritime Regulations and the IMO to load their ships in a way not dramatically different from those reference conditions given in the ship’s loading manual, in the hope that maximum values are not exceeded. Reality far differs from such suggestions as advances in technology have allowed for the application of computerized load distribution procedures, giving captains the freedom to load their vessel in any way they seem fit as long as the maximum loads are within the specified allowable limits. As a result of that the loading manual is less likely to be followed and a large variation of load conditions results from human decisions involved and unfortunately such freedom of choice could also lead to a larger probability of load exceedance due to human errors in the choice of load conditions for which very limited control exists worldwide with decisions laying solely in the hands of the captain and the operator. Moan & Jiao (1988) [47] describe such examples observed during the operation of an offshore production ship. The available statistical results have shown that a normal distribution may be considered as appropriate for the SWBM in conventional ships: cargo ships, containerships and tankers. FPSOs could be considered as ‘modified tankers’ as their construction often results from tanker conversions and even in newly built vessels their hull shape is rather similar to that of tankers. One could then easily suggest that the SWBM statistics for tankers would be applicable for FPSOs. As a result of the special characteristics of FPSOs, in particular due to their frequently changing load distribution, which results from loading and offloading the vessel, the SWBM in FPSOs differs from the one in tankers and other conventional ships. Loading conditions
SWBM
T
13
specified in the loading manual of FPSOs include docking, normal operating and severe storm conditions, but the maximum SWBM is controlled by an on-board computer rather than the load manual, before a new load condition during operation is to be approved [47]. Hence, the captain (also called the Foreman of the FPSO) is able to determine which tanks are to be loaded and unloaded or what the cargo distribution along the ship is to be depending upon his personal judgement or operational convenience. As a result of that, the actual SWBM is, in fact, randomly determined and should thus be described with a probabilistic model. Following such an arbitrary operational procedure it may also be reasonable to assume that individual load conditions are mutually independent. Based on 453 actual still water load condition recorded during the first two operational years of an FPSO, with typical duration of each load condition varying from several hours to one day, Moan & Jiao (1988) [47] found that the cumulative distribution function of the individual sagging SWBM as shown in Fig. 3 is well fitted by a Rayleigh distribution:
� �
�
�
�
�
�
�
�
�
�
�
�
� �
2
exp1�
ssM
MMF
S (1)
Figure 3 Distribution of sagging SWBM of a production ship after Moan and Jiao
in which Ms is the SWBM of an individual load condition, and the scale parameter � is so determined that the specified maximum SWBM, Ms,0 , corresponds to a return period of the design life T0,
� �
00,
11Tv
MFs
sM S�� (2)
in which vs is the mean arrival rate (i.e. the frequency) of one load condition e.g. vs=1.0 day-1.Substituting � solved from Eq. (2) into Eq. (1) we obtain
� � � �
�
�
�
�
�
�
�
�
�
�
�
�
�
�
2
0,0lnexp1
s
sssM M
MTvMF
S (3)
14
While the sagging SWBM follows a Raleigh distribution, the data reported in Moan and Jiao (1988) [47] indicate that the hogging SWBM follows an exponential distribution:
� � � �
�
�
�
�
�
�
�
�
���
0,0lnexp1
s
sssM M
MTvMF
S (4)
According to Dnv (2004) [43], the specified maximum SWBM for conventional ships in a design lifetime of 20 years, Ms,0 is given by
� �� �
� � )(015.01225.0
7.0065.02
20,
hoggingCBLC
saggingCBLCM
BW
BWs
��
���
(5), (6)
In which L, B and CB are the length, breadth and block coefficient of the ship, respectively, and CW is the wave coefficient given by
350for150
35075.10
350300for75.10
300100for100
30075.10
23
23
���
�
��
� �
�
�
���
�
��
� �
�
LL
L
LL
CW
(7), (8), (9)
It should be noted that the rule moments given by Eq. (5) and (6) are not intended for FPSOs. The maximum SWBM for FPSOs should be determined on a case to case basis. As a result of lack of vessel design data that would enable a full analysis of each case to be considered, an alternative solution based on published research had to be found that would enable rule moments. According to the data reported by Moan and Jiao (1988) [47], the maximum midship sagging and hogging moments specified in the load manual are 72.6% and 79.2% of the corresponding Rule moment respectively. Within the two first years of operation, however, the experienced maximum sagging and hogging moments are 75% and 83% of the corresponding Rule moment, respectively. Although these values are all below the Rule moments, the experienced maximum moments in two years have already exceeded the specified maximum moments in the load manual. This situation is allowable since the FPSO of concern was designed to comply with DnV (2004) [43], although smaller moments may be achieved if the load manual is strictly followed. Nevertheless, the Rule moments are used as a convenient reference in this study. It is also noted that the 20-year maximum sagging and hogging moments are 87% and 112% of the corresponding Rule moment, respectively, by extrapolating the experienced maximum moments observed in two years, without accounting for load control. Since load control is required, however, any moments larger than the Rule moments have to be rejected. This implies that a truncation of the distribution functions given by Eq. (3) and (4) at Ms,0 may be considered [47]. Being conservative, no truncation is considered in further analysis.
3.3 MODELLING VERTICAL WAVE BENDING MOMENT (VWBM)
As a result of the random nature of the ocean, the VWBM is a stochastic process. It may be described by either short-term or long-term statistics or probabilistic estimates.
15
The short-term VWBM corresponds to a steady (random) sea state which is considered as stationary with duration of several hours. Within one sea state, the VWBM follows a Gaussian process. The maxima of the VWBM thus follow a Rice distribution. For a narrow-banded Gaussian process, the Rice distribution reduces to the simple Rayleigh distribution. The long-term statistics are based on weighted short-term statistics over the reference period of concern. Typically, design wave loads in classification society rules are specifically based on long-term statistics [48]. The long-term prediction of wave loads is usually based on linear analysis, which is then corrected to account for non-linear effect, based on experiments and/or full-scale measurements. The long-term formulation of the response amplitudes of bending moment may be expressed in many ways, but it is basically a joint probability of x and m0 expressed as
� � � � � �000 |, mpmxpmxq � (10)
where p(x|m0), the probability of x for a given m0, is the conditional density function of x with respect to m0, which is assumed to be Raleigh distributed. Thus,
� �0
2
2
00| m
x
emx
mxp�
��
�
�
��
�
�
� (11)
and p(m0) is the probability density of the response variance in the considered sea states. It depends on several variables such as the wave climate represented by Hs and Tz, the ship heading �, speed v and loading condition c,
� � � � dvdcddTdHcvTHfdrrf zszsR �� ,,,,� (12)
The cumulative long-term distribution is defined by,
� � � � � �� �
� �
��
ixRi drrfmxpxxQ
00| (13)
And since the cumulative Raleigh distribution is
� ��
�
�
�
ix
mx
emxp 0
2
20| (14)
The probability of exceeding amplitude x for a given m0 is given by
� � � ��
�
�
��
0
2 0
2
drrfexxQ Rmx
i (15)
fR(r) may be a complicated function, or group of functions, that defines the probabilities of different combinations of the variables Hs, Tz, �, v and c.The most important of these variables to consider is the ship heading relative to the dominant wave direction. Ship speed, which has relatively small effect on wave bending moment, can be eliminated as a variable by assuming either the design speed or the highest practicable speed for the particular sea condition and the ship heading under consideration. The effect of amounts and
16
distribution of cargo and weights, which in turn affects draft and trim, transverse stability, longitudinal radius of gyration, etc, can be a complicated problem. Usually, however, it can be simplified by assuming two or more representative conditions offloading, such as normal full load, partial load and ballast condition. Then completely independent short and long term calculations can be carried out for all load conditions. With the above simplifications, we are left with the following variables, to be considered in the probability calculation: Hs, Tz and ���The variables are considered to be mutually independent. The probabilities of different combinations of Hs and Tz are given in a wave scatter diagram. The probabilities of each ship heading � have to be established for different cases but in general one can say that they are random. It is generally accepted that the long-term VWBM may be modelled as a Poisson process. The peak of each individual VWBM, Mw, can be well approximated by a two parameter Weibull distribution, as shown in fig().
Figure 4 Long-term distribution of deck stresses resulting from VWBM after Frieze et al.
After Frieze et al. (1991) [49]. Denoting Mw,0 as the maximum VWBM in the reference design period T0, the cumulative distribution function of each individual Mw is then expressed as
� � � �
�
�
�
�
�
�
�
�
�
�
�
�
�
�
W
W
h
w
wwwM M
MTvMF
0,0lnexp1 (16)
in which hw is the shape parameter and vw is the mean arrival rate of one wave cycle. It would be important to note that the above formulation does not account for the dependence between individual peaks, while in reality there are correlated within a single sea-state. Alternatively Mw in Eqn (16) may be considered to refer to the individual maximum moment of each sea state, in which vw then becomes the mean arrival rate of one sea-state, so that mutual dependence of individual moments can be neglected.
17
3.4 SHORT AND LONG TERM RESPONSE
A number of long-term prediction procedures have been proposed [80-82] for the bending moment response estimation for any ship, with several assumptions, for which transfer functions are available. The method used to define suitable sea spectra was the ISSC version of the Pierson-Moskowitz spectrum given by Warnsick (1964),
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
��
�
�
�
�
�
�
�� 45
244.0exp
211.0 ��
� mmms TTTHS (17)
where Tm is the average period and Hs is the significant wave height. Having the Response Amplitude Operators (RAOs), the bending moment response spectra can be calculated by superposition for all of the selected wave spectra. The directional spectrum represents the distribution of wave energy both in frequency of the wave components and in direction��. The analysis of directional buoy records has shown that the spreading function is function of both direction and frequency. Pierson and St. Denis used a directional spreading function that became somewhat generalised because of its simplicity. It is a frequency independent formulation given by:
� � ��
2cos2�
�G ,2�
�� (18)
� � 0��G , 2�
�� (19)
where � is the angle between an angular wave component and the dominant wave direction. The short-term responses were combined with long-term wave statistics to determine the long-term probability of exceedance of different wave induced vertical bending moments. A Weibull distribution was fitted to the resulting distribution. The rationally based design of ships requires the consideration of the largest value (extreme value) of the wave loading, especially the wave-induced bending moment, which is expected to occur within the ship’s lifetime. The prediction of the characteristic value, which is associated with a certain probability of non-exceedance in the time, is of particular interest. The characteristic value is that magnitude of extreme value, which has an appropriate probability of exceedance. There are usually good theoretical grounds for expecting the variable to have a distribution function, which is very close to one of the asymptotic extreme-value distributions. It has been shown that the extreme Vertical Wave Bending Moment (VWBM) can be described by a Type I extreme-value distribution, generally called Gumbel distribution:
� � � �� �� �uMMF weweGumb ���� �expexp (20) Guedes Soares [82] showed that the Gumbel parameters can be estimated from the initial Weibull fit using the following equations:
� �� �bww nku1
ln.� and � �� � bb
ww nkb �
�
1ln.� (21) (22)
where nw is the number of peaks counted in the period Tc given by:
z
cw T
Tn � (23)
18
zT is the average mean zero crossing periods of waves. The parameters uw and w are respectively measures of location and dispersion. uw is the mode of the asymptotic extreme-value distribution. K and b are the Weibull scale and shape parameters. The mean and standard deviation of the Gumbel distribution is related to the uw and w parameters as follows:
wwwe u
�
�
� �� and6w
we�
�
� � (24) (25)
where is Euler’s constant equal to 0.5772.
3.5 STOCHASTIC COMBINATION OF HULL GIRDER BENDING MOMENTS
Considering only the still water induced and wave induced components of the hull girder loads independently and by using the Ferry Borges-Castanheta load combination method [83], the load combination factor, w, for the deep condition can be obtained. As the still water and wave bending moments are stochastic process, the maximum SWBM and VWBM do not necessarily occur simultaneously in a ship’s service lifetime. Moan and Jiao [84] considered both SWBM and VWBM as stochastic process, and, based on a particular solution by Larrabee [85] they introduced a load combination factor, derived by the combination of two stochastic processes:
seewewseswwete MMMMM ���� �� (26)
where sw and w are load combination factors, while Mse and Mwe are the extremes of the still-water and wave induced bending moments.
3.6 FERRY BORGES-CASTANHETA METHOD
The method simplifies the loading process in such a way that the mathematical problems connected with estimating the distribution function of the maximum value of a sum of loading process are avoided by assuming that the loading changes intensity after a prescribed deterministic, equal time interval, during which they remain constant. The intensity of the loads in the different elementary time intervals is an outcome of identically distributed and mutually independent variables. A time interval for still-water loads would typically be one voyage for a conventional ship. For a production ship, the period between different loading conditions must be defined from the operational profile. In the method, the point-in-time distribution for load process {x} is defined as fxi (density function) and the corresponding distribution function is Fxi. The cumulative distribution function of the maximum value in the reference period T is then given by (Fxi)ni , i.e.:
� � iii
niFF )()(max, ��
��� (27)
From load combination theory, we have that the density functions ji
f���
are determined by the convolution integral:
� � � � � �dzzfzffjiji �
�
��
�
�� ������
. (28)
and from basic statistics we have
19
� � � �dzzfFiii �
�
��
� .��
� (29)
by combining the above three formulas we end up with the following expression:
� � � � � � �� � �ij
iji
nni dzzFzfF
�
�
��
�
�� ...max, ������
(30)
Taking i� and j� as Msw and Mw respectively, Eq. (30) may be applied directly to the combination of still water and wave induced bending moments. The total vertical bending moment, Mt, may then be estimated by:
� �� �
�
�
�
�
�
��
�
�
�
��
sw
w
n
ntwswti dzzMFzfMF ).()( (31)
The density distribution function fsw is the still-water bending moment in one year, which is a normal distribution. The number of occurrences of each load condition, nsw, is defined by the operation profile. � �
wnwF is the Gumbel distribution of the extreme wave induced bending
moment in one load condition derived from the Weibull distribution assuming nw wave loads in each load condition.
w
sww J
Jn � (32)
Introducing a load combination factor, the total load might be defined as:
� � � ����� wswt FFF �� (33)
where the extreme distributions are considered at 0.5 exceedance level. The load combination factor can then be determined by the following relationship:
� � � �
� �5.05.05.0
1
11
�
��
�
�
w
swtw
F
FF� (34)
20
21
4. ULTIMATE STRENGTH
4.1 INTRODUCTION
Ship structural design has traditionally been driven by rule-based deterministic procedures. Assumptions are made that all factors influencing the load applied to that structure are known and that the strength-load effects are a known function of these parameters, ignoring uncertainties that might occur such as fluctuations of loads, variability of material properties or uncertainty in analysis models, all of which could contribute to a generally small probability that the structure does not perform as it was originally designed. High-implied margins of safety or load factors generate structures that on average are appreciably stronger than their nominal as-designed ultimate capacity.
4.2 APPROXIMATE FORMULAS FOR LINEAR FIRST YIELD AND FULL PLASTIC BENDING MOMENTS OF SHIP HULLS.
For hull girder bending design, it is of interest to calculate linear first yield and full plastic bending moments of ship hulls under vertical bending, which does not accommodate buckling collapse. An idealised hull section is shown in the Figure 5 with equivalent sectional areas at
Figure 5 Idealised hull section definitions
Figure 6 Hull stress definitions
deck, outer bottom, inner bottom and side shell. All longitudinal stiffeners are smeared into the related plating. Vertical or horizontal sectional areas replace inclined member sections. All inner side plating and longitudinal bulkheads are regarded as side. The section moduli of the equivalent hull can in this case be given as follows:
22
gZ
ZgD
ZZ Z
BZ
D �
�
� , (35), (36)
� � � �
� �
SBIBD
BBISD
SBBIBDZ
AAAADAAAD
g
gDD
ADgAgAgDAZ
2
2122Where
22222
���
��
�
�
�
�
�
�
�
�
�
�
�
���������
(37), (38)
ZD and ZB, are section moduli at deck and outer bottom respectively. AD, AB and ABI are total hull sectional area at deck, outer bottom and inner bottom respectively, while AS is half of the total hull sectional area for side shell including all longitudinal bulkheads. Until the outermost fibre of the hull section in the ship depth direction yields, distribution of longitudinal axial stresses in a hull section is linear elastic as shown in Figure 6. Linear first yield moments (immediately after deck or outer bottom yields) can be calculated by:
oBByBoDDyD ZMZM �� �� , (39), (40)
Where MyD and MyB are first yield hull girder moments at deck and outer bottom, respectively, �oD and �oB are material stresses of deck and outer bottom respectively. After the equivalent hull section reaches the fully plastic state without buckling collapse as shown in the figure below, the full plastic bending moments (denoted by Mp) can approximately be given by,
� � � � � �� �oSLoSUS
oBIBBIoBBoDDp ggDDA
DgAgAgDAM �����
22�������� (41)
� �
DA
AAAAg
oSLoSUS
oBIBIoBBoSSoDD
��
����
�
���
�
22
Where (42)
�oBI, �oSU and �oSL are material yield stresses of inner bottom, upper side shell and lower side shell, respectively. The collapse of the compression flange governs the overall collapse of a ship hull under vertical bending moment as shown by reassessment of actual failures [87] and according to numerical studies [21], [86], [26], [88] there is still some reserve strength beyond collapse of the compression flange as a result of hull cross section neutral axis movement toward the tension flange, after the collapse of the compression flange, and a further increase of the applied bending moment is sustained until the tension flange yields. At later stages of the process, side shells around the compression and the tension flanges will also fail. However in the immediate vicinity of the final neutral axis, the side shells will often remain in the elastic state until the overall collapse of the hull girder takes place. Depending on the geometric and material properties of the hull section, these parts may also fail. Paik and Mansour [15], [89] derived and analytical closed form expression for predicting the ultimate vertical moment capacity of ships with general-type hull sections by using an longitudinal stress distribution approach in the hull cross section at the moment of overall collapse. This approach was later extended to a general hull section with multi-decks and multi-longitudinal bulkheads/side shell as shown in the figure below by Paik et al. (1996) [86]. The figure also shows the presumed longitudinal axial stress distribution over the hull section at the ultimate limit state. The number of horizontal and vertical members is (m+1) and (n=1)
23
respectively. The coordinate y indicates the position of horizontal members (such as bottoms and decks) above the base line and zj shows the position of vertical members (such as side shells and longitudinal bulkheads) from a reference (left/port) outer shell. The sectional area of horizontal members at y=yi is denoted by Ayi, while the sectional area of vertical members at z=zj is denoted Azj. In the sagging condition the longitudinal stress distribution over the hull cross-section can be expressed as:
� �� �
� �� � � �
� �deckyy
miyyHyH
DyH
HyHyH
yy
muD
ioSLioSLuSU
uSU
oSLoSLuSU
ooBx
���
�������
����
�����
���
at
1,,2,1at1at
0at1bottom)(outer0at
�
���
�
���
��
�
(43)
Where subscripts L and U indicate the lower and upper parts of side shell plating, respectively. No net axial force acts on the hull i.e.:
0�� dAu� (44)
And the stress distribution in the vicinity of the neutral axis is elastic linear. The neutral axis gabove the base line may be calculated from
DCDCDCHH
goSLuSU
oSL2
2211 2, ���
�
�
��
�
(45), (46)
� � � �
� �� � �
�
�
��
�
�
�
�
�
��
�
�
���
�
n
j zj
m
j iyi
n
j zjoSLuSU
m
i yioSLyByo
n
j zjuSUuDym
A
yAC
A
AAAAC
0
1
12
0
1
101 ,
��
����
(47), (48)
The ultimate moment capacity of the hull for sagging is then given by
� � � � � �� �
� �� � � �� �� � � �
� � � �� �oSLuSU
n
j zjoByouSUn
j zjm
i iyioSL
m
i iiyioSLuSUuDymVus
gHgH
AD
HgAgHDHDA
DgyA
ygyAH
gDAM
��
���
���
3326
221
1
00
1
1
1
12
����
��������
�������
���
�
��
�
�
�
�
(49) Where g and H are calculated from equations (45) and (46) respectively. Similarly in the Hogging condition, the stress distribution can be expressed as
24
� �� �
� �� � � �
Bottom)Outer(yatBottom)Inner(yat
1,,3,2,yyat1at
0at1(Deck)0at
0
i
i
���
���
������
����
������
���
y
y
miHyH
DyH
HyHyH
yy
uB
uIB
oSUiuSUuSL
uSl
oSUoSUuSL
moDx
�
�
���
�
���
��
�
(50)
It is noted that, for simplicity, the positive direction of the y-axis in hogging condition is taken opposite to that of sagging condition. The ultimate moment capacity of the hull under Hogging is then obtained as:
� � � � � � � � � �� �
� � � �� �
� � � �� �uSUuSL
n
jzj
uSL
n
jzjoDgym
m
iiyioSU
m
iiiyiuSUuSLuIByuByoVuh
gHgHAD
H
gHDHDAD
AgyDA
yDyDgAH
gyDAgDAM
��
���
����
3326
221
1
0
0
1
2
1
2
211
�����
�
�
��
�
�
�
�����
�
�
��
�
�
��
�
�
�
���
�
�
�
����������
�
��
�
�
�
�
�
�
�
(51)
� � � �
� �� �
� �
�
�
�
�
�
�
�
�
�
�
�
�
�
�
��
�
�
�
�
�
�
��
�
�
�
�
�
��
n
jzj
i
m
jyi
n
j zjoSUuSL
oDymoSU
m
j yiuSL
n
j zjuIByuByo
oSUuSL
oSU
A
yDA
CA
AAAAAC
DCDCDCHH
g
0
1
22
0
1
2011
222
11
,
,2,
��
�����
��
�
(52), (53), (54), (55)
25
5. CORROSION MODELLING
5.1 CORROSION WASTAGE MODEL
With an increase in casualties of aged merchant ships under operation in the past decade, several casualties are thought to have occurred as a result of structural failure in rough seas and weather. Age-related degradation of the ship structure resulting from corrosion and fatigue influences the safety of the structure to a large extent and for the authors to research and assess the effect that it might have on the structural reliability and the corresponding probability of global failure, it would be necessary to have a relevant estimate of the corrosion rate for every structural area. In this case a time-dependent corrosion wastage model for FPSO/Tanker structures was used as proposed by Paik, Lee, Hwang and Park [90] and can be seen in Figure 7. The particular model was concerned with localized corrosion as well as general corrosion and is based on statistical analysis of wastage measurements of a population of vessels.
Figure 7 Corrosion process model for marine structures [Paik, Lee, Hwang and Park (2003)]
The structural flexibility characteristics of ocean going single- or double-skin tankers may be different from those of FSUs or FPSOs, which load and unload a lot more frequently, sometimes every week. Such frequent loading/unloading patterns may accelerate the corrosion progress but on the other hand FSUs or FPSOs typically operate at standstill at a specific sea site, and this aspect may likely mitigate the dynamic flexing, keeping the corrosive scale static, compared with that of ocean going vessels. The two counter aspects nay then offset the positive and negative effects of corrosion. A significant amount of research exists in this field by Gardiner and Melchers [91], [92], and [93], IACS [94], DNV [95] and ABS [96] but it is considered that the corrosion models of ocean-going single- or double-skin tanker structures can be applied to corrosion prediction of FPSO or FSU structures as long as the corrosion environment is similar.
26
27
6. THERMAL STRESS MODELLING ON SHIP STRUCTURES
6.1 STRESS DISTRIBUTION IN A COMPOSITE BOX GIRDER DUE TO AN ARBITRARY TWO-DIMENSIONAL CHANGE OF TEMPERATURE.
Figure 8 Idealized hull girder definitions
To determine the stress distribution in a transverse section of an idealized ship girder follows from the assumptions that the stress distribution is the same at every transverse section sufficiently distant from the ends of the beam. As required by equilibrium the net moment, the total longitudinal force and the total transverse shear force in any direction at any transverse section must be zero. With the assumption that the girder acts as if it were made up of separate longitudinal fibres fitted together in the form of the ship girder these are initially allowed to expend freely due to temperature changes. Stress fields are then imposed so as to satisfy the requirements of continuity, geometry, equilibrium and the boundary conditions imposed. By choosing the origin of the coordinates so that the oy and oz axes coincide with the principal centroidal axes of inertia, for beams constructed of more than one material initially a set of stresses �1 is defined preventing all longitudinal fibres of the beam from thermal expansion.
ET�� ��1 (56)
Both E and � can vary over the cross section. Then a constant stress �2 is added which will make the total longitudinal force on the section zero. Now �2 must vary in proportion to E in order to equalize strain in all fibres
�
�� � ����
EdA
EtdAEETdAdA
�
��� 22 0 (57)
A set of stresses �3 and �4, corresponding to pure bending in the xy and xz planes respectively (the x axis being longitudinal), is added to the stresses �1 and �2 so that the net bending moment at the section equals zero. One can achieve this by choosing the oy and oz axes to coincide with the principal centroidal axes of inertia. So initially for a homogeneous beam by taking moments about the oz axis:
28
� �� ���� 04321 ydA���� (58)
This expression is obtained by taking moments about the oz axis, therefore:
� � ycdAy
ydAydAydAy cy�
�
�
�
�
�
�
�
�
� ���
�
�
��� �
�
���
� 32
4123 , (59), (60)
And similarly,
� � zcdAz
zdAzdAzdAy cz�
�
�
�
�
�
�
�
�
� ���
�
�
��� �
�
���
� 42
3124 , (61), (62)
Since the oy and oz axes are chosen to coincide with the principal centroidal axes of inertia,
������ ������ ETzdAIz
ETydAIy
zdAydAydAzdAoyoz
������ 4334 ,,0 (63),
(64), (65)
Now for a composite beam in pure bending,
� � 0dAx� (66)
Also we locate the axes so that bending in the yx and zx axes is uncoupled so that the strain is proportional to y or z respectively,
��� ��� 0EyzdAEzdAEydA (67)
This determines the principal centroidal axis of an area comprised of area elements (EdA),which have the coordinates y, z of dA. So,
EzkEyk 2413 , �� �� (68), (69)
Where k1 and k2 are constants.
� � 04321 ����� ydA����
���� ������ 0and0 22
21 dAEzkETzdAdAEykETydA ��
��
��
�� ETzdAdAEz
EzETydA
dAEy
Ey����
2423 , (70), (71)
The resultant stress distribution (which is the sum of the four � components) is the desired solution inasmuch as it satisfies the requirements of equilibrium, continuity, beam theory and the boundary conditions, except near the ends of the beam. By St. Venant’s principle the
29
solution will be valid at sections distant from the ends of the beam, even though the boundary conditions are not completely satisfied at the ends of the beam.
4321 ����� ���� (72)
��
���
������ ETzdA
dAEz
EzETydA
dAEy
Ey
EdA
ETdAEET ��
�
��
22 (73)
This can be also written as:
� � � ���� ����� ETzdA
Ikz
ETydAIky
ETdAAk
EToyoz
�����
���
(74)
� � � ���� ���
��
�
�
��
�
�
� dAzEE
IdAyEE
IdAEE
AEE
k oyoz22 ,,,
�
�
�
�
�
�
�
With E� an arbitrary constant, which in our case is the Young’s Modulus of mild steel. The
equation (74) provides the stress distribution for any given change T in temperature distribution in a no homogeneous beam but up to this point we have assumed that no transverse restrain has been applied on the beam. With the existence of transverse bulkheads in a ship structure providing such a transverse restraint, which to some extent will prevent the free transverse expansion of the sections during a temperature change, those effects should also be taken into account. To achieve this some additional assumptions have to be made: The walls of the box beam are this so that there are no stresses normal to the thickness of the wall. The cross-sectional shape and dimensions are preserved by closely spaced, imaginary diaphragms, which are rigid in the plane of the diaphragm and free to warp out of their plane. The diaphragms are insulated from the box beam and the temperature of the diaphragms remains constant while the walls of the beam are subjected to temperature changes. To determine again the stresses i.e. �x=0, the stress strain temperature relationships are,
� �� � 0here1����� xzyxx eT
Ee ����� (75)
� �� � TE
e zxyy ����� ����
1 (76)
� �� � TE
e yxzz ����� ����
1 (77)
For horizontal surfaces (parallel to the xz plane, such as the top deck) ez=0 as it occurs from the second assumption, therefore from Equations (75)-(77) by equating ex to ez it is found that,
zx �� � (78)
Using this and the first assumption made (that �y=0) Equation (75) gives,
30
� ��
�
��
�
���
1ET
zx (79)
And with similarly the desired normal stresses �1 in the axial direction and the peripheral normal stresses �8 parallel to the yz plane are given by the expression derived above:
� ��
�
���
�
����
11ET
zx (80)
By substituting the value of �1, which includes full effective lateral restraint, in place of the value previously used we obtain the final relationship for s since none of the remainder of the theory used is affected by the addition of the restraint.
� � � � � ���� ���
�
�� ETzdAIkz
ETydAIky
ETdAAkET
oyoz
���
�
�
�
���
1 (81)
It is noticed that the stress changes induced by temperature changes are greater when a transverse restraint is acting than would otherwise be the case. If, for example Poisson’s ratio v=1/3 is used then the stress variations will be increased by 50 percent. The actual stress in a box beam would be expected to lie somewhere between those obtained for the assumption of zero restraint and full transverse restraint.
31
7. RELIABILITY ANALYSIS
7.1 FUNDAMENTAL CONCEPTS
Structural design could be described as the practice of designing a structure such that it has higher strength or resistance than the effects occurring on the structure from the loads. The parameters affecting both the load and the resistance of the structure are again affected by uncertainties that cannot be easily controlled during the design process. This can be demonstrated by treating the problem in terms of two variables, the resistance of the structure Rand the load S (Figure 9).
Figure 9 Reliability analysis definitions
Their random nature is described by their corresponding probability density functions (fS(s) and fR(r)) and by taking into account the nominal values used during a deterministic safety factor-based approach (SN and RN). Design safety in a deterministic approach is achieved by multiplying SN by a specified margin of safety to ensure that it is greater than RN. Two similar methods are used for deterministic design. The allowable stress design methods compute the allowable stresses in structural members by using a safety factor from the ultimate stress to ensure that the nominal load occurring stresses do not exceed the allowable stresses. The value of RN is multiplied by that safety factor and an allowable resistance Ra occurs. Structural safety will is achieved when SN<Ra. The ultimate strength design method multiplies the loads by certain factors which determine the ultimate loads and hence SN is multiplied by a load factor to obtain the ultimate load Su and safety is achieved by ensuring that Su<RN. As it can be seen from (Figure 9) the probability of structural failure lies in the shaded region under the two probability density function curves. This area of overlap depends initially on the means (�R and �S) of the two variables that control the relative positions of the two curves. A decrease in the distance between the two curves increases the probability of failure. The area is also dependant on the standard deviations (�R and �S) of the variables that describe the dispersion of the two curves. The narrower the two curves are, the smaller the area of overlap and hence the probability of failure is. Finally the area is dependant of the actual PDFs (probability density functions) of the two variables that represent the shape of the curves. Varying such statistical factors is not always very economical as large variation of the resistance mean implies a large increase in the total cost and weight of the structure. Deterministic design methods reduce the area of overlap by shifting the position of the curves with the use of safety factors. By computing the actual probability of failure by taking all the factors that affect the curves into account, it is possible to choose the design variables such that an acceptable probability of failure can be achieved. This is the basic idea behind probabilistic design methods but unfortunately the implementation of such a method by designers is difficult since PDF information is difficult to obtain and the variable means and standard deviations of the variables are the only tools that can be used. Reliability can be defined as “the probability that the system will perform its intended function
32
for a specified period of time under a given set of conditions” [97]. Structural Reliability is concerned with the calculation and estimation of the probability of limit state violation, violation of both serviceability criteria and the ultimate limit state. Initially the relevant load and resistance parameters (basic variables Xi) and the function that relates them are decided. This function can be described as:
� �nXXXgZ ,...,, 21� (82)
The failure surface or performance function of the limit state is then defined as Z and is the boundary between the safe and unsafe regions. A limit state function can be an explicit or implicit function of basic variables and it can be in simple or complicated form. By using Eq. (82) structural failure occurs when Z<0 and the probability of failure pf is then given as:
� �� �� nnxf dxdxdxxxxfp ...,...,,... 2121 (83)
With fx(x1,x2,…,xn) the joint probability density function for X1, X2, …, Xn and the integration performed over the region in which g(.)<0. In the specific case where the random variables are statistically independent, the product of the individual density functions of the integral may replace the joint density function. Calculation of the integral for pf by using Eq. (83) is called the fully distributional approach and is considered to be the fundamental equitation of structural reliability analysis. The join probability density function of all random variables involved in a problem is practically impossible to obtain even in the cases where the PDF is available and therefore methods approximating the integral have been developed. All reliability methods of approximating the integral in Eq. (83) can be grouped into two types, first- and second-order reliability methods (FORM-SORM). The limit state functions of common engineering problems can be either linear or non-linear functions of the basic variables. First-order methods can be used when the limit state function is a linear function of uncorrelated normal variables or when the non-linear limit state function is represented by the first-order (linear) approximation, which is by a tangent at the design point. The second-order methods estimate the probability of failure by approximating the non-linear limit state function, including a linear limit state function with correlated normal variables, by a second-order representation. The first order method originates from the second-moment methods that were developed in the early 70’s [97], [98], [99].
33
8. APPLICATION AND RESULTS
8.1 ULTIMATE STRENGTH AND THERMAL STRESS ANALYSIS
Results obtained during this research have been publicised and extensively reviewed in Moatsos and Das (2003-2004) [100-106]. 2 FPSO structures were analyzed using a procedure based in the theory described in previous chapters of this report. The principal particulars of the vessels can be found in Table 1 and the operational profiles assumed based on a combination of actual and assumed data supplied by SHELL EP Europe and the HSE can be found in Table 2.
Table 1 FPSO Principal Particulars
FPSO 1 FPSO 2 Length BP (m) 233 m. 228.4 m.
Length Overall (m) 230.375 m. 227.756 m. B (m) 42 m. 45 m. D (m) 21.36 m. 27.25 m.
T design (m) 13.6 m. 20 m. T scantling (m) 14.7 m. 20 m.
Cb 0.94 Assumed 0.911 Tonnage 105000 DWT 154000 DWT
Table 2 FPSO Operational Profiles
FPSO 1 FPSO 2 Capacity (bbls) 630000 bbls 1000000 bbls
Production (per day) 105000 bopd 142000 bopd Production (days) 6 days 7 days
Offloading every (days) 5 days 6 days Tsw (days/year) in Full
Loading condition 219 days/year 73 days/year
tsw (hours) 72 hours 24 hours nsw 73 73
In the absence of available recorded air temperature data from FPSOs in the North Sea, data had to be obtained from a different source [107]. Fair Isle is the most southerly island of the Shetland group (the northernmost islands of the British Isles). It lies approximately 25 kilometres south to southwest of the Shetland mainland (Figure 10), and approximately the same distance northeast of North Ronaldsay (the most northerly of the Orkney Islands). Since it is in the North Sea its climate accurately represents the conditions that exist in the area throughout the year. Air temperatures and cloud cover data for the period 1978 to 1998 were used and since the diurnal variation of temperature was required the daily average maximum and minimum temperatures recorded would be the most suitable data set. From all the years a seasonal average and seasonal extreme values were selected to provide an extreme and average model of the climatological conditions.
34
Figure 10 UK Weather Stations
The same procedure was followed for the cloud cover data required again obtaining average and extreme values. In (Figure 11) an example of the air temperature in Fair Isle for the year 1989 (one of the years that an extreme seasonal change in temperature was recorded) can be seen and the annual average change in temperature for the period of time considered.
JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC
-2
0
2
4
6
8
10
12
14
16
18
Diurnal Temperature Change in Fair Isle
Tem
pera
ture
Cha
nge
o C
Month
Air Max. 89 Air Min. 89 Average 89 Average 78-98
Figure 11 Diurnal temperature change in Fair Isle average for 1989
As suggested in the Ship Structures Committee (SSC) Report 240 [108] apart from the average diurnal change in air temperature a number of other parameters had to be taken into account. Deck plating would be subjected to this temperature change plus the change due to insolation (the absorption of radiant heat). The temperature change due to insolation depends on cloud cover and the colour of the deck. As an approximation according to Taggart [109], Table 3 shows maximum differences between air and deck (sun overhead, unshaded) for different colour of the deck. It was assumed in our case that the deck colour is dark grey.
35
Table 3 Temperature differences according to deck colour
Deck Colour Temp. Difference oCBlack 10.00
Red 4.44 Aluminium -12.22
White -12.22
The prediction of average thermal stresses and expected maxima requires also that the frequency of occurrence of different conditions of sun exposure be determined. The assumption of the vessel operating constantly in the North Sea simplified this and monthly cloud cover data was used averaging from 1978 to 1998 and specifically on the dates of extreme temperature changes. A simplified distribution of �T is assumed with the temperature remaining constant over the deck and the sides down to the water line and �t=0 elsewhere (below WL). Longitudinals are assumed to have the same �T as the plating, which they support. Longitudinal bulkheads and associated longitudinals are assumed to have �t decreasing linearly from the deck �t at top to �t=0 at the assumed level of oil inside the tanks. The vessel is also assumed to be initially at a state of zero stress and while on the stocks, when the final welds in the cross-section were made and the temperature of the upper deck exposed structure would be lower than the more sheltered lower parts of the hull.
-200 -150 -100 -50 0 50 100 150
-10000
-5000
0
5000
10000
15000
Thermal Stress 18.37 oC Diurnal Temp. Change
Dist
ance
from
N.A
. (m
m)
Thermal Stress (N/mm2) Outer Shell Tank Top Centerline NWT Girders Stringers
Figure 12 FPSO Midship section thermal stress distribution
A procedure based on the formulation suggested by Paik [6], [110] determining the ultimate strength of the ship hull girder was used. Ongoing research on this area of ship structures provides detailed and accurate analytical methods, such as the one used which have been validated by comparison with experimental results and FE analysis. The thermal stress due to temperature changes was then combined with the ultimate strength theory on a stiffened plate element level by using a load-resistance approach. The thermal stress, used as an imposed load on the ship structure, having been determined for each individual stiffened panel was then subtracted from the ultimate strength, which would more truly represent the resistance of the stiffened plate (Fig. 12 and Fig. 13). Thermal expansion coefficients that were used to model the effect of temperature on sections varied according to steel grade. The theoretical background for the determination of the ultimate strength of a stiffened panel and the ship hull girder as a whole can be found in [110] and Paik and Pedersen [86]. The ultimate strength results for the as built vessels in sagging and in hogging can be seen in Table 4. For the verification of the ultimate
36
strength calculation results the Lloyd’s Register (LR) LRPASS system of programs was used providing an alternative method of calculation. The LR LRPASS [111] programs 20202 and 20203 uses similar type of simplified component approach, where the behaviour of the hull is evaluated based on the behaviour of single structural components. The cross section of the Hull is subdivided into beam column elements, which are assumed to be act independently. Each element is composed of longitudinal stiffeners and an effective breadth of plate. Three collapse conditions are investigated; failure initiated by plate compression, stiffener tension and stiffener compression. In this report, only the case of longitudinal strength of the ship is considered. The code is based on the theory by Rutherford and Caldwell (1990) [21]. Non consistent differences between the two method results, with the LRPASS results in the slightly more conservative side, suggest that more research is required in the field that another method should also be considered when evaluating the ultimate strength of a structure.
Table 4 FPSO Ultimate Strength Results
Paik LRPASS FPSO 1 Sagging 12.754*1012Nmm 13.1366 1012Nmm
Hogging 11.559*1012Nmm 11.9438 1012NmmFPSO 2 Sagging 17.96*1012Nmm 19.33 1012Nmm
Hogging 16.28*1012Nmm 14.372 1012Nmm
20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 340 360
0
5000
10000
15000
20000
25000Ultimate Strength Variation due to Thermal Stress in Outter Shell
Dist
ance
from
Bas
elin
e (m
m)
Ultimate Strength (N/mm2)
No T Effect 11.7 oC Effect 16.01 oC Effect 16.26 oC Effect 17.21 oC Effect 18.37 oC Effect
Figure 13 FPSO Distribution of strength values as affected by temperature variations
8.2 PROBABILITY DISTRIBUTION OF LONG-TERM WAVE INDUCED LOAD EFFECTS
A FORTRAN90 code was developed to calculate the probability distribution of the wave induced load effects that occur during long-term operation of the vessel. The program reads the RAOs and the data from the scatter diagram from data files. The RAOs from the file are given in a non-dimensionalised form, so all the values have to be multiplied by �gaBL2 in order to get consistency in the calculations. The program calculates the average m0 for 7 different headings, and then calculates the probability for exceeding a value of VBM for this particular condition of wave height, wave
37
period, ship speed and load condition. A summation of m0 over 11 Tz and 15 Hs is then performed in order to obtain the total probability of exceeding amplitude x. As explained above ship speed and load condition is taken into account by assuming constant speed and performing load calculations for different load conditions. The calculation of Q(x>VBM) is done for 12 values of VBM and the results are written to an output file, output.dat.
8.3 APPLICATION OF RELIABILITY CONCEPTS TO MARINE STRUCTURES
The largest loads and load affects on a vessel are imposed on the structure in the vertical bending of the hull as a free-free beam. Waves imposing loads on the ship cause variations in the vertical bending moment. Due to their weight distribution, most naval vessels sag even under still water imposed loads which makes possible the fact that hogging bending moment might not occur on the hull as a result of the wave-induced bending moment. In the case of limit state analysis the hull is treated as a box experiencing pure bending. In the case of a simple beam then from classic beam theory:
yu ZM �� (84)
With Mu being the ultimate bending moment, Z the section modulus of the vessel at the section that the problem is treating and �y the tensile yield stress of the vessel material. The limit state for hull girder bending is then expressed as:
LRG �� (85)
Where G is the limit state function, R is the resistance (strength) of the structure and L is the imposed load. When G is less or equal to zero, the structure fails whereas when G is greater than zero, the structure survives. Expressing the limit state function relation is terms of Eq. (84) and by separating the imposed loads into wave and still water bending moments Eq. (85) becomes:
WSWy MMZG ��� � (86)
Where MSW is the still water bending moment and MW the wave induced bending moment. This then produces a non-linear performance function in which each one of the variables above has its own distribution type, mean value and coefficient of variation. Each one of the different types of loads will represent the distribution of the loading in the lifetime of the vessel.
8.4 APPLICATION TO FPSO/TANKER STRUCTURES
Structural reliability analysis of the longitudinal hull girder has been carried out for an FPSO/Tanker structure. Only vertical bending moment has been considered, and the hull girder loads were divided into Stillwater induced and Wave induced components. Mansour and Hovem (1994) [112] evaluated the reliability of ship hull girder strength with the variation in the correlation coefficient between the still water and the wave-induced components and they concluded that the interaction of these two load components might be usually ignored. The two loading components were considered independent and Ferry Borges– Castanheta load combination method was applied to obtain a load combination factor. The failure surface for the ultimate longitudinal strength of the hull girder was defined as;
g(X) = �u Mu - Mse - �w �nl �w Mwe = 0 (87)
38
Mu, represents the ultimate bending moment capacity of the hull girders and a factor �u, (uncertainty on ultimate strength) is introduced in the failure surface equation in order to account for the uncertainties in the ultimate strength model and in the material properties. Teixeira [113] estimated a realistic COV for the total uncertainty on ultimate strength to be 10%, which is in good agreement with Faulkner [2] who proposed a COV of 10-15% for the modelling uncertainty for flat panel collapse. Thus, �u was assumed normally distributed, with a mean value of 1.0 and a standard deviation of 0.10. The most probable extreme still water bending moment, Mse, is calculated based on a mean MSW. The still water bending moment is always hogging for this vessel. A COV of 15 % was applied to this mean value. The load combination factor, �w, is introduced to account for the low probability that the extreme still water and wave bending moments will occur at the same instant. A factor of 0.78 for the structure analysed was obtained respectively, using Ferry Borges–Castanheta load combination method as described before. In addition to the non-linear effects, there are other uncertainties associated with the linear strip theory programs. Different programs will use different procedures for calculating the hydrodynamic coefficients. Dogliani et al [114] showed that the linear strip theory programs generally over-predicted the wave induced bending moments at the 10-8 probability level. Shellin et al [115] identified large variations in long-term distributions of midship induced loads based on transfer functions obtained by different methods. Values of 10% for �w (Uncertainty in wave load prediction) and 1.2 in Sagging and 0.8 in Hogging for �nl (Non-linear effects) were based on the previous research works carried out at Glasgow University [116-118]. Extreme vertical wave bending moment, Mwe, was obtained by combining the short-term responses with the wave statistics for the Fair Isle area. The long-term probability of exceedance of different wave-induced vertical bending moments was obtained, and a Weibull distribution was fitted to the resulting distribution.
8.5 STRUCTURAL RELIABILITY RESULTS
Reliability analysis using a First Order Reliability Method (FORM) and a Second Order Reliability Method (SORM) was performed on the structures by assuming that the ultimate strength of the hull girder follows a Lognormal distribution with a coefficient of variation (COV) of 10%, the wave bending moment following a Gumbel distribution with a COV of 15% and the still water bending moment a Gumbel distribution with a COV of 15%. Although wave bending moment variations could often be in the range of 25% the 15% figure was chosen as an initial starting point for the procedure. Reliability indices � for the structure of the FPSO 2 in the range of temperatures examined and probabilities of failure Pf were obtained for sagging and hogging conditions (Figs. 14~17) and the results showed an obvious decrease in � and an increase in the probability of failure as diurnal change in temperature increased.
39
Prob of Failure Pf in Sagging
1.67E-05 1.67E-05
1.57E-04
6.22E-048.38E-049.16E-04 9.16E-04
5.74E-038.99E-03 1.05E-02
1.77E-02 1.77E-02
4.98E-026.89E-02 7.77E-02
1.00E-05
1.00E-04
1.00E-03
1.00E-02
1.00E-01
1.00E+000 5 10 15 20 25 30
Corrosion in Years
Prob
. of F
ailu
re
0 11.7 18.3 Expon. (18.3) Expon. (11.7) Expon. (0)
Figure 14 Temperature and corrosion effect on Pf in sagging condition
Reliability Index � in Sagging
4.714 4.714
4.235
3.9513.8193.771 3.771
3.388
3.1613.0553.017 3.017
2.710
2.5292.444
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
2.5 7.5 12.5 17.5 22.5 27.5
Corrosion in Years
Rel
iabi
lity
Inde
x b
0 Deg. C. 11.7 Deg. C. 18.3 Deg. C. Poly. (0 Deg. C.) Poly. (11.7 Deg. C.) Poly. (18.3 Deg. C.)
Figure 15 Temperature and corrosion effect on � in sagging
40
Prob of Failure Pf in Hogging
1.49E-03 1.49E-03
1.03E-021.32E-02 1.46E-02
4.71E-02 4.71E-02
7.63E-029.36E-02
1.15E-011.41E-01 1.41E-01
3.52E-01
4.77E-015.35E-01
1.00E-03
1.00E-02
1.00E-01
1.00E+000 5 10 15 20 25 30
Corrosion in Years
Prob
. of F
ailu
re
0 11.7 18.3 Expon. (18.3) Expon. (11.7) Expon. (0)
Figure 16 Temperature and corrosion effect on Pf in hogging
Results also show that � values reduce following a 3rd order Polynomial trend whereas Pf values reduce following and exponential trend. In both cases the effect of temperature and corrosion on the values is quite significant. Partial safety factors for the variables used in reliability analysis could be obtained which would provide a code format for design purposes and allow the design to be even further optimized. If a simple code format such as �R>�sMS+�wMW was assumed where ����s and �w are the partial safety factors for the strength R, the still water bending moment MS and the wave bending moment respectively then the design could be easily optimized. Partial safety factors shown in (Figs.18~19) obtained showed not to be affected by temperature changes in both sagging and hogging conditions whereas in sagging the wave bending moment partial safety factor decreased as diurnal temperature change increased as a result of the sign convention used that made only the wave bending moment to have a negative effect on the limit state function.
Reliability Index � in Hogging
3.420 3.420
3.072
2.8662.7712.736 2.736
2.458
2.2932.2172.189 2.189
1.966
1.8351.773
1.3
1.8
2.3
2.8
3.3
3.8
2.5 7.5 12.5 17.5 22.5 27.5
Corrosion in Years
Rel
iabi
lity
Inde
x b
0 Deg. C. 11.7 Deg. C. 18.3 Deg. C. Poly. (0 Deg. C.) Poly. (11.7 Deg. C.) Poly. (18.3 Deg. C.)
Figure 17 Temperature and corrosion effect on � in hogging
41
0 5 10 15 20
0.0
0.5
1.0
1.5
2.0
Temperature Effect on PSF in Sagging
Parti
al S
afet
y Fa
ctor
Temperature Change oC
Extr. Still Water BM Extr. Wave BM Ultimate Str. Uncertainty Uncert. in Wave Load Prediction Ultimate Strength
Figure 18 Temperature effect on the Partial Safety Factors in sagging
0 5 10 15 20
0.70
0.75
0.80
0.85
0.90
0.95
1.00
1.05
Temperature Effect on PSF in Hogging
Parti
al S
afet
y Fa
ctor
Temperature Change oC
Extr. Still Water BM Extr. Wave BM Ultimate Str. Uncertainty Uncert. in Wave Load Prediction Ultimate Strength
Figure 19 Temperature effect on the Partial Safety Factors in hogging
42
43
9. SUMMARY AND CONCLUSIONS
Two FPSOs are considered in this paper for the structural reliability and ultimate strength analysis. An algorithm has been developed in Fortran90 to compute the longitudinal strength of the hull girders based on two different techniques. These results are also validated using the commercially available software, LRPASS. The simplified method gives a faster and reasonable estimate of the ultimate longitudinal bending moment. A credible stress distribution is assumed in the analysis across the hull cross section to take care of the elastic regions around the neutral axis at failure. The values of plate slenderness ratio and column slenderness ratio of the elements considered have a major influence in the ultimate strength of the deck, side and bottom plating of the hull girder. This in turn influences the ultimate capacity of the FPSOs. The highest values of the slenderness ratio correspond to lower ultimate strength and thus lower ultimate capacity of the hull girder. The program automatically considers the most possible conditions of failure in the analysis by considering different types of stiffeners at different regions of the cross section of the ship. The area of cross section of the deck, sides and bottom also has a major influence in the ultimate capacity of the hull. Although it is not possible to provide a comparison between the results from these three methods because of the nature of the methods and the assumptions taken in order to derive the formulas, most of the corresponding predictions (or results) are close to each other. The loads acting on the hull girder, considered here, are mainly the still water and wave induced bending moments. The two loading components have been considered independently and Ferry Borges-Castenheta load combination method has been applied to obtain load combination factors for both FPSOs. MATHCAD software is used to provide a procedure to describe the distribution of the maximum value of wave induced bending moment based on the underlying Weibull distribution and Gumbel model to describe the extreme models for wave induced load effects. For FPSO2, the annual reliability index in sagging was found to be varying from 3.819 to 4.714. in the un-corroded state. As temperature differences increased and the amount of corrosion in the structure increased as well, a reduction in reliability indices was noticed as expected For FPSO1, the annual reliability index was found to be varying between 2.359 to 2.75 in the un-corroded state. This is rather low value corresponds to approximately one failure in 110 years or one failure per year in 110 structures. From the reliability analysis, it is also found that the safety of the FPSOs is very sensitive to variables associated with the wave bending moment. The effect of temperature change in the reliability based structural design of Tanker/FPSO structures was investigated and a procedure was proposed which took into account diurnal changes of temperature in the North Sea. The significant reduction of the hull girder ultimate strength when thermal stresses were taken into account could be attributed to the way in which thermal stress occurrence was modelled and the extreme nature of the values used which in our case had a 20-year period. Extreme diurnal temperature variations occurred in summer months during which extreme wave bending phenomena would have a lower probability of occurrence, which explains the magnitude of the results. The interesting nature of these phenomena suggests that more research should be made in this field by comparing the method used with other methods existing from other engineering fields such as the aeronautical industry. The procedure could also be easily modified to accommodate vessels travelling between ports that would often encounter changes in temperature due to the climatological characteristics and global position of the ports involved. Reliability analysis provides a stochastic design process that takes into account all random variables involved in ship structural design and provides a better and alternative to deterministic approaches that are currently used extensively by the industry and are only revised when accidents occur and fail to address effectively innovative designs and the rapid change in technologies in the marine industry. Treating design variables in a stochastic manner helps to incorporate in the design procedure the “unexpected” and the random nature of variables that
44
effect ship safety and be able to accommodate for future developments independent of the vessel type or design characteristics such as hull type, structural arrangements, vessel size and type of cargo which should obviously affect the design process but should not restrict it.
45
REFERENCES
1 IACS, “Requirements Concerning the Strength of Ships-S11 Longitudinal Strength”, Revision 3, International Association of Classification Societies, London UK, June 2003.
2 Faulkner, D., Section 1 in “Reliability Based and Assessment of FPSO Structures”, HSE Report Series, OTO 98 164, HSE, December 1998. (Section 2 is by A.C. Walker).
3 Buckley, W.H., “Hull Girder Structural Design – the Case for New Loading Conditions for Extreme Waves”, Naval Engineers Journal, February 1978.
4 Frieze, P.A. et al, “Report on Committee V1, Applied Design”, Proc. 11th ISSC, vol. 1, ed. by P.H. Hsu and Y.S. Wu, Elsevier Applied Science, 1991.
5 Lord Donaldson’s Assessment (DERBYSHIRE), Annex by D. Faulkner, 1995.
6 Paik, J.K., Kim, S.K., Yang, S.H., Thayambali, A.K., “Ultimate Strength Reliability of Corroded Ship Hulls”, RINA Trans., Vol. 140, 1998. (see also Paik and Thayambali, Trans. SNAME, Vol. 146, 1998)
7 Gordo, J.M., Guedes Soares, C., Faulkner, D., “Approximate Assessment of the Ultimate Longitudinal Strength of the Hull Girder”, Journal of Ship Research, Vol. 40, No. 1, March 1996.
8 Dow, R., “Structural Redundancy and Damage Tolerance in Relation to Ultimate Ship Hull; Strength”, Intl Conf., Advances in Marine Structures III, Elsevier Applied Sci., 1997.
9 O.F. Hughes, “Ship Structural Design, A Rationally-Based Computer-Aided Optimization Approach”, John Wiley & Sons, USA, 1983.
10 W.G. John, “On the strength of the iron ship”, Trans. Inst. Naval Arch., Vol. 15, pp.74-93, 1874.
11 J.B. Caldwell, “Ultimate Longitudinal Strength”, Trans. RINA, 1965.
12 M. Maestro, A. Marino, “An Assessment of the structural capacity of damaged ships: The plastic approach in longitudinal symmetrical bending and the influence of buckling”, Int. Shipbuilding Progress, Vol 36:408, pp.(255-265, 1989.
13 S. Nishihara, “Analysis of ultimate strength of stiffened rectangular plate (4th Report)-On the ultimate bending moment of the ship hull girder”, Journal of the Soc. Naval Arch. Of Japan, Vol 154, pp.367-375, in Japanese, 1983.
14 A.E. Mansour, J.M. Yang, A. Thayamballi, “An experimental investigation on ship hull ultimate strength”, Trans. SNAME 98:411-440, 1990.
15 J.K. Paik, A.E. Mansour, “A simple formulation for predicting the ultimate strength of ships”, Journal of Marine Science and Technology, 1995.
16 J.K. Paik, “Hull Collapse of an aging Bulk Carrier under combined Longitudinal Bending and Shearing Force”, Trans. RINA, 1993.
46
17 J.K. Paik, S.K. Kim, S. H. Yang, A.K. Thayamballi, “Ultimate Strength Reliability of Corroded Ship Hulls”, Trans. RINA, 1997.
18 Wei-Biao Shi, “In-Service Assessment of Ship Structures: Effects of General Corrosion on Ultimate Strength”, RINA Spring Meetings, 1992.
19 J.M. Gordo, C. Guedes Soares, “Approximate Load Shortening Curves for Stiffened Plates under Uni-axial Compression”, Integrity of Offshore Structures, EMAS, pp.189-211, 1993.
20 J.K. Paik, “SPINE, A computer program for analysis of elastic –plastic large deflection behaviour of stiffened panels. User’s Manual”, Pusan National Univ., Korea, 1999.
21 S.E. Rutherford, J.B. Caldwell, “Ultimate Longitudinal Strength of Ships: A Case Study”, Trans. SNAME, 1990.
22 T. Yao, “Analysis of ultimate longitudinal strength of a ship’s hull. Text Book of Mini-Symposium on Buckling/Plastic Collapse and Fatigue Strength; State of the Art and Future Problems”, The West-Japan Soc. Naval Arch., pp.154-177, in Japanese, 1993.
23 T. Yao, P. I. Nikolov, “Progressive Collapse Analysis of a Ship’s Hull under Longitudinal Bending”, Journal of the Society of Naval Architects of Japan, Vol. 170, pp.449-461, 1991.
24 T. Yao, P. I. Nikolov, “Progressive Collapse Analysis of a Ship’s Hull under Longitudinal Bending (2nd Report)”, Journal of the Society of Naval Architects of Japan, Vol. 172, pp.437-446, Nov 1992.
25 Y.K. Chen, LM. Kutt, C.M. Piaszczyk & M.P. Binek, “Ultimate Strength of Ship Structures”, Trans. SNAME, 1983.
26 L.M. Kutt, C.M. Piaszczyk, Y.K. Chen, “Evaluation of longitudinal ultimate strength of various ship hull configurations”, Trans. SNAME, Vol. 93, pp.33-55, 1985.
27 S. Valsgaard, L. Jorgensen, A.A. Boe, H. Thorkildsen, “Ultimate hull girder strength margins and present class requirements”, Proc. SNAME Symp. ’91 on Marine Structural Inspection, Maintenance and Monitoring, Arlington, March, B., pp.1-19, 1991.
28 Y. Ueda, S.M.H. Rashed, J.K. Paik (1984), “Plate and stiffened plate units of the idealized structural unit method-Under in-plane loading”, Journal Soc. Naval Arch. Of Japan, Vol. 156, pp.336-375, in Japanese, 1984.
29 J.K. Paik, D.H. Lee, “Ultimate strength-based safety and reliability assessment of ship’s hull girder”, Journal Soc. Naval Arch. Of Japan, Vol. 168, pp.397-409, in Japanese, 1990.
30 J.K. Paik, D.H. Kim, H.S. Bong, M.S. Kim, S.K. Han, “Deterministic and probabilistic safety evaluation for a new double-hull tanker with transverseless system.” Trans. SNAME, Vol. 100, pp.173-198, 1992.
31 J.K. Paik, “Ultimate hull girder strength analysis using Idealized Structural Unit Method: A case study for double hull girder with transverseless system, PRADS ’92, Elsevier, Vol. 2, pp.745-763, Newcastle upon Tyne, 1992.
32 J.K. Paik, J.M. Lee, “An empirical formulation for predicting ultimate compressive strength of plates and stiffened plates”, Journal of Ship Research, 1995.
47
33 Y. Bai, E. Bendiksen, P. Terndrup Pederson, “Collapse Analysis of Ship Hulls””, Marine Structures, Elsevier Science Ltd, Vol. 6, pp.485-507, 1993.
34 Technical Committee III.1, “Ultimate Strength”, Proceedings of the 14th International Ship and Offshore Structures Congress, Nagasaki Japan, Vol. 1, pp.253-321, October 2000.
35 Special Task Committee VI.2, “Ultimate Hull Girder Strength”, Proceedings of the 14th
International Ship and Offshore Structures Congress, Nagasaki Japan, Vol. 2, pp.321-391, October 2000.
36 Lewis, E.V. et al.,”Load criteria for ship structural design”, Report No. SSC-224, Ship Structrure Committee, Washington DC, USA, 1973.
37 Ivanov, L., Madjarov, H., “The statistical estimation of SWBM (still water bending moment) for cargo ships”, Shipping World & Shipbuilder, Vol. 168, pp.759-762, 1975.
38 Mano, H., Kawabe, H., Iwakawa, K., Mitsumune, N., “Statistical character of the demand on longitudinal strength (second report)-long term distribution of still water bending moment”, Journal of the Society of Naval Architects of Japan, Vol. 142, in Japanese, 1977.
39 Dalzell, J.F., Maniar, N.M., Hsu, M.W., “Examination of service and stress data of three ships for development of hull girder load criteria”, Report No. SSC-287, Ship Structures Committee, Washington DC, USA, 1979.
40 Akita, Y., “Lessons learned from failure and damage of ships”, Joint Sessions 1, 8th
International Ship Structures Congress, Gdansk, Poland, 1982.
41 Kaplan, M. Benatar, M., Bentson, J., Achtarides, T.A., “Analysis and assessment of major uncertainties associated with ship hull ultimate failure”, Report No. SSC-322, Ship Structure Committee, Washington DC, USA, 1984.
42 Soares, C.G., Moan, T., “Statistical analysis of Stillwater load effects in ship structures, SNAME Transaction, Vol. 96, pp.129-156, 1988.
43 DnV, “Hull structural design of ships with length 100 metres and above, Rules for Classification of Ships. Part 3.Chapter 1”, Det Norske Veritas, Norway, 1994.
44 IACS, “IACS Requirement S7, Minimum Longitudinal Strength Standards, Revision 3”, International Association of Classification Societies, London, United Kingdom, 1989.
45 Nitta, A., On C. Guedes Soares discussion of paper by A. Nitta et al. “Basis of IACS unified longitudinal strength standard”, Vol. 5, (1992) 1-21, Marine Structures, Vol. 7, pp.567-572, 1994.
46 Soding, H., “The prediction of still-water wave-bending moments in containerships”, Schiffstechnik, Vol. 26, pp.24-41, 1979.
47 Moan, T., Jiao, G., “Characteristic still-water load effects for production ships”, Report MK/R 104/88, The Norwegian Institute of Technology, Trondheim, Norway, 1988.
48 Larrabee, R.D., “Stochastic analysis of multiple load: load combination and bridge loads”, Ph.D. thesis, Department of Civil engineering, MIT, Cambridge MA, USA, 1978.
48
49 M. Fujikubo, T. Yao, B. Varghese, “Buckling and ultimate strength of plates subjected to combined loads”, Proc. 7th Offshore and Polar Engineering Conf., Vol. IV, pp.380-387, 1997.
50 T. Yao, O. Niho, M. Fujikubo, B. Varghese, K. Mizutani, “Buckling/ultimate strength of ship bottom plating”, Proc. Int. Conf. on Advances in Marine Structures III, Paper 28, Dunfermline UK, 1997.
51 T. Yao, M. Fujikubo, B. Varghese, K. Yamamura, O. Niho, “Buckling/plastic collapse strength of rectangular plate under combined pressure and thrust”, Journal Soc. Naval Arch. Of Japan, Vol. 182, pp.561-570, 1997.
52 Steen, E., “Buckling of stiffened plates under combined loads-ABAQUS analysis” DNV Report No. 95-0445, 1995.
53 Steen, E., Balling Engelsen, A., “ABAQUS analysis- Plate buckling/GL code.”, DNV Report No. 97-05606, 1997.
54 T. Yao, M. Fujikubo, K. Kondo, S. Nagahama, “Progressive collapse behaviour of double hull tanker under longitudinal bending”, Proc. 4th Int. Offshore and Polar Engineering Conf., Vol. VI pp.570-577, Osaka, 1994.
55 Mansour, A.E., Lin, Y.H., Paik, J.K, “Ultimate strength of ships under combined vertical and horizontal moments.” Proceedings of the 6th International Symposium PRADS, Seoul Korea, Vol. 2, pp.844-851, 1995.
56 Paik, J.K., Thayamballi, A.K., Jung, S.C., “Ultimate strength of ships under combined vertical bending, horizontal bending and shearing forces.” SNAME Transactions, Vol. 104, pp.31-59, 1996.
57 Gordo, J.M., Guedes Soares, C., “Interaction equation for the collapse of tankers and containerships under combined vertical and horizontal bending moments.” Journal of Ship Research, Vol. 41:3, pp.230-240, 1997.
58 JMT (Japan Ministry of Transport, “Report on the Investigation of Causes of the Casualty of Nakhodka.” The Committee for the Investigation on Causes of the Casualty of Nakhodka, 1997.
59 J.J. Jensen et al., “Report of Committee III.1, Ductile Collapse”, Proc. 12th International Ship and Offshore Structures Congress, Vol. 1, pp.229-387, 1994.
60 C.A. Cornell, “Bounds on the reliability of structural systems”, Journal of the Structural Division, ASCE (ST1), 93, pp.171-200, 1967.
61 O. Ditlevsen, “Narrow reliability bounds for structural systems”, Journal of Structural Mechanics, Vol. 7(4) (453-472), 1979
62 H.S. Ang, H.F. Ma, “On the reliability of structural systems”, Proc. of the Int. Conf. on Structural Safety and Reliability, Trondheim, 1981.
63 B.F. Song, “A numerical method in affine space and amethod with high accuracy for computing structural system reliability”, Computers ans Structures, 42(2), 1992.
49
64 C.G. Bucher, U. Bourgound, “A Fast and Efficient Response Surface Approach for Structural Reliability Problems”, Structural Safety, 7 (57-66), 1990.
65 M.R. Rajashekhar, B.R. Ellingwood, “A new look at the response Surface approach for Reliability Analysis”, Structural Safety, Elsevier Science Publishers B.V., 12 (205-220), 1993.
66 S.H. Kim, S.W. Na, “Response surface method using vector projected sampling points”, Strucutral Saefty, 19(1) (3-19), 1997.
67 Y. Zheng, P.K. Das, “Improved Response Surface Method and its application to Stiffened Plate Reliability Analysis”, Engineering Structures, Elsevier Science Ltd., 22 (544-551), 2000.
68 P.K. Das, Y. Zheng, “Cumulative formation of Response Surface and its use in Reliability Analysis”, Probabilistic Engineering Mechanics, Elsevier Science Ltd., 15 (309-315), 2000.
69 L. Yu, P.K. Das, Y. Zheng, “Stepwise Response Surface Method and its Application in Reliability Analysis of Ship Hull Structure”, Proceedings of OMAE 2001, 20th International Conference on Offshore Mechanics and Arctic Engineering, A.S.M.E., Rio de Janeiro, Brazil, June 2001.
70 Hechtman, R.A., “Thermal Stresses in Ships”, Ship Structure Committee Report Serial No. SSC-95, National Academy of Sciences-National Research Council, Washington, 1956.
71 Hurst, O., “Deflections of Girders and Ship Structures: A Note on Temperature Effects”, Transactions of the Institute of Naval Architects, Vol. 85, pp.74, 1943.
72 Corlett, E.C.B., “Thermal Expansion Effects in Composite Ships”, Transactions of the Institute of Naval Architects, Vol. 92, 1950.
73 Jasper N.H., “Temperature-Induced Stresses in Beams and Ships”, DTMB Report 937, David Taylor Model Basin, Washington, 1955.
74 Meriam, J.L., Lyman, P.T., Steidel, R.F., and Brown, G.W., “Thermal Stresses in the S.S. Boulder Victory,” Journal of Ship Research, Vol. 2, 1958.
75 Gatewood, B.E., “Thermal Stresses-With Applications to Airplanes, Missiles, Turbines and Nuclear Reactors”, McGraw-Hill Book Co., New York, 1957.
76 Mansour, A.E., Thayamballi, A., “Probability-based ship design procedures”, Draft Report SR-1337, Ship Structure Committee, Washington DC, USA, 1993.
77 Moan, T., Wang, X., “Stochastic and Deterministic Combinations of Still Water and Wave Bending Moments in Ships”, Marine Structures, Elsevier Science Ltd, Vol.9, pp.787-810, 1996.
78 Meyerhoff, W.K., Arai, M., Chen, H.H., Day, Y.-S., Isaacson, M., Jankowski, J., Jefferys, E.R., Lee, S.-C., Mathisen, J., Richer, J.-Ph, Riska, K., Romeling, J.U., “Loads, reportof Committee I.2”, 11th International Ship & Offshore Structures Congress, Wuxi, China, 1991.
50
79 Frieze, P.A., Dogliani, M., Huss, M., Jensen, J.J., Kunow, R., Mansour, A.E, Murotsu, Y., Valsgard, S., “Applied Design. Report of Committee V.1”, The 11th International Ship & Offshore Structures Congress, China, 1991.
80 Lewis, E.V., “A study of midships bending moments in irregular head seas, Journal of Ship Research”, Vol.1, 1957.
81 Nordenstrom, N., “Methods for predicting long term distributions of wave loads and probability of failure of ships”, DNV, Research and Development Report 71-2-S, 1971.
82 Guedes Soares, C., “Probabilistic Models for Load Effects in Ship Structures”, Report UR-84-83, Div. Marine Structures, Norwegian Institute of Technology, 1984.
83 Borges Ferry, J., Castanheta, M., “Structural Safety 2nd Edition”, Laboratorio Nacional de Engenharia Civil, Lisbon, Portugal, 1971.
84 Moan, T. and Jiao, G., “Characteristic still-water load effects for production ships”, Report MK/R 104/88, The Norwegian Institute of Technology, 1988.
85 Larrabee, R.D. and Cornell, C.A., “Combination of various load processes”, Journal of Structural Division, ASCE, Vol.107, pp.223-238, 1981.
86 Paik, J.K., Pedersen, T., “A simplified method for predicting ultimate compressive strength of ship panels.” Int. Shipbuilding Progress, Vol. 43, No. 434, pp.139-157, 1996.
87 Faulkner, J.A., Clarke J.D., Smith C.S., and Faulkner D., “The loss of the HMS Cobra-a reassessment.” RINA Transactions, Vol. 126, pp.125-151, 1984.
88 Valsgaard, S. and Steen, E., “Ultimate hull girder strength margins in present class requirements.” Proceedings of the Marine Structural Inspection, Maintenance and Monitoring Symposium, SSC/SNAME, Arlington Virginia, III.B.1-III.B.19, 1991.
89 Paik, J.K., Mansour, A.E., “A simple formula for ultimate strength prediction of hull girders.” Trans. of the Society of Naval Architects of Korea, Vol. 32, pp.83-97, In Korean, 1995.
90 Paik, J.K., Lee, J.M., Hwang, J.S. and Park, Y., “A Time-Dependent Corrosion Wastage Model for the Structures of Single- Double-Hull Tankers and FSOs and FPSOs”, Marine Technology, Vol. 40, No. 3, pp.201-217, 2003.
91 Gardiner, C.P. and Melchers, R.E., “Enclosed Atmospheric Corrosion within Ship Structures”, British Corrosion Journal, Vol. 36, No. 4, pp.272-276, 2001.
92 Gardiner, C.P. and Melchers, R.E., “Corrosion of Mild Steel by coal and iron ore, Corrosion Science”, Vol. 44, No. 12, pp 2665-2673, 2002.
93 Melchers, R.E., “Pitting Corrosion of Mild Steel In Marine Immersion Environment Part 1: Maximum Pit Depth”, Dept. of Civil, Surveying and Environmental Engineering, Research Report No. 238.07.2003, The University of Newcastle, Australia, 2003.
94 IACS, “Bulk Carriers: Guidelines for Surveys, Assessment and Repair of Hull Structure”, International Association of Classification Societies, No. 76, London UK, 2001.
51
95 DNV, “Corrosion Prevention of Tanks and Holds”, Classification Notes No. 33.1, Det Norske Veritas, Hovik, Norway, July 1999.
96 ABS, Rules for Building and Classing Steel Vessels, American Bureau of Shipping, Houston TX, USA, 2003.
97 Melchers, R.E., “Structural Reliability Analysis and Prediction 2nd Edition”, John Wiley & Sons, Chichester, England, 1998.
98 O. Ditlevsen, H.O. Madsen, “Structural Reliability Methods”, John Wiley & Sons, England 1996.
99 Moatsos, I., “Reliability Analysis of Ship Structures”, NAOE-2000-11, Department of Naval Architecture and Ocean Engineering, University of Glasgow, Scotland UK, 2000.
100 Moatsos, I. and Das, P.K., “Reliability based ultimate strength structural design for safety”, Proceedings of the 8th International Marine Design Conference (IMDC 2003), Athens Greece, May 2003.
101 Moatsos, I. and Das, P.K., “Temperature dependent FPSO ultimate strength reliability”, Proceedings of the 13th International Offshore and Polar Engineering Conference (ISOPE 2003, Honolulu Hawaii USA, May 2003.
102 Moatsos, I. and Das, P.K., “Temperature and corrosion effects on FPSO ultimate strength: a reliability based approach”, Proceedings of the 22nd International Conference on Offshore Mechanics and Arctic Engineering (OMAE 2003), Cancun Mexico, June 2003.
103 Moatsos, I. and Das, P.K., “Modelling the effect of extreme diurnal temperature changes on ship structures for the assessment of structural reliability load combination techniques”, 2004-JSC-320, Abstract accepted and paper to be published in the Proceedings of the 14th
International Offshore and Polar Engineering Conference (ISOPE 2004), Toulon France, May 2004.
104 Moatsos, I. and Das, P.K., “Assessment of structural reliability of ships under combined loading including extreme diurnal temperature effects”, OMAE2004-51316, Abstract accepted and paper to be published in the Proceedings of the 23nd International Conference on Offshore Mechanics and Arctic Engineering (OMAE 2004), Vancouver Canada, June 2004.
105 Moatsos, I. and Das, P.K., “Structural Reliability of Ship Structures with Advanced Hull Girder Modelling”, Abstract accepted and paper to be published in the Proceedings of the 2nd ASRANet International Colloquium, Barcelona Spain, July 2004.
106 Moatsos, I. and Das, P.K., “Implementation of combined loading techniques modelling the effects of diurnal thermal loads, slamming and corrosion to determine the reliability of Tanker/FPSO structures.” Journal Paper to be submitted for publication in the Marine Structures Journal, Elsevier Applied Science Publishing
107 Fair Isle Weather Station Web-Site: http://www.zetnet.co.uk/sigs/weather/
108 Lewis, E.V., Hoffman, D., MacLean, W.M., Van Hoof, R., Zubaly, R.B., “Load Criteria for Ship Structural Design”, Ship Structure Committee Report No. SSC-240, 1973.
52
109 Taggart, R. Editor, “Ship Design and Construction”, SNAME, 1980.
110 Paik, J.K., Hughes, O.F., Mansour, A.E., “Advanced Closed-Form Ultimate Strength Formulation for Ships”, Journal of Ship Research, Vol. 45, No. 2, pp.111-132, 2001.
111 LR, “User’ Manual, LR.PASS Personal Computer Programs, Vol. 2, direct calculations”, Lloyd’s Register of Shipping, London, UK, March 1997.
112 Mansour A. E., Hovem L., “Probability-based ship structural safety analysis”, Journal of Ship Research, Vol. 38, no.4, pp.329-339, 1994.
113 Teixeira, A., “Reliability of marine structures in the context of risk based design”, MSc Thesis, University of Glasgow, Scotland UK, 1997.
114 Dogliani, M., Casells, G., Guedes Soares, C., “SHIPREL - Reliability Methods for Ship Structural Design”, BRITE/EURAM Project 9559, Lisbon, 1998.
115 Shellin, T., Ostergaard, C., Guedes Soares, C., “Uncertainty assessment of low frequency wave induced load effects for container ships”, Marine Structures, Vol.9, No.3-4, pp.313-332, 1998.
116 Maerli, A., Das, P.K., Smith, S.N., “A rationalisation of failure surface equation for the reliability analysis of FPSO structures”, International Shipbuilding Progress, No.450, pp.215-225, 2000.
117 Das, P.K., “The application of reliability analysis techniques towards a rational design of ship hull girders”, DERA, Dunfermline Scotland UK, 1988.
118 Das, P.K., Thavalingam, A., Strength and reliability of FPSOs, Third International Shipbuilding Conference, ISC'2002, St. Petersburg, Russia, 2002.
53
BIBLIOGRAPHY
1 J.N. Newman, “Marine Hydrodynamics”, MIT Press, 1977.
2 Chajes, “Principles of Structural Stability Theory”, Civil Engineering and Engineering Mechanics Series, Prentice-Hall, New Jersey, 1974.
3 Hughes, “Ship Structural Design, A Rationally-Based Computer-Aided Optimization Approach”, John Wiley & Sons, USA, 1983.
4 G. Ferro and A.E. Mansour, “Probabilistic Analysis of Combined Slamming and Wave-Induced responses”, Journal of Ship Research, Vol. 29, No. 3, 1985.
5 E. Nikolaidis, P. Kaplan, “Uncertainties in Stress Analysis of Marine Structures”, Ship Structure Committee Report, SSC-363, 1991.
6 P. Friis-Hansen, “On Combination of Slamming and Wave Induced Responses”, Journal of Ship Research, Vol. 38, No. 2, 1994.
7 A.E. Mansour, “Methods of Computing the Probability of Failure Under Extreme Values of Bending Moment”, Journal of Ship Research, June 1972.
8 A.E. Mansour, “Probabilistic Design Concepts in Ship Structural Safety and Reliability”, Trans. SNAME, 1972.
9 M.K. Ochi, “Principles of Extreme Value Statistics and Their Application”, SNAME Extreme Loads Symposium, Arlington VA, 1981.
10 D. Faulkner, “Semi-Probabilistic Approach to the Design of Marine Structures”, SNAME Extreme Loads Symposium, Arlington VA, 1981.
11 A.E. Mansour, “Combining Extreme Environmental Loads for Reliability Based Designs”, SNAME Extreme Loads Symposium, Arlington VA, 1981.
12 M. Chryssanthopoulos, M.J. Baker, P.J. Dowling, “A Reliability approach to the Local Buckling of String-Stiffened Cylinders”, Proceedings of the 5th Offshore Mechanics and Arctic Engineering Symposium in Tokyo, April 1986.
13 A.E. Mansour, H.Y. Jan, C.I. Zigelman, Y.N. Chen, S.J. Harding, “Implementation of Reliability Methods to Marine Structures”, Trans. SNAME, Vol. 92 pp. 353-382, 1984.
14 A.K. Thayamballi, Y.K. Chen and H.H. Chen, “Deterministic and Reliability Based Retrospective Strength Assessments of Ocean-going Vessels”, Trans. SNAME, 1987.
15 J.M. Gordo, C. Guedes Soares, “Approximate Method to Evaluate the Hull Girder Collapse Strength”, Marine Structures, Elsevier Science Ltd, 9 (449-470), 1996.
16 M.K. Chryssanthopoulos, “Response Surface methodology for Probabilistic Analysis and Reliability Assessment”, Notes from a Course on Structural Reliability and Decision Analysis held at the University of Glasgow, May 2000.
54
17 B. Tornsey, “Basic Probability and Some Estimation Theory”, Notes from a Course on Structural Reliability and Decision Analysis held at the University of Glasgow, Nov 2001.
18 D Faulkner, “Introduction to Probability Methods in Design”, Notes from a Course on Structural Reliability and Decision Analysis held at the University of Glasgow, Nov 2001.
19 L. Yu, “Monte Carlo Simulation and its Application in Computational Stochastic Mechanics and Reliability Analysis”, Notes from a Course on Structural Reliability and Decision Analysis held at the University of Glasgow, Nov 2001.
20 P.K. Das, “Decision Analysis”, Notes from a Course on Structural Reliability and Decision Analysis held at the University of Glasgow, Nov 2001.
21 M.K. Chryssanthopoulos, “Session on Structural Reliability Methods and Their Limitations”, Notes from a Course on Structural Reliability and Decision Analysis held at the University of Glasgow, Nov 2001.
22 N. Barltrop, “Code Calibration and Systems Reliability of Ship and Jacket Strutures”, Notes from a Course on Structural Reliability and Decision Analysis held at the University of Glasgow, Nov 2001.
23 R.H. Myers, “Response Surface Methodology”, 1976.
24 P. Thoft-Christensen, M.J. Baker, “Structural Reliability Theory and its Applications”, Springer-Verlag, Berlin Heidelberg New York, 1982.
25 J.I. Ansell, M.J. Phillips, “Practical Methods for Reliability Data Analysis”, Oxford Statistical Sience Series Publications, Clarendon Press, 1994.
26 G.F. Zhao, “Reliability theory and its applications for engineering structures.”, Dalian University of Technology Press, Dalian, China, 1996.
27 A.H-S Ang, W.H. Tang, “Probability Concepts in Engineering Planning and Design”
Printed and published by the Health and Safety ExecutiveC30 1/98
Printed and published by the Health and Safety ExecutiveC0.06 09/04
RR 261
£20.00 9 78071 7 628902
ISBN 0-7176-2890-6