Reliability Engineering Advances [1 ed.] 9781608769711, 9781606923290

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Copyright © 2008. Nova Science Publishers, Incorporated. All rights reserved.

Copyright © 2008. Nova Science Publishers, Incorporated. All rights reserved.

RELIABILITY ENGINEERING ADVANCES

Copyright © 2008. Nova Science Publishers, Incorporated. All rights reserved.

No part of this digital document may be reproduced, stored in a retrieval system or transmitted in any form or by any means. The publisher has taken reasonable care in the preparation of this digital document, but makes no expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions. No liability is assumed for incidental or consequential damages in connection with or arising out of information contained herein. This digital document is sold with the clear understanding that the publisher is not engaged in rendering legal, medical or any other professional services.

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RELIABILITY ENGINEERING ADVANCES

GREGORY I. HAYWORTH

Copyright © 2008. Nova Science Publishers, Incorporated. All rights reserved.

EDITOR

Nova Science Publishers, Inc. New York

Copyright © 2009 by Nova Science Publishers, Inc.

All rights reserved. No part of this book may be reproduced, stored in a retrieval system or transmitted in any form or by any means: electronic, electrostatic, magnetic, tape, mechanical photocopying, recording or otherwise without the written permission of the Publisher. For permission to use material from this book please contact us: Telephone 631-231-7269; Fax 631-231-8175 Web Site: http://www.novapublishers.com NOTICE TO THE READER The Publisher has taken reasonable care in the preparation of this book, but makes no expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions. No liability is assumed for incidental or consequential damages in connection with or arising out of information contained in this book. The Publisher shall not be liable for any special, consequential, or exemplary damages resulting, in whole or in part, from the readers’ use of, or reliance upon, this material. Independent verification should be sought for any data, advice or recommendations contained in this book. In addition, no responsibility is assumed by the publisher for any injury and/or damage to persons or property arising from any methods, products, instructions, ideas or otherwise contained in this publication. This publication is designed to provide accurate and authoritative information with regard to the subject matter cover herein. It is sold with the clear understanding that the Publisher is not engaged in rendering legal or any other professional services. If legal, medical or any other expert assistance is required, the services of a competent person should be sought. FROM A DECLARATION OF PARTICIPANTS JOINTLY ADOPTED BY A COMMITTEE OF THE AMERICAN BAR ASSOCIATION AND A COMMITTEE OF PUBLISHERS.

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Library of Congress Cataloging-in-Publication Data Reliability engineering advances / editor, Gregory I. Hayworth. p. cm. Includes bibliographical references and index. ISBN 978-1-60876-971-1 (E-Book) 1. Reliability (Engineering) I. Hayworth, Gregory I. TA169.R43923 2009 620'.00452--dc22 2009009171

Published by Nova Science Publishers, Inc.

New York

CONTENTS Preface Chapter 1

Review of Studies on the Detrimental Effects of Solid Contaminants in Lubricated Machine Element Contacts George K. Nikas

1

Chapter 2

Statistical Reliability with Applications to Defense Aparna V. Huzurbazar, Daniel Briand and Robert Cranwell

45

Chapter 3

Reliability-Based Design of Railway Prestressed Concrete Sleepers Alex M. Remennikov and Sakdirat Kaewunruen

77

Chapter 4

Reliability Estimation of Individual Predictions in Supervised Learning Zoran Bosníc and Igor Kononenko

Chapter 5

Chapter 6

Chapter 7 Copyright © 2008. Nova Science Publishers, Incorporated. All rights reserved.

vii

A Full Performance Management Policy for a Geographically Disperse System Aimed to Continuous Reliability Improvement Miguel Fernández Temprano, Oscar Duque Pérez, Luis Ángel García Escudero and Ángel Luis Zorita Lamadrid

107

141

Structural Reliability Aspects for Advanced Composite Material Applications Giacomo Frulla and Fabio Borello

167

Meta-modelling Based Approximation Methods for Structural Reliability Analysis Irfan Kaymaz

205

Chaper 8

Seismic Reliability of Nonlinear Structural Systems John W. van de Lindt and Shiling Pei

Chapter 9

Time Dependent Seismic Reliability Analysis of Structure with Uncertain System Parameter Subrata Chakraborty and Abhijit Chaudhuri

259

Response Surface Methodology and Multiple Response Case: Optimization Measures, Developments and Comparisons Rossella Berni

287

Chapter 10

233

vi Chapter 11

Chapter 12

Chapter 13

Contents Seed Development in Castor (Ricinus communis L.): Morphology, Reserve Synthesis and Gene Expression Grace Q. Chen Survivable DWDM Optical Mesh Transport Network Design via Genetic Algorithms Y. S. Kavian, M. S. Leeson, E. L. Hines, W. Ren, H. F. Rashvand and M. Naderi Digital Interferometric Measurement of Heat Dissipation from Microelectronic Devices for Optimal and Reliable Thermal Design C. B. Sobhan and V. Sajith

Chapter 14

Reliability Estimation of Urban Wastewater Disposal Networks Yuri A. Ermolin

Chapter 15

Future Trends on Global Performance Indicators in Industrial Research Livio Corain and Luigi Salmaso

305

331

355 379

399

Short Communication Comparative Distributions of Reliability and Survival Modeling Analysis M. Shuaib Khan and G. R. Pasha

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Index

415 429

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PREFACE Reliability engineering is an engineering field, that deals with the study of reliability: the ability of a system or component to perform its required functions under stated conditions for a specified period of time. It is often reported in terms of a probability. Reliability may be defined in several ways: The idea that something is fit for purpose with respect to time; The capacity of a device or system to perform as designed; The resistance to failure of a device or system; The ability of a device or system to perform a required function under stated conditions for a specified period of time; The probability that a functional unit will perform its required function for a specified interval under stated conditions. The ability of something to "fail well" (fail without catastrophic consequences) Reliability engineers rely heavily on statistics, probability theory, and reliability theory. Many engineering techniques are used in reliability engineering, such as reliability prediction, Weibull analysis, thermal management, reliability testing and accelerated life testing. Because of the large number of reliability techniques, their expense, and the varying degrees of reliability required for different situations, most projects develop a reliability program plan to specify the reliability tasks that will be performed for that specific system. The function of reliability engineering is to develop the reliability requirements for the product, establish an adequate reliability program, and perform appropriate analyses and tasks to ensure the product will meet its requirements. This new book presents the latest research in the field. Chapter 1 – Maximizing the life expectancy of machine elements in relative motion is the ultimate challenge faced by tribology researchers in industry and academia. However, despite the progress in materials science and lubrication methods, a great obstacle remains in achieving this goal, and that is the presence of solid contaminants in lubricants. The entrainment of 1-100 μm debris particles in concentrated contacts such as in bearings, gears, cam-followers or seals, is associated with various damage modes such as surface indentation and abrasion, lubricant starvation and scuffing, high frictional heating etc. All of these refer to plastic deformations and are detrimental to the life of the machine elements involved and, obviously, to the life of the machine. This chapter contains a review of theoretical and experimental studies in the literature on the effects of debris particles in lubricated contacts, exploring the progress made in the last few decades. The studies cover the entrainment, entrapment and passage of debris particles through the contacts, and how this is affected by the operating conditions. Analytical, numerical and experimental studies are discussed in view of understanding the damage mechanisms involved in this process. This helps to improve the designs, depending on application requirements, aiming at minimizing the risks,

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maximizing life expectancy and, thus, improving engineering reliability in industrial, automotive and aerospace applications. Chapter 2 – This chapter discusses selected current trends in engineering reliability modeling with a focus on applications in the defense sector. Our interests are in design for reliability, prognostics and health management (PHM), enterprise level logistics modeling and system-of-systems reliability modeling. It is not unusual for a new weapon system to be developed without fully considering the impact of reliability early in the design period for whatever reason: cost, necessity, technology, and so on. Today, part of that design for reliability can include PHM elements that will predict component failure far enough in advance so that maintenance and operations schedules can be optimized for the purposes of maximizing system availability and minimizing logistics costs. PHM modeling can be part of a real-time maintenance and operations management tool or an enhancement to enterprise level logistics modeling. Part of minimizing the logistics cost at the enterprise level requires that the supply and repair chain processes be simulated with enough detail to capture the realworld failure events. Accurate failure assessment can help determine the correct number of spares, maintenance support equipment, and qualified personnel necessary to complete the repairs. Ultimately, design for reliability, PHM, and enterprise level logistics modeling need to support the system’s ability to operate within a system-of-systems framework, the modeling of which can be quite involved. This chapter will evaluate each of the concepts discussed above, identifying past successes, current applications, and future trends. Chapter 3 – The recently improved knowledge raises a concern in the design manners of prestressed concrete structures. Civil engineers are mostly aware of the design codes for structural prestressed concrete members, which rely on allowable stresses and material strength reductions. In particular, railway sleeper (or railroad tie), which is an important component of railway tracks, is commonly made of the prestressed concrete. The existing code for designing such components makes use of the permissible stress design concept whereas the fibre stresses over cross sections at initial and final stages are limited. Based on a number of experiments and field data, it is believed that the concrete sleepers complied with the permissible stress concept possess the unduly untapped fracture toughness. A collaborative research run by the Australian Cooperative Research Centre for Railway Engineering and Technologies has been initiated to ascertain the reserved capacity of Australian railway prestressed concrete sleepers designed using the existing design code as to develop a new limit states design concept. The collaborative research between the University of Wollongong and Queensland University of Technology has addressed such important issues as the spectrum and amplitudes of dynamic forces applied to the railway track, evaluation of the ultimate and serviceability performances, and reserve capacity of typical prestressed concrete sleepers designed to the current code, and the reliability based design concept. This chapter presents the use of reliability-based approach for endorsing a new design method (e.g. a more rational limit states design) as the replacement of the existing code. The reliability based design approach has been correlated with the structural safety margin provided by the existing prestressed concrete sleepers. The reliability assessment of a prestressed concrete sleeper has been exemplified for better understanding into the sensitivity of dynamic load amplification on the target reliability indices and probabilities of failure. The target safety index and its uses for the reliability design of the prestressed concrete sleepers are later highlighted.

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ix

Chapter 4 – In machine learning, estimation of predictive accuracy for a given model is most commonly approached by analyzing the average accuracy of the model. In general, the predictive models do not provide reliability estimates for their individual predictions. The reliability estimates of individual predictions require the analysis of specific model and instance properties. Measuring the prediction reliability is very important in risk-sensitive areas where acting upon predictions may have significant consequences (e.g. medical diagnosis, stock market, navigation, control applications). In such areas, appropriate reliability estimates may provide additional information about the prediction correctness and can enable the user (e.g. medical doctor) to differentiate between more and less reliable predictions. This chapter summarizes research areas, which provide the motivating ideas for the development of the approaches for individual prediction reliability. Based on these motivations, the authors describe five approaches to reliability estimation (i.e., local sensitivity analysis, bagging variance, local cross-validation, density estimation, local error modeling) and define the corresponding reliability estimates. The empirical evaluation of eight estimates using 28 testing domains and 8 regression models indicated the promising results, especially for the usage with regression trees. The testing results also reveal that the reliability estimates exhibit different performance with different regression models and domains. To deal with this issue, the authors describe two approaches for automatic selection of the most appropriate estimate for a given domain and regression model: (i) the meta-learning approach and (ii) the internal cross-validation approach. The empirical results of both approaches show the advantage of the dynamically selected reliability estimate for a given domain/model pair, when compared to individual reliability estimates in terms of higher positive correlation to the prediction error. The described methodology was applied to a medical prognostic domain, which provided the medical experts with a significant improvement of the prognostic system. Chapter 5 – In this chapter, the authors develop procedures designed to help to apply an effective performance management policy to highly dispersed and complex systems. The proposed methodology is focused on the reliability of the system, taking into account that these systems are usually composed by many subsystems that have to work under different environments and are supervised by different maintenance staffs. In consequence, a complete reliability analysis of the system is developed. The components and installations of the whole system are classified into groups resulting in functional group structures and criticality indices are obtained for these structures. The procedure to obtain the components criticality indices is focused on field reliability data, while for the installations indices a multivariate analysis is applied taking into account local characteristics such as strategic importance, technology particularities and local reliability indices. In both cases, the aim has been the development of non-subjective criteria. These analyses are used to help to design the improvement plans, component replacement policies or the necessary investments with the aim of reducing the failures and improving the availability of the system. The methodology includes several specific novelties. One of them is the design of a procedure to test the performance of the system and the effectiveness of the maintenance tasks establishing warning procedures. Another advance is the use of a methodology based on Duane plot graphical techniques that allows modifying some of the current strategies and even speeding up the replacement of components. This chapter has been developed as part of a project named FIMALAC. The purpose of this project is to study the reliability, availability and maintenance of the 3000 V DC overhead contact line of the Spanish national railway network. The procedures developed

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in this chapter are illustrated with the results obtained in the application of this methodology to this overhead contact line system. Chapter 6 – Variability in composite material property data results from a number of sources, including variability in laying up the material, variability of raw materials, high sensitivity to testing environment and material testing methods. Deterministic mechanical description of composite material may be too penalizing, leading to an increase in structures weight, or may be not conservative, leading to inacceptable safety. Moreover, the use of safety factor in structural design leads to an increase of weight with a not-quantified increase of structural reliability. A way to achieve a more rational use of composite materials, is the development of a probabilistic analysis and design methodology; such methodology, taking in account uncertainties related to geometry, loads, material capabilities, operating environment, should be able to quantify the nominal structural probability of failure (pf), its sensitivities to design variables and the propagation of uncertainty. To achieve such goals, it’s necessary to approach the problem at different levels of growing complexity. At first the propagation of uncertainty from constituent materials to ply level and laminate level should be evaluated to assess which variables contribute the most to mechanical properties dispersion; the propagation of uncertainty analysis can be also adopted as a “virtual testing” tool for defining design values for the considered composite material. Successively a reliability assessment should be carried out for a generic laminate (considered as the basic element of a composite structure) considering appropriate failure modes. Finally, an interface between probabilistic analysis tool and classic structural analysis tool (FEM for example) should be built in order to investigate real world structures. Structural reliability concepts can be used for analysing existing structures as well as a tool for design once appropriate level of probability of failure are defined by standardization authority. Chapter 7 – Approximation methods such as the response surface and kriging are widely used to alleviate the computational burden of engineering analyses. In reliability analysis, however, additional burdens can be brought on since it requires using iterative numerical methods or more costly simulation methods to estimate the probability of failure of the mechanical components/systems. Therefore, using one of the approximation methods for real engineering applications is inevitable, by which a metamodel is used in place of a time consuming analysis. Therefore this chapter begins with the review of the most widely used approximation techniques. However, it should be noted that the approximation methods developed specifically for deterministic analyses are not always suited to the reliability analysis since some points such as approximating the limit state function around the design point better than other regions, using criteria to assess the goodness of the metamodel, etc. must be considered to better estimate the probability of failure. Therefore some points are suggested in this chapter in order to improve the existing approximation methods. One of the most important properties of the approximation methods is to generate experimental points that will be used to form the simple approximate function. Although there are several well known design of experiment methods developed for deterministic analyses in the literature, using the design of experiment in reliability analysis needs a special approach. Thus, this chapter includes a brief review of the classical design of experiment methods as well as indicating how it can be improved for reliability problems in order to better capture the region from which the probability of failure is mostly computed. Thus this chapter will serve as a guidance to understand and use the approximation methods to carry out a reliability analysis more efficiently and quickly. Two selected examples are provided in order to show the

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xi

application of the approximation methods to the reliability analysis of both elementary mathematical functions and an engineering problem. Chapter 8 – This book chapter is divided into two sections: (1) Theory and Methodology, and (2) Illustrative Examples. The theory and methodology section provides the background on seismic excitation and basic nonlinear systems such as the linear oscillator, bi-linear oscillator, complex hysteretic model, and system level model with multi-degree of freedom response to earthquakes. Once the explanation of the methods is completed, a series of illustrative examples is set out and the approach to each explained. This includes four examples which increase in complexity. The first example is the seismic reliability estimates for a linear oscillator subject to a single earthquake (1994 Northridge). The second example looks at a more complex oscillator known as the bilinear oscillator subjected to a suite of 20 earthquakes and determine the reliability of the oscillator to a particular displacement requirement. The third example will focus on a typical North American two-story single family house, with the primary lateral force resisting component (wood shear walls) modeled using a 16 degree-of-freedom hysteretic oscillator. Suites of ground motions all having the same occurrence probability will be used to determine the reliability against exceeding an inter-story drift limit. In the fourth and final example, a method for computing the financial loss during a single earthquake and over a given period of time for a single family home is presented. The simulation procedure illustrated in this last example incorporates the uncertainty sources that impact the seismic reliability related to limiting financial losses. A brief introduction to performance-based seismic design and the application of structural reliability is also discussed. Chapter 9 – The time dependent seismic reliability evaluation of structure is normally performed considering non-stationary models of ground motion neglecting the effect of uncertainty in the structural system parameters. However, the uncertainties result from numerous assumptions made with the geometry and boundary conditions, material behaviors etc. is expected to have significant effect on the overall reliability of the structure. An integrated time dependent unconditional reliability evaluation of structures with uncertain parameter subjected to seismic motion is presented here. The time varying amplitude and frequency content of the ground motion well represented by the family of sigma oscillatory process is developed in double frequency domain. Subsequently, the stochastic dynamic analysis is performed to derive the generalized response power spectral density functions in double frequency domain. The nonstationary response is not uncorrelated with its time derivative. Thus, the reliability is evaluated by conditional crossing rate using the Vanmarcke’s modification based on two states Markov process considering the correlation between response process and its time derivative. The uncertainty of structural system parameters, generally described in spatial coordinate, motivates those to model as random field. The perturbation based stochastic finite element formulation is readily developed to address the parameter uncertainty in the framework of non-stationary dynamic analysis. This involves the important issue of response sensitivity evaluation of structures. The related formulation to obtain the sensitivity of dynamic responses quantities with respect to uncertain parameters has been also briefed. The methodology is elucidated in a focused manner by considering a three dimensional building frame subjected to ground motion due to El Centro 1940 earthquake. The time dependent unconditional reliability is evaluated with respect to desired responses. The results are presented to compare the change in reliabilities of the uncertain system with that of deterministic system. The reliability analysis results are also

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compared with the Monte Carlo Simulation results to study the accuracy and effectiveness of the present approach. Chapter 10 – This chapter focuses on the response surface methodology in the multiple response case by considering the recent issues and the problems linked to the simultaneous optimization of several response variables taking into account the robust design approach. A literature review about dual approach and the multiresponse case with their differences and developments is presented. Furthermore, comparisons with respect to measures and weights are expounded and applied. Finally, an empirical example is shown with transformed or non-transformed response variables. Chapter 11 – Castor (Ricinus communis L.) is a non-eatable oilseed crop producing seed oil comprising 90% ricinoleate (12-hydroxy-oleate) which has numerous industrial uses. However, the production of castor oil is hampered by the presence of detrimental seed storage proteins, the toxin ricin and hyper-allergenic 2S albumins. The authors are developing a safe source of castor oil by two approaches: blocking gene expression of the ricin and 2S albumins in castor seed and engineering a temperate oilseed crop to produce castor oil. To understand the mechanisms underlying the synthesis of ricin, 2S albumins and ricinoleate/oil, the authors conducted a series of seed development studies in castor, including endosperm morphogenesis, storage compound accumulation and gene expression. The entire course of seed development can be divided into four stages, which are recognizable by distinct seed coat color and cellular endosperm volume. Synthesis of ricin, 2S albumins and oil occur during cellular endosperm development. Concomitantly, the authors observed increased transcript levels of 14 genes involved in synthesis of ricin, 2S albumin and oil, but with various temporal patterns and different maximal inductions ranging from 2 to 43,000 fold. The results indicate that gene transcription exerts a primary control in castor reserve biosyntheses. Based on the temporal pattern and level of gene expression, the authors classified these genes into five groups. This transcription-profiling data provide not only the initial information on promoter activity for each gene, but also a first glimpse of the global patterns of gene expression and regulation, which are critical to metabolic engineering of transgenic oilseeds. Since all these studies are based on a well-defined time course, the results also provide integrative information for understanding the relationships among endosperm morphogenesis, reserve biosynthesis and gene expression during castor seed development. Chapter 12 – Acceptable service provision in the presence of failures and attacks is a major issue in the design of next generation dense wavelength division multiplexing (DWDM) networks. Survivability is provided by the establishment of spare lightpaths for each connection request to protect the working lightpaths, which is an NP-hard Problem. This chapter presents a genetic algorithm (GA) solver for the routing and wavelength assignment problem with working and spare lightpaths to design survivable optical networks in the presence of a single link failure. Lightpaths are encoded into chromosomes made up of a fixed number of genes equal to the number of entries in the traffic demand matrix. Each gene represents one valid path, and is thus coded as a variable length binary string. After crossover and mutation, each member of the population represents a set of valid but possibly incompatible paths and those that do not satisfy the problem constraints are discarded. The best paths are then found by use of a fitness function and these are assigned the minimum number of wavelengths according to the problem constraints. The proposed approach has been evaluated for dedicated path protection and shared path protection. Simulation results show that the GA method is efficient and able to design survivable real-world DWDM optical

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mesh networks. The GA solver is further tuned for both a bandwidth optimization scheme (BOS) and a delay optimization scheme (DOS). The performance of the new GA based resiliency model is evaluated for benchmark networks, and simulation results show that the method displays excellent performance in solving this complex, multi-constrained problem for BOS and DOS. Chapter 13 – Effective thermal management of electronic components is essential to avoid their untimely failure, which highly depends on the operating temperature. One of the methods of heat removal from electronic components is to use heat sinks which can increase the surface area for heat transfer, and dissipate heat with the help of fans, by forced convection. The heat dissipation from a component in an electronic package not only influences its own performance, but also affects the performance of neighboring components. Hence, a clear knowledge of the level of heat dissipation from electronic components is essential for evolving reliable and optimal designs of electronic systems and packages. Presently, a comprehensive and exact database for heat dissipation from different electronic components, during their operation, is not available. Further, it is difficult to make intrusive measurements on and around electronic components while they are under operation. To overcome these, a digital interferometric technique, which is a non-intrusive method, has been devised for obtaining the nature and distribution of heat dissipation from various heat dissipating electronic components. Interferometric methods utilize the interference of coherent light beams to measure, for instance, temperature or heat flux fields, using phase differences introduced while they traverse test sections. In the present work, a heat sink consisting of a fin array on a horizontal base, made of aluminum, and fixed to the electronic component, was used for heat removal during experimentation. This heat sink was introduced in the test beam of a Mach-Zehnder Interferometer, and analyzed with an initial parallel wedge fringe pattern. As the electronic component dissipates heat through the fins attached to it, the parallel fringes get deflected due to the change in the refractive index of the medium caused by heating. The resulting distorted fringe patterns are grabbed using a CCD camera interfaced to a computer, at steady state. Digital image processing techniques used in the analysis include fringe thinning using multiple superimposing of images and digital edge detection. From this analysis, the temperature distribution and the local heat dissipation from the surface of the heat sink are obtained using interferometric relations. Various electronic components with different levels of heat dissipation such as fixed and variable voltage regulators, and power silicon transistors were studied, and a database of the heat dissipation level evolved, based on the experimental investigation. Chapter 14 – The ramified head-and-gravity flow sewerage system for receiving and transferring household and industrial sewage typical for a large city is considered. Consideration is restricted to the sub-system of sewage conveyance (sewer network). By a wastewater disposal network is meant the combination of underground pipes (sewers) passing the sewage by gravity, and pumping stations for lifting sewage in the area where gravity flow is impossible of levels. Here the reliability problem of a sewer network is discussed. Practical application of the method is expanding to nonstationary failure flows. The special case when the failure rate acting on an object is an increasing exponential function of time is considered. It is proposed that the nonstationary failure rate be substituted by a stationary one. The formula intended for calculation of this fictitious equivalent stationary failure rate is derived. Numerical examples are used to demonstrate the proposed approach.

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Chapter 15 – Within Research and Development activities, complex statistical problems of hypothesis testing can commonly arise. The complexity of the problem relates mainly to the multivariate nature of the study, possibly to the presence of mixed performance variables (ordinal categorical, binary or continuous), and sometimes to missing values. In this contribution the authors consider permutation methods for multivariate testing on mixed variables within the framework of multivariate randomised complete block design. The novel approach the authors propose has been studied and validated using a Monte Carlo simulation study. Finally the authors propose an application to real data, where several panellists from an R&D division of a home-care company are enrolled to study several possible new fragrances for a given detergent to compare with their own presently marketed product. Short Communication - In this paper, the authors present the comparison among the distributions used in reliability and survival analysis. Simulation technique has been used to study the behavior of reliability and survival distribution modules. The fundamentals of reliability and survival issues are discussed using non-failure criteria. The authors present the flexibility of reliability and survival modeling distribution that approaches to different distributions.

In: Reliability Engineering Advances Editor: Gregory I. Hayworth

ISBN: 978-1-60692-329-0 © 2009 Nova Science Publishers, Inc.

Chapter 1

REVIEW OF STUDIES ON THE DETRIMENTAL EFFECTS OF SOLID CONTAMINANTS IN LUBRICATED MACHINE ELEMENT CONTACTS George K. Nikas1 Imperial College London, Department of Mechanical Engineering, Tribology Group; London, UK.

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Abstract Maximizing the life expectancy of machine elements in relative motion is the ultimate challenge faced by tribology researchers in industry and academia. However, despite the progress in materials science and lubrication methods, a great obstacle remains in achieving this goal, and that is the presence of solid contaminants in lubricants. The entrainment of 1100 μm debris particles in concentrated contacts such as in bearings, gears, cam-followers or seals, is associated with various damage modes such as surface indentation and abrasion, lubricant starvation and scuffing, high frictional heating etc. All of these refer to plastic deformations and are detrimental to the life of the machine elements involved and, obviously, to the life of the machine. This chapter contains a review of theoretical and experimental studies in the literature on the effects of debris particles in lubricated contacts, exploring the progress made in the last few decades. The studies cover the entrainment, entrapment and passage of debris particles through the contacts, and how this is affected by the operating conditions. Analytical, numerical and experimental studies are discussed in view of understanding the damage mechanisms involved in this process. This helps to improve the designs, depending on application requirements, aiming at minimizing the risks, maximizing life expectancy and, thus, improving engineering reliability in industrial, automotive and aerospace applications.

1. Introduction Machine reliability is the ultimate goal of tribology researchers and engineers in industry and academia. The challenge of maximizing the life expectancy of machine elements in relative 1

Correspondence address: Dr G. K. Nikas, 3 Princes Mews, Hounslow, Middlesex, TW3 3RF, England

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2

George K. Nikas

motion such as bearings, gears, seals, cam-followers, CV-joints, etc, involves materials and lubrication science, surface physics, and, naturally, an understanding of phenomena taking place on the micro- and nano-scale. In recent times, numerous theoretical and experimental studies have shown the importance of lubricant cleanliness in achieving the designed life expectancy of machine elements. Unfortunately, despite all precautions, the presence of solid contaminants in lubricating systems is unavoidable. Whether such contaminants are carried to the contact by a lubricant as in the case of transportation cans [1,2], atmospheric air, or because of a deficiency or failure of sealing systems, is unimportant; whether they are generated by wear in a concentrated machine-element contact or in neighbouring contacts is of secondary importance. The fact is that in a normal lubrication system such as that of a passenger car engine, millions of particulates exist at any time during the life cycle of the system. A large volume of the related literature is dedicated to showing that debris particles of 1100 μm in size are responsible for a large proportion of machine element failures. This has been brought into the foreground in recent decades mainly for two reasons: (a) because of the improvements in the cleanliness and homogeneity of bearing steels, and (b) because of the tightening of tolerances and reduction of film thicknesses in lubricated contacts to submicrometre levels. The former reason resulted in reducing bearing failures from contact fatigue initiated below the surface from material inhomogeneity, which is owed to voids or impurities that concentrate stress and are, normally, the sources of cracks. The latter reason (reduction of the average film thicknesses) made lubricated contacts more susceptible to damage from foreign objects because of the additional stress associated with their passage through the narrower contact gaps. The effects of debris particles have gained a lot of attention in the literature, particularly after the 1980s. It is interesting to note that as far in the past as in the 15th century, Leonardo da Vinci [3] described the wear of metal water wheel axles by wear chips embedded in the wooden supporting bush, as reported by Dwyer-Joyce et al [4]. In machine elements such as bearings and gears operating with a combination of rolling and sliding motion, typical contact gap sizes range from zero in dry contacts to a few micrometres in hydrodynamic contacts. In the majority of cases, as for example in oil lubricated rolling bearings, the film thickness is typically in the order of 0.1 to 1.0 μm. Considering the average size of typical debris particles, which is in the order of 1-100 μm, it is obvious that their entrapment and subsequent squeezing in concentrated contacts can cause significant damage. In hard machine element contacts, as in the majority of cases, particle compression causes substantial stressing in the vicinity of the compressed particle. Depending on the size, hardness and fracture toughness of a particle, the hardness and friction coefficients of the contact counterfaces, as well as the type of motion of the concentrated contact (rolling, sliding, spinning) and the magnitude of the counterface velocities, various damage modes can be distinguished. In sliding contacts and with particles harder than at least one of the counterfaces, surface abrasion is to be expected [4-12]. In the majority of contacts, regardless of motion pattern and particle hardness, brittleness or morphology, surface indentation is the most common effect [13-16]. This has been studied experimentally as for example in references [15,16] as well as theoretically, either via a simple stress analysis [13,14] or via a Finite Element Analysis [17,18]. Metallic debris particles have also been proved responsible for scuffing failure of contacts [19-29] via two basic mechanisms: (a) by obstructing the lubricant replenishment of

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a contact [20,22,25,26], and (b) by generating frictional heat when sheared between the relatively moving counterfaces of a contact [19,21,23-29], which is particularly evident in sliding contacts. The extent of such contact failures could be large as in gross scuffing (or galling) [19,22,24,26] or localised. In the latter case, isolated particles squashed and sheared in narrow contact gaps cause frictional heating that can be quite severe (for example, hundreds of degrees Celsius) [21,23,25, 27-29]. It is notable that this can happen not only for particles harder than the contact counterfaces [19,21,24] but softer, too [23,25,27-29], depending on the contact gap thickness and surface speeds. Having examined and understood the basic mechanisms of surface damage from debris particles, the next logical step is to study the effects of the induced damage on the life expectancy of machine elements. Numerous such studies exist in the literature and are based on carefully executed, tedious experiments as in references [30-42] or on often complex mathematical analyses as in references [43-46]. There are also a few review studies on debris particle modelling, experimentation and effects, which are of particular value [25,47-51]. Although the emphasis of this chapter is on the detrimental effects of debris particles, there are applications where the intentional use of particles, usually of very small size, has proved beneficial in the lubrication of machine elements. These are not discussed in this chapter the main focus of which is to discuss the mechanisms of debris-induced damage and to present solutions to minimize the related risks.

2. Sources of Solid Contaminants Generally, the sources of solid contamination can be either internal or external. Specifically, the following categories are distinguished.

2.1. Internal or Built-in

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This involves particles remaining after manufacturing and assembly processes. It is an unfortunate fact that new lubricants contain a large number of debris particles. Kjer [1] in 1980 reported the presence of a large number of metallic and non-metallic particles in new motor oils analysed by a ferrographic technique. The particles were of three types: • • •

spherical particles of a few micrometres and up to 30 μm in size (Figure 1(a)); metallic wear particles of up to 50 μm in size (Figure 1(b)); irregular, non-metallic particles of up to 100 μm in size (Figure 1(c)).

(a) Metallic

(b) Metallic, wear

Figure 1. Debris particles in new motor oils (from Kjer [1]).

(c) Non-metallic, irregular

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Leng and Davies [2] in 1988 used both analytical ferrography and spectrometric oil analysis of unused oils produced and canned in South Africa for diesel engines. They observed debris similar to those reported in Kjer [1]. Specifically, they detected (among others): • • • • •

calcium-rich mineral particles of up to 30 μm in size, which were the most common contaminants (Figure 2(a)); silicon-rich mineral particles of up to 25 μm in size (Figure 2(b)); cutting wear debris (Figure 2(c)); spherical, iron-based debris, 3-20 μm in size, known to be produced by manufacturing processes such as grinding (Figure 2(d)); other particles such as particles made of pure chromium (Figure 2(e)) and large organic debris (Figure 2(f)).

(a) Calcium-rich, mineral

(b) Silicon-rich mineral

(c) Cutting wear debris

(d) Spherical, iron-based

(e) Pure chromium

(f) Large, organic

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Figure 2. Debris particles in unused lubricants for Diesel engines (from Leng and Davies [2]).

All new oils contain solid contaminants in various concentrations and sizes. In 1991, according to the largest bearing manufacturer, SKF [52], a drum of new oil (about 200 litres) contained more than 1.1 billion particles larger than 5 μm. This is equivalent to 5.5 million particles larger than 5 μm per litre of oil. Dwyer-Joyce [51] in 2005 stated that contamination levels in industrial oil supplies vary from around 0.1 to 10 g/l. Particle sizes vary from submicrometre level to 1000 μm [35].

2.2. Generated or Wear Particles This involves particles generated within a system during operation due to various wear processes such as abrasion (roughness asperities or debris particles ploughing material from a surface), adhesion (particles detached from a surface), erosion, contact and pitting fatigue, spalling, scuffing, etc. Internally generated particles can be the product of combustion such as

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soot in diesel engines, metal chips from machining, core sand from castings, paint flakes, rust etc [53]. Such contaminants can be very large, for example, a few millimetres in size.

2.3. Ingested or External This involves particles entering a system from its environment due to insufficient sealing. Examples include dust and sand particles carried by atmospheric air, railway track ballast, rust, swarf, grits from grinding operations, chips from welds, etc. Such contaminants can also be very large, up to a few millimetres in size.

2.4. Introduced during Maintenance This involves particles entering a system during maintenance, for example, when changing lubricants or replacing components such as filters, seals etc. The particles introduced this way, obviously, vary a lot in size, morphology and composition.

3. Types of Debris Particles

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There are various factors used in the characterisation of debris particles for classification and condition monitoring purposes. A systematic approach includes parameters such as shape, size, surface texture, and material. The literature contains many studies in this respect and only a brief review is presented here. The main goal of such studies is to develop automatic classification methods for wear particles in order to monitor the condition of machine parts such as bearings, gears, seals, etc, and decide whether maintenance or repair is due. It has been realised in recent decades that this approach is invaluable for preventative maintenance as well as for establishing potential causes of machine element deterioration or failure [54, 55]. One of the main factors in particle characterisation is “morphology”. This refers to particle appearance, which is a rather vague term, yet still useful for a qualitative description. For example, Roylance et al. [56] provide the classification similar to that shown in Figure 3. Trevor [57] distinguishes the following categories in reference to particle morphology, as reported by Kowandy et al. [58]: • • • • • • •

spheres; distorted smooth ovoids; chunks and slabs; platelets and flakes; curls, spirals and slivers; rolls; strands and fibres.

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Outline shape attributes

Regular

Irregular

Circular

Elongated

Edge detail attributes

Smooth

Rough

Straight

Serrated

Curved

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Figure 3. Particle outline-shape and edge-detail attributes (adapted from Roylance et al. [56]).

In general, principal morphological attributes of wear particles used in establishing the wear process creating the particles are size, shape, surface topography, colour and thickness [59]. Scanning Electron Microscopy (SEM) is normally used for the visual examination of particles but such a visual method requires expert knowledge and is time-consuming, costly and often inconsistent. Therefore, a set of numerical descriptors and protocols had to be developed in order to computerize the classification process, making it faster, more consistent and more formal. Such numerical descriptors involve particle size (apparent area, length, perimeter, equivalent-circle diameter) and outline shape [60] (aspect ratio, shape factor, convexity, elongation, curl, roundness, etc). For further information, the reader is referred to references [56, 57, 61-63] and the references quoted therein. Image analysis systems for wear particles have been developed by several researchers. Stachowiak and co-workers, taking into account the fractal topography of particles, have developed fractal particle-boundary descriptors. Interested readers are referred to Stachowiak [60] and the references quoted therein. The benefit of such techniques is that the destructiveness of particles (for example, their abrasivity) can be quantified when other properties are known, as for example their hardness. One of several interesting classifications in terms of particle shape that are available in the literature can be found in Trevor [57] and is shown in table 1. This is helpful in a preliminary and quick identification of possible origins of the detected particles based on their shape.

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Table 1. Shapes of particles and their possible origins (from Trevor [57] as reproduced by Kowandy et al. [58]). Particle shape

Typical names

Some possible origins

Spheres

Metal fatigue

Distorted smooth ovoids

Quarry dusty; atmosphere dust

Chunks and slabs

Metal fatigue; bearing petting; rock debris

Platelets and flakes

Running-in metal wear; paint; copper in grease

Curls, spirals and slivers

Machining debris produced at high temperature

Rolls

Probably similar to platelets but in a rolled form

Strands and fibres

Polymers, cotton and wood fibres; occasionally metal

Another interesting classification was presented in Bowen and Westcott [64] and Anderson [65]. This is presented in table 2, as reproduced by Roylance et al. [59]. Table 2. Wear particle morphological descriptors. Sizes stated apply to particles obtained from oil or grease samples (from Bowen and Westcott [64] and Anderson [65], as reproduced by Roylance et al. [59]). Particle type Rubbing

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Cutting

Laminar

Fatigue Spheres Severe sliding

Description Particles < 20 μm chord dimension and approximately 1 μm thick. Results from flaking of pieces from the shear-mixed layer. Swarf-like chips of fine wire coils. Caused by abrasive “cutting-tool” action. Thin, bright, free metal particles, typically 1 μm thick, 20-50 μm chord width, with holes in surface and uneven edge profile; emanates from mainly gear and rolling element bearing wear, associated with fatigue action. Chunky, several micrometres thick; for example, 20-50 μm chord width. Typically ferrous, 1-50 μm diameter; generated from microcracks; generated under rolling contact conditions. Large, 50 μm chord width, several micrometres thick. Surface heavily striated with long straight edges. Typically occurs in gear wear.

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If the operating and environmental conditions in a machine are known, it does not require a lot of experience to recognise some characteristic wear debris and the associated wear process that created them. For example, wear debris with a ribbon-like form of machined swarf are characteristic of severe abrasion of a softer surface by a hard, rough counterface (abrasion by roughness asperities); they are also characteristic of a sliding contact contaminated by hard particles with sharp edges [59]. Contact fatigue processes on the other hand tend to produce relatively large wear particles of blocky, irregular shape [59] (see also table 1 and 2); spherical metallic particles are also characteristic of machining processes such as grinding – see for example Figure 2(d). However, caution should be exercised even with automatic pattern-recognition systems because classification errors cannot be ruled out when relying on particle morphology alone. For example different adhesive wear conditions can produce similarly looking particles [66]. Therefore, other factors should be accounted for in order to improve detection accuracy and fault diagnosis. The material of debris particles is established with various methods such as ferrography [54], energy dispersive spectroscopy, IR and X-ray emission analysis. Materials often found in debris particles from lubricant samples include iron (Fe), aluminium (Al), silver (Ag), copper (Cu), carbon (C), zinc (Zn), tin (Tn), silicon carbide (SiC), silica, lead (Pb), chromium (Cr), calcium (Ca), phosphorus (P), and sulphur (S). The last three of those and zinc can also be attributed to lubricant additives. Table 3 shows an example of the materials and particulate concentrations of various lubricant samples. Table 3. Debris particles found in lubricant samples from various sources (from Dwyer-Joyce [49]).

Investigator Dwyer-Joyce [49]

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Dwyer-Joyce [49] SKF-ERC (Beghini et al. [67]) Admiralty Engineering Laboratory Eleftherakis and Khalil [68] Loewenthal and Moyer [31]

Sample source Paper mill Motor vehicle sump Various bearing lubricants

Concentration (estimated) [g/l]

Size range [μm]

Materials present

0.4

0-150

Cu, Sn, silicates

1-2

0-250

C, Fe

0.3-1.5

0-250

Various

0-100

Motor vehicles

0.3

0-120

Aircraft gas turbines

1-2

0-200

Fe, Al, Cu, Sn, SiC, sand Al, Ag, Cr, Zn, Fe Al, Cu, Fe, Pb Fe, C, silicates

One of the most important parameters in assessing the risk of damage from debris particles is the particle hardness. A general description can be found in table 4.

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Table 4. Types of particles in terms of hardness (see for example [49,52] and table 6.1 in [69]). Particle

Hardness [Vickers]

Very soft

Up to 40

Soft, metallic

55-280

Hard, metallic

700 (typical)

Tough material (but brittle)

Usually up to 1300 for ceramics but could be even more for other materials

Type Plastics, paper, wood, textile and vegetable fibres, pure metals such as gold, silver, copper, lead, tin, aluminium and nickel Mild steel, brass, bronze, aluminium, copper Steel (bearings, gears); cast iron Ceramic (silicon carbide, silicon nitride); corundum (aluminium oxide)

Source External for the non-metallic materials; external and internal (e.g., oatings) for the metallic materials Housings, bushings, etc. Hardened surfaces Manufacturing (lapping paste, grinding wheels, etc.

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In the case of wear particles, the source, naturally, is related to the particular application [49]. For example, worn gears may release steel particles. Worn bushes or housings may release aluminium, bronze and copper. Casting moulds may release silicates (dust, sand). Other particles may be brought in from the environment, for example, dust and sand from the atmospheric air, ceramics such as silicon carbide left over from machining operations (e.g., inding), etc. Bearing and other machine element manufacturers are aware of the nature of debris expected in any application related to their products and many have internally-produced research results to estimate the risks and suggest the proper level of filtration or lubricant cleanliness. Most of these internal results are confidential and rarely released. Oil cleanliness standards and protocols have been developed to formalise testing procedures and prevent system failures, particularly in critical applications such as those in the aerospace industry. For further information and a brief introduction, the reader is referred to Sasaki [70].

4. Detrimental Effects of Debris Particles in Lubricated Contacts There is a significant volume of literature devoted to the effect of solid contaminants in machine element contacts. This section begins with general remarks to accustom the reader to the seriousness of this issue and progresses with analysing individual aspects of the problem. The effects of solid contaminants on hydrodynamic bearing performance were quite early recognised and studied. McKee [71] in 1927 measured an increase in running friction of such bearings. Roach [72] in 1951 and Rylander [73] in 1952 reported on a temperature increase resulting from the increased friction in addition to the increased wear. Similar results were reported by Broeder and Heijnekamp [74] in 1965. Fitzsimmons and Clevenger [75] in 1977 found that the wear of tapered roller bearings is proportional to the amount of contaminants in

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George K. Nikas

the lubricant. Ronen et al. [32] in 1980 studied the performance of hydrodynamic bearings with contaminated lubricant. Their test bearing rig simulated an automotive, connecting-rod, engine bearing. Their results showed that both shaft and liner wear increased, typically, by a factor of 20 when running with contaminated oil. Qualitatively similar results were also reported by Ronen and Malkin [76,77]. In 1971, General Motors Corporation (USA) summarized the effects of debris particles on rolling bearing life as follows [49,50,78]: “The presence of dirt or grit in ball bearings is responsible for over 90% of all bearing failures; where bearings are kept clean during mounting, are lubricated with clean oil and are protected by recommended closures, no difficulty should be experienced from this source.” In tests performed by Okamoto et al. [79] in 1974, it was shown that bearing life is reduced by 80 to 90% when ceramic, silica and iron particles are continuously fed to the bearing lubrication system at 12 mg/h. Dalal et al. [80], also in 1974, performed tests, which showed that bearing life is improved several-fold when an ultra-clean lubrication system is used in place of a system with a 10 μm filtration ability. Loewenthal and Moyer [31] in 1979 showed similar results, that is, an increasing bearing life with finer filtration. Loewenthal et al. [33] in 1982 performed fatigue tests on groups of deep-groove ball bearings with two levels of filtration. They concluded that “ultra-clean lubrication produced bearing fatigue lives that were approximately twice that obtained in previous tests [31] with contaminated oil using 3 μm absolute filtration and approximately three times that obtained with 49 μm filtration”. In a study published in 1979 by Wedeven [81], it is recognised that debris is a major factor in component failure of aircraft propulsion systems. According to a study by Cunningham and Morgan [82] in 1979, bearing rejection-causes for aircraft engine, transmission and accessory bearings indicates that the solid contaminants are responsible for approximately 20% of all bearing rejections. In a series of fatigue tests, Bachu [83] in 1980 used wear debris particles generated by a gear machine (see also the report of Sayles and Macpherson [84] and Webster et al. [85]) and varied the filter-element size as follows (sizes in micrometres): 40, 25, 8, 3, 1 and submicrometre. The sub-micrometre filter was a magnetic device and the rest were of the standard, porous-media, cartridge type. The results indicated a 7-fold reduction of the L10 fatigue life (the life at which 10% of the bearing population is expected to fail) under 40 μm filtration compared with 3 μm filtration, the difference attributed to surface indentations from the larger debris passing through the poorer filter. Moreover, running a group of bearings with a 40 μm filtration for 30 minutes, then changing to the 3 μm filter and continuing the test to the remaining of the life of the bearings gave very similar L10 fatigue life to that obtained with 40 μm continuous filtration. This indicates that early surface damage from debris defined the ultimate fatigue life and that subsequent, continued presence of debris in the bearings had little effect on fatigue life. In other words, fine filtration during the running-in period of a lubricated system is critical for its life expectancy. In tests performed by Nilsson et al. [42] in 2005, it was found that “filtration during run-in for one hour with a 3 μm filter can reduce both the mass loss and the number of self-generated particles by a factor of 10”. This is a well-known fact nowadays to, for example, car engine manufacturers who recommend flushing the lubricating system of a new engine after a very short period of time (a few hundred kilometres maximum), as opposed to future oil services.

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In 1991, the largest bearing manufacturer, SKF, categorized the causes of bearing replacement as follows [52]: 36% were owed to poor lubrication, 34% were owed to normal fatigue, 16% were owed to incorrect installation, and 14% were owed to contamination failure. According to Ai [41] (The Timken Company, USA) in 2001, it has been estimated that 75% of bearings that failed before reaching their rated lives have failed from contamination. Naturally, such percentages vary a lot between manufacturers and end users depending on how meticulously lubrication systems are serviced and how efficient the filtration is. In this author’s opinion, such figures cannot be taken as gospel but as indicative of how easy it is to cause damage to machine components if attention is not paid to proper filtration and removal of harmful solid contaminants. The role of proper filtration, as already emphasized, is major in reducing machine element wear from debris particles and avoiding premature failures. Given the high concentrations of harmful particles in new and used lubricants, as explained in sub-section 2.1, using the correct filter for the particular application is of great importance. A common misunderstanding in this area concerns the efficiency rating of a filter and its real consequences on filtration. According to ISO Standard 4572, a filter can be rated using the following ratio:

βx =

number of particles per volume unit (100 ml) larger than x μm upstream the filter number of particles per volume unit (100 ml) larger than x μm downstream the filter

(1)

Obviously, this “beta ratio” rating refers to a specific particle size in micrometres, indicated by subscript x in βx. For example, a filter rating of β14 = 200 means that for every 200 particles of size 14 μm (x = 14) upstream the filter, only one particle is expected to pass all the way through the filter. This is clear to comprehend and take into consideration. However, what is most often quoted commercially is the filtration efficiency, η (as a percentage [%]), which is defined as follows:

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η=

100 number of retained particles = 100 − number of inlet particles β

(2)

For example, the filtration efficiency for the previously quoted example of β14 = 200 is η = 99.5% for 14 μm particles. This may sound sufficient to someone, who might assume that an efficiency of 99.9% in this case is not much better or not really necessary. However, using η = 99.9% and solving for beta from Eq. (2), the result is β14 = 1000, which is five times better (or, equivalently, five times cleaner lubricant) than the original rating β14 = 200. In general, using Eq. (2), it is easily proved that a filter with 99.9% efficiency is ten times more effective than a filter with 99% efficiency. For this reason, caution must be applied when considering efficiency ratings because, as demonstrated with the previous examples, small percentage changes can disguise the extent of lubricant contamination. The level of lubricant contamination and filtration efficiency has been incorporated in bearing life calculations in an attempt to improve the accuracy of bearing life predictions. Ball bearing manufacturer SKF and its New Life Theory for example uses an adjustment

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factor for bearing life rating, which is a function of (among others) a factor ηc for “contamination level” (see page 62 in reference [86]). The calculation of this factor is based on ISO standards for classification of the contamination level in lubrication systems (ISO 4406:1999) as well as on the viscosity ratio κ (in SKF’s terminology) and the size of a bearing – see reference [86] for full details. General guidelines for estimating ηc do exist (see for example table 4 on page 62 of reference [86]), giving value ranges of ηc in cleanliness conditions characterised with adjectives shown in table 5. Table 5. Guideline values for oil contamination factor ηc from SKF’s General Catalogue [86]. Lubricant condition Extreme cleanliness High cleanliness Normal cleanliness Slight contamination Typical contamination Severe contamination Very severe contamination

Description Particle size of the order of the lubricant film thickness. Laboratory conditions. Oil filtered through extremely fine filter. Conditions typical of bearings greased for life and sealed. Oil filtered through fine filter. Conditions typical of bearings greased for life and shielded. Slight contamination in lubricant. Conditions typical of bearings without integral seals, coarse filtering, wear particles and ingress from surroundings. Bearing environment heavily contaminated and bearing arrangement with inadequate sealing. Severe reduction of bearing life (ηc = 0)!

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A characteristic example of the effect of lubricant contamination on bearing fatigue life is provided on page 64 of SKF’s General Catalogue [86] stating the following: “Several 6305 deep groove ball bearings with and without seals were tested in a highly contaminated environment (a gearbox with a considerable number of wear particles). No failures of the sealed bearings occurred and the tests were discontinued for practical reasons after the sealed bearings had run for periods which were at least 30 times longer than the experimental lives of the unsealed bearings. The unsealed bearing lives equalled 0.1 of the calculated L10 life, which corresponds to a factor ηc = 0.”

(a) New oil

(b) Contaminated oil

Figure 4. Vibration waveforms after 60 minutes for new oil (left) and oil contaminated with 40 μm particles (right). The horizontal axis refers to the running time in seconds. (From Akagaki et al. [87].)

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Figure 5. Amplitude of the acoustic emission time signal with three contaminated greases: (a) grease with 0.02% by weight of quartz dust; (b) grease with 0.2% by weight of quartz dust; (c) grease with 2% by weight of quartz dust.

Bearing deterioration following lubricant contamination can be detected acoustically through vibration analysis equipment. Figure 4 shows some of the results of Akagaki et al. [87] on the vibration waveform tests done on new oil and oil contaminated with 40 μm particles. The difference is obvious. Miettinen and Anderson [88] showed similar results for rolling bearings lubricated with contaminated grease (Figure 5). The effects of lubricant solid contaminants are of course not confined to rolling bearings, although they are better known in this sector because of the fine tolerances, thin lubricating films and wide range of applications in industry and transport. Other machine elements such as gears and seals deteriorate when operating in contaminated environments. For example, Sari et al. [89] showed the degradation of spur gears operating with lubricant contaminated with very fine sand particles, in an attempt to simulate operating conditions such as those met in deserts, quarries or mines. They showed that the particles increase the abrasive wear and surface temperature in zones of high sliding, particularly at the teeth roots. Furthermore, the effects of debris are not confined to wear. Mizuhara et al. [90] used a two-disc rig operating in the mixed to hydrodynamic lubrication regime and contaminated the lubricant with spherical, very hard alumina particles. Thus, they showed that friction increased with particle concentration but the effect was diminished at higher sliding velocities, probably due to the thicker hydrodynamic film, which offset the load supported by the particles. They later extended their study with a theoretical analysis of the effects of particles on the friction of journal bearings [91].

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5. Particle Behaviour and Mechanisms of Damage to Lubricated Contacts Particle behaviour in lubricated contacts depends on the size, hardness and fracture toughness of the particles. It also depends on the hardness of the contact counterfaces and on the kinematical conditions of the contact. Extensive experimental studies and theoretical analyses since the 1980s have shown that, in terms of particle material properties, the following four categories can be classified [49,50] (all referring to isolated particles in, predominantly, rolling contacts). (a) Ductile particles passing through a concentrated contact are flattened and become thin platelets with a thickness approximately equal to the average film thickness of the contact. If plastic deformation of the contact counterfaces takes place, the flattened particles appear more like “flying saucer discs” – see Figure 6(a). (b) Brittle particles fracture early in the inlet zone and produce small fragments that imbed surfaces – see Figure 6(b). Surface damage depends on the ultimate fragment size in relation to the film thickness of the contact. (c) Tough particles may fracture late in the inlet zone producing large fragments that imbed surfaces – see Figure 6(c). The fragments are generally larger than in the case of weaker, brittle particles (case (b)) and the immediate damage to the surfaces will be greater. (d) Small, tough particles behave rigidly – see Figure 6(d). If they are larger than the average film thickness of the contact, the contact counterfaces are forced to deform elastically or plastically to accommodate them.

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(a) Ductile particles are flattened and become thin platelets.

(b) Brittle particles fracture early in the inlet zone producing small fragments that imbed surfaces. Figure 6. Continued on next page.

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(c) Tough particles fracture late in the inlet zone producing large fragments that imbed surfaces.

(d) Small, tough particles behave rigidly. Figure 6. Behaviour of particles in rolling contacts depending on particle material properties (from references [49,50]).

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5.1. Soft, Ductile Particles Ductile particles entering an elastohydrodynamic contact with average film thickness smaller than the average particle size will be elastoplastically deformed [15,16,35,49]. Depending on the particle hardness in relation to the contact surface hardness, a particle may or may not cause plastic deformation to the contact counterfaces. If the particle is relatively soft, it will be extruded as it is being squashed between the harder counterfaces of the contact. This extrusion involves some sliding between the particle and the contact counterfaces as well as some sticking, depending on the local friction coefficients. Interfacial friction increases the pressure between the particle and the counterfaces. Depending on the size and hardness of the particle, the counterface deformation may be elastic or elastoplastic. In some cases, the counterfaces can close around the particle before damage (yielding) occurs. Otherwise, plastic deformation in the form of counterface indentation takes place. The process is depicted in Figure 7. In the 1980s, Hamer et al. [13,14] developed an effective analytical model to calculate the contact pressure on a ductile particle and the displacements of the (flat) counterfaces, assuming the particle is a circular disc and the counterfaces are flat. The particle was extruded as a fully plastic solid squeezed between elastic counterfaces. The analysis was axisymmetric and included friction at the particle-counterface interfaces. The model was capable of predicting the onset of surface yield and was subsequently used to construct maps detailing safe and unsafe regions of operation, such as the one shown in Figure 8. It was later extended to calculate the contact pressure from Johnson’s plastic cavity model [92] and, finally, from a Finite Element Analysis in order to account for plastic deformations of the counterfaces [93].

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George K. Nikas

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Figure 7. Mechanisms of indentation by (a) soft, ductile particle and (b) hard, ductile particle (from Dwyer-Joyce [49]).

Figure 8. Safe and unsafe combinations of particle hardness and particle diameter to film thickness ratio (from the work of Hamer and Hutchinson [93]).

A basic rig was constructed to study the deformation of soft, ductile particles squashed between hard anvils, and the development of surface indentations [14,93]. Figure 9 shows a schematic of the rig and some characteristic dents, revealing how softer particles produce wider and shallower dents, and how even a 55 HV aluminium solid can indent a 1010 HV surface (Figure 9(d) – bottom, right).

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Figure 9. Schematic of particle squashing rig (top) and surface indentations (bottom) when thin discs of volume equivalent to 1.5 mm diameter spheres are compressed to closure between lubricated anvils of hardness equal to 1010 HV (from the work of Hamer and Hutchinson [14,93]).

The predictions of the model by Hamer et al. [14,93] agreed reasonably well with experimental results on the shape of surface indentations, although there was discrepancy in terms of dent dimensions, which was speculated that could be reduced if plasticity effects such as strain hardening were included in the model. For a comparison of experimental results and theoretical calculations, the reader is directed to table 3.6 of Dwyer-Joyce’s thesis [49]. The contact pressures were also as expected, with the characteristic “friction hill” as in forging. Figure 10 shows examples of the theoretical predictions on the contact pressure when the contact counterfaces are assumed purely elastic as well as when Johnson’s cavity model is utilised to model surface plasticity.

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The analysis of Hamer et al. [14] lead to the development of a simple equation to predict the critical parameter values during ductile particle squashing that will cause surface plastic deformation. This is expressed as follows [48]:

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D 1 ⎛⎜ H ⎞ = ln 2 3 s ⎟ t H p ⎟⎠ μ ⎜⎝

(3)

where D is the effective diameter of the flattened particle, t is the thickness of the flattened particle, μ is the friction coefficient at the interface between the particle and the contact counterfaces, Hs is the hardness of a counterface, and Hp is the hardness of the particle. Critical particle aspect ratios D/t can thus be calculated from Eq. (3) and maps of safe and unsafe regions of operation can be drawn, such as those presented by Sayles et al. [48]. This analysis is less accurate for higher aspect ratios. Nevertheless, it has drawn the important conclusion that in most bearing applications, particles softer than about 40 HV can be regarded harmless in rolling contacts with typical elastohydrodynamic film thicknesses. However, the situation is different when there is sliding in the contact, as is explained later. As already mentioned, Hamer and Hutchinson [93] used Finite Element Analysis (FEA) to advance the original analytical model of Hamer et al. [13] with plasticity effects (in addition to using Johnson’s cavity model). FEA was also used by Dwyer-Joyce [49]. In the 1980s, Ko and Ioannides [17] used FEA to find the dent shapes and sizes from the squashing of ductile debris between flat surfaces. They developed an axisymmetric model (contact of a sphere on a flat surface) and a plane-strain model (contact of a cylinder on a flat surface). However, instead of solving the complete, three-body extrusion problem, they derived the contact pressure from the work of Hamer et al. [13,14] and applied it directly to one of the surfaces (platens). Their calculated dent shapes were found to be in good agreement with experimentally measured dent shapes. FEA on debris indentation modelling has also been used by Xu et al. [43] but only as a simplification via a rigid spherical indenter. Nevertheless, dent profiles were in reasonable agreement with experimental results. More recently, basic debris indentation modelling has been performed by Kang et al. [18] and Antaluca and Nélias [94]. In the 1990s, Nikas [25] took over the debris modelling work at Imperial College London, following up on the analyses of Hamer, Dwyer-Joyce, Sayles and Ioannides. His work involved the elastoplastic modelling of ductile particles, softer than the counterfaces squashing them. He developed numerical models to calculate the contact forces on plastically deforming particles in line, rolling-sliding, elastohydrodynamic contacts, including the fluid forces on the particle from the surrounding lubricant under pressure [23]. He then proceeded to include frictional heat generated between the particle and the contact counterfaces as the particle was extruded in rolling-sliding contacts (for any slide-roll ratio), adding the heat generated inside the particle during its rapid plastic deformation [27]. For that, he modelled a particle as a group of heat sources and utilized the theory of thermoelasticity and moving sources of heat [95-97] to calculate the transient flash temperatures on the contact counterfaces. His results confirmed some earlier results of Khonsari and Wang [21] (further extended in reference [24]) with a much simpler model of hard, abrasive particles that caused high frictional heating and could be deemed responsible for scuffing damage. Some basic

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work in this respect has also been presented by Hou and Komanduri [98-100] on the polishing of ceramic balls and rollers in the presence of abrasive particles.

Figure 10. Contact pressure distributions on the left, calculated for a 20 μm, 200 HV particle, squashed between hard surfaces with a 1 μm separating film and interface friction coefficients of 0.05, 0.10 and 0.20 from top to bottom; the surfaces are assumed purely elastic. Example of contact pressure on the right using surface plasticity modelling. (From the work of Hamer et al. [14].)

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George K. Nikas

Nikas’ models [23,27] showed quite graphically that, upon its entrapment in a lubricated contact, a ductile, spherical particle sticks to the contact counterface with the higher friction coefficient immediately and enters the contact. As it is squashed between the converging counterfaces, it is extruded and expands laterally. The fluid forces on the particle are dominating at the very early stages of entrapment but are quickly overcome by the normal and frictional forces between the particle and the counterfaces. The crushing pressure on the particle rises as the particle is dragged towards the contact and is maximised when the particle enters the flat contact region known as the Hertzian zone. At the same time, the rapid plastic deformation of the particle and its shearing on the contact counterfaces owed to its extrusion and the sliding speed of the contact (if any) generate heat, which is partitioned between the particle and the counterfaces, and is maximised when the particle enters the Hertzian zone of the contact. Naturally, the amount of frictional heat depends on the material properties, level of pressures and speeds but it was shown that even small metallic particles (e.g. 10 μm) and several times softer than the contact counterfaces are generating enough heat to raise counterface temperature by hundreds of degrees Celsius. The heat is localised in the vicinity of a particle and penetrates the counterfaces, having disappeared at a depth equal to the radius of the extruded particle disc (up to 200 μm for typical particles). In a typical rolling-sliding contact such as in a rolling bearing, the time needed for a particle to pass through the contact is in the order of a few milliseconds or less. Calculation of the subsurface stress field by Nikas et al. [28] showed that the thermal stresses caused by the frictional heat are very high and often capable of causing plastic deformation to the counterfaces. Thus, Nikas called this type of damage “local scuffing” [27]. The theoretical analysis confirmed experimental results and speculations of Chandrasekaran et el. [19] regarding the effect of abrasive contaminants on scuffing, and extended the findings to cover not only hard particles but very soft ones as well. Nikas’ model of ductile debris deformation was subsequently extended in an attempt to verify earlier results and improve its precision and realism. In what was a very complex model, Nikas [29] advanced his modelling to include transient effects such as the thermoelastic displacement of the counterfaces in the vicinity of the particle, the lubricant pressure on the particle, convective heat losses from the particle and the counterfaces to the lubricant, internal heating along plastic shear zones inside the deforming particle, temperature dependency of mechanical and thermal properties of all bodies involved, thermal anisotropy of the counterfaces, a more precise heat partitioning between the particle and the counterfaces, etc. The advanced model not only confirmed earlier results on the level of frictional heating but actually predicted even higher temperatures, in the order of 1,000-2000 ºC and sometimes even more. As a result, thermal stressing is very high. In fact, thermal stresses are as much as ten times higher than the mechanical stresses on the particle, which means that ignoring frictional heating can give completely misleading results in terms of the risk of surface damage. The main conclusions of that work [29] were as follows. (a) A soft, ductile particle becomes a flat, thin disc as it passes through a typical elastohydrodynamic contact. (b) Upon entrapment, the particle sticks to the counterface with the higher friction coefficient, providing that the counterfaces are of equal hardness. (c) The fluid forces on the particle are much weaker than the normal and frictional forces.

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(d) The extrusion and shearing of the particle during its passage through an elastohydrodynamic contact result in frictional heating, which can be severe even for soft and small particles. The maximum flash temperature in the contact can reach hundreds of degrees Celsius and could, thus, cause tempering reactions and local scuffing. The maximum flash temperature is reached within 1 millisecond or less for typical cases, depending on the contact rolling and sliding speeds. (e) The average contact pressure on the particle is maximized just before its entrance to the central (Hertzian) zone of the contact and this affects the location of the maximum flash temperature. (f) Heat partition between the particle and the counterfaces is governed by the thermal properties of the affected bodies. Generally, the greatest amount of heat goes to the surface which the particle sticks to. Generally, the bigger and/or harder the particle, the smaller (as a percentage) the maximum temperature difference between the two counterfaces is. The biggest impact on the level of flash temperatures is caused by the size of the particle instead of its hardness. The maximum flash temperature difference between a 5 μm and a 20 μm particle of the same material is quite substantial (well over 1000 °C). (g) Due to its plastic deformation in an elastohydrodynamic contact, a particle is a heat generator itself. The heat is generated inside the particle but accounts for only about 1% of the maximum flash temperatures encountered in the contact from the overall frictional heating. (h) Convective heat losses from the particle and the counterfaces to the lubricant are negligible because of the small heat convection coefficient (as calculated for typical applications). The frictional heat transferred to the counterfaces is essentially conducted to their interior and is dissipated fast. It is found that the heat has dissipated completely at a depth equal to the semi-width of the Hertzian zone and, thus, deeper regions are not affected by the heat wave. Generally, the penetration depth of the frictional heat generated on the surface was found to be approximately equal to the radius of the particle-disc when fully deformed in the contact. (i) The frictional heating produces thermal stresses, which in many areas are much greater than the mechanical stresses in the contact. Thermal stresses increase significantly the risk of damage, bringing the high-risk zone for plastic deformations very close to (or on) the surface. (j) Following the particle’s presence inside the contact, a hot spot is created on both counterfaces. Such hot-spots, if undergone plastic deformation, would sometimes appear as smooth and shiny indentations, the shiny (white-colour) appearance owing to the high heating followed by fast cooling to a much lower temperature. (There are characteristic similarities between these predictions and, for example, the experimental results of Zantopulos [101] on scuffing of tapered roller bearings, Tallian [102] (section 12.4 – case 4), and the experimental results of Ville et al. [15,16,103] on the appearance of dents caused by debris particles.) (k) When plastically compressed in an elastohydrodynamic contact, soft and ductile particles are flattened and present a much larger interface for friction, as opposed to hard particles, which more-or-less retain their initial shape. Therefore, owing to the

22

George K. Nikas domination of thermal stresses over mechanical stresses, soft particles can cause substantial damage and should never be seen as harmless. (l) The creation of hot spots from the frictional heating of soft, ductile particles could explain some failures in concentrated contacts and resembles the damage characterized as “local scuffing”. (m) Because of the high heat core of the hot spots left behind squashed debris, it is sometimes expected to observe a secondary micro-cavity inside a dent (due to local material softening) and a matching hump on the centre of the bases of the deformed particle-discs.

ian r tz He

ne zo

Counterface 1 Time: 0.52 ms Maximum flash temperature: 1350 °C

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ng lidi s le rtic Pa

n ctio e r di

Counterface 2 Time: 0.52 ms Maximum flash temperature: 846 °C

Figure 11. Flash temperature distributions on the counterfaces of a line, rolling-sliding, elastohydrodynamic contact, 0.52 milliseconds after trapping a 20 μm, 100 HV spherical, ductile particle. The particle is 8 times softer than the counterfaces. Sliding speed = 1 m/s; slide-roll ratio = 1. The particle sticks to counterface 1, which has a higher friction coefficient, and slides on counterface 2. At the instant shown, the particle starts exiting the Hertzian zone of the contact. (From the work of Nikas [25,29].)

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Figure 11 shows an example of the flash temperature distributions on both counterfaces of a typical, rolling-sliding, elastohydrodynamic contact with central film thickness of 0.7 μm, squashing a 20 μm ductile particle that is 8 times softer than the counterfaces, taken from the work of Nikas [25,29]. Maximum flash temperature in excess of 1,300 °C is predicted as the particle, during its extrusion, sticks to the counterface with the higher friction coefficient and slides on the other. The contour maps of those temperature distributions are shown in Figure 12. Table 6 shows the results of the subsurface stresses for this example and it is clear that the thermal stresses are dominating the mechanical stresses (that is the stresses from particle squashing when frictional heating is ignored).

-150

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1300 1200 1100 1000 900 800 700 600 500 400 300 200 100 0

x (μm) Particle sliding direction 800

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y (μ m )

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700 600 500

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-300

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Figure 12. Contour maps of the flash temperature distributions shown in Figure 11. The particle sticks to counterface 1 and slides on counterface 2 (notice the hot trace left there). Temperatures shown are in degrees Celsius. (From the work of Nikas [25,29].)

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Table 6. Overall (mechanical plus thermal) and thermal subsurface stress components for the example of figures 11 and 12 (from the work of Nikas [25,29]). Stress

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σxx σyy σzz τxy τyz τzx

Overall stress [GPa] Body 1 Body 2 –5.75 –5.23 –7.22 –0.07 +1.00 +0.29

–3.30 –3.32 –4.24 +0.04 +0.61 –0.14

Thermal stress [GPa] Body 1 Body 2 –5.29 –5.27 –6.74 +0.07 +1.04 +0.25

–3.28 –3.32 –4.24 –0.04 +0.61 +0.12

It is thus realised that thermal stresses increase the risk of surface damage in the contact significantly. Although the location of thermal stress maxima is, generally, different from the location of mechanical stress maxima, there are many areas where thermal stresses prevail. The strength of thermal stresses over mechanical stresses has been confirmed in several publications, for example Ju and Huang [104]. In an analytical study of thermal versus mechanical effects in high speed sliding surfaces, Marscher [105] speculated that strong compressive thermal loading could explain the occurrence of surface “mud-flat” cracks, which could promote and accelerate wear. Impressive results have also been obtained by Tseng and Burton [106] who wrote that “…the thermal compressive stress is found to be ten times the normal load for the assumed contact conditions and hence it is this stress rather than simple load concentration which causes the trouble”. Moreover, an important aspect of thermal stresses in general is that they bring the high-risk strain zone closer to the surface [107,108], which explains the initiation and propagation of surface thermo-cracks and spalling in scuffed components. Regarding debris particle heat generation, it may be difficult to comprehend that a flash temperature of more than 1000 °C can be reached in half a millisecond and caused by a small and soft particle. Nevertheless, similar results have been obtained in many publications and the interested reader is directed to the detailed discussions, additional analyses and related references contained in Nikas [25,29] on the theoretical and experimental validation of these results. It is now clear that a squashed ductile particle is able to cause severe frictional heating, creating high thermal stresses. Even when the overall stress is below the yield limit of the contact counterfaces, the strong, localised heating can result in tempering reactions (for example martensite-to-austenite transformation at around 700-800 °C), which, followed by fast cooling, will introduce residual stresses and possibly initiate microscopic surface thermocracks. Such surface cracks can propagate fast, particularly when subjected to the high lubricant pressures met in elastohydrodynamic contacts. Table 7 presents a flash-temperature parametric study [25,29]. As is clear in the table, a 20 μm spherical particle half as hard as the counterfaces (400 HV) is capable of causing a flash temperature rise of 1830 °C. For larger particles (e.g. 30 μm), the maximum theoretical temperatures can exceed 2,000 °C, the limitation being the softening of the particle and the yielding of the counterfaces. On the other side, small and very soft particles (e.g. 5 μm and 8

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times softer than the counterfaces) do not cause severe frictional heating (table 7). It must also be emphasized that the results of table 7 were derived using friction coefficients of 0.20 and 0.15 for the two counterfaces, which are rather conservative. It is known that for temperatures higher than about 150 °C, the boundary lubricating film between the particle and a counterface is expected to melt and collapse. This would result in higher friction and, sometimes, the friction coefficient could even be doubled [109,110]. In such an occasion, the flash temperatures will be significantly higher because they very much depend on the friction coefficient. Table 7. Theoretical maximum flash temperatures during ductile debris extrusion in a typical line, rolling-sliding, elastohydrodynamic contact. Counterface hardness is 800 HV. (Adapted from the work of Nikas [25,29].) Particle diameter [μm]

Particle hardness [Vickers]

5 5 5 10 10 10 20 20 20

100 200 400 100 200 400 100 200 400

Maximum flash temperature [°C] Surface 1 Surface 2 93 17 185 34 350 68 211 103 415 182 760 318 1350 846 1560 962 1830 1120

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5.2. Brittle Particles Samples from lubricant systems have shown that a large proportion of debris are ceramics and silicates [48]. Those are friable particles, which often fracture at the inlet zone of concentrated contacts and the fragments then readily enter the contacts. The damage to the contacts is then defined by the ultimate fragment size. In the case of tougher materials, surface indentation may be caused before particle fracture. In a concentrated contact, low-toughness, brittle particles will fracture before the contact counterfaces yield (Figure 13(a); see also Figure 6(b)). The produced fragments enter the contact and, depending on their size, they may or may not be compressed. Eventually, all fragments are fractured below a critical size, depending on their original size, fracture toughness and counterface hardness. Surface damage may follow accordingly. On the other hand, brittle particles of high toughness may plastically deform the counterfaces prior to being fractured (Figure 13(b); see also Figure 6(c)). Upon further compression, those particles may fracture as well and, again, the fragments will enter the contact, possibly causing further damage, depending on their size, fracture toughness and counterface hardness. In practise, almost all ceramic debris will cause damage to hardened steels [50]. Therefore, the important factor is the ultimate fragment size because this is what governs the magnitude of damage. The fracture of brittle materials is caused by the existence of flaws and subsequent propagation of such micro-cracks in the presence of a stress field. Tensile stresses

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George K. Nikas

open the cracks and compressive stresses close them. Thus, brittle materials fail in tension. Based on this observation and using the Hertzian pressure for the contact of a sphere (simulated particle) on a flat surface and a simple equation from fracture mechanics, DwyerJoyce [49] derived an equation relating the critical flaw size, acc, to the fracture toughness, KIC, and the counterface hardness, H, assuming that the maximum Hertzian pressure is equal to the counterface indentation hardness. Assuming that the particle diameter cannot be reduced below acc, the final fragment size is [49,51]:

1 ⎡ 3K IC ⎤ α cc = ⎢ π ⎣ (1 − 2ν )H ⎥⎦

2

(4)

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where ν is the Poisson’s ratio of the counterface. Table 8 shows a comparison of the predictions from Eq. (4) and the experimental results of Dwyer-Joyce [49] on some brittle debris. The agreement is reasonable.

Figure 13. Behaviour of brittle particles during compression: (a) low toughness particle (left) and (b) high toughness particle (right). (From Dwyer-Joyce [49].)

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Table 8. Comparison of ultimate fragment size between theory (Eq. (4)) and experiment (from Dwyer-Joyce [49]). Material Quartz Glass Arizona dust Boron carbide Alumina Silicon carbide

Fracture toughness [MPa·m1/2] 0.5 0.75 1.5 1.7 3 4

Ultimate fragment size (Eq. (4)) [μm] 0.1 0.2 0.7 0.8 2.6 4.7

Experimental fragment size [μm] 0.2-0.5 0.2-0.9 0.2-0.5 2-3 5-10

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5.3. Abrasive Wear Abrasive wear of a surface from a debris particle occurs when the particle digs in, slides and ploughs out grooves of material on the surface (Figure 14). In 1957, Burwell [111] distinguished two kinds of abrasive wear, namely two-body and three-body abrasive wear. From the perspective of particle-caused abrasive wear, two-body wear occurs when a particle remains stuck or imbedded on one counterface of a concentrated contact and scratches the other. Alternatively (three-body wear), a particle may roll or tumble during the sliding motion of the counterfaces and, thus, cause a series of dents on one or both counterfaces. When one counterface is softer than the other, experiments (see for example [4,8,9,49,51]) have shown that particles imbed the softer counterface and scratch the harder. However, when the softer surface is polished, the particles may scratch this surface, too [9]. These phenomena may be explained by the difference in the friction coefficients of the two surfaces, particularly when the ploughing component of friction and the role of surface roughness in raising the friction coefficient are taken into consideration. In fact, Nikas’ theoretical modelling work [23,25,27,29] has shown that a ductile particle will stick to the counterface with the higher friction coefficient as soon as it is trapped. This means that, in many cases, the harder elements suffer from abrasion and the softer elements from indentation. This discovery is useful in establishing which element will fail first. For example, in a study of the abrasive wear of railway tracks by solid contaminants, Grieve et al. [11] found evidence that particles were embedding in the softer wheel and scratching the hard rail surface, causing the rail to wear 2.5 times faster than the wheel. Another interesting observation concerns the effect of the lubricant film thickness and slide-roll ratio of a contact on the nature of debris-related abrasive wear. Particle rolling and tumbling in a lubricated contact depends on the solid pressure exerted on the particle as it passes through the contact. In a typical elliptical, elastohydrodynamic contact such as that of a ball in a groove, the thickness of the lubricating film has the characteristic “horseshoe” appearance (see the bottom part in Figure 15), which means that it is lower in the darker, horseshoe zone and has a central plateau. This may explain the experimental findings of, for example, Dwyer-Joyce [4,49], which indicate that particles are tumbling through the central zone of the contact and plough and roll over at the sides – see the top part of Figure 15. Some grooves appearing in the central zone may be owed to the larger particles in the abrasive size band [4]. The precise morphology of damage very much depends on the particle size to local

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film thickness ratio. In fact, Williams and Hyncica [8] suggested that for tumbling to occur, the aforementioned ratio must be less than about 2. Naturally, the shape of the particles also plays an important role in particle tumbling and ploughing [112]; sharper particles are more likely to plough.

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Figure 14. Photomicrograph (magnification = 100) of ball wear tracks from five tests with diamond abrasives of size bands 0-0.5 μm, 0.5-1.0 μm, 1-2 μm, 2-4 μm, and 3-6 μm. Film thickness = 0.4 μm, slide-roll ratio = 20%. (From Dwyer-Joyce [49].)

Figure 15. Ball wear pattern from ball-on-disc tests with 0.5-1.0 μm diamond abrasive powder; slideroll ratio = 20%. (From Dwyer-Joyce [4,49].)

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In predicting or calculating the wear from abrasive particles, the local surface velocities must be known and accounted for. Specifically, the magnitude of relative sliding between the contact counterfaces must be precisely known in order to estimate the distance a particle will slide and the length of the resulting wear track. This is characteristic in elliptical contacts, which exhibit the so-called “Heathcote slip” (see for example section 8.5 in Johnson [92]), a micro-slip occurring in the contact and attributed to the variable peripheral velocity of the counterfaces in the contact, owed to surface curvature and deformation. Experimental results show that abrasive wear from debris particles follows the Heathcote slip variation, that is, it is greater where the slip velocity is greater (see figures 16 and 17). Chao et al. [37] have speculated that soft, ductile particles, once deformed to sharp platelets, work-hardened and having re-entered a contact ellipse, may shear off surface material as they rotate or spin subjected to the Heathcote differential slip.

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Figure 16. Scratch length for the roller and house washer in debris-related abrasive wear tests on spherical roller thrust bearings (from Nilsson et al. [12]).

Figure 17. Worn inner raceway profile of a ball bearing (transverse direction) and micro-slip distribution. Notice that regions of greatest material removal correspond to those of increased sliding. (From Dwyer-Joyce [10].)

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Concluding this introduction, it is noted that the literature on abrasive wear is very large, spanning many decades. Therefore, it is impossible to present a thorough discussion on this topic. For further information, interested readers are directed to references [412,49,71,72,74,76,77,112], the references quoted therein, as well as to books on tribology, particularly chapter 6 in Hutchings’ book [69].

5.4. Fatigue Life of Debris Dented Surfaces

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The detrimental effects of debris particles in lubricated contacts were presented and discussed in section 4. It was made clear that solid particles, both hard and soft, and both ductile and brittle or tough, are very often responsible for the damage observed on the counterfaces of machine element contacts, particularly those operating with thin lubricating films. Further results are presented in the present section on the effect of debris-caused indentations on contact fatigue, modelled theoretically and backed up by experimental studies. The experimental and theoretical work of Sayles and Ioannides [47] in the 1980s revealed the critical factors causing a reduction of the fatigue life of debris-damaged rolling bearings. In summary, when surface dents are overrolled in elastohydrodynamic contacts, high pressure spikes appear at the edges of the dents or nearby, leading to stress concentration at or near the surface.

Figure 18. Theoretical, dry contact pressure (upper graph) and normalised subsurface shear stress τxy (lower graph) as a roller passes over a 50 μm diameter dent. (From Sayles and Ioannides [47].)

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Figure 18 shows an example of a dry, line, rough contact with dent of about 50 μm in diameter. On the upper graph of Figure 18, the static contact pressure as the dent is overrolled has been plotted, based on a numerical solution of the related contact mechanics problem. The smooth line corresponds to the Hertzian pressure distribution in the absence of the dent and the spiky line corresponds to the contact pressure in the presence of the dent. It can be seen that the pressure spikes at the edges of the dent are in the order of 2.0-2.5 times the maximum Hertzian contact pressure. The distribution of the subsurface shear stress τxy has been plotted on the lower graph of Figure 18. It is clear that, on the left part of the contact, which lies outside the dent, the shear stress distribution is smooth and close to that predicted by the Hertz theory [92]. On the right part though, the spiky contact pressure in the dent has altered the subsurface shear stress distribution significantly, causing stress concentrations to appear near the surface. This may then explain the origination and propagation of surface cracks after a number of overrollings or, equivalently, after a number of stress cycles. An important factor in studying the effect of a dent on the fatigue life of a dented solid is the residual stress field created from the related plastic deformations in the area of the dent. Finite Element Analysis of surface indentations as for example by Ko and Ioannides [17], Lubrecht et al. [46], and Xu et al. [44] have shown that the subsurface stress field during the overrolling of a dent may be combined with the residual stress field from the plastic deformation that created the dent in the first place. Sayles and co-workers [34,48] quite early established that there is a difference between ductile and brittle particle indentations. Ductileparticle indentations are large and shallow; they are associated with a significant subsurface, tensile, residual stress build-up, which, when combined with the stress field from the normal overrolling of the dent, causes a significant reduction of the fatigue life. Brittle-particle dents on the other hand are smaller, with sharp, raised shoulders and associated residual stress fields much closer to the surface. As a result, the residual stresses of those dents are combined with the stresses from the normal overrolling of the dents and cause significant life reduction only in relatively lightly loaded contacts. Heavily loaded contacts with maximum subsurface stresses appearing deep below the surface are thus largely unaffected by brittle-particle induced dents [113]. Another important factor in evaluating the fatigue life reduction from surface indentations is the level of sliding and the direction of traction in a contact between dented surfaces. The direction of traction and the level of friction around a dent during its overrolling are clearly affecting the location of the maximum shear stress. This is important in predicting where around a dent a crack will originate. According to the experimental work of Ville and Nelias [38], the maximum shear stress is located ahead of the dent compared with the sliding direction if the dent is on the slower surface and behind the dent if the dent is on the faster surface. Similar results have been reported by Ville at al. [103]. In elastohydrodynamic contacts, the contact pressure during the overrolling of debris dents is governed by the Reynolds and elasticity equations, i.e. the effect of lubricant should be taken into account. The literature contains many theoretical studies where the elastohydrodynamic problem of a dented surface is numerically solved. For example, Xu et al. [43] used FEA results on the denting of a surface by a spherical indenter, which were subsequently used in a transient, thermal elastohydrodynamic, rolling-sliding, point-contact, numerical solver to analyse the dent effects on contact pressure, film thickness and

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temperature profiles. They found that the dent, which exhibited a horseshoe-shaped material pile-up in the rolling direction, caused high pressure spikes when overrolled, associated with a corresponding film thinning. They also found that the temperature field was strongly affected by the dent and that the maximum temperature calculated was several times higher than that predicted in the absence of the dent. The pressure spikes caused by the dent resulted in high subsurface stresses and the maximum effective stress was brought closer to the surface, increasing the risk of spalling. These results are experimentally verifiable [40]. Ville and Nelias [40] remark that the pressure peak and resulting stress concentration depends on the sliding direction. Specifically, it appears at the leading edge of the dent for driven surfaces and at the trailing edge for driving surfaces. Furthermore, the elastohydrodynamic effect depends on the actual condition of the lubricant in the dented contact and whether it is in a glassy state (owing to the very high pressure) or not. The present author speculates that the latter will influence how compliant or how good of an absorber the lubricating film is during the dynamic incident of dent overrolling. Once a crack near a dent has originated in a lubricated contact, it might fill up with lubricant when overrolled, owing to capillary action and the high hydrodynamic pressure involved [114,115]. Thus, it has been speculated that the crack will open and propagate or shut as it is being overrolled, depending on the direction of traction. If the direction of traction is the same as the direction of contact motion (driver surface – see Figure 19), the crack is squeezed and shut before being overrolled. If, however, the direction of traction is opposite to the direction of contact motion (follower surface – see Figure 19), the crack lips are pulled apart and lubricant enters the crack via hydrodynamic pressurization, the described mechanism promoting the propagation of the crack [116-118].

Figure 19. Inclined crack, direction of contact motion (thick arrow), and direction of traction (distributed arrows). (From Kaneta et al. [116].)

Returning to the issue of contact fatigue, if the residual stress field in the neighbourhood of a debris-induced dent is known and the subsurface stress field generated during the overrolling of the dent has been calculated, the calculation of the fatigue life of the damaged component can be attempted. This is based on several stress criteria, depending on the loading conditions. Moreover, several models have been developed to describe as accurately as possible what is, essentially, a stochastic process.

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33

In this respect, the Ioannides-Harris fatigue life model [119], which improved upon the original Lundberg-Palmgren model [120,121] and is (at the time of writing) the industry standard in rolling bearings, has been used extensively in predicting the life expectancy of debris-dented surfaces [47]. According to that model, the stressed volume of material is subdivided in n elemental volumes ΔVi. Then, the probability of survival, S, of the entire volume is calculated from n (σ − σ ) ⎛1⎞ ln⎜ ⎟ = N e ∑ Ai H (σ i − σ ui ) i h ui ΔVi Zi ⎝S⎠ i =1 c

(5)

where N is the number of stress cycles, σi and σui is the stress and the fatigue limit in a h

volume element, respectively, Z i

is a stress-weighted depth, H is the Heaviside step

function, and Ai, e and c are application-dependent constants. It is obvious from Eq. (5) that because H (σ i − σ ui ) ≠ 0 only if σi > σui, the summation is carried over only on volume elements with stress exceeding the local fatigue limit. Therefore, when the stress distribution is known in the neighbourhood of a dent, the reduction of fatigue life can be estimated by applying Eq. (5). Similar probabilistic models have been developed by other researchers. For example, Tallian’s exhaustive modelling work [30,45,122,123] has led to a combination of various models, not necessarily with direct physical justification, yet with reasonable agreement with experimental results. Ai [41,124] utilised the Weibull weakest-link theory to develop a lifereduction model due to debris denting. He found that fatigue life depends strongly on the indentation area density and slope, particularly for slopes exceeding 5°. It should also be emphasized that such models have satisfactory experimental verification and are backed-up by tedious, costly and time-consuming experimental work (see for example reference [125]). Detailed modelling work on the contact fatigue of dented surfaces has also been published recently by Antaluca and Nélias [94]. The interested reader is also directed to the review contained in section 6.4.1 of Roylance et al. [59].

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5.5. Particle Entrainment and Entrapment in Concentrated Contacts The processes of particle entrainment and entrapment in concentrated contacts determine the probability and nature of potential damage. If particles bypass a lubricated contact, they will obviously cause little or no damage at all. If they are trapped, they may cause surface abrasion, indentation, frictional heating, etc, depending on the surface speeds, film thickness and contact geometry. Thus, the experimental study and theoretical analysis of particle entrainment and entrapment is a vital step in understanding wear mechanisms and assessing the risks of damage. The present author distinguishes particle entrainment and entrapment as follows: entrainment refers to particle transportation by a lubricant to a concentrated contact without the particle being squeezed by the contact counterfaces; entrapment refers to the process where the particle is in contact with both counterfaces and is elastically or plastically

34

George K. Nikas

compressed. Thus, entrainment precedes entrapment (without entrapment being guaranteed to happen). The theoretical analysis of particles translating in viscous fluids is a Fluid Mechanics topic since at least the 19th century. In one of his pioneering theoretical studies, Stokes [126] in 1850 dealt with the translation of rigid spheres through unbounded, quiescent flows at low Reynolds number. The mathematical problem of particle translation and rotation in viscous fluids later received extensive treatment in some quite complex and long studies by Jeffery [127] in 1922 (dealing with ellipsoidal particles), Rubinow and Keller [128] in 1961 (formulating the transverse force on spinning spheres), Bretherton [129] in 1962 and Saffman [130] in 1965 (dealing with flows of low Reynolds number), Leal [131] in 1979 and [132] in [1980] (dealing with low Reynolds number flows and non-Newtonian fluids), Brunn [133135] in 1976-77 (dealing with viscoelastic fluids and single or multiple translating and interacting spheres), Drew [136] in 1978 (dealing with particle forces in a slow flow), and Sugihara-Seki [137] in 1993 (dealing with a single elliptical cylindrical particle in channel flows at low Reynolds number). The aforementioned studies involve rigorous mathematical formulations of the forces acting on single or interacting, rigid particles, mainly in shear flows at low Reynolds number. They also contain long lists of references on the discussed topics, which help readers obtain an accurate picture of the problem and its mathematical treatment. In some cases, they even account for interactions among particles due to fluid disturbances and even electric forces, as in Saville [138]. Of general interest and practical value is also the study of Maxey and Riley [139], which deals with the equation of motion of a rigid particle in a nonuniform flow. However, from an engineering point of view, the formalism, generality and complexity of the previously mentioned studies is often impractical. Applied tribology usually requires simpler and more straightforward solutions or practical formulas.

Particle contact semi-circle

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Trajectory of a particle being rejected

Trajectories of particles bypassing the ball

Trajectory of a particle that will collide on the ball

D = 20 μm

Sr = 1 hin = 100 μm Other data as in Table 1

Figure 20. Theoretical simulation of particle entrainment in a lubricated contact (sphere rolling-sliding on a flat surface) with slide-roll ratio of 1 and oil bath thickness of 100 μm. The figure shows 30 possible trajectories of a 20 μm particle left at 30 randomly chosen positions in the flow in the upper half of the plot and in the inlet zone of the contact. (From Nikas [26].)

Review of Studies on the Detrimental Effects of Solid Contaminants

35

In this respect, Nikas [23,25,26] developed a model to study the entrainment of isolated spherical particles in the inlet zone of an elastohydrodynamic point contact of a sphere rolling-sliding on a flat surface. The analysis concerned particles larger than the minimum film thickness of the contact. The model was based on the solution of the Navier-Stokes equations for the viscous flow of a Newtonian fluid, followed by calculating the fluid drag force on a particle left at a random position in the inlet zone of the contact. Particle trajectories could thus be predicted for thousands of initial particle positions, creating a map of trajectories similar to that shown in Figure 20. The simulation showed how some particles were pushed away via the fluid back-flow and were swept aside the contact while others collided on the rolling element. The effects of the slide-roll ratio, the degree of lubricant starvation, the size and weight of a particle, and the oil viscosity on the risk of damage were examined and found to be significant (this is discussed later). The next step in the analysis involves particle entrapment. A particle approaching a concentrated contact sees two surfaces converging (Figure 21). If the particle touches both counterfaces, normal reaction forces N 1 , N 2 (see Figure 21), and tangential, frictional ~

~

forces T1 , T2 are exerted to it, in addition to the local fluid force F ~

~

(f )

from its interaction

~

with the surrounding fluid. The resolution of all force components on the particle determines the direction the particle will follow and whether it will be entrapped or rejected.

Surface 1: z1

z

z1  x, y u1

y A N 1

T1

O

~

x

D K T2 ~

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Surface 2: z2

B

~

~

~

F

f

~

N2

T2c

~

~

D

z 2  x, y

~

T1c  T1

 T2 ~

u2 ~

Particle centre: K  x0 , y 0 , z 0 Contact points: A x1 , y1 , z1  x1 , y1 and B x 2 , y 2 , z 2  x2 , y 2

Figure 21. Spherical particle of diameter D in contact with the counterfaces of an elliptical, nonconforming contact, at points A and B. Surface speeds

u1 ~

counterface reaction forces

N1 ~

[141].)

and

and

u2 ~

N 2 , and frictional forces T1 ~

, particle fluid force

~

and

T2 ~

F (f ) ~

, particle-

are depicted. (From Nikas

36

George K. Nikas

Elementary yet useful theoretical analyses of spherical particle entrapment can be found in references [10,35,49] for non-conforming, line contacts, and in reference [140] for a ball on a plane (still treated two-dimensionally, as in a line contact). In the aforementioned studies, simple equations were derived to determine whether a particle will be entrapped or rejected, and omit any fluid forces acting on the particle. The present author analysed the problem of spherical-particle entrapment theoretically, both for line and for circular, non-conforming, rolling-sliding, elastohydrodynamic contacts [25,142], including particle-fluid interactions. For example, in reference [142] he derived the following equation for the fluid force on a spherical particle in the x-direction (Figure 21), which lies on the tangent plane of the contact: ( x) Ffluid = 3πη

u1 + u2 2

π ⎡ (D − hc )⎤⎥ (D > hc) h − c ⎢⎣ 4 ⎦

(6)

where η is the fluid dynamic viscosity, u1 and u2 are the tangential velocities of the contact counterfaces, D is the particle diameter, and hc is the central film thickness of the contact. The (x )

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interesting property of Eq. (6) is that it shows that Ffluid < 0 (pushing the particle out of the contact) for D > (1+4/π)hc ≅ 2.3×hc, which means that, in typical elastohydrodynamic contacts (hc < 1 μm) and for all but the smallest particles, the fluid acts in favour of particle rejection from the contact. Nikas [25,142] proceeded in deriving equations for all particle force components. His analysis showed that, as soon as the particle is entrapped, the normal and frictional forces on a particle greatly exceed the fluid forces and dominated the entrapment process. Only at the very beginning of particle entrapment the fluid forces play a significant role. Nikas also developed basic criteria for particle entrapment, based on the direction of the resultant particle force. As an example, Figure 22 shows the maximum particle diameter to enter a line contact for various typical parameters. Similar results were derived experimentally by Wan and Spikes [20] as well as by Cusano and Sliney [143]. In references [25,142], Nikas also derived particle force equations and entrapment criteria for circular, non-conforming, rolling-sliding, elastohydrodynamic contacts, including particlefluid forces. He later extended those models to cover the general case of both conforming and non-conforming elliptical contacts in reference [141] and developed advanced particle entrapment criteria that introduced states other than the classical entrapment and rejection. The improved criteria include states such as “conditional entrapment”, “weak rejection”, “potential rejection”, etc, and the classification is based on the direction of the various particle force components [141,144]. His analysis predict the exact behaviour of a spherical particle at the very early stage of its compression between the contact counterfaces of elliptical contacts and calculates both the maximum particle diameter for entrapment and the minimum particle diameter for unconditional rejection. This way, the effects of the contact geometry, load and speed, lubricant viscosity, and friction coefficients of the contact counterfaces were thoroughly examined for a wide range of operating conditions [144]. It was found that the friction coefficients play the major role in particle entrapment, followed by the contact load and speed, and, lastly, by the lubricant viscosity. The effects were, generally, non-linear and complex, yet supported by engineering experience.

Maximum particle diameter to enter the contact [μm]

Review of Studies on the Detrimental Effects of Solid Contaminants

37

μ1 = 0.10, μ2 = 0.05 μ1 = 0.15, μ2 = 0.10 μ1 = 0.20, μ2 = 0.15 μ1 = 0.25, μ2 = 0.20 μ1 = 0.30, μ2 = 0.25

12 11 10 9 8 7 6 u1 = 1.5 m/s, u2 = 0.5 m/s

5

Slide-roll ratio = 1 η0 = 0.1 Pa·s

4

hc = 0.4 - 0.7 μm (for the range

3

of radii of curvature shown)

2 5

10

15

20

Counterface radius of curvature (R1 = R2) [mm]

Figure 22. Maximum particle diameter to enter a line contact for various friction coefficients μ1 and μ2 of the contact counterfaces. (From Nikas [25,142].) 1 2 1

Small particles

Thin oil bath

Large particles

Very large particles

Thick oil bath

2

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2 1

High sliding

1

Definitions 1

Risk of particle agglomeration, fluid starvation, scuffing.

2

Risk of surface denting.

Figure 23. Schematic showing the risks of damage of a line contact, based on particle size, oil bath thickness and level of contact sliding velocity. (From Nikas et al. [23].)

38

George K. Nikas

Combining the analyses of particle entrainment and entrapment, Nikas et al. [23] created a schematic (Figure 23) showing the risks of damage of a line contact, based on particle size, oil bath thickness and level of contact sliding velocity. In creating such a risk analysis system, it is considered that particles that are difficult to be entrapped may obstruct the lubricant replenishment of a contact by accumulating at the inlet. The subsequent lubricant starvation of the contact could lead to film collapse and even scuffing. Such effects have been formulated and discussed in Nikas [23,25,26,141,142]; they have also been observed experimentally [20,22,145-148] using methods such as optical interferometry, high speed photomicrography, microscopy and video recording to observe real-time particle entry in point contacts. In fact, particle agglomeration has been observed by Oktay and Suh [148] in boundary lubricated, sliding contacts, and related to a rise in friction. Interestingly, wear particle accumulation in the inlet zone of lubricated contacts has been observed by Enthoven and Spikes [22], which led them to postulate that scuffing may be caused by a critical rate of wear debris production and accumulation of the debris in the inlet zone of a contact. In a detailed theoretical analysis, Nikas [25,26] extended his previous model [23] on particle entrainment to point contacts and evaluated a number of risks for contact damage from debris particles. Based on a large number of theoretical simulations, he evaluated probabilities such as that of particle accumulation, particle entrapment, particle accumulation and entrapment (combined), particle-rolling-element collision, etc. The results were then gathered in a table (see table 9) showing which combinations of system operating parameters carry which damage risk.

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Table 9. Combinations of operating conditions and parameters promoting various damage modes in a point contact (ball on plane). (Theoretical analysis from Nikas [23].)

Review of Studies on the Detrimental Effects of Solid Contaminants

39

In the case of a particle smaller than the minimum film thickness, theoretical studies have examined issues related to the disturbance created by the particle as it enters an elastohydrodynamic or hydrodynamic contact. For example, Kang et al. [149] modelled the entry of a rigid, ellipsoidal particle in a rolling-sliding elastohydrodynamic contact and showed that the particle creates a significant pressure build-up in its vicinity. This is associated with a change to the local film thickness. They also showed that the back flow of the contact plays a key role in particle entry. Languirand and Tichy [150] presented a study on the approximate solution of the Stokes equations to determine the effect of a twodimensional, high aspect ratio particle of arbitrary cross-section on the pressure and velocity fields of a plane slider bearing. Their solution showed a pressure drop at the particle location, dependent on particle size, velocity and location. They also showed that the particle has a major effect on the bearing pressure field when it is able to obstruct the flow of lubricant. Experimental studies related to these phenomena have also been conducted, mainly to analyse the effectiveness of lubricant additives such as PTFE (polytetrafluoroethylene) (see for example reference [151]) or the behaviour of colloidal nanoparticles in thin film lubricated contacts (see for example reference [152]).

6. Conclusion Debris particulates are responsible for a great number of failures of engineering components. Their minute size should never be underestimated. Understanding the mechanisms of debrisrelated damage in machine element contacts requires expertise but the fact remains that proper filtration is of paramount importance in avoiding premature failures and achieving the projected service life of engineering components. This chapter was introductory to the detrimental effects of debris. It did not discuss beneficial properties of some particles such as those used as solid lubricants and in powder and granular lubrication. The related topics are quite extensive and a proper discussion requires a separate chapter. The reader must keep in mind that lubricant cleanliness is truly one of the most important factors in achieving projected life expectancies of machine elements and having a machine that operates reliably for as long as it was designed for.

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Acknowledgments The author is grateful to his father, Kostas Nikas, for providing valuable research material on the effects of filtration and lubricant contamination in rolling bearings. The author is also grateful to the Jacob Wallenberg Foundation (Sweden) for a research grant awarded for research in Materials Science in 2007 through the Royal Swedish Academy of Engineering Sciences, part of which was used for his financial support in writing this chapter.

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[24] [25]

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George K. Nikas Leonardo da Vinci. Codex Madrid I. 1493, p. 119. Dwyer-Joyce, R. S.; Sayles, R. S.; Ioannides, E. Wear. 1994, 175, 133-142. Rabinowicz, E.; Mutis, A. Wear. 1965, 8, 381-390. Richardson, R. C. D. Wear. 1968, 11, 245-275. Xuan, J. L.; Hong, I. T.; Fitch, E. C. ASME J. Tribol. 1989, 111, 35-40. Williams, J. A.; Hyncica, A. M. J. Phys. D: Appl. Phys. 1992, 25, A81-A90. Hamilton, R. W.; Sayles, R. S.; Ioannides, E. Proc. 24th Leeds-Lyon Symposium on Tribology, Elsevier Tribology and Interface Engineering Series. 1998, 34, 87-93. Dwyer-Joyce, R. S. Wear. 1999, 233-235, 692-701. Grieve, D. G.; Dwyer-Joyce, R. S.; Beynon, J. H. Proc. IMechE, Part F: J. Rail and Rapid Transit. 2001, 215, 193-205. Nilsson, R.; Dwyer-Joyce, R. S.; Olofsson, U. Proc. IMechE, Part J: J. Eng. Tribol. 2006, 220, 429-439. Hamer, J. C.; Sayles, R. S.; Ioannides, E. Proc. 14th Leeds-Lyon Symposium on Tribology, Elsevier Tribology and Interface Engineering Series. 1987, 12, 201-208. Hamer, J. C.; Sayles, R. S.; Ioannides, E. STLE Tribol. Trans. 1989, 32, 281-288. Ville, F.; Nelias, D. Proc. 24th Leeds-Lyon Symposium on Tribology, Elsevier Tribology and Interface Engineering Series. 1998, 34, 399-409. Ville, F.; Nelias, D. STLE Tribol. Trans. 1999, 42, 231-240. Ko, C. N.; Ioannides, E. Debris denting – The associated residual stresses and their effect on the fatigue life of rolling bearings: An FEM analysis. Tribological Design of Machine Elements (Ed. D. Dowson, C. M. Taylor, M. Godet, and D. Berthe), Elsevier, Amsterdam, The Netherlands, 1989, 199-207. Kang, Y. S.; Sadeghi, F.; Hoeprich, M. R. ASME J. Tribol. 2004, 126, 71-80. Chandrasekaran, S.; Khemchandani, M. V.; Sharma, J. P. Tribol. Intern. 1985, 18, 219222. Wan, G. T. Y.; Spikes, H. A. STLE Tribol. Trans. 1987, 31, 12-21. Khonsari, M. M.; Wang, S. H. Wear. 1990, 137, 51-62. Enthoven, J. C.; Spikes, H. A. Proc. 21st Leeds-Lyon Symposium on Tribology, Elsevier Tribology and Interface Engineering Series. 1995, 30, 487-494. Nikas, G. K.; Sayles, R. S.; Ioannides, E. Proc. IMechE, Part J: J. Eng. Tribol. 1998, 212, 333-343. Khonsari, M. M.; Pascovici, M. D.; Kucinschi, B. V. ASME J. Tribol. 1999, 121, 9096. Nikas, G. K. Theoretical modelling of the entrainment and thermomechanical effects of contamination particles in elastohydrodynamic contacts. Ph.D. thesis, Imperial College London, Mech. Eng. Dept., London, England, 1999. Nikas, G. K. ASME J. Tribol. 2002, 124, 461-467. Nikas, G. K.; Ioannides, E.; Sayles, R. S. ASME J. Tribol. 1999, 121, 272-281. Nikas, G. K.; Sayles, R. S.; Ioannides, E. ASME J. Tribol. 1999, 121, 265-271. Nikas, G. K. ASME J. Tribol., 2001, 123, 828-841. Tallian, T. E. ASME J. Lubr. Technol. 1976, 384-392. Loewenthal, S. H.; Moyer, D. W. ASME J. Lubr. Technol. 1979, 101, 171-176. Ronen, A.; Malkin, S.; Loewy, K. ASME J. Lubr. Technol. 1980, 102, 452-458. Loewenthal, S. H.; Moyer, D. W.; Needelman, W. M. ASME J. Lubr. Technol. 1982, 104, 283-291.

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[34] Hamer, J. C.; Lubrecht, A. A.; Ioannides, E.; Sayles, R. S. Proc. 15th Leeds-Lyon Symposium on Tribology, Elsevier Tribology and Interface Engineering Series. 1989, 14, 189-197. [35] Dwyer-Joyce, R. S.; Hamer, J. C.; Sayles, R. S.; Ioannides, E. Proc. 18th Leeds-Lyon Symposium on Tribology, Elsevier Tribology and Interface Engineering Series. 1992, 21, 57-63. [36] Nixon, H. P.; Zantopulos, H. STLE Lubr. Eng. 1995, 51, 732-736. [37] Chao, K. K.; Saba, C. S.; Centers, P. W. STLE Tribol. Trans. 1996, 39, 13-22. [38] Ville, F.; Nelias, D. STLE Tribol. Trans. 1999, 42, 795-800. [39] Kahlman, L.; Hutchings, I. M. STLE Tribology Trans. 1999, 42, 842-850. [40] Nelias, D.; Ville, F. ASME J. Tribol. 2000, 122, 55-64. [41] Ai, X. Proc. IMechE, Part J: J. Eng. Tribol. 2001, 215, 563-575. [42] Nilsson, R.; Olofsson, U.; Sundvall, K. Tribol. Intern. 2005, 38, 145-150. [43] Xu, G.; Sadeghi, F.; Cogdell, J. D. ASME J. Tribol. 1997, 119, 579-587. [44] Xu, G.; Sadeghi, F.; Hoeprich, M. STLE Tribol. Trans. 1997, 40, 613-620. [45] Tallian, T. E. ASME J. Lubr. Technol. 1976, 251-257. [46] Lubrecht, A. A.; Dwyer-Joyce, R. S.; Ioannides, E. Proc. 18th Leeds-Lyon Symposium on Tribology, Elsevier Tribology and Interface Engineering Series. 1992, 21, 173-181. [47] Sayles, R. S.; Ioannides, E. ASME J. Tribology, 1988, 110, 26-31. [48] Sayles, R. S.; Hamer, J. C.; Ioannides, E. Proc. IMechE, Part G: J. Aerospace Eng. 1990, 204, 29-36. [49] Dwyer-Joyce, R. S. The effects of lubricant contamination on rolling bearing performance. Ph.D. thesis, Imperial College London, Mechanical Engineering Department, London, England, 1993. [50] Sayles, R. S. Proc. IMechE, Part J: J. Eng. Tribol. 1995, 209, 149-172. [51] Dwyer-Joyce, R. S. Proc. 31st Leeds-Lyon Symposium on Tribology, Elsevier Tribology and Interface Engineering Series. 2005, 48, 681-690. [52] SKF Maintenance Products B.V. Oil Cleanliness Control Course. Seminar that took place from 9-9-1991 to 13-9-1991 in SKF-ERC (The Netherlands). [53] Khonsari, M. M.; Booser, E. R. Proc. IMechE, Part J: J. Eng. Tribol. 2006, 220, 419428. [54] Barwell, F. T. Proc. 10th Leeds-Lyon Symposium on Tribology (1983), paper I(i), pp. 3-10, Butterworth, 1984. [55] Rigney, D. A. Proc. 18th Leeds-Lyon Symposium on Tribology (1991), pp. 405-412; Elsevier, 1992. [56] Roylance, B. J.; Albidewi, I. A.; Laghari, M. S.; Luxmoore, A. R.; Deravi, F. Lubr. Eng. 1994, 50, 111-116. [57] Trevor, M. H. Wear Debris Analysis and Particle Detection in Liquids. English edition; Kluwer Academic Publishers, 1993. [58] Kowandy, C.; Richard, C.; Chen, Y.-M.; Tessier, J.-J. Wear. 2007, 262, 996-1006. [59] Roylance, B. J.; Williams, J. A.; Dwyer-Joyce, R. Proc. IMechE, Part J: J. Eng. Tribol. 2000, 214, 79-105. [60] Stachowiak, G. W. Tribol. Intern. 1998, 31, 139-157. [61] Roylance, B. J.; Raadnui, S. Wear. 1994, 175, 115-121. [62] Raadnui, S.; Roylance, B. J. Lubr. Eng. 1995, 51, 432-437. [63] Peng, Z.; Kirk, T. B. Wear. 1999, 225-229, 1238-1247.

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[64] Bowen, E. R.; Westcott, V. C. Wear particle atlas. Final report to Naval Engineering Centre, Lakehurst, New Jersey, USA, Contract NO156-74-C-182, 1976. [65] Anderson, D. P. Wear particle atlas, revised. Report NAEC-92.163, Naval Army engineering Centre, USA, 1982. [66] Stachowiak, G. P.; Podsiadlo, P.; Stachowiak, G. W. Tribol. Letters. 2006, 24, 15-26. [67] Beghini, E.; Dwyer-Joyce, R. S.; Ioannides, E.; Jacobson, B. O.; Lubrecht, A. A.; Tripp, J. H. J. Physics D: Appl. Physics, 1992, 25, 379-383. [68] Eleftherakis, J. G.; Khalil, A. SAE Technical paper No. 900561, SAE International, USA, 1990. [69] Hutchings, I. M. TRIBOLOGY – Friction and Wear of Engineering Materials. Butterworth-Heinemann, Oxford, UK, 1992 (reprinted in 2001). [70] Sasaki, A. Proc. IMechE, Part J: J. Eng. Tribol. 2006, 220, 471-478. [71] McKee, S. A. SAE Trans. 1927, 22, 73-77. [72] Roach, A. E. ASME Trans. 1951, 73, 677-686. [73] Rylander, H. G. Mech. Eng. 1952, 74, 963-966. [74] Broeder, J. J.; Heijnekamp, J. W. Proc. IMechE. 1965-66, 180, 21-31. [75] Fitzsimmons, B.; Clevenger, H. D. ASLE Trans. 1977, 20, 97-107. [76] Ronen, A.; Malkin, S. Wear. 1981, 68, 371-389. [77] Ronen, A.; Malkin, S. J. Lubr. Technol. 1983, 105, 559-569. [78] General Motors Corporation. New Departure Handbook (7th edition), USA, 1971. [79] Okamoto, J.; Fujita, K.; Toshioka, T. J. of the Mechanical Engineering Laboratory (Tokyo), 1972, 26, 228-238 (NASA technical translation; NASA TT F-15, 653; June 1974). [80] Dalal, H.; Cotellesse, G.; Morrison, F.; Ninos, N. Final report on progression of surface damage in rolling contact fatigue. SKF-AL74T002, SKF Industries Inc., February 1974. (AD-780453.) [81] Wedeven, L. D. Diagnostics of wear in aeronautical systems. NASA TM-79185, 1979. [82] Cunningham, J. S.; Morgan, M. A. ASLE Lubr. Eng. 1979, 35, 435-441. [83] Bachu, R. S. The influence of debris on rolling fatigue life. Ph.D. thesis, University of London, England, 1980. [84] Sayles, R. S.; Macpherson, P. B. ASTM STP 771. 1982, 255-275. [85] Webster, M. N.; Ioannides, E.; Sayles, R. S. Proc. 12th Leeds-Lyon Symposium on Tribology (1985), paper VIII(iii), 207-221; Butterworth, 1986. [86] SKF General Catalogue. Catalogue 5000 E. June 2003. [87] Akagaki, T.; Nakamura, M.; Monzen, T.; Kawabata, M. Proc. IMechE, Part J: J. Eng. Tribol. 2006, 220, 447-453. [88] Miettinen, J.; Andersson, P. Tribol. Intern. 2000, 33, 777-787. [89] Sari, M. R.; Haiahem, A.; Flamand, L. Tribol. Letters. 2007, 27, 119-126. [90] Mizuhara, K.; Tomimoto, M.; Yamamoto, T. Tribol. Trans. 2000, 43, 51-56. [91] Tomimoto, M.; Mizuhara, K.; Yamamoto, T. Tribol. Trans. 2002, 45, 47-54. [92] Johnson, K. L. Contact Mechanics. Cambridge University Press, England, 1985. [93] Hamer, J. C.; Hutchinson, J. Denting of rolling element bearings by third body particles. PCS report for SKF. Imperial College London, Mechanical Engineering Department, Tribology Group; England, 1992. [94] Antaluca, E.; Nélias, D. Tribol. Letters. 2008, 29, 139-153. [95] Blok, H. Proc. IMechE, General Discussion on Lubrication, London, 1937, 2, 222-235.

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[96] Carslaw, H. S.; Jaeger, J. C. Conduction of Heat in Solids (second edition, 1957). Oxford University Press, England, (reprint 1993). [97] Barber, J. R. Int. J. Heat and Mass Transfer. 1970, 13, 857-868. [98] Hou, Z.-B.; Komanduri, R. ASME J. Tribol. 1998, 120, 645-651. [99] Hou, Z.-B.; Komanduri, R. ASME J. Tribol. 1998, 120, 652-659. [100] Hou, Z.-B.; Komanduri, R. ASME J. Tribol. 1998, 120, 660-667. [101] Zantopulos, H. ASME J. Tribol. 1998, 120, 427-435. [102] Tallian, T. E. Failure Atlas for Hertz Contact Machine Elements. ASME Press, New York, USA, 1992. [103] Ville, F.; Coulon, S.; Lubrecht, A. A. Proc. IMechE, Part J: J. Eng. Tribol. 2006, 220, 441-445. [104] Ju, F. D.; Huang, J. H. Wear. 1982, 79, 107-118. [105] Marscher, W. D. Wear. 1982, 79, 129-143. [106] Tseng, M.-L.; Burton, R. Wear. 1982, 79, 1-9. [107] Roylance, B. J.; Siu, S. W.; Vaughan, D. A. Proc. 12th Leeds-Lyon Symposium on Tribology (1985), Mechanisms and Surface Distress. Butterworth, 1986, pp. 117-127. [108] Ting, B.-Y.; Winer, W. O. ASME J. of Tribol. 1989, 111, 315-322. [109] Russell, J. A.; Campbell, W. E.; Burton, R. A.; Ku, P. M. ASLE Trans. 1965, 8, 48-58. [110] Lai, W. T.; Cheng, H. S. ASLE Trans. 1985, 28, 303-312. [111] Burwell, J. T. Wear. 1957, 1, 119-141. [112] Fang, L.; Kong, X.; Zhou, Q. Wear. 1992, 159, 115-120. [113] Dwyer-Joyce, R. S.; Hamer, J. C.; Sayles, R. S.; Ioannides, E. Proc. IMechE, Symposium titled “Rolling Element Bearings – Towards the 21st Century”, pp. 1-8, London, England, 1990. [114] Way, S. J. Applied Mechanics. 1935, 2, 49-58. [115] Bower, A. F. ASME J. Tribol. 1988, 110, 704-711. [116] Kaneta, M.; Yatsuzuka, H.; Murakami, Y. ASLE Trans. 1985, 28, 407-414. [117] Murakami, Y.; Kaneta, M.; Yatsusuka, H. ASLE Trans. 1985, 28, 60-68. [118] Kaneta, M.; Suetsugu, M.; Murakami, Y. ASME J. Appl. Mechanics. 1986, 53, 354360. [119] Ioannides, E.; Harris, T. A. ASME J. Tribol. 1985, 107, 367-377. [120] Lundberg, G.; Palmgren, A. Acta Polytechnica, Mech. Eng. Series. Royal Swedish Academy of Engineering Sciences, 1947, 1, 7. [121] Lundberg, G.; Palmgren, A. Acta Polytechnica, Mech. Eng. Series. Royal Swedish Academy of Engineering Sciences, 1952, 2, 96. [122] Tallian, T. E. ASME J. Tribol. 1992, 114, 207-213. [123] Tallian, T. E. ASME J. Tribol. 1992, 114, 214-222. [124] Ai, X.; Nixon, H. P. Tribol. Trans. 2000, 43, 197-204. [125] Lubrecht, A. A.; Venner, C. H.; Lane, S.; Jacobson, B.; Ioannides, E. Surface damage – Comparison of theoretical and experimental endurance lives of rolling bearings. Proc. of Japan International Tribology Conference, Nagoya, Japan, 1990. [126] Stokes, G. G. Proc. Cambridge Philosophical Society. 1850, 1, 104-106. [127] Jeffery, G. B. Proc. Royal Soc. London A. 1922, 102, 161-179. [128] Rubinow, S. I.; Keller, J. B. J. Fluid Mech. 1961, 11, 447-459. [129] Bretherton, F. P. J. Fluid Mech. 1962, 14, 284-304. [130] Saffman, P. G. J. Fluid Mech. 1965, 22, 385-400.

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Leal, L. G. J. Non-Newtonian Fluid Mech. 1979, 5, 33-78. Leal, L. G. Annual Review on Fluid Mech. 1980, 12, 435-476. Brunn, P. Rheologica Acta. 1976, 15, 163-171. Brunn, P. Rheologica Acta. 1976, 15, 589-611. Brunn, P. Rheologica Acta. 1977, 16, 461-475. Drew, D. A. J. Fluid Mech. 1978, 88, 393-400. Sugihara-Seki, M. J. Fluid Mech. 1993, 257, 575-596. Saville, D. A. Annual Review on Fluid Mech. 1977, 9, 321-337. Maxey, M. R.; Riley, J. J. Physics of Fluids. 1983, 26, 883-889. Kusano, Y.; Hutchings, I. M. Proc. IMechE, Part J: J. Eng. Tribol. 2003, 217, 427433. [141] Nikas, G. K. Proc. IMechE, Part J: J. Eng. Tribol. 2006, 220, 507-522. [142] Nikas, G. K. ASME J. Tribol. 2001, 123, 83-93. [143] Cusano, C.; Sliney, H. E. ASLE Trans. 1982, 25, 183-189. [144] Nikas, G. K. Proc. IMechE, Part J: J. Eng. Tribol. 2007, 221, 727-741. [145] Dwyer-Joyce, R. S.; Heymer, J. Proc. 22nd Leeds-Lyon Symposium on Tribology, Elsevier Tribology and Interface Engineering Series. 1996, 31, 135-140. [146] Cann, P. M. E.; Hamer, J. C.; Sayles, R. S.; Spikes, H. A.; Ioannides, E. Proc. 22nd Leeds-Lyon Symposium on Tribology, Elsevier Tribology and Interface Engineering Series. 1996, 31, 127-134. [147] Wan, G. T. Y.; Spikes, H. A. Proc. 12th Leeds-Lyon Symposium on Tribology (Mechanisms and Surface Distress), paper X(i), Butterworth, 1986. [148] Oktay, S. T.; Suh, N. P. Proc. 18th Leeds-Lyon Symposium on Tribology (Wear Particles), Tribology Series 21. Elsevier, 1992, pp. 347-356. [149] Kang, Y. S.; Sadeghi, F.; Ai, X. ASME J. Tribol. 2000, 122, 711-720. [150] Languirand, M. T.; Tichy, J. A. ASME J. Lubr. Technol. 1983, 105, 396-404. [151] Palios, S.; Cann, P. M.; Spikes, H. A. Proc. 22nd Leeds-Lyon Symposium on Tribology, Tribology Series 31. Elsevier, 1996, pp. 141-152. [152] Chiñas-Castillo, F.; Spikes, H. A. Proc. 26th Leeds-Lyon Symposium on Tribology, Tribology Series 38. Elsevier, 2000, pp. 719-731.

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[131] [132] [133] [134] [135] [136] [137] [138] [139] [140]

George K. Nikas

In: Reliability Engineering Advances Editor: Gregory I. Hayworth

ISBN: 978-1-60692-329-0 © 2009 Nova Science Publishers, Inc.

Chapter 2

STATISTICAL RELIABILITY WITH APPLICATIONS TO DEFENSE Aparna V. Huzurbazar1, Daniel Briand2 and Robert Cranwell2 1

Los Alamos National Laboratory, New Mexico, USA 2 Sandia National Laboratories, USA

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Abstract This chapter discusses selected current trends in engineering reliability modeling with a focus on applications in the defense sector. Our interests are in design for reliability, prognostics and health management (PHM), enterprise level logistics modeling and systemof-systems reliability modeling. It is not unusual for a new weapon system to be developed without fully considering the impact of reliability early in the design period for whatever reason: cost, necessity, technology, and so on. Today, part of that design for reliability can include PHM elements that will predict component failure far enough in advance so that maintenance and operations schedules can be optimized for the purposes of maximizing system availability and minimizing logistics costs. PHM modeling can be part of a real-time maintenance and operations management tool or an enhancement to enterprise level logistics modeling. Part of minimizing the logistics cost at the enterprise level requires that the supply and repair chain processes be simulated with enough detail to capture the real-world failure events. Accurate failure assessment can help determine the correct number of spares, maintenance support equipment, and qualified personnel necessary to complete the repairs. Ultimately, design for reliability, PHM, and enterprise level logistics modeling need to support the system’s ability to operate within a system-of-systems framework, the modeling of which can be quite involved. This chapter will evaluate each of the concepts discussed above, identifying past successes, current applications, and future trends.

1. Prognostics and Health Management Prognostics and health management (PHM) provides an ideal data-driven methodology to guide decision-making and manage complex systems. PHM is the process of using one or more parameters to predict the condition of a component at a defined point in future operation.

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PHM is part of larger reliability centered maintenance (RCM). RCM provides methodology through which an optimized maintenance schedule can be formulated incorporating the maintenance practices of preventative (time-based), corrective (reactive), and predictive maintenance as appropriate to the system. Some terms common to RCM are condition monitoring (CM) and condition based maintenance (CBM). CBM is maintenance initiated as a result of knowledge of the condition of an item gained from routine or continuous monitoring. CM is the collection and analysis of data from equipment to ensure that it retains the integrity of the design and can continue to be operated safely. Data-driven PHM attempts to include all available data into the analysis. Raw data can be maintenance or failure data from historical records or real-time sensor data from embedded sensors. Diagnostic techniques include data fusion and data interpretation. Prognostics is concerned with system health and prediction of future system behavior. Health management asks the basic questions: what should be done and when should it be done? PHM considers cost issues at each system health state. For example, when the system is functioning, the method is prescriptive and may suggest performing planned replacement of parts which can be costly in terms of unused residual life of parts. When the system is degrading, the optimum strategy would replace degrading components with maximum usage before failure. This incurs cost associated with time for replacement. If the system is failing, loss of life of the system due to catastrophic failure would be even more costly. The goals of PHM are to reduce scheduled maintenance based on calendar time or component usage, maximize lead times for performing maintenance and getting parts, identify systems or components for impending failures or problems, provide real time notification and understanding of an upcoming event such as maintenance or potential failure at all levels of the logistics chain, and optimize operations and maintenance actions for higher availability at a lower cost. PHM systems must be highly customized to specific equipment such as machinery, structural systems, electrical/electronics systems, and so on. Robust methodology must be developed with general prognostics capability that can be applied to many situations. Statistical system reliability is one tool that is very useful in PHM.

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2. Integrating Statistical System Reliability into PHM Statistical system reliability uses any available and relevant information or data about the system (or its components) to improve the quality of estimation and prediction of system reliability. Full system data is frequently difficult and prohibitively expensive and sometimes impossible to collect in the quantities that are required for sufficient precision. Statistical system reliability provides methods to appropriately incorporate many different types of data. The methods are flexible about using other types of data that may already have been collected. For example, Bayesian statistical methods for prediction are used to incorporate expert knowledge from those who have designed the system. Other types of data that can be included are test set measures of individual components, maintenance records, accelerated testing of some parts, and leveraging information from common components in other systems.

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2.1. Overview of Statistical Model Anderson-Cook and Huzurbazar (2008) describe in detail the implementation of PHM for a hypothetical missile system shown in Figure 1. Their model contains a reliability distribution for each component, as well as how the components are combined into the system. For the basic model, the reliability distribution for each component, typically as a function of age, is estimated from the data along with any expert knowledge available. Components combine into a model of the whole system either serially or with redundancy. Component reliability is estimated by using both full-system and component test set measures. Often systems have different variants. In such a case, this method leverages information across common components. For example, consider the series system of Figure 1 with a total of seven components (Control, Guidance1, Guidance2 Propulsion1, Propulsion2, Target Detection and Armament) and three variants. The first variant is comprised of five components (Control, Guidance1, Propulsion1, Target Detection and Armament), while the second and third variant are also comprised of five components but with some differences of the components included. This formulation of the problem allows for common components across different variants of the system to be estimated with a larger collection of data compared to treating each of the variants as different populations. This leveraging of information across common components can lead to improved precision in estimating reliability, while still taking into account differences between variants of the system. Further details on the methodology can be found in Anderson-Cook et al. (2007, 2008) and Wilson et al. (2006). The term component is used quite generally to refer to a part of the entire system, which could be an individual component or a subsystem of the system. The decomposition that we select is a function of how much understanding and data are available at the various levels of the system. For example, for a relatively simple system with only a few data sources, we might choose to model only the major subsystems, while for a highly complex system with considerable data, a more detailed decomposition would be more appropriate. For this simple example, each of the components mentioned here are clearly subsystems. Determining the appropriate level of decomposition for the system is a key decision for system modeling. Ultimately the reliability for each component in the system and the entire system is estimated. In Figure 1, the blue boxes represent the components of the system, while the yellow boxes below each component represent test set variables associated with that component. For example, the control component is comprised of three continuous test set variables (Cont1, Cont4, and Cont5) and two Pass / Fail (P/F) variables (Cont2 and Cont3). This can be interpreted as the control component having five possible failure mechanisms, which are each tested with a particular measure. Guidance1 consists of two continuous test set variables (G1 1 and G1 2) and one P/F test set variable (G1 3). Guidance2 consists of two continuous test set variables and no P/F test set variables. Propulsion1 and Armament have no associated test set variables while Propulsion2 has only one P/F test set variable (P2 1). Anderson-Cook and Huzurbazar (2008) construct the reliability for each variant of the system as a function of the component reliabilities.

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Aparna V. Huzurbazar, Daniel Briand and Robert Cranwell

Figure 1. Series system with three variants.

For a particular problem involving a real system, there may be tens or hundreds of unknown parameters. Bayesian statistical inference is concerned with computing distributions on the parameters, called posterior distributions, given the data and prior information about the parameters. Sampling from the posterior distribution for required quantities such as system reliability is done via Markov chain Monte Carlo methods. These are computationally intensive methods commonly used with Bayesian analysis. The freely available YADAS

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software (Graves, 2003, http://yadas.lanl.gov) has been developed at LANL specifically for such computations. Ultimately, such reliability computations can be made in some scheduled fashion to update existing estimates or they can be made real-time. This allows decisionmakers to assess the state of the system, and determine when intervention is needed to assure adequate performance of the systems in the future. PHM helps support other techniques such as enterprise level logistics models and support enterprise models discussed next.

3. Enterprise Level Logistics Support Models One of the current trends in defense sector reliability modeling is the focus on enterprise level logistics end-to-end customer support (Morales, 2003). As a result, DoD decision makers and support contractors are depending increasingly on large scale simulations to model the autonomic logistics supply and repair chain processes to help make million to billion dollar business case decisions (Figure 2). Part of minimizing the logistics cost at the enterprise level requires that the supply and repair chain processes be simulated with enough detail to reflect changing real-world component reliability estimates. Enterprise level logistics support models can help determine the correct number of spares, maintenance support equipment, and qualified personnel necessary to complete the maintenance and repairs, but depend upon accurate reliability models of the system and the underlying components. Information Flow Hardware Flow (Transportation)

Autonomic Supply Chain Management

Serviceable Part Unserviceable Part

Local Supply

Regional Supply

Central Supply

Local Repair

Regional Repair

Central Repair

Disposal

Disposal

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Operating Unit

Disposal

Part Manufacturer (May Also Function as a Supply Point & Repair Source)

Figure 2. Enterprise Level Logistics Support Model (Smith et al. 2006)

The remaining paragraphs of Section 3 discuss various aspects of enterprise level logistics modeling. First, the elements of an enterprise level logistics support model are presented. This is followed by a brief discussion of the capabilities of several DoD related enterprise level logistic support models. Then, two current trends within enterprise modeling are presented: 1) the implementation of full lifecycle failure distributions to improve model accuracy and the modeling of PHM to help ascertain its potential value.

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3.1. Elements of an Enterprise Level Logistics Support Model Providing enough detail in the enterprise model is critical to obtaining realistic results. Integrated modeling of supply and repair chain activities for a worldwide support system may include the following activities: 1) 2) 3) 4) 5)

Component demand generation at the unit level. On-system maintenance tasks. Component storage and inventory maintenance. Support equipment and personnel usage. Maintenance and repair of support equipment.

Also included are component repair activities and component production at the original equipment manufacturer (OEM) such as: 1) Transportation of systems, components, and supplies. 2) Changes in operations and maintenance tempo. 3) Changes in the operations and support infrastructure, such as unit deployments. The basic capabilities of an enterprise level logistics model include: 1) Global operations with arbitrary multi-echelon supply/repair capabilities. 2) Integrated supply, repair, manufacturing, and transportation processes. 3) System operations schedules that generate parts repairs and demands. 4) Impact of prognostics and health management (PHM) and support equipment reliability on system sustainment. 5) System activation and deactivation schedules and location plans. 6) Transition from normal to higher or lower operational requirements. 7) Changes in inventory controls, quantity, and location of resources.

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Additional capabilities of an enterprise level logistics model may include: 1) 2) 3) 4) 5) 6) 7) 8)

Full on-and off-system support activities. Random, wear-out, and time change maintenance events. Reliability growth curves. Back shop repairs. Shift schedules. PHM impact on operations, manpower, and support equipment. Optimization of spare part inventories across the enterprise. Integration with standard tools such as Excel and Access.

The analyses using an enterprise level logistics model should identify support system performance, resource requirements, and costs. These may include: 1) System availability.

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2) Mission capable (MC) rates to include fully mission capable (FMC), partially mission capable (PMC), non-mission capable (NMC), non-mission capable-supply (NMCS), and non-mission capable-maintenance (NMCM). 3) Supply fill rates and response times. 4) Spares, personnel, support equipment, and transportation requirements and costs. 5) Cost indicators to include investment and operating costs. The driving force behind an enterprise level logistics model is the components’ demand and repair requirements. Demand and repair requirements at the system, subsystem, and/or components levels depend directly on reliability modeling. A reliability model that encapsulates, accurately, the expected failures over time provides the cornerstone for determining the number of components needed at each level of supply, be it unit, intermediate warehouse, and depot/original equipment manufacturer (OEM). Having the right number of spare components in the pipeline and at the appropriate level in the supply chain can minimize the time the system or equipment is not operating (down) for component replacement or repair, and minimize the total number of components in the entire supply system. In addition, a reliability model can capture other reasons a component is not operating. These other reasons may include preventive maintenance actions where the component has not failed but is merely undergoing an engine wash, lubrication, oil change, inspection, and so on. The time spent in preventive maintenance activities can negatively affect availability and impact cost. In any case, accurate prediction of component failure and maintenance actions is necessary to maximize system availability while minimizing costs.

3.2. Selected Enterprise Level Logistic Support Models

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This section briefly details three enterprise level logistics models in use today by the DoD and illustrates their real-world applicability. The models presented are the Enterprise Logistics Model (ELM), the Weapon System Sustainment Value Stream Model (WSSVSM), and the Support Enterprise Model (SEM). This discussion could be expanded to include examples of models that are used for real-time leadership training or models not used by the DoD. However, it suffices to say that there are many logistics models available today with various levels of complexity and capability to model real world processes.

3.2.1. Enterprise Logistics Model (Elm) Ingraham et al. (2005) states: “…Enterprise Logistics Model (ELM), that supports the analysis of enterprise-level logistics processes. The ELM assists in the iterative process of identifying and analyzing requirements for new logistics systems and also for the transformation of existing (legacy) ones. The main goals are to ensure that the logistics system will perform as intended, optimize the processes, estimate cost and durations, and influence the design for supportability and affordability. The ELM is also intended to assist in the phased transformation of legacy logistics systems and provide supporting economic justification for the changes. The approach considers the dynamic nature of interacting components of logistics operations rather than the traditional supply and demand chain flow. The ultimate goal is to reduce life cycle cost (LCC) and risk prior to logistics operation at any design or implementation stage” (Figure 3).

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Aparna V. Huzurbazar, Daniel Briand and Robert Cranwell

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Figure 3. Enterprise Logistics Model (Ingraham et al., 2005) .

3.2.2. Weapon System Sustainment Value Stream Model The Logistics Management Institute’s (LMI) Weapon System Sustainment Value Stream Model (WSSVSM), http://www.lmi.org/logistics/logisticstools.aspx, is a crosscutting supply chain optimization model developed for the Office of the Secretary of Defense. The purpose of the model is to facilitate acquisition decision making by estimating the total recapitalization and operational cost associated with a desired system level reliability improvement. Additionally, it provides the capability to minimize cost for a certain readiness level or maximize readiness for a constrained budget (Lowder et al., 2006). WSSVSM bridges three models, a system evaluator, a fault tree reliability analysis and optimization model, and a spares optimizer (Figure 4). The system evaluator models the entire logistics support system from the retail echelon to the wholesale echelon. The fault tree system analysis and optimization model provides component level reliability models for each weapon system and provides the capability to optimize improvement options within budget constraints. The spares optimizer minimizes the spares cost of achieving a readiness objective in a multi-indentured, multi-echelon environment (Lowder et al., 2006).

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WSSVSM Manager

System

Reliability

Spares

Evaluator

Evaluator

Optimizer

Module

Module

Module

SE

RE

SO

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Figure 4. Enterprise Logistics Model (Lowder et al., 2006).

3.2.3. Support Enterprise Model The Support Enterprise Model (SEM) is designed to model and simulate the operational and support activities of a multi-echelon global support enterprise by randomizing event occurrences and responses (Figure 5). It provides logistics analysts with the ability to define operational and support environments and ascertain measures of its performance effectiveness. As a robust decision support tool for evaluating operational supply chain, repair chain and on-system maintenance activities, SEM has the capability to examine the sensitivity changes in the support system architecture, processes and business rules, as well as air vehicle reliability and maintainability (RandM) characteristics. Result screens assist the user in identifying support system limitations and conducting tradeoff analyses. SEM operates on standard desktop personal computers with a Windows 2000/XP operational environment to permit widespread distribution among potential users (Smith et al, 2006). As expected of most large scale enterprise logistics models, SEM can model multiple systems, sites, parts demands, etc. (Welch, 2007). A typical problem scale, for example the Joint Strike Fighter, includes:

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Aparna V. Huzurbazar, Daniel Briand and Robert Cranwell • • • • • • • • •

3,000 aircraft 3,000+ parts Multiple aircraft configurations 250 operational, supply, repair, and OEM sites worldwide 500 types of support equipment 50 personnel skill types Arbitrary multi-echelon support structure Up to 50 years of simulation Greater than 1E+6 parts demands

OEM

Repair

Build

Depot

Repair Supply

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Supply Point

CONUS Figure 5. Support Enterprise Model (Cranwell, 2008).

OCONUS

Carrier

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The failures of parts are modeled using the exponential distribution for random failures, the normal distribution for wear-out failures, and a point estimate for time change failures. PHM, modeled only for parts with a wear-out distribution, is used to model prediction and inflight diagnostic of actual failure and the logistics response to that failure. The PHM implementation in SEM is presented in detail in a subsequent section.

3.3. Trends Within Enterprise Level Logistic Support Modeling

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Within the enterprise level logistics support modeling efforts, there are several emerging trends that seek to improve and expand model accuracy and usefulness. One trend is to have the capability to use a component’s full lifecycle distribution. This distribution encompasses all three phases of a component’s time-to-failure distribution: infant mortality, useful life, and wear-out, depending upon the type of component, the level of expertise available, and the availability of failure data. Another trend focuses on adding the capability to model PHM into enterprise level logistic support models for the purpose of estimating the benefit gained by having a PHM system versus its cost of implementation. Both of these trends will be discussed briefly in the next sections.

3.3.1 Full Lifecycle Time-To-Failure Distributions Reliability models depend on failure distributions to characterize the probability of failure over the lifetime of a component. Many types of components typically will have a bathtubshaped failure rate life distribution. This distribution is characterized by a decreasing failure rate during the early portion of its life, a constant failure rate during the useful portion of its life, and an increasing failure rate during the wear-out portion of its life (Figure 6). During the early portion of its life, failures are typically caused by manufacturing defects. During the useful portion of its life, component failures are usually caused by chance, perhaps as a result of overstress or a shock to the system. The wear-out portion of its life is characterized by wear or accumulated damage that exceeds allowable limits for normal operation (Wolstenholme 1999). In many logistics models and analyses, the failure rate distributions for components model only the useful life, constant failure rate period which does not take into account the aging process and the wear out problems that will occur (Caruso 2005). As a result, this component failure representation may not capture enough of the characteristics of the failure processes to provide for accurate modeling. Bowles (2002) extends that shortfall suggesting that using a constant failure rate model can give very misleading guidance for system design. Typically, only one or two of the phases of the bathtub curve are considered simultaneously (Jaisingh, Lolarik and Dey, 1987). However, when optimizing supply/repair chain processes in a large scale logistics support models, being able to use the failure characteristics across a component’s lifetime provides greater accuracy and usefulness of the model. Unfortunately, trying to use common failure distributions, such as the Gamma or Weibull, may be quite involved since converting expert opinion to a parameter value would most likely require several iterative steps (Booker, et al., 2000). Although there is a considerable body of literature on bathtub shaped failure distributions, few practical models are available (Xie and Lai, 1995).

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Bathtub Curve

Failure Rate

Infant Mortality

Random

Wear-out

(Useful Life)

(End of Life)

Time

Figure 6. The Bathtub Curve (Wilkins 2002).

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Some reliability and prognostic analyses attempt to model relatively complex systems (e.g., aircraft, aircraft carriers, etc.) or multiple systems (e.g., different aircraft models/configurations that may or may not share components) in the early design or early life cycle phases. In these early life cycle phases, reliability data may not be abundant, as opposed to later in the systems lifecycle, where at least some specific testing has been accomplished or operational failure data become available. Where data are not abundant, expert opinion is solicited. Sometimes data from similar components can be used but must be updated with expert opinion. The expert opinion may come from engineers and technicians who are typically not statisticians or reliability experts so it helps immensely to be able to elicit the necessary information using more common terms and concepts, i.e., how long is the burn-in phase, what percent of total component failures are a result of burn-in, what is the mean life expectancy of the component given it makes it past burn-in, etc. 3.3.1.1 Combined Lifecycle Distribution Swiler et al. (2003) propose the CoMBined Lifecycle (CMBL) distribution as a practical alternative for representing component failure lifecycle. With a bathtub shaped hazard function, the CMBL probability density function (pdf) provides an application friendly method for characterizing a component’s failure or lifecycle distribution. Its five parameters, 1) the probability that the component will fail during infant mortality (F1), 2) the duration of the infant mortality phase (t1), 3) the probability that failure will occur randomly (F2), 4) the mean of the wear-out phase (μ), and 5) the standard deviation of the wear-out phase (σ), are chosen specifically to make it relatively easy to elicit the probability of failure distribution from subject matter experts. The CMBL distribution can help simplify the component failure characterization process early in a system’s life cycle. While most currently available methods focus on modeling the hazard function, the uniqueness of the CMBL distribution is that the focus is on modeling the density function with easily interpretable parameters that can be converted to a hazard function as necessary. The CMBL distribution assumes a linearly declining failure rate during

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infant mortality, a constant failure rate during normal life, and a normally distributed TTF as the component begins to wear out (Figure 7). This failure distribution represents the entire component’s lifetime with parameters that make it relatively easy to elicit the probability of failure distribution from subject matter experts or operational data.

(mt + b) e

−(

mt 2 + bt ) 2

A σ 2π

− (t −μ )2

e

2σ2

f(t)

λ c e −λ ct Time Monotonic Increasing

Linear Decreasing Failure Rate

Constant

Infant t1 Mortality

Random

t2

Time

Wear-out

Figure 7. CMBL Density and Hazard Functions.

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The explicit form of the probability density f(t) as a function of time is: 2 ⎧ −( mt2 +bt ) + ( mt b ) e ⎪ ⎪⎪ λc e −λct f (t ) = ⎨ (t − μ )2 ⎪ − 2 A ⎪ ( ) e 2σ ⎪⎩ σ 2π

0 ≤ t ≤ t1 t1 ≤ t ≤ t 2

(1)

t2 ≤ t ≤ ∞

where t1 = infant mortality or burn-in duration (known) μ = mean of the normally distributed portion of the TTF distribution (known) σ = standard deviation of the normally distributed portion of the TTF distribution (known) λc = failure rate for the constant failure rate portion t2 = transition from constant failure rate to the normal TTF portion A = area multiplier for the truncated normal portion In addition

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Aparna V. Huzurbazar, Daniel Briand and Robert Cranwell F1 = fraction of failures occurring in the infant mortality portion (known) F2 = fraction of failures occurring in the random failure portion (known) λd = mt+b, failure rate for the linearly decreasing failure rate portion

The values of the CMBL distribution parameters F1, t1, F2, μ, and σ, are provided by subject matter experts, data from similar components, or limited test data. The unknowns are the model parameters A, λc, and λd, and the CMBL distribution parameter t2. They are estimated using an iteration scheme that increments λc and adjusts A and t2 appropriately so that f(t) integrates “essentially” to 1. This distribution has been used successfully in a system of systems logistics effectiveness simulation and a prototype PHM consequence analysis model at Sandia National Laboratories (Briand et al. (2007), Briand and Huzubazar (2008)). 3.3.1.2. Closed-Form Full Lifecycle Distribution Like the CMBL distribution, the Closed-form Full Life-Cycle (CFLC) distribution probability density function is a probability density function which represents all three sections of a component’s lifecycle: infant mortality, useful life, and wear-out. Briand et al, (2006) present the closed-form or analytic solution written in terms of known functions and constants, but with a slight change in its parameterization from the CMBL distribution described above. As with the CMBL distribution, there are five intuitive and directly estimable input parameters:

μ = mean of the normally distributed wear-out portion of the TTF distribution (known) σ = standard deviation of the wear-out portion of the TTF distribution (known) λ = failure rate due to random failures Λ = failure rate due to manufacturing or installation defects α = proportion of items that have a manufacturing or installation defect that would eventually result in a failure if not precluded by another type of failure event. where μ should be at least 3σ or a lognormal distribution may be more appropriate. The explicit form of the CFLC probability density f(t) as a function of time is:

⎛ − ( t − μ2 ) ⎞ ⎜ e 2σ ⎟ 2 − e tΛ (−1 + α ) + α π ⎜ ⎟ + 1 ⎟ σ f (t ) = e −t ( λ + Λ ) ⎜ ⎜ ⎟ 2 ⎛t − μ ⎞ ⎛ μ ⎞⎞⎟ ⎜ tΛ − + α λ − α λ + Λ ⎛⎜ − + ⎟⎟ + Erf ⎜⎜ ⎟⎟ ⎟⎟ ⎟ ) ( ) ⎜ 2 Erf ⎜⎜ ⎜e ( 1 ⎝ 2σ ⎠ ⎝ 2σ ⎠ ⎠ ⎠ ⎝ ⎝

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2

(

(

)

(2)

)

where Erf is the error function encountered in integrating the normal distribution. It is differentiable everywhere on the interval [0,∞), and integrates to unity on the same interval. A demonstration of the CFLC distribution can be accessed at http://demonstrations.wolfram. com/ClosedFormFullLifeCycleDistribution. These two full lifecycle representations should be valuable in enhancing maintenance planning and real-time situational awareness processes. Following additional exploration and validation, these methods, used in enterprise level logistics and PHM modeling, should more accurately help assess component system failures by ensuring the timing of the failures is

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captured more accurately, a capability not attained when exponential distributions with constant failure rates are used. This improved accuracy will provide timely feedback on the current status of equipment; provide tactical assessment of the readiness of equipment for the next campaign; identify parts, services, etc. that are likely to be required during the next campaign; provide a realistic basis for scheduling and optimizing equipment maintenance schedules; and help ensure that the useful life of expensive components is maximized while reducing the incidence of unplanned maintenance.

3.3.2. PHM in Enterprise Level Logistics Support Models Statistical reliability methods used with PHM can model how the failure rate will change over time. This is important for predicting a failure of an individual component early enough to be able to modify operations and maintenance scheduled in order to maximize availability. The prognostic capability relies on knowing how an individual component’s failure rate deviates from the “average” or expected component’s failure rate distribution. The change can be analyzed to determine or predict the component’s remaining useful life. This prognostic capability relies on sensors operating in real-time and/or inspections to detect changes in a component’s health, and data fusion algorithms that use that information to predict the change in time to failure or remaining useful life. In the SEM, described above, the improvement in supply and repair chain processes due to PHM is modeled (Welch, 2007). The results of modeling PHM can impact the logistics response to a component’s impending failure. This impact should directly alter the NMCS time, the time the repair process spends waiting for a part. One approach to modeling PHM is to assume that a specific PHM capability is available, and given that, determine the effect that level of PHM capability has on overall supply and repair chain efficiency. For example, assume there are three types of component replacements:

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1) Random failures that are most commonly modeled with an exponential distribution with a constant failure rate. 2) Wear-out failures that are commonly modeled with a normal or Weibull distribution. 3) Time change failures in which a component is replaced at a specified time interval of its operational use irrespective of its condition. Without any additional simulated real-time sensor data or maintenance inspection results, modeling random and time change failures can be an involved process. However, modeling wear-out failures where the wear-out process is represented by a normal distribution can be accomplished a bit more easily. The following definitions are needed to describe the PHM modeling approach: • • • • •

Lifetime Mean: The mean of the Actual Failure normal distribution. Lifetime St Dev: The standard deviation of the Actual Failure distribution. Predict Lead Time: The mean time (flight hours) ahead of the Predicted Failure the support network will know of the failure. Predict Lead Time SD: The standard deviation of the time the Predicted Failure is known ahead of the failure. This can change over time. Late Prediction Probability: The proportion (0-100%) of the predictions that will be late (past the Actual Failure time).

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•

Run to Failure: If yes, then the failed part is replaced the first opportunity after the failure occurs. If no, the failed part is replaced the first opportunity after the ordered part arrives. Order Lead Time: earliest time (simulation hours) prior to failure at which a part can be preordered

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In SEM for the wear-out part shown in Figure 8, obtaining information about the impending component failure as early as possible allows the replacement part to be ordered and delivered to the repair facility when it is needed. The order lead time is the earliest time a part can be ordered prior to a component failure being anticipated by the simulation. If the failure prediction is not accurate or not early enough, the system may be waiting idle while the spare component is being obtained and transported to the repair facility.

Figure 8. Component Failure Distribution Due to Wear-Out (Lockheed Martin, 2007).

The prediction capability can be created as a normal distribution that has a different mean but the same distribution. Thus, the predicted failure is just a shift in the original time-tofailure distribution based on the Late Predict Probability. Predicted Failure Time is shifted

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from the Actual Failure Time according to the Predict Time SD and the Late Prediction Probability as shown in Figure 8. The Predict Lead Time is the time at which the logistics support system will know of the Predicted Failure, so that time decisions can be made as to what actions need to be taken (Lockheed Martin, 2007). Predicted Failure Time = Actual Failure Time + Normal(Alpha*Predict Time SD, Predict Time SD), where Alpha = -erf -¹(2* Late Prediction Probability – 1). When component failure detection is made (at Predict Lead Time) before the predicted failure, SEM attempts to order the spare part. If it is earlier than Order Lead Time, then the part will not be ordered until Order Lead Time prior to predicted failure to order the part. If it is within Order Lead Time, then the part can be ordered immediately (Lockheed Martin, 2007).

4. System-of-Systems Modeling and Simulation

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4.1. Background Although just beginning to take hold and gain popularity in the last decade, the study of “System-of-Systems” (SoS) concepts is not necessarily new. Studies of SoS constructs date back to the mid 1900s, but were then more often referred to as hierarchical systems, embedded systems, and, more recently, families of systems (Sousa-Poza, Kovacic, and Keating, 2008). Within the Department of Defense (DoD), the need to succeed in a new multilateral, asymmetric threat environment has led to a radical transformation in operations to promote agility and enhance responsiveness. The transformation process, as well as the resulting new order of operations, relies heavily on SoS solutions to effectively bridge existing gaps in operations (Brown and Flowe, 2005). The emergence of SoS has resulted in a trend away from a platform centric view of systems towards a capabilities and effects view. Capabilities and effects are typically based on collections of systems, some of which are very complex, working together synergistically, with some systems supporting multiple capabilities and sharing functionality with other systems within the SoS. This trend has required that systems be manufactured in ways that facilitate their operation in a SoS environment. This concept has led to the discipline referred to as “System-of-Systems Engineering (SoSE).” Although SoSE is beginning to filter its way into the curriculum of various universities, it is generally agreed that the theory, nature, meaning and application of SoSE is in its early stages of development (USAF SAB, 2005). Figure 9 illustrates a maturing of the SoSE field, with the split along with approaches based on either a ‘technical’ or ‘inquiry’ based approach (Sousa-Poza, Kovacic, and Keating, 2008). While particular views vary on the definition and meaning of a SoS and SoSE, it is widely agreed that both of these are critical disciplines for which frames of reference, thought processes, quantitative analysis, tools and design methods for these problems are incomplete. Wikipedia defines a System-of-Systems as: “…a moniker for a collection of task-oriented or dedicated systems that pool their resources and capabilities together to obtain a new, more complex, ‘system’ which offers increased functionality and performance than simply the sum of the constituent systems.”

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Figure 9. Evolution of SoSE.

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Commonly proposed descriptions, not necessarily definitions, of Systems-of-Systems and System-of-Systems Engineering (SoSE), are listed below, with primary focus and application domain: 1. Sage and Cuppan (2001), and Maier (1998) describe SOS for military applications: “Modern systems that comprise System-of-Systems problems are not monolithic, rather they have five common characteristics: Operational independence of the individual systems; Managerial independence of the individual systems; Geographical distribution; Emergent behavior; and Evolutionary development”. Their primary focus is on evolutionary acquisition of complex adaptive systems. 2. Manthorpe (1996) describes SoS as “Linking systems into joint System-of-Systems allows for the interoperability and synergism of Command, Control, Computers, Communications, and Information (C4I) and Intelligence, Surveillance and Reconnaissance (ISR) Systems.” The primary focus is on information superiority in military applications. 3. Kotov (1997) is focused on communication structures and information systems in private industry and describes SoS as “System-of-Systems are large-scale concurrent and distributed systems the components of which are complex systems themselves.”

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4. Luskasik (1998) is focused on the education of engineers on the importance of systems and their integration: “System-of-Systems Engineering (SoSE) involves the integration of systems into Systems of Systems that ultimately contribute to evolution of the social infrastructure.” 5. Pei (2000) concerns the integration of information intensive systems: “System of Systems Integration is a method to pursue development, integration, interoperability, and optimization of systems to enhance performance in future battlefield scenarios.” 6. Carlock and Fenton (2001) also focus on information intensive systems: “Enterprise Systems-of-Systems Engineering is focused on coupling traditional systems engineering activities with enterprise activities of strategic planning and investment analysis.” 7. DeLaurentis (2005) and DeLaurentis and Callaway (2004) are concerned with the national transportation system with integrated military and space exploration applications. They state “System-of-Systems problems are a collection of transdomain networks of heterogeneous systems that are likely to exhibit operational and managerial independence, geographical distribution, and emergent and evolutionary behaviors that would not be apparent if the systems and their interactions are modeled separately.” 8. The Department of Defense (DoD) defines a System-of-Systems as: “A set or arrangement of systems that result when independent and useful systems are integrated into a larger system that delivers unique capabilities” (Defense Acquisition Guidebook, 2008). When integrated, the independent systems can become interdependent, which is a relationship of mutual dependence and benefit between the integrated systems. The SoS definition conforms to the accepted definition of a system in that it consists of parts, relationships, and a whole that is greater than the sum of the parts; however, although a system-of-systems is a system, not all systems are system-of-systems (DoD System of Systems, Systems Engineering Guide, 2006).

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4.2.Examples of Systems-of-Systems Systems-of-Systems, while still being applied predominantly in the defense sector, are also seeing applications in other fields such as homeland security, space exploration and our complex infrastructure such as air and ground transportation, communications networks, and energy networks. Some examples of complex Systems-of-Systems are discussed below. The purpose of this discussion is to demonstrate SoS complexity and the difficulty it raises in terms of the modeling, simulation and analysis of these constructs.

4.2.1 Army’s Future Combat System (FCS) as an SoS Within the defense sector, one of the most visible SoS constructs is the Army’s Future Combat System (FCS) (Figure 10).

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Figure 10. FCS Brigade Combat Team SoS.

The FCS Brigade Combat Team (FBCT) SoS consists of a mix of six Manned Ground Vehicles (MGVs), two classes of Unmanned Aerial Systems (UASs), three classes of Unattended Ground Systems (UGSs), three classes of Unmanned Ground Vehicles (UGVs), the soldier, and the network, which provides the communication link between all systems within the FBCT SoS. The FCS clearly fulfills Maier’s five characteristics of a SoS (Maier, 1996). Operational independence exists in the elements of the SoS. If the FCS is disassembled into its component systems, they can still operate independently in a useful manner. Managerial independence in the elements of the SoS also exists. Component systems within the FCS are separately acquired and can be managed independently. The FCS evolves over time, with component systems capabilities added, removed, or modified as needs change and experience is gained. The FCS has emergent capabilities and properties that do not reside in the component systems. The FCS SoS component systems are geographically distributed but have the ability to readily exchange information.

4.2.2 U.S. National Airspace as an Sos Another example of a SoS is the National Airspace System shown in Figure 11 (Pyster and Gardner, 2006). The U.S. National Airspace System demonstrates characteristics that are consistent with an SoS. Air space management and safety are achieved by all component systems working together. Individual systems provide useful services such as navigation to a pilot. Individual systems come and go routinely and are acquired independently with different contractors. They are operated by FAA, airports, airlines, NOAA, and so on.

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Figure 11. U.S. National Airspace System as a SoS.

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Although geographically dispersed, they are highly network centric with standard protocols and interfaces. Overarching CONOPS and architecture provide for evolutionary development. Despite extensive modeling, system complexity leads to emergent behavior. Extensive coordination is central to achieving high levels of efficiency and safety.

4.2.3. The Nation’s Infrastructure As An Sos Figure 12 gives an example of the nation’s infrastructure depicted as a SoS. In this case, the infrastructure is defined as, “The framework of interdependent networks and systems that provides a continual flow of goods and services essential to the defense and economic security of the United States” (Hoyt, 2004). The U.S. National Strategy Identifies 14 sectors and key resources that must be protected as part of the Critical Infrastructure Protection. These are agriculture, information and telecommunications, food, energy, water, transportation, public health, banking and finance, emergency services, chemical industry and hazardous materials, government, postal and shipping, and the defense industrial base. In addition, a number of key resources are considered. These include national monuments and icons, nuclear power plants, dams, government facilities, and commercial key assets. Interdependencies add to the infrastructure protection challenge – “Failure in one asset or infrastructure can cascade to disruption or failure in others, and the combined effect could prompt far-reaching consequences affecting government, the economy, public health and safety, national security, and public confidence” (National Strategy for the Physical Protection of Critical Infrastructures and Key Assets, February 2003). “A ‘System of Systems’ perspective is needed for analyzing infrastructure interdependencies” (Hoyt, 2004).

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Figure 12. The Nation’s Infrastructure as an SoS.

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4.3. System-of-Systems Modeling and Simulation There are a few SoS modeling and simulation capabilities that have been developed, and are currently being used to analyze existing or proposed complex SoS. The approach generally varies from tool to tool, and each has its advantages and disadvantages. We will discuss a couple of the tools in existence in this section. Pyster and Gardner (Pyster and Gardner, October 2006) in Figure 13 below lay out a proposed role of modeling and simulation in SoS Engineering. Their hypothesis is that SoS design and integration are complex activities that can be effectively supported by continuous verification and validation (VandV) using modeling and simulation tools. Two SoS modeling and simulation tools in use today include the System-of-Systems Availability Model (SoSAM) and the System of Systems Analysis Toolset (SoSAT).

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Figure 13. The Role of Modeling and Simulation in SoSE.

4.3.1. The System-of-Systems Availability Model (SoSAM). The System-of-Systems Availability Model (SoSAM) is based on a model developed by TRADOC (UAMBL) using Arena simulation software. The original TRADOC model, which met the Increment 1 Objective of FCS, and depicted one Combined Arms Battalion (CAB), was expanded by AMSAA to include three CABs and a unit of action (UA) support slice. AMSAA is currently in the process of further enhancing SoSAM by adding unmanned aerial vehicles (UAVs) and incorporating greater complexity into the events modeled. SoSAM simulates the failure, recovery, and maintenance activities that are expected to be conducted in an FCS UA during a three-day combat pulse. SoSAM provides, in addition to various maintenance metrics, transient operational availability for each system type at various echelons with the UA. SoSAM also provides data that permits one to calculate the joint probability that a required quantity of systems of each type will be operationally available when needed at any point during the combat pulse. The next version of SoSAM will represent all ground vehicles, as well as unmanned aerial vehicles, the FCS Increment One Threshold URS (Resourced) Organization Design. SoSAM analyses have shown reliability to be the biggest driver on operational availability (75th MORS Symposium, June 2007). 4.3.2. System-of-Systems Analysis Toolset (SoSAT) In the pursuit of modeling and analyzing complex SoS capabilities, a multi-system time simulation capability called the System of Systems Analysis Toolset (SoSAT) has been developed by Sandia National Laboratories (SoSAT User Manual, October 2007). SoSAT development was driven by the need to support the Army’s Future Combat Systems Brigade Combat Team (FCS (BCT)); however, SoSAT can be applied to almost any systems of systems problems. As mentioned above, the FCS (BCT) is the Army’s modernization program consisting of a family of manned and unmanned platform systems, connected by a common network.

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SoSAT is a time step stochastic simulation tool designed to model and simulate multiechelon operation and support activities as projected to be conducted by programs such as FCS (BCT). It provides logistics analysts with the ability to define operational and support environments and ascertain measures of its performance effectiveness based on multiple trials. SoSAT characterizes sensitivity changes to all platforms, support systems, processes and decision rules as well as vehicle reliability and maintainability (RandM) characteristics. It is designed to be a robust decision support tool for evaluating the readiness and sustainment of complex Systems of Systems such as the FCS (BCT), and can include fuel, water, ammunition and maintenance operations. Simulation output results assist the user in identifying platform, as well as SoS level performance and logistics support issues. 4.3.2.1 State Model Objects and State Modeling Key to the multi-system simulation capability within SoSAT has been the development of State Model Objects (SMOs) that enable a system, its elements, and its various functionalities to be encapsulated for use in simulations. Every system in a simulation is represented by an SMO which has a defined composition of items that help define the system’s functionality. The systems are the central objects of the model and are the entities that march through a simulation (Campbell et al, 2005). The framework for State Model Objects (SMOs) is State Modeling, which is based on traditional dynamic state modeling found in state charts (Harel and Naamad, 1996), also called activity charts. State charts are used to define a hierarchy of states and a means of moving from state to state. The status of system(s) and their functions can be determined based on which states are occupied. The states of a system can be designed such that if, for example, a system encounters a failure, a function associated with that failure moves from one state to another. The basics of state charts include:

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•

• •

•

A state model that contains a hierarchical system of states. Each state is either a parent state or a leaf state, that is, one with no children. Each parent state is decomposed into its children either as an AND configuration or an OR configuration. If the system occupies an AND parent state, it must occupy each child state. If the system occupies an OR parent state, it must occupy exactly one of the children. So the OR in this case is an exclusive OR. If the system does not occupy a parent state, it cannot occupy any of its child states and vice versa. One state, called the root state, which is a parent of every state. It is the only state that does not have a parent. A subset of states defined as initial states. The system initially occupies each of these states and they cannot be conflicting. For example, two children of an OR parent state cannot both be initial states as the children would be conflicting. User-defined transitions, which transition systems from one set of states to another set one step at a time. Figure 14 shows a simple example. Transition X is characterized by a source state S, a destination state D, a guard state G, and a trigger T. If at some step, the system occupies state S, the trigger T is true, and the guard state G is true then the system will transition from state S to state D. In general terms a trigger activates a transition and a guard state allows the transition. Both have to be true for the transition to fire. A transition can have a trigger, a guard state, or both.

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Figure 14. A Transition from State S to State D.

• •

• • •

•

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•

The source state for a transition, which is the state the transition emanates from and can be a parent state or a leaf state. Destination states for a transition, all of which must be leaf states. In traditional state charts, if a destination state is a parent state, then the transition enters the destination state at a predefined default entry state. A trigger, which is a Boolean expression of events. Events, which point to a failure mode that has either a failure probability or failure rate, and a downtime. A guard, which is a Boolean expression of states and external elements. If a system occupies a state in a guard expression, then effectively that state variable is set to true. External elements, which are variables that can appear in guard expressions. They are assigned a Boolean value prior to simulation run. An example might be a sandstorm. For one run it could be assigned to true and for another it could be assigned to false. A goal state, which is a special state. If the system reaches this state, there is a particular meaning or consequence. The primary function of state modeling is to determine what combinations of events must occur for the system to reach a goal state.

State modeling adds the concept of functions to traditional state charts. Functions are limited to goal states so that each goal state indicates some degree of functionality of the systems in the state model. Each function is evaluated for a standard set of performance measures. It is the responsibility of the user to provide the interpretation of these performance measures in the context of the function. 4.3.2.2 Benefits of State Modeling State modeling offers several benefits: •

A state model is quite flexible concerning the definition of states and transitions, in particular the trigger expressions for the transitions. In general the more states that are included the simpler the trigger expressions. On the other hand, the number of states can be reduced by defining more complex trigger expressions. If the detailed states are required because it is anticipated that they could be a goal state, it is

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•

•

• • • •

important to include them in the state model detail. Otherwise, their inclusion becomes optional. A state model can have multiple goal states. When the state model builds a solution, the solution contains results for every goal state in the model. Results for all goal states can be displayed simultaneously. A state model can have different sets of initial states. Typically results are desired for the case when every system is initially in its fully operational state. On the other hand if some systems are inoperable or are partially operable, initial states can be so defined. Goal states are not restricted to inoperable states. The state model can contain partially operable conditions. A state model can contain multiple systems of the same or different type. It is easy to incorporate dependencies and shared functionality between systems in a state model. It is simple to incorporate external elements into a state model. The occurrence of bad weather, rough terrain, or turbulence, for example can be defined as an external element and incorporated into guard expressions.

4.4. PHM in System-of-Systems Modeling and Simulation As discussed earlier in this chapter, prognostics and health management (PHM) is the concept of utilizing real-time data from embedded sensors and/or maintenance or failure data from historical records to analyze and predict system health and future system behavior. The goals of PHM are basically to:

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• • • •

Minimize or eliminate calendar and component usage-based scheduled maintenance Maximize lead times for maintenance and parts procurement Identify systems or components for impending failures or problems Provide real-time notification and understanding of an upcoming maintenance event at all levels of the logistics chain

The end result of a successful PHM program is to optimize operations and maintenance actions to obtain higher system performance at lower costs. PHM, also commonly referred to as Condition-based Maintenance Plus (CBM+) within the DoD, is being mandated throughout the majority of military programs. For example, PHM is a key component of the F-35 Autonomic Logistics concept – “JSF Autonomic Logistics Vision – A highly reliability aircraft which encompasses PHM.” Within the Army’s FCS (BCT) program, FCS requirements state that “Each FCS platform…shall prognose system abort failures…far enough in advance…to avoid failures during a mission pulse.” A Department of Army memo dated August 2005 states, …“The U.S. Army will implement CBM+ across all new and existing systems…” A concern with blanket statements about implementation of PHM/CBM+ is the need to show in advance the actual benefits to a weapon system, or system-of-systems from PHM/CBM+. Because of the use environment or cost of implementation, the benefit from PHM/CBM+ to certain weapon systems may not offset the cost of implementation. That is,

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PHM/CBM+ needs to buy its way onto a weapon system. Therefore, there is a need to be able to model the impacts of PHM/CBM+ in tools such as SoSAM and SoSAT. Within SoSAT, PHM is modeled as follows (Swiler et al., 2003): During the initialization of a simulation in SoSAT, PHM enabled elements (that is, those elements within a system that have PHM capability) will be grouped for each system. All instances of the element will be considered PHM capable. Also defined for a PHM capable element is the inspection interval duration in units of time steps. The inspection intervals are defined at the element level such that different elements can be inspected at different rates. This is important since elements can have very different Time-to-Failure distributions and rates. At each time step during the simulation, systems are aged, elements fail and consumable are used, states are changed and requisitions are placed. Prior to the logic that checks for element failures and consumable usage, an inspection routine will occur. This inspection routine will execute for each element in the PHM capable group if and only if the particular time step is a multiple of the inspection interval. For example, if Element A is a PHM capable part and its defined inspection interval is 5 time steps, then at the 5th, 10th, 15th, etc. time steps, an inspection process will be executed for Element A. The inspection routine (Figure 15) will check to see if the element fails before the next inspection time.

Figure 15. Inspection Routine for PHM in SoSAT (Welch, 2007).

The determination of whether or not an ill fail is made using the Time-to-Failure distribution of the element and the current utilization and usage of the element. If it is determined that the element will fail within the next inspection interval, then the false negative probability (user input value) will then determine whether or not the detected failure is a false negative. A false negative is defined as the failure to detect a pending element failure. If it is determined that the element will not fail within the next inspection interval, then the false positive probability

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(user input value) will be used to decide whether or not the undetected failure is a false positive. A false positive is defined by a reporting of a non-existent failure. Based upon this logic, the inspection routine has four possible outcomes:

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1) If it is determined that a failure will occur within the next inspection interval and the failure detection is NOT a false negative, then: a) The simulation will report a pending failure b) The component failure will be scheduled c) A PHM request will be generated for any required spares and services d) The simulation will record a successful PHM detecting for statistics output 2) If it determined that a failure will occur within the next inspection interval and the failure detection IS a false negative, then: a) The simulation will not report a pending failure b) The simulation will record a false negative for statistics output c) The component failure will be scheduled 3) If it is determined that NO failure will occur before the next inspection time and the determination is NOT a false positive, then: a) Inspection routine is done b) No reporting will occur (4) If it is determined that NO failure will occur before the next inspection time and the determination IS a false positive, then: a) The simulation will record a False Positive for statistical output b) No failure will be scheduled for the component c) A PHM request will be generated for any required spares and services. A successful PHM detection in the simulation (outcome #1) will result in a PHM request that will attempt to order any spares and services prior to the actual failure. Note that the request is an attempt since SoSAT has ordering rules defined for spares and services. The PHM request cannot override any of these applied rules. For example, if an inspection of Element A is done in the Field and a successful PHM detection occurs, then a PHM request is generated for Spare A. However, if Element A is only allowed to order spares at the Repair Facility, then the order will not be placed until the platform is in a mission segment that is at the Repair Facility. Once the spare and services arrive, an attempt will be made to perform the repair on the PHM capable element. Again, if there are specific rules already defined (e.g. repair in field, repair at repair facility, etc), then the repair will only occur when and where it is allowed based upon the user-defined rules. If the repair occurs prior to the actual “failure”, then the simulation will record it as a PHM removed prior to failure. If the actual “failure” occurs prior to receiving spare and services required, then the simulation will record it as a PHM that ran to failure. In the case of a false negative (outcome #2), a record of a false negative statistic will be created for output purposes. The detected failure will be scheduled, but the support system within the simulation will be unaware of the pending failure. Therefore no spares or services will be requisitioned and the result will be a PHM failure with a longer (i.e., “normal”) down time.

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If no failure is detected (outcome #3), then the simulation will continue on without any action. The last case (outcome #4) is one such that failure detection is reported falsely. In this case the simulation reports a pending failure although no failure is scheduled for the element. Any spares and services required are requisitioned. When the spare and/or service arrive, it will attempt the repair of the element. The reason these items are still requisitioned and a repair is done is to simulate the resource and spare contention and unnecessary down time that would occur in the case of a false positive.

Conclusion This chapter provides an introduction to many recent concepts and developments in statistical reliability with applications to the defense sector. We have discussed design for reliability including PHM, several enterprise level logistics models and their relationship to PHM and many applications of SoS modeling and simulation. We have chosen specific defense applications that are in use in the DoD community and have practical implementation in industry. The future of PHM /CBM+ is rich in all aspects of the engineering and business community.

References [1]

[2]

[3]

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Anderson-Cook, C.M., and Huzurbazar, A.V. (2008). Data-Driven Reliability Estimation for Prognostics and Health Management (PHM). Los Alamos National Laboratory Technical Report, LA-UR-08-2609. Anderson-Cook, C.M., Graves, T., Hamada, M., Hengartner, N., Johnson, V., Reese, C.S., Wilson, A.G. (2007) “Bayesian Stockpile Reliability Methodology for Complex Systems” Journal of the Military Operations Research Society, 12, 25-37. Anderson-Cook, C.M., Graves, T., Hengartner, N., Klamann, R., Wiedlea, A.K., Wilson, A.G., Anderson, G., Lopez, G. (2008) “Reliability Modeling using Both System Test and Quality Assurance Data” Journal of the Military Operations Research Society (in press). Booker, J. M., Bement, T.R., Meyer, M.A., and Kerscher W.J. III. (2000), “PREDICT: A New Approach To Product Development and Lifetime Assessment Using Information Integration Technology,” in Handbook of Statistics 22: Statistics in Industry, eds. R. Khatree and C.R. Rao, Amsterdam: Elsevier Science B.V., 499-521. Bowles, J. B. (2002), Commentary—Caution: Constant Failure-Rate Models May be Hazardous to Your Design,” IEEE Transactions on Reliability, 51, 375-377. Briand, D., Campbell, J. E. and Huzurbazar, A. V. (2007), “Updating a User Friendly Combined Lifetime Failure Distribution,” Reliability and Maintainability Symposium, RAMS '07, p. 311 - 316. Briand, D. and Huzurbazar, A.V. (2008). “Bayesian Reliability Application of a Combined Lifecycle Failure Distribution'', Journal of Risk and Reliability, Vol. 222, No. 4.

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Aparna V. Huzurbazar, Daniel Briand and Robert Cranwell Briand, D., Lowder, K. and Shirah, D. (2006), “Updating Combined Lifecycle Time-toFailure Distributions,” SAND2006-6890. Sandia National Laboratories, Albuquerque, New Mexico. Brown, M. M., and Flowe, Rob (2005), “Joint Capabilities and System-of-Systems Solutions,” Defense AR Journal (April-July 2005), pp.139–153. Caruso, H. (2005), “Too Much Time in the Bathtub (Curve) – An Aging Aircraft Paradigm that Doesn’t Hold Water,” Joint Council on Aging Aircraft, Aging Aircraft Conference Technical Paper. Carlock, P.G., and R.E. Fenton (2001), “System of Systems (SoS) Enterprise Systems for Information-Intensive Organizations,” Systems Engineering, Vol. 4, No. 4, pp. 242261. Campbell, J.E., Dennis Longsine, Don Shirah and Dennis Anderson (2005), “System of Systems Modeling and Analysis,” SAND2005-0020, January 2005. Cranwell, R. (2008), “Ground Vehicle Reliability – Reducing Maintenance and Repair Cycles,” Defense Maintenance and Repair Conference, April 2008, Washington, DC. Defense Acquisition Guidebook (2008), “System Engineering Guide for System of Systems,” Ver. 1.0, p. 3. DeLaurentis, D. (2005), “Understanding Transportation as a System of Systems Design Problem,” 43rd AIAA Aerospace Sciences Meeting, Reno, Nevada, AIAA-2005-0123. DeLaurentis, D. A. and R.K. Callaway (2004), “A System-of-Systems Perspective for Future Public Policy,” Review of Policy Research, Vol. 21, No. 6, pp. 829-837. DoD System of Systems (2006), “Systems Engineering Guide: Considerations for Systems Engineering in a System of Systems Environment,” Ver. .9, p. 12. Graves, T. (2003). An Introduction to YADAS. Available at www.yadas.lanl.gov. Hoyt, J. (2004), “Critical Infrastructure Protection (CIP) Science and Technology,” Technologies for Public Safety in Critical Incident Response, September 27, 2004. Ingraham, L., Solomon, C., and Lowe, B. (2005), “Analysis and Improvement of Enterprise Logistics Processes using Simulation-Based Methodologies,” Autotestcon 2005, IEEE, Sep 2005, 690-697. Jaisingh, L. R., Kolarik, W. J. and Dey, D. K. (1987), “A Flexible Bathtub Hazard Model for Non-Repairable Systems with Uncensored Data,” Microelectronics Reliability, 27, 87-103. Kotov, V. (1997), “Systems of Systems as Communicating Structures,” Hewlett Packard Computer Systems Laboratory Paper HPL-97-124, pp. 1-15. Lockheed Martin Aerospace and Sandia National Laboratories, “SEM Detail Design Documentation-01016_SEM_SSDD,” SEM Model Documentation, Albuquerque, New Mexico. Logistics Management Institute: http://www.lmi.org/logistics/modeling.aspx Lowder, K., Culosi, S., Wightman, D., (2006), “Weapons System Sustainment Value Stream Model (WSSVSM) Brief,” Logistics Management Institute Briefing to ADUSD(LMandR)/MRandMP, 2 May. Luskasik, S.J. (1998), “Systems, Systems of Systems, and the Education of Engineers,” Artificial Intelligence for Engineering Design, Analysis, and Manufacturing, Vol. 12, No. 1, pp.55-60. Maier, M. W. (1998), “Architecting Principles of Systems-of Systems,” Systems Engineering, Vol. 1, No. 4, pp. 267-284.

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[28] Manthorpe, W.H. (1996), “The Emerging Joint System of Systems: A Systems Engineering Challenge and Opportunity for APL,” John Hopkins APL Technical Digest, Vol. 17, No. 3, pp. 305-310. [29] Morales, D. K. (2003), “Implementing the Future Logistics Enterprise End-to-End Customer Support,” Memorandum for Deputy Chief of Staff, G-4, U.S. Army et al, Deputy Under Secretary of Defense for Logistics and Materiel Readiness, Mar 6, 2003. [30] Pei, R.S. (2000), “Systems of Systems Integration (SoSI) – A Smart Way of Acquiring Army C$I2WS Systems,” Proceeding of the Summer Computer Simulation Conference, pp. 574-579. [31] Pyster, A. and Gardner, P. (2006), “Critical Success Factors in Systems of Systems Engineering,” SAIC briefing, October 25, 2006. [32] Sage, A.P., and C.D. Cuppan (2001), “On the Systems Engineering and Management of Systems of Systems and Federations of Systems,” Information, Knowledge, Systems Management, Vol. 2, No. 4, pp. 325-345. [33] Smith, V. D., Searles, D. G., Thompson, B. M. and Cranwell, R.M. (2006), “SEM Enterprise Modeling of JSF Global Sustainment,” Winter Simulation Conference ‘06, Monterey, California. [34] SoSAM, 75th MORS Symposium, June 2007. [35] Sousa-Poza, A., Kovacic, S., and Keating, C. (2008), “System of Systems engineering: An emerging multidiscipline,” Int. J. System of Systems Engineering, Vol. 1, Nos. 1/2, pp. 1-17. [36] Swiler, L. P., Campbell, J. E., Lowder, K. S., and Doser, A. B. (2003), Algorithm Development for Prognostics and Health Management (PHM). SAND2003-3820. Sandia National Laboratories, Albuquerque New Mexico. [37] System of Systems Analysis Toolset (SoSAT) User Manual (2007), Sandia National Laboratories, Ver. 1.0, October 2007. [38] USAF SAB (2005), “System of Systems Engineering for Air Force Capability Development,” SAB-TR-05-04. [39] Welch, K. M. (2007), “Modeling Prognostics and Health Management in Large-Scale Sustainment and Logistic Simulations,” INFORMS Annual Meeting, Nov 4-7, 2007. [40] Wilson, A.G., Graves, T., Hamada, M. and Reese, C.S. (2006) “Advances in Data Combination, Analysis, and Collection for System Reliability Assessment. Statistical Science, 21, 514-531. [41] Wolstenholme, L. C. (1999), Reliability Modeling, A Statistical Approach, Chapman and Hall/CRC, New York. [42] Xie, M., and Lai, C. C., (1995), “Reliability Analysis Using an Additive Weibull Model with Bathtub-Shaped Failure Rate Function,” Reliability Engineering and System Safety, 52, 87-93.

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In: Reliability Engineering Advances Editor: Gregory I. Hayworth

ISBN: 978-1-60692-329-0 © 2009 Nova Science Publishers, Inc.

Chapter 3

RELIABILITY-BASED DESIGN OF RAILWAY PRESTRESSED CONCRETE SLEEPERS Alex M. Remennikov1 and Sakdirat Kaewunruen*, 1, 2 1

School of Civil, Mining, and Environmental Engineering, Faculty of Engineering The University of Wollongong, Wollongong 2522 NSW, Australia 2 RailCorp – Track Engineering Level 13, 477 Pitt Street, Sydney 2000 NSW, Australia

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Abstract The recently improved knowledge raises a concern in the design manners of prestressed concrete structures. Civil engineers are mostly aware of the design codes for structural prestressed concrete members, which rely on allowable stresses and material strength reductions. In particular, railway sleeper (or railroad tie), which is an important component of railway tracks, is commonly made of the prestressed concrete. The existing code for designing such components makes use of the permissible stress design concept whereas the fibre stresses over cross sections at initial and final stages are limited. Based on a number of experiments and field data, it is believed that the concrete sleepers complied with the permissible stress concept possess the unduly untapped fracture toughness. A collaborative research run by the Australian Cooperative Research Centre for Railway Engineering and Technologies has been initiated to ascertain the reserved capacity of Australian railway prestressed concrete sleepers designed using the existing design code as to develop a new limit states design concept. The collaborative research between the University of Wollongong and Queensland University of Technology has addressed such important issues as the spectrum and amplitudes of dynamic forces applied to the railway track, evaluation of the ultimate and serviceability performances, and reserve capacity of typical prestressed concrete sleepers designed to the current code, and the reliability based design concept. This chapter presents the use of reliability-based approach for endorsing a new design method (e.g. a more rational limit states design) as the replacement of the existing code. The reliability based design approach has been correlated with the structural safety margin provided by the existing prestressed concrete sleepers. The reliability assessment of a prestressed concrete sleeper has been exemplified for better understanding into the sensitivity of dynamic load amplification on the target reliability *

E-mail: [email protected] or [email protected]. Tel: 02 4221 5574 Fax: 02 4221 3238. Corresponding Author: Sakdirat Kaewunruen: School of Civil, Mining, and Environmental Engineering, Faculty of Engineering, University of Wollongong, Wollongong, 2522 NSW. Australia

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Alex M. Remennikov and Sakdirat Kaewunruen indices and probabilities of failure. The target safety index and its uses for the reliability design of the prestressed concrete sleepers are later highlighted.

Keywords: Prestressed concrete sleeper; Reliability based design; Safety index; Permissible stress; Limit states.

1. Introduction Railway is commonly believed as the world’s safest transportation system for either passengers or merchandise across distant areas. Track structures guide and facilitate the safe, cost-effective, and smooth ride of trains. Figure 1 illustrates the main components constituting typical ballasted railway track (Steffens, 2005). Its components can be subdivided into the two main groups: superstructure and substructure. The visible components of the track such as the rails, rail pads, concrete sleepers, and fastening systems form a group that is referred to as the superstructure. The substructure is associated with a geotechnical system consisting of ballast, sub-ballast and subgrade (formation) (Esveld, 2001; Indraratna and Salim, 2005). wheel axle

rail rail pad & fasteners sleeper ballast bed

subgrade

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Figure 1. Typical ballasted railway tracks.

Both superstructure and substructure are mutually vital in ensuring the safety and comfort of passengers and a satisfactory quality of ride for passenger and freight trains. Note that in Australia, UK, and Europe, the common term for the structural element that distributes axle loads from rails to the substructure is ‘railway sleeper’, while ‘railroad tie’ is the usual term used in the US and Canada. This chapter will adopt the former term hereafter. The main duties of sleepers are to: (1) transfer and distribute loads from the rail foot to underlying ballast bed; (2) hold the rails at the proper gauge through the rail fastening system; (3) maintain rail inclination; and (4) restrain longitudinal, lateral and vertical movements of the rails (Esveld, 2001). Remennikov and Kaewunruen (2008) reviewed the typical load conditions on railway track structures as well as common design procedures for ballasted railway tracks. It has been found that the design method for railway sleepers is based on

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permissible fibre stresses (Standards Australia, 2003). The permissible stress design approach makes use of an empirical function taking into account the static wheel load ( P0 ) with a dynamic impact factor ( φ ) to account for dynamic vehicle/track interactions:

PD = φ P0

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where PD is the design wheel load, P0 is the quasi-static wheel load, and

(1)

φ is the dynamic

impact factor (>1.0). Recently, significant research attention has been devoted to the forces arising from vertical interaction of train and track as these forces are the main cause of railway track problems when trains are operated at high speed and with heavy axle loads. It has been found that wheel/rail interactions induce much higher-frequency and much higher-magnitude forces than simple quasi-static loads. These forces are referred to as ‘dynamic wheel/rail’ or ‘impact’ forces. The summary of typical impact loadings due to train and track vertical interaction has been presented elsewhere with particular reference to the shape, magnitude and duration of impact loads found in railway track structures (Remennikov and Kaewunruen, 2007; 2008). As aforementioned, current Australian and international design standards for prestressed concrete (PC) sleepers are based on the permissible stress concept where various limiting values or reduction factors are applied to material strengths and load effects (Standards Australia, 2003; Kaewunruen and Remennikov, 2007a; 2007b). Empirical data collected by railway organisations suggests that railway tracks, especially railway PC sleepers, might have untapped strength that could bring potential economic advantage to track owners. The permissible stress design approach does not consider the ultimate strength of materials, probabilities of actual loads, risks associated with failure, and other short- and long-term factors which could lead to overdesigning the PC sleepers. A research programme to investigate the actual load carrying capacity of PC sleepers was initiated as a collaborative project between UoW, QUT and the industry partners (QR, RailCorp, Austrak, Rocla) within the framework of the Australian Cooperative Research Centre for Railway Engineering and Technologies (Rail-CRC). The main objective was the conversion of the existing Australian design code for PC sleepers into limit states design format, in order to account for the statistical nature, probability and risk of failure. The by-product consequence is also the target safety index for the reliability-based design concept of such track components. Murray and Leong (2005; 2006) proposed a limit states design concept and load factors for a revamped standard AS1085.14. The expressions for predicting the impact loads at different return periods (based on long-term field data from impact detectors at two sites) were proposed. It was suggested that a simple pseudo-static (using factored load) approach can be used in the design procedures of PC sleepers under routine traffic. For concrete sleepers under non-routine traffic, a dynamic analysis was suggested as part of a design process. The research team of the Rail-CRC Project has undertaken statistical, probabilistic and experimental studies to investigate the ultimate resistance of the PC sleepers in a manner required by a limit states design approach (Leong, 2007; Kaewunruen and Remennikov, 2005a; 2005b). In addition to experimental investigations in this project, conversion of the existing design standard into new limit states design format required a comparative examination of the

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safety margin and probability of failure of PC sleepers designed in accordance with both permissible stress and limit states provisions. It is well known that the performance of structural systems depends on the weakest element with lowest reliability (Remennnikov et al., 2007). To achieve uniform performance and reliability in structural designs for different design principles, the reliability-based approach is the most suitable, in order to either maintain consistent levels of desirable structural reliabilities or overcome the differences of such reliabilities (Kaewunruen and Remennikov, 2006a; 2009a; 2009b). From a review of the literature, very few studies were found devoted to the development of the limit states design method for PC sleepers. This chapter proposes the use of reliability-based approach in the conversion of the existing design code for PC sleepers to limit states design format. Experimental results complementing the reliability concepts for the impulsive response and ultimate resistance of PC sleepers are also presented in this Chapter. An example of the reliability assessment of an Australian-manufactured PC sleeper is presented to evaluate the influence of dynamic load amplification on the target reliability indices and probabilities of failure. The reliability-based design methodology of prestressed concrete sleepers is also discussed. The target safety index and its uses for the reliability design of the prestressed concrete sleepers are the highlight in this Chapter.

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2. Current Design Practice Codes of practice including Australian Standard AS1085.14-2003 (2003) prescribe the design methodology for PC sleepers. The life cycle of the sleepers based on this standard is 50 years. The design process relies on the permissible or allowable stress of materials. A load factor is used to increase the static axle load ‘as if’ to incorporate dynamic effects. The design load is then termed ‘combined quasi-static and dynamic load’ which has a specified lower limit of 2.5 times static wheel load. Load distribution to a single sleeper, rail seat load, and moments at rail seat and centre can be interpolated using tables provided in AS1085.14 (2003). It should be noted that the ballast pressure underneath sleepers is not permitted to exceed 750 kPa. Factors to be used for strength reduction of concrete and steel tendons at transfer and after losses can be found in the standards, ranging between 40% to 60% reduction. However, the minimum pre-camber compressive stress at any cross-section through the rail seat area is set at about 1 MPa after all losses (loaded only from prestress). It should be noted that 25% loss of prestress is to be assumed for preliminary design or when there is no test data. A lower level of 22% loss has been generally found in final design of certain types of sleepers (see details in the Australian Standard AS1085.14 Appendix E, 2003). The standard testing procedures in AS1085.14 (2003) have been recommended for strength evaluation of PC sleepers. Past practice has shown that uses of this standard are adequate for flexural strength design and there is no need for any other consideration to checking stresses other than flexural stresses, because the permissible stress design concept limits the strengths of materials to relatively low values compared to their true capacity. Despite under the design loads, the material is kept in the elastic zone so there is no permanent deformation. In general, the sleepers that comply with permissible stress design concept have all cross sections fully in

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compression, under either pre-camber or design service loads. This approach ensures that an infinite fatigue life is achieved and no cracking occurs. Sleepers designed in this manner therefore have a significant reserve of strength within their 50 year life cycle under normal service loads. In reality, impact forces due to wheel/rail interactions may subject the sleepers to dynamic loads that are much larger than the code-specified design forces. Large dynamic impact forces may initiate cracking in the concrete sleepers; indeed, testing at the University of Wollongong has shown shear failure can also occur at or near the ultimate flexural limit. However, concrete sleeper flexural failures have rarely been observed in railway tracks, showing the conservative nature of the existing design process. To develop an ultimate limit states design approach, a study of the response of concrete sleepers to high-magnitude shortduration loading is required. The earlier proposal of allowing cracks in sleepers (by Wakui and Okuda, 1999) could also be considered in a limit states design approach.

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3. Dynamic Track Forces A maximum allowed impact force of 230 kN to be applied to the rail head by passing train wheels has been prescribed in The Defined Interstate Network Code of Practice in Volume 5, Part 2 - Section 8, 2002 (Australasian Railway Association, 2002). That impact force may come about from a variety of effects, including flats worn on the wheel tread, out-of-round wheels, and defects in the wheel tread or in the rail head. Leong (2007) showed that the largest impact forces are most likely from wheel flats; because such flats strike the rail head every revolution of the wheel, severe flats have the potential to cause damage to track over many kilometres. Despite the Code of Practice requirement, there is little published data able to be found showing the actual range and peak values of impact for normal operation of trains, and certainly none were found for the defined interstate network. The value of 230 kN is therefore a desired upper limit rather than a measure of real maximum forces encountered on track. A comprehensive investigation of actual impact forces was undertaken by Leong (2007) as part of the Rail CRC project at QUT. Over a 12 month period, track force data have been gathered from two Teknis Wheel Condition Monitoring stations located on different heavy haul mineral lines. The forces from a total of nearly 6 million passing wheels were measured, primarily from unit trains with 26 to 28 tonne axle loads, in both the full and empty states. An analysis of Leong’s data from one of those sites is shown as a histogram Figure 2. The vertical axis shows the number of axles on a log scale, while on the horizontal axis is the measured impact force from the Teknis station. Note that the impact force in Figure 2 is the dynamic increment above the static force exerted by the mass of the wagon on a wheel (about 60-140 kN). Over 96% of the wheels created impact forces less than 50 kN. However, that small percentage still comprised over 100,000 wheels throughout the year of the study, and they caused impact forces as high as 310 kN. The sloping dashed line in the graph represents a line of best fit to the data for these 100,000 wheel forces.

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Impact Force VS No of Axles (Combined Full & Empty Wagons) 2005-2006 10000000 1000000

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Figure 10. Meta-decision tree for prediction of the most appropriate reliability estimate • average density of the problem space, estimated by Parzen windows and sampled in points, given by learning examples (avg.dens), • average distance to the 5 nearest neighbors, averaged across all learning examples (avg.DA), • average difference between prediction of an example and the predictions for the 5 nearest neighbors, averaged across all learning examples (avg.DK).

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Each meta-learning example we assigned a class value, representing reliability estimate which achieved maximum positive correlation with the prediction error (irrespective of whether the correlation was statistically significant or not) for a given regression model. To assure the unbiasedness of the testing procedure, we removed the learning example from the meta-learning set which represented the domain/model combination we were metapredicting. Meta-classifier. Using the above set of the meta-learning examples, we constructed a decision tree meta-classifier, which is shown in Figure 10 (the tree was pruned using the 1-SE rule and cost-complexity pruning algorithm (Breiman et al., 1984; Torgo, 2003)). We can see, that the most informative attribute, which constitutes the root node of the tree, is cv.rmse (the relative mean squared error, achieved by the given regression model on a domain using the tenfold cross-validation). Based on this, we can also see that the tree leads to selection of estimates BAGV, BVCK, SAvar and CNK-a for the usage with better performing models (lower cv.rmse) and to selection of DENS, CNK-s and SAbias-s for the usage with models which perform worse (higher cv.rmse). Performance comparison with the individual estimates. The testing results of the automatic selection of the best performing estimate using the meta-learning approach are shown in Table 3 and in Figure 11. From comparison with the most successful (the largest num-

Reliability Estimation of Individual Predictions in Supervised Learning

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ber of significant positive correlations with the prediction error) individual reliability estimate BVCK we can see that we achieved better average results using the meta-learning approach. The proposed meta-learning approach therefore shows the potential for predicting the domain/model-based best performing reliability estimate. Further improvement of meta-learning. In the previous experiments, we defined the meta-model to predict any of nine possible reliability estimates as a class value, without considering which of the predicted estimates achieved good average performance. This leads to a question whether our meta-learning approach could achieve good results if it would be allowed to predict only a subset of class values (reliability estimates). Namely, by narrowing down the number of classes we could still achieve good performance, while preventing the meta-predictor to predict an estimate which performs poorly on average. The further analysis (Bosni´c & Kononenko, 2008a) showed that for every domain/model pair usually more than one estimate achieved good performance (i.e. had significant positive correlation to the prediction error). Using the percentage of estimates with the good performance we were therefore able to order the domains with respect to their difficulty of the reliability estimation. Similarly, if selecting only a subset of nine possible reliability estimates, we were also able to compute the percentage of testing domains, in which these estimates performed well (i.e. the domain coverage) and use it for improvement process as follows. Based on this idea, new meta-predictors were constructed and were allowed to predict only the number of reliability estimates selected in advance (from two to all nine possible estimates). Each meta-predictor, modified in this way, was allowed to predict only those estimates which achieved the greatest domain coverage. In this way, the meta-predictor for selection among two possible reliability estimates was allowed to predict only CNK-s and LCV (since no two other estimates achieved greater domain coverage); the meta-predictor for selection among three estimates was allowed to predict CNK-s, LCV and SAvar, etc. The testing results exhibited improvement in the performance of many modified metapredictors, compared to the original meta-predictor. The combination of seven reliability estimates (BAGV, CNK-s, BVCK, DENS, LCV, SAbias-s and SAvar) turned out to give the best results: the meta-predicted estimate significantly positively correlated with the prediction error in 60% of tests and significantly negatively correlated with the prediction error in 1% of tests. The testing results confirmed our expectations that the meta-predictor could be improved by restricting the model to predict only estimates which perform well on average. For more details, see (Bosni´c & Kononenko, 2008a). The results of the most improved meta-predictor (which predicts only seven estimates) are shown in Table 3. We can see that using the meta-learning approach we managed to outperform the average results, achieved by nine individual reliability estimates, and even improve them further. The results therefore show the potential of the proposed metalearning approach in the context of estimating the reliability of individual predictions.

6.2. Internal Cross-Validation If we lack relevant problem-specific knowledge, cross-validation methods may be used to empirically select a learner (Schaffer, 1993). When faced with a number of possible learning strategies and having no prior knowledge about the data, a natural idea is to allow the

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data itself to indicate which method will work best. Using the cross-validation approach we therefore divide the data into two parts, use one part as an input to a number of classification algorithms and then choose the algorithm which produces the most accurate model on the second part. In contrast to selecting the learning algorithm, we adapt the general cross-validation approach for selection of the most appropriate reliability estimate for a given domain and model (Bosni´c & Kononenko, 2008a). The adapted internal cross-validation approach divides the available examples to n equally-sized subsets, as in the typical cross-validation approach. Each subset (selection set) is used for performance evaluation (correlation to the prediction error) of all testing reliability estimates. Based on acquired n correlation coefficients for each reliability estimate, the final (most appropriate) reliability estimate is selected as the one with the highest average correlation. This estimate is then used to estimate the reliability of all examples in that particular model and domain. The testing of this approach was also performed by correlating the automatically selected reliability estimate with the prediction error of examples. The summarized testing results, displayed in Table 3, show that the estimates, selected by the internal cross-validation approach, achieved better average performance than any of the individual estimates or the meta-learning approach. We can see that with the internal cross-validation 73% of estimates on average significantly positively correlated with the prediction error, while 0% of estimates significantly negatively correlated with the prediction error. Figure 11 displays performance comparison of the individual estimates and both approaches for the most appropriate estimate selection. The graphs show the percentages of experiments with the significant positive correlations and the percent of experiments with the significant negative correlation. We can see that internal cross-validation performed better than any of the other estimates in 7 out of 8 regression models. For more details, see (Bosni´c & Kononenko, 2008a).

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6.3. Comparison of Meta-Learning and Internal Cross-Validation The comparison in Figure 11 shows that internal cross-validation performed better than any of the other estimates with the majority of used regression models. The meta-learning approach has performed good as well, since its performance was ranked in the top half with all regression models. The achieved results indicate that it is reasonable to handle the problem of automatic estimate selection using the proposed approaches. By comparing how frequently was each of nine reliability estimates selected with the both approaches (illustrated in Figure 12), we can see that the meta-learning approach most frequently selected those estimates, which perform well on average, i.e. BVCK, BAGV and CNK-a. In contrast, the internal cross-validation approach most frequently selected estimates which perform well in specific domains. We explain this phenomenon based on the main characteristics of the both approaches, as follows: 1. In contrast to the meta-learning approach, which is restricted with descriptive power of the meta-attributes, the internal cross-validation operates directly with the available data and tests the candidate estimates on the subset of examples. Since it is likely that the estimate which correlates best with the prediction error on a subset of examples

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Figure 11. Comparison (ranking) of meta-learning (META) and internal cross-validation (ICV) performance to the performance of individual reliability estimates. Graphs show the percentages of experiments with the significant positive correlations (desired) and the percentages of experiments with the significant negative correlations (undesired) to the prediction error.

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Table 3. The performance comparison of the most successful individual estimate BVCK, the meta-predicted optimal estimate (basic and improved meta-learning) and the estimate, selected with the internal cross-validation. The table shows the percentage of experiments exhibiting significant positive/negative correlations between the reliability estimates and the prediction error.

model RT LR NN BAG SVM LWR RF GAM average

BVCK +/− 71/4 50/0 54/0 61/0 46/0 43/0 50/0 54/0 54/1

meta-learning (basic) +/− 79/0 54/0 46/0 61/0 54/0 43/4 64/0 54/0 57/1

meta-learning (improved) +/− 82/0 71/0 61/4 57/0 57/0 43/0 50/0 61/0 60/1

internal cross-validation +/− 87/0 73/0 73/0 67/0 67/0 73/0 60/0 80/0 73/0

will also correlate well on the rest of the examples, this allows the internal crossvalidation approach to select estimates which are more tailored to the domain itself. 2. Aiming to induce a general rule for selection of the most appropriate estimate and avoiding overfitting, the meta learning approach would most often predict estimates which perform best on average.

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7.

Application on a Real Domain

Performance of nine individual reliability estimates (see Section 4.) and the methodology for automatic selection of the best performing estimate were preliminarily applied on a real domain. The data consisted of 1035 breast cancer patients, who had surgical treatment for cancer between 1983 and 1987 in the Clinical Center in Ljubljana, Slovenia. The patients were described using standard prognostic factors for breast cancer recurrence. The goal of the research was to predict the time of possible cancer recurrence after the surgical treatment. The analysis showed that this is a difficult prediction problem, because the possibility for recurrence is continuously present for almost 20 years after the treatment. Furthermore, the data presents a mixture of two prediction problems, which additionally hinders the learning performance: (i) yes/no classification problem, whether the illness will recur at all, and (ii) the regression problem for the prediction of the recurrence time. In our study, we chose the locally weighted regression model for the use with this prediction problem, due to its low RMSE on this domain. After computing all nine reliability estimates, the statistical evaluation showed that only the estimate BVCK significantly positively correlates to the prediction error, which confirmed our expectation that this is a dif-

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0

5

10

15

20

META ICV

BVCK

CNK−a

BAGV

SAbias−s

SAvar

CNK−s

DENS

LCV

SAbias−a

Figure 12. The percentage of automatically selected reliability estimates using the metalearning and internal cross-validation approach (estimates are ordered in the decreasing order with respect to the frequency of selection using the meta-learning approach). ficult prediction and reliability estimation problem. Nevertheless, both, the meta-learning approach and the internal cross-validation approach, selected the estimate BVCK as the most appropriate estimate, which therefore confirms the utility of this methodology in practice. To conclude, our study resulted in complementing the bare recurrence predictions with reliability estimate BVCK, helping the doctors with the additional validation of the predictions’ accuracies. For further details, see (Bosni´c & Kononenko, 2008a).

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8.

Conclusion

The chapter presents the problem of reliability estimation for individual predictions and describes the needs for its application in supervised learning. Reviewing the related work, we divide the approaches into two families: model-dependent and model-independent. Through their implementations in the related work, we show that the key properties which they differ on are: probabilistic interpretability of estimates, usability with different models, and capability to be analytically analyzed. The summary of approaches, which provide motivating ideas for the development of model-independent approaches, related this field to the following research fields: perturbations of examples, the usage of unlabeled data in supervised learning, the sensitivity analysis, the minimum description length principle, active learning, transductive reasoning, and others. Based on the motivations, stemming from these fields, we described five approaches to reliability estimation (local sensitivity analysis, bagging variance, local cross-validation, density estimation, and local error modeling) and defined eight individual prediction reliability estimates. The empirical evaluation of eight estimates using 28 testing domains and 8 regression models, indicated the promising results. Estimates SAbias-s and CNK-s exhibited the most outstanding performance with the regression trees, where these two estimates significantly positively correlated with the signed prediction error in 82% and 86% of tests, respectively. The best average performance was achieved by estimate BAGV, which

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turned out to be the best choice for the usage with neural networks, bagging and locally weighted regression. By combining pairs of estimates, we defined a ninth (combined) estimate BVCK, which on the average achieved better performance than every other individual estimate. Testing revealed that the reliability estimates exhibit different performance in different regression domains and with different regression models. To deal with this problem, we described two approaches for automatic selection of the most appropriate estimate for a given domain and regression model: the meta-learning approach and the internal cross-validation approach. The empirical results of both approaches showed the advantage of dynamically selected reliability estimate for a given domain/model pair when compared to individual reliability estimates in terms of higher positive correlation to the prediction error. The best results were achieved using the internal cross-validation procedure. With this approach, the selected reliability estimates significantly positively correlated with the prediction error in 73% of domains and significantly negatively correlated with the prediction error in none. The testing of the proposed methodology on a medical prognostic domain also showed the potential for its usage in practice. By supplementing the predictor output with prediction reliability estimates, we provided the medical experts with significant improvement of their prognostic system. The achieved results in the field of estimating reliability for individual predictions offer the challenges for further work. Some of the ideas include: • The signed reliability estimates (SAbias-s and CNK-s) with the signed prediction error implies the potential for the usage of reliability estimates for the correction of regression predictions. We shall therefore explore whether these two reliability estimates can be utilized to reduce the error of regression predictions.

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• Different performances of reliability estimates in different testing domains indicate that the potential for estimation of prediction reliability is in some domains more feasible than in the others. The domain characteristics, which lead to a good performance of reliability estimates, shall be analyzed in more detail. • The preliminary experiments showed that the selection of a regression model with low RMSE and the selection of a model on which reliability estimates perform well, is a trade-off criteria. This phenomenon shall be analyzed in further theoretical and empirical work. • It shall be tested, whether the meta-learning estimate can be adapted to select the best performing estimate on an example-basis, thus improving the results further. Note that this introduces another issue: different reliability estimates have to be calibrated in order to allow the comparison of the reliabilities of predictions for different examples in the same problem domain. • A feasibility study and empirical evaluation of the usage of reliability estimates with data streams shall be performed.

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Reviewed by: prof. dr. Vladislav Rajkoviˇc, University of Maribor, Faculty of Organizational Sciences, Kidriˇceva 55a, 4000 Kranj, Slovenia, e-mail: [email protected].

In: Reliability Engineering Advances Editor: Gregory I. Hayworth

ISBN: 978-1-60692-329-0 © 2009 Nova Science Publishers, Inc.

Chapter 5

A FULL PERFORMANCE MANAGEMENT POLICY FOR A GEOGRAPHICALLY DISPERSE SYSTEM AIMED TO CONTINUOUS RELIABILITY IMPROVEMENT Miguel Fernández Temprano1, Oscar Duque Pérez2, Luis Ángel García Escudero1 and Ángel Luis Zorita Lamadrid2 1

2

Dpt. Statistics and O.R. Universidad de Valladolid, Valladolid, Spain Dpt. Electrical Engineering. Universidad de Valladolid, Valladolid, Spain

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Abstract In this chapter we develop procedures designed to help to apply an effective performance management policy to highly dispersed and complex systems. The proposed methodology is focused on the reliability of the system, taking into account that these systems are usually composed by many subsystems that have to work under different environments and are supervised by different maintenance staffs. In consequence, a complete reliability analysis of the system is developed. The components and installations of the whole system are classified into groups resulting in functional group structures and criticality indices are obtained for these structures. The procedure to obtain the components criticality indices is focused on field reliability data, while for the installations indices a multivariate analysis is applied taking into account local characteristics such as strategic importance, technology particularities and local reliability indices. In both cases, the aim has been the development of non-subjective criteria. These analyses are used to help to design the improvement plans, component replacement policies or the necessary investments with the aim of reducing the failures and improving the availability of the system. The methodology includes several specific novelties. One of them is the design of a procedure to test the performance of the system and the effectiveness of the maintenance tasks establishing warning procedures. Another advance is the use of a methodology based on Duane plot graphical techniques that allows modifying some of the current strategies and even speeding up the replacement of components. This chapter has been developed as part of a project named FIMALAC. The purpose of this project is to study the reliability, availability and maintenance of the 3000 V DC overhead contact line of the Spanish national railway network. The procedures developed in this chapter

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M. Fernández Temprano, O. Duque Pérez, L. Á. García Escudero et al. are illustrated with the results obtained in the application of this methodology to this overhead contact line system.

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1. Introduction This chapter is motivated by the study of a highly dispersed and complex system. The procedures developed are designed to help to apply an effective performance management policy aiming to improve the functionality of this kind of system. Examples of systems of this type could be a power electrical system, a telecommunication system, a railway network or any infrastructure system. In other words, we are dealing with systems dispersed along hundreds or thousands of kilometres, usually composed by different subsystems that have to work under different environments, supervised by many maintenance teams and with large populations of relatively uniform and not too sophisticated components. Therefore, there must be significant differences from management tasks in an industrial system where we would deal with a small number of complex machinery in a uniform and stable environment. For instance, as dispersed systems are usually scattered over zones with different environmental conditions and different service requirements, maintenance tasks should be adapted to these specific conditions. We must consider too, that we will deal with systems with increasing functionality requirements and, in many cases in many countries, subject to liberalization processes, which oblige to improve their performance, offering a more efficient service, integrating into the management policy the concepts of reliability (the ability to perform a required function under stated conditions for a stated period of time), maintainability (ability to be maintained preventively and correctively), availability (ability to function in a given moment) and security (ability to operate without causing harm) [1]. Besides, these goals must be achieved reducing the overall costs. In this chapter we intend to demonstrate both the potential and benefits that a “full performance management policy” will bring to geographically dispersed systems, increasing their reliability and availability. This management policy has the aim of facilitating decisionmaking with regard to improving policies and resource sharing, adapting maintenance tasks to be performed not only to reduce costs but also and especially to improve reliability indices and ensure high levels of availability. A proper maintenance of these geographically dispersed systems will reduce the frequency of service interruptions and the many undesirable consequences of such interruptions [2], increasing the quality of service offered to customers While there is an extensive literature on reliability management for classical industrial systems, see [3], [4] or [5], there is not too much for systems as the ones considered here. Among these we can cite [6] and references therein. For these systems, the traditional approach has been basically based on the experience of the maintenance staff, working in a geographically dispersed structure, with different environment conditions, which often leads to implement different methods in changeable working conditions. In this way, the procedures are very dependent on the specific local conditions and on the experience and judgment of technicians and maintenance engineers. Therefore, one of the main objectives of the proposed methodology is to reduce the subjectivity in the results by obtaining them from the analysis of the system's behaviour instead of mainly relying on an “expert” opinion. Consequently, we elaborate a dynamic plan,

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flexible enough to be changed according to the variations or improvements that are introduced in the system, not so dependent on the staff in charge. Moreover, we develop tools to analyze the influence of the operating environment, which is very important in the kind of systems we are interested in, to analyze the reliability trend of the components and to determine the criticality of components and systems sections. One of the key features of the proposed methodology is the definition and computation of indices, obtained from failures database, reflecting the behaviour of the system and the quality of service offered to users. These indices will allow us to undertake the management of the system taking into account its reliability, maintainability and availability. These indices can be used in different ways such as, to compare alternatives of management, to accept or reject an alternative depending on acceptable minimum levels of reliability, to detect reliability weaknesses or to quantify the influence of a variety of sometimes contradictory factors To achieve these objectives it has been developed a methodology of analysis, as follows: Section 2 explains the identification of the system in physical (components) and logical (areas) units. Section 3 describes component reliability analysis, while Section 4 analyzes the performance of the different areas of the system. In Section 5 we use the tools appearing in the preceding sections to describe a full management policy that is applied to a case study corresponding to the overhead contact line system of the Spanish railways. Section 6 concludes the chapter.

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2. System Identification The first step to take when defining a performance management policy for a specific system is to develop a complete analysis of the system allowing us to identify its installations, the elements that compose it, its functional standards and its operational environment (specially important as we are dealing with geographically dispersed systems). Next, we must make a very important decision, the establishment of the basic units of analysis considering that these are systems with large populations of relatively uniform and not too sophisticated elements. When defining a long-term maintenance policy, element clustering will reduce the number of objects into a manageable amount. From literature [7] and infrastructure management staff sources it becomes clear that there is no common methodology for clustering elements. This grouping can be made considering the use of objects, their lifecycle, location and so on. Considering maintenance conditions as well as system reliability and functionality, we propose to make the clustering in two basic levels: one related with the elements that compose the system, and one related with the way the system is operated. In both cases, previous considerations have to be made before defining the basic units of analysis: • • •

The most important consideration when defining these basic units must be their functional aspect. The goal is not to return all the components to an “as new” condition, but to deliver the best possible service to customers, reducing the overall cost. We must take into account the level of detail provided, especially in failure reports, since if data are not precise enough, what is the point of wanting to go beyond?

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To take into account geographical and environmental particularities, it is necessary to divide the system into homogenous units.

Then, to begin the analysis of such a system, we consider the following units: • Physical unit: the component, according to the function performed and to the typology of elements established in each system. For example, all the isolators in a power system or the tracks in a railway transport system are considered as single components. We will use index “i” to refer to this type of units. • Logical unit: the area, considering the different zones into which the system is divided from maintenance point of view, that is, the zone for which each maintenance team is responsible. These units could be subdivided into smaller logical units when studying environmental conditions that might vary within a maintenance team zone, or grouped in bigger zones for other management considerations. Examples of these situations will appear in the case study developed in Section 5. We will use index “j” when referring to these units. When comparing performance and maintenance conditions of different areas, to put them on an equal foot, it could be convenient to consider its size. The unit to measure the size will depend on the analyzed system. For example, the area size can be measured in kilometres or in number of installed components according to the system considered.

3. Component Reliability Analysis

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3.1. Failure Distribution Determination In order to develop the statistical reliability model for each component of the system, we need to determine the failure distribution. We will obviously choose a distribution that, being as simple as possible, fits reasonably well the component failure data. Once the underlying model is fixed, we will use it for the estimation of the parameter of interest such as, for example, the component failure rate. As the complexity of the geographically dispersed system we are considering is quite high, the assumption of a constant failure rate for the components would simplify a lot the task. On a first thought, this hypothesis does not seem to be very reasonable as some components of the system will be usually wearing away. However, we also have to take into account that in our system there are also maintenance tasks that can compensate that wearing so that, in many cases, the failure rate of many components can be considered as constant (at least in a not too long period of time). Notice that we are considering each of the components of the system as a repairable system [8]. This constant failure rate hypothesis will be checked using Duane Plots [9]. These plots are a very useful tool when studying repairable systems. They will allow us to see whether the failure rate can be assumed to be constant or if there is an improvement or worsening on the studied component. We will show an example of its practical usefulness when we develop our case study in Section 5.

A Full Performance Management Policy…

145

Duane plots are obtained by representing the following points:

{(log (ti ,l ) , log( N (ti , l ) / ti , l ))}lL=i 1

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where ti,l are the failure times of component i, N(ti,l) the number of cumulative failures of component i until time ti,l and Li the total number of failure of component i in the period considered. If these points are aligned, the data will fit well a power law model. In this case the failure rate is proportional to tβ, where β is the slope of the line that best fits the points in the plot. Therefore, a slope close to 0 will lead to a constant failure rate (the situation known as homogeneous Poisson process), while β>0, an increasing line, tells us that the component performance is deteriorating (the mean time between failures is shortening). Obviously, β 0 as shown in Figure 1 in the transformed U-space. In order to get the probability of failure for a given problem by means of Eq. (1), the joint density function f X (x) must be constructed but this is often very difficult

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in practice because of a lack of statistical data. Moreover, for problems whose statistical data is sufficient to form a joint distribution function, it may be almost impossible to perform the multidimensional integration over the generally irregular domain D [2], although some progress is being made in this respect [3][4]. For these reasons, many methods have been developed to find reliability data such as the probability of failure of a component, its reliability index and so on, which are based on numerical analysis [2] as well as simulation such as Monte Carlo Simulation (MCS) [5], or more advanced simulation methods [6][7].

Figure 1. Regularly shaped design CCD (a), and space filing design (b).

For some problems, including implicit performance functions or time consuming analyses, methods such as the First Order Reliability Method (FORM) or the Second Order Reliability Method (SORM) and MCS become very time consuming or in some cases may suffer convergence problems [8]. Therefore, approximation methods based on metamodeling

Meta-modelling Based Approximation Methods for Structural Reliability Analysis 207 techniques become inevitable for such problems to reduce the computing costs as well as to find the correct result for problems with convergence issues.

2. Overview of the Metamodeling Techniques Approximation methods such as design of experiments combined with metamodels are commonly used in engineering design to minimize the computational expense of structural reliability assessment. The basic approach is to form a simplified mathematical approximation of the computationally expensive analysis code, which is then used in place of the original code to facilitate structural reliability analysis. Since the approximation model is used as a surrogate for the original code, it is often referred to as a surrogate model, approximation model, or metamodel which is a “model of a model” [9]. A variety of approximation models exist such as polynomial response surfaces, kriging models, radial basis functions, neural networks, multivariate adaptive regression splines [10]. In these metamodeling techniques, a functional relationship between design variables (input) x and responses (outputs) y is constructed. If true response is y = f (x)

(2)

Then a metamodel to approximate the true function y is defined as yˆ = g (x)

(3)

And the relationship between the true response and metamodel is stated as y = yˆ + ε

(4)

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Where ε denotes the vector of all errors, i.e. the observation/measurement errors as well as approximation errors. Constructing a metamodel involves the following main steps: • • •

Choosing the experimental points (design of experiment) Choosing the type of model function (model choice) Evaluation of the modeling error (model validation)

Although there are several options for each of these steps [11], only the most widely used methods are given in detail as outlined in the following sections.

2.1. Experimental Design An experimental design or mostly called Design of Experiment (DoE) is a structured, organized method that is used to determine the relationship between the factors (inputs) set at specified levels (values each factor takes and the output of that process. A classification of

208

Irfan Kaymaz

DoE can be given according to whether experimental points are selected in a regular pattern such as a rectangular shape or in an irregular pattern by spreading the sample points out in the design space, where the former is called a regularly shaped design space, and the later is a space filling design. In a regularly shaped space, the allowable range of the variable is arranged as equally-spaced levels as is shown in Figure 2(a). By contrast, the experimental points in a space filing design are spread out to fill the design space as illustrated in Figure 2(b). For a regularly shaped design space, the most widely used DoE approaches are factorial designs, Central Composite Design (CCD) and the Box-Behnken method [12]. For an irregularly shaped design space, a computer-generated design is used, of which the most popular designs are D-optimal design and Latin Hypercube Design.

Figure 2. Regularly shaped design CCD (a), and space filing design (b).

2.1.1. Regularly Shaped Designs The most basic DoE for regularly shaped designs is a full factorial design such as 2 k and 3 k

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designs where k denotes the number of the factors; i.e. design variables [12]. 2 k design is used to evaluate main effects and interactions whereas 3 k design is formed to evaluate main and quadratic effects and interactions. These factorial designs generates q sample values for each variable, thus the total number of the experiments is p = q k . For two independent variables 2 k and 3 k factorial designs are depicted in Figure 3(a) and (b) in terms of coded variables ξ which are usually defined to be dimensionless with zero mean and the same spread or standard deviation. A 2 k factorial design is often used to fit a first order metamodel while 3 k is sufficient to fit a second-order model.

Meta-modelling Based Approximation Methods for Structural Reliability Analysis 209

3

Figure 3. (a) 2 design, (b) 33 design and (c) CCD.

It should be noted that the number of the experiments dramatically increases with the number of the independent variable k. For instance, for k = 7 , two factorial design requires 128 design where for three factorial design this number becomes 2187. Therefore, a CCD is utilized to fit a second order model, which consists of 2 k factorial design augmented with 2k axial and kc central points as depicted in Figure 3(c) for k = 2 . Thus CCD generates

p = 2 k + 2k + kc experiments. The axial points are specified by α which shows a distance from the centre point and its value is generally chosen as k . For different DoE, Table 1 shows the number of the experimental points as well as the number of the terms to be determined in both linear and quadratic model with interactions.

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Table 1. The numbers of experimental points for different DoE k

Linear model

Quadratic model

1 2 3 4 5

2 3 4 5 6

3 6 10 15 21

2k 2 4 8 16 32

3k 3 9 27 81 243

CCD 5 9 15 25 43

As seen from Table 1, when the number of the terms to be determined in the metamodel is considered, the factorial designs become quite unattractive. Although having a DoE in which the number of the experiments is more than the number of the terms to be determined helps validate the metamodel, a moderate number of the experiments can be achieved using optimal designs [13], which are outlined in the following section.

2.1.2. Computer-Generated Optimal Designs The DoE method discussed so far, such as the factorial design, the CCD, are widely used because they are quite general and flexible designs. If the experimental region from which the experimental points are to be generated is either a cube or sphere , these DoE methods will be applicable to the problem. However, for the reason listed below by Montgomery [12] they

210

Irfan Kaymaz

may not be the obvious choice, thus computer-generated designs can be an alternative to obtain the experimental points. An irregular experimental region: if the region from which the experimental points to be generated is not a cube or a sphere, standard designs may not be the best choice. Irregular regions of interest occurs for problems where there are some constraints on the design variables. A nonstandard model: usually linear or quadratic model are selected as a metamodel. On the other hand, if there is some knowledge on the process to be modelled, a nonstandard model can be chosen to fit the response of the process. Unusual sample size requirements: for the problems in which the runs are extremely expansive or time-consuming, the classical DoE such as CCD may not be suitable in terms of the computation-time. Therefore, computer-generated design can be used in this regard to reduce the number of the experiments. The computer-generated design is usually called as optimal design that means a design that is best with respect to some criterion. In order to generate such a design, computer programs are required. Among these designs, D-optimality criterion is the most widely used one and a design is said to be D-optimal if ( X ′X ) −1

(5)

is minimized, which means that a D-optimal design minimizes the volume of the joint confidence region on the vector of regression coefficients.

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2.1.3. Experimental Designs for Computer Experiments As most DoE methods are developed for physical experiments, they include some features that are questionable to use in a computer-based design, which would normally be the case in a structural reliability analysis, owing to the absence of random errors in the computer model. Unlike physical experiments, running the code with the same inputs will always produce the same output in computer-based experiments. Therefore, the use of repeated experimental points becomes meaningless for computer-based experiments. For instance, the central runs in any DoE method become meaningless because the same response is obtained in each run for the same experimental points. Thus, computer experiments should have the following two principals [14]: 1. Designs should not take more than one observation at any set experimental points. 2. Since the true relation between the response and inputs are not known, designs should provide information about all portions of the experimental region. Therefore a design that places points evenly over the region of interest is desirable, and such design is called as space filling designs. There are several methods that can be used to fill a given experimental region, which can be classified mainly into two groups, i.e. design based on sampling methods and design based on measure of distance.

Meta-modelling Based Approximation Methods for Structural Reliability Analysis 211 2.1.3.1. Designs Based on Sampling Methods In order to spread experimental points evenly in the experimental region using a sampling strategy, there are several methods such as simple random sampling, stratified sampling and Latin hypercube sampling. Among these methods, designs based on Latin hypercube sampling are the most widely used in the computer experiment literature [33, 16, 17]. thus it will be explained in detail in the following section. 2.1.3.2. Latin Hypercube Design (LHD) Based on Sampling Latin hypercube Sampling (LHS) was initially developed by McKay [18], and has been further developed for different purpose by several researches [19, 20]. LHS can be formed utilizing the following steps for two input variables [14]: Let the experimental region be a unit square [0,1]2 . In order to obtain a design with n ⎡ 0,1 ⎞

⎡n −1 ⎤

points, each axis [0,1] is divided into n equally spaced intervals such as ⎢ ⎟,…, ⎢ ,1⎥ . ⎣n ⎠ ⎣ n ⎦ Thus it forms n 2 cells of equal size. Then, these cells are filled with integers 1,2, … , n to form a Latin square in which each integer only appears once in each row and in each column. One of the integer is selected at random. In each of the n cells containing this integer, a point is selected at random, and thus a LHD of size n is formed. For k input variables, a LHD is constructed as follows [21]: 1. the range of each variable X i , i = 1,…, k , is divided into n intervals, each of which

1 . n 2. for the ith interval, the sample cumulative probability is obtained as: has equal probability with

Probi =

1 i −1 ru + n n

where ru is a uniform random number ranging from 0 to1. 3. The probability is transformed into the sample value xi using the inverse of the distribution function as: xi = F −1 ( Probi ) Copyright © 2008. Nova Science Publishers, Incorporated. All rights reserved.

4. The n values obtained for each variable xi are paired randomly to form a LHD of size p. An example of LHD for two variables with standard normal distribution is illustrated in Figure 4. The cumulative distribution of both X 1 and X 2 is divided into five equal interval with the probability of 0.2. Within each interval, a random sample is taken, which is represented by black dot in the figure. Five samples from each variable are then paired in a random manner to form a Latin square as depicted in Figure 4.

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Irfan Kaymaz

Figure 4. Example of LHD for two variables with normal distribution [21].

2.1.3.3. Designs Based on Measure of Distance Another type of space-filling designs, which is widely used for computer experiments, is based on a measure that quantifies how a set of experimental points is spread out into experimental region. One measure can be stated such that no two points in the design to be too close together, one of which is the maximim distance criterion. If the distance between two points x1 and x2 is measured by

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k

d ( x1 , x2 ) = x1 − x2 =

∑ (x

1, i

− x2 , i ) 2

(6)

i =1

then a maximim design attempts to maximize the shortest distance between any two points of a design D: max min d ( x1 , x2 ) (7) x1 , x2∈D

in which the outer maximization is over all possible designs. This criteria can be combined with LHD by performing the maximization over a set of LHD designs:

Meta-modelling Based Approximation Methods for Structural Reliability Analysis 213 max

D∈LHD ( p ,k ,n )

min d ( x1 , x2 )

x1 , x2 ,∈D

(8)

where LHD ( p, k , n) is a set of p randomly generated LHD designs for k variables with n points for each variable. Matlab’s Statistics Toolbox [22] provides two functions to generate experimental points for LHD; i.e. lhsdesign and lhsnorm. The former utilizes maximin criterion to generate the experimental points while the latter generates the experimental points with a normal distribution. These two LHDs are generated for two variables with n = 10 samples as shown in Figure 5.

Figure 5. Example of LHD for 2 variables (a) LHD with sampling, (b) LHD with measure of distance.

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2.2. Model Choice and Model Fitting After having the experimental points utilizing one of the DoE and the responses to the corresponding to the experimental points, the next step is to choose the type of metamodel and fitting method. Several methods exist to obtain the metamodel, such as response surface method, neural networks, inductive learning, kriging etc. However, among them, two most widely used metamodeling methods; i.e. response surface and kriging are explained in detail in this chapter.

2.2.1. The Response Surface Method The Response Surface Method (RSM) can be described as a collection of statistical tools and methods for forming and exploring an approximate functional relationship between a response variable and a set of design variables [23]. The most widely used form of the functional relationship is a low order polynomial that is referred to as a Response Surface Function (RSF).

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Irfan Kaymaz

In structural analysis, the RSM are used to approximate (estimate) the complex relationship between the performance of a structure and the variables that affect the performance. Hence, for many applications, the problem is to estimate a response (output) function or mechanism: g = g (ξ )

(9)

with the input k-vector ξ = (ξ1 ,…, ξ k )′ , and an unknown (to be estimated) function g . For the estimation of the unknown response function  g , observations or estimates η (i ) ≈ g (ξ i ) of the g = g (ξ ) ,

response

ξ (1) , ξ ( 2) ,…, ξ ( p ) are

i = 1,…, p corresponding to p input k-vectors

available. The unknown function g is estimated by approximating g by a polynomial of a certain (low) order s. Hence, if s = 1 , i.e., if a linear approximation is used, then g (ξ ) ≈ b0 + b1ξ1 + … + bk ξ k

with

k +1

(10)

b j , j = 0,1,2,…, k . Consequently, the observations

unknown coefficients

η (i ) , i = 1,…, p may be represented by

η (i ) = b0 + b1ξ1(i ) + … + bk ξ k(i ) + ε i Here, ε i , i = 1,…, p , are error terms including observation/measurement errors as well as approximation errors. The above p equations are then represented by the matrix equation

η = Xb + ε

(11)

η := (η (1) ,…,η ( p ) )T

(12)

where

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is the p-vector of all observations of g at the p input vectors ξ (i ) , i = 1,…, p b := (b0 , b1 ,

, bk )T

(13)

is the (k+1)-vector of the unknown coefficients, X is the p × (1 + k ) matrix with the p rows X = (1, ξ (i ) ,…, ξ (i ) ) , i = 1,…, p which is called design matrix, and ε = (ε ,…, ε )′ denotes the i

1

k

1

p

vector of all errors, i.e. the observational or measurement errors as well as the (analytical) errors from the approximation of g by a (first order) polynomial. Having no more information, the vector b of unknown coefficients is determined by LSQ-techniques [24, 25], i.e. the estimate bˆ of b is defined by minimizing the function

Meta-modelling Based Approximation Methods for Structural Reliability Analysis 215 L(b) := η - Xb

2

(14)

Under corresponding rank conditions, the estimator of b is then given by −1 bˆ = X T X X T η

(

)

(15)

For a non-linear RSF, however, Eq.(15) cannot be used to get the coefficients of the RSF. Instead, one of the non-linear regression methods that are explained in detail in the literature [26] should be used.

2.2.2. Kriging Method The computer analysis codes are deterministic and thus not subjected to measurement error since we get the same output (the response matrix) for the same input (experimental points). Hence, the usual measures of uncertainty derived from least-squares residuals have no statistical meaning, and some statisticians [11, 27] have argued to use it for deterministic analysis. Consequently, the following model is suggested by Sacks et al. [28] to model the deterministic computer response Y(x) as a realization of a stochastic process Y: Y(x) = f T (x)β + Z(x)

(16)

With f(x) = [ f1 (x),…, f m (x)]T and β = [β1 ,…, β m ]T where m denotes the number of the basis function in regression model, Y(x) is the unknown function of interest, f(x) a known function of x, β is the regression coefficient vector, and Z(x) is assumed to be a Gaussian stationary process with zero mean and covariance: Cov(x i , x j ) = σ 2 R(x i , x j ) i, j = 1,…, p

(17)

where p denotes the number of the experiments, R(.,.) is a correlation function and σ 2 is the process variance. The term f T (x)β in Eq.(16) indicates a global model of the design space,

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which is similar to the polynomial model in the response surface method. The second part in Eq.(16) is used to model the deviation from f T (x)β so that the whole model interpolates the experimental points generated according to one of the DoE approach. The construction of a kriging model can be explained as follows: For a certain experimental design such as LHD, a set of experimental points are generated as: x = {x1 , x 2 ,…, x p } xis ∈ ℜ k (18) where k is the number of the design variables. The resulting outputs from the function to be modeled are given as:

{

}

Y = y1 (x), y 2 (x),…, y p (x)

(19)

216

Irfan Kaymaz From these outputs the unknown parameters β and σ 2 can be estimated:

(

βˆ = F T R −1F) −1 F T R −1Y

σˆ 2 =

(

)

(

)

(20)

)

T 1 Y − Fβˆ R −1 Y − Fβˆ n

(21)

where F is a vector including the value of f(x) evaluated at each of the experimental points and R is the correlation matrix which is composed of the correlation function evaluated at each possible combination of the experimental points: ⎡ R(x1 , x 2 ) … R (x1 , x p ) ⎤ ⎢ ⎥ R=⎢ ⎥ ⎢ R(x p , x1 ) … R(x p , x p )⎥ ⎣ ⎦

(22)

However, before calculating βˆ and σˆ 2 , first the unknown parameters of the correlation function have to be estimated. Using maximum likelihood estimation, they result from the minimization of [27]: 1 / 2 n ln σˆ 2 + ln det R (23)

(

)

which is a function of only the correlation parameters and the response data. For the estimation of these parameters, the best linear unbiased prediction of the response is: yˆ (x) = f T (x)β + r T (x)αˆ

(24)

(

(25)

with the column αˆ defined by:

αˆ = R −1 Y − Fβˆ

)

where r T (x) is a vector representing the correlation between an unknown set of point x and

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all known experimental points:

{

}

r T (x) = R(x, x1 ),…, R(x, x p )

(26)

The second part r T (x)αˆ of Eq.(24) is in fact an interpolation of the residuals of the regression model f T (x)β . Thus, all response data will be exactly predicted. 2.2.2.1. Correlation Functions The stochastic process part of the model given in Eq.(16) includes a correlation function which affects the smoothness of the model [29]. In all literature concerning the method of Sacks et al. [7], a correlation function of the type R(x i , x j ) = R(x i − x j ) is generally selected, and a product correlation rule is used for mathematical convenience:

Meta-modelling Based Approximation Methods for Structural Reliability Analysis 217 p

R(x i , x j ) =

∏ R(x

s i

− x sj )

(27)

s =1

where x is and x sj denotes sth component of the experiments. The correlation function R (x i , x j ) is specified by the user, and several correlation functions exist in the literature.

Among them, however, the following function is mostly used: ⎡ R (x i , x j ) = exp ⎢− ⎢⎣

p

∑

2⎤

θ k x is − x sj ⎥

k =1

⎥⎦

(28)

which permits control of both the range of influence and the smoothness of the approximation function.

2.3. Model Validation The metamodels described in the previous sections of this chapter is only an approximation of the functional relationship between the structural response and the design variables. Thus, there is always some lack of fit between the actual value of the structural response and the predicted response from the metamodel. It is important to determine that the meta model constructed sufficiently represents the true response of the structural system. Therefore, several statistical criteria proposed in the literature are used in the evaluation of the prediction capabilities of the metamodels, which is based on the following measure:

ei = yˆ i − yi

(29)

where yˆ i and yi are the ith response value from the metamodel and true function,

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respectively. In the following, a short review of the most common statistical criteria is given for both regression and interpolating type metamodel formed using the kriging method.

2.3.1. Model Validation for a Regression Type Metamodel The common modeling error assessment utilized to assess the goodness of a regression type metamodel is the Coefficient of Determination (R2) that represents the proportion of the variability in the response data that is accounted for by the response surface approximation. After carrying out regression analysis to obtain the metamodel, R2 is computed as follows [30]: R2 =

SS R SS = 1− E S yy S yy

(30)

in which SSR denotes the regression sum of squares, SSE the error or residual sum of squares, and Syy denotes the total sum of squares, for which more explicit forms of the expression are as follows:

218

Irfan Kaymaz ⎛ ⎜ ⎜ SS R = b T XT y − ⎝

⎞ yi ⎟ ⎟ i =1 ⎠ p p

∑

2

(31)

SS E = y T y − b T X T y

S yy

⎛ ⎜ ⎜ = yT y − ⎝

(32)

⎞

p

∑ y ⎟⎟ i

i =1

⎠

(33)

p

in which p is the number of experiments, b is the vector for the coefficients in the RSF, X is the design matrix including the experimental points, and y is the vector showing the results obtained from true function for the experimental points in the design matrix. The value of R2 is between 0 and 1, and a large value of R2 may indicate that the selected model can result in a good approximation to the function. However, this measurement is not independent on the number of terms in the RSF because adding a variable to the model will always increase R2, regardless of whether it is significant or not. In order to overcome this deficiency related to the measurement, the adjusted R2 approach can be used whose definition is stated below: ⎛ p −1 ⎞ SS /( p − l ) ⎟⎟ 1 − R 2 AdjR 2 = 1 − E (34) = 1 − ⎜⎜ S yy /( p − 1) p l − ⎠ ⎝

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(

)

where l shows the number of parameters in the RSF. AdjR2 indicates the goodness of the fit of the selected model to the performance function. A value of AdjR2 = 1 means that the RSF passes through every experimental point in the design matrix while a value of AdjR2 = 0 means that the RSF does not represent the experimental points any better than a horizontal line. Therefore, an RSF having a higher value of AdjR2 means that the RSF is a good choice for the experimental point used to form that RSF. For practical purposes it is generally chosen very close to 1. Although the R2 and AdjR2 explain how well the metamodel fits to the experimental points, it does not, however, shows the prediction potential of the metamodel to other points that are not used to generate model. In order to verify the overall accuracy of the metamodel, statistical tests at additional experimental points in the design space are usually performed in the literature, which include the Average Absolute Error (%AvgErr), the Maximum Absolute Error (%MaxErr), and the Root Mean Square Error (%RMSE) [31]. These measures are defined as follow:

% AvgErr = 100

1 N

N

∑ y − yˆ i

i

i =1

1 N

N

∑y

i

i =1

(35)

Meta-modelling Based Approximation Methods for Structural Reliability Analysis 219 ⎡ ⎤ ⎢ ⎥ yi − yˆ i ⎥ ⎢ % MaxErr = maxi ⎢100 ⎥ N 1 ⎢ yj ⎥ ⎢ N j =1 ⎥ ⎣ ⎦

(36)

∑

% RMSE = 100

1 N

N

∑ ( y − yˆ ) i

i

2

i =1

1 N

(37)

N

∑y

i

i =1

where N is the number of additional validation points. While %RMSE provides good estimates of the “global” error over the region of interest, %MaxErr gives a good estimate of the “local” error by measuring the worst error within the region of interest, where a good approximation will have low %RMSE and low %MaxErr values. [32].

2.3.2. Model Validation for Interpolating Type Metamodels The relationship between the true response and metamodel given in Eq.(4) can be expanded as y = g (x) + ε bias + ε random (38) where

ε bias indicates

the

error

of

approximation,

and

ε random represents

observation/measurement error. In regression analysis, ε random is assumed to have identical and independent normal distribution with mean zeros and standard deviation of σ ; i.e.

ε randomi.i.d .N (0,σ 2 ) [11]. However, for deterministic computer analyses ε random has mean zero and variance zero since the computer experiments generates the same output for the same input. Thus the relationship in Eq.(38) is written as

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y = g (x) + ε bias

(39)

Since most standard tests for model and parameter significance for regression type metamodels are based on computations of ε random , these model validation tests are questionable to use for the metamodel generated using the computer experiments. It is also supported by literature in the statistics community; as Sacks et al. [14,13] pointed out, because deterministic computer experiments lack random error: • • •

response surface model adequacy is determined solely by systematic bias, the usual measures of uncertainty derived from least-squares residuals have no obvious statistical meaning, the classical notions of experimental blocking, replication and randomization are irrelevant.

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Irfan Kaymaz

Therefore model validation for interpolating type metamodels are usually carried out using additional test samples with RMSE measure. However, this approach requires a large number of computationally expensive computations to obtain the additional structural responses that are not used to fit the metamodel. As a more computationally efficient alternative, cross-validation (CV) is used as to determine the accuracy for predicting the original model [29]. In CV, an observation i from p experiment used to fit the metamodel is selected, and a new metamodel is obtained from the remaining p − i design. This metamodel is used to predict the withheld observation y i and the predicted value is denoted as yˆ i . Thus, the prediction error for point i is computed as e(i ) = yi − yˆ i which is called the i th PRESS residual. This procedure is repeated for each response, thus producing a set of p PRESS residuals. Then PRESS statistics is computed using p

PRESS =

∑

e(2i ) =

i =1

p

∑ [y

i

− yˆ i ]2

(40)

i =1

2

PRESS can be used to compute an approximate R for the prediction capability of the metamodel as PRESS (41) R 2prediction = 1 − S yy 2

However, R prediction statistic is not independent from the number of terms in the 2

metamodel because adding a variable to the model will always increase R prediction , regardless 2

of whether it is significant or not. In order to overcome this deficiency the R prediction can be adjusted as

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AdjR 2prediction = 1 −

p −1 (1 − R 2prediction ) p−l

(42)

where p is the number of the observations and l is the number of the term in the metamodel. For a kriging model l = number of β + number of θ + 1 [33].

3. Special Metamodeling Techniques for Structural Reliability Analysis Metamodeling techniques developed specifically for reliability problems aims to fit a metamodel around a region that might include the design point shown in Figure 1 since it contributes most significantly to the probability of failure. Therefore, the metamodeling techniques described in the previous sections in this chapter have been modified to obtain a metamodel that is formed in a sub space rather than taking into account the whole design

Meta-modelling Based Approximation Methods for Structural Reliability Analysis 221 space. Thus, this metamodel can be used in place of the actual performance function for the calculation of the probability of failure. Bucher [34] first applied a modified version of the traditional RSM to reliability problems, which is explained in detail in the following section.

3.1. Bucher’s Response Surface Method The aim of Bucher's RSM is to replace a performance function g ( x) by an equivalent response surface function g (x) which is mostly given as a second order polynomial as shown in Eq.(43) below: k

g (x) = a +

∑ i =1

k

bi xi +

∑c x

2 i i

(43)

i =1

in which Xi are the basic variables, i =1,2,...,k shows the number of the basic variables, and the parameters a,bi, ci are to be determined. As the number of these free parameters is 2k+1, only a few calculations are needed to obtain the RSF. In Bucher’s RSM, the suggested way of obtaining these parameters is interpolation using the points along the axes xi which are chosen to be of the form: xi = x i ± f i σ i (44) in which xi and σi are the mean value and standard deviation of Xi, respectively, and fi is an

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arbitrary factor, generally chosen as 3. The interpolation points are selected as indicated in Figure 6.

Figure 6. The first (a) and second (b) stage of Bucher's RSM.

The values of these points are then substituted into the performance function to get the response of g (x) , which forms the response matrix. After g (x) is formed, this RSF is used along with the information xi and σi to obtain an estimate of the design point x D . One of the key steps is to find the design point because the next iteration will be based on this point

222

Irfan Kaymaz

which significantly affects the accuracy of the second and final RSF. After x D1 is found the new centre point xM for interpolation is obtained on a straight line from x to x D1 as shown in Figure 6(b), and an explicit formulation of xM is given as x M = x + (x D1 − x)

g ( x) g ( x) − g (x D1 )

(45)

Once xM is found, the new interpolation points are established according to the new centre point xM to find a new RSF which is supposed to be closer to the limit state. Although this method of updating the polynomial RSF is assumed to result in a sufficiently accurate approximation to the limit state function after two iterations, as Rajashekhar and Ellingwood [35] pointed out, the accuracy of the reliability results from application of Bucher’s RSM depends on the characteristics of the limit state. Therefore, one cycle of updating may not be sufficient. Thus, a criterion to carry out further updating was proposed by the authors, based on an observation of the distance between the design point and the centre points obtained in Eq.(45). The iteration process is repeated until the distance is acceptably small. In their improved RSM, the authors also studied the effect of the selection of the experimental points from one side of the distribution function of random variables in order to choose the experimental points to be as close to the region of interest as possible.

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Another criterion on whether mixed terms such as

x1 x2 should be added or left out from the

RSF was also proposed, in which the necessity of the terms is determined by comparing the result of the performance function at the centre point and the experimental point. The selection of the experimental points is very important in terms of having a RSF close the limit state since it defines the failure state or safe state of the component. Therefore, an improved sequential RSM which is mostly based on the selection of experimental points close to the original failure surface was proposed by Kim and Na [36]. In this method, a linear RSF is used to get the design point and the reliability index rather than the commonly used quadratic RSF, by projecting the experimental points on the response surface obtained in the preceding iteration. Further research was carried out by Das and Zheng [37], in which the RSF is formed in a cumulative manner in order to account for second order effects. In basic response surface methods normal regression is commonly used which gives equal weight to the coefficients of the RSF formed. However, the main effort in the application of the RSM is to form a RSF as close as possible to the design point. Therefore, Kaymaz and McMahon [38] replaced normal regression with a weighted regression method in which the RSF is formed by giving higher weight to the points closer to the limit state, as explained in the following section.

3.2. Weighted Regression The main aim in the formation of the RSM is to fit a RSF as closely as possible to the limit state function. In the standard RSM, see above, the coefficients of the RSF using least square method are given by

Meta-modelling Based Approximation Methods for Structural Reliability Analysis 223

(

)

−1

bˆ = X T X

X Tη

(46)

Here, X denotes the design matrix comprising the experimental points, and η represents the response vector obtained from the performance function corresponding to the experimental points. In this method the estimation errors are equally weighted. However, a good RSF must be formed such that it describes the performance function well, especially close to the limit state surface given by g (x) = 0

(47)

Therefore, the weighted regression method [30] is utilized to find the coefficients of the RSF, for which the weights are usually determined by allowing the uncorrelated residuals to have different variance, unlike the normal regression, for the error term ε as ⎡σ 12 ⎢ V [ε ] = ⎢ ⎢0 ⎣

0⎤ ⎥ ⎥ σ k2 ⎥⎦

(48)

The above equation indicates that the error terms independently follow probabilistic distributions of different variance. Thus, a weight wi to each observation is assigned so that w1σ 12 =

= wk σ k2 = 1σ 02 where σ 02 is termed the standard deviation of unit weight. The

estimate bˆ of b is defined by minimizing the function L(b) := (η - Xb )W (η - Xb )

(49)

Under corresponding rank conditions, the estimator of b is then given by

(

bˆ = X T WX

)

−1

X T Wη

(50)

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where W is an p × p diagonal matrix of weights as: ⎡ w1 ⎢0 W =⎢ ⎢ ⎢ ⎢⎣ 0

0 w2 0

0 ⎤ 0 ⎥⎥ ⎥ ⎥ w p ⎥⎦

(51)

In general the weights are assigned to observations so that the weight of an observation is proportional to the inverse expected (prior) variance of that observation, wi ∝ 1 / σ i2, prior . However, in this study, a different approach is proposed to select the weights for the

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Irfan Kaymaz

observations since we are seeking to find a RSF close to the limit state function where g (x) = 0 , which is achieved as follows: Among the responses of the performance function corresponding to the design matrix the best design is selected based on closeness to a zero value, which indicates that the experimental point is close to the limit state: yˆ best = min g (x) x∈X

(52)

where min indicates the minimum response value of the performance function evolutions obtained according to the design of the experiments, thus yˆ best indicates the value of the absolute minimum performance function response. The following expression is found to be suitable to obtain the weight for each experiment: wi =

where

⎛ η ( i ) − yˆbest ⎜− ⎜ yˆbest e⎝

⎞ ⎟ ⎟ ⎠

(53)

η (i ) indicates the ith response from the ith experiment designed according to the design

of experiment selected, where i corresponds to the number of the experiment. The obtained weights are used in the weighted regression to estimate bˆ of b as:

(

bˆ = X T WX

)

−1

X T Wη

(54)

where W is a diagonal matrix as given in Eq.(51). Thus, the RSF to be formed from the weighted regression will have coefficients with greater weights for the points closer to the limit state, thus leads to a better estimate for the probability of failure as shown in the examples.

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3.3. The Kriging Method for Reliability Problems Although the kriging method has gained popularity especially in the field of deterministic optimization, the application of the kriging method to reliability problems has not been realized until recently by Romero et al. [39], in which they compared several data fitting and interpolation techniques including the kriging method based on Progressive Lattice Sampling. More recently, Kaymaz [40] outlined how the kriging method can be used for reliability problems, in which a kriging Toolbox called DACE: A MATLAB Kriging Toolbox [41], which is developed for deterministic problems, was modified to use with well-known structural reliability methods such as FORM/SORM or Monte Carlo Simulation. The approach proposed is schematically shown in Figure 7. As seen from Figure 7, the approach can be divided mainly into two stages as in Bucher’s RSM. The following explanation can be extended for the second stage as well. First a design of experiment method can be selected to form the necessary experimental points. Instead of choosing the RSF type for the first stage, the parameters of the kriging method such as the

Meta-modelling Based Approximation Methods for Structural Reliability Analysis 225

θ initial and correlation function are selected, and the kriging model is formed from the DACE toolbox. Then, one of the structural reliability methods such as FORM/SORM or MCS can be chosen to find the design point corresponding to the reliability index. Based on the design point found in the first stage, a new center point xM in Eq.(45) is calculated. The

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experimental points for the second stage are formed based on the new center point, applying one of the design of experiment methods. The same procedure explained above is followed for the second stage.

Figure 7. Flowchart of the kriging method modified for the use of reliability problems.

As pointed out in the literature [40], on the condition that the parameters of the kriging method are adjusted properly, the kriging method gives better approach than that of Bucher’s RSM. The other main advantage of the kriging method for reliability analysis is that since there is no need to specify the type of the RSF, the kriging method is independent from the type of the RSF to be fitted. On the other hand, it should be noted that the implementation of the kriging method is much more difficult than that of Bucher’s RSM.

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Irfan Kaymaz

3.4. Application of the Metamodeling Techniques to Reliability Problems 3.4.1. Example 1:Cantilever Beam As a first example, a cantilever beam with a rectangular cross section subjected to uniformly distributed loading is considered [35], for which the performance function is given as g ( x) = 18.46154 − 7.47692 × 1010

x1 x23

where X 1 denotes the load and X 2 is the depth of the beam, and they are assumed to be random variables whose values are given in Table 2. Table 2. Statistical properties of the random variables in Example 1 Random Variables X1

Distribution Normal

Mean Value 0.001

Standard deviation 0.0002

X2

Normal

250

37.5

The results obtained using adaptive Monte Carlo simulation with 10000 samples given in Table 3 is considered as exact or reference solution of the problem when comparing to the results obtained from the metamodeling techniques given in Table 3. Four metamodeling techniques were used to approximate the performance function, two of which were based on regression type metamodel; i.e. Bucher’s RSM and the weighted regression while the other two metamodeling techniques were based on the kriging method. For regression type metamodels, CCD was utilized as DoE, and two stages approach described in Bucher’s RSM was performed, and Table 3 includes the reliability results that were obtained after the second stage. For the kriging metamodel, two different LHD were generated, one of which was based on sampling and the other measure of distance. The number of the experiments for both CCD and LHD are the same, i.e. for two random variables both designs generate 9 experimental points for each stage.

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Table 3. Reliability results for Example 1 Methods applied Adaptive Monte Carlo Simulation (10000 samples) Bucher’s RSM with CCD Weighted regression with CCD Kriging with LHD (sampling) Kriging with LHD (measure of distance)

Beta 2.351 1.277 2.230 2.337 2.352

Pf 0.00936 0.10069 0.01284 0.00971 0.00933

As can be seen from Table 3, the interpolating type metamodels gives better results than the regression type metamodels, which can also be observed from Figure 8 which graphically shows the approximation to the performance function by four metamodeling methods utilized for this example. The kriging models in Figure 8(c) and (d) approximate the performance function better around the design point. As pointed out by Kaymaz [40], the kriging model

Meta-modelling Based Approximation Methods for Structural Reliability Analysis 227 can better follow the shape of the function to be approximated since it interpolates the experimental points. In order to validate the metamodels constructed, two data sets are utilized. Data Set A denotes the experimental points used to construct the metamodels while Data Set B includes randomly selected extra points. The model validation measures based on both Data Set A and Data Set B are presented in Table 4 for the metamodels constructed to approximate the performance function. Table 4. Model validation for Example 1 Data Set A (9 Points)

AdjR 2

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Bucher’s RSM with CCD Weighted Regression with CCD Kriging model with LHD (sampling) Kriging model with LHD (measure of distance)

Data Set B (30 Points)

AdjR 2prediction

% AvgErr

% RMSE  

% MaxErr

0.840 0.881

0.191 0.757

71.112 18.428

23.730 12.080

208.222 71.234

-

0.738

44.800

18.835

161.885

-

0.764

67.598

23.137

297.761

(a)

(b)

(c)

(d)

Figure 8. Performance function (solid line) and metamodels (dotted line) for Example1 together with the experimental points generated both in the first stage (o) and in the second stage (*). indicates the design point.

228

Irfan Kaymaz AdjR 2 values were not computed for the

Since kriging models interpolate the data,

interpolating type metamodels. Although AdjR 2prediction is higher for the metamodels that give better reliability results, the model validation criteria for Data Set B, especially % RMSE , do not indicate the goodness of the metamodel when the reliability results are considered. This can be explained such that these criteria take into account the extra points generated in the whole design space. However, the experimental points near the design point contribute the value of the probability failure most. Thus, a new model validation criterion must be considered to evaluate the metamodels especially in the vicinity of the design point.

3.4.2. Example 2: Connecting Rod The second example of this chapter includes the reliability analysis of a connecting rod schematically shown in Figure 9. The geometrical parameters are given in Table 5. The connecting rod is subjected to a force acting on the inner right hole, and is constrained at the left hole.

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Table 5. Geometrical parameters of the connecting rod R1

R2

R3

R5

D1

D2

A

50.26

35.02

35.41

37.41

90.01

191.39

26.01

Figure 9. The connecting rod.

The force applied and young modulus of the connecting rod were assumed to be random, and their statistical properties are given in Table 6. Table 6. Statistical properties of the random variables for Example 2 Random Variables F

distribution Normal

Mean Value 200 

Standard deviation 20

E

Normal

2 × 10 5

2 × 10 4

Meta-modelling Based Approximation Methods for Structural Reliability Analysis 229 The performance function of the connecting rod is specified as: g ( x) = 0.0025 − ε eqv

where ε eqv denotes the equivalent von Mises elastic strain which was computed using finite element analysis for which ANSYS [42] was utilized. Due to symmetry, half of the connecting rod was considered for the Finite element model as shown in Figure 10.

Figure 10. Finite element model of the connecting rod.

The probability of failure was computed as 0.01 using Importance Sampling with 10000 samples, which serves as an exact or reference solution. However, it should be noted that for each simulation ANSYS needs to be called to obtain ε eqv , which takes a significant amount of computation time. For the reliability analysis of the connecting rod, the computation time was 2325.16 seconds. Therefore, four metamodeling techniques were applied to approximate the performance of the connecting rod, and the reliability analysis was carried out in just 17.49 seconds. The reliability results obtained are presented in Table 7.

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Table 7. Results for Example 2 Methods applied Importance Sampling (10000 samples) Bucher’s RSM with CCD Weighted regression with CCD Kriging with LHD (sampling) Kriging with LHD (measure of distance)

Beta 2.326 2.289 2.303 2.336 2.314

Pf 0.0100 0.0102 0.0106 0.00974 0.0103

As can be seen from Table 7, all the metamodels considered here give very close results to the exact solution obtained from importance sampling. However, the kriging models constructed using the experimental points generated by both LHD sampling and LHD measure of distance are slightly better than the regression type metamodels. This conclusion can be verified by the model validation criteria presented in Table 8. The validation measures for the kriging models are slightly better than that of the regression type metamodels for both Data set A and Data set B.

230

Irfan Kaymaz Table 8. Model validation for Example 2 Data Set A (9 Points)

Bucher’s RSM with CCD Weighted Regression with CCD Kriging model with LHD (sampling) Kriging model with LHD (measure of distance)

Data Set B (30 Points)

AdjR 2

AdjR 2prediction

% AvgErr

% RMSE

% MaxErr

0.972 0.969 -

0.860 0.950 0.995

100.021 100.001 99.994

28.144 28.141 28.140

134.269 134.249 134.242

-

0.999

99.997

28.140

134.245

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4. Conclusion This chapter outlined the basic steps of the metamodeling techniques which are inevitable to use for reliability analysis of time consuming engineering problems. In this respect, the methods used to generate the experimental points, called Design of Experiment, were reviewed for both regular design space and computer experiments. As shown in the examples, space filling designs such as Latin hypercube design are most suitable method to generate experimental points for deterministic computer models. However, it should be noted that regular design such as factorial design or central composite design are easy to implement and repeatable. Two most common techniques for the model fitting; i.e. the response surface method and the kriging method were explained in detail in this chapter. In order to assess the goodness of the metamodel in terms of approximation to the performance function, model validation measures were also reviewed for both regression and interpolating type metamodel. Although the extra points for a model validation is the most appropriate approach, it is impractical for time consuming engineering problems. Thus, cross-validation based on the experimental points used to generate the model was also explained. As pointed out in the examples, the measures outlined in this chapter needs to be improved for the metamodels constructed for reliability analysis. For reliability problems, however, metamodeling techniques developed specifically for deterministic computer experiments are not the best choice to approximate the performance function since the classical metamodels aim to cover the whole design space. As pointed out, a metamodel for reliability analysis should fit the region where the design point is likely to exist since it contributes most significantly to the probability of failure. Thus, metamodeling techniques specifically developed for reliability analysis such as Bucher’s RSM, weighted regression and the kriging method modified for reliability analysis were also given in this chapter. Two selected examples were provided in order to show the application of the approximation methods to the reliability analysis of both elementary mathematical functions and an engineering problem. As can be seen from the results of these examples, the kriging method with a space filling design gives better results than the regression type metamodels with a regular DoE. However, it should be pointed out that application of the kriging method

Meta-modelling Based Approximation Methods for Structural Reliability Analysis 231 to reliability problems needs a special attention in terms of adjusting the parameters of the kriging model [40]. In summary, metamodeling based approximating methods reviewed in this chapter can be used for the reliability problems requiring time consuming analysis to reduce the computational time. However, special metamodeling methods are needed for reliability analysis when compared to its applications to deterministic computer problems. Therefore, further researches are needed to develop more efficient metamodels for reliability analysis.

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]

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[15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26]

Ditlevsen, O.; Madsen H.O. Structural Reliability Methods; John Wiley: Chichester, 1996. Shinozuka, M. J.Struct.Eng.ASCE. 1983, 109(3), 721-740. Tvedt, L. J.Eng.Mech.ASCE. 1990, 16(6), 1183-1197. Bargmann, H; Rustenberg, I.; Devlukia, J. Fatigue.Fract.Eng.M. 1994, 17(12), 14451457 Bjerager, P. J.Eng.Mech.ASCE. 1988, 114(8), 1285-1301. Bucher, C.G. Struct.Saf. 1988, 5(2), 119-126. Melchers, R.E. Struct.Saf. 1990, 9(2), 117-128. Wang, L.; Grandhi, R.V. Comput.Struct. 1996, 59(6), 1139-1148. Kleijnen, J.P.C. Interfaces. 1975, 5(1), 21–23. Simpson, T.W.; Booker, A.J.; Ghosh, D.; Giunta, A.A.; Koch, P.N.; Yang, R.J. Struct. Multidiscip.O. 2004, 27, 302–313. Simpson, T.W.; Peplinski, J.D.; Koch, P.N.; Allen, J.K. Eng.Comput. 2001, 17, 129150. Montgomery, D.C. Design and Analysis of Experiments. John Wiley & Sons: NY, 2005. Bucher, C.; Macke, M. In Engineering Design Reliability Handbook; Nikolasid, E.; Ghiocel, D.M.; Singhal, S. CRC Press: NY, 2004. Santner, T.J.; Williams, B.J.; Notz, W.I. The Design and Analysis of Computer Experiments. Springer Verlag: NY, 2003. Martin, J.D.; Simpson, T.W. J.AIAA, 2005, 43, 853-863. Wang, G.G., J.Mech.Design. 2003, 125, 210-220. Zou, T.; Mahadevan, S.; Mourelatos, Z.; Meernik, P. Relib.Eng.Syst.Safe. 2002, 78, 315–324. McKay, M.; Conover, W.; Beckman, R. Technometrics. 1979, 21, 239–245. Olsson, A.M.J.; Sandberg, G.E. J.Eng.Mech.ASCE. 2002, 128(1), 121–5. Stein, M. Technometrics. 1987, 29(2), 143–51. Minasny, B.; McBratney, B. Comput.Geosci. 2006, 32, 1378-1388. MATLAB Reference Guide. Natick: The Math Works Inc., 2007. Yao, T. H. J.; Wen, Y. K. J. Struct.Eng. 1996, 122, 193-201. Kleinbaum, D. G.; Kuper, L. L.; Muller K. E. Applied Regression Analysis and Other Multivariable Methods. PWS-KENT Publishing Company: Boston, 1987. Kreyszig, E. Advanced Engineering Mathematics. John Wiley & Sons: Singapore, 1993. Ryan, T. Modern Regression Methods. John Wiley & Sons: NY, 1997.

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[27] Welch, W.J.; Buck, R.J.; Sacks J.; Wynn, H.P.; Mitchell, T.J.; Moriss, M.D. Technometrics. 1992, 34(1), 15-25. [28] Sacks, J.; Schiller, S.B.; Welch, W.J. Technometrics. 1989, 31(1), 41-47. [29] Martin, H.D.; Simpson, T.W. Proceedings of DETC’03 ASME 2003 Design Engineering Technical Conferences and Computers and Information in Engineering Conference, Chicago, Illinois USA, September 2-6, 2003. [30] Myers, R. H.; Montgomery, D. C. Response Surface Methodology: Process and Product Optimisation using Designed Experiments. John Wiley & Sons:NY, 1995. [31] Venter, G.; Haftka, R. T.; Chirehdast, M. Proceedings of the 38th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Material Conference and AIAA/ASM/AHS Adaptive Structures Forum, Kissimmee, FL., 1997. [32] Simpson, T. W.; Lin, D. K. J.; Chen, W. J.Reliab.App. 2001, 2, 209-240. [33] Martin, J.D.; Simpson T.W. Proceedings of DETC’04: ASME 2004 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference Salt Lake City, Utah USA, September 28 - October 2, 2004. [34] Bucher, C.G. Struct.Saf. 1990, 7(1), 57-66. [35] Rajashekhar, M.R.; Ellingwood, B.R. Struct.Saf. 1993, 12(3), 205-220. [36] Kim, S.; Na, S. Struct.Saf. 1997, 19(1), 3-19. [37] Das, P.K.; Zheng, Y. Probabilist.Eng.Mech. 2000, 15, 309-315. [38] Kaymaz, I.; McMahon, C.A. Probabilist.Eng.Mech. 2005, 20 (1), 11-17. [39] Romero, V.J.; Swiler, L.P.; Giunta, A.A. Struct.Saf. 2004, 26(2), 201-219. [40] Kaymaz, I. Struct.Saf. 2005, 27, 133-151. [41] Lophaven, S.N.; Nielsen, H.B.; Søndergaard, J. DACE: A Matlab kriging toolbox. 2003. [42] ANSYS, User's Manual (Version 11.0), Swanson Analysis Systems Inc., Houston, TX, USA, 2008.

In: Reliability Engineering Advances Editor: Gregory I. Hayworth

ISBN: 978-1-60692-329-0 © 2009 Nova Science Publishers, Inc.

Chaper 8

SEISMIC RELIABILITY OF NONLINEAR STRUCTURAL SYSTEMS John W. van de Lindt and Shiling Pei Colorado State University, USA

Copyright © 2008. Nova Science Publishers, Incorporated. All rights reserved.

Abstract This book chapter is divided into two sections: (1) Theory and Methodology, and (2) Illustrative Examples. The theory and methodology section provides the background on seismic excitation and basic nonlinear systems such as the linear oscillator, bi-linear oscillator, complex hysteretic model, and system level model with multi-degree of freedom response to earthquakes. Once the explanation of the methods is completed, a series of illustrative examples is set out and the approach to each explained. This includes four examples which increase in complexity. The first example is the seismic reliability estimates for a linear oscillator subject to a single earthquake (1994 Northridge). The second example looks at a more complex oscillator known as the bilinear oscillator subjected to a suite of 20 earthquakes and determine the reliability of the oscillator to a particular displacement requirement. The third example will focus on a typical North American two-story single family house, with the primary lateral force resisting component (wood shear walls) modeled using a 16 degree-offreedom hysteretic oscillator. Suites of ground motions all having the same occurrence probability will be used to determine the reliability against exceeding an inter-story drift limit. In the fourth and final example, a method for computing the financial loss during a single earthquake and over a given period of time for a single family home is presented. The simulation procedure illustrated in this last example incorporates the uncertainty sources that impact the seismic reliability related to limiting financial losses. A brief introduction to performance-based seismic design and the application of structural reliability is also discussed.

Keywords: Seismic reliability; earthquake engineering; light-frame wood structures.

Design of structural systems for earthquake loading is perhaps best characterized as a problem with competing objectives. On one hand, we, as engineers, want a stiff strong building capable of resisting strong ground motions, but at the same time we need energy dissipation to control damage. Linear systems are idealized and in virtually all cases do not

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John W. van de Lindt and Shiling Pei

exist since they would be cost prohibitive except in very special cases such as nuclear power plants. Most systems that are designed in seismic regions of the world exhibit some level of ductility. Ductility is defined as the ratio of the ultimate deformation at fracture to the deformation at yield. Within this chapter we will look at both types of systems.

1. Seismic Response of a Structural System The seismic response of a structural system can be described with four terms. The mass of the system, M, the stiffness of the system, K, the damping in the system, C, and the forcing function, F. These can be expressed as scalars for a single degree of freedom system as pictured in Figure 1, or in matrix form for a complex structural system.

Figure 1. Analog of a Single Degree of Freedom System.

1.1. Linear Oscillator The response of a linear oscillator such as the one pictured in Figure 1 can be determined by solving the equation of motion, namely

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m

d 2x dx + c + kx = −mag ( t ) 2 dt dt

(1)

where m is the oscillator mass, c is the damping coefficient, and k is the stiffness. The ag ( t ) term is the ground acceleration as a function of time and dt indicates a derivative with respect to time. The solution to this second order homogenous differential equation when the forcing function is harmonic can be found in any graduate level dynamics text (e.g. Chopra, 2005). However, Figure 2 shows a typical ground acceleration record from Southern California. Note that the ground motion is non-stationary which is defined, albeit as a simplification, as a change in the variance over the time series. The part with the highest variance is often termed the strong ground motion portion of the earthquake. Early methods of response prediction which utilized closed form solutions to the equation of motion (discussed below) relied on the assumption of Gaussianity. This assumption is that all points within the acceleration record are samples from a normal distribution. Figure 3, excerpted from van de Lindt and Niedzwecki (2005), shows normalized histograms for three well-known very damaging

Seismic Reliability of Nonlinear Structural Systems

235

earthquakes versus a normal distribution curve. One can see that for the entire record the fit is not good, but if only the strong ground motion portion of the record is utilized then the fit is much better. The Kurtosis, which is the fourth statistical central moment, can be computed for both the full records and the strong ground motion and are shown in Figure 3. A value of Kurtosis approaching 3 is typical of a normally distributed random variable.

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Figure 2. Example of an Earthquake Ground Acceleration Record,

FIgure 3. Histograms of the 1999 ChiChi, 1940 El Centro, and 1995 Kobe earthquakes: The entire record versus the strong ground motion.

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John W. van de Lindt and Shiling Pei

Because of the complexities associated with earthquake records and the seismic response of all types of oscillators, a numerical integration of the equation of motion in equation (2) must be performed if accuracy is desired, which for a linear single degree of freedom oscillator can be done using Duhamel’s integral. Duhamel’s integral is a simple convolution integral and the interested reader is referred to Clough and Penzein (1993) for the details of the method. Another well-known approach within structural engineering is Newmark-Beta method (Newmark, 1956). The Newmark method is more general and allows both the solution of nonlinear oscillators, multi-degree of freedom structural systems, and full nonlinear systems. It is utilized within numerical commercial software as are other approaches to numerical integration of the equation of motion.

1.2. Nonlinear Oscillator

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Determining the response of a nonlinear oscillator requires one to numerically make the assumption that it is linear over a very small time increment. Thus, the same approach such as Newmark-Beta method can be used and the time increment shortened to the point that this assumption becomes valid. The procedure is to then step through time and maintain equilibrium at the end of each time step by adjusting the computed acceleration. The simplest and most commonly used nonlinear oscillator is the elasto-plastic oscillator, which as the name implies, has the idealistic property of being perfectly plastic once it yields. A slight variation on the elasto-plastic oscillator is the bilinear oscillator which has historically been used to model steel building components, and is pictured in Figure 4. Many systems have a unique hysteresis that is not well modeled by idealizations such an elasto-plastic or bilinear oscillator. One such structural assembly is the wood shear wall, which is used as the primary lateral force resisting system in light-frame wood buildings.

Figure 4. Hysteresis for a Bilinear Oscillator.

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Figure 5 presents the experimental hysteresis for a typical wood shearwall, which by inspection, one can see has a significantly different shape than the previously discussed idealized oscillators. Many models have been developed for wood shear walls and the interested reader is referred to a comprehensive review by van de Lindt (2004) for a list of these models from 1982 through 2003.

Figure 5. Experimental Hysteresis for a Wood Shear Wall.

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1.3. Structural System Seismic response of a complex structural system can be estimated using a system dynamic model. Similar to structural analysis under a static load, a wide range of methods can be used in developing such a model. System models with discrete degree of freedom (such as a finite element model and models based on generalized degrees of freedom, e.g. Rayleigh’s method) are highly favored in seismic analysis because they are relatively straight forward to program and to numerical integration. Typically a stiffness matrix will be formed based on the discrete degrees of freedom selected to represent structural response. A mass matrix is also established according to the distribution of seismic mass over the system. A general equation of motion in matrix form can be expressed as

MX + CX + KX = M (rX g )

(2)

where M is the mass matrix; K is the stiffness matrix; C is the damping matrix; X is the generalized degree of freedom vector; Xg is the ground motion vector, which usually has three components in the orthogonal directions; and r is the transformation matrix which operates on

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the ground acceleration vector in order to ensure the acceleration vector corresponds to the system degree of freedom vector. The damping matrix for a linear system is usually constructed based on the mass and stiffness matrix in order to preserve the orthogonality of the matrix, which helps to make the decoupling of the equation of motion possible using the orthogonal mode shapes. Another method to obtain the damping matrix is to assign different damping ratios to each vibration mode and then back calculate and assemble the overall damping matrix. However, the decoupling of the equation of motion cannot be applied to a nonlinear system. When the structural behavior becomes nonlinear, the stiffness matrix will change during the loading and the modal analysis results based on the initial system stiffness will no longer be valid. The nonlinear damping matrix can be derived using variational methods (e.g. Hamilton’s principal) if the mechanism behind the damping forces is known. Since seismic analysis usually deals with the nonlinear behavior of structures, numerical integration methods are the most commonly used tools to solve the equation of motion. In these procedures, an incremental form of the equation of motion is often used and can be expressed as

MΔX + CΔX + K s ΔX = M (rΔX g )

(3)

where K s is the tangential or secant stiffness matrix of the system. The analysis is often carried out using a well established integration procedure such as Newmark-Beta or the Wilson-Theta method. The stiffness matrix must be updated at every time step based on the tangent or secant stiffness of the components in the structural system. In order to obtain accurate results, the incremental time step for these analyses is usually quite small (on the order of 0.01 second).

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2. Illustrative Examples Applied reliability analysis in the field of earthquake engineering is quite complex because of the variability in the earthquake events themselves as well as highly nonlinear structural responses. Due to the wide range of sources of uncertainty and the types of uncertainty involved in seismic engineering analyses, simulation methods are often applied primarily because of their robustness. Depending on the focus of a study, different levels of uncertainty can be included in the analysis. For example, a study on structural reliability against a particular earthquake record may only take into account the uncertainty in structural properties (this analysis is quite elementary, but serves little purpose, since a particular earthquake ground motion will never occur again). On the other hand, another study might neglect the randomness of structural properties (or deem it to be negligible compared to seismic uncertainty [van de Lindt and Niedzwecki, 2000]) and use a suite of earthquake ground motions to introduce the uncertainty from the seismic hazard events. The uncertainty of earthquake occurrence over time may also be included in studies concerning life cycle reliability of the structure (Pei and van de Lindt, 2008). The results from a simulation study can also be used to derive simplified empirical relationships or formula for practical engineering use. In this section, four illustrative examples are presented in increasing order of

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complexity and are intended to provide the reader with an overview of the state-of-art in earthquake engineering seismic reliability analysis circa 2008.

2.1. Single Degreeof Freedom Oscillator under A Given Ground Motion

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As was discussed earlier, the simplest realistic system that is modeled in earthquake engineering or general structural dynamics is a linear elastic single degree of freedom oscillator. This example considered the response of a linear oscillator subjected to a single earthquake ground motion. With this deterministic loading, the uncertainty within the system is solely the result from the uncertainty associated with the oscillator properties. The linear oscillator was assumed to have concentrated mass normally distributed with a mean of 500 kg and coefficient of variation (COV) of 0.05 (recall the COV is defined as the ratio of the standard deviation to the mean). The stiffness of the system was assumed to follow a lognormal distribution with mean equal to 100 kN/m and COV equal to 0.1. The damping ratio of the system was estimated to be in the range of 4% to 7% critical damping, assuming a uniform distribution. The system is shown in Figure 6. The ground motion time history used in this example was recorded at the Canoga Park station in California and is from the 1994 Northridge earthquake. The acceleration and displacement time history for the linear oscillator at its mean structural values are shown in Figure 7.

Figure 6. Linear oscillator with structural uncertainty.

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Figure 7. Time history record for Northridge earthquake (Canoga Park station).

Even with the simple linear system described above, accurate closed-form reliability estimation is not available due to the complexity of the loading. An approximate estimation through the response spectra may be conducted, but the development of response spectrum for a particular ground motion requires time history analyses to be performed in the first place. In this example, time history analysis with “samples” of the structural system from the respective random variable distributions can be generated and used within Monte-Carlo simulation (MCS) to obtain the samples of the response of interest. Depending on the sources of uncertainty involved (three sources of uncertainty in this example) and the correlation between their variability, the number of samples needed to obtain a comprehensive result can easily go into the thousands. At the same time, the level of precession needed for the output may also affect the number of simulation samples needed. For example, a simulation with 100 samples will be biased if the goal is to approximate the 98th percentile value, but will likely be adequate if the median value is of primary concern. In this example, one thousand samples from each of the three random variables were generated for the time history analysis. Figure 8 shows the response from one of the simulation runs with randomly generated system parameters.

Figure 8. Time history response of the oscillator.

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241

The maximum forces and displacements of the oscillator from these simulations were collected and fit to lognormal distributions which are shown in Figure 9. The lognormal distribution is often used in civil engineering because it automatically excludes negative values and its relationship to the normal distribution provides numerical convenience. The Type I or Type II extreme value distribution (EV) could also be used and in some cases may provide a better fit (indicated by goodness of fit). However, as one can see from Figure 9, the fit to the lognormal distribution is quite good. With enough samples, even the empirical distribution based on the samples can be used to obtain statistics and perform reliability assessment. However, the examples in this Chapter will use the lognormal distribution, i.e. parametric distribution, for illustrative purposes.

Figure 9. Simulation results for linear oscillator.

The reliability for the reaction force of the linear oscillator was evaluated against a deterministic force limit state and a displacement limit state. These are expressed as

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F ≤ 12kN and Δ ≤ 0.12m

(4)

By fitting the maximum responses (force and displacement) obtained from the simulation to a lognormal distribution, the exceedance probability PrE against the target requirements was calculated as 0.0923 for displacement and 0.0292 for force. The approximate reliability index can then be found by calculating the inverse of the cumulative distribution function (CDF). The reliability indices of the system against the proposed deterministic limit states are

β F = 1.89 and β Δ = 1.33

(5)

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2.2. Bilinear Oscillator under a Suite of Earthquake Ground Motions In the previous section, the reliability of a linear single degree of freedom oscillator was evaluated for a given earthquake ground motion time history, which means the earthquake excitation is deterministic and only the uncertainty in the structural properties was included in the analysis. This type of evaluation is not very useful for prevention and design purposes since existing (or generated) earthquake ground motions will never occur again. In order to evaluate the reliability of a system against future earthquakes, the uncertainty in the ground motion events must be accounted for in the analysis. For non-stationary non-ergodic processes such as the earthquake ground motion acting on nonlinear systems, theoretical approaches that have the potential to directly relate the uncertainty in excitation to the uncertainty in responses (usually based on the theory of random vibration) do not typically apply without significantly simplified assumptions. Unfortunately, these assumptions are good for illustration but not realistic enough for engineering studies. The most reliable and robust approach for this type of problems is still a combination of Monte-Carlo simulation and time history analysis. For ground motion uncertainty, the most widely used approach is to incorporate a suite of earthquake ground motions in the simulation. Such earthquake suites often consist of historical ground motions or synthetic motions, acting as a “sample pool” or family of earthquakes (assumed to be) representing all possible future ground motions in the reliability analysis. As one would imagine, this approximation will never be comprehensive since the number of ground motions in a suite is limited, but it is currently an accepted method within earthquake engineering research and practice. Further, with the help of synthetic motion generation techniques and local historical ground motion information, the representation of the ground motion characteristics for a particular site can be quite satisfactory, especially for civil engineering purposes. The ground motions in a suite can often be scaled to different seismic intensity levels in order to represent earthquakes of different seismic intensity. The simulation with a scaled ground motion suite is often adopted in earthquake engineering to obtain fragility curves, which is the distribution of an engineering demand parameter such as inter-story drift or base shear conditioned on ground motion intensity. Currently, there is no standard procedure to scale a ground motion suite since the scaling is often correlated with the records in the suite. However, scaling for a target spectral acceleration value at either an elastic period or range of periods of interest is often used for fragility curve development. An earthquake suite consisting of 20 recorded ground motions from five different events was developed as part of the CUREE-Caltech project (Krawinkler et al, 2000) and is listed in Table 1. The peak ground acceleration (PGA) values corresponding to three predefined seismic hazard levels are also provided in Table 1, namely 50/50, 10/50, and 2/50 level corresponding to the probability of exceedance of 50%, 10%, and 2% in 50 years, respectively. These intensity levels were also adopted in ASCE 41 (2006) to investigate with structural performance at the Immediate Occupancy (IO), Life Safety (LS), and Collapse Prevention (CP) levels. This ground motion suite consists of only historical ground motions and is intended to be representative of shallow crustal earthquakes in California.

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243

Table 1. Suite of earthquake records

Earthquake Event and Year

File SUP1

Superstition Hills(1987)

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2/50 0.985

0.255

0.584

0.973

0.174

0.398

0.643

0.205

0.470

0.759

0.261

0.599

0.967

0.206 0.210 0.266

0.472 0.482 0.609

0.762 0.778 0.840

0.212

0.485

0.783

0.206 0.185 0.206 0.227 0.179 0.181 0.262 0.231

0.472 0.423 0.473 0.520 0.410 0.415 0.600 0.530

0.762 0.683 0.764 0.840 0.662 0.670 0.969 0.856

Rio Dell Overpass

0.232

0.532

0.859

LAN1 LAN2

Desert Hot Springs Yermo Fire Station

0.237 0.174

0.542 0.399

0.875 0.644

NOR4 NOR5 NOR6 NOR9

Landers (1992)

10/50 0.604

CM2

NOR3

Cape Mendocino (1992)

50/50 0.264

NOR10 LP1 LP2 LP3 LP4 LP5 LP6 CM1

SUP2

NOR2

Loma Prieta (1989)

Peak Ground Acceleration(g)

Brawley El Centro Imperial County Center Plaster City Beverly Hills 14145 Mulhol Canoga Park – Topanga Canyon Glendale – Las Palmas LA – Hollywood Storage LA – (North) Faring Road North Hollywood – Coldwater Sunland – Mt Gleason Ave Capitola Gilroy Array #3 Gilroy Array #4 Gilroy Array #7 Hollister Differential Array Saratoga – West Valley Fortuna Boulevard

SUP3

Northridge (1994)

Station

The purpose of this example is to develop fragility curves for the displacement of a bilinear single degree of freedom oscillator using the ground motion suite listed in Table 1 and assess the reliability of the system displacement for several different hazard levels. A seismic mass of 500 kg was used. The initial stiffness of the system was set equal to 100kN/m. The yielding displacement was 0.05 m and post-yielding stiffness was set equal to 10% of the initial stiffness. A constant damping ratio of 0.05 was used for the analysis. With the assumption that the uncertainty of the bilinear system is negligible compared to the uncertainty associated with the ground motion records in the suite, there was no variation in model parameters in the simulation, thus resulting in 20 maximum displacement values for each intensity level. Figure 10 shows the displacement time history of the system under the CM1 earthquake from Table 1 scaled to the 2/50 seismic intensity level as well as the hysteretic response of the bilinear oscillator.

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Figure 10. Time history response for bilinear oscillator,

Table 2. Maximum force and displacement results for bilinear oscillator

0.025

50/50 Disp (mm) 14.7

Force (kN) 1.47

10/50 Disp (mm) 33.6

Force (kN) 3.36

2/50 Disp (mm) 54.1

Force (kN) 5.04

0.075

15.6

1.56

35.6

3.56

54.4

5.04

0.125

17.1

1.71

39.1

3.91

58.3

5.08

0.175

17.5

1.75

40.2

4.02

62.0

5.12

0.225

19.1

1.91

43.8

4.38

62.6

5.13

0.275

19.2

1.92

43.8

4.38

66.7

5.17

0.325

20.0

2.00

45.8

4.58

68.7

5.19

0.375

20.8

2.08

47.7

4.77

69.5

5.20

0.425

21.4

2.14

49.2

4.92

71.9

5.22

0.475

21.6

2.16

49.5

4.95

73.6

5.24

0.525

22.9

2.29

52.5

5.03

76.6

5.27

0.575

24.0

2.40

53.9

5.04

79.4

5.29

0.625

24.0

2.40

55.2

5.05

80.6

5.31

0.675

25.1

2.51

58.1

5.08

84.4

5.34

0.725

25.4

2.54

58.8

5.09

86.1

5.36

0.775

28.3

2.83

59.4

5.09

87.1

5.37

0.825

29.9

2.99

60.2

5.10

90.4

5.40

0.875

30.1

3.01

62.7

5.13

102.3

5.52

0.925

32.9

3.29

67.6

5.18

110.0

5.60

0.975

33.1

3.31

73.4

5.23

144.7

5.95

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Empirical CDF

The maximum displacement and force of the system from the simulation were listed in Table 2 for all three hazard levels. These maximum values were fit to the lognormal distribution and are shown in Figure 11. These CDF curves are the displacement and force fragility curves for the oscillator response at three seismic intensity levels listed in Table 1. It can be seen from Figure 11 that unlike the linear system, the simulated restoring force of the bilinear spring did not increase as much as the displacement when the seismic hazard level

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245

increases. This is mainly due to the yielding of the system. As a result, the maximum restoring force may be less useful in the performance evaluation of a nonlinear system compared to maximum displacements.

Figure 11. Fragility curves for bilinear oscillator displacement and force.

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By using fragility curves, the uncertainty of the seismic responses become conditioned only on a seismic intensity indicator such as spectral acceleration or peak ground acceleration. However, this may introduce significant additional modeling uncertainty since ground motions do differ based on fault type, soil type, and other seismological parameters. This problem may be addressed through the development of a localized earthquake suite using synthetic ground motion generation techniques. In the present case, three discrete hazard levels were used as the intensity indicators; in other cases, a continuous indicator such as spectral acceleration can be used as mentioned. It can be viewed as an extension of a more traditional seismic design scenario where only one seismic hazard level is considered (e.g. the design basis earthquake). However, the benefit of this method will become more apparent when one needs to consider the impact of the full spectrum of possible seismic events with different intensities. In this example, the reliability indices for a deterministic displacement limit state of 80 mm. This gave reliability indices of

β 50 / 50 = 5.24 β10 / 50 = 2.16 β 2 / 50 = 0.16

(6)

Of course, it is more logical to propose different displacement limit states for the different seismic intensity levels since the consequences are quite different depending on the seismic intensity. This variation in displacement limit state is consistent with the concept of performance based seismic design (PBSD), a new emerging design philosophy which enables one to explicitly consider the structure’s performance for various combinations of seismic hazard and performance expectation. For example, one can propose displacement limits for three hazard levels as

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John W. van de Lindt and Shiling Pei

Δ 50 / 50 < 40mm Δ 10 / 50 < 90mm Δ 2 / 50 < 180mm

(7)

Then the reliability indices for these hazard levels will be

β 50 / 50 = 2.38 β10 / 50 = 2.71 β 2 / 50 = 3.38

(8)

Reliability levels similar to those shown in Equation (8) are often the final results in a reliability analysis.

2.3. Two-Story Single Family Home Subjected to Different Seismic Hazard Levels

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The reliability assessment of a single family home is computed in this example in order to illustrate the level of complexity in a more realistic case, i.e. a full structure. The house is a traditional North American light frame wood construction, with approximately 130 m2 area. All the shearwalls in the house were assumed to have 8d common nails connecting the 12mm (15/32”) oriented strand board to the framing, with spacing of 0.15m (6”) on the panel perimeter and 0.3m (12”) for the interior studs (termed a 6/12 nail schedule). The floor plan for this two story house is shown in Figure 12.

Figure 12. Floor plan for the two-story single family home structure.

As long as the dynamic responses of the structure can be predicted with good accuracy, any type of numerical model can be used in the seismic reliability analysis simulations. However, several characteristics of the model are essential to an efficient and accurate assessment. The model should be able to include the effect of all components within the

Seismic Reliability of Nonlinear Structural Systems

247

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structure that might have an impact on the structural reliability estimate. In light frame wood structural modeling, for example, the inclusion of non-structural finishing panels (drywall/gypsum wall board) are critical because they can significantly reduce the dynamic response of the structure. The model prediction should achieve consistent and acceptable accuracy levels for all of the earthquake hazard levels of interest. In most cases, a linear model will not be applicable for high seismic intensities unless the structure is highly overstrengthened and even then, for wood buildings, this may not be appropriate. Finally, due to the fact that a seismic reliability simulation usually requires thousands of nonlinear time history analyses, the model of choice should be computationally efficient. This example used a simplified biaxial shear model (Folz and Filiatrault, 2001) incorporated in the SAPWood software package (Pei and van de Lindt, 2007) which was designed to perform nonlinear time history analysis for light-frame wood structures. Wood shearwalls and drywall partition walls were represented in this model as nonlinear hysteretic spring elements. An example of the behavior of these hysteretic springs was given in Figure 13, where the hysteretic responses of a standard 2.44m x 2.44m structural shearwall (two 1.22m x 2.44m OSB panels) and drywall partition wall are illustrated. The accuracy of this model has been verified through shake table tests (Folz and Filiatrault, 2004; van de Lindt et al, 2008). The SAPWood software (available free of charge at www.engr.colostate.edu/NEESWood/SAPWood/ ) also has automated features to run series of analyses using suites of ground motions and models, making it a suitable candidate for seismic reliability analyses of light frame wood structures.

Figure 13. Hysteretic responses for shearwall and drywall partition wall.

The ground motion suite listed in Table 1 was used in this example. The uncertainty of the structure itself was neglected in this particular example but can be included easily once the distribution of the components strength/stiffness is known, although it would be very computationally intensive, i.e. samples for the entire structural system need to be generated and used in the time history analysis. The maximum inter-story displacements resulting from the simulations are listed in Table 3.

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Table 3. Maximum inter-story displacements of the single family dwelling Empirical Displacement (mm) CDF 50/50 10/50 2/50 0.025

6.4

20.3

36.8

0.075

7.0

22.0

53.8

0.125

7.9

27.8

95.3

0.175

7.9

33.9

99.1

0.225

8.5

43.5

111.8

0.275

8.5

44.2

142.9

0.325

9.2

45.0

156.7

0.375

9.3

46.6

163.9

0.425

9.4

49.3

224.4

0.475

9.6

50.6

294.1

0.525

9.6

53.3

372.9

0.575

9.7

58.4

398.0

0.625

9.8

58.8

440.2

0.675

9.9

64.0

700.5

0.725

10.4

72.9

742.4

0.775

10.8

86.7

766.1

0.825

12.2

102.9

1148.8

0.875

12.2

133.1

1255.0

0.925

12.7

135.5

1354.8

0.975

16.0

174.2

1458.7

In the time history analysis of nonlinear structures, it is quite common to see extremely large responses caused by numerical instability if the loading significantly exceeds the typical possible capacity of the structure. For example, in the CP level simulation of the example building, the inter-story displacements from the numerical analysis are greater than 1 m, which is not realistic for a single story level. A cut-off point of 7% inter-story drift (interstory displacement/story height) was used in this example to identify the failure (collapse) of the structure, essentially right censoring the data. Assuming the story height is 2.75m, the cutoff drift level corresponds to approximately 190mm. It can be seen from the table that most of the CP level simulation led to deformations in excess of the allowable 190mm for the structure (the underlined bolded displacement numbers in Table 3). The numbers above the failure threshold are then used as an indication of failure but are not used in the fitting of drift fragility distributions. As was shown in Figure 14, the displacements from IO and LS levels were fitted to lognormal distribution. While the CP level data was used to construct an empirical CDF curve.

Seismic Reliability of Nonlinear Structural Systems

249

Figure 14. Fragility curves for the single family home building inter-story displacement.

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2.4. Life Cycle Loss Due to Earthquakes of a Two Story Single Family Home With a structural model capable of performing nonlinear time history analysis, the assessment of reliability related to structural responses such as resistance, drift, and acceleration are relatively straightforward using the Monte-Carlo simulation procedures illustrated in the earlier examples. However, the impact of earthquake hazard must be accounted for since it is a very large source of uncertainty. With the improvement in numerical models and computational efficiency of modern computers, seismic engineering research has seen increasing focus on damage or loss mitigation beyond only life safety concerns. Of course, life safety is still provided by these new design approaches and thus these techniques provide an opportunity for seismic design to mitigate losses. This type of research often deals with multiple performance requirements under earthquake loading where some of the performance requirements are not structural responses, but rather related to the life cycle costs which have been studied for multiple structural types (Liu et al., 2003, Ang et al., 2001 etc.). In this section, loss analysis for residential woodframe structures exposed to seismic hazard is illustrated through a numerical example. In addition to time history analysis and the fragility concept included in the previous examples, unique elements for loss simulation are introduced, including the seismic loss estimation framework, component damage fragilities, and the assembly based vulnerability (ABV) method (Porter et al., 2001; Pei and van de Lindt, 2008). Generally speaking, earthquake-induced losses are determined by properties of structures and seismic events. As one considers the expected financial loss over a future time period for a new or existing design, a significant amount of uncertainty is introduced into the assessment. Given a time period, t, beginning immediately following the completion of the construction of a new structure, if earthquakes in this period are considered Poisson events, and the structure is assumed to be repaired immediately to its initial state following every earthquake, the total loss, L(t), can be modeled as a random variable following the compound Poisson distribution as,

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John W. van de Lindt and Shiling Pei N

L(t ) = ∑ C s (i )

(9)

i =1

N ~ Poisson(λt )

(10)

C s (i ) (subscript “s” indicates a single earthquake ground motion event) is a random th variable representing the loss given the i earthquake which occurs in time t, and these where

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values for different earthquakes are assumed to be independent of one another; N is a random variable that follows the Poisson distribution, representing the total number of earthquakes in time t; λ is the rate parameter of the Poisson distribution, which can be interpreted as the average number of earthquake-induced ground motion events at the site in a unit of time (usually a year). The time period t is assumed to start immediately following the construction of the building. The components of this loss estimation framework, i.e. the model for the uncertainty sources in long term seismic induced loss, are summarized in Figure 15.

Figure 15. Financial loss simulation framework for seismic hazard.

The time history analyses from the earthquake motion suite to the response distribution conditional on seismic intensity is essentially the procedure used to obtain the fragility curves introduced earlier. Once the structural responses are determined, simulation of repair/replacement costs based on structural responses is the next step. This simulation, of course, requires a response-cost relationship for every damageable component, which essentially includes anything in the structure that might result in losses during earthquakes such as structural walls, non-structural finishing materials, furniture and other personal contents. This relationship is not deterministic because of the uncertainties involved,

Seismic Reliability of Nonlinear Structural Systems

251

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including but not limited to, the variability in the cost of repairing material, workmanship, repair method, and influence of repair to adjacent undamaged components, etc. A fragilitybased categorization system termed as damage fragility was adopted in this example to incorporate these uncertainties. A conceptual diagram for this approach is shown in Figure 16. In Figure 16, four damage categories/levels are defined as (I) non-detectable, (II) repairable, (III) borderline repairable, and (IV) not repairable-replace to capture the components’ damage status. Damage fragilities (probability distributions conditional on the damage level) of the structural responses (inter-story drift in the case of wood shearwalls) and repair cost (loss) are then assigned for each damage category based on the available information. These fragilities characterize the amount of damage sustained by components and the resulting financial implications. Later in the simulation framework, these distributions will be the basis to generate samples for repair cost given structural response from the time history analysis. Once the cost samples are generated based on time history analysis results, the total loss due to a single earthquake ground motion can simply be calculated as the summation of the cost of all damageable components. This is the concept of the assembly based vulnerability (ABV) method adopted in recent loss related seismic engineering studies (Porter et al, 2001; Pei and van de Lindt, 2008). Note that the vulnerability analysis procedure illustrated in Figure 15 was conditioned on seismic intensity. As a result, the total loss obtained through the ABV method is also conditioned on seismic intensity. Thus a distribution model can be used to represent the total financial loss caused by a single earthquake, also conditioned on seismic intensity. If this process was repeated for all intensity levels of interest, a vulnerability model can be established. The concept of a vulnerability model is illustrated in Figure 17.

Figure 16. Damage fragility system.

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It is an empirical model that characterizes the relationship between the single earthquake loss distribution parameters and seismic intensity. For example, the single earthquake loss distribution adopted in this example is a four parameter distribution controlled by zero loss probability (Pr0), collapse probability (Prc), and two parameters for the lognormal distribution ( μ ln and σ ln ). Then the vulnerability model is the set of curves representing these four

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parameters as function of seismic intensity. From another perspective, vulnerability model is an ensemble of fragilities in the sense that a sample from each can reproduce a fragility curve for any given intensity level. The seismic vulnerability model is a comprehensive representation of the structures “vulnerability to loss” (or loss potential), as the name implies. Ideally, it should only be controlled by a structure’s design, including structural component arrangement and properties and the arrangement of damageable components within the structure. However, since the simulation procedure involves the use of the ground motion suite in order to account for the uncertainty within earthquake ground motions themselves, the influence of the earthquake ground motions on the buildings seismic vulnerability is also present. The sensitivity of using different ground motion suites may be reduced by introducing a widely adopted standard for ground motion suite selection, as the need for seismic reliability analysis grows in the profession. The two-story single family home used in the previous example was also used in this example. The damageable components considered in the analysis include all structural shearwalls, drywall partition walls, doors and windows, and general contents.

Figure 17. Concept of a seismic vulnerability model.

Seismic Reliability of Nonlinear Structural Systems

253

The first three component types were assumed to be displacement sensitive, which means their damage and related losses are associated with component displacement (deformation) during earthquakes in the damage fragility model setup. The general contents, on the other hand, were used to (approximately) represent all contents in the structure that can move and fall down due to the acceleration caused by the earthquake, including furniture and appliances. The consideration of only four component types is obviously not comprehensive, but is felt to be enough for illustration purposes. The summation of all the components replacement costs for this structure was termed herein as the “collapse loss” ($75,775 USD for this example), which refers to the loss that would result from the complete collapse of the building. Since the inclusion of damageable components in the example was not fully comprehensive, but rather illustrative, this collapse loss represents some portion of the true loss that would result from collapse. By comparing the results of the loss analysis to this collapse loss, a normalized cost can be determined which indicates what percentage of the structure value is lost due to the seismic events. The results from the vulnerability simulation are listed in Table 4

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Table 4. Vulnerability analysis results (collapse loss=$75,775)

$146 $777 $3,298 $6,458 $9,791 $15,048 $23,094 $28,138 $36,505 $37,631 $42,432 $46,456 $53,327 $54,415 $56,916 $57,766 $59,004 $61,346 $66,952 $67,870

Normalized Mean (%) 0.2 1.0 4.4 8.5 12.9 19.9 30.5 37.1 48.2 49.7 56.0 61.3 70.4 71.8 75.1 76.2 77.9 81.0 88.4 89.6

3.32 2.24 1.39 1.01 0.77 1.03 0.98 0.86 0.74 0.69 0.62 0.56 0.47 0.45 0.42 0.39 0.37 0.32 0.24 0.22

0.847

$68,413

90.3

0.20

0

0.877

$69,689

92.0

0.17

0.26

0

0.892

$70,452

93.0

0.15

0.22

0

0.947

$72,158

95.2

0.11

Intensity (Sa)

μ ln

σ ln

Pr0

Prc

Mean (USD)

0.05 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.8 2 2.2

5.04 6.02 7.33 8.23 8.82 9.13 9.36 9.56 9.67 9.8 9.9 9.96 10 10.06 10.05 10.16 10.21 10.34 10.34 10.37

1.03 1.09 1.27 1.12 0.98 0.82 0.72 0.64 0.6 0.59 0.53 0.5 0.5 0.48 0.46 0.42 0.44 0.37 0.4 0.42

0.458 0.053 0.002 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0.05 0.15 0.2 0.324 0.314 0.389 0.454 0.583 0.593 0.648 0.648 0.664 0.693 0.828 0.847

2.6

10.48

0.35

0

3

10.53

0.29

3.5

10.6

4

10.57

COV

254

John W. van de Lindt and Shiling Pei

The vulnerability analysis results for this example have some interesting characteristics that are common for this type of structure. The probability of zero loss tends toward zero as the seismic intensity increases as one would expect. The mean loss value approaches the collapse loss as the probability of collapse increases. The COV for the loss distribution decreases with seismic intensity (as the maximum loss is set to be the collapse loss). The vulnerability model for this example can be summarized as the vulnerability curve shown in Figure 18.

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Figure 18. Vulnerability curves for seismically induced financial loss.

The vulnerability curves shown in Figure 18 were obtained empirically by connecting the distribution parameter values listed in Table 4. Another option is to use a regression model to fit these points to a functional form. The empirical curves were used for simulations throughout this example. They are first used to generate fragility curves for several intensity levels and compared with the empirical CDF curves obtained directly from the simulated data. Very good agreement was observed (see Figure 19) between the sample data generated and the model used in the analyses, verifying the accuracy of using the vulnerability model to represent all of the simulated results. With the vulnerability model in place, evaluation of seismic losses, which one can treat as a subset of seismic reliability, can be performed easily once the performance requirements, in terms of losses, are given for specific seismic intensity levels. Furthermore, the loss in the long term can be evaluated following a simulation procedure with the information on earthquake occurrence given. For locations in the U.S., the seismic hazard curve can be

Seismic Reliability of Nonlinear Structural Systems

255

obtained from the United States Geological Survey (USGS) with the seismic intensities and their corresponding annual probability of exceedance. This information can then be used to obtain the distribution model for seismic intensity as function of location. Figure 20 (Pei, 2007) shows the hazard curve for San Francisco, CA. with the derived intensity distribution.

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Figure 19. Single earthquake loss samples and vulnerability model generated fragilities.

Figure 20. Hazard curve for San Francisco, CA area and derived intensity CDF (Pei, 2007).

256

John W. van de Lindt and Shiling Pei

Then, a long term loss simulation (lifetime loss analysis) can be conducted for any given time period in the future. Firstly, the total number of earthquake events can be generated based on the earthquake frequency in the region (determined based on historical data or probabilistic estimation based on local seismic activity). Then the same number of intensity samples was generated from the distribution derived from hazard curve. For each intensity sample, a single earthquake loss sample can be generated from the vulnerability model. By adding all the single earthquake loss samples together, a long term loss sample may be obtained. This process can be repeated for a desired number of long term loss samples. Note that this simulation does not require time history analysis, which was already conducted to develop the vulnerability model and analysis. The long term seismic induced loss for the example structure is listed in Table 5 for various exposure periods in the San Francisco area. In the long term (75 years for example), the expected loss for this building is approximately $35,000 or less than $500 per year (annualized) on average. Not accounting for time value of money for simplicity even though it would be significant for 75 years, one can see that an insurance rate of $45 to $50 per month would, in theory, provide a break even point. Table 5. Statistics of long term loss samples Time duration (year)

Zero loss probability

Probability of exceedance 80%

50%

10%

5%

2%

1%

1

0.797

$0

$0

$653

$2,070

$9,023

$23,506

3

0.547

$0

$0

$3,670

$10,210

$26,739

$73,561

5

0.384

$0

$241

$9,246

$22,107

$73,362

$75,132

10

0.165

$95

$1,007

$20,691

$60,508

$75,484

$79,543

20

0.031

$1,031

$5,254

$73,321

$76,378

$91,550

$104,060

50

0.000

$6,039

$21,924

$96,323

$124,010

$181,370

$200,830

75

0.000

$15,181

$35,234

$114,040

$141,150

$164,170

$217,200

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Normalized loss (%) 1

0.797

0

0

0.9

2.7

11.9

31

3

0.547

0

0

4.8

13.5

35.3

97.1

5

0.384

0

0.3

12.2

29.2

96.8

99.2

10

0.165

0.1

1.3

27.3

79.9

99.6

105

20

0.031

1.4

6.9

96.8

100.8

120.8

137.3

50

0.000

8

28.9

127.1

163.7

239.4

265

75

0.000

20

46.5

150.5

186.3

216.7

286.6

The simulated samples for the long term loss in this example are shown in Figure 21. One can see from the shape of the empirical CDF that the distribution for long term cumulative loss is not similar to parametric statistical distribution models such as the Normal or lognormal distribution. Thus, the reliability assessment of long term losses could be carried out with empirical distribution models if necessary and not necessarily an assumed simple parametric model.

Seismic Reliability of Nonlinear Structural Systems

257

Figure 21. Empirical CDF curves for long term losses.

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References Ang, A.H. and Lee, J.C. Cost optimal design of R/C buildings. Reliability Engineering and System Safety. 2001; 73(3): 233-238. Folz, B. and Filiatrault, A. Seismic analysis of woodframe structures. I: Model formulation. Journal of Structural Engineering 2004; 130(9): 1353-1360. Krawinkler, H., F. Parisi, L. Ibarra, A. Ayoub, and R. Medina. Development of a Testing Protocol for Woodframe Structures. CUREE Publication No. W-02, Richmond, CA. 2000. Liu, M., Burns, S.A. and Wen, Y.K. Optimal seismic design of steel frame buildings based on life cycle cost considerations. Earthquake Engineering and Structural Dynamics. 2003; 32(9): 1313-1332. Pang W., Rosowsky D.V., Pei S., and van de Lindt J.W. (2007). “Evolutionary parameter hysteretic model for wood shear walls” ASCE Journal of Structural Engineering, 133(8), 1118-1129. Pei S., and van de Lindt J.W. 2008. Methodology for earthquake-induced loss estimation: An application to woodframe buildings. Structural Safety, In press, 2008. Pei S. 2007. Loss analysis and Loss-based design for Woodframe structures. Doctoral dissertation. Colorado State University, CO. Pei S., and van de Lindt J.W. 2007. User’s manual for SAPWood for Windows, Version 1.0. SAPWood program user’s manual, Colorado State University, Fort Collins CO

258

John W. van de Lindt and Shiling Pei

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Porter, K.A. 2000, Assembly-based vulnerability of buildings and its uses in seismic performance evaluation and risk-management decision-making, Doctoral dissertation, Stanford CA. van de Lindt, J.W. (2004). “Evolution of Wood Shear Wall Testing, Modeling, and Reliability Analysis: A Bibliography.” ASCE Practice Periodical on Structural Design and Construction, 9(1), 44-53.

In: Reliability Engineering Advances Editor: Gregory I. Hayworth

ISBN: 978-1-60692-329-0 © 2009 Nova Science Publishers, Inc.

Chapter 9

TIME DEPENDENT SEISMIC RELIABILITY ANALYSIS OF STRUCTURE WITH UNCERTAIN SYSTEM PARAMETER Subrata Chakraborty1,* and Abhijit Chaudhuri2 1

Department of Civil Engineering, Bengal Engineering and Science University, Shibpur Howrah -711103, India, 2 Department of Civil, Environmental and Architectural Engineering, University of Colorado, Boulder, CO, USA

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Abstract The time dependent seismic reliability evaluation of structure is normally performed considering non-stationary models of ground motion neglecting the effect of uncertainty in the structural system parameters. However, the uncertainties result from numerous assumptions made with the geometry and boundary conditions, material behaviors etc. is expected to have significant effect on the overall reliability of the structure. An integrated time dependent unconditional reliability evaluation of structures with uncertain parameter subjected to seismic motion is presented here. The time varying amplitude and frequency content of the ground motion well represented by the family of sigma oscillatory process is developed in double frequency domain. Subsequently, the stochastic dynamic analysis is performed to derive the generalized response power spectral density functions in double frequency domain. The nonstationary response is not uncorrelated with its time derivative. Thus, the reliability is evaluated by conditional crossing rate using the Vanmarcke’s modification based on two states Markov process considering the correlation between response process and its time derivative. The uncertainty of structural system parameters, generally described in spatial coordinate, motivates those to model as random field. The perturbation based stochastic finite element formulation is readily developed to address the parameter uncertainty in the framework of non-stationary dynamic analysis. This involves the important issue of response sensitivity evaluation of structures. The related formulation to obtain the sensitivity of dynamic responses quantities with respect to uncertain parameters has been also briefed. The methodology is elucidated in a focused manner by considering a three dimensional building frame subjected to ground motion due to El Centro 1940 earthquake. The time dependent *

E-mail address: [email protected]

260

Subrata Chakraborty and Abhijit Chaudhuri unconditional reliability is evaluated with respect to desired responses. The results are presented to compare the change in reliabilities of the uncertain system with that of deterministic system. The reliability analysis results are also compared with the Monte Carlo Simulation results to study the accuracy and effectiveness of the present approach.

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1. Introduction Earthquake is a source of critical loading condition for most of the land-based structures located in the seismically active region of earth. Due to its disastrous consequence in recent past, it has attracted more importance. Many cities have been developed in earthquake prone zone due to their industrial or agricultural importance or to facilitate transport problem of the country. These have led various civil-engineering constructions like buildings, bridges, dams, harbor etc. The community is well aware that it is the improper building design and construction that cause the destruction of properties and human lives, not the earthquakes. Thus, the earthquake resistance analysis has been introduced for safe and reliable design and construction of such structures to withstand the effect of earthquake to minimize disaster. The earthquake phenomena are associated with a large number of uncertainties, which can be viewed in two different scales. The first category of uncertainties is the long time feature of earthquake such as number, size, and relative location of the earthquake. While the second type of uncertainties are associated with the details of ground motion at structure’s site during single episode of seismic action. The second type of uncertainty is an important aspect in structural engineering. The problem of analyzing structures under earthquake motion essentially deals in two separate steps. The first step is the so-called seismic risk analysis at the site of the structure. The outcome of the risk analysis is obtained as a set of probability estimates for peak ground motion parameters in typical design period. It is based on the historical earthquake records in the neighborhood of the structural site, the geological condition of ground between the earthquake sources and the structural site and the classical probability model of the Poisson process. The second step is to select a time varying ground motion function from a historical strong earthquake record and scale up (or down) the magnitude of the ground acceleration based on the acceptable design risk from the first step. This design ground motion with modified scale is used as input excitation to the structure. The above approach is clearly partial stochastic in nature, as the first step of risk analysis for peak acceleration is stochastic in nature whereas the second step is entirely a deterministic dynamic response analysis. As earthquake is a random phenomenon, the ground motion during earthquake can be modeled as a random process. It is obvious that a structure subjected to earthquake produces random response. Thus, a probabilistic treatment of earthquake engineering problem provides a rational and consistent basis for aseismic design. In fact, a new branch of mechanics deals with the treatment of uncertainties inherent with parameters defined the behavior of a structure is emerged as probabilistic structural mechanics. The behavior of structure under excitation can be mathematically expressed by a set of equation with suitable prescribed boundary and initial conditions as:

Ω [Y (x, t ) ] = F ( x, t )

(1)

Time Dependent Seismic Reliability Analysis of Structure...

261

Where, Y ( x, t ) is the output of the system due to input F ( x, t ) and Ω is the differential operator, representing the mechanical system. The model as defined by Eqn. (1) involves two independent variables i.e. the position vector x and the time variable t. Depending on the nature of the variables involve in a typical problem, various type of analysis problems may arise. The broad classification of various problems is depicted in table 1. Table 1. Classifications of structural analysis problem

1

System Operator(Ω) Deterministic

Deterministic

Output vector (Y) Deterministic

2 3 4

Deterministic Random Random

Random Deterministic Random

Random Random Random

Level

Input (F)

Remark Conventional structural analysis Random vibration Stochastic system Ideal model

From above it is understood that the analysis of a structure assuming deterministic parameters subjected to earthquake loading is a problem of 2nd category i.e. random vibration problem. Extensive contributions are found on this field. Furthermore, the uncertainties results from the numerous assumptions made with the geometry, boundary condition, and constitutive behavior of materials can have significant effect on the dynamic response and hence the reliability of the structure. Obviously, the analysis deal in such case is a problem of 4th category.

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2. The Background of Seismic Reliability Analysis The present chapter focuses on the time dependent seismic reliability analysis of structure characterized by uncertain parameters. The analysis presented in the framework of spectral modeling of earthquake force. Conceptually, the development can be studied in three broad categories: (i) the spectral modeling of earthquake forces as input to the structure, (ii) stochastic dynamics analysis and extreme value response evaluations and (iii) the effect of parameter uncertainties on structural dynamic response analysis and reliability evaluation. The stochastic analysis of structure under random earthquake deals with the modeling of load as random process. The complete description of random response required the probability density function (pdf) of the earthquake motion over any given set of points in the time domain. If a large number of ground motion records are available for a particular site, the parameters of a stochastic process model can be determined directly by statistical analysis. As it is rarely possible to get large number of such data, considerable judgment becomes necessary to construct and validate stochastic models of ground motion on the basis of few records available at the site or at comparable location, coupled with the seismological data and local site condition. The analysis is quite convenient in the frequency domain. In this approach the power spectral density (PSD) functions, representing the distribution of energy over frequency are most commonly used to describe the behavior of the random process. The most renowned among these descriptions is the description of the PSD function of the

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262

Subrata Chakraborty and Abhijit Chaudhuri

stochastic process. Various such models of ground motion due to earthquake are available in the literatures. If a lightly damped structure whose response is very high at its natural frequency, is excited by a slowly varying wide band excitation process, it may be assumed, without loss of much accuracy, that the excitation is a white-noise process with the PSD function equal to the value of the PSD function at natural frequency of the structure. The stationary white-noise model was first proposed by Housner [1947]. Tajimi [1960] based on the works of Kanai [1957] proposed a stationary non-white model for ground acceleration. The earthquake acceleration at the bed rock is modeled as a stationary white noise and the soil layer overlying the bedrock is modeled as a linear viscously damped single degree of freedom (SDOF) system. Thus, the ground acceleration at the site is response of a SDOF system due to the white noise excitation at bed rock. The PSD function of the ground acceleration has a peak at the natural frequency of soil layer. However, the effect of natural seismicity on structural responses is markedly non-stationary due to non-stationary random excitation process as well as due to the transition nature of responses of the structure. Typically the seismic motions have a sudden beginning and slow decay. To incorporate this non-stationary character of ground motion various models are proposed in the literatures. Some of such important models are: deterministic modulating function [Shinozuka and Sato 1967, Jennings et al. 1969], evolutionary power spectrum [Priestly 1967], the energy spectra, double frequency spectrum, time series model, Weigner process, random pulse train model [Lin and Yong 1987], sigma oscillatory process [Battaglia 1979] etc. As mentioned, the analysis of a structure subjected to earthquake loading is a random vibration problem. The well-established random vibration theory [Bolotin 1961, Lin 1967, Nigam 1983, Lutes and Sarkani 1997] can be readily applied to obtain the probabilistic description of the responses of the structure. Extensive contributions are made on this field [Nigam and Narayanan 1994, Lin and Cai 1995]. The reliability evaluation of system under stationary stochastic process is well established, but not so for the non-stationary case [Rackwitz 1997]. Majority of the literatures mainly emphasize on the extreme random response computation considering different ground motion model. The probabilities that these extreme responses have ever been exceeded some critical levels during the time interval are mainly tried with simplified assumptions primarily restricted to stationary model or that of an evolutionary model. But the evolutionary earthquake models normally neglects the frequency non-stationary character. Moreover, it is well verged that the maximum response and its likely duration can be determined approximately from the shape of the envelope function without carrying out the nonstationary response analysis [Gupta and Trifunac 1996). The reliability analysis for non-stationary cases are mainly restricted to simplified Poisson’s assumption applied to simple SDOF model [Crandal et al. 1966, Racicot and Moses 1971, Yang and Cheng 1993, Cai and Lin 1998). Micholov et al. [1999a, 1999b] proposed analysis of spectral characteristic and the probability distribution of envelope process of nonstationary process seems to be useful in time dependent seismic reliability evaluation. The stochastic dynamic response analysis and reliability evaluation as discussed above considers the structural parameters as deterministic. But uncertainties in structural parameters are bound to occur from numerous assumptions made with the geometry, boundary condition and constitutive behavior of materials. As a result, the variation of structural parameters due to uncertainty may lead to considerable random fluctuations in the response. The stochastic finite element method (SFEM) [Vanmarcke et al. 1986, Benorya and Rehak 1988, Spanos and Ghanem 1989, Klieber and Hien 1992, Schuëller1997] has been developed to incorporate the

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Time Dependent Seismic Reliability Analysis of Structure...

263

effect of randomness in structural parameters. Application of random vibration theory with SFEM to incorporate structural parameters uncertainty is noteworthy [Kotulski and Sobczyk 1986, Igusa and Kiureghian 1988, Chang and Chang 1997, Manohar and Ibrahim 1999, Muscolino et al. 2000]. Interestingly in most of the cases [Rahaman and Hwang 1990, Ambros et al.1996, Chakarborty and Dey 2000, Li and Liao 2001, Gao et al. 2004, Falsone and Ferro 2005], the computations are primarily aimed to obtain the second order statistical quantities such as the PSD function and mean square value of the responses. It is recognized that incorporation of system parameter uncertainties creates an interaction between the probabilistic descriptions of the loading and uncertain parameters [Jensen 2000]. The reliability analysis under uncertain structural system parameters is tried for simple structural system and dynamic load model [Pradlwarter and Schuller 1990, Wall and Bucher 1990, Zhang et al 1998]. The time variant reliability under parameter uncertainty is limited in existing literature particularly for realistic non-stationary earthquake model. Balendra et al. [1991] studied the time variant reliability of linear oscillator subjected to nonstationary earthquake model accounting for uncertainty in structural parameters. Through numerical study on SDOF system, they pointed out that the uncertainties in structural parameters affect the reliability significantly. Zhao et al.[1999] proposed a response surface base procedure to obtain the time-variant reliability of multi degree of freedom (MDOF) non-linear structures with stochastic parameters. The perturbation based SFEM is adopted by Venini and Mariani [1999] in the reliability analysis of uncertain structural systems under stochastic excitation for both the controlled and uncontrolled cases. Li and Chen (2004) proposed a pdf evolution method for dynamic response analysis of structures with random parameters. A framework for an analytical approximation to the reliability of linear dynamic systems with higherdimensional output under Gaussian excitation is presented for both certain and uncertain systems [Taflanidis and Beck 2006]. The safety evaluation of structures under earthquake e.g. the time varying reliability evaluation due to nonstationary earthquake in usual random vibration theory normally considers the structural parameters as deterministic. Various attempts are found in literature to combine the random vibration theory with SFEM to consider structural parameter uncertainty. It is generally felt that the reliability analysis of structure under random earthquake considering the uncertainties in structural parameters has been neglected or tried with simplified stationary assumptions and follow the Poisson independent model for reliability evaluation. A more complete stochastic analysis is felt essential to integrate the earthquake load model, stochastic dynamic response analysis and reliability estimation, so that the design values of response parameters of interest for specified reliability and excitation can be obtained. A time variant reliability analysis of MDOF linear system under generalized nonstationary seismic excitation incorporating uncertainty in structural system parameters is proposed in Chaudhuri and Chakraborty [2006]. However, the work neglects the effect of cross correlation of responses. The work is further extended here to consider the effect of cross correlation as well. The time varying reliability is evaluated using conditional crossing rate following Vanmarcke’s modification based on the two states Markov process [Corotis et al. 1972, Vanmarcke 1975] rather than the commonly used unconditional crossing rate based on Poisson’s assumption. The perturbation approach is adopted to tackle the parameter uncertainties. The unconditional reliability analysis results are further compared with the results of the Monte Carlo Simulation to study the accuracy and effectiveness of the proposed

264

Subrata Chakraborty and Abhijit Chaudhuri

approach. A typical three dimensional reinforced concrete frame subjected seismic load due to El Centro 1940 earthquake is taken up to illustrate the time dependent unconditional reliability computation procedure.

3. The Ground Motion Model The fully non stationary earthquake model based on the family of sigma oscillatory process as described in Conte and Peng [1997] has been readily extended here to derive the generalized PSD function in double frequency spectrum. The non-stationary character of the ground motion both in amplitude and frequency content is described as a sigma-oscillatory process. The closed form expression of the autocorrelation function of response of a linear structure subjected to the ground motion modeled as a sigma oscillatory process was derived in Peng and Conte [1998]. The model views the earthquake ground motion as superposition of the component process described by the individual arrival time, frequency content and time intensity function. The ground acceleration is expressed as, p

p

k =1

k =1

u g ( t ) = ∑ X k ( t ) = ∑ Ak ( t ) Z k ( t )

(2)

Here, Ak ( t ) is the time modulating function of the k-th component process defined by,

Ak ( t ) = α k ( t − ζ k ) Where,

βk

exp ( −γ k ( t − ζ k ) ) H ( t − ζ k )

(3)

α k and γ k are the positive constants, β k is a positive integer and ζ k is the

arrival time of the k-th sub-process and H ( t − ζ k ) is the unit step function. The k-th Gaussian stationary process Z k (t ) characterized by its autocorrelation function defined as,

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∞ Z k Z k (τ ) = RZ k Z k ( t1 − t2 ) = ∫−∞ φZ k Z k

In above, the PSD function

φZ k Z k ( ω ) = Where,

(ω ) e (

i ωt1 −ω t 2

) dω

(4)

φZ k Z k (ω ) is described by the following:

υk 2π

⎡ ⎤ 1 1 ⎢ ⎥ + ⎢υ k 2 + ( ω + η k ) 2 υ k 2 + ( ω − η k ) 2 ⎥ ⎣ ⎦

(5)

υk and ηk are the two parameter representing the frequency band width and

predominant or central frequency of the sub-process Z k (t ) . The model parameters are

generally determined by adaptive least square method. The analytical evolutionary PSD

Time Dependent Seismic Reliability Analysis of Structure...

265

function of the model is fitted to that estimated from the actual target earthquake to determine the model parameters. Extended Thomson’s multiple window spectrum estimation method is used for this purpose. Note that if p is the number of component process then total 6p parameters need to be estimated. More the component included better the model will fit the estimated data. Further details of the method, estimation capability etc. with useful references may be found in Conte and Peng [1997]. The autocorrelation function and the evolutionary PSD function of the ground motion u g (t ) can be expressed as, p

Ru g u g ( t1, t2 ) = ∑ Ak ( t1 ) Ak ( t2 ) RZ k Z k ( t1 − t2 )

(6)

k =1

p

2

φu g u g (ω , t ) = ∑ Ak ( t ) φZ k Z k (ω )

(7)

k =1

The PSD function as described above is evolutionary type. However, the generalized non-stationary cross-spectral density matrix for the ground motion can be represented in double frequency domain as a function of two frequencies. For this, the double Fourier transform is applied [Holman and Hart, 1974] to the associated auto covariance function to obtain the following:

Su g u g (ω1, ω2 ) =

1

∞ ∞

∫ ∫ Ru u ( t1, t2 ) exp ( −i (ω1t1 − ω2t2 ) )dt1dt 4π 2 −∞ −∞ g g

(8)

Now, use of equation (6) in (8) yields, Su g u g (ω1, ω2 ) =

∞ ∞ ⎛ p ⎞ ∑ Ak ( t1 ) RZ k Z k ( t1 − t2 ) Ak ( t2 ) ⎟ exp −i (ω1t1 − ω2t2 ) dt1dt2 2 ∫ ∫ ⎜ 4π −∞ −∞ ⎝ k =1 ⎠

1

(

)

(9)

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With the strength of equation (5), equation (9) further yields,

Su g u g (ω1, ω2 ) =

⎛∞ ⎞ i (ωt1 −ωt2 ) A t φ ω e d ω ∑ ( ) ( ) ∫ ∫ ∫ ⎜ ⎟ k Z A 1 k k 4π 2 k =1 −∞ −∞ ⎝ −∞ ⎠ × Ak ( t2 ) exp ( −i (ω1t1 − ω2t2 ) ) dt1dt2 1

p ∞ ∞

(10)

By simplifying Eqn. (10), the generalized PSD function of the ground motion in double frequency spectrum is finally obtained as: p ∞

Su g u g (ω1, ω2 ) = ∑ ∫ M k (ω − ω1 ) φZ k Z k (ω ) M k* (ω − ω2 )dω k =1 −∞

(11)

266

Subrata Chakraborty and Abhijit Chaudhuri

α k eiωξk 1 ∞ iωt Where, M k (ω ) = Γ( β k + 1) ∫ Ak (t ) e dt = 2π −∞ (γ k − iω )( β k +1)

(12)

Here Γ(⋅) is the gamma function and * denotes its complex conjugate. In subsequent numerical study, the six parameters needed for each component process are taken from Conte and Peng [1997] corresponding to the El Centro (1940) earthquake record. The parameters and associated PSD function of the ground motion in double frequency domain are furnished in Table 1 and Figure 1, respectively.

Figure 1. The Generalized power spectral density Function (El Centro, 1940 Earthquake).

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Table 1. Parameters correspond to El Centro 1940 Earthquake data k 1 2 3 4 5 6 7 8 9

αk 37.2434 104.024 31.9989 43.8375 33.1958 41.3111 4.2234 19.9802 2.4884

βk 8 8 8 9 9 9 10 6 10

γk 2.7283 2.9549 2.6272 3.1961 3.1763 3.1214 2.9904 1.895 2.6766

ζk -0.5918 -0.9857 1.7543 1.686 -0.0781 -7.096 -0.9464 1.402 5.3123

νk 1.4553 2.4877 3.3024 2.1968 3.1241 6.7335 2.6905 7.2086 6.1101

ηk 6.7603 11.0857 7.3688 13.5917 14.3825 25.1532 48.0617 37.6163 19.4612

Time Dependent Seismic Reliability Analysis of Structure...

267

Table 1. Continued k 10 11 12 13 14 15 16 17 18 19 20 21

αk 24.1474 2.5916 2.2733 24.2732 41.3111 1.3697 15.4646 0.0174 2.9646 0.0007 0.8092 16.7115

βk 10 2 3 3 9 10 2 10 10 10 4 2

γk 3.3493 0.224 0.5285 1.0361 3.1214 2.5936 0.7044 1.8451 3.1137 1.3686 0.5969 0.7294

ζk 8.8564 3.2558 16.2065 17.5331 -7.096 21.683 27.2979 -2.4168 1.5751 2.5173 6.4396 12.493

νk ηk 1.9862 9.04 2.4201 9.3381 1.5244 14.1067 1.7141 24.0444 6.7335 25.1532 1.9362 12.9198 1.7897 12.0205 4.9373 98.628 1.9726 61.8316 3.2479 43.90675 3.6749 26.3365 1.7075 37.1139

4. The Stochastic Dynamic Analyses The dynamic equilibrium equation for a MDOF system subjected to ground motion can be readily expressed as,

[ M ]{u ( t )} + [C ]{u ( t )} + [ K ]{u ( t )} = − [ M ][ L ]{u g ( t )}

(13)

[ M ] , [C ] and [ K ] are the global mass, damping and stiffness matrix, respectively. {u ( t )} is the displacement due to ground motion {u g ( t )} . The jth column of the influence coefficient matrix [ L ] represents the pseudo-elastic response in all degrees of Where

freedom (dof) due to unit translation of support motion at jth dof. The displacement of the linear MDOF system subjected to unit amplitude ground motion u g ( t ) = eiωt can be assumed in the form of u ( t ) = Uu (ω ) .eiωt . Thus, the equation of Copyright © 2008. Nova Science Publishers, Incorporated. All rights reserved.

{

}

{

}

motion transform to

([ K ] − ω 2 [ M ] + iω [C ]){Uu (ω )} = − [ M ]{L}U g (ω ) i.e. ⎡⎣ D (ω )⎤⎦ {Uu (ω )} = {F (ω )} {

(14)

}

Where, [ D ( ω )] is the dynamic stiffness matrix and F (ω ) is the forcing vector. All these matrix and vectors are the functions of either of any design variable or in combination, which may be Young’s Modulus (E) and/or Poisson’s ratio, mass density (m), cross-sectional properties, boundary conditions etc. Representing the uncertain properties by the vector, say {d}, Eqn. (14) can be explicitly re-written as,

268

Subrata Chakraborty and Abhijit Chaudhuri

{

} {

} {

}

⎡⎣ D (ω, {d} ) ⎤⎦ Uu (ω, {d} ) = F (ω, {d} ) or Uu (ω,{d} ) = ⎡⎣ D (ω, {d} ) ⎤⎦

−1

{F (ω,{d})}

(15)

For linear systems with known complex frequency response function, the evolutionary PSD function of response under evolutionary non-stationary ground motion is normally obtained based on the assumption that the amplitude of forcing function is a slowly varying function of time such that it can be treated as being nearly independent of time (Lin 1967). The response statistics is then computed using the pdf approximation of the non-stationary process under the assumptions that a non-stationary process behaves like a stationary process at each instant of time (Gupta and Trifunac 1996). When the non-stationary input is described as a function of time ' t ' and frequency ' ω ' the transient nature of response can only be included through convolution integral. The conventional evolutionary PSD function can be easily expressed as function of two frequencies as described earlier and the analysis can be done in double frequency domain avoiding the double integration. When the forcing function on the right hand side of Eqn. (15) is non-stationary, the cross correlation matrix of any responses variable {u(t)} can be expressed as (Holman and Hart 1974), T

∞ ∞

⎡⎣ Ruu ( t1, t2 , {d } ) ⎤⎦ = ∫ ∫ ⎡⎣U u (ω1, {d } ) ⎤⎦ Su u (ω1, ω2 ) ⎡U u* (ω2 , {d } ) ⎤ ei (ω1t1 −ω2t2 ) dω1dω2 g g ⎣ ⎦

(16)

−∞ −∞

Thus, for a linear MDOF system with known frequency response function, the PSD function ⎡⎣ Suu

(ω1, ω2 , {d })⎤⎦

{

}

of displacement u ( t ) for known PSD function of ground

motion can be readily obtained as:

{

}

{

}

⎡⎣ Suu (ω1, ω2 , {d } ) ⎤⎦ = U u (ω1, {d } ) ⎡ Su u (ω1, ω2 ) ⎤ U u* (ω2 , {d } ) ⎣ g g ⎦

T

(17)

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The PSD function for ground motion is described elaborately in section 3. The PSD function of any other response quantities such as stress or combination of stresses can be evaluated simply utilizing the constitutive relationship i.e.

{ f ( t )}i = [ k ]i {u ( t )}i Where, [k] and

{u ( t )}i are

(18)

the element stiffness matrix and displacement vector,

respectively for ith element. In frequency domain the above equation will be as following:

{F (ω )}i = [ k ]i {U (ω )}i Where, { F (ω )} , {U (ω )} are the Fourier transform of i

i

Thus, the PSD function for force can be readily obtained as,

(19)

{ f ( t )}i , {u ( t )}i , respectively.

Time Dependent Seismic Reliability Analysis of Structure... T ⎡ ⎤ ⎡ ⎤ ⎣ S ff (ω1 , ω2 , {d } ) ⎦ i = [ k ]i ⎣ Suu (ω1, ω2 , {d } ) ⎦ i [ k ]i

(

269 (20)

)

Where, ⎡⎣ S uu ω1 , ω 2 , {d } ⎤⎦ is the PSD function of displacement vector of the i-th i element.

5. Sensitivity of Stocahstic Responses

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The sensitivities of structural responses under static and dynamic load represent an essential ingredient for SFEM, structural reliability analysis, structural optimization, structural identification etc. The analysis and design sensitivity computations are very simple for static problems and from early seventies; analytical formulations for design sensitivity analysis are being reported in the literature for deterministic system [Zienkiewcz and Campbell 1977, Haftka and Adelman 1988, Klieber et al. 1997] as well as for stochastic system [Benfratello et al. 2000, Bhattacharya and Chakraborty2002]. The sensitivity analysis of structure under transient dynamic loads is also reported [Hsieh and Arora 1984, Dutta and Ramakrishnan1989, Bogomolni et al. 2006]. However, the analyses are restricted to deterministic dynamic load and not much research seems to have been carried out for computation of design derivative under stochastic dynamic environment. Cacciola et al. [2005] presented response sensitivity of structural systems subjected to non-stationary random loads. Jensen [2005] presented a sensitivity analysis with respect to the optimization variables and general system parameters for optimization of building under stochastic earthquake excitation. The evaluation of response sensitivity of both classically and nonclassically damped discrete linear structural systems under stochastic actions is presented. An algorithm to evaluate the sensitivities of dynamic responses of structures subjected to stochastic earthquake load modeled as a fully non-stationary sigma oscillatory process is presented in Chaudhuri and Chakraborty [2004]. The formulation computes the analytical sensitivity gradient of the stochastic dynamic responses and reliability with respect to the structural design parameters. If the sensitivity computation is performed with respect to uncertain property of kth element, differentiation of equation (15) with respect to parameter dk of kth element gives, ⎤ −1 ⎡ ∂ ∂ ∂ ⎡⎣ D (ω , {d } ) ⎤⎦ ⎥ U (ω , {d } ) = ⎡⎣ D (ω , {d } ) ⎤⎦ ⎢ F ( ω , {d } ) − U ( ω , { d } ) ∂d k ∂d k ⎣ ∂d k ⎦

{

}

{

} {

}

(21)

The sensitivity of PSD function of displacements at nodes with respect to ‘dk’ is obtained directly from equation (17) i.e.

270

Subrata Chakraborty and Abhijit Chaudhuri

⎡ ∂ *T ∂ ⎡⎣ Suu (ω1, ω2 , {d } ) ⎤⎦ = ⎢ U (ω1, {d } ) Sug ug (ω1, ω2 ) U (ω2 , {d } ) ∂dk ⎣ ∂dk

{

{

}

}

+ U (ω1, {d } )

{

}

*T ⎤ ∂ Sug ug (ω1, ω2 ) U (ω2 , {d } ) ⎥ ∂dk ⎦

{

}

(22)

The sensitivity of the RMS value of the response, its derivatives with respect to desired parameter can be directly obtained by differentiating appropriate equation with respect to the parameter considered. Further details may be found in Chaudhuri and Chakraborty [2004].

6. Parameter Uncertainty and Total Reliability The analysis described so far assumes all the input structural system parameters as deterministic. If the parameters uncertainties are included in the analysis, the dynamic stiffness matrix [D (ω )] as described by Eqn. (14) will involve the random variables di (i=1, 2,…, N) (N is the number of discretized element of the random field), which represents the spatial variation of any structural parameters.

{U (ω )} will

also involve the random

variables, di . These are explicitly shown in Eqn. (15). However, for ease of presentation, now onwards the parameter {d} is dropped in the expression of dynamic stiffness and frequency response function. The random parameter di can be viewed as the superposition of the deterministic mean component ( d i ) with a zero mean deviatoric component ( di′ ) . Any response variable that depends on the system parameters will be uncertain due to uncertainty in system parameters. Now, these can be expanded in Taylor series about the mean value of uncertain system parameter (with the assumptions that the random variation is small). The dynamic stiffness matrix [D(ω )] can be also decomposed into a mean and a random part with zero mean i.e.

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⎡⎣ D (ω ) ⎤⎦ = ⎡⎣ D (ω ) ⎤⎦ + ⎡⎣ D (ω ) ⎤⎦

′

(23)

Substituting Eqn. (23) in Eqn. (15), the response vector in frequency domain is obtained as

{U (ω )} = ⎝⎜⎛ [ I ] + ⎣⎡ D (ω )⎦⎤ −1 ⎡⎣ D (ω )⎤⎦′ ⎠⎟⎞

(

= [ I ] − ⎣⎡ D (ω ) ⎦⎤ ⎣⎡ D (ω ) ⎦⎤

−1

−1

−1

⎡⎣ D (ω ) ⎤⎦ + ⎣⎡ D (ω ) ⎦⎤

′ ⎣⎡ D (ω ) ⎦⎤ −

⎡⎣ D (ω ) ⎤⎦

−1

−1

⎡⎣ D (ω ) ⎤⎦

⎞ H ω U ω ⎟ { ( )} g ( ) ⎠

′

[ M ]{L}U g (ω ) (24)

Time Dependent Seismic Reliability Analysis of Structure... Where,

{H (ω )} = ⎡⎣ D (ω )⎤⎦ −1 [ M ]{L} ,

271

is the deterministic part of the frequency

response vector. Now, the random dynamic stiffness matrix [D (ω )] is expanded in first order

{}

Taylor series about the mean d as, N ′ I I ⎡ ∂D (ω ) ⎤ ⎡⎣ D (ω ) ⎦⎤ + ⎣⎡ D (ω ) ⎦⎤ = ⎣⎡ D (ω ) ⎦⎤ + ∑ ⎣⎡ D (ω ) ⎦⎤ di′ where, ⎣⎡ D (ω ) ⎦⎤ = ⎢ ⎥ (25) i i i =1 ⎣ ∂di ⎦

Using this and neglecting the higher order terms in equation (24), the expression of response is obtained as:

{U (ω )}

N N N ⎛ −1 −1 I = ⎜ [ I ] − ∑ ⎡⎣ D (ω ) ⎤⎦ ⎡⎣ D (ω ) ⎤⎦ di′ + ∑ ∑ ⎡⎣ D (ω ) ⎤⎦ i i =1 i =1 j =1 ⎝

× ⎡⎣ D (ω ) ⎤⎦

I i

⎡⎣ D (ω ) ⎤⎦ N

−1

I

⎡⎣ D (ω ) ⎤⎦ di′d ′j j

= {U (ω )} + ∑ {U (ω )} di′ + i I

i =1

){H (ω )}U

g

(ω )

(26)

1 N N II ∑ ∑ {U (ω )}ij di′d ′j 2 i =1 j =1

Where,

{U (ω )} = {H (ω )}U g (ω ) {U (ω )}iI = − ⎣⎡ D (ω )⎦⎤ −1 ⎡⎣ D (ω )⎤⎦iI {H (ω )}U g (ω ) and

(27)

{U (ω )}ijII = 2 ⎣⎡ D (ω )⎦⎤ −1 ⎡⎣ D (ω )⎤⎦iI ⎣⎡ D (ω )⎦⎤ −1 ⎡⎣ D (ω )⎤⎦ Ij {H (ω )}U g (ω )

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In frequency domain the force displacement relations as describe by Eqn. (19) can also be expressed as,

{F (ω )}k = ⎜ ⎣⎡ k ⎦⎤ k + ∑ [ k ]klI dl ⎟ ({U 0 (ω )}k + ∑ {U (ω )}kiI di′ + ⎛

⎝

N

N

⎞

N

l =1

⎠

i =1

N N

= { F (ω )} + ∑ { F (ω )} di′ + ∑ ∑ { F (ω )} di′d ′j ki kij k i =1

Where,

I

j =1 i =1

II

N N ⎞ II ∑ ∑ {U (ω )}kij di′d ′j ⎟⎟ j =1 i =1 ⎠

(28)

272

Subrata Chakraborty and Abhijit Chaudhuri

{F (ω )}k = ⎡⎣k ⎤⎦ k {U (ω )}k {F (ω )}kiI = ⎡⎣k ⎤⎦ k {U (ω )}kiI + [ k ]kiI {U (ω )}k , [ k ]kiI = and { F (ω )}

II kij

= ⎡⎣ k ⎤⎦

∂ [ k ]k

(29)

∂di

II I I I I U (ω )} + [ k ]ki {U (ω )} + [ k ]k j {U (ω )} { kij kj ki k

In above expression the third order term is neglected. Taking the expectation on equation (26) and (28), the mean response can be obtained. As ground motion u g (t ) is a zero mean random process and it is uncorrelated with the

structural parameters, the responses are also zero mean random processes. Let x(t ) be any

generalized response of the structure. The corresponding PSD function of x(t ) can be readily written as, *

S xx (ω1, ω2 ) = X (ω1 ) Su g u g (ω1, ω2 ) X (ω2 ) Where

{ X (ω )} 1

and

{ X (ω )} 2

(30)

are the well known transfer functions at two different

frequencies. To consider the effect of uncertainty in the structural system parameters, above PSD function can also be expressed using Taylor series expansion as is done in case of displacement response in Eqn. (26). Subsequently, the expression of PSD function of any such response quantity considering the effect of system parameter uncertainty can be expressed as: *

N N

(

*

S xx (ω1, ω2 ) = X (ω1 ) Su g u g (ω1, ω2 ) X (ω2 ) + ∑ ∑ X iI (ω1 ) Su g u g (ω1, ω2 ) X Ij (ω2 ) i =1 j =1

)

(31)

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* * + X (ω1 ) Su g u g (ω1, ω2 ) X ijII (ω2 ) + X ijII (ω1 ) Su g u g (ω1, ω2 ) X (ω2 ) E ⎡⎣ di′d ′j ⎤⎦

[

]

Where, E d i′d ′j is the covariance of the random properties. The covariance properties are obtained by random field discretization. This is further discussed specifically in numerical example section. Note that the last three terms in equation (31) are contributed by the randomness of the structural properties. The cross and auto PSD function of x(t ) are readily obtained as,

S xx (ω1, ω2 ) = −iω2 S xx (ω1, ω2 ) and S xx (ω1, ω2 ) = ω1ω2 S xx (ω1, ω2 )

(32)

Following Michaelov et al. [1999a,b], various spectral moments which are used to determine the reliability of structure based on the first passage failure criteria are readily obtained as:

Time Dependent Seismic Reliability Analysis of Structure...

273

C00 ( t ) = ∫0∞ ∫0∞ S xx (ω1, ω2 ) exp ( i (ω1 − ω2 ) t ) d ω1dω2 C01 ( t ) = ∫0∞ ∫0∞ −ω2 S xx (ω1, ω2 ) exp ( i (ω1 − ω2 ) t ) d ω1dω2

(33)

C11 ( t ) = ∫0∞ ∫0∞ ω1ω2 S xx (ω1, ω2 ) exp ( i (ω1 − ω2 ) t ) dω1dω2 Subsequently, the unconditional variances of the nonstationary process, its derivative and the cross correlations between those are obtained from the following:

σ x2 ( t ) = C00 ( t ) , σ x2 ( t ) = C11 ( t ) , ρ xx ( t ) =

Im ( C01 )

σ xσ x

, q x2 ( t ) = 1 −

Re ( C01 )

σ x2σ x2

(34)

It can be pointed out here that the variances and the cross correlations between those as obtained above consider the effect of parameter uncertainty. For linear structural behaviour and Gaussain input ground motion, the output responses will be also Gaussian and the reliability for response x(t ) can be expressed as [Lutes and Sarkani 1997],

(

t

R ( xB , t ) = R ( xB , 0 ) exp − ∫0 α ( xB , s ) ds

)

(35)

Where, for nonstationary or zero start random process R (xB, 0) is the probability that the process is below the barrier level at time t = 0 , xB is the barrier level. For a structure subjected to earthquake ground excitation, the responses are initially zero and hence the initial

(

)

reliability R ( xB , 0 ) is taken as unity. And α xB , s, {d } is the so called hazard function

i.e. crossing rate of D-type barrier level x ( t ) = xB at time‘ t ’. The better approximation of the phenomenon of barrier crossings can be obtained assuming that the crossings follow a two-state Markov process [Corotis et al. 1972, Vanmarcke 1975]. The time dependent

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(

)

reliability is normally evaluated by computing the crossing rate α xB , s, {d } by assuming all the structural parameters as deterministic. However, knowing the unconditional second order information of response quantities as defined by Eqn. (34), the unconditional time dependent reliability can be obtained by computing the unconditional up-crossing rate of barrier level x B for the nonstationary response considering the effect of parameter uncertainty from the following [Michaelov et al. 1999]:

⎛ ⎛ 2 qx2 ( t ) − ρ xx ( t ) Ψ A ⎞⎟ ⎟⎞ ⎜1 − exp ⎜ − π xB ( )⎟⎟ 2 ⎜⎜ ⎜ 2 σ x (t ) 1 − ρ xx t) ⎟ ( t σ ( ) 1 ⎝ ⎠⎠ 2 1 − ρ xx α ( xB , t ) = (t ) x × ⎝ π σ x (t ) ⎛ ⎛ x2 ⎞ ⎞ B ⎟ − 1⎟ ⎜ exp ⎜ ⎜ 2σ 2 ( t ) ⎟ ⎟ ⎜ ⎝ x ⎠ ⎠ ⎝

(36)

274

Subrata Chakraborty and Abhijit Chaudhuri

(

Where, Ψ ( A) = exp − A

2

)+

π A(1 + erf ( A)) and A =

1

x B ρ xx (t )

2σ x (t ) q x2 (t ) − ρ x2x (t )

The reliability is finally obtained by evaluating Eqn. (35) using the expression of unconditional crossing rate

α ( xB , t ) from Eqn. (36).

7. Numerical Example

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A two storied unsymmetrical R.C. building idealized as a space frame (shown in Figure 2) subjected to earthquake ground motion correspond to El Centro 1940 earthquake is taken up to elucidate the present unconditional time dependent seismic reliability analysis procedure. The sizes of the columns and beams are 0.35m×0.35m and 0.25m×0.40m, respectively. The density, Poisson’s ratio and proportional damping ratio (Rayleigh type) of concrete members are taken as 2400 kg/m3, 0.2 and 0.05 respectively.

Figure 2. The idealized space frame model.

The Young’s modulus ( E ) having mean value of 2×107 kN/m2 is considered as random parameters to demonstrate the proposed numerical procedure. It is assumed to be varying randomly along the length of each beam and column (structural element) and the cross correlation among the structural elements are further taken into account. The Young’s modulus is considered to follow lognormal distribution within an element and it is expressed as:

E ( z ) = EG exp ( f ( z ) )

(37)

Time Dependent Seismic Reliability Analysis of Structure...

275

e position of the point considered, f ( z ) is log

Young’s modulus. The geometric mean and standard deviation of f ( z ) are expressed as:

(

)

EG = E − σ 2f 2 and σ f = log 1 + σ E2 E 2 .

(38)

The correlation function of the random Young’s modulus is decomposed in two parts. The first part corresponds to the variability inside the beam and or column while the second part is due to the variability among the beams and columns. In this study each beam and column is divided into four elements to take care of the variability inside the beam and/or column. The 1-D random field within the structural element follows a squared exponential correlation function defined as:

(

ρ1D ( z ) = exp z 2 λ 2

)

(39)

Where, z is the distance between any two points in the discretized structural element and λ is the correlation length. The properties of different beams or column of the space frame are also correlated and follow a 3-D random correlation function:

(

ρ 3 D ( z ) = exp z 2 λ2L

)

(40)

Here, z is the distance between centers of any two structural elements and

λ L is the

correlation length for 3D random field. The variance of logarithmic of Young’s modulus is decomposed into two components as:

σ 2f = σ 2f

1D

+ σ 2f

3D

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The correlation coefficient

, where σ 2f = p1Dσ 2f and σ 2f = p3Dσ 2f 1D 3D

(41)

ρ fi f j1D of discertized 1-D random filed and ρ fi f j 3D for

3-D random filed can be readily obtained by using the local averaging method [Vanmarcke 1983, Chakraborty and Bhattacharya 2002]:

ρ fi f j1D = ρ fi f j 3 D

(

)

1 ∫ ∫ ρ1D zi − z j dzi dz j Li L j L L i j

(

)

1 = ∫ ∫ ∫ ρ1D zi − z j dzi dz j dzk Li L j Lk L L L i j k

(42)

The covariance of log Young’s modulus is obtained by combining the 1D and 3D random field and it is expressed as:

276

Subrata Chakraborty and Abhijit Chaudhuri

E ⎡⎣ fi′ f j′ ⎤⎦ = ρ fi f j 1Dσ 2f 1D + ρ fi f j 3Dσ 2f 3 D

(43)

Finally, the covariance between E i and E j is obtained as,

( ) ( exp ( E ⎣⎡ fi′f j′ ⎦⎤ ) − 1) .

E ⎣⎡ Ei′E ′j ⎦⎤ = EG2 exp σ 2f

(44)

To obtain the unconditional PSD function of response as described by Eqn. (31), the results of Eqn. (44) is used. In present numerical study, the coefficient of variation (cov) of Young’s modulus is taken as 0.25. The results of the following three cases with different sets of ( p1D and p3 D ) are presented:

Case A: p1D = 1.0, p3D = 0.0 , Case B: p1D = 0.5, p3D = 0.5 Case C: p1D = 0.0, p3D = 1.0 In order to compare the unconditional time dependent reliability analysis results of the proposed approach with the results of the Monte Carlo Simulation (MCS), the reliability are also computed based on the MCS. For this, the discretised stochastic fields are simulated and their sample function values are obtained by the method based on covariance matrix decomposition. The method makes it possible to transform a set of independent random variables into a set of correlated random variables with prescribed covariance matrix. In order to simulate, the stochastic filed is discretised into N number of discrete elements. If there are ‘N’ finite elements in the total structure, then there is ‘N’ property values associated with each zero mean homogeneous random field. Then, for MCS the auto-correlated random vector ( { E} ) of size N (where N is the number of discretised elements) is obtained as (Yamazaki and Shinozuka 1990):

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{E} = [L]{Z },

[L]

(45)

In which {Z } is a vector comprising of N independent lognormal random variables and is the lower triangular matrix derived through the Cholesky decomposition of the

associated covariance matrix of the Young’s modulus described in equation (44). For each realization of the MCS, the PSD function of response is determined through random vibration analysis. In the present study the ensemble average of PSD function of response is obtained for 500 realizations. The ensemble average of PSD function is then used for estimation of the reliability of uncertain system by using Eqn. (35) through Eqn. (31) to (34). The root mean square displacement (RMSD) value ( σ x (t ) ) at node 16 along horizontal direction and root mean square of shear force (RMSF) along the horizontal direction at base

Time Dependent Seismic Reliability Analysis of Structure...

277

of the column number 4 of a deterministic system and uncertain system for case B are shown in figure 3 and 4, respectively.

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Figure 3. The comparison of RMSD at node 16 along horizontal direction

Figure 4. The comparison of RMSF value along horizontal direction at base of the column number 4.

The associated results of the uncertain system as obtained by the MCS procedure are also depicted in the same figures to compare the consistency and accuracy of the proposed approach. Further results for case A and C are compared in Figure 5 and 6. It can be observed from the figures that the results are in good agreement with the results of the MCS study.

278

Subrata Chakraborty and Abhijit Chaudhuri

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Figure 5. The comparison of RMSD at node 16 along horizontal direction for different cases of parameter uncertainty.

Figure 6. The comparison of RMSF value along horizontal direction at base of the column number 4 for different cases of system uncertainty.

The comparison of reliability of a deterministic structural system and uncertain system for Case B obtained by the present approach and MCS are shown in Figure 7 and 8, respectively. The time varying reliability of the deterministic structural system is computed by fixing the value of the random parameters at the mean values. It can be seen that there is a definite change in reliability of uncertain system with that of a deterministic system. It can be

Time Dependent Seismic Reliability Analysis of Structure...

279

further observed that though the difference is small in RMS values, it results significant difference when compared for reliability with that of a deterministic system.

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Figure 7. The comparison of reliability of structure based on RMSD at node 16 along horizontal direction for x B = 3 max(σ x (t )) .

Figure 8. The comparison of reliability of structure based on RMSF value along horizontal direction at base of the column number 4 when x B = 3 max(σ x (t )) .

Further results of reliability are obtained for all three cases of uncertainty and two different barrier levels to define the failure. Figure 9 and 10 shows the reliability with respect to horizontal displacement at node 16 of the building considering the barrier levels as 3 times

280

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and 4 times of the maximum RMS values of the responses of the deterministic system, respectively.

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Figure 9. The comparison of reliability of structure based on RMSD at node 16 along horizontal direction for different cases of system uncertainty when x B = 3 max(σ x (t )) .

Figure 10. The comparison of reliability of structure based on RMSD at node 16 along horizontal direction for different cases of system uncertainty when x B = 4 max (σ x (t )) .

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Figure 11 and 12 shows the reliability with respect to the shear force at base of column 4 considering the barrier levels as 3 times and 4 times of the maximum RMSF values of the responses of the deterministic system, respectively. Comparing Figure 9 and 11 with Figure10 and 12, it is clearly seen that the uncertainty effect is less when the time varying reliability is evaluated based on the base shear force failure criteria. This is due to the fact that the base shear is directly related to the ground acceleration and mass of the structure rather than the stiffness of the structure. Thereby, the uncertainties in excitation process dominate the reliability when computed with respect to base shear.

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Figure 11. The comparison of reliability based on RMSF value along horizontal direction at base of the column number 4 for different cases of system uncertainty when x B = 3 max(σ x (t )) .

Figure 12. The comparison of reliability based on RMSF value along horizontal direction at base of the column number 4 for different cases of system uncertainty when x B = 4 max (σ x (t )) .

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It is generally observed that for a comparatively smaller values of correlation among the elements random properties representing relatively less uncertainty in the system, the change in reliability is small due to the fact that the uncertainty contained in the excitation process dominates in such case, but the random variability of structural parameters have significant contribution for larger uncertainty (larger correlation among the elements uncertain properties). As expected, it is seen that the reliability decreases with level of uncertainty in system parameters, but it is dependent on the barrier level as well.

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8. Conclusion The fully nonstationary random vibration analysis of linear MDOF system under generalized earthquake motion model and system parameter uncertainties are integrated in time varying unconditional reliability evaluation. The conventional seismic response analysis is based on the assumption of slowly varying function of time such that the response can be treated as being nearly independent of time. When the non-stationary input is described as a function of time and frequency the transient nature of response can only be included through convolution integral. This integration can be avoided if the analysis is done in double frequency domain. The analysis in double frequency domain is justified as it takes care of transient nature of the response. In general, for narrow band process crossings may occur in clump i.e. crossing does not behave always as a Poisson process. The first passage failure probability or reliability is determined using the expression given by Vanmarcke based on the assumption that the crossing by the random process is a two state Markov process. Though the uncertainty contained in the excitation process dominates, the random variability of the structural parameters contributes significantly on the reliability of the structure. In general, the increasing level of uncertainty in the structural parameters produce larger variation of the reliability, hence reduced the safety margin of the system. The change of reliability considering parameter uncertainty depends on the variation of uncertain parameters within any structural element and the variation among the structural elements as well. The reliability evaluated based on the base shear force criteria is less affected by the system uncertainty compare to that of displacement criteria. The change of reliability of uncertain system with that of deterministic one depends on the barrier level. The effect of uncertainty is seen to be less for higher barrier level as the reliability for deterministic system becomes almost unity with increase in the barrier level. It possibly indicates that the structure remains almost safe under the uncertainties associated with the load as well as structural properties. The numerical study presented here for parameter uncertainty restricted to Young’s modulus only. However, the methodology can be readily extended to analyze the effect of other uncertain parameters associated with the system.

Acknowledgement The financial support receives in OYS Scheme No. SR/FTP/ET-40/2000 dated 3.10.2000 from the Department of Science and Technology, Govt. of India to initiate this study is gratefully acknowledged. Authors are also thankful to Dr. Giuseppe Carlo Marano of Technical University of Bari, Italy for his suggestions and critical comments about the article.

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In: Reliability Engineering Advances Editor: Gregory I. Hayworth

ISBN: 978-1-60692-329-0 c 2009 Nova Science Publishers, Inc.

Chapter 10

R ESPONSE S URFACE M ETHODOLOGY AND M ULTIPLE R ESPONSE C ASE : O PTIMIZATION M EASURES , D EVELOPMENTS AND C OMPARISONS Rossella Berni∗ Department of Statistics, University of Florence, Florence, Italy

Abstract This chapter focuses on the response surface methodology in the multiple response case by considering the recent issues and the problems linked to the simultaneous optimization of several response variables taking into account the robust design approach. A literature review about dual approach and the multiresponse case with their differences and developments is presented. Furthermore, comparisons with respect to measures and weights are expounded and applied. Finally, an empirical example is shown with transformed or non-transformed response variables.

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1.

Introduction

In the past few decades, Response Surface Methodology (RSM) has been largely applied in order to improve the Taguchi’s approach, especially considering the concept of a robust design and the two-step procedure (dual approach). Initially, the dual approach has been faced through RSM [1] and the class of Generalized Linear Models (GLMs) [2]; nevertheless, the sequential nature of the RSM method, i.e. experimental design and statistical model, has certainly represented an advantage for the application of the RSM methodology [3]. In order to improve both the two-step procedure and the robust approach, Combined Array (CA) [4], is a further suitable alternative. On the other hand, GLMs confirm their important role in this field and they have been integrated with the local optimal experimental design theory [5]. A further issue is the consideration of several response variables. In this case, the main practical problem faced by the researcher appears during the optimization process, since it ∗

E-mail address: [email protected]

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Rossella Berni

is practically unfeasible to reach an ideal optimum simultaneously. In this context, many authors, starting from the methods suggested by [6] and [7], have proposed methods to synthetize and to optimize the response variables [8], [9] and [10]. In addition, a further issue is that of the multiresponse case and the dual approach [11], [12] and [13]. Optimization must take care of a double feature: the simultaneous optimization of several variables, jointly with the consideration of two surfaces, according to the primary and secondary response values linked to the problem of robust design. A literature review is carried out in order to explain the differences about the meaning of multiple response when the dual approach or the combined-array is performed or, alternatively, when a true multiple response case has been applied. In both cases, developments about optimization methods and related issues, such as weighting and scaling, are considered. Finally, different measures suggested in the literature are applied; an optimization measure [14] is compared with [15], which is based on only one surface with a revised optimization method, initially defined in the dual approach context [15] and extended to the multiple response case. Transformation and weighting for response variables are also taken into account. Methods are discussed also through an empirical example in the orthodontic field, in order to improve the accuracy in the measurements of a numerical control machine (N/C machine). This chapter is organized as follows: a literature review is presented in the second section; in the third section, optimization procedures are discussed in detail. Section four presents the data and results about the empirical example, while the concluding remarks are outlined in the final section.

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2.

Literature Review

Many developments and improvements in off-line quality control and statistical methods have been implemented during the 1980’s and 1990’s, when RSM and GLMs were definitively introduced as alternative methods to the Taguchi’s two-step procedure and to the robust design approach [1] and [16]. As was said hereinabove, a different evolution was observed for these two alternative approaches to the Taguchi’s method, given that RSM is a complete and sequential technique based on the experimental design and polynomial statistical models, while the class of GLMs has a lower flexibility and consideration with respect to the experimental planning and design. Nevertheless, through GLMs, theory on estimation methods, as QuasiLikelihood (QL) and Extended Quasi-Likelihood (EQL), and response variable distribution in the Exponential Family, are proper tools in order to improve the analysis of heteroschedasticity, as well as the multiplicative relation between mean and variance and the consideration of the robust approach in the model phase [17] and [18]. The last developments in improving the GLMs approach by the distributional point of view may be found in [19] by Generalized Linear Mixed Models (GLMMs), while in [12] a Bayesian optimization, jontly with the consideration of a specific distribution for the noise factors, is considered.

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In the dual approach case, two statistical models for location and dispersion effects are defined; starting from this simple case of the multiresponse problem, a large debate appeared in the literature in order to find a unique optimum value in the presence of two surfaces, through RSM or, in general, by two statistical models, as in [20] where GLMs and a non-linear programming optimization are applied on defects data according to a Poisson distribution. In addition, the problem of reaching a unique optimum in the dual approach, as an alternative to the Taguchi’s two-step procedure, can also be viewed as a particular case of the general multiresponse situation, when two or several dependent variables are taken into account. In particular, many authors have suggested methods and improvements in order to study the dual approach and the related optimization issues, as in [21], [22]; a very subtle distinction exists when the multiple response case is studied jointly with the dual approach. In fact, in this instance, optimization problems are linked to the need of defining a simple measure. The main authors [6] and [7] introduced two general and recognized methods for achieving a unique optimum value in the multiresponse case of RSM. Subsequently, many other authors have suggested further developments in this specific field, such as [8]. Detailed reviews are found in [3], [23]. We prefer to make a distinction between interesting and applied studies (see [24] and [25]) and theoretical efforts, which have improved or increased the existing methods. We note that the proposal about a unique measure of simultaneous optimization or, alternatively, a revision of existing methods was largely introduced jointly with case studies [26] [27] and [28]. As to the theoretical efforts, research works may be grouped according to: (1) dual approach, (2) simultaneous optimization, (3) simultaneous optimization and the dual approach. These two last topics may be further divided according to the optimization of only one measure or by means of two measures (see section 2.1). Furthermore, comparisons in optimization, also considering weighting and constrained optimization linked to computational problems, are shown in section 3.

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2.1.

Basic Theory

The set of control factors is defined which influence the measurement process: [x1, .., xk, .., xK ]; as well as the set of noise variables: [z1 , .., zh, .., zH ]. Following the theory previously introduced about RSM, the general model for the i response can be written as: 0

0

0

0

0

Yi (x, z) = βi0 + x βi + x Bi x + z δi + z ∆i z + x Λiz + i

i = 1, .., I

(1)

where x is the set of the control variables, z is the set of the noise variables; βi , Bi , δi , ∆i , and Λi are vectors and matrices of the model parameters for each Yi , which is the vector of observations for the i-th dependent variable, i ∼ iidN (0, σi2) is the vector of the random error, Λi is a [KxH] matrix which plays an important role for the robust design approach since it contains the parameters of the control x noise interaction effects. Alternatively, the same full statistical model, up to 2nd order, may be defined according to a general single observation yijt j = 1, ..., J; t = 1, .., T ; where j and t are the j-th run and the t-th replicate of the experimental plan (J xT observations), respectively:

290

Rossella Berni yijt = βi0 + βi1 x1j + .. + βiK xKj + δi1 z1j + .. + δiH zHj + +βi11 x21j

+ .. +

βiKK x2Kj

+

2 δi11z1j

+ .. +

2 δiHH zHj

(2)

+

+λi11 x1j z1j + .. + λiKH xKj zHj + ijt

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i = 1, .., I; j = 1, ..., J; t = 1, .., T Here, xk , k = 1.., K, is the vector of k-th control factor, and xkj is the relative value of the level at j-th run; zh , h = 1, .., H, is the vector of h-th noise factor, and zhj is the level value at j-th run; βik , δik , βikk , δihh , λikh are the generical parameters of the linear, quadratic, and interaction effects related to control, noise and control ×noise factors, respectively. The term ijt is the random error for the single observation yijt . Note that if T = 1 the experimental design is planned without replicates. The general response surface as formulated in (1) may be considered as a surface which can be differently specified when different experimental situations are defined; in general, polynomial models are obtained by an approximation of the original surface of the 2nd order, according to a specific 2nd order experimental design. Nevertheless, optimization may be performed also by considering statistical models defined out of the context of the RSM but covering the requirements of a robust design approach. Therefore, in the dual response case, when the multiple strategy is reduced to optimizing two polynomial models, (1) identifies two measures: one for the location effect ( Yµ ) and one for the dispersion effect (Yσ ). In this case, the two response variables may be computed on T replicates as in the generalization of the Taguchi’s two-step procedure, introduced by [1] or alternatively, in the no-replicates case (T = 1), i.e. the combined-array approach, as introduced by [4], may be formulated; thus the two response variables for the location and dispersion effects become: E(Yˆ ) and var(Yˆ ), computed through E(Yˆ − E(Yˆ ))2; see a synthesis in table 1, cases (a) and (b); see, for example [29], [30]. Furthermore, studying in depth the multiresponse case, the dual approach situation can be generalized by considering the dual approach and several Yi ; i = 1, ..., I. In detail, we have the following general expressions for the location and dispersion effects, respectively: Yi;µ and Yi;σ , according to the general surface model (1) with T ≥ 2; therefore, we must simultaneously optimize Ix2 surfaces taking care of the experimental situations: Nominal the Best (NTB-a specific target value:τntb = τo ), Larger the Better (LTB-maximization) and Smaller the Better (STB-minimization); see table 1, case (d). Note that STB and LTB may be viewed as specific and ideal target values: τstb = 0 or min, τltb = max, respectively. In the literature, many authors have attempted to improve the dual approach optimization, also from the computational point of view: see [31], [32], [33], [34]; less relevance has obtained the consideration of the more complex case, when the dual approach is viewed in a natural multiple response case, as in [34], [35], [9]. However, also when several Yi are planned without replicates, T = 1, the combined-array can be applied by evaluating the expected value and variance for each Yi , as shown previously, see table 1 case (c). An alternative to the solution of the dual approach in the presence of multiple Yi could be the desirability approach [6], or similar and subsequent approaches as in [10], or the definition of a unique function in order to optimize only I surfaces and, at the same time, to perform the two-step procedure of minimizing variability and adjustment to I ideal targets shown in the following section.

Response Surface Methodology and Multiple Response Case

291

Table 1. Summary of surfaces and data in the multiresponse case

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3.

No. Yi \No. rep. I=1

T =1 case (a)-combined E(Yˆ );E(Yˆ − E(Yˆ ))2

I≥2

case (c)-combined array: E(Yˆi);E(Yˆi − E(Yˆi))2

array:

T >1 case (b)-dual Yµ ;Yσ case (d)-dual Yi;µ ;Yi;σ

approach: approach:

Optimization in the Multiresponse Case

In this section, taking into account the previous remarks, we go deeply into the optimization topic, relating to the critical aspects which must be faced when several Yi are evaluated. Starting from [33] which summarizes a very articulated discussion in the literature about the dual approach, we note that the main aim is the effort to solve issues on weighting and on alternative computational solutions and algorithms. In general, almost all of the studies concern the case of multiresponse and the dual approach (cases (c),(d) of table 1), where the presence of replicates is irrelevant for the optimization step. Less frequently, in the literature, multiple dependent variables without the joint consideration of a robust design approach are analyzed. However, general multiresponse problems, like linear dependencies between the Yi variables,[36], are also considered in the specific issue of a robust design approach, [37]. In the following, we are mainly interested in studying in depth the multiple case jointly with a dual response context. In the literature, further developments are related to the introduction and application of nonlinear programming, as genetic algorithms (GA), neural networks and the fuzzy approach. In general, these three methods may be specifically formulated and applied separately; however, further developments, in the scientific world, show the joint application of neural networks with fuzzy sets and genetic algorithms. In this specific field, the genetic algorithms were more generally applied with respect to the other two approaches. However, neural networks may be closer to the multiresponse optimization, in particular for weighting and location of the design space. As regards GA algorithms, an initial work may be found in [38]; more recently, in [13] the performance of these algorithms was analyzed in comparison with the Generalized Reduced Gradient (GRG) approach. In [13], the performance of the genetic algorithms is evaluated in order to identify feasible and unfeasible design points and subsequently the optimal set of factors level; these define a function based on the desirability approach, which has been modified by a penalty function at measuring the degree of non-feasibility for the design space. The choice of the desirability approach is linked to the nature of GA, given that maximization is the goal of the GA approach. A distinction must be made when genetic algorithms are applied in order to discriminate between models in finding the optimal design, as in [39] or in [40]. A neural network [41] is applied to solving the dual response approach with respect to a previous research [28], while a fuzzy approach is suggested in [42] and,

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more recently, in [43]. Here, a formulation is shown of a fuzzy approach defining a unique optimization measure for a weighting general multiresponse case. Weighting is also evaluated in [44] and [45]; in [44] a weighted MSE is proposed as a system of weights for the dual approach; therefore, a procedure in order to estimate two weights for the location and dispersion effects is defined. Note that a couple of weights according to a dual response surface approach and not the consideration of several Yi is searched for. Focusing on other critical aspects of optimization, the transformation of the response variables concerned and the scaling of the factors are the main issues which can be traced back to the general issues of quality of prediction (or optimal value) and of the data. In optimization, an important problem is the Yi ’s range, ∀i. This issue must be considered jointly with the most evident problem of measure unit. The standardization of each Yi , with respect to mean (ˆ µi ) and to standard deviation (ˆ σi), is a compulsory solution when the Yi are not expressed by the same unit of measure. At the same time, when this is the same for each Yi , the standardization must be performed carefully. More precisely, a distinction must be made by evaluating the range of each Yi and the order of magnitude of the corresponding ˆi σ ˆi . In fact, when data are very close to the average value and the order of magnitude of σ is very low, the standardization could imply some problems in achieving good results in simultaneous optimization, in particular, when starting hypotheses or for some diagnostic results, as convergency. Alternatively, when ranges of Yi are different or slightly different and σ ˆi are low, an alternative could be the standardization with respect to the maximum value of Yi . In this way, we obtain data independent of the unit of measure and, at the same time, the arbitrary enlargement of the Yi range is avoided. In the empirical example (section 4) results are reported according to different kinds of weights and standardization approaches. As to this specific topic, in [30] two approaches are described in order to improve the quality of predicted data and the search of the best value of noise variables according to a specific decomposition of the process variance. Finally, a last remark is about the evaluation of weighting for each Yi expressed through their relative cost and correlation with the other response variables. Then, a loss function is introduced as a decomposition of the MSE, as in [45], [46]. In [46], this proposal is based on the identification of a loss function which can be defined by some properties, such as bias, robustness and quality of prediction; an analysis of previous and related works, [34] and [35], is included. Note that in the two references ([30] and [46]), noise variables play an obvious and central role in a robust design context. In [47], the attention is focused on robust design and categorical noise variables when all values of control factors are not acceptable taking into account cost and time issues. Nevertheless, the consideration of a decomposition of the MSE, as in [35], according to several error components related to external elements, costs or consumer loss, is also analyzed through a combined array approach in the multiresponse case [48].

3.1.

Optimization Measures

In this section, as mentioned previously, some issues about optimization are further discussed in the multiresponse case jointly with the consideration of the dual approach and, therefore, of a robust design approach. Our proposal, for details see [14], considers the optimization measures suggested as an alternative to the methods already mentioned and based on the relation between location and dispersion effects. In fact, the central role of

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this relation is well known when the Taguchi’s two-step procedure is performed (see [49]). Therefore, the function shown herein is obtained by considering the concept of performance measure as defined in the Taguchi’s approach and subsequently improved in [50]. In this case, the experimental situation takes care of a multiplicative relation between the location and dispersion effects. Considering a response surface model as defined in (1), a general risk function is defined as follows: R(x, z) = (µ − τ )2 + f (µ(x))σ 2 (3) where f (µ(x)) is in this case equal to E(Yˆ ); therefore, we assume the function f equal to the identity function and σ 2 is defined, for each Yi , as: σ ˆi2 = E(Yˆi − E(Yˆi))2 = var(Yˆi). Thus, the final expression is: Fm,i = σˆi 2 (−σˆi 2/4 + τi )

(4)

The objective function to be minimized is the following, where the weights and the multiresponse case are evaluated: M IN

X

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i

wi Fm,i conditionally on :

X

wi = 1; i = 1, .., I

(5)

i

with τLi ≤ τi ≤ τU i and −1 ≤ xj ≤ +1, where i is for the response variable and m for the multiplicative case. Note that different situations may be assessed according to expression (5) and weights. In fact, two kinds of weights may be applied; case (i): weights selected a priori; case(ii): insertion of the weights in the optimization procedure as parameters in (0, 1) with the constraint in expression (5). Case (i) could be performed when the information a priori, by means of the estimated models or by subjective evaluations on data, such as costs of maintaining optimal Yi ’ s value, are useful. In addition, and jointly with the two weighting cases, transformation about Yi must be taken into account when homogenizing the original response values. As to the previous observaµi , σ ˆi), or, tions of section 3, the standardization of Yi can be performed with respect to (ˆ alternatively, with respect to max Yi . A particular attention is paid to the three experimental situations: NTB, STB and LTB mentioned in section 2.1. More precisely, expression (5) is apparently defined according to the NTB case, when a specific target value is identified; nevertheless, the distinction between the three situations becomes less relevant according to the following observations: 1. the ideal target values in the STB and LTB cases may be identified by real values, when minimizing or maximizing Yi , such as, for example, 0 for shrinkage and 100% for a percent yield are used as Yi ; 2. the ideal target values are not involved in (5) where τi is identified by a specific target defined by the manufacturer; e.g. an eccentricity which can not be set to zero and a very low value close to zero is established as a target; 3. the E(Yˆ ) is computed by the estimated model and it depends on the moment value and the parameter estimates.

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Finally, note that expression (5) allows the optimization of I response variables according to a robust design approach through only one weighted function. 3.1.1. The Other Optimization Approaches In the empirical example of section 4, the optimization already shown is compared with the method suggested in [15] and subsequently analyzed in [33]. The method proposed by these authors takes into account two surfaces, one for the mean and one for the dispersion, for only one response. Therefore, this is case (b) in table 1. Furthermore, the following measure is initially defined as an optimization measure for the NTB case; however, as is pointed out hereinafter, the objective function proposed is a valid measure also in the STB case, according to a definition of an ideal target for the dispersion surface, τσ , equal to zero. Starting from the definition of two statistical models as defined in (2), where the two responses Yµ and Yσ are computed on the T replicates for each experimental run, the objective function may be viewed as a decomposition of the MSE. This measure evaluates simultaneously the deviation from the target value and the dispersion model, as in the following: δµ2 + δσ2 = (ˆ yµ (x, z) − τ )2 + yˆσ2 (x, z)

(6)

which has to be minimized when performing the dual approach for only one response variable. Then, a weighted decomposition is defined as: yµ (x, z) − τ )2 + wσ yˆσ2 (x, z) δµ2 + δσ2 = wµ (ˆ

(7)

where wµ and wσ are the weights for the two surfaces of location and variability. A natural extension for several response variables could be (i = 1, ..., I) :

and:

X

∆i = δµ2i + δσ2i

(8)

X

(9)

∆i =

i

[δµ2i + δσ2i ]

i

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As regards a double step of weighting during optimization, expression (9) may be formulated with two sets of weights: the weights as defined in expression (7), computed for each i response, jointly with the consideration of a global weight wi for each Yi . Therefore, the complete objective function, to be minimized, becomes (i = 1, ..., I) : X i

wi ∆i =

X

wi [w(µ;i)δµ2i + w(σ;i)δσ2i ]

(10)

i

Our weights choice has been to set the following constraints: X

wi = 1

(11)

i

and w(µ;i) + w(σ;i) = 1 ∀i

(12)

In addition, the weights for the mean and the dispersion may be computed by considering ˆi , with a small tolerance value. In the the nominal target values τi and the real values of σ empirical example of section 4, optimization is performed taking into account the double step of weights and the transformation about response variables, as explained in section 3.

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295

An Empirical Example

The application concerned is based on experimental data described in detail in [14]; in the following, (section 4.1) a brief description of the data useful for the interpretation of results is presented. Section 4.2 describes results about models, while section 4.3 shows the optimization results computed through the theory described in sections 3.1 and 3.1.1; a comparison with the desirability approach is also included. Statistical models and optimization are computed by the Statistical Analysis System (SAS) software.

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4.1.

The Experimental Data

The aim of planning is the improvement of the precision in measurements of a N/C machine. The piece considered for the measurement process is a dental implant. The response variables are four quantitative variables Yi , i = 1, .., 4; I = 4 related to the different positioning of the feeler pin on the piece and to the NTB case; the Yi are (target values between brackets): the half-angle cone (τ1 : 1.50mm), the maximum circle diameter (τ2 : 3.00mm), the minimum circle diameter (τ3 : 2.79mm), the neck outside circle diameter (τ4 : 4.10mm). As regards the differentiation about control and noise factors, and the further distinction on the noise factors, (controllable and/or measurable), our aim is to obtain a robust design taking care of the specific location of the machine. In fact, this machine is located in a proper room, where the room temperature (z1 ) may be controlled and set at fixed levels, so this internal noise factor may be evaluated in the inner array as control factor, just because it is closely linked to the measurement process through the temperature of the N/C machine. The inner variables involved in planning are: iteration number (x2 ), drift speed (x1) and measurement speed (x3). In addition, the experiment is performed with three replicates (T = 3) in improving the problem linked to the positioning of the piece on the 2 2 clamp. We apply a 34−1 IV fractional factorial design with word generator: I = z1 x1 x2 x3 . We point out the only one confounding situation involving two first order interaction effects: x1 x2 = z1 x3 . In this case, the x1 x2 interaction can be supposed as negligible, while the z1 x3 interaction could be relevant during the analysis because the room temperature may influence the measurement speed, according to a robust design approach. The coded factor levels are (−1, 0, +1); the experimental region χ, where we are interested to perform our optimization, is connected to the operational conditions of each factor in [−1, +1].

4.2.

Model Results

Surface estimates are shown in tables 2-4. For brevity’s sake, estimates about nonstandardized data (see [14]), the application of desirability function and standardized data by max(Yi ), are not shown. In table 2, estimates of the following model for each standardized Yi are reported: yijt = βi0 + βi1x1j + βi2x2j + βi3x3j + δi1 z1j + +βi33 x23j

+

2 δi11 z1j

+ λi31x3j z1j + ijt

i = 1, .., 4; j = 1, ..., 27; t = 1, .., 3; k = 1, .., 3; H = 1;

(13)

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Rossella Berni Table 2. Fitted surfaces for the Yi responses- yijt standardized by (ˆ µi , σ ˆi) Parameter Intercept z1 x1 x2 x3 x3z1 z12 x23 R2

Y1 0.2994 0.2932 0.1264 -0.0215 -0.1560 -0.3469 -0.2286 -0.2205 0.1624

Y2 -0.5485 1.0369 0.0428 0.0086 -0.1114 -0.1157 0.7970 0.0257 0.8846

Y3 -0.7257 0.9665 -0.0375 0.0375 -0.0657 0.0282 0.9665 0.1220 0.8492

Y4 0.5184 -1.0982 -0.0160 -0.0561 0.0160 0.0601 -0.7134 -0.0641 0.9336

Table 3. Fitted surfaces for the Yi responses- y¯ij standardized by (ˆ µi , σ ˆ i) Parameter Intercept z1 x1 x2 x3 x3z1 z12 x23 R2

Y1 0.2994 0.2932 0.1264 -0.0215 -0.1560 -0.3469 -0.2286 -0.2205 0.3877

Y2 -0.5485 1.0369 0.0428 0.0086 -0.1114 -0.1157 0.7970 0.0257 0.9751

Y3 -0.7257 0.9665 -0.0375 0.0375 -0.0657 0.0282 0.9665 0.1220 0.9625

Y4 0.5184 -1.0982 -0.0160 -0.0561 0.0160 0.0601 -0.7134 -0.0641 0.9859

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d ) standardized by (ˆ Table 4. Fitted surfaces for the Yi responses- var(y µi , σ ˆi) ij Parameter Intercept z1 x1 x2 x3 z12 R2

Y1 0.8609 0.1715 -0.1648 0.0164 -0.3293 0.1581

Y2 0.1983 -0.0357 -0.0238 -0.0516 -0.0396 -0.0912 0.2897

Y3 0.2758 -0.0285 -0.0238 0.0190 0.0095 -0.1522 0.3130

Y4 0.0786 0.0312 0.0069 -0.0173 0.0e-10 0.1194

As we can see, the first dependent variable, Y1 , has a bad good-of-fit; this result confirms the result obtained through the original data. In fact, the optimization on original data by means of expression (5) excludes Y1 . The estimated four models (13) are used in applying the optimization measure (5) when data are standardized. The same surface is applied when the y¯ij data are considered, (table 3), for the application of

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the optimization measure (10). Each y¯ij is computed through the three replicates; note that estimates do not change with respect to the corresponding values in table 2. The explanation of this result may be found by considering that in this case each single observation, for each Yi , is very close to its nominal target value; therefore, these very low discrepancies may be observed only through the R2 values and after the 6th decimal digit. Table 4 shows the estimates related to the variance models, where each dependent variable d ) is computed on the three replicates, for each trial. In this case, the four surfaces are var(y ij different: the models of the first and fourth response variables do not include the quadratic term of the measurement speed and the interaction z1 x3. In the following, the two fitted surfaces for Y1 and Y4 , and, below, for Y2 and Y3 , are shown. d )= β +β x +β x +β x +δ z + var(y ij i0 i1 1j i2 2j i3 3j i1 1j ij

(14)

i = 1, .., 4; j = 1, ..., 27; k = 1, .., 3; H = 1

d ) = β + β x + β x + β x + δ z + δ z2 +  var(y ij i0 i1 1j i2 2j i3 3j i1 1j i11 1j ij i = 1, .., 4; j = 1, ..., 27; k = 1, .., 3; H = 1

(15)

In table 4, the goodness of the four fitted surfaces is not so satisfactory.

4.3.

Objective Functions and Optimization Results

The objective functions (5) and (10) are applied as defined in section 3. Considering expression (5), for each Yi , the E(Yˆi) is computed through the parameter estimates (table 2) and the expected values of the moments. Weights in case (i) are determined on the basis of the relevance of dependent variables. The optimization measure (10) is applied by building a single function, for each Yi , based on the two fitted surfaces, one for the location effect and one for the dispersion effect, as shown in tables 3 and 4. Note that the weights for location and dispersion are included in this single function; then a unique objective function, as (10), is formulated. For this measure, weights may be formulated as:

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• no weighting: this is equivalent to wi = 1 and w(µ;i) = 1; w(σ;i) = 1 ∀i; • case (i): weights wi are determined a priori and w(µ;i) = 1; w(σ;i) = 1 ∀i; • case (ii): weights wi included as parameters in (0,1) in the objective function and w(µ;i) = 1; w(σ;i) = 1 ∀i; • case (iii): double weighting including weights wi as parameters in (0,1) in the objective function and (w(µ;i) = 1/τ(µ;i)); (w(σ;i) = 1/τ(σ;i)) ∀i. Tables 5-10 show the optimization results obtained through the objective functions mentioned previously and described in section 3.1. Some diagnostic measures about optimization are also included, such as: the objective function value (of), the infinity norm of the gradient (kxk∞ ), the determinant of the Hessian matrix (|H|). We have also checked the max-step, i.e. a specified limit for the step length of the line search algorithm, during the

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Table 5. Simultaneous optimization results- measure (5); non-transformed data Results τˆi Best trial Weights of; kxk∞; |H|

measure(5); case (i) τˆ2 = 2.990; τˆ3 = 2.780; τˆ4 = 4.060 z1 = 1; x1 = 1; x2 = −1; x3 = 1 w2 = 0.3e − 6;w3 = 0.9900; w4 = 0.7e − 6 6.51; 1.2e − 4; > 1.0e − 8

measure(5); case (ii) τˆ2 = 3.000; τˆ3 = 2.790; τˆ4 = 4.100 z1 = 0; x1 = 1; x2 = 1; x3 = 0.50 w2 = −5.30e − 9; w3 ≈ 1.0000; w4 = −5.30e − 9 −8.8e3; -0.32; > 1.0e8

Table 6. Simultaneous optimization results- measure (10); non-transformed data Results τˆi

Weights: wµ;i , wσ;i

no wt.

measure(10); case (ii) τˆ2 = 3.000; τˆ3 = 2.790; τˆ4 = 4.084 z1 = 0.79; x1 = 0; x2 = 1; x3 = 0.76 w2 = 0.2897; w3 = 0.4946; w4 = 0.2157 no wt.

of; kxk∞ ; |H|

< 6.0e − 10; 8.0e − 11; > 1.2e − 6

1.3e − 7; 1.0e − 4; < 1.1e − 10

Best trial Weights: wi

measure(10); case no−wt τˆ1 = 1.480; τˆ2 = 2.990; τˆ3 = 2.790; τˆ4 = 4.080 z1 = 0.45; x1 = 1; x2 = 1; x3 = 1 no wt.

measure (10); case (iii) τˆ1 = 1.487; τˆ2 = 2.995; τˆ3 = 2.788; τˆ4 = 4.085 z1 = 0.53; x1 = 0.67; x2 = 0.93; x3 = 0.01 w1 = 0.4269; w2 = 0.0014; w3 = 0.1891; w4 = 0.3826; wµ;1 = 0.9987;wσ;1 = 0.0013; wµ;2 = 0.9997;wσ;2 = 0.0003; wµ;3 = 0.9996;wσ;3 = 0.0004; wµ;4 = 0.9998; wσ;4 = 0.0002; < 6.3e − 7; 6.4e − 4; < 1.0e − 10

Table 7. Simultaneous optimization results- measure (5); standardization by (ˆ µi , σ ˆi) Results τˆi Best trial

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Weights of; kxk∞; |H|

measure (5); case (i) τˆ1 = 1.468; τˆ2 = 2.987; τˆ3 = 2.788; τˆ4 = 4.082 z1 = −0.53; x1 = 1; x2 = −1; x3 = 0.29 w1 = 0.2500; w2 = 0.2500; w3 = 0.4000; w4 = 0.1000 -1.47; 3.6e − 10; > 4.0e − 14

measure (5); case (ii) τˆ1 = 1.475; τˆ2 = 2.988; τˆ3 = 2.788; τˆ4 = 4.087 z1 = −0.50; x1 = 0; x2 = 1; x3 = 0.33 w1 = −1.37e−10;w2 = −1.47e−10; w3 ≈ 1.0000; w4 = −1.58e − 10 -1.92; 0.1e − 3; > 1.9e − 13

first n iterations. Initially, the results are compared within each measure; then, our discussion is extended to the comparisons between the two functions: (5) and (10), and to the desirability approach. In table 5, the optimization measure (5) is applied by considering case (i) and case (ii), respectively with a priori weights or weights involved in the optimization procedure as parameters in (0,1). In addition, in case (i), factors x1 and x2 are included in the model as covariates; this means that during the optimization we set x1 and x2 at fixed levels. Results are very satisfactory for case (ii) where target values are perfectly achieved and tempera-

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Table 8. Simultaneous optimization results- measure (10); standardization by (ˆ µi , σ ˆ i) Results τˆi

Best trial Weights:wi Weights:wµ;i, wσ;i of; kxk∞ ; |H|

measure (10); case (i) τˆ1 = 1.330; τˆ2 = 2.990; τˆ3 = 2.780; τˆ4 = 4.040 z1 = 0.70; x1 = 1; x2 = −1; x3 = 0.40 w1 = 0.9900; w2 = 0.0004; w3 = 0.0003; w4 = 0.0003 no wt.

measure (10); case (ii) τˆ1 = 1.330; τˆ2 = 2.995; τˆ3 = 2.649; τˆ4 = 4.071

measure (10); case (iii) τˆ1 = 1.330; τˆ2 = 2.990; τˆ3 = 2.780; τˆ4 = 4.094

z1 = 0.68; x1 = 1; x2 = −1; x3 = −0.39 w1 ≈ 1.0000; w2 = −1.31e − 9; w3 = 1.31e−9; w4 = 1.31e−9 no wt.

z1 = 1; x1 = 1; x2 = −1; x3 = −1

2.81; 2.3e − 8;> 4.5e − 13

2.73; 8.2e − 4; 8.7e − 4

w1 ≈ 1.0000; w2 = −1.48e − 10; w3 = −1.58e−10; w4 = −1.68e−10 wµ;1 = 0.9987;wσ;1 = 0.0013; wµ;2 = 0.9997;wσ;2 = 0.0003; wµ;3 = 0.9996;wσ;3 = 0.0004; wµ;4 = 0.9998; wσ;4 = 0.0002; 1.52; 1.9e − 9; 3.3e − 11

Table 9. Simultaneous optimization results- measure (5); standardization by max(Yi ) Results τˆi Best trial Weights of; kxk∞; |H|

measure (5); case (i) τˆ1 = 1.330; τˆ2 = 2.990; τˆ3 = 2.780; τˆ4 = 4.040 z1 = 0.37; x3 = 0.38 w1 = 0.8000; w2 = 0.0100; w3 = 0.0100; w4 = 0.1800 3.5e − 10; 6.4e − 11; > 8.0e − 13

measure (5); case (ii) τˆ1 = 1.415; τˆ2 = 2.995; τˆ3 = 2.785; τˆ4 = 4.070 z1 = 0.51; x3 = 0.08 w1 = 0.1450; w2 = 0.2729; w3 = 0.1379; w4 = 0.4441 5.3e − 8; 3.5e − 7; < 4.0e − 15

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Table 10. Simultaneous optimization results- desirability approach- standardization ˆi ) by (ˆ µi , σ Results Best trial of; kxk∞; |H|

max D z1 = 0.50; x1 = −1; x2 = 1; x3 = 0.40 41.86; 6.6e − 13; < 0.0e − 10

ture z1 has a good level. In addition, case (i) presents a temperature value at a high level and this is not a desirable result because the N/C machine increases the measurement errors when temperature is high. Note that, in this part of the application Y1 is not included in the optimization. Measure (10), computed on original data (table 6), is applied by considering the three cases of weighting as described previously. In this context, we evaluate all the Yi , but in case (ii), as in table 5, Y1 is not relevant for the optimization. Targets are not perfectly achieved, with the exception of τ3 in cases (i) and (ii), and τ2 in case (ii). As to diagnostic measures, all the three weighting cases are quite satisfactory, even though problems on |H| in case (iii)

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Rossella Berni

are encountered. At the same time, the temperature achieves a good level with no weights or with the double weighting; on the contrary, factor x2, i.e. the iteration number, is always set at a high value and factor x3, i.e. the measurement speed, is at a high level only in case (i). Therefore, in table 6, a good compromise, taking into account the diagnostic measures, the target values achieved and the setting of factor levels, is obtained in case (ii). If we compare table 5 and table 6, the best result is obtained in case (ii) of table 5. In general, by considering the results on standardized data, optimization problems are encountered; these problems, on convergency or about the violation of initial hypotheses, are substantially related to the issues about standardization discussed in section 3. When convergency is not been reached, the corresponding result is eliminated, while the violation of initial hypotheses will be pointed out, in the following, when this occurred. Tables 7-8 show the results obtained by standardized data through (ˆ µi , σ ˆi), corresponding to those obtained in tables 5-6, respectively. In table 7, the convergency is obtained in both cases, but, in case (i) problems on violation of initial hypotheses were found. However, the results of case (ii) are generally better. As to diagnostic measures, a notable improvement has been achieved (table 8), related to measure (10), where weighting is always performed; in fact, either convergency or initial hypotheses are always met. Nevertheless, a distinction is made about computed target values and the optimal set of factor levels; thus, cases (ii) and (iii) are satisfactory. In case (iii), the only bad result is the temperature at the high level. If we compare table 7 and table 8, case (ii) is the best in both tables, and case (ii) in table 7 is slightly better. In table 9, the results obtained with (5) and data standardized through max(Yi ) are shown; note that the estimated surfaces include only z1 and x3 as independent variables. As was said in section 3, our effort consists in improving the diagnostic results when data are transformed. We point out that case(i) presents the same problems as those of case (i)- table 7; on the contrary, a good result is obtained in case (ii). However this result does not greatly improve the corresponding result of table 7 and, in addition, the information about two factors (x1 and x2) is lost. Finally, the desirability approach applied through maximization of D=

Y

di

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i

on standardized data, is shown in table 10. In [14] we applied the desirability approach on original data with satisfactory results. Unfortunately, the same situation is not presented here; violation of the initial hypotheses is always found. The satisfactory results in table 10 are obtained by imposing a smaller range for temperature; in this case, posing −1 ≤ z1 ≤ 0.5, the diagnostic measures are improved and the optimal setting of factors of table 10 is achieved. A last consideration as a conclusion of this empirical example is about the relevance of weighting in optimization; measures (5) and (10) achieve good results when weights as parameters in (0,1) are included in the optimization procedure; by the double step of weights applied to measure (10), general good results are achieved but the optimal value of temperature is too high. The best result is obtained by case (ii) and original data (table 5).

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301

Conclusion

In this chapter, RSM and the multiresponse case were discussed, together with optimization issues and the robust design approach. The main theoretical differences and some of the most relevant problems were also considered when a robust design approach with two response location and dispersion surfaces is applied in a multiresponse field. An empirical example has shown the difficulties in optimizing an only one weighting objective function, according to different definitions of weights and the assessment of standardized response variables.

References [1] Vining, G.; Myers, R.H. Combining Taguchi and response surface philosophies: a dual response approach J. Qual. Tech., 1990, 22, 38-45 [2] Nelder, J.A.; Lee, Y. Generalized Linear Models for the analysis of Taguchi-type experiments Appl. Stoch. Mod. Data Anal., 1991, 7, 107-120. [3] Myers, R.H.; Montgomery, D.C.; Vining, G.G.; Borror, C.M.; Kowalski, S.M. Response Surface Methodology: a retrospective and literature survey J. Qual. Tech., 2004, 36, 53-77 [4] Myers, R.H.; Khuri, A.I.; Vining, G. Response surface alternatives to the Taguchi robust parameter design approach, Am. Stat., 1992, 46, 131-139 [5] Dror, A.H.; Steinberg, D.M. Robust design for Generalized Linear Models Tech., 2006, 48, 520-529 [6] Derringer, G.; Suich, R. Simultaneous optimization of several response variables J. Qual. Tech., 1980, 12, 214-219

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[7] Khuri, A.I.; Conlon, M. Simultaneous optimization of multiple responses represented by polynomial regression functions Tech., 1981, 23, 363-375 [8] Del Castillo, E.; Montgomery, D.C.; McCarville, D.R. Modified desirability functions for multiple response optimization J. Qual. Tech., 1996, 28, 337-345 [9] Kros, J.F.; Mastrangelo, C.M. Comparing methods for the multi-response design problem Qual. Rel. Eng. Int., 2001, 17, 323-331. [10] Kros, J.F.; Mastrangelo, C.M. Comparing multi-response design methods with mixed responses Qual. Rel. Eng. Int., 2004, 20, 527-539 [11] Ames, A.E.; Mattucci, N.; Macdonald, S.; Szonyi, G.; Hawkins, D.M. Quality loss functions for optimization across multiple response surfaces J. Qual. Tech., 1997, 29, 339-346 [12] Rajagopal, R.; Del Castillo, E.; Peterson, J.J. Model and distribution-robust process optimization with noise factors J. Qual. Tech., 2005, 37, 210-222

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[13] Ortiz, Jr. F.; Simpson, J.R.; Pignatiello, Jr. J.J.; Heredia-Langner, A. A genetic algorithm approach to multiple-response optimization J. Qual. Tech., 2004, 36, 433-450 [14] Berni, R.; Gonnelli, C. Planning and optimization of a numerical control machine in a multiple response case Qual. Rel. Eng. Int. Sp.Is., 2006, 22, 517-526 [15] Lin, D.K.J.; Tu, W. Dual response surface optimization J. Qual. Tech., 1995, 27, 34-39 [16] Lee, Y.; Nelder, J.A. Robust design via Generalized Linear Models J. Qual. Tech., 2003, 35, 2-12 [17] Engel, J.; Huele, A.F. A Generalized Linear Modelling approach for qualityimprovement experiments Tech., 1996, 38, 365-373 [18] Myers, W.R.; Brenneman, W.A.; Myers, R.H. A Dual-response approach to robust parameter design for a Generalized Linear Model J. Qual. Tech., 2005, 37, 130-138 [19] Robinson, T.J.; Wulff, S.S.; Montgomery, D.C.; Khuri, A.I. Robust parameter design using Generalized Linear Mixed Models, J. Qual. Tech., 2006, 38, 65-75 [20] Brinkley, P.A.; Meyer, K.P.; Lu, J.C. Combined Generalized Linear Modelling-nonlinear programming approach to robust process design-a case study in circuit board quality improvement J.R.S.S. Appl. Stat., 1996, 45, 99-110 [21] Copeland, K.F.A.; Nelson, P.R. Dual response optimization via direct function minimization J. Qual. Tech., 1996, 28, 331-336 [22] Del Castillo, E.; Fan, S.-K. S.; Semple, J. The computation of global optima in dual response system J. Qual. Tech., 1997, 29, 347-353 [23] Myers, R.H. Response surface methodology-current status and future directions J. Qual. Tech., 1999, 31, 30-44

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[24] Berget, I.; Næs, T. Sorting of raw materials with focus on multiple end-product properties J. Chem., 2002, 16, 263-273 [25] Wong, W.K.; Ng, S.H.; Xu, K. A statistical investigation and optimization for an industrial radiography inspection process for aero-engine components Qual. Rel. Eng. Int., 2006, 22, 321-334 [26] Reddy, P.B.S.; Nishina, K.; Babu, A.S. Unification of robust design and goal programming for multiresponse optimization- a case study Qual. Rel. Eng. Int., 1997, 13, 371-383 [27] Persson, P.; Kammerlind, P.; Bergman, B.; Andersson, J. A methodology for multicharacteristic system improvement with active expert involvement Qual. Rel. Eng. Int., 2000, 16, 405-416 [28] K¨oksoy, O.; Doganaksoy, N. Joint optimization of mean and standard deviation using response surface methods J. Qual. Tech., 2003, 35, 239-252

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[29] Kunert, J.; Erdbr¨ugge, M.; Ewers, R. An experiment to compare Taguchi’s product array and the combined array J. Qual. Tech., 2007, 39, 17-34 [30] Mir´o-Quesada, G.; Del Castillo, E. Two approaches for improving the dual response method in robust parameter design J. Qual. Tech., 2004, 36, 155-168 [31] Fan, S.-K. S. A generalized global optimization algorithm for dual response system J. Qual. Tech., 2000, 32, 444-456 [32] Del Castillo, E.; Montgomery, D.C. A nonlinear programming solution to the dual response problem J. Qual. Tech., 1993, 25, 199-204 [33] Tang, L.C.; Xu, K. A unified approach for dual response surface optimization J. Qual. Tech., 2002, 34, 437-447 [34] Pignatiello, Jr. J.J. Strategies for robust multiresponse quality engineering IIE Trans., 1993, 25, 5-15 [35] Vining, G.G. A compromise approach to multiresponse optimization J. Qual. Tech., 1998, 30, 309-313 [36] Box, G.E.P.; Hunter, W.G.; MacGregor, J.F.; Erjavec, J. Some problems associated with the analysis of multiresponse data Tech., 1973, 15, 33-51 [37] Chiao, C.-H.; Hamada, M. Analyzing experiments with correlated multiple responses J. Qual. Tech., 2001, 33, 451-465 [38] Carlyle, W.M.; Montgomery, D.C.; Runger, G.C. Optimization problems and methods in quality control and improvement J. Qual. Tech., 2000, 32, 1-17 [39] Heredia-Langner, A.; Montgomery, D.C.; Carlyle, W.M.; Borror, C.M. Model-robust optimal design: a genetic algorithm approach J. Qual. Tech., 2004, 36, 263-279

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[40] Drain, D.; Carlyle, W.M.; Montgomery, D.C.; Borror, C.; Anderson-Cook, C. A genetic algorithm hybrid for constructing optimal response surface design Qual. Rel. Eng. Int., 2004, 20, 637-650 [41] K¨oksoy, O.; Yalcinoz, T. A Hopfield neural network approach to the dual response problem Qual. Rel. Eng. Int., 2005, 21, 595-603 [42] Kim, K.J.; Lin, D.K.J. Dual response surface optimization: a fuzzy modelling approach J. Qual. Tech., 1998, 30, 1-10 [43] Xu, K.; Lin, D.K.J.; Tang, L.C.; Xie M. Multiresponse systems optimization using a goal attainment approach IIE Trans., 2004, 36, 433-445 [44] Jeong, I-J.; Kim; K.-J.; Chang, S.Y. Optimal weighting of bias and variance in dual response surface optimization J. Qual. Tech., 2005, 37, 237-247 [45] Ding, R.; Lin, D.K.J.; Wei, D. Dual-response surface optimization: a weighted MSE approach Qual. Eng., 2004, 16, 377-385

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[46] Ko, Y-H.; Kim K-J.; Jun C-H. A new loss function-based method for multiresponse optimization J. Qual. Tech., 2005, 37, 51-59 [47] Robinson, T.J.; Brenneman, W.A.; Myers, W.R. Process optimization via robust parameter design when categorical noise factors are present Qual. Rel. Eng. Int., 2006, 22, 307-320 [48] Romano, D.; Varetto, M.; Vicario, G. Multiresponse robust design: a general framework based on combined array J. Qual. Tech., 2004, 36, 27-37 [49] Nair, V.N. Taguchi’s parameter design: a panel discussion Tech., 1992, 30, 127-161

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[50] Leon, R.V.; Shoemaker, A.C.; Kacker, R.N. Performance measure independent of adjustment Tech., 1987, 29, 253-285

In: Reliability Engineering Advances Editor: Gregory I. Hayworth

ISBN: 978-1-60692-329-0 © 2009 Nova Science Publishers, Inc.

Chapter 11

SEED DEVELOPMENT IN CASTOR (RICINUS COMMUNIS L.): MORPHOLOGY, RESERVE SYNTHESIS AND GENE EXPRESSION Grace Q. Chen* U.S. Department of Agriculture, Agricultural Research Service, Western Regional Research Center, 800 Buchanan Street, Albany, CA 94710, U.S.A

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Abstract Castor (Ricinus communis L.) is a non-eatable oilseed crop producing seed oil comprising 90% ricinoleate (12-hydroxy-oleate) which has numerous industrial uses. However, the production of castor oil is hampered by the presence of detrimental seed storage proteins, the toxin ricin and hyper-allergenic 2S albumins. We are developing a safe source of castor oil by two approaches: blocking gene expression of the ricin and 2S albumins in castor seed and engineering a temperate oilseed crop to produce castor oil. To understand the mechanisms underlying the synthesis of ricin, 2S albumins and ricinoleate/oil, we conducted a series of seed development studies in castor, including endosperm morphogenesis, storage compound accumulation and gene expression. The entire course of seed development can be divided into four stages, which are recognizable by distinct seed coat color and cellular endosperm volume. Synthesis of ricin, 2S albumins and oil occur during cellular endosperm development. Concomitantly, we observed increased transcript levels of 14 genes involved in synthesis of ricin, 2S albumin and oil, but with various temporal patterns and different maximal inductions ranging from 2 to 43,000 fold. The results indicate that gene transcription exerts a primary control in castor reserve biosyntheses. Based on the temporal pattern and level of gene expression, we classified these genes into five groups. This transcriptionprofiling data provide not only the initial information on promoter activity for each gene, but also a first glimpse of the global patterns of gene expression and regulation, which are critical to metabolic engineering of transgenic oilseeds. Since all these studies are based on a welldefined time course, the results also provide integrative information for understanding the relationships among endosperm morphogenesis, reserve biosynthesis and gene expression during castor seed development.

*

E-mail address: [email protected]. Tel, 510-559-5627; Fax, 510-559-5768.

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Introduction Castor oil is the only commercial source of ricinoleate, 12-hydroxyoleic acid. The hydroxy group imparts unique chemical and physical properties that make castor oil a vital industrial raw material for numerous products such as lubricants, cosmetics, paints, coatings, plastics and anti-fungal products [10]. Now with the growth of biodiesel, castor oil has an expanding new use. It is the best substance for producing biodiesel because it is soluble in alcohol and does not require the heat energy input of other vegetable oils in transestering them into fuel [16]. In addtion, the castor oil biodiesel eliminates the need for adding sulfur-based lubricity components in conventional diesel fuel, significantly reducing air pollution [24]. However, the presence of ricin toxin and hyper-allergenic 2S albumins in seed poses a health concern during its cultivation and processing. In order to develop a safe source of castor oil, we conducted a series of seed developmental studies in castor. This chapter illustrates our results and findings which are critical to metabolic engineering of ricinoleate production in transgenic oil seeds, as well as genetic suppression of ricin and 2S albumin in castor.

Morphlogical & Physiological Changes during Seed Development

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The basic structures of a mature castor seed include testa, caruncle, endosperm and embryo (Figure1). The seed consists of a mass of endosperm and an embryo with two thin papery cotyledons lying in the center of the endosperm. The endosperm is not absorbed by the embryo until seed germination. Castor plants produce a racemic, monoecious inflorescence, with male and female flowers blooming asynchronously, so pollination must be used to provide a common starting point for determining seed developmental age. Mature female flower buds just before opening were hand-pollinated and labeled as 0 day after pollination (DAP). Developing seeds harvested at 7-day intervals from 12 to 61 DAP were dissected out from the seed capsule. Morphological and physiological characteristics of the seeds were recorded immediately upon collection. For biochemical analysis, separate sets of seeds were frozen immediately in liquid nitrogen after dissection and stored at –80°C.

Figure 1. Sections through a castor seed. Left, section of cotyledons; middle, transverse section; right, longitudinal section.

Seed Development in Castor (Ricinus communis L.) No cellular Cellular endosperm (% filled) endosperm 30% 70% 100%

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Figure 2. The morphological changes of castor seeds during development. Upper, transverse section. Cellular endosperm is shown in opaque color. Lower, whole seeds. DAP, days after pollination. A 100

Lipid

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Water

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Endosperm Development

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20

12 19 26 33 40 47 54 61 Developing Seed (DAP)

ar ion at cle z i u r n ula ee ell Fr ge C a st

n n tio tio ra icca ta u s M De

Figure 3. The physiological changes of castor seeds during development. Each data point represents the mean (± SD) of three measurements and at least 20 seeds.

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Grace Q. Chen A Precursor protein

S

A 145 bp

L

B

ClaI/444 bp

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Ricin cDNA 352 bp

RCA cDNA

EcoRI/1015 bp

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Scale for 100 bp:

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ND RI ClaI ND RI ClaI ND RI ClaI

2000 1200 800

Ricin

400 200 C 2000 1200 800 400 200

RCA

(bp) Marker

26 DAP

40 DAP

54 DAP

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Figure 4. Ricin and RCA genes and RT-PCR-based restriction fragment analysis. A, a scale diagram showing a precursor protein with positions of the signal peptide (S), A-chain (A), linker (L), and Bchain (B), and the cDNAs of ricin and RCA with the primer sites (arrows) and unique restriction enzyme sites indicated. B, the RT-PCR products of ricin gene and C, RCA gene were loaded for undigested samples (ND), and samples digested with EcoRI (RI) and ClaI restriction enzymes.

Figure 2 illustrates the morphology of a developing seed, and Figure 3 showed the physiological changes, including seed weight and size [12], lipids and water content, and nonlipid dry weight [14]. Among all morphological features we examined, the testa color and the cellular endosperm volume are most distinctive in determining the developmental age of castor seed. Based on changes of the testa color and the cellular endosperm volume, we divided seed development into four stages (I to IV, Figure 2). During stage I of seed development, the seed had ivory white testa color and grew rapidly to full size at 19 DAP ([12], Figure 3B,C). The majority of space in the seed was filled with inner integument tissue and the endosperm tissue remained in free nuclear stage (Figure 2). The seeds contained about 90% water, 5% lipids and 5% non-lipid dry weight (Figure 3A). These small amounts of lipids and dry mass are probably structural components of the cell, such as membranes and cell walls for maintaining basic cell function. In stage II, the volume of cellular endosperm increased dramatically (Figure 2, 26 to 40 DAP). The growing endosperm ultimately displaced the inner integument, and accounted for about 90% of the seed mass. The expansion of endosperm volume coincided with the changes of testa color which started from the

Seed Development in Castor (Ricinus communis L.)

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caruncle end and spread to the opposite chalazal end. By the end of this phase, the endosperm tissue had reached full expansion. The testa was covered with uneven shades of purple and brown color and partially sclerified. Seeds in stage III (47 to 54 DAP) can be distinguished by fully filled endosperm and sclerified, dark pigmented testa. During the course of cellular endosperm development (26 to 54 DAP), the seeds gradually lost water and gained storage lipid to 60% when it reached to the end of the maturation stage (Figure 3A). At 61 DAP (desiccation phase), capsules senesced and seeds desiccated showing a reduction in seed weight (Figure 3B). It is known that a developing castor cellular endosperm consists of living cells that synthesize and store oil and protein and produce enzymes necessary for the mobilization of these reserves; a mature seed contains up to 64% lipids, 18% proteins, and a negligible level of carbohydrate [6]. Our measurements of the major fractions are similar to the previous described. There was about 20% nonlipid dry mass accumulated at the end of seed development in addition to the 5% base dry mass for structural components (Figure 3A). The additional nonlipid dry mass includes storage proteins, such as ricin and 2S albumins (Figure 5 and 7). A 2000 800

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De sic ca tio n

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Ce llu ;ar

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Fr e sta e n ge ucl ea r

Marker

Endosperm development

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E

Figure 5. RT-PCR and Northern analysis of ricin and RCA gene expression. D, approximately 5 µg of total RNA was loaded on each lane. The blot was exposed to film for 10 seconds. E, a ethidium bromide stained gel as a control for equal RNA loading.

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Leaf

Developing Seed 12 19 26 33 40 47 (DAP) 1.3 kb

A B C

Figure 6. Northern analysis of 2S albumin gene. Approximately 5 µg of total RNA was loaded on each lane. The 2S albumin blot was exposed to film for 10 seconds (A), or 10 minutes (B) to show dramatic differences of the transcript levels. Bottom (C), a ethidium bromide stained gel as a control for equal RNA loading. RNA size (kb) is indicated. DAP stands for day after pollination.

A 30

prepro2S albumin (29.3 kDa)

20 14 6.5

(kDa) RicC3 (4.2 kDa)

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45 30 20 14

Endosperm development

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6.5 (kDa)

Marker

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Developing Seed ( DAP)

N

th row g o

Ex

n

sio

n pa

n

tio

M

ra atu

n

tio

ca sic

De

Figure 7. Expression profiles of 2S albumins and total proteins during seed development. A, immunodetection of the 2S albumins by using an anti-2S albumin antibody, which detects both the prepro2S albumin (29.3 kDa) and small subunit of RicC3 (4 kDa). B, visualization of total proteins by using Coomassie staining.

Seed Development in Castor (Ricinus communis L.)

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Expression of Ricin and Ricinus Communis Agglutiinin Genes during Seed Development Among the factors limiting domestic production of castor oil is the presence of the toxin ricin and its less toxic homologue Ricinus communis agglutinin (RCA) in seeds. Ricin is a dimeric glycoprotein composed of a toxic A-chain and a lectin B-chain linked by disulfide bonds. The A-chain is a ribosome-inactivating enzyme which depurinates a specific adenine residue of 28S ribosomal RNA, thereby inactivating eukaryotic protein synthesis [20]. The B-chain, which contains two galactose-binding sites, binds specifically to cell surface glycoproteins or glycolipids and facilitates the movement of A-chain into cells [64]. RCA is a tetramer consisting of two ricin-like dimers held together by non-covalent bonds. Mature RCA is less toxic than ricin and causes agglutination of red blood cells in mammals [41, 52]. Both ricin and RCA genes encode a precursor protein containing an N-terminal signal sequence, an A chain, a 12 amino acid linking region and a B chain [40, 50, 67]. The A chains of ricin and RCA differ in 18 of their 267 residues and are thus 93% identical at the amino acid level, whereas the B chains differ in 41 of 262 residues giving 84% identity [50]. Although the sequences of ricin and RCA genes are known, their transcriptional expression patterns have not been distinguished due to their high degree of sequence similarity. To assess the successful development of a ricin-free crop it is necessary to distinguish the expression of these two genes. We have designed a gene specific reverse transcription-polymerase chain reaction (RT-PCR) assay to differentiate expression of the ricin and RCA genes in developing seeds [13]. Figure 4A illustrates a scaled diagram showing a precursor protein, cDNAs of ricin and RCA genes, PCR primers, and the unique restriction enzyme sites. Using the selected primers, RT-PCR should amplifies a 1,562 bp ricin cDNA, which generates 1,263 and 299 bp fragments after ClaI digestion. In the case of RCA, the amplified cDNA is 852 bp and generates 663 bp and 189 bp fragments after EcoRI digestion. With this design, we examined the expression of ricin and RCA genes during the seed development. As shown in Figure 5A and B, no ricin and RCA cDNA was detected in young seeds of 12 and 19 DAP, but the expression of each increased significantly in 26 DAP seeds, and the upward trend continued into later stages until 54 DAP. A trace amount of RCA transcripts could still be detected in desiccating seeds at 61 DAP and in two-year old dormant seeds. The identity of the RT-PCR products was further validated by sequencing (data not shown). However, the different unique restriction enzyme sites allow simple verification for the authenticity of RTPCR products by restriction fragment analysis. The RT-PCR products from 26, 40 and 54 DAP developing seeds were digested with ClaI and EcoRI. The resulting restriction fragments are shown in Figure 4B and C. As predicted, ClaI digestion produced 1263 and 299 bp fragments for ricin cDNA, while EcoRI digestion produced 663 and 189 bp fragments for RCA cDNA. To confirm the expression patterns of ricin and RCA genes revealed by the RTPCR method, a Northern analysis was performed on total RNA from the same set of seeds. The result showed a pattern similar to that obtained using RT-PCRs (Figure 5D). The results from this study indicate that the expression of ricin and RCA genes corresponds to the developmental profile of endosperm. The mRNA signal is not detectable before the endosperm begins to cellularization (12 and 19 DAP), but becomes significant from 26 to 54 DAP when the cellular endosperm expands to occupy most of the seed volume. When seeds enter the desiccation stage (61 DAP), the expression of ricin and RCA genes

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drop to a trace level. These observations are similar to those of lectin and ribosome inactivating proteins from various species including soybean, common bean, wheat, and barley [22]. In soybean, the expression of a lectin gene is transcriptionally activated and spatially regulated during the mid-maturation phase of embryogenesis, and repressed in late maturation when seeds enter the desiccation phase and begin dormancy [23, 48]. In castor seed, the ricin and RCA mRNA levels are correlated with the dimension of endosperm, suggesting that they are also spatially and temporally regulated. Our RT-PCR results supported by our Northern analysis not only present the specific temporal expression patterns for ricin or RCA genes, but also define their developmental profiles based on ages and factors more directly related to distinct stages during castor seed development. In addition, we have demonstrated that an RT-PCR assay and restriction fragment analysis can be used to determine the transcriptional expression patterns of ricin and RCA genes. Although ricin and RCA share substantially similar biochemical characteristics, they are distinctive genes and may have different functions. We are using genetic engineering to generate a series of plants with altered expression of ricin and RCA genes. The RT-PCR assay will provide a method to screen individual transgenic lines to assess the efficacy of different gene silencing constructs.

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Castor 2S Albumins Castor 2S albumin proteins were identified as the primary allergenic components based on fractionation studies [17, 44, 65, 70]. By analyzing the sequences of the 2S albumin gene and proteins, it was found that a single preproprotein produces two heterodimeric 2S albumin proteins through post-translational modification [32, 33], RicC1 [59] and RicC3 [4, 17, 44]. As part of a genetic approach to eliminating 2S albumin in castor, we characterized the 2S albumins in castor, including gene expression [12], protein accumulation [1] and phylogeny analysis [1, 12, 15]. 2S Albumin Transcripts in Developing Seeds. Figure 6 shows the 2S albumin messenger RNA levels in seeds collected at 12, 19, 26, 33, 40 and 47 DAP. A sample of total RNA from young leaves was included in the same blot to test for 2S albumin gene expression in vegetative tissue of castor. During the course of castor seed development, the expression of the 2S albumin was very low in young seeds at 12 and 19 DAP. The mRNA levels rose sharply during 26 to 33 DAP when the endosperm underwent rapid expansion. Once the endosperm tissue had fully expanded (40 DAP), the level of 2S albumin mRNA decreased. The mRNA level continued to decrease as shown for mRNA from 47 DAP (Figure 6A). After 47 DAP, the mRNA degraded largely and did not show its signal integrity (data not shown). No 2S albumin mRNA was detected in leaf tissue even after prolonged exposure of the Northern blot (Figure 6B). A similar bell-shaped pattern was observed when we used quantitative RT-PCR analysis in the developing seeds from cultivar, Hale, where we observed an increase at 26 DAP, a maximum induction of about 100-fold at 40 DAP and decreases thereafter [1]. The observed temporal expression pattern of 2S albumin gene is similar to the expression profiles of 2S albumins from oilseed rape [7] and Arabidopsis [26]. In castor, the expression of 2S albumin gene exhibits a temporal pattern coinciding with cellular endosperm development. The timing suggests that endosperm development and the 2S albumin gene expression are spatially and temporally co-regulated. Our results support the observations that castor endosperm is the compartment for synthesizing and storing various seed proteins

Seed Development in Castor (Ricinus communis L.)

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.

313

including 2S albumins [51, 70]. Furthermore, we present a defined developmental profile for the 2S albumin gene expression during seed development. 2S Albumin Proteins in Developing Seeds. To examine temporal profile of 2S albumin protein in developing seeds, we developed an antibody using a peptide whose sequence was derived from the small subunit of RicC3 [1]. As stated earlier, a single prepro2S albumin undergoes post-translational modification to produce two mature proteins, RicC1 and RicC3. This antibody allowed us to examine not only the accumulation of the two different forms of 2S albumins but also the timing of the post-translational modification. As shown in Figure 7A, the anti-2S albumin antibody detected only the pre-pro2S albumin (29.3 kDa) and the small subunit of the mature RicC3 (4.2 kDa) in developing seed, with high specificity. The pre-pro2S albumin started synthesis at 26 DAP. The levels of the protein increased to a plateau at about 40 to 47 DAP, when the endosperm reached its maximum volume (Figure 2), and remained at the same high level thereafter. Whereas, the small subunit of the mature RicC3 appeared at 40 DAP, 2 weeks after the synthesis of the pre-pro2S albumin and its level reached a plateau after 47 DAP. The timing of the appearance of mature RicC3 at 40 DAP coincided with that of a change in total protein profile (Figure 7B). In the seeds at early developmental stages between 19 and 33 DAP, high molecular weight proteins from 40 to 60 kDa were predominant. These proteins degraded rapidly from 33 to 47 DAP. Concomitantly, smaller proteins including a 29-30 kDa and a 4 kDa protein started accumulation and became predominant during of maturation (47 to 61 DAP). We observed that both forms of the 2S albumin proteins continued to accumulate during the rest of the stages of seed development. The steady levels of the pre-pro2S albumin and the mature protein during the last stage of seed development are due to the stability of the proteins, not constant protein synthesis, considering the decrease in the amount of the 2S albumin transcript after 40 DAP (Figure 6). The accumulation of castor 2S albumins took place from the middle stage of the seed development, which consisted with the temporal patterns of seed storage protein synthesis in many plants [30]. Similarities Of Castor 2S Albumin To Those From Different Plant Species. Forty best-hit sequences, representing 37 plants and 3 metazoan species, were retrieved from a BLASTP search on the GenBank protein database with castor 2S albumin protein sequence (gi:227594) as a query [15]. Among 37 plant sequences, 28 were unique 2S albumin-like sequences, representing 25 dicots, 1 monocot and 2 conifer species. The remaining 9 plant sequences were 3 redundant sequences (A37931 from Brassica napus subspecies, CAA11026 and CAA46705 from Hordeum vulgare subspecies), 1 incomplete sequence (A59346 from Lycopersicon esculentum) and 5 monocot sequences, including high molecular weight glutenins (AAL82616 from Aegilops umbellulata and CAC40676 from Secale cereale), Globulins (AAM77580 from Aegilops tauschii and BAA09308 from Oryza sativa) and Gliadin (D22364 from Triticum aestivum). These 9 sequences and the metazoan sequences were eliminated from further data analysis. To achieve an optimal alignment for mature 2S albumin proteins, we first removed the signal peptide portion of sequences. The castor and sunflower precursors have two proteins divided into RicC1and RicC3, BA1 and BA2, respectively. Pairwise comparison and multiple sequence alignment were then generated by ClustalW. Based on this alignment, a phylogenetic tree was constructed using the PHYLIP program [21]. As shown in Figure 8, thirty 2S albumins were divided into 6 clades. Clade A and B each contains a single sequence, Hvul and Fesc. Clade C, D and F comprise multiple sequences from plants sharing the same taxonomic families, Pinaceae, Fabaceae and

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Brassicaceae, respectively. The castor 2S albumin belongs to clade E that contains 10 sequences from 8 eudicot families. Protein name

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0.1

Hvui Fesc Ahyp Lang Gmax Cmas Atha Rsat Bcar Bnig Brap Bnap Salb Bjun Bole Ghir Lusi BA1 RicC3 Bexc BA2 Aocc Sind RicC1 Jreg Cill Ckur Mcha Pstr Pgla

Plant name Barley (Hordeum vulgare) Buckwheat (Fagopyrum esculentum) Peanut (Arachis hypogaea) Blue lupine (Lupinus angustifolius) Soybean (Glycine max) Maninlang (Capparis masaikar) Arabidopsis (Arabidopsis thaliana) Radish (Raphanus sativus) Ethiopian mustard (Brassica carinata) Black mustard (Brassica nigra) Field mustard (Brassica rapa) Oilseed rape (Brassica napus) Yellowe mustard (Sinapis alba) Oriental mustard (Brassica junce) Cabbage (Brassica oleracea) Cotton (Gossypium hirsutum) Flax (Linum usitatissimum) Sunflower (Helianthus annuus) Castor (Ricinus communis) Brazil nut (Bertholletia excelsa) Sunflower (Helianthus annuus) Cashew (Anacardium occidentale) Sesame (Sesamum indicum) Castor (Ricinus communis) English walnut (Juglans regia) Pecan (Carya illinoinesis) Pumpkin (Cucurbita cv. kurokawa) Balsam pear (Momordica charantia) White pine (Pinus strobus) White spruce (Picea glauca)

NCBI protein gi # Clade 123972 17907758 15418705 19141 2305020 13124692 68853 81711 1523806 17728 349402 468018 2129817 17721 17878 167359 20502192 27526481 227594 17713 27526481 24473800 13183175 227594 1794252 28207731 459405 21327881 20645 3550129

A B C

D

E

F

Figure 8. Unrooted phylogenetic tree of 2S albumins. Each 2S albumin is given a protein name. The corresponding plant name is displayed as both common name and scientific name (Latin, in parenthesis). The protein sequences used for constructing the tree were obtained from NCBI database, their corresponding accession numbers are given as protein gi number. The scale bar on the left indicates 0.1 nucleotide substitutions per site.

Seed Development in Castor (Ricinus communis L.) Name RicC3 Ckur RicC1 Lusi Cill Jreg Bexc Sind Bole Bjun Cmas Aocc

10 20 | | - - - - E S K GE RE GS S S QQCRQE V QRK D- L - - - V E V E E N R QGR E - E R C R QMS A R E E - L - - - - - - - - - - - - P S QQGCRGQI QE QQNL DT NQGRGGQGGQGQQQQCE K QI QE QDY L ME I D E D I D N P R R R GE S - C R E QI QR QQY L ME I D E D I D N P R R R GE G- C R E QI QR QQN L T L E E E QE E N P R GR S E QQC R E QME R QQQL - - - - T A I DDE A NQQS QQCRQQL QGRQ- F F DE DDA T NP A GP F RI P K CRK E F QQA QHL F DE DDA T DS A GP F RI P K CRK E F QQA QHL V D E E E D N - - - - - - Q L WR C Q R Q F L Q H Q R L V E E DS GR- - - - - - - E QS CQRQF E E QQRF C

RicC3 Ckur RicC1 Lusi Cill Jreg Bexc Sind Bole Bjun Cmas Aocc

60 70 | | S P GE E V L R MP GD E N - QQQE S QQL - - - R D V L Q M R G I E N P WR R E G G S F - - GQGP - - - - - - - RRS DNQE RS L - - K GGRS Y Y Y NQGRGGGQQS QHF - - S GGY DE D- - - - - - - - NQRQHF - - S GGY DE D- - - - - - - - NQRQHF - - E S P Y QNP R- - - P L RRGE E P HL P Y GGE E D E V L E MS T GN QQS E QS L D S E F D F E D D ME N P QGP QQR P P L L D GE F D F E D D ME N S QGP QQR P P L L E DE V E DDNDDE N- - - - QP RRP A L - - - - - - - - - - - - - - - - - QRQE S L

RicC3 Ckur RicC1 Lusi Cill Jreg Bexc Sind Bole Bjun Cmas Aocc

110 120 | | - - - I QQGQL HGE E - - S E RV A QRA - - - - E QR QA R GQE - - GR QML QK A - - - QS QGQL QGQD- - V F E A F RT A Q D I Q Q Q G Q Q Q E V E R WV Q Q A K Q V A - - - QQE E GI R GE E - - ME E MV QC A - - - QQQQGL R GE E - - ME E MV QS A - - - R E E ME L QGE Q- - MQR I MR K A - - - QQE GGY QE GQ- - S QQV Y QRA QQI QQQGQQQGK QQMV S R I Y QT A QQI QQQGQQQGK QQMV S R I Y QT A QRQI I QGP QQ- - - - - L RRL F DA A QQE QI K GE E V - - - - - - RE L Y E T A

30 | SSCERYL RS CE QY L RQCQE Y I RS CQQF L NRCQDY L NHCQY Y L N H C R MY L RS CQRY L R A C Q Q WL R V C Q Q WL RA CQRF I RNCQRY V C

40 | RQS S S RRQQS - - K QQV S - WE K V Q - RQQCR- RQQS R- R QQME - S QGRS - H K QA MQS H K QA MQS HRRA QF K QE V QR-

80 90 | | QQCCNQV K QV RDE CQCE A I D E C C R E L K N V D E E C R C D ML R GC C D H L K QMQS QC R C E GL DS CCDDL K QL RS E CT CRGL R QC C QQL S QME E QC QC E GL R QC C QQL S QMD E QC QC E GL D E C C E QL E R MD E MC R C E GL RDCCQQL RNV DE RCRCE A I L QCCNE L DQE E P L CV CP T L QQCCNE L HQE E P L CV CP T L RQCCNQL RQV DRP CV CP V L RE CCQE L QE V DRRCRCQNL CC C C

GE I RNL ANL RDL SDL RDL ENL RDL T HL T HL RNL SEL

V P P P P P L P P P P P

315

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - GGGP S GGGL S - GGQP - GGRY

KYI A EEI A RQA I ERAI RQA V RQV V R MML RQA V K GA S K GA S RQA A E QMV

50 | R L L L -

100 | E DQRE- E QQGQMR RQQRRQRRQRQQKAVK KAVK QQV L RQL Q

130 140 | | S S C GV R - - - - C MR QT - - - - S MC GI R P QR - C D F - - - - - - S MC GV S P T E - C R F - - - - - - GQCGT QP S R- CQL QGQQQS A K E CGI S S RS - CE I R- - - - RS NE CGI S S QR- CE I R- - - - RS S R C N L S P QR - C P MGGY - - T A R R C N MR P QQ- C QF R V I F V - K V C N I P QV S V C P F QK T MP GP K V C N I P QV S V C P F QK T MP GP N I C N I P N I G A C P F R - A WP - RI CS I S P S QGCQF QS S Y - - C

- - - - - - - - WT WT DE N-

- - - WF WF WF WL - SY SY - - -

C

Copyright © 2008. Nova Science Publishers, Incorporated. All rights reserved.

Figure 9. Multiple sequence alignment of 2S albumins. The 2S albumin protein names and their sequence sources are described in the Figure 8. The dark- and light-shaded sequences are designated for identical and similar amino acids, respectively. The conserved eight cysteines are marked under with "C".

To further characterize the common amino acid pattern shared in 2S albumins, we selected a subset of 12 sequences showing >30% amino acid identity to RicC1 or RicC3. Their multiple sequence alignment was re-generated by ClustalW, and manually adjusted for improvement (Figure 9). As shown in Table 1 of the percentage identities, the Jreg from English walnut and Ckur from pumpkin have the highest sequence identity to RicC1 and RicC3, showing 45.1% and 33.9% respectively, while the identity between RicC1 and RicC3 is only 25.5%. With the exception of Ckur, RicC1 is more homologous to all the other 2S albumins than RicC3 (top and bottom lines of Table 1). Nonetheless, as shown in the shaded box presentation of the alignment (Figure 9), these sequences share a common pattern of eight cysteines, i.e., …C…C…CC…CXC…C…C…, which is consistent with previous findings . These 8 conserved cysteines are also present in many other seed proteins, such as nonspecific lipid transfer proteins [61]. It has been postulated that the invariant 8 cysteines

316

Grace Q. Chen

forming 4 disulfide bonds have been retained possibly due to some structural and metabolic constraints of these proteins for seed metabolism [61]. Table 1. Percentage identity among different plant 2S albumins. RicC1

Jreg

Lusi

Cill

Aocc

Sind

Bexc

Cmas

Bjun

Bole

Ckur

RicC3

RicC1

100

45.1

43.1

43.1

38.2

36.3

33.3

32.4

31.4

31.4

30.4

25.5

Jreg

39.3

100

33.3

86.3

27.4

35

38.5

28.2

27.4

27.4

29.1

23.9

Lusi

32.4

28.7

100

28.7

25.7

27.2

19.9

22.8

27.9

27.9

24.3

20.6

Cill

37.6

86.3

33.3

100

29.1

34.2

33.3

26.5

29.9

29.9

25.6

23.1

Aocc

35.5

29.1

31.8

30.9

100

39.1

27.3

39.1

30.9

30.9

31.8

20.9

Sind

30.1

33.3

30.1

32.5

35

100

27.6

30.9

28.5

27.6

30.1

30.9

Bexc

27.2

36

21.6

31.2

24

27.2

100

22.4

20.8

20.8

31.2

21.6

Cmas

26

26

24.4

24.4

33.9

29.9

22

100

39.4

40.2

22

18.9

Bjun

21.5

21.5

25.5

23.5

22.8

23.5

17.4

33.6

100

94.6

18.1

18.8

Bole

21.5

21.5

25.5

23.5

22.8

22.8

17.4

34.2

94.6

100

18.1

18.1

Ckur

27.7

30.4

29.5

26.8

31.2

33

34.8

25

24.1

24.1

100

35.7

RicC3

22

23.7

23.7

22.9

19.5

32.2

22.9

20.3

23.7

22.9

33.9

100

Copyright © 2008. Nova Science Publishers, Incorporated. All rights reserved.

The protein names are described in the Figure 8, their sequences are displayed in the Figure 9. Each number in the table is the percentage of identity between two sequences listed in the corresponding row and column. Depending on which one is the reference sequence, the percentage identity values could be different.

We found that the 2S albumin gene was conserved among different castor genotypes (Sudanese, Peru and Hawaii origins) and abundantly expressed in the endosperm [12]. The conserved nature of castor 2S albumin suggests it has certain important structural features serving biological functions other than that of food reserve for seed germination. A number of in vitro biochemical activities have been reported for Brassicaceae 2S albumins, including antifungal activity, trypsin inhibitory activity and calmodulin antagonist activity [62]. Recently, trypsin inhibitory activity was demonstrated in vivo for Bjun 2S albumin from Brassica juncea [45]. Although RicC1 and RicC3 2S albumins share limited sequence identity with Bjun, 32% and 24%, respectively (Table 1), we have identified a trypsin/αamylase inhibitor Pfam family domain in both RicC1 and RicC3 [12], indicative of possible roles for castor 2S albumins serving as trypsin inhibitors for insect resistance. The trypsin/αamylase Pfam inhibitor family includes 33% of well-known allergenic seed proteins [60]. The molecular basis of the allergenicity of some proteins from this family has been analyzed by determining the allergen-specific IgE-binding sequence or epitopes, as well as their threedimensional structures [8]. An epitope core sequence, RGEE, identified in the hypervariable region of the Jreg 2S albumin from English walnut [53] is comparable to HGEE also located in the hypervariable region of castor RicC3 (Figure 9) [49]. Although the Jreg and RicC3 share overall 23.7% identity (Table 1), it is possible that the HGEE may function as a castor epitope involved in IgE-RicC3 interaction. The three-dimensional structure of castor RicC3 has been determined by NMR [49]. The RicC3 belongs to a single fold class described in

Seed Development in Castor (Ricinus communis L.)

317

SCOP (Structural Classification of Protein) database [31]. Available structures from the trypsin/α-amylase Pfam inhibitor family members show that they have a similar fold and fall into the same SCOP class [60]. There is not yet sufficient information to conclude common structural characteristics of 2S albumin allergen, but this could change as more epitope mapping and crystallographic or NMR studies are completed. Genetic engineering has proven successful in manipulating the expression levels of seed proteins[62], including the antisense suppression of 2S albumins in oilseed rape and Arabidopsis [25, 38]. We examined the expression profiles of 2S albumin in developing castor seeds. These results provide critical information in developing and implementing a genetic silencing approach to eliminate 2S albumin from castor seed. In addition, the sequence conservation observed in our studies on the phylogenetic relationship among 30 2S albumins from 28 plant species, along with observations of biological activities of other 2S albumins suggest a possible role for castor 2S albumins as defense proteins.

Copyright © 2008. Nova Science Publishers, Incorporated. All rights reserved.

Expression Profiles of Genes Involved in Fatty Acid and Triacylglycerol Synthesis in Castor Castor seeds contain 60% oil in the form of triacylglycerol (TAG) that also serves as a major energy reserve for seed germination and seedling growth. To understand how castor oil synthesis is regulated, scientists have sought to identify key genes responsible for the ricinoleate synthesis in order to develop temperate oilseeds to produce ricinoleate. A castor fatty acid hydroxylase gene (FAH) for the hydroxylase which is directly responsible for synthesis of ricinoleate was successfully isolated [68], but transgenic expression of the FAH in the model oilseed Arabidopsis produced only 17% ricinoleate [9], which was too low to be useful. There are many examples of transgenic production of other unusual fatty acids by over-expression of a key gene in oilseeds. With the exception of lauric acid and gammalinolenic acid, the amount of desired fatty acids in the transgenic oilseed has been considerably lower than in the wild species from which the transgene was obtained [66]. These results suggest that expressing a single key gene required for unusual fatty acid biosynthesis in transgenic plants is insufficient to produce a large amount of unusual fatty acids in seed. Current efforts on metabolic engineering of new oilseeds have been shifted towards searching for additional genes [42] or general transcription factors [11] that may upregulate multiple activities or entire pathways leading to oil biosynthesis. Therefore, knowledge of expression of multiple genes and their regulation during castor oil biosynthesis is needed to further understand the regulatory mechanisms controlling castor oil metabolism. In general, castor oil biosynthesis mostly follows the common biosynthetic pathways for fatty acids in plastid as well as TAG synthesis in the endoplasmic reticulum (ER), although there is a modification step for ricinoleate formation in the latter pathway [3]. Based on the available castor sequences at GenBank in year 2005, we identified 12 genes that participate in different steps of the pathways leading to fatty acid and TAG synthesis. We characterized the expression of these genes in developing seeds during ricinoleate/TAG accumulation and found that these genes can be classified into five groups with distinct temporal patterns [14]. Changes In Fatty Acid Content During Seed Development. Table 2 shows the changes in ricinoleate and other fatty acid content during seed development. The ricinoleate and total

318

Grace Q. Chen

fatty acids contents showed accumulation patterns parallel to each other, and the increase of the total fatty acids was attributed predominantly to the increase of ricinoleate. At early stages (12 and 19 DAP) in free-nuclear endospermic seeds, the ricinoleate was not detectable (Figure 1C). However by 26 DAP when 30% of seed’s volume filled with cellular endosperm (Figure 1D), the ricinoleate accumulated immediately to 9.2% of seed dry weight (Figure 1B) and accounted for about 77% of fatty acid content. During the remaining stages of cellular endosperm development, the ricinoleate kept increasing and reached plateaus of 55% of seed dry weight at 54 DAP and 90% of total fatty acids at 40 DAP (Table 2). Besides ricinoleate, there were about seven minor fatty acids detected in seeds at all stages of the development, all of them accumulated at low background levels in a total amount of about 3 to 6% of dry seed weight (Table 2). These fatty acids were probably components of structural lipids for maintaining the cell membrane and seed coat, in addition to being minor components of the oil. By using the same sets of developing seeds, similar results were observed by measuring changes in lipid classes, including acylglycerols, free fatty acids, phosphatidylcholine (PC) and phosphatidylethanolamine (PE) [29]. Before 19 DAP, there were considerable amounts of PC and PE present in seed lipid, and the acylglycerols contain less than 7% TAG. After 26 DAP, the relative amount of PC and PE dropped to negligible levels, and the fatty acids were almost exclusively in acylglycerols of which 77-89% was TAG [29]. It is known that ricinoleate accumulates almost exclusively as storage TAGs rather than as membrane lipids in castor seeds [3], therefore, the appearance of ricinoleate indicates the initiation of storage TAG synthesis. The accumulation pattern of ricinoleate (Table 2) consists with that of storage lipids or oil/TAG (Figure 3A), ricin (Figure 5) and 2S albumins (Figure 7). These results all suggest that the time between 19 and 26 DAP is a critical switchover stage for synthesizing both oil and storage proteins in castor. Moreover, these results also demonstrate that our series of developing seeds allows us to identify the initial timing and temporal pattern of different biochemical and cellular activities during seed development and to draw accurate comparisons between experiments. Table 2. Fatty acid composition in developing seeds. Fatty acid

Copyright © 2008. Nova Science Publishers, Incorporated. All rights reserved.

(% Seed Dry Weight)

Developing Seed (Days After Pollination) 12

19

26

33

40

47

54

61

C16:0

1.6

1.6

0.3

0.6

0.4

0.4

0.6

0.4

C18:0

0.0

0.1

0.1

0.4

0.4

0.5

0.7

0.7

C18:1(9)

0.0

0.0

0.6

1.3

1.1

1.1

1.5

1.4

C18:1(11)

0.0

0.0

0.2

0.3

0.2

0.2

0.5

0.3

C18:2(9,12)

0.7

0.6

1.0

2.2

1.9

2.2

2.2

2.0

C18:3(9,12,15)

0.0

0.0

0.3

0.4

0.3

0.3

0.3

0.3

C20:1(11)

2.1

2.0

0.3

0.3

0.2

0.2

0.7

0.1

C18:1(9)OH(12)

0.0

0.0

9.2

33.9

38.6

49.4

54.8

55.2

Total fatty acid

4.4

4.4

12.0

39.3

43.1

54.4

61.2

60.4

Seed Development in Castor (Ricinus communis L.)

319

Table 3. Selection of lipid genes, primer sequences, size of amplification products, and PCR efficiencies.

Copyright © 2008. Nova Science Publishers, Incorporated. All rights reserved.

Lipid gene AbbreviaGenBank cellular location ted ID and activity name Plastid location Acetyl-CoA carboxylase biotin carboxylase subunit Acyl carrier protein 46 kDa βKetoacyl-ACP synthase 50 kDa βKetoacyl-ACP synthase Enoyl-ACP reductase Enoyl-reductase domain of type I-like FAS Stearoyl-acylcarrier protein desaturase Cytoplasm location Acyl-CoA-binding protein Endoplasmic reticulum location Oleate 12hydroxylase Diacylglycerol acyltransferase Oleosin1

Ole1

Oleosin2

Ole2

Constitutive control gene Actin

Act

Forward primer Reverse primer

Amplicon PCR size (bp) efficiency

BC

L39267

TTCCTACGACGATAGAATACC AACTCTGACCTTTCAAATGTG

230

94.6

ACP

T15016

142

93.8

KASA

L13241

AAGGTCGTGGCATAGTG TGATCCCAAATTCCTCCTC GGGTAGGGAAAGGAGAATATGC GCCACAATGATGCGGAGAG

102

103.6

KASB

L13242

GAGTTGCTTGCTTATAGAG AGATAGACTTGATACTGAAATG

102

101.3

CF98122 ACTCCTGCCACACAGATGAC TCTAAGCCTAGTCCAAGATGCC 9 FASI-like* T15158 AATTCTGTGTAACACCATCAG ATTGTCCGGCAACCATTC

120

96.4

204

97.0

EAR

SAD

M59857

GAGTCTACACAGCAAAGGATTATG 150 TCTCTTCCAGCCTTCTAATTCTTG

97.4

ACBP

Y08996

ACAAGCAAGCCACCGTTG CTTCCTCCGTAGATTTCCCTTC

114

95.6

FAH

U22378

155

93.5

144

101

205

97.2

209

95.5

176

95.4

DGAT1

TAACCAGCAACAACAGTGAG ATAGGCAACATAGGAGAATGAG AY36649 GACACCATTCATAAGGAAG CTTTCTAATAAATGCTGTGC 6 AY36021 CTGCTGCCGTTGTTATTG 8 ATGCTTGTCCCACTTCC AY36021 AGTCTCCTATTTCTTTCTGG 9 TGCTTTCCTGTAACATACC

AY36022 GAATCCACGAGACTACATACAAC 1 TTATGAAGGTTATGCTCTC * Tentative assignments of the gene name and the cellular activity.

320

Grace Q. Chen A

Relative Copy Number

140 120

BC

B

C

250

350 ACP

80

150

200

60

100

150 100

40

50

50

0

0

12 19 26 33 40 47 54 Leaf

D

0 12 19 26 33 40 47 54 Leaf

E 200

140 120

KASB

EAR

6000 100

60

4000

40

50

2000

20

0

0

Relative Copy Number

12 19 26 33 40 47 54 Leaf

G 250

H

80 40

0

0

3x105 2x105 1x105 0 12 19 26 33 40 47 54 Leaf

K 3x105 DGAT1

250

2.5x105

100

1x105

50

5x104

0

0 12 19 26 33 40 47 54 Leaf

Developing Seed (DAP)

(7.3)

12 19 26 33 40 47 54 Leaf

Ole2

Ole1

3x105

1.5x105

150

(15) (24)

L 4x105

2x105

200

FAH

5x105 4x105

150

12 19 26 33 40 47 54 Leaf

Relative Copy Number

I

120

50

(0.4)

12 19 26 33 40 47 54 Leaf

7x105 6x105

ACBP

SAD

100

(7.1) (8)

0 12 19 26 33 40 47 54 Leaf

160

J 350

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FASI-like

8000

80

300

12 19 26 33 40 47 54 Leaf

F 10000

150

100

200

KASA

250

100

20

Relative Copy Number

300

200

2X105 1x105 (9.1) (17)

(9.6) (8.5)

(1.6)

(1)

0 12 19 26 33 40 47 54 Leaf

Developing Seed (DAP)

12 19 26 33 40 47 54 Leaf

Developing Seed (DAP)

Figure 10. Expression of lipid genes in developing seeds and leaf. Abbreviated names for the genes are described in Table 3. Each data point represents the mean ± SD of three replicates. DAP, days after pollination. Values in parentheses indicate relative copy number.

General Expression Profiles Of Lipid Genes And Lipid Gene Clusters. To examine expression profiles of castor lipid biosynthetic genes involved in fatty acid and TAG synthesis biosynthesis, we searched the Genbank nucleotide database (http://www.ncbi.nlm.gov) by using compound queries each consisting of Ricinus communis and the nomenclature of genes involved in synthesis of fatty acid and TAG from a comprehensive lipid gene catalog [5]. We noticed that all hits were for cDNA clones from

Seed Development in Castor (Ricinus communis L.)

321

developing castor seeds, indicating their functional expression during seed lipid biosynthesis. In some cases, more than one accession exists because of multiple partial sequences submitted by different laboratories. We examined the candidate genes by multiple sequence alignment and selected the most complete sequence to represent the gene. The search also found a partial sequence (GenBank ID T15158) containing an enoyl-reductase domain of Type I fatty acid synthase (FAS). This sequence is tentatively designated as FASI-like gene in Table 3. Quantitative PCR is one of the most sensitive and quantitative methods for measuring mRNA levels [2]. We optimized all assay conditions [14], and the primer sets and PCR efficiencies are listed in Table 3. In addition, PCR products were designed to have similar sizes between 100 and 230 bp (Table 3). These optimized conditions allow simultaneous analysis of multiple genes on a 96-well plate and permit accurate comparison of relative copy numbers among genes. Using the quantitative RT-PCR technology, we examined the steady state mRNA levels of lipid genes in seed samples from various developmental stages. For convenience, these mRNA levels are referred to as ‘expression’ in this chapter. Expression profiles of the lipid genes are shown in Figure 10. With the exception of the diacylglycerol acyltransferase 1 (DGAT1), we detected low background expression for the majority of lipid genes in leaf tissue and in young seeds at 12 and 19 DAP when endosperm was at the free-nuclear stage. When the seeds progressed to cellular endosperm development (26-54 DAP), the expression of the majority of lipid genes was induced to higher levels, displaying various temporal patterns; the maximum induction also varied dramatically, ranging from 4 to 43,000 fold (Figure 10, Table 3). The results revealed that a major transcriptional activation of lipid gene expression occurred at the onset of cellular endosperm development, coinciding with the beginning of the storage TAG accumulation (Figure 3A). It indicates a primary role of gene transcription in regulating castor oil biosynthesis. Table 4. Summary of transcript profiles of lipid genes. Temporal pattern during seed development Cluster 1, decline

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Cluster 2, flat rise

Genea DGAT1

Maximum ratio of expression seed/seedb seed/leafc 2.4 0.4

BC KASB EAR ACBP

4 12 21 7.8 7 28 6 4.7

Cluster 3, concave rise, ACP KASA SAD

22 20 70 22 23 23

Cluster 4, bell-shaped Cluster 5, linear rise,

Ole1 Ole2

24,172 140,400 27,150 252,230

FASI-like 940 18,288 FAH 43,083 86,990 a, the gene short names are described in the Table1. b, ratio of the maximum to the minimum expression in seed. c, ratio of the maximum expression in seed to the expression in leaf.

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Grace Q. Chen

To examine relationships among the temporal expression patterns of the lipid genes, we performed clustering analysis [14], which classified the lipid genes into five groups based on their pattern similarities. As expected, the DGAT1 itself formed a cluster showing a specific declining pattern (cluster 1 in Figure 11). The majority of the lipid genes showed various upregulated patterns during seed development including flat-rise (cluster 2), concave-rise (cluster 3), bell-shaped (cluster 4) and linear-rise (cluster 5). The clusters are summarized in Figure 11 and Table 4, together with their gene members and normalized mean pattern description, the changes in expression (maximum ratio) for each gene as well as the maximum ratio between the developing seed and mature leaf. Endosperm Development No growth Expansion/Maturation

Endosperm Development No growth Expansion/Maturation

A 96

96

64

64

32

32

0

Normalized Mean Expression (% maximum)

B

Cluster 1: DGAT1

C 96

0 12 19

26

33

40

47

54

12 19

D

Cluster 3: ACP, KASA, SAD

96

64

64

32

32

0

96

26

33

40

47

54

Cluster 4: Ole1, Ole2

0 12 19

E

Cluster 2: BC, KASB, EAR, ACBP

26

33

40

47

54

Cluster 5: FASI-like, FAH

12 19 26 33 40 47 54 Developing Seed (DAP)

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64

32 0 12 19 26 33 40 47 54 Developing Seed (DAP)

Figure 11. Expression patterns of lipid gene clusters during seed development. Abbreviated names are described in the Table 3. Each cluster is represented by the mean expression pattern over all the genes assigned to it. Error bars denote ± SD. DAP, days after pollination.

Expression profiles of lipid genes involved in Fatty Acid Biosynthesis. According to our current knowledge of castor seed TAG synthesis, oleic acid is synthesized in the plastid and then exported to cytoplasm following the standard fatty acid biosynthesis pathway [63]. Oleic

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Seed Development in Castor (Ricinus communis L.)

323

acid is activated to oleoyl-CoA in the cytoplasm and imported into ER with help of acyl-CoA binding protein (ACBP) for TAG synthesis [63]. We have examined seven castor lipid genes involved in oleic acid biosynthesis and transport and found they belong to either cluster 2 or cluster 3. Cluster 2 has a flat-rise pattern (Figure 11B) with 4 members: acetyl-CoA carboxylase biotin carboxylase subunit (BC), 50 kDa ß-ketoacyl-ACP synthase (KASB), enoyl-ACP reductase (EAR), and acyl-CoA-binding protein (ACBP); while cluster 3 (Figure 11C) includes 3 genes, acyl carrier protein (ACP), 46 kDa ß-ketoacyl-ACP synthase (KASA), and stearoyl-acyl-carrier protein desaturase (SAD), all with a concave-rise pattern. Further comparisons among gene clusters reveal that cluster 2 and 3 share certain expression characteristics. First, the maximum induction ratios between seed and leaf in cluster 2 and 3 (4.7-28 fold) are much smaller than those in cluster 4 and 5 (18,288-252,230 fold). Second, during the time-course of seed development, cluster 2 and 3 have moderate maximum induction ratios (4-70 fold) compared to cluster 4 and 5 (940-43,083 fold). Third, the temporal patterns of cluster 2 and 3 are less dynamic than those of cluster 4 and 5; cluster 2 has a simple flat-rise, whereas cluster 3 has a quick rise that reaches its half-maximum at the beginning of cellular endosperm development, and increases slowly (> λ , and, then,

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Yuri A. Ermolin

λ+μ ≈μ .

(13)

Eq. (12) can be written, in view of Eq. (13), as:

λ A− B −C ≈

(λ A + λ C ) q A + ( λ B + λ C ) q B . q A + qB

(14)

Thus, the Y-like sewer system shown in Figure 1a) is superseded formally with an equivalent fictitious sewer ABC having a failure rate λ A− B −C and sewage flow rate at the inlet

(q A + qB ) (Figure 1b)). We emphasize that the sewer network fragment of Figure 1a is a structure-forming component in the sense that any arbitrary complicated dendritic sewer network may be thought of as a combination of such components that substantially reduces and simplifies a body of calculations in estimating raw sewage discharged from the network. Below we give a numerical illustrative example of how to apply this approach.

Applications. Example 1 Consider the network in Figure 4a) consisting of 15 sewer sections, each determined by the values λi (i = 1 – 15) and μ . The sewage flow rates at the network inlets ( q1 − q8 ) will be considered to have constant values. It is necessary to estimate the raw sewage volume discharged from the network throughout the year as a consequence of possible failures. To carry out the calculations we need some data. Consider, for the sake of definiteness, that we have at present the following data: λ1 = 0.52 1/yr, λ 2 = 0.68 1/yr, λ3 = 0.792 1/yr,

λ 4 = 0.91 1/yr, λ5 = 1.344 1/yr, λ6 = 0.83 1/yr, λ7 = 0.75 1/yr, λ8 = 0.025 1/yr, λ9 = 0.852 1/yr, 1/yr,

λ10 = 0.62 1/yr, λ11 = 0.837 1/yr, λ12 = 1.1 1/yr, λ13 = 0.025 1/yr, λ14 = 0.5

λ15 = 0.025 1/yr; and μ = 365 1/yr. The inlets sewage flow rates are: q1 = 0.3 m 3 /s,

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q 2 = 0.9 m 3 /s, q3 = 0.6 m 3 /s, q 4 = 0.4 m 3 /s, q5 = 0.5 m 3 /s, q 6 = 0.1 m 3 /s, q 7 = 0.5 m /s, q8 = 0.7 m /s. We assume that and T = 1 yr = 31.536 ⋅ 10 s. 3

3

6

In addition, without loss of generality, we assume that the length of each sewer section is equal to 1 km. We note also that all values are hypothetical, convenient for calculations only. First we consider the contours I, II and III in Figure 4a). Either contour includes the Ylike system, and, consequently, can be substituted by one equivalent sewer with its associated value of failure rate calculated according to the method proposed above. Using Eq. (14) where now, taking account of new notations, λ A = λ 2 , λ B = λ3 , λC = λ9 , q A = q 2 and

q B = q9 , we have for contour I:

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389

q2

I

2 λ2 3

λ3

9 λ9

5 λ5

4

6

11

14

1

q1

λ14

λ1

II

q4

λ4

10 λ10

q5

q3

λ11

13

q6

7

12

λ12

λ13

15 λ15 outlet

λ6

q7

λ7 8 λ8 q8

a)

III

(q2 + q3 )

IV

I λI

4

10

1

q1

λ10

λ4

II λ II 13

14

λ14

λ1

(q5 + q6 )

q4

15 λ15

V III

λ III

λ13

(q7 + q8 )

outlet

b)

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(q2 + q3 + q4 )

VI q1

IV λ IV

VII

1

14

λ1

λ14

V λV 15 λ15

outlet

c)

8

∑q ∑q i

i =5

i =1

V

VI

4

i

λVI

8

15 λ15

λV

∑q i =1

8

∑q i =5

i

i

VII λVII

outlet

outlet

d)

e)

Figure 4. Initial sewer network a) and its sequential transformations b), c), d), e).

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Yuri A. Ermolin

λ I = λ 2 − 3− 9 =

(λ 2 + λ 9 ) q 2 + (λ 3 + λ 9 ) q 3 = 1.577 1/yr. q 2 + q3

Similarly, for contour II (here

λ II = λ5−6−11 = and for contour III (where

λ A = λ5 , λ B = λ6 , λC = λ11 , q A = q5 , q B = q6 )

(λ5 + λ11 )q 5 + (λ6 + λ11 )q 6 = 2.095 1/yr, q5 + q 6

λ A = λ7 ; λ B = λ8 ; λC = λ12 ; q A = q 7 ; q B = q8 ) :

λ III = λ7 −8−12 =

(λ7 + λ12 )q 7 + (λ8 + λ12 )q8 = 1.427 1/yr. q 7 + q8

The results obtained enable one to present the initial network in the form shown in Figure 4b). But here are the Y-like systems (contours IV and V) again. Using Eq. (14) we have the failure rate λ IV for contour IV:

λ IV = λ I − 4−10 = and

(λ I + λ10 )(q 2 + q3 ) + (λ 4 + λ10 )q 4 = 2.057 1/yr, q 2 + q3 + q 4

λV for contour V: λV = λ II −13− III =

(λ II + λ13 )(q5 + q 6 ) + (λ III + λ13 )(q 7 + q8 ) = 1.675 1/yr. q5 + q 6 + q 7 + q8

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In consequence of this the structure shown in Figure 4b) substitutes by structure depicted in Figure 4c) where the Y-like sub-system (contour VI) may be besides selected. Equivalenting this contour again by one sewer section with the failure rate

λVI = λ1− IV −14 =

(λ1 + λ14 )q1 + (λ IV + λ14 )(q 2 + q 3 + q 4 ) = 2.347 1/yr, q1 + q 2 + q3 + q 4

we going to the Figure 4d). But structure shown in Figure 4d) is the Y-like fragment (contour VII) in itself that may be substituted by one sewer (see Figure 4e)). Thus, we have finally the failure rate λVII of one equivalent sewer substituting the initial network depicted in Figure 4a):

Reliability Estimation of Urban Wastewater Disposal Networks

λVII = λVI −15−V =

391

(λVI + λ15 )( q1 + q 2 + q3 + q 4 ) + (λV + γ 15 )(q5 + q 6 + q 7 + q8 ) = q1 + q 2 + q3 + q 4 + q5 + q 6 + q 7 + q8

=2.07 1/yr. Now we can calculate the volume Qd of raw sewage discharged from the network during one year. Taking account of the formula (10) and its physical sense we have:

Qd =

λVII λVII + μ

(q1 + q 2 + q3 + q 4 + q5 + q 6 + q 7 + q8 )T = 0.711 ⋅ 10 6 m 3 ,

that is 0.56 % of the total sewage volume ( q1 + q 2 + q3 + q 4 + q 5 + q 6 + q 7 + q8 )T = 126.144 ⋅ 10 m that entered the network inlets. As may be seen from this example, unlike the state-enumeration method here, there is no need to solve an unwieldy set of equations. The problem reduces to a sequence of simple computations using, at every stage, the results of a preceding step. 6

3

Applications. Example 2

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The method developed in this work may be used to solve one more problem considered below in a simplified statement. Let us assume that specialists analyzing the results obtained in preceding example 1 come to the conclusion that the raw sewage discharge from the sewer network (Figure 4a)) is much too large, and, consequently, the network needs in reliability increasing. A question concerning replacement of some outdated components by a new sewer pipe is discussed. But, by reason of conditions, it is possible to replace one sewer only because the funds available are limited. On present evidence it may be argued that the failure rate for a new sewer (manufacturer’s data) is λ N = 0.025 1/yr. It is desired to identify the preferential alternative. As before, we will take the discharged sewage volume as an efficiency index of the alternative to be accepted. Calculate this quantity assuming that the replacement of sewer section 1 in the initial network (Figure 4a)) has just been made. For this purpose, we substitute the input data (associated with the sewer 1) of the example 1 considered in preceding section, for one another (corresponding to the new sewer), namely λ1 = λ N = 0.025 1/yr. Carrying out the relevant calculations, we obtain the discharged sewage volume (Qd )1 expressed as a percentage of the total sewage entered the network during the year: 0.55 %. By repeating the similar calculations with respect to each network section (Figure 4a)) we come to results represented in Table 1. We call attention to the fact that Table 1 is not included the columns relating to sewers 8, 13 and 15. The reason is that these pipes have the failure rates identical with new pipe (see initial data), and their replacement is virtually meaningless.

392

Yuri A. Ermolin Table 1.

Referring to Table 1, it is seen that the smallest volume of sewage to be discharged from the network throughout the year occurs when the network’s section 12 is replaced (in Table 1 this case is boldly printed). It is obvious that, under otherwise equal conditions, this alternative is preferable from the viewpoint of the reliability index accepted in this work. The examples considered are simple as well; for this reason, the results seem to be quite trivial. Note, however, that the simplicity of the examples make it possible to see the potential of proposed method for practical use.

Case of Aging Elements

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With time, for some real objects, however, from observations, it is established that the hypothesis of failure flow stationarity cases to be true. For example, the mean number of failures in a unit of time for sewer network components tends to increase, and the failure rate λ for which is, consequently, an increasing function of time, i. e. λ = λ (t ) . In reliability theory, such elements known as aging elements [7]. But, when such is the case, calculating reliability indexes by formulas implied from the stationarity hypothesis is incorrect mathematically. Meanwhile, such calculations are necessary, for example, under decision making about the opportunity of either repairing or replacing some sections of a sewer network. In the case of nonstationary failures, attempts to obtain expressions intended for engineering calculations lead either to cumbersome formulas inconvenient for practical implementation or to completely impenetrable mathematical difficulties. In practice therefore, it is generally believed that the failure flow acting on an object is stationary with a certain (most often, arithmetically averaged over data observed) failure rate. In so doing, the reliability index assessments are, naturally, in error. Below a more rigorous approach that provides a way of calculating the equivalent stationary failure rate when the real failure rate is a increasing function of time is proposed. The value of equivalent failure rate is calculated from a condition in which the value of a

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393

survival function at a certain time (time of prediction) of an object being considered remains fixed under such substitution. From theory, it is apparent that all reliability time characteristics are interrelated, i. e. if only one among them is given in an analytical form, all others, in principle, can be obtain by some mathematical transformations. In particular, the survival function (reliability function) pi (t ) of object i is related to the corresponding failure rate λi (t ) by the expression [3]:

⎧ t ⎫ pi (t ) = exp⎨− ∫ λi (t )dt ⎬ . ⎩ 0 ⎭

(15)

Let us suppose that we have a certain stationary flow acting on an object i with the unknown constant failure rate (λi ) s . Then, according to Eq. (15), the survival function of this object ( pi ) s (t ) is:

⎧ t ⎫ ( pi ) s (t ) = exp⎨− ∫ (λi ) s dt ⎬ = exp{− (λi ) s t}. ⎩ 0 ⎭

(16)

Assuming t = T , where T is a prediction time, we require the fulfillment of the equality:

pi (t = T ) = ( pi ) s (t = T ) .

(17)

Taking account of the Eqs. (15) and (16), we have from (17): T

∫ λ (t )dt = (λ ) T , i

i

s

(18)

0

and solving this equation for the unknown quantity we fit (λi ) s so that the condition (17)

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would be fulfilled:

(λi ) s =

T

1 λi (t )dt . T ∫0

(19)

Thus, the real nonstationary failure flow is superseded formally by an equivalent stationary flow having the failure rate the value of which is calculated according to Eq. (19). From the observations in actual practice, it follows that the aging process for an element i can be adequately approximated by an increasing exponential function of the time as:

λi (t ) = (λi ) 0 eα t , (α i ≥ 0) ; i

(20)

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Yuri A. Ermolin

where (λi ) 0 is the failure rate of ith element at present (t = 0) , and

α i is an aging

coefficient relating to ith element. Substituting Eq. (20) in Eq. (19) and integrating its we obtain the expression:

e α iT − 1 , (λi ) s = (λ i ) 0 α iT

(21)

that makes it possible to “stationarize” the initial nonstationary failure flow by a stationary one over the prediction interval T. The relationship (λi ) s /(λi ) 0 = f (α iT ) is plotted in Figure 5.

(λi )s / (λi )0 7 6 5 4 3 2 1 0 0

0,5

1

1,5

2

2,5

3

α iT

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Figure 5. Plot of stationarized failure rate versus prediction time (in relative units).

In closing this section we note that the proposed stationarization procedure is applicable not only for flows having exponentially increasing failure rate, but for those with different types of a nonstationarity as well. In [8], for example, treats the special case when the failure rate acting on an object is a periodic piecewise constant function of time. The implementation of the obtained result we can demonstrate with the following Example 3.

Applications. Example 3 The problem statement and bounding conditions are the same as those in Example 2. But, this case differs from the above in that the failure rate of all sections (see Figure 4a)), with an

Reliability Estimation of Urban Wastewater Disposal Networks

395

exception of sewers 8, 13 and 15 that are new, tend to increase with time. We assume that a processing of observational data obtained over a period of the several preceding years, suggested that the increasing tendencies are adequately describable by functions of the form (20) with parameters: (λi ) 0 = λi where λi is the value of failure rate ith sewer at the time of

α1 = 0.1 1/yr; α 2 = 0.5 1/yr; α 3 = 0.6 1/yr; α 4 = 0.5 1/yr; α 5 = 0.7 1/yr; α 6 = 0.9 1/yr; α 7 = 0.2

decision making (t =0), and aging coefficients having following values: 1/yr;

α 8 = 0 ; α 9 = 1.5 1/yr; α10 = 0.4 1/yr; α11 = 1.5 1/yr; α12 = 0.2 1/yr; α13 = 0 ;

α14 = 0.27 1/yr and α15 = 0 . By formula (21), assuming t = T = 1 year, where T is, as before, the prediction time interval, we calculate the stationarized failure rates for all sewers of the network shown in Figure 4a). We have: (λ1 ) s = 0.547 1/yr; (λ2 ) s = 0.882 1/yr; (λ3 ) s = 1.085 1/yr;

(λ4 ) s = 1.181 1/yr; (λ5 ) s = 1.946 1/yr; (λ6 ) s = 1.346 1/yr; (λ7 ) s = 0.830 1/yr; (λ8 ) s = 0.025 1/yr; (λ9 ) s = 1.978 1/yr; (λ10 ) s = 0.762 1/yr; (λ11 ) s = 1.943 1/yr; (λ12 ) s = 1.218 1/yr; (λ13 ) s = 0.025 1/yr; (λ14 ) s = 0.574 1/yr and (λ15 ) s = 0.025 1/yr. Then, carrying out the calculations, as was done for the Example 1 (but in respect of these stationarized failure rates), we are led to the results summarized in Table 2.

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Table 2.

The decision following from Table 2 is: the network section 9 must be replaced. Comparison analysis of Tables 1 and 2 indicates that: -

as would be expected, in case of aging sewer network components the sewage volumes discharged to the environment are considerable more; when the aging processes take into account, the sewer recommended for a replacement is different from that obtained in Example 2; this testifies that an ignoring this factor may cause a faulty decision.

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Yuri A. Ermolin

Conclusion Although sewer reliability depicts a fairly complete reliability measure of the sewer network, it is convenient to use a single index to represent the composite effect of the component reliabilities. We propose to assess sewer network reliability as a whole by a volume of raw sewage discharged from the system because of failures of its components for an appreciable length of time. The traditional method for solving such problems is the so-called stateenumeration method. But, for the multicomponental networks, this generates a need to solve a set of equations having very high order, which renders the method unsuitable for many practical applications. The approach proposed in this work makes it possible to circumvent these difficulties by using the concept of equivalent sewer. As a result, the problem reduces to a sequential consideration of elementary subproblems the solution of which is easily accomplished. Classical reliability theory, in principle, provides a way of finding all numerical and time reliability characteristics. However, in the case that the failure rate acting on an object is nonstationary, the fulfillment of prescribed formal procedures brings results, which, as a rule, contain integrals involving expressions of very complicated functions. The calculation of such integrals is possible only by means of numerical calculations, which is not always convenient. The approach proposed in this work solves this problem in analytical form by using the concept of equivalent stationary failure rate. In practical engineering applications approximate analytical solutions, which are of simpler form and convenient for analysis, are frequently acceptable. Such an approach allows one to find the solution of reliability problems appearing, for example, in exploring rehabilitation needs and strategies for water distribution systems [9], and so forth, in a convenient analytical form. The computational examples cited above demonstrate these procedures. In our view, it may be suggested that similar problems exist also in the course of maintenance of oil, gas and other pipeline systems. It seems, such a setting and solving of problems may also be of interest for specialists working with general reliability issues.

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References [1] Lansey, K. E.; Mays, L. W.; Woodburn, J.; Wunderlich, W. O. In Reliability Analysis of Water Distribution System; Mays, L. W.; Ed.; ASCE; New York, NY, 1989; pp 433 – 471. [2] Mays, L. W. In Proceedings of the Seventh IAHR International Symposium, Stochastic Hydraulics’96; Tickle, K. S.; Goulter, I. C.; Xu, C.; Wasimi, S. A.; Bouchart, F.; Eds.; Mackay, Queensland, Australia, 1996; pp 53 – 62. [3] Ventzel, E. S. Reliability Theory; “Vysshaja Shkola”, Moscow, 1999; 576 pp. (in Russian). [4] Mays, L. W.; Tung, Y. K.; Cullinane, M. J. In Reliability Analysis of Water Distribution Systems; Mays, L. W.; Ed.; ASCE; New York, NY, 1989; pp 163 – 189. [5] Ventzel, E. S.; Ovcharov, L. A. Reliability Theory: Tasks and Exercises; “Vysshaja Shkola”, Moscow, 2000; 366 pp. (in Russian).

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[6] Ventzel, E. S. Operations Analysis: Problems, Principles and Methodology; “Vysshaja Shkola”, Moscow, 2001; 208 pp. (in Russian). [7] Gnedenko, B. V.; Beliaev, Yu. K.; Soloviov , A. D. Mathematical Methods in Reliability Theory; “Nauka”, Moscow, 1965; 524 pp. (in Russian). [8] Ermolin, Yu. A. ISA Tranctions. 2007, vol 46, 127 - 130. [9] Herz, R. K. J Water SRT – Aqua. 1998, vol 47, 275 – 283.

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In: Reliability Engineering Advances Editor: Gregory I. Hayworth

ISBN: 978-1-60692-329-0 © 2009 Nova Science Publishers, Inc.

Chapter 15

FUTURE TRENDS ON GLOBAL PERFORMANCE INDICATORS IN INDUSTRIAL RESEARCH Livio Corain and Luigi Salmaso Department of Management and Engineering, University of Padova, Vicenza, Italy

Abstract Within Research and Development activities, complex statistical problems of hypothesis testing can commonly arise. The complexity of the problem relates mainly to the multivariate nature of the study, possibly to the presence of mixed performance variables (ordinal categorical, binary or continuous), and sometimes to missing values. In this contribution we consider permutation methods for multivariate testing on mixed variables within the framework of multivariate randomised complete block design. The novel approach we propose has been studied and validated using a Monte Carlo simulation study. Finally we propose an application to real data, where several panellists from an R&D division of a homecare company are enrolled to study several possible new fragrances for a given detergent to compare with their own presently marketed product.

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Keywords: permutation tests, RCB design.

1. Complex Designs for Industrial Research Within the evaluation of industrial products and processes and in particular for Research and Development (R&D) activities, complex statistical problems of hypothesis testing connected with global performance indicators may often arise. The complexity of the study relates mainly to the presence of mixed performance variables (ordinal categorical, binary or continuous) and missing values. When performance evaluation takes more than one aspect into account the problem can be complicated and some methodological and practical issues arise: standardization, multivariate structure of data, accuracy of partial indicators, distance with respect to target (highest satisfaction level), stratification in presence of confounding

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Livio Corain and Luigi Salmaso

factors (Bird et al., 2005). Hence, it is clear that some methods may be more effective and robust than others. For example, in industrial experimentation one of the most crucial and common R&D problems is the definition of the best product among all prototypes being studied on the basis of experimental performances achieved in the R&D laboratory. Especially in the food and body care industry, useful experimental performance indicators are individual sensory evaluations provided by trained people (panellists) during a so-called sensory test (Meilgaard et al., 2006). These kinds of measures are in fact thought to be comparable with the assessment that final consumers will make when using the new product. Within this framework the experimental design typically handles panellists as blocks. If the evaluations of panellists are multivariate in nature, the statistical problems become hard to cope with following the traditional parametric approach and a suitable inferential approach should be developed in order to build up any effective global performance indicator.

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2. Permutation Tests for Multivariate Randomised Complete Block Design When dealing with complex designs, conditional nonparametric methods can provide a reasonable approach. Remember that traditional unconditional parametric testing methods (such as t test or F test) may be available, appropriate and effective only when a set of restrictive conditions are satisfied. Accordingly, just as there are circumstances in which unconditional parametric testing procedures may be proper from a related inferential result interpretation point of view, there are others in which they may be improper or even impossible. In conditional testing procedures, provided that exchangeability of data in respect of groups is satisfied in the null hypothesis, permutation methods play a central role. This is because they allow for reasonably efficient solutions, are useful when dealing with many difficult problems, provide clear interpretations of inferential results, and allow for weak extensions of conditional to unconditional inferences. For a detailed discussion on the topic of the comparison between permutation conditional inferences and traditional unconditional inferences see Pesarin (2002). Let us consider the experimental design where there are n blocks and, within each block, experimental units are randomly assigned to the C treatments and exactly one experimental unit is assigned to each of the C treatments (C>2). Let Y be the multivariate variable related to a p-dimensional vector of responses and let us assume, without loss of generality, that high values of each Yk univariate element, k=1,…,p, correspond to better performance and therefore to a higher degree of treatment preference. The experimental design is developed with the aim of comparing the C treatments with respect to p different response variables. The statistical model (with fixed effects) for the multivariate randomised complete block (RCB) design can be represented as follows: Yij = μ + βi + τj + εij, εij∼IID(0,Σ), i = 1,...,n , j = 1,...,C,

(1)

where βi, τj and Yij, are respectively the effect of the i-th block, the effect of the j-th treatment and the p-dimensional multivariate response variable for the i-th block and the j-th treatment.

Future Trends on Global Performance Indicators in Industrial Research

401

The random term εij represents a p-vector of experimental errors with zero mean, variance/covariance matrix Σ and unknown continuous distribution P. The usual sideconditions for effects are given by the constraints ∑i βi = ∑j τj = 0. Model (1) is called “effect model” (Montgomery, 2004). If we define μj = μ + τj, j = 1,...,C, an alternative representation of model (1) is the so called “mean model”, i.e. Yij = μj + βi + εij.

(2)

The resulting inferential problem of interest is concerned with the following hypotheses 1. H0: {τj = 0, ∀j}, against H1: {∃j: τj ≠ 0}; note that this hypothesis refers to a multivariate global test; if H0 is rejected, it is of interest to perform inference on each individual univariate response variable, that is 2. H0k: {τjk = 0, ∀j}, against H1k: {∃k: τjk ≠ 0}, k=1,…,p; if H0k is rejected, it is usually of interest to perform pair-wise comparisons between couples of treatments, i.e. 3. H0k(jh): τjk = τhk, j,h= 1,...,C, j≠h, against H1k(jh): τjk ≠ τhk.

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Note that in the framework of RCB designs there is usually no interest in testing the block effect which is handled as a nuisance factor. Note also that, since no interaction effect between treatments and blocks is here supposed to exist, expressions (1) and (2) do not consider any interaction effect. In the framework of traditional parametric methods, when assuming the hypothesis of multivariate normality for random error components, the inferential problem can be solved by means of any MANOVA procedure (for hypothesis H0), ANOVA F test (for hypotheses H0k) and a further set of pair-wise tests using for example Fisher’s LSD or Tukey procedures (for hypotheses H0k(jh)), which are two of the most popular multiple comparison procedures (Montgomery, 2004). In this paper we propose a novel solution for the whole set of hypotheses of interest within the nonparametric framework of NonParametric Combination (NPC) of dependent permutation tests (Pesarin, 2001; Corain and Salmaso, 2004). In order to better explain the proposed approach let us denote an (n×C×p) data set with Y:

⎡Y11 ⎢… ⎢ Y =[Y1 ,..., Yj , ..., YC ]= ⎢ Yi1 ⎢ ⎢… ⎢Yn1 ⎣

… Y1 j … Y1C ⎤ … … … … ⎥⎥ … Yij … YiC ⎥ , ⎥ … … … …⎥ … Ynj … YnC ⎥⎦

where Yij = [Yij1,…,Yijk,…,Yijp]′, represents the ij-th observed p×1 response for the i-th block and j-th variable, i = 1,...,n, j=1,...,C, (C>2). The effect model for the k-th univariate component of Yij can be written as

402

Livio Corain and Luigi Salmaso Yijk = μk + βik + τjk + εijk, i = 1,...,n , j = 1,...,C, k=1,…,p.

(3)

In the framework of NonParametric Combination (NPC) of dependent permutation tests we suppose that, if the global null hypothesis H0 is true, the hypothesis of exchangeability of random errors within the same block holds. Hence, the following set of mild conditions should be jointly satisfied: i)

ii)

we suppose that for Y=[Y1,...,YC] an appropriate probabilistic p-dimensional distribution structure P exists, Pj∈F, j=1,...,C, belonging to a (possibly non-specified) family F of non-degenerate probability distributions; the null hypothesis H0 states the equality in distribution of the multivariate distribution of the p variables in all C groups:

⎡ d d ⎤ H 0 : [ P1 = ... = PC ] = ⎢ Y1 = ... = YC ⎥ . ⎣ ⎦ Null hypothesis H0 implies the exchangeability, within each i-th block, of the individual data vectors with respect to the C groups. Moreover H0 is supposed to be properly decomposed into p sub-hypotheses H0k, k=1,...,p, each appropriate for partial (univariate) aspects, thus H0 (multivariate) is true if all the H0k (univariate) are jointly true: p

d

p

d

H 0 :[∩ Y1k = ... = YCk ] = [∩ H 0 k ] . k =1

k =1

H0 is called the global or overall null hypothesis, and H0k, k=1,...,p, are the partial null hypotheses. Finally, H0k, k=1,...,p, is supposed to be properly decomposed into C×(C−1)/2 sub-hypotheses H0k(jh), j,h= 1,...,C, j≠h, each one related to the jh-th pairwise comparison between couples of treatments: p

H 0 :[∩

C

∩Y

k =1 j , h =1 j≠h

iii)

p

d

jk

=Yhk ] = [∩

0 k ( jh )

].

k =1 j , h =1 j≠h

The alternative hypothesis H1 is represented by the union of partial H1k(jh) subalternatives: p

p

H1 :[∪ H1k ] = [∪ Copyright © 2008. Nova Science Publishers, Incorporated. All rights reserved.

C

∩H

k =1

C

∪H

1k ( jh )

],

k =1 j , h =1 j≠h

so that H1 is true if at least one of sub-alternatives is true. In this context, H1 is called the global or overall alternative, and H1k, k=1,...,p, are called the partial alternatives. iv) let T=T(Y) represent a p-dimensional vector of test statistics, p>1, whose components Tk(jh), k=1,...,p, j,h= 1,...,C, j≠h, represent the partial univariate and non-degenerate partial test appropriate for testing the sub-hypothesis H0k(jh) against H1k(jh). Without loss of generality, all partial tests are assumed to be marginally unbiased, consistent and significant for large values (for more details see Pesarin, 2001).

Future Trends on Global Performance Indicators in Industrial Research

403

At this point, in order to test the global null hypothesis H0 and the p univariate hypotheses H0k, we perform the partial (univariate) tests which are focused on jh-th partial aspects, and then, we combine them with an appropriate combining function, firstly to test H0k, k=1,...,p, and finally in order to test the global (multivariate) test which is referred to as the global null hypothesis H0. However, we should observe that in most real problems, when the number of blocks is great enough, there is a clash over the problem of computational difficulties in calculating the conditional permutation space. This means it is not possible to calculate the exact p-value of observed statistic Tk(jh)0. This is overcome by using the Conditional Monte Carlo Procedure (CMCP). The CMCP on the pooled data set Y is a random simulation of all possible permutations of the same data under H0 (for more details refer to Pesarin, 2001). Hence, in order to obtain an estimate of the permutation distribution under H0 of all test statistics, a CMCP can be used. Every resampling without replacement Y* from the data set Y actually consists of a random attribution of the individual block data vectors to the C treatments. In every Yr* resampling, r=1,...,B, the K=k×(k−1)/2 partial tests are calculated to obtain the set of values [Tir*=T(Yir*), i=1,..,K; r=1,…,B], from the B independent random resamplings. It should be emphasized that CMCP only considers permutations of individual data vectors within each individual block, so that all underlying dependence relations which are present in the component variables are preserved. From this point of view, the CMCP is essentially a multivariate procedure. As can be seen, under the general null hypothesis, the CMC procedure allows a consistent estimation of the permutation distributions, both marginal and combined, of the K partial tests. In the nonparametric combination procedure, Fisher’s combination function is usually considered, principally for its good properties which are both finite and asymptotic (Pesarin, 2001). Of course, if it were considered appropriate, it would be possible to take into consideration any other combining function (Folks, 1984; Pesarin, 2001). The combined test is unbiased and consistent. A general characterization of the class of combining functions is given by the following three main features for the combining function ψ: a) it must be non-increasing in each argument:

Copyright © 2008. Nova Science Publishers, Incorporated. All rights reserved.

ψ (..., λi ,...) ≥ ψ (..., λi′,...) if λi < λi′ , i ∈{1,…,K}; b) it must attain its supreme value

ψ , possibly non finite, even when only one

argument reaches zero:

ψ (..., λi ,...) → ψ if λi → 0 , i ∈{1,…,K}; c) ∀α > 0, the critical value of every ψ is assumed to be finite and strictly smaller than the supreme value:

Tα′′ < ψ .

404

Livio Corain and Luigi Salmaso

The above properties define the class C of combining functions. Some of the functions most often used to combine independent tests (Fisher, Lancaster, Liptak, Tippett, Mahalanobis, etc.) are included in this class. For a detailed description on how to build partial and global permutation tests refer to Pesarin (2001) and Corain and Salmaso (2004).

3. Simulation Study In this section, in order to validate the proposed method and to evaluate its performance in comparison with the traditional approach (F and t test), we perform a comparative simulation study. The simulation study is designed in two steps: the first focuses on the multivariate global test while the second is concerned with univariate tests and related pair-wise comparisons. For the first simulation study let us consider the following settings: ƒ ƒ ƒ ƒ ƒ

ƒ

1,000 independent simulations; number of blocks: n=6,10,20; number of treatments: C=3,5,7; block effect βi, i = 1,...,n, is generated from standard normal distribution (independently for all univariate components); number of response variables (all numeric continuous): p=3; number of response variables where treatment effect is active: 1,2,3; by “active variable” we mean that for the k-th variable, k=1,2,3, true means differ from each other (following the pattern detailed below) so that ∃ k: τjk ≠ 0, j = 1,...,C, k=1,2,3; instead, we say that the treatment is not active for the k-th variable when all τjk are set equal to zero; with reference to model (2), at univariate level the treatment effects are set as follows

0

0.25

0.5

0.75

1

1.25

1.5

treatment scale C (No. of treat.)

μ1

μ2

μ3

Copyright © 2008. Nova Science Publishers, Incorporated. All rights reserved.

3

μ1

μ2

μ3

μ1

μ2

μ3

μ4

μ5

5

7

μ4

μ5

μ6

μ7

Future Trends on Global Performance Indicators in Industrial Research ƒ

405

three types of random errors: normal, exponential (as an example of an asymmetric distribution) and Student’s t with 2 degrees of freedom (as an example of a heavytailed distribution); each one of the p=3 random component εijk, i = 1,...,n , j = 1,...,C, k=1,2,3, are generated

○ independently (with σij2 =1), or ○ with a variance/covariance structure as follows 1 Σ = -0.5 0.5

-0.5 0.5 1 -0.25 -0.25 1

We perform a multivariate permutation test (1,000 conditional Monte Carlo iterations, CMC) for each simulation using the Fisher combining function and we also consider the parametric Wilk’s Lambda F-test (MANOVA with two factors; Johnson and Wichern, 2007). The considered significance α-level is equal to 0.05. Tables 1 and 2 summarize the rejection rates under the hypothesis of independent random errors and correlated random errors respectively. In order to better underline possible differences in performance between the F and permutation test, we highlight in grey the rejection rates where the F or permutation test shows an estimated increase of power of at least 5%. The goal of the second simulation study is focused on the univariate test H0k and related treatment pair-wise comparisons (H0k(jh)). Let us consider the following setting:

Copyright © 2008. Nova Science Publishers, Incorporated. All rights reserved.

ƒ ƒ ƒ ƒ ƒ

1,000 independent simulations; number of blocks: n=6,10,20; number of treatments: C=3,5,7; block effect βi, i = 1,...,n, is generated from standard normal; the treatment effects are set as in the previous simulation study; three types of random errors (σij2 =1): normal, exponential and Student’s t with 2 degrees of freedom.

For each simulation we perform the permutation tests (1,000 CMC) using the Fisher combining function and we consider the traditional F-test for RCB design. The considered significance levels are α=0.05 and 0.1. In case of rejection of the null hypothesis H0k, and in order to perform the treatment pair-wise comparisons, we consider permutation tests for two paired samples (difference of means, as test statistic) and Tukey’s test. Remember that the Tukey post-hoc pair-wise comparisons are defined in order to maintain the first type error probability of the main global hypothesis H0k at the desired α-level. Similarly, we adopt a multiplicity correction strategy for permutation tests by using the closed testing approach (Marcus et al., 1976) via Tippett’s combining function (the so-called minP procedure, Westfall et al., 1999) which is particularly suitable to implementation within the framework of permutation tests (Finos and Salmaso, 2007). Tables 3 and 4 summarize the rejection rates for α=0.05 and α=0.1 respectively. Note that, in order to properly compare the performances of the F and permutation test with different values of C (i.e. no. of treatments), rejection rates of pair-wise comparisons are presented in terms of delta (δ), i.e. of the true differences between treatment effects: δjh =τj − τh, j,h= 1,...,C, j≠h. For example, we have δ=0.5 for C=3

406

Livio Corain and Luigi Salmaso

from the difference between μ2 and μ1 while we get δ=0.5 for C=5 from the differences μ3 − μ1, μ4 − μ3 and μ5 − μ4. With reference to the first simulation study, under the null hypothesis both the F and permutation test seem to exactly respect the nominal level in case of independent random errors (Table 1, last 3 columns) while they are somewhat conservative when random errors are correlated (Table 2). From a general point of view, as expected, power increases when increasing the number of blocks and the number of active variables. Obviously, the F-test is optimal under normality, but in case of exponential errors and particularly of Student’s t errors, the permutation test shows greater power except when the number of blocks is equal to 6. This is probably due to the relative small cardinality of the support of the test statistic. Note that results in Table 1 are generally strengthened in the case of correlated random errors (Table 2). This remark demonstrates the robustness of the permutation test in comparison to F test when the multivariate response variables are correlated as we expected in most of the real experimental situations. With reference to the second simulation study, under the null hypothesis both the F and permutation test seem to respect the nominal level (Tables 3 and 4, last 3 columns; for sake of simplicity we consider only the case where C=5). Findings of the second simulation study are more or less the same as the first study. Note that when we use a significance level of 0.1 instead of 0.05, there is a relative benefit for the permutation test and this is due to the fact that the rejection threshold is further from the minimum achievable value of the permutation test statistic. As a final remark for the second simulation study, note that when increasing the number of treatments, i.e. C is increasing, the F test seems to gain some relative advantage compared to the permutation test.

Copyright © 2008. Nova Science Publishers, Incorporated. All rights reserved.

4. Application to a Real Case Study The R&D division of a home-care company is studying 5 possible new fragrances (labeled r, s, t, v, w) of a given detergent to compare with their own presently marketed product (labeled x). The experiment is designed as follows: after testing one given product (using sense of smell), the panelist assigned three different scores to it, describing the three most important aspects of the product: Strength – S (1-5 points), Pleasantness – P (1-5), Appropriateness – A (Yes, No). The same experiment is replicated under different assessment conditions (Bloom B, Dry - D, Long - L, Neat - N and Wet - W), which should represent the situations in which the final customers will make use of the product. The graph in Figure 1 shows the described complex RCB design with its comparison reference between fragrance r and x. Figure 1 also displays the idea behind the solution to this problem that we propose using NPC Test methodology: for comparison between fragrance r and x, first of all a set of 15 univariate permutation tests is computed, where each test iTj, i=B,D,L,N,W, j=S,P,A, takes into account for comparison of one given aspect (Strength – S, Pleasantness – P, Appropriateness – A) within a given condition (Bloom - B, Dry - D, Long L, Neat - N and Wet - W). The next step relates to a multivariate comparison of fragrance r and x for each condition, which can be resolved by test iT, i=B,D,L,N,W, obtained via nonparametric combination of the univariate test for the three aspects. The final step is a global multivariate test T, obtained by a final further nonparametric combination of the five tests iT, i=B,D,L,N,W.

Copyright © 2008. Nova Science Publishers, Incorporated. All rights reserved.

Test Permut. F Permut. F Permut.

Student's T (2 df)

Exponential

Normal

F

Error

Table 1. Rejection rates for the first simulation study (multivariate global hypothesis H0) under the hypothesis of independent random errors

6

C=3 no. of active var. 1 2 3 .090 .340 1.00

H1 C=5 no. of active var. 1 2 3 .040 .428 1.00

C=7 no. of active var. 1 2 3 .035 .320 1.00

3 5 7 .065 .035 .050

10

.660

1.00

1.00

.600

1.00

1.00

.570

1.00

1.00

.045 .070 .035

20

.975

1.00

1.00

.990

1.00

1.00

.975

1.00

1.00

.030 .045 .065

6

.090

.255

.985

.065

.199

.985

.065

.175

.995

.025 .040 .025

10

.385

1.00

1.00

.405

.995

1.00

.370

1.00

1.00

.065 .060 .040

20

.895

1.00

1.00

.915

1.00

1.00

.930

1.00

1.00

.040 .065 .045

6

.075

.250

.865

.015

.227

.880

.030

.200

.905

.030 .025 .060

n

H0

C

10

.178

.720

1.00

.154

.747

.995

.110

.795

1.00

.050 .040 .075

20

.443

.975

1.00

.405

.980

1.00

.370

1.00

1.00

.035 .040 .060

6

.080

.220

.720

.030

.207

.745

.055

.240

.785

.015 .025 .040

10

.165

.740

1.00

.188

.780

1.00

.125

.845

.995

.035 .040 .055

20

.453

.985

1.00

.450

.995

1.00

.435

1.00

1.00

.035 .045 .045

6

.066

.152

.435

.070

.134

.340

.080

.155

.370

.085 .065 .065

10

.072

.287

.630

.090

.199

.510

.090

.195

.455

.030 .065 .065

20

.112

.517

.810

.130

.403

.805

.145

.305

.735

.060 .075 .075

6

.044

.116

.300

.030

.105

.250

.030

.085

.320

.030 .030 .005

10

.082

.311

.715

.099

.244

.640

.105

.285

.655

.030 .075 .025

20

.128

.575

.870

.145

.572

.915

.155

.515

.920

.035 .045 .030

Copyright © 2008. Nova Science Publishers, Incorporated. All rights reserved.

Test Permut. F Permut. F Permut.

Student's T (2 df)

Student's T (2 df)

Normal

F

Error

Table 2. Rejection rates for the first simulation study (multivariate global hypothesis H0) under the hypothesis of correlated random errors

6

C=3 no. of active var. 1 2 3 .025 .310 1.00

H1 C=5 no. of active var. 1 2 3 .015 .355 1.00

C=7 no. of active var. 1 2 3 .005 .220 1.00

3 5 7 .035 .025 .005

10

.645 .995 1.00

.595 1.00 1.00

.415 1.00 1.00

.015 .025 .005

20

.980 1.00 1.00

.980 1.00 1.00

.985 1.00 1.00

.015 .010 .010

6

.045 .255 .995

.035 .295 .985

.025 .350 1.00

.010 .000 .000

10

.275 .995 1.00

.255 1.00 1.00

.170 1.00 1.00

.030 .020 .000

20

.910 1.00 1.00

.890 1.00 1.00

.905 1.00 1.00

.015 .000 .010

6

.056 .260 .928

.007 .204 .899

.014 .194 .982

.029 .024 .054

10

.187 .725 1.00

.157 .766 1.00

.106 .830 1.00

.049 .039 .072

20

.468 1.00 1.00

.424 1.00 1.00

.406 1.00 1.00

.034 .038 .057

6

.075 .234 .786

.032 .246 .810

.044 .291 .915

.014 .024 .039

10

.148 .764 1.00

.189 .789 1.00

.106 .955 1.00

.033 .037 .052

20

.473 1.00 .968

.464 1.00 1.00

.472 1.00 .985

.034 .042 .043

6

.050 .100 .360

.015 .140 .310

.045 .100 .280

.020 .020 .055

10

.045 .260 .555

.020 .180 .475

.060 .140 .395

.025 .015 .015

20

.085 .445 .880

.055 .310 .720

.060 .295 .675

.020 .010 .010

6

.030 .110 .220

.015 .070 .185

.005 .040 .145

.020 .000 .000

10

.045 .280 .615

.035 .240 .595

.028 .175 .610

.025 .025 .010

20

.100 .520 .905

.060 .445 .860

.062 .400 .920

.015 .005 .010

n

H0

C

Copyright © 2008. Nova Science Publishers, Incorporated. All rights reserved.

Test

Error

Table 3. Rejection rates for the second simulation study (univariate hypothesis H0k and pair-wise comparison H0k(jh)) with α=0.05

n

Permut. F Permut. F Permut.

Student's T (2 df)

Exponential

Normal

F

6

C=3 delta Glob. 0.5 1 1.5 .547 .045 .239 .503

H1 C=5 delta Glob. 0.5 1 1.5 .542 .032 .144 .333

C=7 delta Glob. 0.5 1 1.5 .617 .035 .042 .124

H0 C=5 delta Glob. 0.5 1 1.5 .050 .007 .003 .000

10

.766

.114 .368 .741

.801

.041 .241 .642

.826

.103 .082 .204

.075

.010 .010 .000

20

.980

.209 .702 .975

1.00

.133 .649 .970

.995

.232 .249 .527

.030

.003 .003 .000

6

.343

.030 .055 .055

.373

.015 .027 .035

.380

.007 .012 .010

.030

.010 .003 .000

10

.736

.095 .289 .672

.776

.020 .129 .383

.433

.038 .017 .055

.065

.005 .008 .000

20

.980

.289 .756 .975

1.00

.118 .570 .940

.965

.181 .177 .353

.035

.000 .000 .000

6

.582

.055 .294 .542

.572

.051 .172 .453

.637

.060 .077 .159

.050

.005 .012 .010

10

.816

.134 .478 .801

.836

.055 .331 .672

.896

.113 .119 .259

.065

.007 .005 .005

20

.985

.229 .746 .970

1.00

.133 .654 .940

.995

.264 .311 .592

.040

.008 .003 .010

6

.423

.045 .139 .149

.468

.012 .067 .119

.095

.018 .027 .035

.020

.013 .005 .005

10

.846

.189 .488 .756

.856

.061 .279 .493

.637

.081 .080 .149

.070

.007 .010 .010

20

.990

.323 .796 .960

1.00

.164 .624 .886

.965

.257 .294 .503

.050

.002 .000 .000

6

.159

.020 .050 .119

.149

.005 .022 .085

.119

.003 .005 .020

.040

.008 .003 .010

10

.279

.035 .110 .234

.199

.005 .042 .110

.259

.023 .015 .015

.040

.000 .003 .000

20

.468

.045 .214 .373

.458

.023 .097 .254

.448

.038 .027 .040

.030

.000 .000 .005

6

.134

.025 .015 .040

.105

.003 .017 .050

.055

.003 .003 .010

.010

.013 .005 .000

10

.373

.065 .164 .313

.289

.015 .042 .095

.274

.012 .012 .020

.035

.003 .000 .005

20

.522

.06 .249 .438 5

.582

.032 .154 .393

.532

.051 .055 .110

.035

.002 .003 .005

Copyright © 2008. Nova Science Publishers, Incorporated. All rights reserved.

Test

C=3 delta Glob. 0.5 1 6 .657 .090 .328 10 .896 .199 .517 20 1.00 .274 .871 n

Permut.

1.5 .014 .010 .010

.587

.085 .194 .443

.642

.028 .085 .164

.704

.027 .035 .050

.084

.010 .011 .009

10

.866

.219 .522 .816

.846

.043 .241 .547

.884

.066 .075 .179

.090

.010 .010 .015

20

.995

.413 .896 .990

1.00

.196 .704 .985

1.00

.284 .316 .562

.095

.017 .020 .005

F

1.5 .154 .304 .652

6 10 20

.707 .891 1.00

.149 .368 .657 .159 .572 .851 .388 .861 .990

.721 .920 1.00

.085 .286 .532 .093 .438 .776 .204 .714 .990

.731 .891 .995

.093 .114 .174 .156 .159 .353 .318 .403 .662

.075 .090 .105

.010 .015 .010 .008 .017 .025 .010 .015 .015

Permut.

1.5 .537 .711 .995

C=7 delta Glob. 0.5 1 .707 .073 .077 .920 .153 .162 1.00 .332 .358

H0 C=5 delta Glob. 0.5 1 .091 .013 .010 .080 .003 .012 .100 .015 .020

6

.632

.179 .338 .488

.697

.046 .087 .144

.721

.032 .042 .075

.095

.010 .015 .005

10

.910

.254 .642 .776

.935

.100 .403 .652

.856

.121 .129 .254

.100

.003 .012 .005

20

1.00

.537 .906 .990

1.00

.265 .692 .960

.990

.333 .383 .597

.095

.010 .012 .045

F

1.5 .617 .861 .995

H1 C=5 delta Glob. 0.5 1 .687 .058 .246 .866 .070 .341 1.00 .191 .736

6

6 10 20

.249 .443 .602

.030 .105 .199 .065 .194 .358 .090 .234 .537

.289 .403 .537

.023 .062 .154 .027 .097 .214 .046 .154 .313

.299 .313 .532

.018 .020 .045 .032 .015 .020 .043 .052 .085

.075 .090 .110

.008 .012 .000 .012 .012 .020 .015 .005 .000

6

.279

.030 .139 .209

.304

.018 .062 .065

.335

.025 .020 .040

.070

.005 .012 .010

10

.488

.119 .209 .413

.522

.037 .119 .254

.418

.023 .030 .055

.119

.013 .012 .020

20

.667

.159 .333 .622

.667

.061 .219 .403

.711

.093 .077 .174

.124

.010 .012 .010

Permut.

Student's T (2 df)

Exponential

Normal

F

Error

Table 4. Rejection rates for the second simulation study (univariate hypothesis H0k and pair-wise comparison H0k(jh)) with α=0.1

Copyright © 2008. Nova Science Publishers, Incorporated. All rights reserved.

Table 5. Results for comparison between 5 new fragrances (r, s, t, v, w) with the presently marketed product (x) Bloom Comparison

S

P

Dry A

S

P

Long A

S

P

Neat A

S

P

Wet A

S

P

A

x vs r

.368 .513 .757 .334 1.00 .119 .201 .654 .221 .056 .087 .250 .170 .523 .250

x vs s

.123 .316 1.00 .247 .881 .119 .189 .898 .111 .167 .429 .250 .078 .484 .493

x vs t

.271 .250 1.00 .154 .604 .296 .078 .039 .111 .149 .168 .812 .412 .492 .498

x vs v

.507 .393 .757 .063 .035 .119 .032 .244 .231 .056 .055 .733 .078 .217 .250

x vs w

.015 .017 .485 .034 .882 .119 .096 .064 .231 .056 .030 .250 .039 .795 .250 Combin.

x vs r

.344

.164

.131

.010

.090

.027

x vs s

.090

.174

.072

.056

.052

.042

x vs t

.205

.171

.010

.124

.377

.059

x vs v

.453

.024

.017

.043

.055

.015

x vs w

.001

.010

.059

.017

.029

.000

Global test

.011

Livio Corain and Luigi Salmaso

dataset

412

Fragrance r S P A 5 4 1 … … … 5 5 0

Bloom vs.

Fragrance x S P A 3 4 0 … … … 2 3 0

B TS BPP B TA BT

Dry

Long

Neat

Wet

… … … … DTS DPP DTA

… … … … LTS LPP LTA

… … … … NTS NPP NTA

… … … … WTS WPP WTA

DT

LT

NT

WT

T Figure 1. Experimental design description and solution scheme via NPC Test methodology.

As an application of the proposed solution to real data, Table 5 displays the whole set of results from a real experiment with 7 panellists where we compare five fragrances with fragrance x and univariate permutation p-values have been corrected by multiplicity. Figure 2 provides a graphical p-value representation of NPC Test results for comparison of w and x. Bloom

Dry

Long

Neat

Strength Pleasant. Appropriat.

Wet