Optimization and Anti-optimization of Structures Under Uncertainty

Optimization and Anti-Optimization of Structures Under Uncertainty

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Optimization and Anti-Optimization of Structures Under Uncertainty

Isaac Elishakoff Florida Atlantic University, USA

Makoto Ohsaki Kyoto University, Japan

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Imperial College Press

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Published by Imperial College Press 57 Shelton Street Covent Garden London WC2H 9HE Distributed by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

OPTIMIZATION AND ANTI-OPTIMIZATION OF STRUCTURES UNDER UNCERTAINTY Copyright © 2010 by Imperial College Press All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

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ISBN-13 978-1-84816-477-2 ISBN-10 1-84816-477-7

Printed in Singapore.

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To Aron Lev, Chana Batsheva, Yehudit Lea, Menachem Mendel, Moshe Chaim, and Tamar Devorah I.E.

To Noriko, Takeshi, and Satsuki M.O.

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Preface

‘Uncertainties appear everywhere in the model. When using a mathematical model, careful attention must be given to the uncertainties in the model.’ (Feynman, 1988) ‘New methods for treating uncertainty will become important in virtually all branches of mechanics.’ (Oden, Belytschko, Babuˇska and Hughes, 2003) ‘Scientific progress usually can only be achieved by experts of different disciplines working together.’ (Eschenauer and Olhoff, 2001) ‘The ethics of our profession today does not allow any design for a structure without optimization.’ (Mungan, 2001)

Researchers in applied mechanics, whether in the field of aerospace, architecture, civil, mechanical or ocean engineering, invariably adopt the either/or style. Namely, they devote themselves either to linear or to nonlinear analysis of the structure they are dealing with, they are engaged in analyzing it either in the elastic or in the inelastic range; they deal either with its static or with its dynamic behavior. Along the same lines, researchers classify themselves according to the deterministic or non-deterministic nature of the problems they are tackling. The former stipulate that the geometry, loads, boundary conditions, material characteristics involved in a problem are fully specified; they stress that they are concerned with understanding the phenomenon at hand, and then providing methods of solution. To achieve such a basic comprehension of what is going on, they make (often farreaching) assumptions: that the parameters involved are fully specified, that they take on large or small values, etc. Such an approach, which may seem unjustifiable at first glance, often leads to breakthrough discoveries in understanding the phenomenon under study. The difference between the deterministic and non-deterministic approaches lies in the fact that the deterministic design attempts to retain its determinism until the very last stage of design, whereas the non-deterministic analysts maintain that there vii

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are no two structures with identical material and geometric properties, boundary conditions, or loading and, therefore, uncertainty analysis is called for. In reality, this deterministic versus non-deterministic attitude is a fallacy. To justify this undiplomatic statement, we need deeper insight into the essence of deterministic analysis. In it, after careful evaluation of displacements, strains, and stresses (often with many significant digits – since computer outputs consist of numbers), the engineers compare the latter against the allowable stress; i.e., the experimentally measured yield- or ultimate-level multiplied by some fudge coefficient called the safety factor (as in ‘better safe than sorry’). Use of this factor in textbooks on engineering mechanics is invariably motivated by insufficient knowledge (don’t we always have to make analytic assumptions of various sorts?), or by uncertainty (yes, deterministic analysts do pay lip service to the property they claim to neglect!) in loads, geometric characteristics, or scatter in material properties. Thus, deterministic analysis, strictly speaking, is not as deterministic as advertised. Within it, uncertainty is introduced as dessert or, as in Bolotin’s words (Bolotin, 1961), via the back door. By contrast, in non-deterministic design, uncertainty figures as a legitimate ingredient throughout the whole design process, specifically during the thinking, analysis and design stages. One may conclude, therefore, that non-deterministic analysis is more honest than its so-called deterministic counterpart as it places its cards on the table from the very beginning. As for the latter, in view of the above reasoning, it would be appropriate to dub it pseudo-deterministic rather than deterministic. If experience enables us to establish the precise value of a safety factor, then the pseudo-deterministic unknowns are nothing less than the lumped, integral equivalent of the inherently present uncertainty in engineering problems. Naturally, the advent of the safety factor was at the time a major breakthrough in design. Despite its obvious arbitrariness, it allowed us to put up a wall against failure. At the same time, this dogmatism implies that if the actual stress turns out to be below the allowable level, then there can be no failure, and the structure will be safe throughout its exploitation. But how can we establish the safety factor, except by experience or trial-anderror (one could sarcastically ask at this stage: is this a trial or an error?). According to Norton (2000), ‘choosing the safety factor is often a confusing proposition for the beginning engineer,’ and not only for the beginning one, we may add. According to Bruhn (1975), ‘[the safety] factor is designed to arbitrarily account for items such as material properties, manufacturing differences, since no two parts can be made exactly the same; uncertainties in the loading environment; and unknowns in the internal load and stress distributions.’

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As we observe, even with the manner of explanation of the safety factor, this concept is quite critical within the semi-deterministic framework for it utilizes adjectives like arbitrary (Bruhn, 1975) and confusing (Norton, 2000). Freudenthal (1968) emphasized: ‘It seems absurd to strive for more and more refinement of methods of stress analysis, if in order to determine the dimension of the structural elements, its results are subsequently compared with a so-called working stress, derived in a rather crude manner by dividing the values of somewhat dubious material parameters obtained in a conventional material test by a still more dubious empirical number, called a safety factor.’ Bolotin (1961) had written, ‘The values of safety factors, as well as closely associated values of design loads and design resistances, were imposed and modified mainly empirically, by way of generalization of long-term experience in exploitation of structures. Yet, as is seen from the essence of the problem, there are in principle, also theoretical approaches possible, with wide application of the theory of probability and mathematical statistics.’ We must immediately note that probability theory and mathematical statistics are not the only avenues for the theoretical approach to the safety factors. Non-deterministic analysis enables us to recast the safety factor approach in terms of other quantities. For example, Freudenthal (1957) and Rzhanitsyn (1947) pioneered an interpretation of the safety factor in terms of reliability based on the probability theory and mathematical statistics, and introduced the so-called central safety factor. (It was shown by Elishakoff (2004) that probabilistic mechanics offers four possible interpretations of the safety-factor concept.) Probabilistic analysis is not the only game in town, in the non-deterministic context. While the probability densities of the random variables involved in it are often not known, simpler characteristics like the sets within which the parameters vary – may be specified. Such a description of uncertainty (Bulgakov, 1942) is referred to as an unknown-but-bounded one. In the West it was independently developed by Boley (1966a) for static thermoelastic problems; and by Drenick (1968) and Shinozuka (1970) for dynamic ones. Some of these and other results in applied mechanics were summarized by Ben-Haim and Elishakoff (1990), who also developed some new material. They concentrated their attention on models of uncertainty in which the variations were treated as convex sets, and yielded the most and least favorable stresses, strains or displacements over such a convex set. This approach has been extended by Elishakoff, Lin and Zhu (1994e) by combining it with probabilistic analysis; and by Elishakoff, Li and Starnes (2001) via systematic comparison

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with the probabilistic methodology. Elishakoff (1994) interpreted the safety factor within unknown-but-bounded uncertainty as the interval [nL , nU ], between lower ‘L’ and upper ‘U’ bounds representing the ratios of the yield stress to the least and most favorable response, respectively, in which case the yield stress can safely serve as the design parameter. Likewise, Elishakoff and Ferracuti (2006a, 2006b) provided interpretation of the safety factor within fuzzy sets. Here, too, as in probabilistic analysis, there are four possible interpretations of the fuzzy safety factor. This book is exclusively concerned with the unknown-but-bounded uncertainty. Chapter 1 explains in detail why this approach was chosen. Here, we content ourselves with noting that in the case of random variables with bounded support, it is shown that probabilistic design and the unknown-but-bounded approach yield coincident results, when the required reliability tends to unity. Accordingly the second version was chosen as the less complicated of the two, in keeping with Albert Einstein’s sage precept: ‘Everything should be made as simple as possible, but not simpler.’ In the past decade or so, many papers were devoted to unknown-butbounded uncertainty in applied mechanics. Several books were published as well. Regrettably, some of the authors, instead of concentrating on the safety factor as the main tool of structural design and professional communication, turned to alternative concepts or actually introduced new (and hardly needed) ones, thereby diverting research activities from the really relevant problems (see the instructive title of the book by Sokal and Bricmont (1998)). We would like next to consider the optimization/anti-optimization problem, but before proceeding further it would be instructive to reproduce the text of a letter sent by Dr. Rudolf F. Drenick (2001), Professor Emeritus of Brooklyn Polytechnic Institute, to Dr. Izuru Takewaki, Professor of Kyoto University: ‘You might be amused by the story of how I came to do the work on the critical excitation. In (I think) 1965 I participated in a Japan/U.S. workshop on applied stochastics. After two days of lectures, one of the observers at our meeting made some comments which I found very interesting. He said something like this. “My name is Ozawa and I work for the Tokyo Building Licensing Bureau. I am forever visited by architects who show me plans of buildings they plan to construct and I am supposed to tell them whether their plans are good or bad, and I have never really known what to tell them. Now I have listened to you gentlemen for two days and I still don’t know what to tell them.” There followed a great deal of discussion among the workshop participants but in the end the discussion leader, Professor Bogdanov, said “I am very sorry, Mr. Ozawa, but we can’t help you.” I went home from the workshop and began to think of how I could

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deal with Mr. Ozawa’s problem. I concluded that any method I should suggest to him would have to be, first of all, practical. I also felt that it should be as distribution-free as possible. After some thought the concept of the critical excitation came to me. (The term critical excitation was not my idea. It was suggested by Prof. Penzien.) My first results were largely distribution-free, as you know, but they were not practical. They were much too conservative to be useful.’ This worst-case design was never adopted by the nuclear industry because of its extreme conservatism, as Drenick informed one of us (I.E.). He, accordingly, became very pessimistic about the likelihood of the engineering profession ever adopting it. He was however very encouraging, and so kind as to write the foreword to the book by Ben-Haim and Elishakoff (1990) in which he stated perhaps with some measure of exaggeration: ‘Their approach is novel and highly welcome. In my opinion, it is inevitable that it, and its extensions, will dominate the future practice of engineering.’ In consequence of the above-mentioned book, it was realized by one of us (I.E.) that in order to make this worst-case scenario research practical, it needed an infusion of new blood. (It was realized that unknown-but-bounded uncertainty analysis is a defacto as anti-optimization process, the reverse of searching for the best solution.) It became also apparent that it is impractical to content oneself with the least favorable static or dynamic response, or with the worst possible buckling load. Rather a structure should be so designed as to minimize the first and maximize the second – in other words, adherence to the time-honored precept of ‘making the best out of the worst.’ It is gratifying to read that this sentiment is now shared by other investigators, for example, Ben-Tal and Nemirovski (2002): ‘The worst-case oriented interval model of uncertainty looks too conservative. Indeed, when the perturbations in coefficients of an uncertain linear inequality are of stochastic nature, it is highly improbable that they will simultaneously take the most dangerous values.’ These considerations eventually gave rise to the optimization and antioptimization terminology or better yet optimization with anti-optimization. If there is an uncertainty, one should, as it were, participate in two dances concurrently, doing both optimization and anti-optimization steps. Such a hybrid approach would be less conservative than its worst-scenario counterpart, hence is much more likely to gain credibility and acceptance among researchers and practicing engineers. Indeed, it is often said that if a researcher cannot explain his or her research to a lay person, then that researcher does not yet fully understand the subject

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under discussion (Einstein, here too, comes handy: ‘you do not really understand something unless you can explain it to your grandmother’). Indeed, would not it be an engineering sin, as it were, not to improve the worst scenario and leave it intact – especially if one has to save expenditure? Only very rich companies, or those in positions of exceptional responsibility, can afford exclusive reliance on anti-optimization. One argument against optimization is that if it is applied to a specific loading condition, the result may prove unsatisfactory under a different condition. In such a case, it suffices to add the latter and repeat the procedure for the new circumstances. Alternatively, one can introduce uncertainty and assign bounds so as to obtain the optimal solution under the worst-case scenario. Another criticism concerns optimization against nonlinear buckling. As explained in detail in Chap. 5, optimization against buckling does not always yield imperfection-sensitive structures, the resulting sensitivity being in fact lower than that of a non-optimal design (Ohsaki, 2002c). However, the buckling load of a perfect system is drastically increased by optimization, and its counterpart for an imperfect system is little affected by sensitivity changes. The combined optimistic-pessimistic approach can be characterized as follows: optimists build ships; pessimists build lifeboats. Both are needed for safe sailing. Recall the overly optimistic view on the Titanic, which was claimed to be so robust that even God couldn’t sink it! As a result, an insufficient number of lifeboats was on board during its maiden voyage. This complacency cost over 1,500 lives. We advocate a combination of healthy doses of optimism (represented by optimization) and pessimism (anti-optimization). The above discussion gave rise to the notion that uncertainty analysis is reducible to one of the three vertices of the uncertainty triangle (Elishakoff, 1990, 1998b), shown below. These three approaches do not represent three non-interesting magistrata! Although the first paper co-authored by one of us (Ben-Haim and Elishakoff, 1989b) was presented at two conferences in 1988 and appeared in 1989, we had been able, on an earlier occasion (Elishakoff, 1983, p. 42), to compare the probabilistic and worst-case designs on a simple example included on the ‘Statistical Methods in Elasticity’ course at the Technion - Israel Institute of Technology since 1973. The monograph by Ben-Haim and Elishakoff (1990) did not correlate probabilistic and convex modelings. Only later, in the paper by Elishakoff, Cai and Starnes (1994a), was a direct comparison made between probabilistic and anti-optimization techniques on the nonlinear boundary-value problem (see also its generalization by Qiu, Ma and Wang (2006a)). It turned out that if the probabilistic information is available and the required reliability is not excessively high, anti-optimization may be conservative. However, if near-unity reliability is required, the two approaches tend to yield close or coincident results. It was also realized that these seemingly competing approaches are nevertheless

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amenable to a certain measure of cooperation. An instructive example is the case of the turbulent excitation experienced by the space shuttle weather protection systems at the Kennedy Space Center. This excitation is inherently random and must be analyzed probabilistically, but inadequate data is available on its crossspectral density. This shortage was remedied (Elishakoff, Lin and Zhu, 1994e) by means of the anti-optimization approach, which yielded the least favorable meansquare displacement of the weather protection panels. In another combination, Kosko (1990), Maglaras, Nikolaidis, Haftka and Gudney (1997), Chiang, Dong and Wong (1987), and Wood, Antonnson and Beck (1990) watched the fuzzy-sets approach against the probabilistic one, theoretically and/or experimentally. Kwakernaak (1978), Haldar and Reddy (1992), Savoia (2002), and M¨ oller and Beer (2004) combined the two approaches. Finally, Fang, Smith and Elishakoff (1998) and Tonon, Bernardini and Elishakoff (2001) utilized all three approaches for their analyses. Bernardini (1999) and Tonon, Bae, Grandhi and Pettit (2006) claim that all three methodologies stem from a single theory of random sets (see Matheron (1975) and Dubois and Prade (1991)). Recently, Ben-Haim (2001) also adopted combined probabilistic and non-probabilistic analyses. As is clearly seen from the above, probabilistic and non-probabilistic analyses of uncertainty are not irreconcilable and can complement each other when deemed useful. Such reconciliation was in fact advocated by Elishakoff (1995b): ‘Convex modeling of uncertainty, and in general the set-theoretic modeling of it (the constraints should not necessarily be treated as convex), do not support probabilistic ideas. Convex modeling rather complements both the probabilistic approach and the fuzzy subsets based treatment.’

Theory of Probability and Randon Processes

Fuzzy Sets

Anti-Optimization with Set Theoretical Aproach (Least Favorable Responses)

Uncertainty triangle (Elishakoff, 1990).

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One should constantly bear in mind that just as pseudo-deterministic analysis incorporates uncertainty via a lumped number (namely the safety factor), the probabilistic approach is partly deterministic, as not all parameters are considered variable. Does one have to consider all parameters of a structure as uncertain, thereby covering the general case? Some research establishments take this line and present on their slides perhaps tens of random variables with arbitrary probabilistic distributions to choose from! Its Excellency the Computer then provides the desired output in seconds. Such excessive zeal is unjustifiable, seeing that no probabilistic dependencies are either known or incorporated into the programs! As Babuˇska, Nobile and Tempone (2005) emphasize, ‘whenever the input data belong to an infinite dimensional space (they might be functions of position and/or time), their probabilistic characterization must include knowledge of the cross-correlation of the values that the data can take at different points in space and time. In this case, the solution of a stochastic model becomes quickly too costly.’ Still, we should be thankful that the innocent number π, for example, is not declared a random variable with mean value 3.14... and zero variance! One should refrain from considering all of them as random variables unless they have sufficiently large variability to justify the non-deterministic approach. Likewise, one should abandon the idea (attractive for beginners) of treating all parameters as uncertain, even if some of them have sufficiently small variability. In conclusion, one has to maintain a delicate and healthy balance in modeling a system. Bolotin (1969), for example, is against both underestimation and overestimation of uncertainty analysis. We hope that this book will serve as a beacon for practicing engineers and researchers, and help them reevaluate their thinking and practices on uncertainty analysis. The authors are indebted to the Japan Society for Promotion of Science (JSPS) established by the late Emperor Showa (Hiro-Hito) in 1932. The JSPS, long before the modern trends of globalization, made it possible for non-Japanese researchers to spend time in the Land of the Rising Sun with their host counterparts, thereby enabling both sides to elevate their horizons through collaboration. I.E. is thankful to Kyoto University which provided the hospitable atmosphere for the joint research. We record our appreciation to the co-authors of some joint papers that found their way into our monograph. These are Professor Dan Givoli of the Department of Aerospace Engineering, Technion - Israel Institute of Technology, for his kind agreement for us to reproduce the shortened version of the paper ‘Stress concentration at nearly circular hole with uncertain irregularities,’ by D. Givoli and I. Elishakoff, 1992; Professor Nobuhiro Yoshikawa of the Institute of Industrial Science, University of Tokyo for his kind agreement for us to reproduce the contents of the paper ‘Worst case estimation of homology design by convex analysis,’ by N. Yoshikawa,

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I. Elishakoff and S. Nakagiri, 1998; Professor Alessandro Baratta and Professor Giulio Zuccaro of the Department of Scienza delle Costruzioni of the University of Naples Federico II, for their kind agreement to draw portions from two papers: ‘A generalization of the Drenick–Shinozuka model for bounds on the seismic response of a single-degree-of-freedom systems,’ by A. Baratta, I. Elishakoff, G. Zuccaro and M. Shinozuka, 1998, and ‘Antioptimization of earthquake excitation and response,’ by G. Zuccaro, I. Elishakoff and A. Baratta, 1998 (we also thank Hindawi Publishers for allowing us to reproduce the results and figures of the paper); Professor Massimiliano Zingales of the Dipartimento di Ingegneria Strutturale e Geotecnica, Universit` a degli Studi di Palermo, for his kind agreement to draw materials from the following joint papers: ‘Contrasting probabilistic and anti-optimization approaches in an applied mechanics problem,’ by I. Elishakoff and M. Zingales, 2003, ‘Antioptimization versus probability in an applied mechanics problem: Vector uncertainty,’ by M. Zingales and I. Elishakoff, 2000, and ‘Hybrid aeroelastic optimization and antioptimization,’ by M. Zingales and I. Elishakoff, 2001; Professor Kiyohiro Ikeda of the Department of Civil Engineering, Tohoku University for his kind agreement to include materials in the paper ‘Imperfection sensitivity analysis of hill-top branching with many symmetric bifurcation points,’ by M. Ohsaki and K. Ikeda, 2006; Dr. Jingyao Zhang of the Department of Architecture and Urban Design, Ritsumeikan University, for his kind agreement for presenting summaries of the papers ‘Optimization and anti-optimization of forces in tensegrity structures,’ by M. Ohsaki, J. Y. Zhang and I. Elishakoff, 2008, and ‘Optimal measurement positions for identifying stress distribution of membrane structures using cable net approximation,’ by J. Y. Zhang, M. Ohsaki and Y. Araki, 2004. The authors would also like to thank Prof. Yoshihiro Kanno of the Department of Mathematical Informatics, University of Tokyo, for checking the details of the manuscript; Dr. Takao Hagishita of Mitsubishi Heavy Industries (former graduate student of Kyoto University), Mrs. Juxi Hu of Beijing University of Aeronautics and Astronautics, and Ms. Melanie Dolly of the IFMA–French Institute of Advanced Mechanics for typing some parts of the book; Ms. Shawn Pannel and Mr. Joshua Kahn of Florida Atlantic University for redrawing the figures; Mr. Eliezer Goldberg of the Technion - Israel Institute of Technology, for checking the preface and Chapter 10 of this book; Ms. Sarah Haynes of Imperial College Press for her assistance in bringing the manuscript to its final form. We appreciate the support from the National Science Foundation over several years, as well as the National Center for Earthquake Engineering Research. Any opinions, findings, and conclusions or recommendations expressed by this monograph are those of the authors and do not necessarily reflect the views of these agencies. May 2009

Isaac Elishakoff, Makoto Ohsaki

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Contents

Preface

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1. Introduction 1.1 1.2 1.3

1

Probabilistic Analysis: Bad News . . . . . . . . . . . . . . . . . . Probabilistic Analysis: Good News . . . . . . . . . . . . . . . . . Convergence of Probability and Anti-Optimization . . . . . . . .

2. Optimization or Making the Best in the Presence of Certainty/Uncertainty 2.1 2.2 2.3 2.4

2.5 2.6

2.7 2.8 2.9

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . What Can We Get from Structural Optimization? . . . . Definition of the Structural Optimization Problem . . . . Various Formulations of Optimization Problems . . . . . 2.4.1 Overview of optimization problems . . . . . . . . 2.4.2 Classification of optimization problems . . . . . . 2.4.3 Parametric programming . . . . . . . . . . . . . 2.4.4 Multiobjective programming . . . . . . . . . . . . Approximation by Metamodels . . . . . . . . . . . . . . . Heuristics . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.1 Overview of heuristics . . . . . . . . . . . . . . . . 2.6.2 Basic approaches of single-point search heuristics 2.6.3 Simulated annealing . . . . . . . . . . . . . . . . Classification of Structural Optimization Problems . . . . Probabilistic Optimization . . . . . . . . . . . . . . . . . . Fuzzy Optimization . . . . . . . . . . . . . . . . . . . . .

17 . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

3. General Formulation of Anti-Optimization 3.1 3.2 3.3

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Models of Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . Interval Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii

1 10 13

17 19 20 23 23 24 26 28 30 31 31 32 35 36 39 41 47 47 49 50

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3.4

3.5 3.6

3.7

3.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 3.3.2 A simple example . . . . . . . . . . . . . . . . . . . 3.3.3 General procedure . . . . . . . . . . . . . . . . . . . Ellipsoidal Model . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Definition of the ellipsoidal model . . . . . . . . . . 3.4.2 Properties of the ellipsoidal model . . . . . . . . . . Anti-Optimization Problem . . . . . . . . . . . . . . . . . . Linearization by Sensitivity Analysis . . . . . . . . . . . . 3.6.1 Roles of sensitivity analysis in anti-optimization . . 3.6.2 Sensitivity analysis of static responses . . . . . . . 3.6.3 Sensitivity analysis of free vibration . . . . . . . . 3.6.4 Shape sensitivity analysis of trusses . . . . . . . . . Exact Reanalysis of Static Response . . . . . . . . . . . . . 3.7.1 Overview of exact reanalysis . . . . . . . . . . . . . 3.7.2 Mathematical formulation based on the inverse of modified matrix . . . . . . . . . . . . . . . . . . . . 3.7.3 Mechanical formulation based on virtual load . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . the . . . . . .

4. Anti-Optimization in Static Problems 4.1 4.2 4.3 4.4 4.5

4.6

A Simple Example . . . . . . . . . . . . . . . . . . . . . . . . . . Boley’s Pioneering Problem . . . . . . . . . . . . . . . . . . . . . Anti-Optimization Problem for Static Responses . . . . . . . . . Matrix Perturbation Methods for Static Problems . . . . . . . . Stress Concentration at a Nearly Circular Hole with Uncertain Irregularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.2 An asymptotic solution . . . . . . . . . . . . . . . . . . . 4.5.3 A worst-case investigation . . . . . . . . . . . . . . . . . Anti-Optimization of Prestresses of Tensegrity Structures . . . . 4.6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.2 Basic equations . . . . . . . . . . . . . . . . . . . . . . . 4.6.3 Anti-optimization problem . . . . . . . . . . . . . . . . . 4.6.4 Numerical examples . . . . . . . . . . . . . . . . . . . . .

Introduction . . . . . . . . . . . . . . . A Simple Example . . . . . . . . . . . Buckling Analysis . . . . . . . . . . . Anti-Optimization Problem . . . . . . Worst Imperfection of Braced Frame Loads . . . . . . . . . . . . . . . . . . 5.5.1 Definition of frame model . . .

70 74 77

5. Anti-Optimization in Buckling 5.1 5.2 5.3 5.4 5.5

50 51 52 57 57 58 61 63 63 64 67 68 69 69

77 80 85 86 90 90 91 97 102 102 103 107 109 113

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . with Multiple . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . Buckling . . . . . . . . . . . .

113 114 116 117 118 118

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Contents

5.5.2 5.5.3 5.5.4

5.6 5.7

5.8

Worst imperfection of optimized frame . . . . . . . . . . Mode interaction . . . . . . . . . . . . . . . . . . . . . . Worst-case design and worst imperfection under stress constraints . . . . . . . . . . . . . . . . . . . . . . . . . . Anti-Optimization Based on Convexity of Stability Region . . . Worst Imperfection of an Arch-type Truss with Multiple Member Buckling at Limit Point . . . . . . . . . . . . . . . . . . . . . . . 5.7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 5.7.2 Hilltop branching point of perfect system . . . . . . . . . 5.7.3 Imperfection sensitivity of hilltop branching point . . . . 5.7.4 Worst imperfection . . . . . . . . . . . . . . . . . . . . . 5.7.5 Worst imperfection of an arch-type truss . . . . . . . . . Some Further References . . . . . . . . . . . . . . . . . . . . . . .

6. Anti-Optimization in Vibration 6.1 6.2

6.3 6.4

6.5

6.6

6.7

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A Simple Example of Anti-Optimization for Eigenvalue of Vibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Anti-optimization for forced vibration . . . . . . . . . . . Bulgakov’s Problem . . . . . . . . . . . . . . . . . . . . . . . . . Non-probabilistic, Convex-Theoretic Modeling of Scatter in Material Properties . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.2 Basic equations for vibrating viscoelastic beam . . . . . . 6.4.3 Application to a simply supported beam . . . . . . . . . 6.4.4 Least and most favorable responses . . . . . . . . . . . . 6.4.5 Numerical examples and discussion . . . . . . . . . . . . Anti-Optimization of Earthquake Excitation and Response . . . 6.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.2 Formulation of the problem . . . . . . . . . . . . . . . . . 6.5.3 Maximum structural response . . . . . . . . . . . . . . . 6.5.4 Ellipsoidal modeling of data . . . . . . . . . . . . . . . . A Generalization of the Drenick–Shinozuka Model for Bounds on the Seismic Response . . . . . . . . . . . . . . . . . . . . . . . . 6.6.1 Preliminary comments . . . . . . . . . . . . . . . . . . . 6.6.2 Credible accelerograms . . . . . . . . . . . . . . . . . . . 6.6.3 Application . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6.4 Discussion and conclusion . . . . . . . . . . . . . . . . . Aeroelastic Optimization and Anti-Optimization . . . . . . . . . 6.7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 6.7.2 Deterministic theoretical analysis . . . . . . . . . . . . . 6.7.3 Stability analysis within two-term approximation . . . .

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120 123 126 128 133 133 134 135 136 139 144 145 145 146 148 149 150 150 151 155 157 160 163 163 164 166 166 175 175 180 184 188 190 190 191 193

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6.7.4 6.7.5 6.7.6

6.8

6.7.7 6.7.8 6.7.9 Some

Convex modeling of uncertain moduli . . . Anti-optimization problem: polygonal region of uncertainty . . . . . . . . . . . . . . . . . Anti-optimization problem: ellipsoidal region of uncertainty . . . . . . . . . . . . . . . . . Minimum weight design . . . . . . . . . . . . A numerical example . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . Further References . . . . . . . . . . . . . . . .

. . . . . . . 195 . . . . . . . 197 . . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

7. Anti-Optimization via FEM-based Interval Analysis 7.1 7.2 7.3 7.4

7.5 7.6

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . Interval Analysis of MDOF Systems . . . . . . . . . . . . . Interval Finite Element Analysis for Linear Static Problem Interval Finite Element Analysis of Shear Frame . . . . . . 7.4.1 Basic equations . . . . . . . . . . . . . . . . . . . . 7.4.2 A numerical example . . . . . . . . . . . . . . . . . Interval Analysis for Pattern Loading . . . . . . . . . . . . Some Further References . . . . . . . . . . . . . . . . . . . .

211 . . . . . . . .

. . . . . . . .

. . . . . . . .

8. Anti-Optimization and Probabilistic Design 8.1 8.2

8.3

200 201 206 208 208

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Contrasting Probabilistic and Anti-Optimization Approaches . . 8.2.1 Problem formulation . . . . . . . . . . . . . . . . . . . . 8.2.2 Probabilistic analysis . . . . . . . . . . . . . . . . . . . . 8.2.3 Uniformly distributed random initial imperfections . . . 8.2.4 Random initial imperfections with truncated exponential distribution . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.5 Random initial imperfection with generic truncated distribution . . . . . . . . . . . . . . . . . . . . 8.2.6 Buckling under impact load: Anti-optimization by interval analysis . . . . . . . . . . . . . . . . . . . . . . . 8.2.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . Anti-Optimization Versus Probability: Vector Uncertainty . . . 8.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.2 Deterministic analysis . . . . . . . . . . . . . . . . . . . . 8.3.3 Probabilistic analysis . . . . . . . . . . . . . . . . . . . . 8.3.4 Initial imperfections with uniform probability density: Rectangular domain . . . . . . . . . . . . . . . . . . . . . 8.3.5 Initial imperfections with general probability density: Rectangular domain . . . . . . . . . . . . . . . . . . . . .

211 212 214 217 217 220 221 225 227 227 229 229 231 232 240 244 245 246 247 247 248 250 251 262

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8.3.6 8.3.7 8.3.8 8.3.9

Initial imperfection with uniform probability density function: Circular domain . . . . . . . . . . . . . . . . . Initial imperfections as interval variables: Interval analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . Initial imperfections as convex variables: Circular domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . Concluding remarks . . . . . . . . . . . . . . . . . . . . .

9. Hybrid Optimization with Anti-Optimization under Uncertainty or Making the Best out of the Worst 9.1 9.2 9.3 9.4

9.5

9.6 9.7

9.8

9.9

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A Simple Example . . . . . . . . . . . . . . . . . . . . . . . . . . Formulation of the Two-Level Optimization–Anti-Optimization Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Algorithms for Two-Level Optimization–Anti-Optimization . . . 9.4.1 Cycle-based method . . . . . . . . . . . . . . . . . . . . . 9.4.2 Methods based on monotonicity or convexity/concavity . 9.4.3 Other methods . . . . . . . . . . . . . . . . . . . . . . . . Optimization against Nonlinear Buckling . . . . . . . . . . . . . 9.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 9.5.2 Problem formulation . . . . . . . . . . . . . . . . . . . . 9.5.3 Numerical examples . . . . . . . . . . . . . . . . . . . . . Stress and Displacement Constraints . . . . . . . . . . . . . . . Compliance Constraints . . . . . . . . . . . . . . . . . . . . . . . 9.7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 9.7.2 Optimization problem and optimization algorithm . . . 9.7.3 Numerical examples . . . . . . . . . . . . . . . . . . . . . Homology Design . . . . . . . . . . . . . . . . . . . . . . . . . . 9.8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 9.8.2 Deterministic loading . . . . . . . . . . . . . . . . . . . . 9.8.3 A numerical example . . . . . . . . . . . . . . . . . . . . 9.8.4 Convex model of uncertain loading . . . . . . . . . . . . 9.8.5 Worst-case estimation . . . . . . . . . . . . . . . . . . . . 9.8.6 A numerical example of worst-case estimation of homology design . . . . . . . . . . . . . . . . . . . . . . . 9.8.7 Concluding remarks . . . . . . . . . . . . . . . . . . . . . Design of Flexible Structures under Constraints on Asymptotic Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 9.9.2 Definition of asymptotic stability . . . . . . . . . . . . . 9.9.3 Optimization problem . . . . . . . . . . . . . . . . . . . . 9.9.4 Numerical examples . . . . . . . . . . . . . . . . . . . . .

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264 269 269 272

273 273 274 276 277 277 278 279 280 280 281 285 288 291 291 294 295 298 298 300 302 303 304 305 306 307 307 308 309 310

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9.10 Force identification of prestressed structures . . . . . . . . . . 9.10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 9.10.2 Equations for self-equilibrium state . . . . . . . . . . 9.10.3 Formulation of identification error . . . . . . . . . . 9.10.4 Sensitivity analysis with respect to nodal coordinates 9.10.5 Optimal placement of measurement devices . . . . . 9.10.6 Numerical examples . . . . . . . . . . . . . . . . . . . 9.11 Some Further References . . . . . . . . . . . . . . . . . . . . .

. . . . . . . .

. . . . . . . .

10. Concluding Remarks 10.1 10.2 10.3 10.4 10.5

Why Were Practical Engineers Reluctant to Adopt Structural Optimization? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Why Didn’t Practical Engineers Totally Embrace Probabilistic Methods? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Why Don’t the Probabilistic Methods Find Appreciation among Theoreticians and Practitioners Alike? . . . . . . . . . . . . . . . Is the Suggested Methodology a New One? . . . . . . . . . . . . Finally, Why Did We Write This Book? . . . . . . . . . . . . . .

316 316 317 318 320 322 323 325 327 327 331 333 334 339

Bibliography

343

Index

387

Author Index

391

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Chapter 1

Introduction

‘May the best ye’ve ever seen, Be the warst ye’ll ever see.’ (Scottish blessing) ‘It is not enough to do your best; you must know what to do, and then do your best.’ (W. Edwards Deming) ‘It’s only the man who can look at the same problem from many different aspects that will make a true leader.’ (Takao Fujisawa)

In this chapter, we explain why it is preferable to deal with set-theoretical rather than probabilistic or fuzzy uncertainty. It is demonstrated that probabilistic analysis with unbounded support is prone to high sensitivity of the results to the behavior of the tail of the distribution. This bad news is balanced by good news: a probability distribution with bounded support yields the same results as the simple and transparent anti-optimization technique. We argue that since there are apparently no physical parameters with unbounded support, one has to resort to probability densities with bounded ones or directly to the anti-optimization technique. This provides, in a nutshell, the main justification for this book to deal with bounded uncertainty and anti-optimization. 1.1

Probabilistic Analysis: Bad News

Apparently, the first study in which probabilistic methods were applied to structures was conducted by Mayer (1926). Since then, probabilistic methods have developed beyond the age of adolescence (Cornell, 1981). Usually, probabilistic reliability studies involve assumptions on the probability densities, whose knowledge regarding the relevant input quantities is central. Given this, the probabilistic properties of the output quantities can be determined. As Wentzel (1980) stresses, probabilistic methods are ‘frequently regarded as a kind of magic wand which produces in1

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formation out of a void. This is a fallacy; the theory of probability only enables information to be transformed, and conclusions on inaccessible phenomena to be drawn from data on observable ones.’ Probabilistic methods must yield a central quantity–reliability of the structure, namely, the probability that the structure will perform its mission satisfactorily. Modern society rightfully expects extremely high reliabilities and, consequently, extremely small probabilities of failure. The deterministic mechanical theories represent a cornerstone of probabilistic mechanics. First, the phenomenon should be understood qualitatively; next, it should be understood sufficiently well quantitatively. Then the output quantities in their intricate dependence on their input counterparts are developed as a set of equations, algebraic expressions, or numerical codes of varying complexity. At this stage the parameters to be treated as uncertain variables or functions are identified. The deterministic relation or numerical code then serves as a transfer function determining the probabilistic characteristics of the output and the reliability of the structure. Our goal being extremely high reliabilities, it is immediately understood that the deterministic relations or numerical codes must be of supreme accuracy, in order to avoid a GIGO (‘garbage in–garbage out’) situation. However, since each deterministic relation is based on simplifying assumptions with a specified degree of accuracy of their own, legitimate questions arise: Are less accurate tools suitable for determining the extremely accurate desired probabilities of failure? What happens to the reliability of structures designed on the basis of increasingly accurate theories? No less central is the input data. In most cases, accurate information is unavoidable. Probabilistic analysts maintain that their theoretical analyses, presently devoid of experimental inputs, still are of values; and once information becomes available, it can be incorporated into a ready-made theoretical framework. But in the meantime these analyses have taken on lives of their own: to make up for the lacking information, researchers resort to such assumptions as specific patterns of normal distribution, time-wise stationarity of a random process, space-wise homogeneity of a random field, ergodicity of the process, or the Markov property. Thus the questions ‘What can go wrong with probabilistic methods?’ and ‘To use or not to use them?’ have become extremely relevant. Their various aspects have been addressed in monographs by Ben-Haim and Elishakoff (1990) and BenHaim (1996), and in papers, e.g., Elishakoff and Hasofer (1996). The need for a major revision of our understanding of the role of probabilistic methods is stressed in the monograph by Ben-Haim and Elishakoff (1990) and in the articles by Kalman (1994) and Elishakoff (1995b). In this section, we elucidate difficulties inherent in structural reliability due to imperfect information, in conjunction with the ever increasing desire to allow only extremely small probability of failure. First of all, we address this cardinal question: Why does one need probabilistic

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3

methods? The reply to this question is best given in terms of the design of a structure. In a deterministic design process, one requires σ ≤ σy (1.1) where σ is an actual stress that is assumed to be positive in tensile state, and σy is an yield stress. Here, uncertainty enters the picture in the sense that we may not know the loads precisely, or there may be imprecision in measuring the geometric parameters of the cross-section, or we may have built an imperfect mechanical model to describe the behavior of the structure. Accordingly, a required safety factor k req is introduced, and Eq. (1.1) is replaced by σy (1.2) σ≤ kreq Once the structure is designed, one can introduce the actual safety factor σy (1.3) kact = σmax where σmax is the maximum actual stress occurring in the structure. Thus the design requirement can be formulated as kact ≥ kreq , or, in other words, the actual safety factor should not be less than the required one. Can one quantify that the actual safety factor of the uncertainty is not hidden, but is directly introduced into the scene? Let us attempt to answer this question. Let the random force p, acting on a tension-compression element (bar) with crosssectional area a, have a Weibull (1951) distribution with the following probability distribution function "  k # p − p0 FP (p) = 1 − exp − , k > 0, w > p0 , p ≥ p0 (1.4) w − p0 For p < p0 , FP (p) ≡ 0. Equation (1.4) is called the 3-parameter Weibull distribution, which is a generalization of the exponential distribution. It has been developed originally to describe failure strength of metals. More recently it has been utilized in connection with fracture, as well as life distribution of mechanical compounds. For the special case p0 = 0, the distribution is called 2-parameter Weibull distribution; in this case k = 1 yields the exponential distribution, whereas k = 2 is associated with Rayleigh distribution. The shape parameter k < 1 is typical of wearing phenomena, whereas k > 1 is typical of aging effects. The 3-parameter Weibull distribution is useful in describing phenomena for which some minimum value p0 exists for the random variable P , so that P takes on values greater than or equal to p0 . In the following, we denote random variables by upper-case, and lower-case notation is reserved for their possible values. We are interested in the reliability of the structure, i.e., the probability of the structure performing its intended mission satisfactorily. Such a performance is identified with relationship holding Eq. (1.1) true. Thus, the reliability becomes: R = Prob(Σ ≤ σy ) = Prob(P/a ≤ σy )

= Prob(P ≤ σy a)

= FP (σy a)

(1.5)

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Thus, we have

"  k # σy a − p 0 R = 1 − exp − w − p0

(1.6)

How can we define the safety factor in the context of probabilistic design? One natural way is to relate it to some characteristic load, say the average load. The latter equals E(P ) = p0 + (w − p0 )Γ(1 + 1/k)

(1.7)

where Γ( · ) is the Gamma function. The variance of the load is V (P ) = (w − p0 )[Γ(1 + 2/k) − Γ2 (1 + 1/k)]

(1.8)

The central safety factor n is defined as the ratio of the yield stress to its average counterpart E(P )/a: σy a σy a = (1.9) n= E(P ) p0 + (w − p0 )Γ(1 + 1/k)

Let us design the structure probabilistically. Probabilistic design requires that the reliability be not less than a codified value r: R≥r Thus, in view of Eq. (1.6), we obtain "  1 − exp −

σy a − p 0 w − p0

(1.10) k #

≥r

(1.11)

The design value of the cross-sectional area adesign is found from the equality R = r, and reads adesign =

p0 + (w − p0 )[ln 1/(1 − r)]1/k σy

(1.12)

Substitution of adesign for a in Eq. (1.9) enables us to write the central safety factor explicitly in terms of the codified required reliability r: n=

p0 + (w − p0 )[ln 1/(1 − r)]1/k p0 + (w − p0 )Γ(1 + 1/k)

(1.13)

1 + 2[ln 1/(1 − r)]1/4 1 + 2Γ(1.25)

(1.14)

For the set of parameters w = 3p0 , k = 4, we have n=

Therefore, from Eq. (1.14), n = 1.2314 for r = 0.9; n = 1.3971 for r = 0.99; n = 1.5082 for r = 0.999; n = 1.5942 for r = 0.9999; n = 1.6653 for r = 0.99999; n = 1.7263 for r = 0.999999, etc. As we see, a probabilistic model allows us to associate the required reliability directly with the safety factor. Let us see now how a small error can affect the reliability calculations. Say that the actual values are w1 , p1 and k1 , while those used in the analysis are w, p0 and

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5

k. The actual reliability Ract and the actual probability of failure Pf,act are related as Pf,act = 1 − Ract

(1.15)

The actual reliability is given by Eq. (1.6) with the actual values substituted: "  k # σy a − p 1 1 (1.16) Ract = 1 − exp − w1 − p 1 and the actual probability by Pf,act

"  k # σy a − p 1 1 = exp − w1 − p 1

(1.17)

If the design of the structure has been performed using the values w, p0 and k, the appropriate value of the cross-sectional area is given by Eq. (1.12). To calculate the actual probability of failure corresponding to the design value adesign , we substitute the expression adesign into Eq. (1.17) and have "  k # σy adesign − p1 1 Pf,act = exp − (1.18) w1 − p 1

or

Pf,act

"   k1 # p0 − p1 + (w − p0 )[ln 1/(1 − r)]1/k = exp − w1 − p 1

(1.19)

Let us consider some particular cases. In the simplest case p1 = p0 , w = w1 , but k1 6= k, "  k1 /k # 1 Pf,act = exp − ln (1.20) 1−r Since r is the required reliability, 1 − r is recognized in Eq. (1.20) as the allowed probability of failure Pf,all . Thus, Eq. (1.20) can be rewritten as: "  k1 /k # 1 Pf,act = exp − ln (1.21) Pf,all Let Pf,all = 10−6 . Then Pf,act = exp[−13.81551056k1/k ]

(1.22)

This function of the ratio k1 /k is shown in Fig. 1.1. As is seen, when k1 /k = 1, Pf,act = Pf,all = 10−6

(1.23)

is satisfied. By contrast, when k1 6= k, the actual probability of failure may differ from the allowed one. Remarkably, there can be a serendipitous situation, namely, when an error in measurement of k1 may be of a favorable nature: for k1 /k > 1, the actual

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0.00002

Pf , act

0.000015 0.00001 0.000005 0 0.9

0.95

1

1.05

1.1

k1 / k Fig. 1.1 Actual probability of failure as a function of the ratio k1 /k (solid line); for k1 = k it coincides with the required one (dotted line); for k1 /k > 1 it is less than the required one; for k1 /k < 1 it may well exceed the allowed value, resulting in a detrimental state.

probability of failure is less than the allowable one. However, when k1 /k > 1, the effect of a small error in evaluating k may be detrimental: If k1 /k = 0.95, the actual probability is about five times the allowed one; if k1 /k = 0.93, the actual probability is approximately ten times as large as the one which was permitted! We conclude that the probability of failure is too sensitive a parameter to make do with imprecise input characteristics, and that accurate determination of the probabilistic characteristics of input must be an integral part of a rigorous probabilistic analysis. Regarding the effect of a small deviation in the probability density on the reliability estimate, we consider a bar with cross-sectional area a subjected to a load P which is a random variable with the probability density fP (p), p being a possible value of the load. The material of the bar is assumed to be perfectly elastic in compression and has a yield stress in tension σy . We also assume that the bar cannot lose its stability, or undergo any other form of failure. To recapitulate, its reliability is defined as the probability of the stress Σ = P/a not exceeding the yield stress: R = Prob(Σ ≤ σy )

(1.24)

Let us consider the situation in which the data is suggestive for the analyst to assume the probability density of the load in the form of the log-normal variable     1 −(ln p − b)2  √ , for p > 0 exp 2c2 fP (p) = pc 2π (1.25)   0, otherwise where b and c characterize the probability density. The mean value E(P ) and the variance V (P ) are expressed as   1 2 E(P ) = exp b + c 2 (1.26) 2 2 V (P ) = exp(2b + c )[exp(c ) − 1]

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The reliability is thus given by R = Prob(P ≤ σy a)

= Prob(ln P ≤ ln σy a)   1 ln σy a − b = + erf 2 c

where

erf(x) =

1 2π

Z

(1.27)

x

exp(−t2 /2)dt

(1.28)

0

since ln P is a normal variable with mean b and variance c2 . Let us assume now that the actual probability density differs slightly from the true one and reads, for p > 0:   1 (ln p − b)2 (ε) {1 + ε sin[2π(ln p − b)]} (1.29) fP (p) = √ exp − 2c2 pc 2π where ε is a constant belonging to an interval [−1, 1]. For p < 0, the probability (ε) density vanishes. It can be shown that fP (p) is indeed a probability density, nonnegative and satisfies the equality Z ∞ (ε) fP (p)dp = 1 (1.30) −∞

To prove this property, we have to demonstrate that Z ∞ fP (p) sin[2π(ln p − b)]dp = 0

(1.31)

and further substitution t − b = u leads to   Z ∞ u2 1 √ exp − 2 sin(2πu)du I= 2c −∞ c 2π

(1.33)

−∞

To do this, we make a substitution ln p = t. Thus, the integral to be calculated reads   Z ∞ 1 (t − b)2 √ exp − I= sin[2π(t − b)]dt (1.32) 2c2 −∞ c 2π

which vanishes since the integrand is an odd function of u. Thus, the function defined in Eq. (1.29) represents a probability density of some random variable, (ε) denoted by P (ε) . Obviously, if ε = 0, fP is equal to fP as per Eqs. (1.25) and (1.29). Stoyanov (1987, pp. 89–91) demonstrates that for any k = 1, 2, . . . , we have E(P (ε) )k = E(P k ) (ε)

(1.34)

i.e., the perturbed random variable P has the same moments as those of an unperturbed variable P . Let us calculate now the true reliability associated with f (ε) : Z σy a (ε) R = Prob(P ≤ σy a) = fP (p)dp (1.35) 0

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0.02

Pf , act

0.015 0.01 0.005 0 −1

Fig. 1.2

−0.5

0 ε

0.5

1

Relation between ε and Pf,act for c = 0.1, Pf,all = 0.01 (r = 0.99).

0.002

Pf , act

0.0015 0.001 0.0005 0 −1

Fig. 1.3

or

−0.5

0 ε

0.5

1

Relation between ε and Pf,act for c = 0.2, Pf,all = 0.001 (r = 0.999).

  (ln p − b)2 1 √ exp − R= {1 + ε sin[2π(ln p − a)]}dp (1.36) 2c2 pc 2π 0 Introducing again the new variable of integration ln p = t, we reduce reliability to   Z ln σy a (t − b)2 1 √ exp − {1 + ε sin[2π(t − b)]}dt (1.37) R= 2c2 c 2π 0 and with t − b = u, we have     Z ln σy a−b 1 ln σy a − b ε 1 u2 √ exp − 2 sin(2πu)du (1.38) R = + erf + √ 2 c 2c c 2π 0 c 2π Evaluation of this integral yields r    π 1 ln σy a − b ε −2π 2 c2 R = + erf + √ −e 2 c 2 c 2π (1.39)       2πc2 − i(b + ln[σy a]) 2πc2 + i(b + ln[σy a]) √ √ × erfi + erfi 2c 2c Z

σy a

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0.0002

Pf , act

0.00015 0.0001 0.00005 0 −1

−0.5

0

0.5

1

ε

Fig. 1.4

Relation between ε and Pf,act for c = 0.2, Pf,all = 0.0001 (r = 0.9999).

where erfi(z) is an imaginary error function, defined as (Wolfram, 1996, p. 745) erfi(z) = erf(iz)/i

(1.40)

√ where i = −1. One can visualize that the reliability calculations have been per(0) formed with the probability density of the load set at fP (p). Let required reliability be r, so that the design is performed using the expression in Eq. (1.27). Figures 1.2–1.4, associated with r = 0.99, r = 0.999 and r = 0.9999, respectively, depict the actual situation in which the control parameter ε describing the deviation from reality does not vanish identically. Thus, the actual probability of failure Pf,act varies with ε. Naturally, when ε = 0, Pf,act takes the value 1 − r, or 0.01, 0.001 and 0.0001, respectively, in Figs. 1.2–1.4. The integrand in Eq. (1.36) is an oscillatory function. Therefore, depending on the value of c, the value of the integral is either positive or negative; hence, different behaviors are exhibited in Figs. 1.2–1.4. The diagrams demonstrate the possibility of safe errors; i.e., deviation of the model from the reality may turn out to be on the safe side. Indeed, if ε in Fig. 1.2 tends to −1 from above, the actual probability of failure tends to zero. Likewise, in Fig. 1.3, associated with c = 0.2 and Pf,all = 10−3 , if ε tends to unity from below, the actual reliability too approaches unity from below. Analogously, in Fig. 1.4, with c = 0.2, Pf,all = 10−4 , and ε tending to unity from below, the actual probability of failure tends to zero from above. It is prudent, however, to consider what can go wrong when the theoretical model differs from reality. As Fig. 1.2 demonstrates, when ε tends to unity, the actual probability of failure becomes double the allowable one. This implies, through the frequency interpretation of the probability notion, that the number of unsuccessful performances may be double the allowed. Nearly analogous situations are depicted in Figs. 1.3 and 1.4 when ε tends to −1. For a critical appraisal of methods to determine the possibility of failure, the reader may like to consult, for example, the paper by Schu¨eller and Stix (1987) and the monograph by Elishakoff (2004).

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Probabilistic Analysis: Good News

The somewhat discouraging picture painted above by no means justifies boycotting the probabilistic approach. In fact, the latter makes it possible to substantiate the mysterious fudge factor, or factor of safety. Taking an example from a monograph (Elishakoff, 2004) on safety factors, consider an element subjected to a stress modeled as a random variable. In a similar manner as in previous section, random variables are denoted with capital letters. Hence the random stress is denoted by Σ with its realization being denoted by lower-case notation σ. We also assume that the yield stress σy is a deterministic quantity. Both the actual and yield stresses can be treated as random variables, but for greater convenience only one therein is so treated. We also adopt the simplest possible assumption of a continuously distributed Σ, i.e., a uniform probability density   0, for σ < σL , σ > σU 1 (1.41) fΣ (σ) = , for σL ≤ σ ≤ σU  σU − σ L where σL and σU are the lower and upper bounds of the stress. Now, the reliability R, i.e., the probability that the actual stress does not exceed the yield stress σy R = Prob(Σ ≤ σy ) equals

  0, for σy < σL  Z σy  σ −σ y L , for σL ≤ σy ≤ σU fΣ (σ)dσ = R=  σU − σ L −∞   1, for σy > σU

(1.42)

(1.43)

Let us concentrate on the range σL ≤ σy ≤ σU . We first note that the mean value E(Σ) of the actual stress equals 1 (1.44) E(Σ) = (σL + σU ) 2 whereas the standard deviation dΣ is given by σU − σ L √ (1.45) dΣ = 2 3 By Eqs. (1.44) and (1.45), the lower and upper bounds of the stress are expressed, respectively, as √ σL = E(Σ) − 3dΣ (1.46) √ σU = E(Σ) + 3dΣ Thus, the reliability is rewritten, in the range σL ≤ σy ≤ σU as √ σy − E(Σ) + 3dΣ √ R= 2 3dΣ

(1.47)

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At this juncture it is instructive to introduce the central safety factor as the following ratio σy (1.48) n= E(Σ) along with the coefficient of variation νΣ of the stress dΣ (1.49) νΣ = E(Σ) Dividing the numerator and denominator of Eq. (1.47) by E(Σ), we obtain √ n − 1 + 3νΣ √ (1.50) R= 2 3νΣ This formula relates reliability, safety factor, and the coefficient of variation of the stress. Expressing the safety factor from Eq. (1.50), we have   √ 1 n = 1 + 2 3νΣ R − (1.51) 2 Probabilistic design is based on the requirement R≥r

(1.52)

where r is the required, codified reliability, and the required safety factor nr is obtained from Eq. (1.51) by setting R = r:   √ 1 (1.53) nr = 1 + 2 3νΣ r − 2 This is a simple formula connecting the required safety factor nr with the required reliability r and coefficient of variation νΣ of the stress. We can deduce several useful conclusions: (i) The safety factor, so often criticized by practitioners and researchers alike, is actually a powerful concept, which can be given a probabilistic interpretation. (ii) Probability theory strips the mystery from the safety factor, which instead of being assigned at the will of the designer, or out of the sky as it were, gains an analytical framework for its determination. (iii) If one can quantify the required reliability, say through legislation, one can quantify the safety factor as well. (iv) If the required reliability r is greater than 0.5, i.e., if we do not tolerate nearly half of our products being defective (and, hopefully, we don’t!), the required safety factor exceeds unity. This is in agreement with all textbooks and designs, where the required safety factor is in excess of unity. (v) The higher the coefficient of variation of the stress, the higher the required safety factor. This matches one’s anticipation, as it should be. Moreover, Eq. (1.53) tells us exactly how much the safety factor must be, depending on the magnitude of uncertainty νΣ . Were the coefficient of variation negligibly small, i.e., νΣ = 0+ , the safety factor would be nr = 1+ . This implies that the truly deterministic design is valid only when νΣ tends to zero. This fact clearly indicates that purely deterministic design is contained as a particular case in probabilistic design.

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Consider now a closely related example from a textbook (Elishakoff 1983, 1999) on the probabilistic theory of structures. Again, our objective is a tension– compression element (bar), in this case a uniform one subjected to random load P which is uniformly distributed between the lower bound pL and the upper bound pU . The bar must be designed so that its reliability R equals the required value r or exceeds it. The design itself amounts to choosing the cross-sectional area a of the bar. The reliability equals R = Prob(P/a ≤ σy ) = Prob(P ≤ σy a) = FP (σy a)

(1.54)

where FP (p) is the probability distribution function of the load. Their reliability formula maintains that the reliability of the bar equals the value of the probability distribution of the load function computed at the product of the yield stress and the cross-sectional area. Calculation of the reliability yields   0, for σy a < pL  σ a−p y L , for pL ≤ σy a ≤ pU (1.55) R= pU − p L    1, for σ a > p y

U

As noted, the range of P is pL ≤ σy a ≤ pU . The design criterion stipulates σy a − p L ≥r pU − p L

(1.56)

σy a r − p L pU − p L

(1.57)

The minimum required cross-sectional area ar corresponds to r=

which leads to the needed cross-sectional area (pU − pL )r + pL ar = σy

(1.58)

We observe that when r tends to unity, the numerator tends to pU and the design value ar approaches the value pU ar = (1.59) σy This expression is also obtainable from the third alternative in Eq. (1.55) with unity reliability being obtained for σy a ≥ pU . It is remarkable that the expression Eq. (1.58) can also be obtained without any probabilistic analysis. Indeed, since the load varies in the interval [pL , pU ], the stress also is an interval variable σI = [pL /a, pU /a], which must be less than the yield stress σy . This requirement is satisfied if pU ≤ σy (1.60) a which leads to the minimum required value of the area as given in Eq. (1.58). We arrive at the elegant conclusion: The probabilistic and anti-optimization approaches

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lead to the same result! Not only is there no antagonism between them, but they both tell us the same thing! This implies that we can interchangeably utilize either of the two approaches! Those who prefer the probabilistic sophistification, must be allowed (does anyone need permission, really?) to continue development and usage of probabilistic mechanics. Those who prefer simplicity will stick to the anti-optimization technique. Naturally, this begs the question: Is one of the approaches preferable to the other? It appears that the anti-optimization approach, as the simpler of the two, has a certain advantage. Moreover, as one of the respondents stressed in the poll conducted by Elishakoff (2000b): ‘Engineers, especially those at the top, do not trust statements and predictions of probabilistic character. They must make decisions on important projects, large structures, and big investments, and they prefer to be completely sure that their decisions are true. Of course, proper education in the theory of probability and mathematical statistics helps here, but there is a more profound, maybe subconscious cause of such an attitude. Most engineers are more happy to look at samples of the behavior of a system, at its time signatures, than at probability density functions, and cumulative distribution functions that all seem alike to practicing engineers. In addition, they become suspicious when a probabilist talks about probabilities of failure, say 3.14 × 10−6 and all that. In my own experience, it is expedient to show a set of samples of the system behavior, in particular the worst sample, the best one and an average or a typical one...’ At the same time, it should be borne in mind that in its pure form, without proper improvements in the structure via optimization, anti-optimization is likely to yield ultra-conservative results. This is why this monograph opts for a combination of anti-optimization and optimization.

1.3

Convergence of Probability and Anti-Optimization

Another natural question may pop up: if the probability approach is so good as to provide the same answer as its intellectual competitor, anti-optimization, why was it necessary to talk, in the previous section, about it being (sometimes) bad? It is important to have a reply to this question, because the variously named non-probabilistic approaches are presented as sound alternatives to the probabilistic description. It appears that this happens because probability theory often uses

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random variables which stretch from minus infinity to plus infinity. Such is the case with the best-known discussion – the normal or Gaussian; do the exponential and Gamma distributed random variables take on values from zero to infinity? Are any of these random variables, as mental constructs, good for describing geometric or material characteristics? Whereas the exponential and Gamma distributions with their only positive values can be utilized for describing positive quantities, it is doubtful that there exists any physical parameter that can take on values beyond physical limits. Likewise, use of the normal distribution to describe positive quantities appears to be doubtful, if not outright abnormal (pun intended). At best, these distributions are convenient analytical approximations. The above implies that engineers ought to use truncated distributions, i.e., distributions with bounded support. It is gratifying that awareness of this fact grows amongst stochastic analysts. For example, Grigoriu (2006) mentions: ‘All distributions with bounded support in (0, ∞) are consistent with available information.’ Ma, Leng, Meng and Fang (2004) write: ‘Bounded random parameters ... are more reasonable for engineering structures than the unbounded Gaussian random ones.’ In their study, Minciarelli, Gioffre, Grigoriu and Simiu (2001) write: ‘Unless otherwise indicated, all uncertainty variables . . . will be assumed to have truncated normal distribution.’ Cai and Wu (2004) note: ‘When investigating a dynamical system under random excitation, it is important that each existing process should be modeled properly to resemble its measured or estimated statistical and probabilistic properties. In many cases, Gaussian distribution is assumed for convenience of analysis. However, the range of Gaussian distribution is unbounded; namely, there exists probability of having very large values. This violates the very nature of a real physical quantity, which is always bounded.’ Likewise, Cai and Lin (2006) write: ‘Physically realistic random procedures are bounded, and they may deviate far from being Gaussian.’ At the end of their paper these authors are even more forceful: ‘Physically realistic stochastic processes must be bounded.’ (see also Cai (2003)). Ang and Tang (1975) realized this fact quite long time ago: ‘In engineering, the information may often have to be expressed in terms of the lower and upper limits of the variable. For example, when judgment is necessary it is often convenient and perhaps more realistic to express judgmental information in the form of a range of possibilities given the range of possible values of a random variable and the underlying uncertainty may be evaluated by prescribing a suitable distribution within the range.’

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In their study, Simiu and Heckert (1996) analyzed the wind speed model and fitted an approximate extreme value distribution. They concluded that the reverse Weibull distribution is more appropriate than those of Gumbel or Fr´echet. Their conclusion is supported by the physical fact that non-tornadic extreme winds are expected to be bounded: ‘In our opinion, the analysis provided persuasive evidence that extreme wind speeds are described predominantly by reverse Weibull distributions, which unlike the Gumbel distribution have finite upper tail and lead to reasonable estimates of wind load factors.’ They also stress ‘It is a physical fact that extreme winds are bounded. Their probabilistic model should reflect this fact. To the extent that an extreme value distribution would be a reasonable model of extreme wind behavior, one would intuitively expect that the best fitting distribution to have finite tail...’ Kanda (1994) also showed that extreme winds are best fitted by distributions with limited tails; see also Walshan (1994). Holmes (2002) notes that extreme winds have a physical upper limit (which may differ for different storm types); hence, they have a bounded distribution with data from nearly all stations in Australia using various fitting methods. He reached a bounded generalized extreme value (GEV) distribution in over 80% of the cases. He stresses: ‘The approach ... in substituting one third of the observation by “unbounded”, “randomly generated” normal variables is not convincing.’ Thus, the pragmatic engineer cannot agree with Lindley (1987) who states: ‘Probability is the only satisfactory description of uncertainty.’ Neither can one agree with the notion implied by the title of Taleb’s 2001 book that we are ‘fooled by randomness.’ (Readers are also advised to read insightful articles on the different sides of the issue by Klir (1989, 1994).) Berleant and Goodman-Strauss (1998) used interval analysis to bound the results of arithmetic operations on random variables of unknown dependency. Thus, randomness and boundedness may pragmatically interact with each other. Kim, Ovseyevich and Reshetnyak (1993) and Elishakoff, Cai and Starnes (1994a) compared probabilistic and anti-optimization approaches. Ferson and Ginzburg (1996) advocate for application of different methods to describe ignorance and variability. We opt in this book for the anti-optimization, or worst-scenario approach because it is simpler than the probabilistic one, not because we are against as Kosko (1994) calls it ‘the probability monopoly’. Moreover, it yields the same results as a more complicated probabilistic methodology (according to Hans Hoffmann, ‘the ability to simplify means to eliminate the unnecessary so that the necessary may speak’). We just add a spice to the approach: anti-optimized, worst-scenario re-

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sults need to be optimized so as to yield cheaper, less voluminous designs. This is why this book is fully dedicated to marrying these two concepts: optimization and anti-optimization.

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Chapter 2

Optimization or Making the Best in the Presence of Certainty/Uncertainty

‘Make the best of things.’ (Russian proverb) ‘Good, better, best. Never let it rest until your good is better and your better is best!’ (Anonymous) ‘We have to do our best and leave the rest to Providence.’ (Japanese proverb)

In this chapter, we first address the motivation for optimization of structures in the presence of certainty or in its absence. We try to provide some thoughts on both the philosophical and pragmatic question: ‘What can we get from structural optimization?’ We also provide an overview of optimization problems, and various, but as expected, not all methodologies. Then we provide a touch on probabilistic (or stochastic) optimization, and complete the chapter with optimization via fuzzy sets. Reviews of these deterministic and non-deterministic techniques provide a glimpse into what will be the main scope of the book, namely, a specific nondeterministic technique which uses neither the concept of randomness nor the idea of fuzziness. This is done not because we feel that these approaches are without merit: a whole volume can be written on each of them. By discussing stochastic and fuzzy optimization techniques, we hope that the reader can have at least a fragmentary picture of topics which will not be central in our subsequent exposition.

2.1

Introduction

In the design process of structures in various fields of engineering, the design variables such as the cross-sectional geometries, nodal locations, material properties, etc., are modified by the designer to improve structural performance within the limited cost for production and construction. Hence, the whole process of structural design may be regarded as that of seeking the best set of design variables under given design requirements. 17

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The first paper on structural optimization is ascribed to Galileo Galilei who in 1638 discussed the optimal shapes of beams. In this respect it is instructive to give a podium to Venkayya (1993): ‘The notion of an optimum solution to an engineering problem is intriguing and has been investigated for a long time. The strongest cantilever beam in bending and constant shear as formulated by Galileo Galilei was also an optimum design for minimum weight under a uniform stress constraint. Galileo’s problem was probably one of the earliest optimization problems.’ The various types of minimum principles, such as minimum potential energy and minimum kinetic energy, were extensively investigated in the 18th century to be established as calculus of variation. The minimization problem for finding the solutions governed by a minimum principle is called variational problem. The constrained variational problem called isoperimetric problem was solved by Leonhard Euler (1707–1783) to show that a circle maximizes the interior area for a specified length of a closed smooth curve on a plane. The shape of a hanging cable called a catenary that minimizes the potential energy was found by John Bernoulli (1667–1748). Variational methods are applied in structural mechanics to find the equilibrium states of an elastic conservative system based on the minimum principle of total potential energy. The origin of modern optimization based on computational methods is generally credited to the development of linear programming during the period of World War II. Since then, many problem formulations and algorithms have been developed to establish the field called mathematical programming (Luenberger, 2003) in applied mathematics, and operations research in management science. Structural optimization is regarded as an application of optimization strategies to the field of structural design (Arora, 2004; Haftka, G¨ urdal and Kamat, 1992; Vanderplaats, 1999). The paper by Prager and Taylor (1968) may be cited as the first contribution on modern structural optimization, where the optimality criteria are derived for several problems under constraints on linear buckling load, eigenvalue of vibration, external work against static loads called compliance, and plastic collapse load. For topology optimization problems, where the locations of members of framed structures are optimized (see Sec. 2.7 for details), the so-called Michell truss (Michell, 1904; Hemp, 1973) is considered to be the first contribution and is extensively cited for verification of recently developed numerical methods. However, the paper by Dorn, Gomory and Greenberg (1964), which initiated the widely used method called ground structure approach, may be cited as the first work of modern computer-based topology optimization. In the 1970s, the analytical approaches based on optimality conditions were investigated (Rozvany, 1976). Since the 1980s, with rapid development of computer technology, many research papers have been

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published on computational methods and their industrial applications. Now, in the 21st century, optimization methods have matured, and we can optimize realworld structures with moderately large numbers of degrees of freedom and design variables. 2.2

What Can We Get from Structural Optimization?

The basic strategy of structural design without resort to optimization algorithm may simply be stated as follows: • Find the best design iteratively modifying the design variables by trial-and-error process based on the experience of the designer, under requirements on the structural responses and mechanical properties, e.g., the stresses and displacements against static/dynamic design loads, eigenvalues of free vibration, liner/nonlinear buckling loads, etc. However, the following questions may arise: • How can we define the best design? • What is the best strategy to modify the design variables if the current design is not the best? Optimization can answer these questions. In the traditional design process, arbitrary initial (trial) values are given for the design variables, structural analysis is carried out for evaluating the responses, and the variables are modified intuitively if the design requirements are not satisfied; i.e., there is no general procedure or guideline for design modification. Furthermore, the design process is often terminated if all the requirements are satisfied, and no effort is made to find better solutions. On the other hand, in view of code-based design, where the upper and lower bounds of the responses as well as the bounds of design variables are given by the design code, structural optimization provides us with the following benefits: • A design satisfying all the constraints (feasible design) can be found automatically, and efficiently, while simultaneously minimizing the objective function, such as the total structural volume, if the problem is appropriately formulated so that the set of feasible designs is nonempty. • The optimization tool helps the decision making of the designer, i.e., it is not an automatic design tool that gives a negative impression to the structural engineers and designers. If the optimal design is not acceptable to the designer, the upperlevel parameters, such as the geometry and material properties of the structure, can be modified, or additional constraints can be assigned for the optimization problem, to obtain a more reasonable optimal solution. Therefore, the designers can spend more time on the synthetic jobs, rather than structural analysis, if

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Design parameter (geometry, topology)

Design load Structural analysis

Structural optimization

Response (stress, displacement)

A step of conventional design process Fig. 2.1

Design variable

A step of optimization-based design process

Illustration of conventional and optimization-based design processes.

optimization tools are effectively used for decision making. The conventional design process and optimization-based design process are illustrated in Fig. 2.1. • Optimization is very helpful for designing complex structures, for which even an experienced designer cannot easily find a feasible design. • Even if the optimal solution cannot be used directly in design practice, the optimal design gives insight into a better design. • If we start with a feasible and nearly optimal design found by an expert, the design cannot be worse after optimization, and usually a better design can be found. Furthermore, in view of decision making, • The trade-off relation between the structural cost and responses can be made clear, if optimization is carried out several times by modifying the input parameters such as the cost coefficients and upper bounds of responses. The most positive ways of using optimization may be • To find new structural systems and shapes by optimization (Bendsøe and Sigmund, 2003). • To generate innovative structure or material that cannot be found without optimization, e.g., compliant mechanisms. Figure 2.2 shows an example of a compliant bar-joint structure utilizing snapthrough behavior (Ohsaki and Nishiwaki, 2005).

2.3

Definition of the Structural Optimization Problem

The structural optimization problem can be formally formulated as • Minimize objective function (cost, weight, volume) subject to constraints on mechanical performances.

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21

Input load (forced displacement)

Input node

Output node Output displacement Fig. 2.2

Illustration of a compliant bar-joint system.

where subject to means under constraints on, and mechanical performances include member stresses, nodal displacements, etc., for which the upper and lower bounds are given. The bounds for the mechanical performances are determined from the requirements given by the codes (regulations) in practical design process. The total structural volume (or weight) is usually assigned as the objective function, because it is a critical requirement to reduce the weight of the aerospace and mechanical structures. For the long-span structures in architectural and civil engineering, the reduction of the self-weight generally leads to reduction of the design loads, and hence leads to lower cost. It is also important that, for any acceptable definition of the objective function, a solution satisfying all the constraints, called feasible design, is obtained after optimization as discussed in Sec. 2.2. If the concept of minimum weight is not under consideration, the following alternative formulation may be valid:

• Maximize mechanical performance subject to upper bound for cost (weight, volume).

Note that the two formulations stated above lead to similar optimal solutions if the problem parameters are appropriately assigned. Consider, for instance, the problem of minimizing the structural volume of a truss under constraints on stresses and displacements against static loads. The design variables are the cross-sectional areas of the members. Let σ = (σ1 , . . . , σm )> and U = (U1 , . . . , Un )> denote the vectors of member stresses and nodal displacements that are called state variables, respectively, where m is the number of members and n is the number of degrees of freedom. All vectors are column vectors, and a subscript is used for indicating a component of a vector throughout this book. The vector of member cross-sectional areas are denoted by A = (A1 , . . . , Am )> , and let Li denote the length of the ith member.

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The structural optimization problem under stress and displacement constraints is formulated as minimize subject to

m X

A i Li i=1 σiL ≤ σi (A) ≤ σiU , (i = 1, . . . , m) UiL ≤ Ui (A) ≤ UiU , (i = 1, . . . , n) ALi ≤ Ai ≤ AU i , (i = 1, . . . , m)

(2.1a) (2.1b) (2.1c) (2.1d)

where the upper and lower bounds are denoted by the superscripts ( · )U and ( · )L , respectively. In Problem (2.1), the stress of the ith member is written as a function of A only as σi (A). However, for a general statically indeterminate structure, the responses are defined implicitly through the vector of nodal displacements U (A), which is also a function of A. The responses may also depend explicitly on A, e.g., the axial force Ni of member i is defined as σi (U )Ai . Next, we formulate the problem in a more general form. Let x = (x1 , . . . , xm )> denote the vector of design variables representing the cross-sectional areas, nodal coordinates, etc. Suppose the objective function such as the total structural volume is defined as an explicit function of x as F (x). Then, the structural optimization problem is formulated as follows: minimize

F (x)

(2.2a) I

(2.2b)

E

(2.2c)

subject to Gi (x, U (x)) ≤ 0, (i = 1, . . . , N )

Hi (x, U (x)) = 0, (i = 1, . . . , N )

where Gi and Hi are generally nonlinear functions of x and U (x), and N E and N I are the numbers of equality and inequality constraints, respectively. If the design variables x can take continuous values, then the structural optimization problem (2.2) is classified as a nonlinear programming problem, for which various algorithms such as sequential quadratic programming and method of feasible directions are available. On the other hand, if the variables take integer values or are selected from a list or a catalog of available values, then the problem is classified as a mixed integer nonlinear programming problem with continuous state variables U (x) (see Sec. 2.4.2 for a detailed classification of optimization problems). The standard approach to nonlinear programming problem is based on the gradient information of the objective and constraint functions. Suppose the vector U of state variables is defined in a general form by the state equations such as the equilibrium (stiffness) equations as Zi (x, U (x)) = 0, (i = 1, . . . , n)

(2.3)

Then the differential coefficients, which is called sensitivity coefficients, ∂Uk /∂xi of Uk with respect to xi can be computed from a set of n simultaneous linear

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equations (Choi and Kim, 2004) n

∂Zj X ∂Zj ∂Uk + = 0, (j = 1, . . . , n) ∂xi ∂Uk ∂xi

(2.4)

k=1

The details of sensitivity analysis can be found in Sec. 3.6. In order to apply nonlinear programming algorithms, the constraint function Hi (x, U (x)) is regarded as a function of x only: e i (x) = Hi (x, U (x)) H

(2.5)

e i (x) are obtained by using the sensitivity Then the differential coefficients of H coefficients of U (x): n

ei ∂Hi X ∂Hi ∂Uk ∂H = + , (j = 1, . . . , m) ∂xj ∂xj ∂Uk ∂xj

(2.6)

k=1

e i (x) = Gi (x, U (x)) are obtained similarly. The differential coefficients of G

2.4

2.4.1

Various Formulations of Optimization Problems Overview of optimization problems

If the objective and constraint functions are smooth and continuously differentiable with respect to the design variables, the optimal solutions can be found by the so-called gradient-based mathematical programming approach, in which analysis and sensitivity analysis are consecutively performed and the design variables are modified in accordance with the optimization algorithm. However, this approach cannot be used if the variables have integer values. The global optimal solution for such problems can be found by methods of integer programming such as the branch-and-bound method (Horst and Tuy, 1990), which requires computational time of exponential function of the problem size. Heuristic approaches have been developed to obtain approximate optimal solutions within reasonable computational cost (Reeves, 1995) (see Sec. 2.6 for the details). In this section, we classify optimization problems based on the continuity of the functions and variables. Then the methodologies that are closely related to anti-optimization, and optimization with anti-optimization, such as approximation methods, multilevel optimization and multiobjective programming, are described. Text books, e.g., Luenberger (2003), may be consulted for the well-developed methods for linear and nonlinear programming problems. We first summarize the terminologies used in the problem formulation and algorithms of optimization. Note that a minimization problem is assumed for simplicity. Feasible region: A solution that satisfies all the constraints is called a feasible solution, and the region of feasible solutions in the space of variables is called a feasible region.

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Local optimal solution: The solution that has the smallest objective value among those in its neighborhood feasible solutions is called a local optimal solution. Global optimal solution: The solution that has the smallest objective value among those of all the feasible solutions is called a global optimal solution. Active/inactive constraints: An inequality constraint that is satisfied with equality is said to be active. The constraint is said to be inactive if it is satisfied with strict inequality. Convex set: Let x1 and x2 denote two solutions belonging to a set I. If αx1 + (1 − α)x2 is contained in I for any 0 ≤ α ≤ 1, then the set I is said to be a convex set. Convex function: Let x1 and x2 denote two solutions belonging to a convex set I. If αF (x1 ) + (1 − α)F (x2 ) ≥ F (αx1 + (1 − α)x2 ) for any 0 ≤ α ≤ 1, then F (x) is said to be a convex function. Convex programming problem: If the objective function is convex, and the feasible region is a convex set, then the optimization problem is called a convex programming problem, for which the solution satisfying the conditions of local optimality is also a global optimal solution. Positive semidefinite matrix: An n × n symmetric matrix S is positive semidefinite, if all the eigenvalues of S are non-negative, which is equivalent to the nonnegativeness of the bilinear form b> Sb for any n-vector b. Furthermore, S is positive definite if b> Sb is positive for any b 6= 0. 2.4.2

Classification of optimization problems

The lower and upper bounds for the variable xi are denoted, respectively, by xLi and xU i . The optimization problem for minimizing the objective function F (x) is generally formulated as minimize

F (x)

(2.7a)

subject to Gi (x) = 0, ≤ xi ≤

(2.7b)

I

(2.7c)

(i = 1, . . . , N )

Hi (x) ≤ 0, xLi

E

(i = 1, . . . , N )

xU i ,

(i = 1, . . . , m)

(2.7d)

The constraints (2.7d) are called side constraints, bound constraints, or box constraints, which are treated independently of the general inequality constraints (2.7c) in most of the optimization algorithms. If the variables take only integer values, the problem is called an integer programming (IP) problem. Since the optimal combination of the variables is found by IP, it is equivalently called combinatorial optimization problem. Among various formulations of IP, the problem with 0–1 variables only is called a 0–1 programming problem. A problem that has real and integer variables is called a mixed integer programming (MIP) problem.

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Consider next the case where all the variables can take real values. If F (x), Gi (x) and Hi (x) are all linear functions of x, the problem is called a linear programming (LP) problem, for which the global optimal solutions can be found by well-established methods such as the simplex method and interior point method. If at least one of F (x), Hi (x) and Gi (x) is a nonlinear function of x, the problem is called a nonlinear programming (NLP) problem. For LP and NLP, the Lagrangian for Problem (2.7) is given as E

L(x, λ, µ) = F (x) +

N X

I

λj Gj (x) +

j=1

N X

µj Hj (x)

(2.8)

j=1

where λ = (λ1 , . . . , λN E )> and µ = (µ1 , . . . , µN I )> are the vectors of Lagrange multipliers corresponding to the equality and inequality constraints, respectively. Note that the side constraints (2.7d) are included in the general inequality constraints (2.7c) for simplicity. Suppose the feasible region of Problem (2.7) is nonempty and has a sufficiently smooth boundary. Then the necessary conditions to be satisfied at an optimal solution, called the optimality conditions or Karush–Kuhn–Tucker (KKT) conditions, are written as NE

NI

X ∂Gj X ∂Hj ∂F + λj + µj = 0, (i = 1, . . . , m) ∂xi j=1 ∂xi ∂xi j=1

µj ≥ 0, µj Hj = 0, (j = 1, . . . , N I )

(2.9a) (2.9b)

with the constraints (2.7b) and (2.7c), where the conditions (2.9b) are called a complementarity conditions. Consider a simple problem called quadratic programming (QP) problem, where the objective function is a quadratic function of x and we have only linear equality constraints as minimize

x> Dx + c> x >

subject to C x − b = 0

(2.10a) (2.10b)

where b = (b1 , . . . , bn )> and c = (c1 , . . . , cm )> are constant vectors, C is an n × m constant matrix, D is an m × m constant matrix, and we assume rank C < m. If D is positive definite, then Problem (2.10) is called convex QP. From the KKT conditions, the optimal solutions are explicitly written as 1 x = − D −1 (C > λ + c), 2 (2.11) λ = −(CD−1 C > )−1 (2b + CD−1 c) In this book, we often formulate an anti-optimization problem of maximizing (minimizing) a linear objective function under a quadratic constraint on x as maximize c> x >

subject to x Dx ≤ b

(2.12a) (2.12b)

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where b is a positive constant, and D is a positive definite weight matrix. The antioptimal solutions for Problem (2.12) are explicitly derived from the KKT conditions as √ CD −1 C 1 −1 ∗ √ (2.13) D c, µ = µ = x= 2µ 2 b where µ is the Lagrange multiplier for the constraint (2.12b), and µ = −µ∗ corresponds to the optimal solution that minimizes c> x. The optimization problem containing constraints such that the variable matrix is positive semidefinite is called a semidefinite programming (SDP) problem (Wolkowicz and Vandenberghe, 2000; Ohsaki and Kanno, 2007). The objective function and other constraints are linear in the standard form of SDP problem. Since the constraints of positive semidefiniteness of matrices include linear and convex quadratic constraints, the SDP is an extension of LP and convex QP. Let X  O indicate that the symmetric matrix X is positive semidefinite. The inner product X • Y of the n × n matrices X = (Xij ) and Y = (Yij ) is defined as X •Y =

n X n X

Xij Yij

(2.14)

i=1 j=1

The standard SDP problem is formulated as minimize

C •X

subject to Ai • X = bi , (i = 1, . . . , m) XO

(2.15a) (2.15b) (2.15c)

where C and Ai are n × n constant matrices, bi (i = 1, . . . , m) are scalars, and X is the variable matrix. Various structural optimization problems including the truss topology optimization problem under frequency constraint (Ohsaki, Fujisawa, Katoh and Kanno, 1999), linear buckling constraint (Kanno, Ohsaki and Katoh, 2001), and robust optimization problem (Ben-Tal and Nemirovski, 1997) can be formulated as SDP problems. 2.4.3

Parametric programming

An optimization problem that has parameters in addition to the variables is called a parametric programming problem (Ant´ onio, 2002; Gal, 1979). For a two-level optimization problem formulated for structural optimization considering the worst-case scenario, the lower-level anti-optimization problem is conceived as a parametric programming problem, for which the design variables of the upper-level optimum design problem are regarded as parameters (see Sec. 9.3 for the problem formulation). Parametric programming approaches have been widely applied to structural optimization. For example, for a simple cross-sectional optimization of trusses, the geometrical variables such as nodal coordinates are considered to be parameters.

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Nakamura and Ohsaki (1988) presented a parametric programming approach for generating a set of optimal trusses under multiple frequency constraints. Ohsaki and Arora (1993) presented a general parametric programming approach to structural optimization. In the field of structural optimization, the sensitivity of the optimal design with respect to the problem parameter is called the optimum design sensitivity, which has been extensively studied for application to two-level structural optimization problems (Sobieszczanski-Sobieski, James and Dovi, 1985; Vanderplaats and Yoshida, 1985; Barthelemy and Sobieszczanski-Sobieski, 1983; Bloebaum, Hajela and Sobieszczanski-Sobieski, 1992). Let p = (p1 , . . . , pN P )> denote a parameter vector, where N P is the number of parameters. A parametric optimization problem is stated as minimize

F (x, p)

(2.16a) I

(2.16b)

E

(2.16c)

subject to Gi (x, p) ≤ 0, (i = 1, · · · , N )

Hi (x, p) = 0, (i = 1, · · · , N )

where the side constraints are included in the general inequality constraints (2.16b). Let R(x, p) denote the vector of constraint functions consisting of the equality constraints (2.16c) and the active inequality constraints in Eq. (2.16b) at the optimal solution. The ith component of R is denoted by Ri . Then the KKT conditions for Problem (2.16) are written as NA

X ∂Rj ∂F + λj = 0, (i = 1, . . . , m) ∂xi j=1 ∂xi

(2.17)

where N A is the number of constraints satisfied in equality, and λj is the Lagrange multiplier for the constraint Rj = 0. Note that λj ≥ 0 should be satisfied if Rj is related to the active inequality constraint, whereas no restriction in sign exists for λj related to an equality constraint. The design variables and the objective value at the optimal solution is conceived as functions of p, because they can be obtained for each specified value of p. A function of p only is denoted by a hat (b· ). The derivatives of the optimal objective value Fb(p) = F (b x(p), p) with respect to pk is computed from m

X ∂F ∂b ∂ Fb ∂F xi = + , (k = 1, . . . , N P ) ∂pk ∂pk i=1 ∂xi ∂pk

(2.18)

Suppose that the active constraints in

bj (p) = 0, (j = 1, . . . , N A ) R

(2.19)

at an optimal solution remain active for a small variation of pk , i.e., bj ∂R = 0, (j = 1, . . . , N A ; k = 1, . . . , N P ) ∂pk

(2.20)

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Then the following relation is obtained from Eqs. (2.19) and (2.20): m

xi ∂Rj X ∂Rj ∂b + = 0, (j = 1, . . . , N A ; k = 1, . . . , N P ) ∂pk ∂x ∂p i k i=1

(2.21)

By multiplying ∂b xi /∂pk to the both sides of Eq. (2.17) and taking summation over i, we obtain A

N m m X xi X X ∂Rj ∂b xi ∂F ∂b + λj = 0, (k = 1, . . . , N P ) ∂x ∂x ∂p i ∂pk i k j=1 i=1 i=1

(2.22)

The following relation is derived by multiplying λj to Eq. (2.21) and taking summation over j corresponding to the equality constraints and the active inequality constraints: A

N X

NA m

∂Rj X X ∂Rj ∂b xi λj + λj = 0, (k = 1, . . . , N P ) ∂p ∂x ∂p k i k j=1 j=1 i=1

(2.23)

From Eqs. (2.18), (2.22) and (2.23), NA

X ∂Rj ∂ Fb ∂F = + λj , (k = 1, . . . , N P ) ∂pk ∂pk j=1 ∂pk

(2.24)

is derived. We can see from Eq. (2.24) that the derivative of the optimal objective value with respect to pk can be obtained without computing the derivatives of the variables and the Lagrange multipliers with respect to the problem parameter. Differentiation of Eq. (2.17) with respect to pk leads to   NA NA m 2 2 2 X X X ∂ R ∂b x ∂ F ∂ 2 Rj ∂ F j l   + λj + + λj ∂xi ∂xl j=1 ∂xi ∂xl ∂pk ∂xi ∂pk j=1 ∂xi ∂pk l=1 (2.25) NA X bj ∂Rj ∂ λ + = 0, (i = 1, . . . , m; k = 1, . . . , N P ) ∂x ∂p i k j=1 b1 , . . . , λ bN P and x The derivatives of λ b1 , . . . , x bm with respect to pk are computed A from a set of m + N linear equations (2.21) and (2.25), where the terms in the parentheses of the first term on the left-hand-side of Eq. (2.25) consist of the Hessian of the Lagrangian. 2.4.4

Multiobjective programming

So far, we have assumed that a single objective function is minimized or maximized through optimization. However, in the practical situation of designing structures, it is very natural to assume that we have more than one performance measure to consider. Thus, the optimization problem turns out to be a multiobjective programming problem that has more than one objective functions (Cohon, 1978).

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F2

A

Pareto optimal set F1 Fig. 2.3

Pareto optimal set in objective function space for N F = 2; gray region: feasible region.

Let F1 (x), . . . , FN F (x) denote the objective functions to be minimized. The multiobjective programming problem is formulated as minimize

F1 (x), . . . , FN F (x)

(2.26a) I

(2.26b)

E

(2.26c)

subject to Gi (x) ≤ 0, (i = 1, · · · , N )

Hi (x) = 0, (i = 1, · · · , N )

where the side constraints are included in the inequality constraints (2.26b) for simplicity. Contrary to the single-objective problem, generally there is no solution that simultaneously minimizes all objective functions. Consider two feasible solutions x(1) and x(2) satisfying all the constraints (2.26b) and (2.26c). Dominance of solution is defined as • If Fi (x(1) ) ≤ Fi (x(2) ) for i = 1, . . . , N F and Fj (x(1) ) < Fj (x(2) ) for one of j ∈ {1, . . . , N F }, then x(2) is said to be dominated by x(1) . The Pareto optimal solution is defined as • If there is no feasible solution that dominates a feasible solution x∗ , then x∗ is called a nondominated solution, noninferior solution, compromise solution or Pareto optimal solution. The set of Pareto optimal solutions is called a Pareto optimal set, or Pareto set for brevity. The thick line in Fig. 2.3 illustrates the set of Pareto optimal solutions in the objective function space for the case of N F = 2, where the gray region is the set of feasible solutions. For example, the solution ‘A’ is a Pareto optimal solution, because there is no solution in the region surrounded by the two dotted lines, where both of the two objective functions are improved. The approaches to selecting the most preferred solution from the Pareto optimal set are classified as a priori articulation of preference and a posteriori articulation of

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preference (Marler and Arora, 2004). In the a priori approach, the objective functions are combined or transformed to constraints to formulate a single-objective problem. The most popular and convenient approach for this purpose is the linear weighted sum approach, where the objective functions F1 (x), . . . , FN F (x) are combined as F

F (x) =

N X

wi Fi (x)

(2.27)

i=1

with positive weight coefficients w1 , . . . , wN F . This approach is suitable for the case where the Pareto optimal solutions form a smooth convex curve or hyper-surface in the objective function space, and the preference of the designers or the decision makers are rather fixed. Note that the single-objective problem for minimizing F (x) should be solved iteratively modifying the weight coefficients wi , if the solution is not acceptable to the decision maker. In the a posteriori approach, on the other hand, a set of Pareto optimal solutions is first generated, and the most preferred solution is selected by the decision maker from the set based on additional preference functions or trade-off relations among the objective functions. Even for a problem with continuous variables and smooth differentiable functions, an NLP approach is not suitable for the purpose of generating many Pareto optimal solutions with enough diversity, because it generates a single optimal solution for the given set of problem parameters. The genetic algorithm (GA) is one of the most suitable method for this purpose (Goldberg, 1989; Ohsaki, 1995), because it is classified as a population-based approach that has many solutions at each step of iteration called generation. On the other hand, the simulated annealing (SA) and tabu search (TS) are categorized as single-point-search heuristics that have only one solution at each step of optimization (see Sec. 2.6 for details of SA for single-objective problems). Recently, SA has been extended to multiobjective programming problems (Whidborne, Gu and Postlethwaite, 1997). TS has also been shown to be very effective with multiobjective problems (Baykasoglu, Owen and Gindy, 1999; Ohsaki, Kinoshita and Pan, 2007). 2.5

Approximation by Metamodels

One of the main difficulties in structural optimization for large-scale, real-world structures is that substantial computational cost is required for function evaluation (structural response analysis). Also for anti-optimization, structural analysis should be carried out many times to find the worst-case scenario. Therefore, several approximation methods such as response surface approximation (RSA) (Myers and Montgomery, 1995; Venter, Haftka and Starnes, 1998), kriging method (Lee and Jung, 2007; Sakata, Ashida and Zako, 2003), radial basis function (Park and Sandberg, 1991) and artificial neural networks (Rosenblatt, 1962) have been used for optimization of complex structures. Among them, RSA, also called the response

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surface method (RSM), has been widely applied to structural optimization problems due to its simplicity of implementation (Roux, Stander and Haftka, 1998). Let F (x) denote the response quantity to be approximated by a quadratic function of the variables x = (x1 , . . . , xm )> as F (x) ' F a (x) = c0 +

m X i=1

c i xi +

m X m X i=1 j=i

cij xi xj + · · ·

(2.28)

where c0 , ci , cij , . . . are the coefficients. Suppose the values of F (x(k) ) at the set of data points x(k) (k = 1, . . . , N D ) are known. The coefficients are determined by minimizing the square error e given as D

e=

N X

k=1

(F a (x(k) ) − F (x(k) ))2

(2.29)

The differential coefficients of e with respect to cij vanishes as   ND m m X m X X X ∂e (k) (k) (k) (k) (k) c0 + =2 c i xi + cij xi xj − F (x(k) ) xi xj = 0 (2.30) ∂cij i=1 i=1 j=i k=1

Similar equations are obtained for differentiation with respect to ci . Thus, the coefficients are found by solving a set of linear equations based on the well-known method of least squares. Suppose the variables are normalized to the range −1 ≤ xi ≤ 1 (i = 1, . . . , m). The sampling values for xi may be given by the two-level model (−1, 1), three-level model (−1, 0, 1), etc. In the three-level model, however, the number of data points is 3m if the set of all possible combinations is to be considered. The number increases exponentially as m is increased. Therefore, the method called design by experiment (Myers and Montgomery, 1995) is effectively used to select the data points of a moderately small number, while a priori estimation of the approximation error is minimized to obtain the set of data points. 2.6 2.6.1

Heuristics Overview of heuristics

Optimal designs of moderately large structures can be obtained easily by a nonlinear programming approach, if the objective function and the constraints are continuous functions of the design variables such as cross-sectional properties of a structure. In the various fields of engineering, however, especially in structural design in civil engineering, the design variables are often selected from the lists or catalogs of the standard sections. Also for an anti-optimization problem, the vertices of the feasible region of the parameters are often searched to find the worst-case scenario. Therefore, in these cases, the anti-optimization problem is formulated as a combinatorial problem. Furthermore, we have state variables such as nodal displacements,

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which are generally nonlinear functions of the design variables. Thus, the problem turns out to be a mixed-integer nonlinear programming (MINLP) problem (Floudas, 1995). It is very easy to solve a combinatorial optimization problem by using the integer programming approaches, e.g., branch-and-bound method and branch-and-cut method (Horst and Tuy, 1990), if the number of variables is small and the subproblem formulated by relaxing some integer variables to real variables is convex. However, the computational cost increases as an exponential function of the problem size, and it is not possible to solve a practical problem within a practically admissible computational time. Furthermore, for the structural optimization problem, the relaxed nonlinear programming problems are usually non-convex problems, for which the global optimality is not guaranteed. Recently, with the rapid development of computer hardware and software technologies, we can carry out structural analysis many times to obtain optimal solutions. Another important point is that the global optimality need not be strictly satisfied in the practical design process. Heuristic approaches (or heuristics for simplicity) have been developed to obtain approximate optimal solutions within reasonable computational time, although there is no theoretical proof of convergence (Reeves, 1995). The most popular approach is the genetic algorithm (GA) (Goldberg, 1989), which can be categorized as a multipoint search or population-based method that has many solutions at each iterative step called generation. Since computational cost for evaluating the objective and/or constraint functions at each step can be very large for structural optimization problems, a multipoint strategy may not be appropriate especially for optimization/anti-optimization of large structures. Therefore, single-point search heuristics such as simulated annealing (SA) (Aarts and Korst, 1989) and tabu search (TS) (Glover, 1989) can be effectively used. In the following, we present basic algorithms of single-point search heuristics.

2.6.2

Basic approaches of single-point search heuristics

Single-point search heuristics are based on local search (Aarts and Lenstra, 1997), where the solution is consecutively updated to a neighborhood solution if it improves the value of the objective function. The neighborhood solutions are generated by slightly modifying the value of one or several variables. The constraints are incorporated by penalty functions, or the solutions violating the constraints are simply rejected. Since it is not always possible to find an optimal solution by simple local searches, various heuristic approaches have been proposed to improve the convergence property to a good approximate optimal solution, e.g., update to a non-improving solution is allowed in some heuristics including SA.

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2.6.2.1 Neighborhood solutions The neighborhood solutions should be defined before introducing the strategies of heuristics. Let J = (J1 , . . . , Jm )> denote the vector of integer variables, and denote by J (k) the current solution at the kth iterative step of optimization or local search. The neighborhood of J (k) is defined as the set of solutions for which the distance from J (k) is sufficiently small. The distance of two solutions J A and J B may be defined, e.g., as D1 =

m X i=1

D2 =

q

|JiA − JiB |

(J A − J B )> (J A − J B )

(2.31)

D∞ = max |JiA − JiB | i

where D1 is the Manhattan norm (L1 -norm), D2 is the Euclidean norm (L2 -norm), and D∞ is the Chebychev norm (L∞ -norm). For an integer variable defined by a binary bit string, the humming distance is defined by the number of different bits, e.g., the humming distance between the binary numbers 1010110 and 1001101 is 4. However, the definition of a neighborhood for heuristics is less strict than those based on distances. For example, the following operations can be used for generating a neighborhood solution: • Increase or decrease the value of one variable Ji by a specified amount. • Modify the values of a set of variables by a predefined rule. • Exchange the values of a pair of variables (Ji , Jj ). 2.6.2.2 Basic algorithm of single-point search heuristics Let F (J ) denote the objective function to be minimized. The values corresponding to the kth iteration is indicated by the superscript ( · )(k) . We only consider the inequality constraints Hj (J ) ≤ 0 (j = 1, . . . , N I ) for simplicity. The basic algorithm of single-point search heuristics can be summarized as Step 1 Assign the initial solution J (0) , and set the iteration counter k = 0. Step 2 Randomly generate a neighborhood solution J ∗ of the current solution J (k) , or select the best solution J ∗ from randomly generated neighborhood solutions based on the criterion defined by the values of the objective function F (J ∗ ) and the constraint function Hj (J ∗ ). Step 3 Update the solution as J (k+1) = J ∗ or reject J ∗ as J (k+1) = J (k) in accordance with the given strategy. Step 4 Output the best solution satisfying the constraints and terminate the process, if the solution converged, or the stopping criteria are satisfied; otherwise, let k ← k + 1 and go to Step 2.

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We can use either specified or randomly generated values as the initial solution. An optimal solution obtained by another optimization approach can be used, with possible modification, as the initial solution; e.g., the nearest solution with integer variables from the optimal solution with real variables can be used. For a constrained optimization problem, the constraints can be incorporated to the objective function by using the penalty function approach. For example, for a problem with inequality constraints Hj (J ) ≤ 0 (j = 1, . . . , N I ), the following penalty function can be used to penalize the solution with violated constraints: I

Fe (J ) = F (J ) +

N X

αj [max(Hj (J ), 0)]2

(2.32)

j=1

where αj is a sufficiently large penalty coefficient. It is seen from Eq. (2.32) that no penalty is given if the constraint is satisfied as Hj (J ) ≤ 0, and the penalty proportional to the square of Hj (J ) is given if the constraint is violated. The penalty function of this type is called the exterior penalty function. On the other hand, the interior penalty function can be defined by the logarithmic barrier function as NI X e F (J ) = F (J ) − βj log(−Hj (J )) (2.33) j=1

where the penalty coefficient βj should be sufficiently small. However, the interior penalty function approach cannot be used for the case where a solution can be infeasible during the search process, because log(−Hj (J )) is not defined for Hj (J ) > 0. Therefore, only the feasible solutions should be searched, which is very difficult for a heuristic approach to optimization of large-scale complex structures. 2.6.2.3 Greedy method Heuristics can be classified into deterministic and probabilistic approaches. The simplest deterministic approach is the greedy method. The procedure of the greedy method for a simple structural optimization problem such as minimum weight design under stress constraints is described as follows:

Step 1 Assign the initial solution J (0) that does not satisfy all the constraints Hj (J ) ≤ 0 (j = 1, . . . , N I ), e.g., choose the smallest value for all variables for the case where F (J ) is an increasing function, and H1 (J ), . . . , HN I (J ) are decreasing functions. Set the iteration counter k = 0. Step 2 Select a neighborhood solution J ∗ that most efficiently improves the performance of the solution, and update the solution as J (k+1) = J ∗ . The ratio of decrease of the maximum violated value of H1 (J ), . . . , HN I (J ) to the increase of F (J ) can be used for the definition of the efficiency of improvement. Step 3 Let k ← k + 1 and go to Step 2 if at least one of the constraints is violated; otherwise, output the best solution satisfying all the constraints, and terminate the process.

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The approach that is the reverse of the greedy method, which starts from a solution satisfying all the constraints and reduces the objective value consecutively, is called the stingy method. The greedy method and the stingy method work fairly well if F (J ) and Hj (J ) are monotonic functions of J. However, the algorithms strongly depend on the initial solution, and will often reach a local optimal solution.

2.6.3

Simulated annealing

Instead of defining complicated rules for moving to a neighborhood solution, we can use a probabilistic approach to improve the possibility of obtaining the global optimal solution. The simplest probabilistic approach is the random search (Ohsaki, 2001). Although the random search is not efficient in view of computational cost for reaching the global optimal solution, it is simple and very easy to implement. Another approach that incorporates probabilistic process is the simulated annealing (SA) that has been developed to prevent convergence to a local optimal solution by allowing the move to a solution that does not improve the objective value, where the probability of accepting such a non-improving solution is defined by the amount of increase (for minimization problem) of the objective value. The term simulated annealing comes from the fact that it simulates the behavior of the metals in the annealing process. The penalty function approach is usually used for constrained optimization problems. The basic algorithm of SA for a minimization problem is given as follows (Aarts and Korst, 1989; Kirkpatrick, Gelatt and Vecchi, 1983): Step 1 Randomly generate the initial solution J (0) . Initialize the temperature parameter as T (0) = T0 , where T0 is the specified value, and set the iteration counter k = 0. Step 2 Let J ∗ denote a randomly generated neighborhood solution, and define ∆F = F (J ∗ ) − F (J (k) ). If ∆F < 0, let J (k+1) = J ∗ ; otherwise, accept J ∗ with the probability P defined by the temperature parameter T (k) as   ∆F P = exp − (k) T

(2.34)

Step 3 Decrease T (k) to T (k+1) based on the prescribed rule, e.g., the Metropolis rule T (k+1) = ηT (k) , where η is the specified parameter that is slightly less than 1. Step 4 Set k ← k + 1 and go to Step 2 if the termination condition is not satisfied; otherwise, output the best solution satisfying the constraints, and terminate the process.

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Optimum design Shape optimization Cross-sectional optimization

Fig. 2.4

Topology optimization

Geometry optimization

Classification of optimization of trusses and frames with respect to design variables.

P Fig. 2.5

2.7

P Topology optimization.

Classification of Structural Optimization Problems

There are several approaches to the classification of structural optimization problems. One of the traditional classifications for trusses and frames is as shown in Fig. 2.4. The cross-sectional properties are optimized in cross-sectional optimization (sizing optimization). The shape optimization includes geometry optimization (configuration optimization) and topology optimization, where the nodal locations and connectivity between nodes by members are optimized, respectively (Kirsch, 1989; Nakamura and Ohsaki, 1992; Ohsaki, 1997; Ohsaki, Nakamura and Isshiki, 1998a; Ohsaki and Katoh, 2005). Simultaneous optimization of geometry and topology is also called layout optimization (Rozvany, 1997). For continuum structures such as plates and shells, the optimization of external and/or internal boundary shape is called shape optimization, and the optimization allowing addition and removal of holes is called topology optimization (Bendsøe and Sigmund, 2003; Eschenauer, 2003). A process of topology optimization of a truss is illustrated in Fig. 2.5, where the dashed lines indicate the removed members after optimization. This approach of removing unnecessary members from the highly connected initial structure is called the ground structure approach. Thus, topology optimization is conceived as

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P Fig. 2.6

37

P

Feasible region Geometry optimization.

Ceiling Wall Cable H θ Rod P = 1000 N W = 1000 mm

Fig. 2.7

A model of coat-hanger problem.

an extension of cross-sectional optimization. Figure 2.6 illustrates the process of geometry optimization of a truss, where the gray squares indicate feasible regions of the nodal locations, which are assigned so that the shape does not change drastically. Note that the feasible regions of the nodes should also be appropriately assigned to prevent the existence of too short members or overlap of nodes called melting nodes that results in singularity of the stiffness matrix, i.e., the topology does not change in the process of geometry optimization. Therefore, simultaneous optimization of topology and geometry is very difficult (Ohsaki, 1997). In the following, effectiveness of topology optimization and geometry optimization is demonstrated using a simple rod supported by a cable as shown in Fig. 2.7. The objective is to design a structure to hang a object with the weight of 1000 N at a place 1000 mm from the wall. This very small illustrative problem is called the coat-hanger problem. We use mm and N for the units of length and force, respectively, which are omitted for brevity. The constraints for the stresses σc and σr of the cable and rod, respectively, are

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( ×10 4 ) 8.0

Total structural volume

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7.0

6.0

5.0 30

40

50

60

70

Angle of cable (deg.) Fig. 2.8

Relation between the angle θ of the cable and the optimal total structural volume.

W

W

H

H

P

P (a)

Fig. 2.9

(b)

Examples of topologies for H/W = 0.3.

given as σc ≤ 50, −50 ≤ σr

(2.35)

The objective function is the total structural volume V that is to be minimized. Note that the cable and rod are modeled by the truss elements, and the buckling of the rod is not considered. Let θ denote the angle between the cable and the rod. Then the length of the cable is 1000/ cos θ, and the axial forces of the rod and cable are −1000/ tan θ and 1000/ sin θ, respectively. We first consider the sizing optimization problem, where θ is fixed. If we decrease the cross-sectional areas, then the structural volume decreases; however, the absolute values of the stresses increase. Consequently, there exist optimal crosssectional areas under stress constraints, where σc = 50 and σr = −50 are satisfied, i.e., the optimal solution of this statically determinate truss is fully stressed. Hence, the optimal cross-sectional areas of the cable and rod are 20/ tan θ and 20/ sin θ, respectively, and the value of V of the optimal solution is obtained as   1 1 + (2.36) V = 2.0 × 104 tan θ sin θ cos θ

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Fig. 2.10

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39

Optimal cross-sectional areas with four units for H/W = 0.2.

Table 2.1 Relation between the number of units and the optimal structural volume. Number of units 2 3 4 5 6 Optimal structural volume (×105 ) 1.620 1.533 1.530 1.560 1.607

If we can modify θ in addition to the cross-sectional areas, i.e., if we consider geometry optimization, the relation between the angle and the optimal structural volume is as shown in Fig. 2.8. As is seen, V takes the minimum at θ ' 55◦ However, in a practical situation of the coat-hanger problem, θ may not attain ◦ 55 due to the restriction on the location of the ceiling. Obviously, if the height H is small, the configuration as shown in Fig. 2.9(a) is not recommended, and a truss as illustrated in Fig. 2.9(b) will be better. Therefore, we next consider the topology optimization problem, where all the members are supposed to consist of truss element, and the upper bound for the absolute values of the stresses is 50. Since the truss in Fig. 2.9(b) is statically determinate, the optimal design for specified topology can be obtained by assigning the cross-sectional areas so that the truss is fully stressed, i.e., the absolute value of the stress of each member is equal to the upper bound. Optimal solutions have been found for the truss with the number of units 2, 3, 4, 5 and 6, respectively, for the fixed values of W = 1000 and the aspect ratio H/W = 0.2. The relation between the number of units and the optimal structural volume is listed in Table 2.1. As is seen, the structural volume takes its minimum when the number of units is 4. The optimal cross-sectional areas for this case are as shown in Fig. 2.10, where the width of each member is proportional to its cross-sectional area. This way, the objective value can be effectively reduced by optimizing the topology. 2.8

Probabilistic Optimization

Optimization of structures in the random environment was pioneered apparently by Hilton and Feigen (1960). It attracted the attention of early investigators; the reader may consult the works by Switzky (1964), Khachaturian and Haider (1966), Moses and Kinser (1967), Vanmarcke, Diaz-Padilla and Roth (1972), Kapur (1975), Parimi and Cohn (1975, 1978a, 1978b), Frangopol (1995, 1998), Frangopol and Corotis (1996), and many followers thereafter. A review of current state-of-the-art was given by several researchers. Hilton and Feigen (1960) defined the task of the

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probabilistic optimization as the ‘proportioning of probabilities of failure among structural components in terms of a preassigned probability of failure of the entire structure, such that the total structural weight is a minimum.’ The papers by Pochtman and Khariton (1976) and Frangopol and Maute (2005) give a careful glimpse of early and recent developments, respectively. Modern state-of-the-art is summarized by Rackwitz and Cuntze (1987), Marti (2005), and Tsompanakis, Lagaros and Papadrakakis (2007). The problem of optimal design can be posed in one of the following two alternative forms: • Minimize objective function (usually cost, weight or volume) subject to the constraint on probabilistic measure on mechanical performance. Such a probabilistic measure usually is taken as the overall probability of failure of the structure, i.e., Pf (A) ≤ Pfa

(2.37)

R(A) = 1 − Pf (A)

(2.38)

where A = (A1 , . . . , Am )> is the vector of design variables, Pf (A) is the probability of failure, and Pfa is the acceptable probability of failure, the value of which ought to be available in codes. • Maximize the reliability of the structure R(A) which is the complement of the probability of failure to unity, subject to the upper bound for the cost, weight or volume denoted by C(A). In other words, this constraint reads C(A) ≤ Ca

(2.39)

where Ca is the allowable maximum value of C(A). There are also two equivalent approaches: (a) reliability index approach, and (b) performance measure approach (Frangopol and Maute, 2005). Often the probabilistic optimization is posed as the problem of minimizing the overall cost of the structure C = C 0 + P f Cf

(2.40)

where C0 is the initial cost of the structure, the product Pf Cf is the cost of the failure, and Cf is the projected cost incurred by the failure (Sarma and Adeli, 2002). As Gasser and Schu¨eller (1997) note, ‘The decision making process may be improved when the risk of failure in terms of expected failure consequence costs are balanced with the structural costs. For this purpose, an objective function should contain the following types of costs: the initial construction cost, and the expected consequences related to partial and total system failure.’

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In Eq. (2.40), Cf is a deterministic quantity. However, some researchers treat it as a function depending on random variables. Then the reliability-based optimization reads minimize E[C(A, Y )] = C0 (A) +

m X

E[Cf,j (A, Y )]

(2.41)

j=1

where Y is the vector describing random parameters of the system, and E[C] and E[Cf ] are the mean values of C and the expected cost Cf due to failure with regard to the jth criterion, respectively. In his review article, Schu¨eller (1998) emphasizes that ‘... irrespective of the route of optimization which is followed, its treatment within efficient software environment is an indispensable requirement.’ A pertinent question arises concerning the assignment of the acceptable probability of failure. Freudenthal (1956), anticipating the reliability-based optimization, stressed the following about acceptable risk, which ought to be evaluated: ‘... on the basis of economic balance between the cost of increasing safety and the cost of failure...’ Hence, the cost and probability of failure should be accurately defined for reliable formulations of probabilistic or reliability-based optimization (see further discussion in Chap. 10). 2.9

Fuzzy Optimization

It appears that we ought to describe all (that are known to us, obviously) approaches to optimization of structures, so as not to leave an impression that we include only what we think is important in this field. The first study on fuzzy sets was apparently pioneered by a British-American philosopher, Max Black (1909–1988) over 70 years ago. He started by quoting the famous British philosopher Bertrand Russell (1872–1970), who stated that ‘Vagueness and accuracy are important notions, which are very necessary to understand.’ Black (1937) introduces the essence of his argument as follows: ‘It is a paradox, whose importance familiarity fails to diminish, that the most highly developed and useful scientific theories are ostensibly expressed in terms of objects never encountered in experience. The line traced by a draftsman, no matter how accurate, is seen beneath the microscope as a kind of corrugated trench, far removed from the ideal line of pure geometry. And the “point planet” of astronomy, the “perfect gas” of thermodynamics, or “pure species” of genetics are equally remote from exact realization.’ Further,

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‘While the mathematician constructs a theory in terms of “perfect” objects, the experimental scientist observes objects of which the properties demanded by theory are and can, in the very nature of measurement, be only approximately true.’ He further discusses ‘the vagueness of symbols’ as a symptom of the degree of deviation of the ‘model’ language from the empirically discoverable linguistic habits in the corresponding speech community. Zadeh (1965) independently introduced fuzzy sets, and coined the term fuzzy. In an insightful article Hirota (1991) explains: ‘The word “fuzzy” has the meaning “vague, not clear”, and is used ... in contrast to the precision normally expected of mathematics ... with fuzzy sets, a mathematical model can be made of properties or quantities that are imprecisely defined, such as everyday statements.’ According to Kosko (1992), ‘fuzzy theory holds that all things are matters of degree ... Fuzzy theory reduces black-white logic and mathematics to special limiting cases of gray relationships.’ In applications of the fuzzy set theory, one first identifies the so-called universal set (Klir, St. Clair and Yuan, 1997), which is a set that consists of all the members that are of interest in the problem at hand. For example, if one is dealing with classifying loads applied at a structure by various criteria, then the universal set is composed of all the values that the loads can take on. Every subset A of a set X can be uniquely represented by a function, called characteristic function µA (x), defined as  1, if x ∈ A (2.42) µA (x) = 0, if x ∈ /A for any x ∈ X . Examples of characteristic functions are shown in Fig. 2.11. The function µc (x) in Fig. 2.11(a) corresponds to a crisp set or a classical non-fuzzy set defined by 0 ≤ x ≤ x0 , where µc (x) = 1 for 0 ≤ x ≤ x0 ; otherwise, µc = 0. However, membership of the load in a fuzzy set may allow some uncertainty; therefore, it is a matter of degree (Kosko, 1992; Klir, St. Clair and Yuan, 1997). One can state that the degree of membership constitutes the degree of compatibility of introduced operating concepts with the fuzzy set. Figure 2.11(b)–(d) illustrates three types of characteristic functions, where the function value 0 ≤ µF (x) ≤ 1 indicates the corresponding value of x is possibly the member of the set; hence, the characteristic function is also called the membership function. For example, consider the uniform bar under tensile load, which is chosen as variable x. Then X can be identified with the set of all nonnegative real numbers, assuming the material strength of the bar is unbounded. Let N be a set of loads that are real values and range from 10 kN to 20 kN. Then, if we use the framework

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µF

µc

1

1

0

x0

0

x

(a) crisp set

x0

x

(b) fuzzy set (type 1)

µF

µF

1

1

0

xU

xL (c) fuzzy set (type 2) Fig. 2.11

x

0

xL

xU

x

(d) fuzzy set (type 3) Examples of characteristic functions.

of a non-fuzzy set, N is a subset of X with the characteristic function  1, if 10 ≤ x ≤ 20 µN (x) = 0, otherwise

(2.43)

However, if we use the concept of a fuzzy set, the characteristic function µN (x) may be defined, e.g., as shown in Fig. 2.11(c) with xL = 10 and xU = 20. As is seen, the definition of a fuzzy set is a generalization of that of a classical set; in the former, the characteristic function is allowed, as it were, to take any value between zero and unity. In the latter, the value of the membership function is either zero or unity, which follows the Greek philosopher Aristotle’s (384–322 B.C.E.) maxim: ‘Everything must either be or not be, whether in the present or in the future.’ Fuzzy set theory follows Bertrand Russell (1872–1970) who stated in his book (Russell, 1985) ‘Everything is vague to a degree you do not realize till you have tried to make it precise.’ Fuzzy optimum design of structures and machining processes was apparently pioneered by Dubois (1987), Rao (1987a, 1987b, 1987c), Wang and Wang (1984, 1985), and was studied extensively by Arakawa and Yamakawa (1993), M¨ oller, Grab and Beer (2000), Rao, Sundararaju, Prakash and Balakrishna (1992a, 1992b), Shih, Chi and Hsiao (2003), Xu (1989), Yeh and Hsu (1990), and others. Zheng and Lewis (1993) conducted optimization of grey systems. Bernardini (1999) presented a multiobjective programming approach based on fuzzy theory. Suppose we have N F (x) objective functions F1 (x), . . . , FN F (x), for which the membership function is given as µ1 , . . . , µN F . Then the membership

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function µ(x) for the intersection of the objective functions is given as µ(x) = min µi (x) i

(2.44)

Therefore, the multiobjective optimization problem is formulated as Find max min µi (x) x

i

(2.45)

with appropriate deterministic constraints. The optimization problem in a fuzzy environment can be formulated as the following min–max problem with deterministic inequality constraints: max x

min

j=1,...,h+1

subject to

µDj (x)

(2.46a)

Hi (x) ≤ 0, (i = 1, . . . , N I − h)

(2.46b)

where µDj is the membership function of fuzzy decision, and x is the design variable vector. The fuzzy environment is represented by the objective function and h constraints out of N I inequality constraints; the rest represent the crisp (non-fuzzy) deterministic constraints. The fuzzy constraints are incorporated into the objective function; therefore, we have h + 1 membership functions in the objective function. We must not hide the fact that many researchers criticize the fuzzy-sets based approach in strong terms. Here are several quotes (see Kosko (1993)): • ‘Fuzzy theory is wrong, wrong and pernicious. What we need is more logical thinking, not less. The danger of fuzzy logic is that it will encourage the sort of imprecise thinking that has brought us so mush trouble. Fuzzy logic is the cocaine of science.’ (William Kahan) • ‘ “Fuzzification” is a kind of scientific permissiveness. It tends to result in socially appealing slogans unaccompanied by the discipline of hard scientific work and patient observation.’ (Rudolf Kalman) • ‘Fuzziness is probability in disguise. I can design a controller with probability that could do the same thing that you could do with fuzzy logic.’ (Myron Tribus) It appears noteworthy to quote Ditlevsen (1980): ‘... the sources of fuzzy information are non-objectivistic and nonreproductive in their very nature like the process of perception in the human brain.’ Zadeh (1994) vehemently disagrees with these criticisms in his tellingly titled paper ‘Why the success of fuzzy logic is not paradoxical.’ The following definition is proposed: ‘... the skeptics will find it hard to understand why they failed to realize that fuzzy logic is a phase in a natural evolution of science – an evolution brought by the need to find an accommodation with the pervasive impression of the real world.’

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In their paper Dubois and Prade (1993) ‘try to overcome misunderstandings’ between the proponents of the stochastic and fuzzy-sets based methodologies (see also Dempster (1967); Blockley (1985), Bezdek (1994)). Kosko (1990) strongly defends fuzzy sets: ‘... a hundred years from now, a thousand years from now... no one... will believe that there was a time when a concept as simple, as expressive as a “fuzzy set” met with such impassional denial.’ To conclude this section, we note that researchers performed comparison of fuzzy and probabilistic modelings. The interested reader may consult, for example, with papers by Chiang, Dong and Wong (1987), Lodwick, Jamison and Russell (2000), Oberguggenberger and Russo (2001), and Moens and Vandepitte, (2005, 2006). Maglaras, Nikolaidis, Haftka and Cudney (1997) performed a definitive analytical-experimental comparison between these two methods. These two approaches were also contrasted by Sophie (2000). A fuzzy safety measure was introduced by Blockley (1977), Brown (1979), Kam and Brown (1983), Brown, Johnson and Loftus (1984), Shiraishi and Furuta (1985), Brown and Yao (1987), and others. Safety factor in the fuzzy environment was introduced by Elishakoff and Ferracuti (2006a, 2006b). Optimization of structures in fuzzy environment was studied by Rao (1987a, 1987b, 1987c), Rao, Sundararaju, Prakash and Balakrishna (1992a, 1992b), Arakawa and Yamakawa (1992, 1993), and Ohta and Haftka (1997). Possibility theory was employed by Utkin and Gurov (1996), Cremona and Gao (1997), Gurov and Utkin (1999), Jensen (2001), Stroud, Krishnamurthy and Smith (2001), Jensen and Maturana (2002), Mourelatos and Zhou (2005), and Du, Choi and Youn (2006) among many others. Nikolaidis, Chen, Cudney and Haftka (2004) provided a systematic comparison of probabilistic and possibilistic methodologies. Safety analysis of structure under hybrid uncertainty was performed by Chakraborty and Sam (2006). Some authors lump both concepts to yield fuzzy reliability (Cai, 1996) or fuzzy randomness (M¨ oller and Beer, 2004). Fang, Smith and Elishakoff (1998) combined anti-optimization and fuzzy-sets based analyses. Adali (1992) and Pantelides and Ganzerli (2001) compared these two models. Neumaier (2004) introduced a new concept, namely, that of a ‘cloud’ as a ‘mediator’ as it were, between the concept of a fuzzy set and that of a probability distribution. He states ‘A cloud is to a random variable more or less what as interval is to a number.’ Elishakoff and Li (1999) combined probabilistic and ellipsoidal analyses, whereas Qiu, Yang and Elishakoff (2008b) presented a hybrid probabilistic-interval analysis study. Wang, Qiu and Elishakoff (2008b) proposed a non-probabilistic set-theoretic model for structural safety measure. Good (1995), Elishakoff (1995a), Guo and L¨ u (2003), and Qiu, Chen and Wang (2004b) discussed a concept of ‘non-probabilistic reliability’. The book edited by Ross, Booker and Parkinson (2002) is devoted to bridging the gap between fuzzy logic and probability applications. Natke and Ben-

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Haim (1997), Elishakoff (1990, 1998c), Pal (1999), and Oberguggenberger (2004) discuss various possible modeling techniques of uncertainty. Finally, a mention should be made of random sets that combine the probabilistic and set-based models (Matheron, 1975; Dubois and Prade, 1991; Bernardini, 1999; Tonon, 2004). Optimization of uncertain structures in the context of the random sets was performed by Tonon and Bernardini (1998). M¨ oller and Beer (2008) studied optimization in conjunction with non-probabilistic uncertainty modeling. Popular and highly readable books on the fuzzy sets and logic are those by McNeill and Freiberger (1993), Kosko (1993, 1999), and Sangalli (1998). Pertinent books that employ the fuzzy sets in the structural analysis context are those by Blockley (1980), Yao (1985), Ayyub (1998), Vick (2002), M¨ oller and Beer (2004), Hanss (2005), and Fellin, Lessmann, Oberguggenberger and Vieider (2005). Nikolaidis and Haftka (2001) revised theories of uncertainty for risk assessment when data are scarce. Tsompanakis, Lagaros and Papadrakakis (2007) edited a definitive collection of papers on various methods of uncertainty analysis in the optimization context. Nikolaidis, Ghiocel and Singhal (2005) edited Engineering Design Reliability Handbook which provides much more than the title promises; the handbook covers both the traditional probabilistic reliability as well as various non-probabilistic treatments of uncertainty.

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Chapter 3

General Formulation of Anti-Optimization

‘We have to do the best we can. This is our sacred human responsibility.’ (Albert Einstein) ‘By contrast to the well-known probabilistic approach, the guaranteed approach does not require precious knowledge of probability distributions (which are seldom available in practical problem) and yields reliable estimates for the system behavior.’ (Felix L. Chernousko, 1999) ‘We are not suggesting that all computation should be carried out using interval techniques but only that interval methods provide another set of tools in applied mathematics...’ (Ramon E. Moore, 1979)

In this chapter, a general formulation of anti-optimization is described briefly. We first provide a motivation, accompanied by a simple example of interval analysis of a bar subjected to loads that vary within intervals. Pros and cons of interval analysis are presented. Ellipsoidal analysis is described next, along with sensitivity analysis and exact reanalysis of static response.

3.1

Introduction

Anti-optimization represents an alternative and a complement to traditional probabilistic methods, and also is a generalization of the mathematical theory of the interval analysis. It is instructive, as it often is, to start with a story. This one is by Hayes (2003): ‘On February 25, 1991, a Patriot missile battery assigned to protect a military installation at Dhahran, Saudi Arabia, failed to intercept a Scud missile, and the malfunction was blamed on an error in computer arithmetic. The Patriot’s control system kept track of time by counting tenths of a second; to convert the count 47

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into full seconds, the computer multiplied by 1/10. Mathematically, the procedure is unassailable, but computationally it was disastrous. Because the decimal fraction 1/10 has no exact finite representation in binary notation, the computer had to approximate. Apparently, the conversion constant stored in the program was the 24-bit binary fraction 0.00011001100110011001100, which is too small by a factor of about one ten-millionth. The discrepancy sounds tiny, but over four days it built up to about a third of a second. In combination with other peculiarities of the control software, the inaccuracy caused a miscalculation of almost 700 meters in the predicted position of the incoming missile. Twenty-eight soldiers died. Of course it is not to be taken for granted that better arithmetic would have saved those 28 lives. (Many other Patriots failed for unrelated reasons; some analysts doubt whether any Scuds were stopped by Patriots.) And surely the underlying problem was not the slight drift in the clock but a design vulnerable to such minor timing glitches. Nevertheless, the error in computer multiplication is mildly disconcerting. We would like to believe that the mathematical machines that control so much of our lives could at least do elementary arithmetic correctly.’ Two other examples belong to Moore (2006), a recognized founding father of modern interval analysis: (i) ‘A physics problem at Lockheed (ca. 1960): Q: Is the strange behavior of the computer model due to round-off errors? A: After converting the program to run in interval arithmetic with outward rounding, it was determined that round-off error was very small in the original program. Result: the physicist took another look at the model equations and found that there was a missing term.’ (ii) ‘A long controversy between research groups at MIT and Caltech concerned whether the observed behavior of computer simulations was due to roundoff error of defects in the mathematical model, in the case of computer solutions of the Birkhoff-Rota complex partial differential/integral equations modeling the onset of turbulence in wind-shears. My graduate student Jeffrey Ely wrote a program for variable-precision interval arithmetic with outward rounding, and finally by using about 300 decimal place (nearly 1000 bits) in outwardly rounded interval arithmetic, was able to settle the controversy. By carrying enough bits, the model realistically determined the onset of turbulence at a reproducible, finite time after the initial appearance of the wind-shear. I have always found it odd to suppose that anything we want to compute can be done carrying only some fixed number of digits or bits. Is 40 bits enough? 80? A thousand? It depends on what we are trying to compute...’

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The natural question arises: What is interval analysis? As Hayes (2003) informs us, ‘Interval analysis is not a new idea. Invented and reinvented several times, it has never quite made it into the mainstream of numerical computing, and yet it has never been abandoned or forgotten either.’ The first paper on interval analysis apparently belongs to Rosalind Cicely Young (1931) who published a paper on an algebra of many-valued quantities and gave rules for calculating with intervals. Paul S. Dwyer (1951) described arithmetic with intervals, calling it range numbers. The consequent three works appeared nearly simultaneously: by Mieczyslaw Warmus (1956) in Poland, Teruo Sunaga (1958) in Japan, and Ramon E. Moore (1966) in the United States. 3.2

Models of Uncertainty

There are several concepts for modeling uncertainty based on the non-probabilistic approach. Ladev`eze, Puel and Romeuf (2006) introduced the concept of lack of knowledge to take into account uncertainty of the parameters and modeling errors. They proposed a method for reducing errors through additional information, and applied it to static and free vibration problems. Oden, Babuˇska, Nobile, Feng and Tempone (2005) classified the errors of numerical analysis into modeling error, discretization error and error due to uncertainty. Soize (2005) classified the uncertainty into data uncertainty and model uncertainty. Data uncertainty concerns the structural parameters defining geometry, stiffness, material properties, and so on, which can be successfully represented by the parametric probabilistic approach. Model uncertainty, on the other hand, is related to approximation in analysis and the details that are actually unknown. They applied their methods to dynamic problems. Probabilistic and non-probabilistic approaches to anti-optimization are compared by Elishakoff and Zingales (2003). Various models of uncertainty are reviewed by Elishakoff and Fang (1995). Let a = (a1 , . . . , aN )> denote the vector of uncertain parameters, such as nodal loads and material properties. The types of the feasible regions or the bounds of the uncertain parameters are classified as follows (Qiu, M¨ uller and Frommer, 2001): • The parameters are linearly bounded by the intervals: aLi ≤ ai ≤ aU i , (i = 1, . . . , N )

(3.1)

where aLi and aU i are the specified lower and upper bounds for ai (see Sec. 3.3 and Chap. 7 for details of the interval analysis). • A quadratic bound is given by the ellipsoidal model: 2 N  X ai ≤1 (3.2) gi i=1 where gi is the specified positive parameter that represents the semi-axis of ai .

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Equation (3.2) can be rewritten in a vector-matrix form: a> W a ≤ 1

(3.3)

aLi (t) ≤ ai (t) ≤ aU i (t), (i = 1, . . . , N )

(3.4)

where W = diag(1/g1 , . . . , 1/gN ) is the weight matrix. Note that a has been translated to have zero central values, and rotated from the original axes so that W is diagonal with respect to a1 , . . . , aN (see Sec. 3.4 for details of the ellipsoidal model). • For transient responses, the uncertain parameter ai (t) of time t has the envelope bound: aLi (t)

aU i (t),

where the lower and upper bounds and respectively, are the specified deterministic functions. • An integral of a quadratic norm of ai (t) in the time interval [t1 , t2 ] is bounded as Z t2 [ai (t)]2 dt ≤ α, (i = 1, . . . , N ) (3.5) t1

where α is the specified upper bound. • The vector of Fourier coefficients κi for a function ai in space- or time-domain is bounded as κ> i W i κi ≤ βi , (i = 1, . . . , N )

(3.6)

where W i is the specified positive definite weight matrix, and βi is the specified upper bound.

3.3 3.3.1

Interval Analysis Introduction

In the theory of structural analysis, we often encounter the situation that once the problem is formulated and the governing differential equations are written down, the exact solution is either unavailable or demands elaborate symbolic analysis with special functions. In these circumstances, we usually resort to approximation techniques like the Rayleigh–Ritz or Galerkin method. For the eigenvalues, some enclosure methods are developed. Thus, intervals which contain the exact solution of the formulated problem are to be found. Archimedes, in the third century before the common era (circa 287–212 B.C.E.) derived such bounds for the transcendental number π, showing that it belongs to the interval 1 10 <π<3 (3.7) 3 71 7 by approximating the circle with the inscribed and circumscribed 96-side regular polygons. In this section, we present an example of interval analysis of simple static problems.

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[12.5, 14.7]

[2.32, 2.41] [12.5, 14.7]

M = [mL , mU] Fig. 3.1

A cantilever beam.

N2 R

N2

Fig. 3.2

3.3.2

N1

N1

A uniform bar (type-1).

A simple example

In structural engineering, the intervals appear as the means of representing uncertainty in the values of the physical quantities. We will now introduce basic operations with interval variables. Let us consider a cantilever beam as shown in Fig. 3.1, which is subjected to a load that varies in the interval [12.5, 14.7] kN at the location that is also not known exactly and varies in the interval [2.32, 2.41] m. Then the reaction moment is also an interval quantity whose bounds [mL , mU ] kN · m are mL = 12.5 × 2.32 ' 28.9, mU = 14.7 × 2.41 ' 35.4

(3.8)

Next, we consider the uniform bar (type-1) as shown in Fig. 3.2, which is clamped at its left end, and is subjected to two loads of uncertain magnitudes. Still, we can state with confidence that the values that the loads can take lie within certain bounds. Let the first load N1 lie between 17.4 kN and 21.5 kN, whereas the second load N2 lies somewhere between 12.5 kN and 14.7 kN. The reaction R = N2 + N1 then lies in the interval [12.5, 14.7] + [17.4, 21.5] = [12.5 + 17.4, 14.7 + 21.5] = [29.9, 36.2] kN

(3.9)

The difference of interval variables appears if the direction of the load N1 is reversed as shown in Fig. 3.3. Then the reaction turns out to be the difference of

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N2 R

N2

Fig. 3.3

N1

N1

A uniform bar (type-2).

two interval valued loads as R = N2 − N1 = [12.5, 14.7] − [17.4, 21.5]

= [12.5 − 21.5, 14.7 − 17.4] = [−9.0, −2.7]

(3.10)

Since the upper and lower bounds are negative, we conclude that the direction of the force R should be reversed. Then it takes values in the interval [2.7, 9.0]. The division of two interval variables occurs, for example, in the following problem. Suppose the cross-sectional area A of the bar is also an interval variable A = [0.05, 0.06] m2

(3.11)

Then the normal stress σ = N/A for the uniform bar (type-1) in Fig. 3.2 is also an interval variable. The maximum stress σ U occurs when the loads N1 and N2 take on the maximum values 21.5 and 14.7, respectively, resulting in the maximum axial force N U = 36.2 kN; and the cross-sectional area takes on the minimum value AL = 0.05 m2 in the region between the clamped end and the cross-section where the force N2 is applied: 36.2 NU 2 = = 724 kN/m (3.12) AL 0.05 The minimum value σ L of the stress near the clamped end is achieved when the loads take on their minimum values resulting in the minimum axial force N L = 29.9 kN, whereas the cross-sectional area takes on the maximum value AU = 0.06 m2 : NL 29.9 2 σL = U = = 498 kN/m (3.13) A 0.06 Thus, although the stress near the clamped end is uncertain due to the uncertainty of the loads and the cross-sectional area, we are sure that it takes values in the 2 interval [498, 724] kN/m . σU =

3.3.3

General procedure

Interval analysis is a set theoretical approach that is closely related to fuzzy theory (Mullen and Muhanna, 1999) (see Sec. 2.9 for the details of fuzzy sets). The principles of interval algebra is discussed in a number of books such as Alefeld and Herzberger (1983), Ferson (2002), Hansen and Walster (1992), Keafott (1996), Moore (1962), Moore, Kearfott and Cloud (2009), Neumaier (1990), and Rohn (2006).

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An interval X of a real number x is defined with the upper bound xU and the lower bound xL as X = [xL , xU ] = {x|xL ≤ x ≤ xU }

(3.14)

The following operations are defined for the addition and subtraction of the variables in the intervals X = [a, b] and Y = [c, d]: Addition: X + Y = [a + c, b + d] Subtraction: X − Y = [a − d, b − c]

(3.15)

X − Y = X + (−1) × Y

(3.16)

Note that the subtraction can be reduced to the sum as

Likewise, the division of the non-interval variable x by an interval variable Y , provided that Y does not contain zero, is reduced to the multiplication x/Y = x × [1/y U, 1/y L ]

(3.17)

In general, a single formula provides a definition of the four standard arithmetic operations to intervals. Let us denote by ◦ any of the operations +, −, × and / (Muhanna, Mullen and Zhang, 2005), and the division by an interval variable that includes zero is prohibited. Then X ◦ Y is defined as X ◦ Y = [min xi ◦ y j , max xi ◦ y j ]

for i ∈ {U, L}, j ∈ {U, L}, ◦ ∈ {+, −, ×, /}

(3.18)

which is alternatively written as

[xL , xU ] ◦ [y L , y U ]

= [min(xL ◦ y L , xL ◦ y U , xU ◦ y L , xU , y U ), L

L

L

U

U

L

(3.19) U

U

max(x ◦ y , x ◦ y , x ◦ y , x , y )]

In other words, one has to compute the four possible combinations of the lower and upper bounds. The next step is to choose the smallest of the four values as the lower bound, and to take the largest of them as the upper bound. We are sure that every possible combination of x ◦ y lies within these limits. It must be stressed that the interval analysis processes information obtained for input data to provide the interval for the output. Moore (2006) writes: ‘Suppose we have measured that input parameters fall within certain upper and lower limits, then we can use interval inputs for them, and interval computation is designed for just that sort of thing. For example, if we measure a width as w = 7.2 ± 0.1, length as l = 14.9 ± 0.1, and height as h = 405.6 ± 0.02, then the volume V = w × l × h is within the interval ([7.1, 7.3] × [14.8, 15.0]) × [405.4, 405.8]

= [42599.432, 44435.1] = 43517.266 ± 817.834

(3.20)

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If desired, we could use outward rounding to retain containment with numbers having only one digit after the decimal point, and find the volume is contained in the interval of numbers, [42599.4, 4435.1] = 43517.3 ± 817.9. All I am saying in the above example is that IF all we know about w, l, and h is that their values lie in the intervals, THEN all we can say about V is that it is in the interval obtained by the given endpoint of the input variables w, l, and h. If we need V more successfully, then we have to measure w, l and h more accurately.’ A question arises if the definition of the four algebraic operations and their use is sufficient to obtain tight bounds of a function of an interval variable. Consider the function (Alefeld and Claudio, 1998; Alefeld and Mayer, 2000) x f (x) = for x 6= 1 (3.21) 1−x with x being an interval variable x = [2, 3]. The derivative of this function equals f 0 (x) =

1 − 2x (1 − x)2

(3.22)

It vanishes when x = 1/2. Moreover, this value is outside the interval [2, 3]. Therefore, the extreme values of f (x) are reached at the endpoints. Namely, the lower bound is f L = f (2) = −2 whereas the upper bound is f U = f (3) = −3/2. Let us now calculate straightforwardly f (x) by replacing x with the interval [2, 3]: f ([2, 3]) =

[2, 3] [2, 3] [2, 3] [2, 3] = = = = [−3, −1] (3.23) 1 − [2, 3] [1, 1] − [2, 3] [1 − 3, 1 − 2] [−2, −1]

As we see, the true interval of the function f (x) is included in the one obtained by the direct evaluation as follows: [−2, −3/2] ⊂ [−3, −1]

(3.24)

As we observe, overestimation has taken place. Another simple example is the uniform bar (type-2) subjected to two loads N1 and N2 , which are equal in their magnitude N , but have opposite directions as shown in Fig. 3.3. The reaction equals R = N − N = 0. However, if N is an interval value [N L , N U ], then the expression for the reaction R = N − N , evaluated via direct use of interval algebra, leads to R = [N L , N U ] − [N L , N U ] = [N L − N U , N U − N L ]. Since N U > N L , the lower bound of the reaction is negative as N L − N U < 0, and the upper bound is positive as N U − N L > 0. Thus, the true zero value is contained in the interval result [N L − N U , N U − N L ]. If, for example, N L = 1, N U = 2, then [N L − N U , N U − N L ] = [−1, 1]. Such an overestimation in the interval estimation is called catastrophic by Muhanna and Mullen (2001). What is the remedy for such an error? Interval arithmetic cannot recognize the multiple occurrence of the same variable. It evaluates the function R = N − N as a function of two variables, namely,

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R = N1 − N2 , instead of evaluating the interval as a function of a single variable R = N − N . In other words, interval arithmetic is testing N − N as if evaluating N1 − N2 with N2 equal to but independent of N1 . If possible, one has to take care with such cases. This dependency problem can be eliminated if the reaction is rewritten as R = N (1 − 1). Likewise, in the example of Eq. (3.21), f (x) = x/(1 − x) can be rewritten as 1 x = for x 6= 0, 1 (3.25) f (x) = 1−x 1/x − 1

Direct use of interval arithmetic to Eq. (3.25) leads to the correct result 1 = [−2, −3/2] (3.26) 1/[2, 3] − 1 The reason for the overestimation is the fact that the rules that are valid for the arithmetic for real numbers do not hold for interval arithmetic. The distributivity property for real numbers x(y + z) = xy + xz is not valid for interval variation. If, for example, X = [−3, 3], Y = [1, 3], Z = [−3, 1]

(3.27)

then X(Y + Z) = [−3, 3]([1, 3] + [−3, 1]) = [−3, 3][−2, 2] = [−6, 6]

(3.28)

XY + XZ = [−3, 3][1, 3] + [−3, 3][−3, 1] = [−9, 9] + [−9, 9] = [−18, 18]

(3.29)

but We can confirm that the interval number X(Y + Z) is contained in the interval [−18, 18]; however, they are not equal. For interval numbers, the following property of subdistributity generally holds: X(Y + Z) ⊆ XY + XZ

(3.30)

0 ∈ Y − Y, 1 ∈ Y /Y

(3.31)

q(X, Y ) = max{|xL − y L |, |xU − y U |}

(3.32)

For real numbers, cancellation takes place, namely, x − x = 0; likewise x/x = 1 for x 6= 0. However, for interval numbers, as we have seen, X − X 6= 0; moreover X/X 6= 1. Indeed, Y −Y = [1, 3]−[1, 3] = [−2, 2], and Y /Y = [1, 3]/[1, 3] = [1/3, 3]. For interval numbers, the subcancellation property reads For the example of Y = [1, 3], the interval number Y − Y = [−2, 2] is enclosing but not equal to zero, whereas in the case of division, the interval Y /Y = [1/3, 3] is enclosing the true number 1 but not equal to it. The distance q of two intervals X = [xL , xU ] and Y = [y L , y U ] is defined as a real number L

U

The absolute value of an interval X = [x , x ] is defined as the distance of X from 0, i.e., |X| = q(X, 0) = max{|xL |, |xU |}

(3.33)

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The diameter, or width w of an interval X = [xL , xU ] is defined as w(X) = xU − xL

(3.34)

and the radius is given as w/2. The following rules are valid for two interval numbers: w(X ± Y ) = w(X) + w(Y ), w(XY ) ≥ max{|X|w(Y ), w(X)|Y |}

(3.35)

Consider the following example of Alefeld and Claudio (1998). Let f (x) = x−x2 and X = [1/2 − r, 1/2 + r]

(3.36)

where r is a real number taking values in the range 0 ≤ r ≤ 1/2. The function attains its maximum value 1/4 at x = 1/2. The minimum value is attained at the lower and upper bounds of the interval, namely at x = 1/2 ± r, and equals 1/4 − r 2 . Thus, the exact range of the function f (x), denoted as R(f ; X) is R(f ; X) = [1/4 − r 2 , 1/4]

(3.37)

For the interval evaluation, we obtain f (X) = [1/2 − r, 1/2 + r] − [1/2 − r, 1/2 + r][1/2 − r, 1/2 + r] = [1/4 − 2r − r 2 , 1/4 + 2r − r 2 ]

(3.38)

Thus, the distance between R(f ; X) and f (X) is q(R(f ; X), f (X)) = max{|1/4 − 2r − r 2 − 1/4 + r2 |, |1/4 + 2r − r 2 − 1/4|} = max{2r, 2r − r 2 }

= 2r

Now the diameter of the interval X is w(X) = xU − xL =

1 +r− 2

(3.39) 

1 −r 2



= 2r

(3.40)

As we see, the distance between R(f, X) and f (X) equals 2r, or coincides with the width of the interval X. Moore (1966) has demonstrated that, under reasonable assumption, the following inequality holds for the general interval number X: q(R(f ; X), f (X)) ≤ γw(X), γ ≥ 0

(3.41)

Note that γ = 1 in the example above. The inequality signifies that the overestimation of R(f ; X) by f (X) approaches zero linearly with respect to the diameter of interval number X. Interval analysis found many applications in applied mechanics. First applications are apparently by Dimarogonas (1993), Elishakoff (1994), Dimarogonas (1995), and Dessombz, Thouverez, Lain´e and J´ez´equel (2003). Numerous other references such as Chen, Qiu and Liu (1994), Chen, Qiu and Song (1995), Qiu, Chen and Jia (1995b), Qiu, Chen and Elishakoff (1995a, 1996a, 1996b), Jensen (2000) should also be mentioned.

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Ellipsoidal Model

3.4.1

Definition of the ellipsoidal model

In this section, we summarize what was apparently the first method of generating the multi-dimensional ellipsoidal model by Schweppe (1973). This method was referred to by him as unknown-but-bounded. Further definitive contributions were made by Chernousko (1994, 1999) and others who dubbed it as a guaranteed approach. The ellipsoidal model of uncertainty was utilized in the theory of control by Schweppe (1973), and in modeling geometric imperfections in structures by BenHaim and Elishakoff (1990) (see also an essay by Elishakoff (1991b)). Suppose we have a vector of uncertain parameters a = (a1 , . . . , aN )> and M experimental data points a(r) (r = 1, . . . , M ). The ellipsoidal convex model assumes that all these points belong to an ellipsoid (a − a0 )> W (a − a0 ) ≤ 1

(3.42)

b = Ta

(3.43)

where a0 is the center of the ellipsoid, and W is a positive definite constant N × N weight matrix. The best ellipsoid with minimum volume is achieved by rotating the axes of the ellipsoid. Let b = (b1 , . . . , bN )> denote the new coordinate transformed from a by an N × N matrix T as where the transformation matrix T is the function of the rotation angles θ1 , . . . , θN −1 with respect to the axes a1 , . . . , aN −1 , respectively. T can be constructed by any pair of orthogonal vectors, e.g., the Gram–Schmidt orthogonalization. By using T , the experimental points are transformed as b(r) = T a(r) , (r = 1, . . . , M )

(3.44)

To obtain the ellipsoid, we first consider the N -dimensional box (r)

bk − b0k ≤ dk , (k = 1, . . . , N ; r = 1, . . . , M ) 0

(3.45)

(b01 , . . . , b0N )>

that contains all M experimental points. The center b = and the vector d = (d1 , . . . , dN )> are defined so that the volume of the box is minimized. The ellipsoid is then defined to enclose the box as !2 N (r) X bk − b0k ≤ 1, (r = 1, . . . , M ) (3.46) gk k=1

where gk is the kth semi-axis of the ellipsoid. We assume that gk is nonzero and bounded. The following condition is to be satisfied so that the ellipsoid passes through the corners of the box: 2 N  X dk =1 (3.47) gk k=1

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The volume Ve is given as Ve = C

N Y

gk

(3.48)

k=1

where C is a constant depending on the dimension N of the ellipsoid. By minimizing Ve under constraint (3.47), we obtain (Pantelides and Ganzerli, 1997) √ (3.49) gk = Ndk , (k = 1, . . . , N ) Then the ellipsoid is written as (b − b0 )> D(b − b0 ) ≤ 1

with

2 D = diag 1/g12 , . . . , 1/gN

(3.50)


(3.51)

If there exists a corner of the box without experimental point, the semi-axis is further reduced to ηgk , where v !2 u N (r) u X bk − b0k t ≤1 (3.52) η = max r gk k=1

Hence, the ellipsoid finally is given as

e − b0 ) ≤ 1 (b − b0 )> D(b

where

e = diag 1/(ηg1 )2 , . . . , 1/(ηgN )2 D

(3.53)


(3.54)

The volume Ve is further minimized by considering the angles θ1 , . . . , θN −1 as variables. For this purpose, any method of nonlinear programming can be used. 3.4.2

Properties of the ellipsoidal model

The ellipsoidal model has been extensively investigated and extended by Chernousko (1994), Kurzhanski and Valyi (1997), and others. Hereinafter, we closely follow Chernousko (1999); however, we use the positive-definite N × N weight matrix W instead of W −1 for consistency with other parts of this book. Consider the N -dimensional ellipsoid E defined using the inner (scalar) product as 0

(W (a − a0 ), (a − a0 )) ≤ 1

(3.55)

which is denoted as E(a , W ). Thus, E(a0 , W ) = {a|(W (a − a0 ), (a − a0 )) ≤ 1}

(3.56)

b = Ta + h

(3.57)

The ellipsoid in Eq. (3.56) depends on N (N + 1)/2 parameters in W and N parameters in a0 . Consider in general an affine transformation in N -dimensional space as

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where T is a nonsingular N ×N transformation matrix and h is a constant vector. If a belongs to an ellipsoidal set, b too belongs to an ellipsoidal set. Solving Eq. (3.57) to obtain a = T −1 (b − h) and substituting it into Eq. (3.55) result in (W (T −1 (b − h) − a0 ), (T −1 (b − h) − a0 ))

= ((T −> W T −1 )(b − (T a0 + h)), (b − (T a0 + h))) ≤ 1

(3.58)

where T −> = (T −1 )> . We observe that b belongs to the ellipsoid with the center T a0 + h and weight matrix T −1 W T −> . In other words, which is written as 0

0

b = T a + h ∈ E(T a0 + h, T −1 W T −> )

(3.59)

b ∈ E(b0 , D)

(3.60)

−1

−>

with b = T a + h and D = T W T . As we have seen in Sec. 3.4.1, we can make its axes parallel to the principal axes of the ellipsoid (3.59) with 2 D = diag(1/g12 , . . . , 1/gN )

(3.61)

2

where gi is the ith semi-axis of the ellipsoid. If D = r I, where r > 0 is a scalar, and I is the N × N identity matrix, then the ellipsoid (3.56) is a ball in N -dimensional space with center b0 and radius r. If W is positive definite and its eigenvalues are bounded, then W −1 is also positive definite and its eigenvalues are bounded. However, sometimes one encounters a degenerate ellipsoid for which W −1 is not positive definite but positive semidef2 inite. In this case, D −1 = diag(g12 , . . . , gN ) is positive semidefinite, and some of the semi-axes of the ellipsoid are zero (gi = 0 in Eq. (3.61)), and its volume is also equal to zero. A disk in the three-dimensional space represents an example for this case. The second case, namely, when W is not positive definite but positive semidefinite, indicates that some of the semi-axes of the ellipsoid are infinite (gi = ∞, in Eq. (3.61)), with the volume of the ellipsoid also being unbounded. A cylinder represents an example for this case. When W −1 approaches zero, i.e., D −1 → O, all semi-axes vanish, and we obtain a point E(a0 , W ) = {a|a = a0 }. Hereinafter we concentrate on non-degenerate ellipsoids, i.e., we assume W is positive definite and bounded. Ellipsoid (3.56) can be transformed into the unit ball E(0, I) by the affine transformation (3.57). Indeed, according to Eq. (3.59), such a transformation is obtained if T −> W T −1 = I, T a0 + h = 0

(3.62)

These conditions are satisfied if 1/2

T = W 1/2 , T −1 = W −1/2 , h = −W 1/2 a0

where W is a square root of W . The existence of non-singular W is guaranteed if W is a positive definite matrix. The following theorems hold (Chernousko, 1999):

(3.63) 1/2

and W −1/2

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Theorem 1: For an arbitrary convex set D in N -dimensional space, there exists an ellipsoid E(a0 , W ) ⊆ D such that E(a0 , (1/N 2 )W ) contains D, i.e., E(a0 , W ) ⊆ D ⊆ E(a0 , (1/N 2 )W )

(3.64)

Theorem 2: Let D be a convex set in N -dimensional space, which has the property of central symmetry with respect to some point a0 . There exists an ellipsoid E(a0 , W ) ⊆ D such that E(a0 , (1/N )W ) contains D, i.e., E(a0 , W ) ⊆ D ⊆ E(a0 , (1/N )W )

(3.65)

The proof of Theorem 1 is given in the book by Leichtweiss (1980), whereas the proof of Theorem 2 is given by John (1998). For approximating sets, Chernousko (1999) suggests using inner ellipsoid E L of the maximum volume and outer ellipsoid E U of the minimal volume. The following theorems (Zaguskin, 1958) are relevant: Theorem 3: For any bounded set D in N -dimensional space, there exists a unique ellipsoid E U of the minimum volume containing D as E U ⊃ D. Theorem 4: For any closed convex set D in N -dimensional space, there exists a unique ellipsoid E L of the maximum volume contained in D as E L ⊆ D. The set of all points a = a1 + a2 such that a1 ∈ A, a2 ∈ B is called the direct sum (or the Minkowski sum) of the sets A and B, and is denoted by C = A ⊕ B. We treat the following ellipsoids in N -dimensional Euclidean space: E(a01 , W 1 ) = {a|(W 1 (a − a01 ), (a − a01 )) ≤ 1}

E(a02 , W 2 ) = {a|(W 2 (a − a02 ), (a − a02 )) ≤ 1}

(3.66)

Here a01 and a02 are the N -dimensional vectors of the centers of ellipsoids, and W 1 and W 2 are the N × N positive definite matrices. The Minkowski sum of these ellipsoids reads a = a1 + a2 ∈ S ≡ E(a01 , W 1 ) ⊕ E(a02 , W 2 ),

a1 ∈ E(a01 , W 1 ), a2 ∈ E(a02 , W 2 )

(3.67)

where the set S is bounded, closed, and convex, and in general is not an ellipsoid. Chernousko (1980) posed the following pertinent problems: Problem 1 Find the ellipsoid E(aU , W U ) of the minimal volume containing the sum S of two ellipsoids (3.67), i.e., an ellipsoid such that S ⊆ E(aU , W U ), det W U → min. Problem 2 Find the ellipsoid E(aL , W L ) of the maximal volume contained in the sum S of two ellipsoids (3.67), i.e., an ellipsoid such that E(aL , W L ) ⊆ S, det W L → max. Chernousko (1980) solved both problems. For the details, see Chernousko (1999, p. 143). Chernousko (2000) writes:

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‘Dynamical systems with unknown, uncertain, or perturbed parameters appear in numerous applications. It is important to obtain bounds on the reachable sets of such systems, in other words, bounds on all possible motions of these systems when they are subjected to unknown bounded perturbations.’ Such reachable sets were derived by Chernousko (1988, 1994, 1999, 2005). The apparently first applications of ellipsoidal analysis to applied mechanics was the work by Borkowski (1983). Ellipsoidal analysis methods have been considered by Attoh-Okine (2002, 2004), Au, Cheng, Tham and Zheng (2003), Ben-Haim (1992, 1999), Calafiore and El Ghaoui (2004), Chernousko (1980), Elishakoff (1991b, 1991c, 1994, 1995b, 1998a, 1998c, 2000b, 2004), Elishakoff and Ben-Haim (1990b), Elishakoff and Colombi (1993), Elishakoff, Gana-Shvili and Givoli (1991), Elishakoff, Li, and Starnes (2001), Elishakoff, Lin and Zhu (1994e), Elseifi and Khalessi (2001), Kurzhanski (1977), Kurzhanski and Valyi (1992, 1996), Lindberg (1992a, 1992b), Pantelides and Booth (2000), Pantelides and Ganzerli (2001), Pantelides and Tzan (1996), Qiu (2003, 2005), Qiu, Chen and Wang (2004a), Qiu, Ma and Wang (2006a), Qiu, M¨ uller and Frommer (2004e), Schweppe (1973), Tzan and Pantelides (1996a, 1996b), Vinot, Cogan and Lallement (2003), Wang, Elishakoff and Qiu (2008a), Yoshikawa (2002, 2003), Yoshikawa, Nakagiri and Kuwazuru (1998b), Zhu and Elishakoff (1996), and Zuccaro, Elishakoff and Baratta (1998). Ben-Haim (1985) observed that ellipsoidal analysis is a particular case of convex analysis. A first application of convex analysis to applied mechanics are apparently the works by Moreau (1966, 1976, 1979) and Panagiotopoulos (1976, 1985). The comprehensive presentation of convex analysis in the applied mechanics context was made by Ben-Haim and Elishakoff (1990). In closing this section, it must be noted that in recent work by Kreinovich, Neumaier and Xiang (2008) a combination of interval and ellipsoidal uncertainties is treated. Specifically, they consider the interesting case in which the actual values of uncertain variables belong to the intersection of interval and ellipsoid. 3.5

Anti-Optimization Problem

The response property in question is denoted by f (a), which is supposed to give the worst value if it is maximized with respect to the uncertain parameters a. Uncertainty may exist in material parameters, cross-sectional geometry, external loads, and so on. If an ellipsoidal bound is given for a = (a1 , . . . , aN )> , the antioptimization problem is formulated as maximize f (a) 2 N  X ai ≤1 subject to gi i=1

(3.68a) (3.68b)

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where gi is the specified positive parameter that represents the semi-axis of the parameter ai . Note that a has been appropriately transformed so that the ellipsoid has its center at the origin 0 and the direction of each semi-axis coincides with a coordinate axis. The anti-optimal solution of Problem (3.68) can be found from the stationary condition of the Lagrangian Q defined by " 2 # N  X ai Q(a, µ) = f (a) + µ 1 − (3.69) gi i=1

where µ ≥ 0 is the Lagrange multiplier. The stationary condition of Q with respect to a leads to ∂f 2µai ∂Q (3.70) = − 2 = 0, (i = 1, . . . , N ) ∂ai ∂ai gi The details of the Lagrange multiplier approach can be found in Sec. 2.4.2. If f (a) is a linear function of a, or has been linearly approximated with respect to a as N X f (a) = ci a i (3.71) i=1

with the coefficient ci , then ai /gi is expressed from Eq. (3.70) as ai gi ci = (3.72) gi 2µ Since the objective function is linear and the feasible region of the variables a is convex, the anti-optimal solution exists at the boundary of the feasible region. Therefore, the constraint (3.68b) is satisfied in equality as 2 N  X ai =0 (3.73) 1− gi i=1 Incorporation of Eq. (3.72) into Eq. (3.73) results in v uN uX 1 ∗ µ = µ = t (gi ci )2 2 i=1

(3.74)

Note that µ = −µ∗ corresponds to the minimum value of f (a). This way, the anti-optimal solution can be obtained by a simple arithmetic operation. If f (a) is a nonlinear function, then only locally anti-optimal solution can be found by using a method of nonlinear programming. The global anti-optimality is guaranteed when f (a) is a concave function. Alternatively, the bound constraints can be assigned to the anti-optimization problem as maximize f (a) subject to

aLi

≤ ai ≤

(3.75a) aU i ,

(i = 1, . . . , N )

(3.75b)

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where aLi and aU i are the lower and upper bounds for ai . Problem (3.75) is a linear programming problem, if f (a) is a linear function of a. For this simple problem with bound constraints only, the solution is found at ai = aLi or aU i depending on the sign of the coefficient for ai in f (a). For the case where f (a) is a nonlinear function, the globally anti-optimal solution can be found by a nonlinear programming approach, if f (a) is concave. On the other hand, if f (a) is a convex or monotonic function of a, then the globally anti-optimal solution can be found by enumerating the vertices of the feasible region defined by Eq. (3.75b) (see Secs. 4.6 and 5.6 for examples of anti-optimization by vertex enumeration). If ai ≥ 0 (i = 1, . . . , N ), the bounds for a are generalized as " N   # p1 X ai p ≤ 1, (p = 1, . . . , ∞) (3.76) gi i=1 Note that p = 2 corresponds to the ellipsoidal bound, and p = ∞ corresponds to Eq. (3.75b) with aLi = 0 and aU i = gi (i = 1, . . . , N ) (see Eq. (2.31) in Sec. 2.6 for various definitions of the norm). Special cases with N = 2 are discussed in Adali, Lene, Duvaut and Chiaruttini (2003) (see also Kurzhanski (1977), Kurzhanski and Valyi (1992, 1996, 1997), and Kurzhanski and Varaiya (2002)). 3.6

Linearization by Sensitivity Analysis

3.6.1

Roles of sensitivity analysis in anti-optimization

Sensitivity analysis is mathematically defined as a process of computing the derivatives of the solution to governing equations with respect to the parameters defining the equations. In the field of structural analysis and design, sensitivity analysis has been developed as a tool for evaluating the derivatives (gradients or sensitivity coefficients) of the structural responses with respect to variation of the design or shape parameters (Haug, Choi and Komkov, 1986; Choi and Kim, 2004). Sensitivity analysis also serves as a basic tool in various fields of mathematics and engineering, e.g., inverse problem and stability analysis. The procedures of sensitivity analysis for optimization are classified as follows by the type of variables: • Design sensitivity analysis to find the sensitivity coefficients of the responses with respect to the sizing design variables such as the cross-sectional areas of trusses and thicknesses of the plate elements. • Shape sensitivity analysis to find the sensitivity coefficients of the responses with respect to the shape design variables such as the nodal locations of the frames and the plates discretized to finite elements. The sensitivity coefficients of the responses with respect to the uncertain param-

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4

y

Y

3 2

Node 2

1 X

x

θi Node 1

Fig. 3.4

Coordinates, displacement numbers and node numbers of a plane truss member.

eters are to be computed in the process of anti-optimization. The role of sensitivity analysis in anti-optimization is two-fold: • Linearly approximate the responses using sensitivity coefficients to formulate a convex anti-optimization problem. • Evaluate the gradients of the objective and constraint functions with respect to uncertain parameters to carry out anti-optimization using a nonlinear programming approach. The sensitivity coefficients can also be obtained by the perturbation equations (see Sec. 4.4 for the static perturbation). However, in the field of numerical structural optimization, sensitivity equations are derived by differentiation of the governing equation as presented in this section. 3.6.2

Sensitivity analysis of static responses

In order to present the basic ideas of sensitivity analysis, the direct differentiation method for static responses of plane trusses is briefly introduced. The global coordinates (X, Y ), local coordinates (x, y), local node numbers, local displacement numbers, and angle θi of the member i to the X-axis are defined in Fig. 3.4. The displacement vector uli = (ul1 , ul2 , ul3 , ul4 )> and the nodal force vector l f i = (f1l , f2l , f3l , f4l )> of member i with respect to the local coordinates are transformed, respectively, to the displacement vector ugi = (ug1 , ug2 , ug3 , ug4 )> and the nodal force vector f gi = (f1g , f2g , f3g , f4g )> with respect to the global coordinates by the transformation matrix T i as

where

ugi = T i uli , f gi = T i f li

(3.77)

  cos θi − sin θi 0 0  sin θi cos θi 0 0   Ti =   0 0 cos θi − sin θi  0 0 sin θi cos θi

(3.78)

Let Ai and Li denote the cross-sectional area and the length of the ith member, respectively. The 4 × 4 member stiffness matrix k li with respect to the local

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coordinates is given as kli =

Ai E > dd Li

(3.79)

where d = (−1, 0, 1, 0)>

(3.80)

and E is Young’s modulus. From Eqs. (3.77) and (3.79), the member stiffness matrix k gi with respect to the global coordinates is given as kgi =

Ai E > > T dd T i Li i

(3.81)

and kgi of all the members are assembled to construct the n × n global stiffness matrix K, where n is the number of degrees of freedom of the structure. Let m denote the number of members. For design sensitivity analysis, where the cross-sectional areas are considered as design variables, the stiffness matrix is a function of A = (A1 , . . . , Am )> , which is written as K(A). If the self-weight is considered, the nodal load vector P = (P1 , . . . , Pn )> is also a function of the design variable vector A, which is written as P (A). The displacement vector U (A) = (U1 , . . . , Un )> is obtained by solving the stiffness equation as K(A)U (A) = P (A)

(3.82)

In the following, the argument A is omitted for brevity. Differentiation of Eq. (3.82) with respect to Ai leads to ∂K ∂U ∂P U +K = ∂Ai ∂Ai ∂Ai Then the sensitivity coefficient of U with respect to Ai is obtained from

(3.83)

∂P ∂K ∂U = − U (3.84) ∂Ai ∂Ai ∂Ai Therefore, the sensitivity coefficients of P and K are needed to compute those of the displacements. By differentiating Eq. (3.81) with respect to Ai , we obtain the explicit relation K

E > > ∂kgi = T dd T i (3.85) ∂Ai Li i Hence, for a truss, the sensitivity coefficient of K with respect to Ai is easily obtained by an arithmetic operation, because K is an assemblage of k gi . Sensitivity of P is also easily computed, when the self-weight is considered, because P is an explicit function of A. Note that K is decomposed to a product of triangular and diagonal matrices by Cholesky decomposition in the process of computing U from Eq. (3.82). Hence, the computational cost of solving Eq. (3.84) is very small even for the case where the sensitivity coefficients with respect to all m design variables A1 , . . . , Am are to be computed.

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U1 P1 A2

A1

Fig. 3.5

U2 P2

A 2-bar structure.

Consider, for example, a 2-bar structure as shown in Fig. 3.5 subjected to axial forces P1 and P2 at nodes 1 and 2, respectively. The displacements of nodes 1 and 2 are denoted by U1 and U2 , respectively. The two bars have the cross-sectional areas A1 and A2 , and the same length L. Then Eq. (3.82) is written as      E A1 + A2 −A2 U1 P1 = (3.86) −A2 A2 U2 P2 L

which is easily solved for U1 and U2 as

(P1 + P2 )L P2 L (P1 + P2 )L , U2 = + (3.87) A1 E A1 E A2 E Consider A1 as the design variable, and suppose that P1 and P2 are independent of A1 . Equation (3.83) is then written as   ∂U1        E 1 0 E A1 + A2 −A2  U1 0 ∂A1    + = (3.88)   ∂U2 U2 −A2 A2 0 L 0 0 L ∂A1 By incorporating Eq. (3.87) into Eq. (3.88), we obtain U1 =

∂U2 (P1 + P2 )L ∂U1 = =− ∂A1 ∂A1 A21 E

(3.89)

which agrees with the direct differentiation of U1 and U2 in Eq. (3.87) with respect to A1 . The sensitivity coefficients can also be computed by the finite-difference approach. Let ∆Ai denote the m-vector whose ith component is ∆Ai and the other components are 0, where ∆Ai is the small variation of Ai . The approximate sensitivity coefficients by the central finite-difference approach is given as ∂U U (A + ∆Ai ) − U (A − ∆Ai ) ' (3.90) ∂Ai 2∆Ai The approximate sensitivity coefficients can also be found by the forward finitedifference approach U (A + ∆Ai ) − U (A) ∂U ' ∂Ai ∆Ai or the backward finite-difference approach ∂U U (A) − U (A − ∆Ai ) ' ∂Ai ∆Ai

(3.91)

(3.92)

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Although these finite-difference approaches are very simple, the number of response analysis (solution process of Eq. (3.82)) for sensitivity evaluation with respect to m design variables is m for the forward or backward finite-difference approach and 2m for the central finite-difference approach. Therefore, it is seen that the computational cost for sensitivity analysis is drastically reduced by direct differentiation of the governing equations. 3.6.3

Sensitivity analysis of free vibration

We next consider the sensitivity analysis of eigenvalues and eigenmodes of free vibration. Let M (A) denote the n × n mass matrix of the structure, which is a function of the design variable vector A. The eigenvalue problem of free vibration is formulated as KΦr = Ωr M Φr , (r = 1, . . . , n)

(3.93)

where Ωr and Φr are the rth eigenvalue and eigenmode, respectively, which are functions of A. The eigenmode Φr is ortho-normalized by Φ> r M Φs = δrs , (r, s = 1, . . . , n)

(3.94)

where δrs is the Kronecker delta. By differentiating both sides of Eq. (3.93) with respect to Ai , we obtain ∂Φr ∂Ωr ∂M ∂Φr ∂K Φr + K = M Φr + Ω r Φr + Ωr M (3.95) ∂Ai ∂Ai ∂Ai ∂Ai ∂Ai Premultiplying Φ> r to both sides of Eq. (3.95), using Eqs. (3.93), (3.94), and the symmetry of M and K, we obtain   ∂Ωr ∂K ∂M > = Φr − Ωr Φr (3.96) ∂Ai ∂Ai ∂Ai Note that K and M are explicit functions of Ai . Therefore, when the eigenvalue Ωr and eigenmode Φr are known after solving the eigenvalue problem (3.93) with Eq. (3.94), the sensitivity coefficients of eigenvalues are computed from Eq. (3.96) by simple arithmetic operation without resort to the sensitivity coefficients of the eigenmodes. Differentiation of Eq. (3.94) with respect to Ai leads to ∂Φr ∂M Φr + 2Φ> =0 (3.97) Φ> r M r ∂Ai ∂Ai From Eqs. (3.95) and (3.97),      ∂Φr ∂M ∂K   Ω − Φr K − Ω r M M Φr  ∂Ai   r ∂Ai  ∂Ai   = (3.98)    ∂Ωr  1 > ∂M Φ> 0 r M − Φr Φr 2 ∂Ai ∂Ai is derived. Therefore, the sensitivity coefficients of each pair of eigenvalue and eigenmode can be found from Eq. (3.98), which is a set of n + 1 simultaneous linear equations.

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Shape sensitivity analysis of trusses

We next consider variation of nodal locations to compute the shape sensitivity coefficients. A plane truss member as shown in Fig. 3.4 is used, for simplicity, also in this section. The 4 × 4 member stiffness matrix kgi with respect to the global coordinates is given as Eq. (3.81). Since Li , sin θi and cos θi depend on the nodal coordinates, the sensitivity coefficients of these values are needed to compute the shape sensitivity coefficients of kgi . Let (X1 , Y1 ) and (X2 , Y2 ) denote the global coordinates of nodes 1 and 2, respectively, of member i. Then Li , sin θi and cos θi are given as p (3.99) Li = (X2 − X1 )2 + (Y2 − Y1 )2 X2 − X 1 Y2 − Y 1 cos θi = , sin θi = (3.100) Li Li Note that (cos θi , sin θi )> is a unit vector directed from node 1 to 2. Rewriting Eq. (3.99) as L2i = (X1 − X2 )2 + (Y1 − Y2 )2

(3.101)

and differentiating it with respect to X1 and X2 , respectively, we obtain 1 ∂Li = − (X2 − X1 ) = − cos θi , ∂X1 Li ∂Li 1 = (X2 − X1 ) = cos θi ∂X2 Li

(3.102)

where Eq. (3.100) has been used. The sensitivity coefficients of cos θi are derived as   1 ∂Li sin2 θi ∂ cos θi = −1 − cos θi =− ∂X1 Li ∂X1 Li (3.103)   ∂ cos θi 1 ∂Li sin2 θi = 1 − cos θi = ∂X2 Li ∂X2 Li The sensitivity coefficients of sin θi with respect to Y1 and Y2 are derived in a similar manner. Finally, the sensitivity coefficients of T i are easily obtained from Eq. (3.78), and those of kgi with respect to the nodal coordinates; e.g., X1 are computed, as follows, by differentiating Eq. (3.81): Ai E ∂Li > > Ai E > > ∂T i ∂kgi =− 2 T dd T i + T dd ∂X1 Li ∂X1 i Li i ∂X1  > ∂T i Ai E + dd> Ti ∂X1 Li

(3.104)

which are assembled to the total structure and simply incorporated into Eq. (3.84) to compute the shape sensitivity coefficients of the displacements.

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Uj reanalysis

sensitivity analysis Ai Fig. 3.6

3.7 3.7.1

Ai +∆Ai

Estimation of displacement by sensitivity analysis and reanalysis.

Exact Reanalysis of Static Response Overview of exact reanalysis

The differential coefficients of the responses with respect to the design variables are computed by design sensitivity analysis as presented in Sec. 3.6. Then the responses such as nodal displacements can be linearly estimated by the sensitivity coefficients as shown in the straight tangent line in Fig. 3.6 that illustrates the variation of the displacement component Uj with respect to the increment ∆Ai of the design variable Ai . On the other hand, the method for accurately estimating the responses after modification of the variable by the specified amount is called exact reanalysis as illustrated by the arrow in Fig. 3.6. Note that the term reanalysis does not mean that responses are simply computed again after modification of the variable, but that accurate responses are found by computationally more inexpensive manner using the information of the structure before modification. In an anti-optimization problem, the solution often exists at the boundary of the feasible region of the uncertain parameters, if the objective and constraint functions have monotonicity and/or convexity properties; see, e.g., Sec. 9.6. In this case, the anti-optimal solution can be found by vertex enumeration of the feasible region, and the methods of exact reanalysis can be effectively used. There are several approaches to the approximate or exact reanalysis of responses of structures subjected to static loads. Melosh and Luik (1968) presented a simple approach for the reanalysis of trusses. Argyris and Roy (1972) presented a more general approach allowing changes in the number of degrees of freedom of displacements and support conditions. Kirsch and Rubinstein (1972) discussed convergence properties of the iterative process of reanalysis. Kirsch (1994) presented an efficient reanalysis method combining the reduced basis method and the expansion method. His method has been shown to lead to exact responses of trusses (Kirsch and Liu, 1995), and is applicable to any finite dimensional structures. Application to shape modification and topological changes has also been demonstrated (Kirsch and Liu,

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1996). Kirsch (2000) extended his method to the combined reanalysis for nonlinear static analysis and eigenvalue analysis. The virtual distortion method has been developed for static and dynamic analyses of trusses (Putresza and Kolakowski, 2001). Deng and Ghosn (2001) developed a pseudo force method for nonlinear analysis and reanalysis. Bickford (1987) presented a refined higher-order perturbation method for eigenvalues and eigenvectors of a generalized eigenvalue problem. Chen, Yang and Lian (2000) compared several reanalysis methods for eigenvalue analysis. Chen, Wu and Yang (2006) presented a reanalysis method for eigenvalue analysis allowing large parameter modification. Makode, Corotis and Ramirez (1999) presented a pseudodistortion method for static nonlinear analysis of trusses. From a mathematical point of view, structural reanalysis can be regarded as a process of finding the inverse of the stiffness matrix of the modified structure. Calculation of the inverse of a modified matrix has been discussed in many fields of engineering and mathematics. Developments in general form of inverting modified matrices are summarized in the review article by Henderson and Searle (1981). However, it is important to note, for application to the static analysis problem of structures, that the inverse of the stiffness matrix of the initial structure is not usually known; i.e., the matrix is only decomposed to a product of triangular and diagonal matrices in the process of structural analysis. Kavlie, Graham and Powell (1971) developed a stiffness-based method for computing the displacements of the modified design based on the Sherman–Morrison–Woodbury (SMW) formula (Sherman and Morrison, 1950; Woodbury, 1950). Akg¨ un, Garcelon and Haftka (2001) extended the SMW formula to the nonlinear reanalysis problem. Ohsaki (2001) showed that the displacements of the modified structures are found without computation of the inverse of the stiffness matrix of the initial system; i.e., only the Cholesky decomposition is necessary. We present below a method of exact reanalysis for the static responses of trusses.

3.7.2

Mathematical formulation based on the inverse of the modified matrix

The equations for exact reanalysis is first formulated as a purely mathematical process of solving a set of linear equations with respect to a modified matrix. Consider a truss subjected to static nodal loads P . Let n denote the number of degrees of freedom, and K denote the n×n stiffness matrix. The nodal displacement vector U 0 of the initial (reference) system before modification is computed from KU 0 = P

(3.105)

The matrix K is a function of the vector A = (A1 , . . . , Am )> of the cross-sectional areas, where m is the number of members. Since the components of K of a truss

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are linear functions of Ai , K is written as K=

m X

Ai K i

(3.106)

i=1

where K i is the n×n stiffness matrix for unit cross-sectional area of the ith member. Note that the member properties are defined by the matrices and vectors of the size n for the convenience of presenting the matrix operation. Computation is carried out, however, by element-size matrices and vectors. Consider the case where the cross-sectional area of the kth member is increased by ∆Ak . Equation (3.105) for the modified truss is formulated as (K + ∆Ak K k )U 1 = P

(3.107)

The objective here is to find the displacement vector U 1 of the modified truss for the given load vector P . In the following, the superscripts ( · )0 and ( · )1 denote the values of the initial and the modified structures, respectively. There have been a variety of studies on computing the inverse of the modified matrix from the known inverse of the initial matrix. For our purpose, however, those formulas cannot be directly used, because the inverse of K is not usually known; i.e., K is only decomposed for computing U 0 . Therefore, a more explicit formula is needed for computing U 1 of the modified truss. Let N = (N1 , . . . , Nm )> denote the vector of axial forces. The relation between 0 Ni and U 0 is written by using an n-vector ci as 0 Ni0 = Ai c> i U

(3.108)

Let E and Li denote Young’s modulus and the length of the ith member, and define vector bi as Li bi = ci (3.109) E Then the relation between Ni0 and the n-vector of equivalent nodal forces F 0i of the ith member is written as F 0i = Ni0 bi

(3.110)

Hence, bi represents the nodal load vector corresponding to unit axial force of the ith member. The matrix K i is defined by using ci and bi as K i = b i c> i

(3.111)

Note from Eqs. (3.109) and (3.111) that the rank of K k is 1. The inverse of the modified matrix K + ∆Ak K k , when rank K k = 1, is obtained from (Henderson and Searle, 1981)
−1 −1 (K + ∆Ak K k ) = K + ∆Ak bk c> k −1 (3.112) ∆Ak K −1 bk c> kK = K −1 − −1 > 1 + ∆Ak ck K bk

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By post-multiplying P to Eq. (3.112), the following relation is derived: U1 = U0 −

0 ∆Ak K −1 bk c> kU −1 1 + ∆Ak c> bk kK

(3.113)

where Eqs. (3.105) and (3.107) are used. Let superscript k∗ denote a value corresponding to the initial truss subjected to the load vector bk that generates the unit axial force in member k. Then Eq. (3.113) is rewritten as U1 = U0 − = U0 −

0 ∆Ak U k∗ c> kU k∗ 1 + ∆Ak c> kU

(3.114)

∆Ak Eε0k 1 + ∆Ak Eεk∗ k

where Eq. (3.108) is used, and εk is the strain of member k. Note that U k∗ and εk∗ k can be computed through simple arithmetic operation, because K has been already decomposed in the process of computing U 0 . This way, U 1 can be obtained without using K −1 . The strain of the member with null cross-sectional area may also be obtained from the displacements of the nodes connected to the member. Therefore, Eq. (3.114) can be used for addition or removal of a member between two existing nodes. Next, we consider a case where the cross-sectional areas of q (≥ 2) members are modified simultaneously. It is assumed without loss of generality that Ak (k = 1, . . . , q) are modified. The stiffness matrix of the modified truss is formulated using Eqs. (3.109) and (3.111) as K+

q X

k=1

∆Ak K k = K +

q X ∆Ak E

k=1

Lk

bk b> k

(3.115)

= K + BDB >

where the ith column of the n × q matrix B is bi , and D is a q × q diagonal matrix defied as D = diag(∆A1 E/L1 , . . . , ∆Aq E/Lq )

(3.116)

The inverse of the modified stiffness matrix is written as (Henderson and Searle, 1981; Sherman and Morrison, 1950; Woodbury, 1950) (K + BDB > )−1 = K −1 − K −1 B(D −1 + B > K −1 B)−1 B > K −1

(3.117)

By post-multiplying P to Eq. (3.117), the following relation is derived: U 1 = U 0 − K −1 B(D −1 + B > K −1 B)−1 DB > U 0 = U 0 − U ∗ (D−1 + B > K −1 B)−1 y

(3.118)

where the ith component of the q-vector y is ε0i Li , and the ith column of the n × q matrix U ∗ is U i∗ .

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A temporary variable vector z is introduced as z = (D −1 + B > K −1 B)−1 y

(3.119)

which is found by solving a set of q linear equations (D −1 + B > K −1 B)z = y

(3.120)

(D −1 + Y ∗ )z = y

(3.121)

which is simply rewritten as

∗

where the (i, j)-component of Y is equal to εi∗ j Lj . Therefore, the inverse matrix −1 K need not be explicitly computed also for this case. Finally, Eqs. (3.119) and (3.121) lead to U 1 = U 0 − U ∗z

(3.122)

It may be easily seen that Eq. (3.114) for the modification of a single variable is a special case of Eqs. (3.121) and (3.122) for multiple variable modification. As an illustrative example, consider the 2-bar structure in Fig. 3.5 in Sec. 3.6.2. Let P1 = 0, P2 = 1, E = 1, and L = 1, for simplicity. The matrices and vectors for the design A1 = A2 = 1 are derived as       2 −1 1 1 1 K= , K −1 = , U0 = (3.123) −1 1 1 2 2 Suppose A1 is increased to 2; i.e., ∆A1 = 1. Then we have     1 0 3 −1 K1 = , K + ∆A1 K 1 = , 0 0 −1 1   1 1 1 −1 , (K + ∆A1 K 1 ) = 2 1 3       1 1 1 1 b1 = , c1 = , U1 = 0 0 2 3 From Eq. (3.112), we obtain       1 1 1 1 0 1 1 1 1 (K + ∆A1 K 1 )−1 = − 0 0 1 2 1 2 1+1 1 2     1 1 1 1 1 = − 1 2 2 1 1   1 1 1 = 2 1 3 which agrees with the result in Eq. (3.124). Furthermore, we obtain         1 1 1 1 1 1 1 = U 1∗ = , ε1∗ = 1, U = − 1 1 2 2 1 2 3 which also agrees with the result in Eq. (3.124).

(3.124)

(3.125)

(3.126)

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Ak , Nkk 1

Ak , Nk

Fig. 3.7

Initial truss subjected to P .

αk

Ak , Nk + αk Nkk

Fig. 3.8 Initial truss subjected to virtual k load P .

Ak+∆ Ak , Nk + αk Nkk+ αk

αk

Fig. 3.9 Initial truss subjected to P and k virtual load αk P .

3.7.3

1

Fig. 3.10 load.

Modified truss without virtual

Mechanical formulation based on virtual load

We next investigate the reanalysis process through mechanical formulations using the virtual loads due to modification of the cross-sectional area of a member. Consider again the case where the cross-sectional area Ak of member k of a truss is to be modified. The initial truss subjected to the nodal load vector P is illustrated in Fig. 3.7 for a 7-bar truss. Figure 3.8 shows the state where the load k vector P representing the unit virtual axial force of member k is applied to the k k k k truss. The axial force vector against P is denoted by N = (N 1 , . . . , N m )> . Suppose the axial force of member k under P increases as the result of modification of Ak to Ak + ∆Ak . The axial force of member i of the initial truss under the k sum of P and the additional load αk P corresponding to the incremental virtual axial force αk of member k is given as k

Nik = Ni + αk N i

which is illustrated in Fig. 3.9. The strain εki corresponding to as Li k (Ni + αk N i ) εki = Ai E k

(3.127) k Nik + αk N i

is given (3.128)

Note that the virtual load αk P does not actually exist, and it should be supplied as the result of an increment of the axial force due to the modification ∆Ak of the cross-sectional area of member k. Therefore, the deformation against the set of

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k

loads P and αk P shown in Fig. 3.9 should be the same as that of the modified truss without the additional virtual loads. Hence, the pair of loads are transformed back to the axial force αk as shown in Fig. 3.10, and we obtain the following relation from the equivalence of the strain of member k: Lk Lk k k (Nk + αk N k ) = (Nk + αk N k + αk ) Ak E (Ak + ∆Ak )E

(3.129)

which results in αk = k

∆Ak Nk k

Ak − N k ∆Ak

(3.130)

Since N k is the axial force of member k against a pair of loads corresponding to the unit axial force in member k, the following relation holds for a statically indeterminate truss: k

−1 < N k < 0

(3.131)

and the denominator of Eq. (3.130) is positive for ∆Ak ≥ −Ak ; i.e., the value of αk for Ak + ∆Ak ≥ 0 that includes removal of member k is obtained from Eq. (3.130). Accordingly, the displacements, axial forces, strains, etc., can be comk puted by applying the loads P + αk P to the initial truss with cross-sectional areas A. It is easily observed that αk in Eq. (3.128) is equal to the coefficient for U k∗ in Eq. (3.118) derived from the formula of the inverse of a modified matrix. Hence, the mathematical and mechanical formulations lead to the same result. k For a statically determinate truss, N k = −1 and the strain of member k of the modified truss is obtained from Eq. (3.128) as 1 k (Nk + αk N k ) Ak E 1 = (Nk − αk ) Ak E Nk = (Ak + ∆Ak )E

εkk =

(3.132)

which corresponds to the fact that the axial force is independent of the crosssectional areas. As an illustrative example, consider again the 2-bar structure in Fig. 3.5 in Sec. 3.6.2, with the same parameter values in the example in Sec. 3.7.2. The crosssectional areas of the initial truss is A1 = A2 = 1, and A1 is increased to 2. Then we have   1 1 1 1 1 P = , N1 = N2 = 1, N 1 = −1, α1 = = (3.133) 0 1+1 2 and the strains of the modified truss are obtained from Eq. (3.128) as ε11 = 1 −

1 1 = , ε12 = 1 2 2

(3.134)

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which is consistent with the displacements in Eq. (3.125) obtained by the mathematical formulation. In closing this section, it should be noted that the exact reanalysis method can be very effectively used for enumerating the responses of the trusses corresponding to the vertices of feasible regions of the uncertain cross-sectional areas. Direct application of the reanalysis method to truss topology optimization can be found in Ohsaki (2001).

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optimization

Chapter 4

Anti-Optimization in Static Problems

‘Hope for the best but prepare for the worst.’ (English proverb) ‘If you think the worst, you won’t be far wrong.’ (Spanish proverb) ‘The worst is not always certain but it’s very likely.’ (French proverb) ‘If anything can go wrong, it will, and at the worst possible time.’ (Popular version of Murphy’s law)

In this chapter, we first present a simple static anti-optimization problem so that the analytically minded researcher may follow it with pen, pencil, and of course with the head. Later we explore the apparently first static anti-optimization study by Boley (1966a) devoted to the worst temperature distribution through the thickness of the plate, and we present several anti-optimization problems including the evaluation of the worst possible hole that produces the maximum stress concentration factor. The worst distribution of prestress of a tensegrity structure is found by vertex enumeration of the feasible region of the uncertain parameters. 4.1

A Simple Example

√ Consider a 2-bar truss as shown in Fig. 4.1, where H = 1, W1 = 3, W2 = 1. The cross-sectional area and Young’s modulus of the members are denoted by A and E, respectively. The nodal loads and displacements in (x, y)-directions are denoted by P = (P1 , P2 )> and U = (U1 , U2 )> , respectively, which are related by √ √ 1 √ [(4 + 2)P1 + (−4 + 6)P2 ] U1 = AE(2 + 3) (4.1) √ √ 1 √ [(−4 + 6)P1 + (4 + 3 2)P2 ] U2 = AE(2 + 3) which are simply written as > U1 = c > 1 P , U2 = c2 P

77

(4.2)

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P2 , U2 P1 , U1

H x y Fig. 4.1

W1

W2

An asymmetric 2-bar truss (W1 6= W2 in general).

with

√  √    1 1 4 + √2 −4 + √ 6 √ √ , c2 = c1 = AE(2 + 3) −4 + 6 AE(2 + 3) 4 + 3 2 We assume uncertainty in P as P = P0 + a 0

(P10 , P20 )>

(4.3)

(4.4) >

where P = is the nominal load vector and a = (a1 , a2 ) represents the uncertainty. Suppose, for simplicity, U2 is chosen as the performance measure, and our purpose is to find the worst value of a that maximizes U2 , which is assumed to be positive. Consider the case where the bound of a is given by the quadratic inequality constraint a> a ≤ D

(4.5)

as the simplest ellipsoidal model presented in Sec. 3.4, where D is a prescribed value. Then the anti-optimization problem is formulated in the following form of a quadratic programming problem: maximize U2 (a) >

(4.6a)

subject to a a ≤ D

(4.6b)

Q(a, µ) = U2 (a) + µ(D − a> a)

(4.7)

It is seen from Eqs. (4.2) and (4.4) that U2 is a linear function of a. Therefore, the inequality constraint (4.6b) is satisfied with equality at the anti-optimal solution, which is found explicitly as follows. The Lagrangian Q of Problem (4.6) is defined as where µ ≥ 0 is the Lagrange multiplier. The stationary condition of Q with respect to a leads to ∂U2 − 2µai = 0, (i = 1, 2) (4.8) ∂ai from which we obtain 1 a= c2 (4.9) 2µ

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a2 Anti-optimal solution 1

1

−1

a1

−1

Fig. 4.2

Illustration of anti-optimal solution by quadratic programming.

where Eqs. (4.2) and (4.4) have been used. Since the constraint (4.6b) is satisfied with equality at the anti-optimal solution, the Lagrange multiplier is obtained as follows by incorporating Eq. (4.9) into Eq. (4.5), and using µ ≥ 0: r 1 c> 2 c2 µ = µ∗ = (4.10) 2 D Note that µ = −µ∗ corresponds to the optimum (minimum) value of U2 . If EA = 1 and D = 1, for simplicity, then c2 = (−0.4155, 2.2086)>, and the anti-optimization problem is formulated as maximize U2 = −0.4155a1 + 2.2086a2 subject to

a21

+

a22

≤1

(4.11a) (4.11b)

From Eq. (4.11a), a2 is written in terms of U2 and a1 as a2 = 0.4528U2 + 0.1881a1

(4.12)

The gray area in Fig. 4.2 is the feasible region defined by Eq. (4.11b), and each dashed line represents the solution with a constant value of U2 . The anti-optimal solution is found as a feasible solution that attains the maximum intercept of the a2 -axis. Since µ = 1.1237 is obtained from Eq. (4.10), the worst load-case is found from Eq. (4.9) as a = (−0.1849, 0.9828)>. If the bounds are given for a by an interval aL ≤ a ≤ a U

(4.13)

with the lower bound aL and upper bound aU , the anti-optimization problem turns out to be a linear programming problem: maximize U2 (a) L

(4.14a) U

subject to a ≤ a ≤ a

(4.14b)

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a2 Anti-optimal solution (−1, 1)

(1, 1)

a1

(−1, −1)

Fig. 4.3

(1, −1)

Illustration of anti-optimal solution by linear programming.

L For the case EA = 1, D = 1 and aU i = −ai = 1 (i = 1, 2), Problem (4.14) is explicitly written as maximize U2 = −0.4155a1 + 2.2086a2 (4.15a)

subject to − 1 ≤ ai ≤ 1, (i = 1, 2) (4.15b) The gray area in Fig. 4.3 is the feasible region defined by Eq. (4.15b), and each dashed line represents the solution with a constant value of U2 . The anti-optimal solution is obtained as the intersection of the feasible region and the dashed line with maximum intercept of the a2 -axis. Therefore, the anti-optimal solution is attained at the vertex (−1, 1) of the feasible region, which can be easily obtained in view of signs of the coefficients of ai in the linear expansion of the objective function U2 .

4.2

Boley’s Pioneering Problem

Boley (1966a,b) pioneered a study on the bounds of the thermoelastic stresses and deflections in a beam or a plate. Its derivations will be reproduced here briefly as the first work of anti-optimization. The normal thermoelastic stress σ in a beam of arbitrary cross-section, e.g., rectangular cross-section as shown in Fig. 4.4, is given by the formula (Boley and Weiner, 1960) MT y PT + (4.16) σ = −αET (y, z) + A I where T (y, z) is the temperature, which is a function of the coordinates (y, z) in the cross-section, PT is the thermal force, MT is the thermal moment, defined, respectively, as Z Z PT =

αET dA, MT =

A

αET ydA

A

(4.17)

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Anti-Optimization in Static Problems

y

y

c c

81

x

z

L Fig. 4.4

A beam with rectangular cross-section.

and A is the cross-sectional area, I is the moment of inertia about the z-axis, E is Young’s modulus, and α is the thermal coefficient. Although Eq. (4.16) is valid whether principal axes are used or not, we will assume that (y, z) is the principal coordinate system for simplicity. In his paper, Boley (1966b) poses the following question: ‘What is the maximum value of σ that can be caused in a given beam by arbitrary temperature distributions?’ He presumes that the temperature T (y, z) is bounded in the interval −Tm ≤ T (y, z) ≤ TM

(4.18)

where Tm and TM are the constants such that TM ≥ −Tm . The problem is to determine the constant k such that |σ| ≤ kα(Tm + TM )

(4.19)

for all possible temperature distributions. The coefficient k has a physical sense of the factor of proportionality between the maximum possible stress and the maximum temperature excursion Tm + TM . The following question arises: ‘Why is it useful to know the coefficient k?’ It turns out that the temperature excursion can often be straightforwardly estimated, especially in problems in which the body surface temperature is known. Therefore, Eq. (4.19) permits the easy estimation of the maximum normal stresses. For simplicity, we will deal with the beam of the rectangular cross-section as shown in Fig. 4.4, which was studied by Boley (1966b) in an earlier paper, presented at a conference in 1964. Accounting that the product αE is constant, we rewrite Eq. (4.16) as follows: Z Z σ(η) 1 1 3η 1 = −T (η) + T (η0 )dη0 + T (η0 )η0 dη0 (4.20) αE 2 −1 2 −1 where η = y/c with c being the half-depth of the beam as shown in Fig. 4.4. For a special case  δ δ  T0 , for η0 − < η < η0 + (4.21) T = 2 2  0, elsewhere

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the stress, denoted as σ0 (η; η0 ), is obtained as  δ δ δ σ0 (η; η0 )  −1 + (1 + 3ηη0 ), for η0 − < η < η0 + = 2 2 2  0, αET0 otherwise

(4.22)

The function σ0 (η; η0 ) is a Green function or a fundamental solution. Then, for the arbitrary temperature distribution, one may write Z 1 1 σ(η) = −T (η) + T (η0 )(1 + 3ηη0 )dη0 (4.23) αE 2 −1

with an observation of the fact that σ(η) vanishes identically if and only if the temperature is the linear function of the y-coordinate, i.e., if T = c0 + c1 η. We now realize that in the realistic problems the exact temperature distribution along the cross-sectional area is not known. In Boley’s (1966b) words, ‘an approximate temperature distribution is available with a known bounded error Te ,’ such that −Tm ≤ Te (η) ≤ TM

(4.24)

We are interested in finding the maximum stress that Te (η) can cause. We consider the case 0 < η < 1, without loss of generality. The examination of the integral in Eq. (4.23) reveals   1    ≤ 0 for − 1 < η < −  0  1 3η   if ≤ η < 1 1  3 (4.25) (1 + 3η0 η) = ≥ 0 for − < η0 < 1   3η    1  ≥ 0 for all η if 0 ≤ η ≤ 0 3 Boley (1966b) observes: ‘Now, clearly the largest possible positive stress that can occur is that causes by a temperature which is at each point of the same sign (and equal to its largest positive or negative value, as the case may be) as the quantity (1 + 3η0 η).’ Therefore, the expression for Te in the integrand  1  −Tm for − 1 < η0 < −   3η   1 Te (η0 ) = TM for − < η0 < 1  3η     T for all η M 0

of Eq. (4.23) ought to be    1 if ≤ η < 1  3  (4.26)

1 3 whereas T (η) is set equal to −Tm in the non-integral term. To find the maximum negative stress, one has to interchange all TM and (−Tm ) in Eq. (4.26). The final result turns out to be the same in both cases. Thus,    1 1 1   2 + 3η + , for < η ≤ 1 |σ|max 4 3η 3 = (4.27) 1 αE(Tm + TM )   1, for 0 ≤ η ≤ 3 if 0 ≤ η ≤

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83

with a symmetric result for negative values of η. It represents the maximum value, either positive or negative, which can be expected in the stress. A comparison of Eqs. (4.19) and (4.26) reveals that    1 1 1   2 + 3η + , for < η ≤ 1 3η 3 (4.28) kmax = 4 1   1, for 0 ≤ η ≤ 3 A less accurate inequality for kmax is    3 , for 1 < |η| < 1 3 kmax ≤ 4 (4.29) 1   1, for 0 ≤ |η| ≤ 3 or even 4 kmax ≤ (4.30) 3 The above formulas allow us to estimate the maximum stress that can develop in a beam of rectangular cross-section under any temperature distribution that satisfies the inequality (4.18). If the temperature is a positive interval value, such that 0 ≤ T ≤ Tmax

(4.31)

then 4 αETmax (4.32) 3 Boley (1966b) also derived a more refined bound for the case when the temperature distribution is symmetric or antisymmetric with respect to y. For the symmetric case, T (η) = T (−η), and Eq. (4.23) turns out to be Z 1 σ = −T (η) + T (η0 )dη0 (4.33) αE 0 |σ| ≤

The following formula for the maximum stress is obtained |σ|max ≤ αE(Tm + TM )

(4.34)

kmax ≤ 1

(4.35)

Thus, in this case

instead of the inequality kmax ≤ 4/3 in Eq. (4.30). Clearly, by gaining additional information on the symmetry in distribution of the temperature, a smaller upper bound is derived as expected. The situation is analogous if the information is obtainable on the antisymmetry of the temperature distribution, T (η) = −T (−η). Then the stress becomes Z 1 σ = −T (η) + 3η T (η0 )η0 dη0 (4.36) αE 0

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Instead of Eq. (4.34), we obtain |σ|max ≤ αETM



3 1 + |η| 2



(4.37)

Since in this case Tm = TM , the inequality (4.18) can be rewritten as follows: −TM ≤ T (η) ≤ TM , T (η) = −T (−η)

(4.38)

Boley (1966b) derives the following final expression for this case: |σ|max ≤

5 αETM 2

(4.39)

In order to compare with Eq. (4.27), Eq. (4.39) is rewritten as |σ|max ≤

5 αE(2TM ), T (η) = −T (−η) 4

(4.40)

Therefore, for the antisymmetric case, the inequality for kmax results in kmax ≤

5 4

(4.41)

for which the upper bound is again smaller than its counterpart 4/3 in Eq. (4.30). In the second paper, Boley (1966a) derived the expression for kmax for the general case of the cross-section. For example, for the solid circular cross-section and arbitrary temperature distribution √   1 1 1 11 15 + sin−1 ' 1.428 (4.42) kmax ≤ + 2 16π π 4 For the triangular cross-section one arrives at kmax ≤

27 ' 1.6875, 16

(4.43)

whereas for the thin circular tube kmax

√ 3 2 ' 1.218 ≤ + 3 π

(4.44)

In his paper, Boley (1966b) also derived the expression for maximum axial and transverse deflections. It should also be noted that in earlier papers, Boley (1963, 1964) and Wu and Boley (1965) dealt with the upper and lower bounds for the solution of melting problems. To the best of our knowledge, the pioneering works of Boley remained largely unknown and thus unappreciated despite the fact that they appeared in the prime journals.

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4.3

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85

Anti-Optimization Problem for Static Responses

The general formulations presented in Chap. 3 are readily used for simple static responses such as nodal displacements and member stresses of trusses. Pantelides and Ganzerli (1997) found anti-optimal solutions of trusses considering uncertainty in the direction and magnitude of nodal load vector using ellipsoidal bounds on parameters as 2 N  X ai ≤1 (4.45) gi i=1 where a = (a1 , . . . , aN )> is the vector of uncertain parameters, and the value of the semi-axis gi is defined by the minimum volume method as shown in Sec. 3.4. The static response Z(a) that represents nodal displacement or member stress is linearly estimated from the nominal values a0 = (a01 , . . . , a0N )> with ∆ai = ai − a0i as N X ∂Z(a0 ) Z(a0 + ∆a) = Z(a0 ) + ∆ai (4.46) ∂ai i=1

using the sensitivity analysis in Sec. 3.6.2 or the perturbation approach in Sec. 4.4. The worst parameter values are obtained from the stationary conditions of the Lagrangian of the anti-optimization problem as demonstrated in Sec. 4.1. The elastic responses such as member stress and nodal displacement are often chosen as the objective function to be maximized. The compliance defined as the external work (twice of the strain energy) is also often taken as the global performance measure (see Secs. 9.6–9.8 for examples of anti-optimization for static responses considering stress, compliance and homology design, respectively, which are demonstrated as the lower-level problem of hybrid optimization–anti-optimization). Cherkaev and Cherkaev (2003) defined principal compliance as the maximum value of the compliance against the admissible loading scenarios. Barbieri, Cinquini and Lombardi (1997) used second-order approximation for anti-optimization of trusses. Other studies on static anti-optimization include papers by Alotto, Molfino and Molinari (2001), Attoh-Okine (2002, 2004), Babuˇska, Nobile and Tempone (2005), Banichuk (1975, 1976), Banichuk and Neittaanm¨ aki (2007), Barbieri, Cinquini and Lombardi (1997), Ben-Haim (1992, 1999), Ben-Tal and Nemirovski (1998, 2002), Chen, Lian and Yang (2002), Chen and Ward (1990), Cho and Rhee (2003, 2004), Corliss, Foley and Keafott (2007), Dzhur, Gordeyev and Shimanovsky (2001), Elishakoff, Gana-Shvili and Givoli (1991), Hlav´ aˇcek (2002a, 2007), Hlav´ aˇcek, Chleboun and Babuˇska (2004), Kanno and Takewaki (2006b,c, 2007a,b), K¨ oyl¨ uoˇ glu, C ¸ akmak and Nielsen (1995), Kulpa, Pownuk and Skalna (1998), Liu, Chen and Han (1994), Lombardi (1995, 1998), Lombardi and Haftka (1998), Lombardi, Mariani and Venini (1998), McWilliam (2001), Mullen and Muhanna (1999), Muhanna and Mullen (2001, 2005), Neumaier and Pownuk (2007a), Popova, Datcheva, Iankov and Schanz (2003, 2004), Qiu (2003), Qiu, Chen and Song (1996c), Qiu, Chen and

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Wang (2004a), Qiu, Ma and Wang (2004d), Qiu, Wang and Chen (2006b), and Skalna (2003, 2006). Anti-optimization in the context of comparison of alternative structural models was proposed by Haftka and his co-workers; see, e.g., the paper by Gangadharan, Nikolaidis, Lee and Haftka (2003) and Lee, Haftka, Griffin, Watson and Sensmeier (1994).

4.4

Matrix Perturbation Methods for Static Problems

In this section, we follow Qiu, Chen and Song (1996c) for the perturbation method to solve the static problem in the matrix, or finite element setting KU = P

(4.47)

where K is the stiffness matrix, P is the load vector, and U is the displacement vector. The perturbation theory deals with the behavior of a structure that is subjected to a small perturbation in its variables. The perturbation may occur in the stiffness matrix as K + ε∆K and/or in the load vector, which takes the form P + ε∆P . We are interested in the change of behavior of the system under perturbations. The perturbation terms ε∆K and ε∆P will represent for us the ever-present uncertainties. We suppose that the solution U = K −1 P of the unperturbed structure are known. Note that Eq. (4.47) is solved without computing K −1 ; however, the form U = K −1 P is used for the expression purpose. Note that the sensitivity analysis in Sec. 3.6.2 is equivalent to the first-order perturbation. The solution in the presence of perturbation is denoted as U new . Equation (4.47) is replaced by (K + ε∆K)U new = P + ε∆P

(4.48)

For sufficiently small ε, U new is sought as a series U new = U + ∆U = U + εU (1) + ε2 U (2) + · · · + εn U (n) + · · ·

(4.49)

Equation (4.48) becomes (K + ε∆K)(U + ∆U ) = P + ε∆P

(4.50)

Grouping terms in powers of ε, we obtain the following set of equations: KU (1) = ∆P − ∆KU

(4.51a)

KU (2) = −∆KU (1) .. .

(4.51b)

KU (n) = −∆KU (n−1)

(4.51c)

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We derive the terms U (j) of series (4.49) as follows: U (1) = −K −1 (∆KU − ∆P ) U (2) = −K .. .

−1

(4.52a)

∆KU (1)

(4.52b)

U (n) = −K −1 ∆KU (n−1)

(4.52c)

−1

If the norm of the matrix K ∆K is less than unity, or, more rigorously, if and only if the spectral radius of K −1 ∆K is less than unity, Eq. (4.49) represents a convergent series. This result is considered by Deif (1986, 1991). We now proceed in studying Eq. (4.47) subject to inequalities KL ≤ K ≤ KU, P L ≤ P ≤ P U

(4.53)

or componentwise,

L

L U Kij ≤ Kij ≤ Kij , PjL ≤ Pj ≤ PjU

U

(4.54) L

where P and P are the lower and upper bounds for P , and K is the stiffness matrix composed of the lower bounds of components Kij , whereas K U is its counterpart composed of the upper bounds of components. The uncertainty is represented by Eq. (4.54), indicating that we possess practical information, in the form of bounds alone. Equation (4.53) can be replaced by the statement I

L

U

K ∈ K I, P ∈ P I

(4.55) I

where K = [K , K ] is a real symmetric interval matrix and P = [P , P U ] is an interval vector. Thus, Eq. (4.47) is replaced by KIU = P I

L

(4.56)

Equation (4.55) can be rewritten in the following form: K c − ∆K ≤ K ≤ K c + ∆K, P c − ∆P ≤ P ≤ P c + ∆P

(4.57)

(K c + ∆K I )U = P c + ∆P I

(4.59)

∆K I = [−∆K, ∆K], ∆P I = [−∆P , ∆P ]

(4.60)

where K c and ∆K contain the elements 1 U 1 L U L c + Kij − Kij ), ∆Kij = (Kij ) (4.58) Kij = (Kij 2 2 with similar expressions valid for P c and ∆P . We assume hereafter that every matrix contained in K I is nonsingular. We employ the central interval notation, following Alefeld and Herzberger (1983), Deif (1991), and Moore (1979):

where

It is remarkable that Eq. (4.59) is analogous to Eq. (4.50) in its form. Qiu, Chen and Song (1996c) pose the following question: ‘If each element Kij of K, and each

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component Pi of P , are allowed to exhibit perturbation around mid-points (denoted c by Kij and Pic ) within certain bounds, that is c c Kij − ∆Kij ≤ Kij ≤ Kij + ∆Kij , Pic − ∆Pi ≤ Pi ≤ Pic + ∆Pi

(4.61)

then what would the expected variable in U be?’ The answer to this inquiring is provided by Eqs. (4.52a–c); namely, U I(1) = −(K c )−1 (∆K I U c − ∆P I )

(4.62a)

U I(2) = −(K c )−1 ∆K I U I(1)

(4.62b)

U I(n) = −(K c )−1 ∆K I U I(n−1)

(4.62c)

.. .

Hence, ∆U L = εU L(1) + ε2 U L(2) + · · · + εn U L(n) + · · · ,

2 U n U ∆U U = εU U (1) + ε U (2) + · · · + ε U (n) + · · ·

(4.63)

From Eq. (4.49), we deduce that U I = [U L , U U ] = U c + ∆U I

(4.64)

with the following notations KcU c = P c, L

c

U = U − ∆U , U

(4.65a) U

c

= U + ∆U

(4.65b)

Qiu, Chen and Song (1996c) considered a 5-bar truss as shown in Fig. 4.5 with two pin-supports, i.e., the total number of degrees of freedom is 4. Young’s modulus 2 and the length are fixed at E = 2.1 × 1011 N/m and L = 1.0 m, respectively. The cross-sectional areas of the bars 1–4 are considered to have a deterministic value 1.0 × 10−2 m2 . The cross-sectional areas of diagonal bars 5 and 6 are interval variables AI = [1.0 × 10−2 , 1.02 × 10−2 ] m2 . Likewise, the external load parameter P is treated as an interval variable P I = [2.0998 × 104 , 2.1002 × 104 ] N. Then the interval load vector is given as   [2.0988, 2.1002]  [4.1996, 4.2004]  4  (4.66) PI =  [−3.1403, −3.1497] × 10 [5.2495, 5.2505] where the displacement numbers are indicated in Fig. 4.5. The interval stiffness matrix reads   [4.26, 4.27] [−1.03, −1.01] [−3.5, −3.5] [0, 0] [−1.03, −1.01] [3.97, 4.0] [0, 0] [0, 0]   × 108 KI =  (4.67)  [−3.5, −3.5] [0, 0] [4.26, 4.27] [1.01, 1.03] [0, 0] [0, 0] [1.01, 1.03] [3.97, 3.97]

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2P 4

1 2

P

3

(2)

1.5P

(6)

(3)

0.8 L

(5)

(1)

y x (4) 0.6 L Fig. 4.5

A 5-bar truss.

The interval displacements after a single perturbation turn out to be

L U(1),1 L U(1),2 L U(1),3 L U(1),4

= 0.00087, = 0.00032, = 0.00091, = −0.00033,

U U(1),1 U U(1),2 U U(1),3 U U(1),4

= 0.00093, = 0.00034, = 0.00097, = −0.00031

(4.68)

U L are the lower and upper bounds for the first-order term and U(1),i where U(1),i of Ui , respectively. As is seen, the uncertainty in interval displacements is very U L c )/2 = 0.00090, whereas + U(1),1 = (U(1),1 small. Indeed, the central value U(1),1 U L the uncertainty parameter ∆U(1),1 = (U(1),1 − U(1),1 )/2 equals 0.00003, with the c deviation parameter ∆U(1),1 /U(1),1 being 0.03. Other examples of interval analysis of structures were studied by Chen, Lian and Yang (2002), Dimarogonas (1995), K¨ oyl¨ uoˇ glu, C ¸ akmak and Nielsen (1995), Kulpa, Pownuk and Skalna (1998), McWilliam (2001), Mullen and Muhanna (1999), Neumaier and Pownuk (2007a, 2007b), Qiu, Chen and Song (1996c), Qiu and Gu (1996), Qiu, Chen and Wang (2004a), Qiu, Ma and Wang (2004d), Qiu, Wang and Chen (2006b), Rao and Berke (1997), and Wang and Li (2003). Ellipsoidal modeling was employed by Kanno and Takewaki (2006a), Rao and Majumder (2008), Qiu, Ma and Wang (2006a), Qiu, M¨ uller and Frommer (2001), and Qiu, Wang and Chen (2006b). Comparison of static response estimation by interval and ellipsoidal models was conducted by Qiu (2003).

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4.5

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Stress Concentration at a Nearly Circular Hole with Uncertain Irregularities Introduction

Stress concentrations in the vicinity of holes of various profiles have been treated in a number of publications. For a pertinent bibliography one may consult Neuber (1958), Peterson (1974), and Savin (1961). Traditionally, the stress concentration factors (SCF) around holes of ideal shapes have been considered. However, in practice the manufacturing process of structural parts inevitably produces imperfections which may affect the actual value of the SCF. Irregularities in ideally shaped holes have been modeled using a probabilistic description by Palmov ´ (1964). The profile of the hole as a random function was also treated by Sayles (1982). Lomakin and Sheinin (1970, 1974) obtained stress concentration at a boundary of a randomly inhomogeneous elastic body. Lomakin (1968) developed a double perturbation scheme for a hole with a rapidly oscillating boundary. The boundary was characterized as a random stationary Gaussian function with zero mean and with a specified autocorrelation function. The stress concentration factor κ was determined, as suggested by Palmov ´ (1964), from the condition that σθθ < κ E(σθθ ) holds in the region 0 ≤ θ ≤ 2π with specified probability. Here σθθ is the circumferential stress at the boundary (which turns out to be a random function too), and E(σθθ ) is its mathematical expectation. If one has a complete probabilistic description of the irregularity of the hole boundary, Lomakin’s approach is an attractive method to examine its effect on the stress concentration. Yet in many cases only a limited amount of information on the irregularities may be available. In this section, we present the results in Givoli and Elishakoff (1992) for stress concentration at the boundary of a nearly circular hole in an infinite elastic plane under uniform radial tension at infinity. We consider a realistic situation, where we do not possess the full probabilistic information, but only have a partial knowledge with regard to the shape of the imperfection around the hole boundary. We define an admissible imperfection profile as one which is bounded in some sense, and which may be required to have (depending on the boundedness restriction of the profile) a finite number of harmonics in its Fourier expansion. Then we find the worst-case possible imperfection profile that gives, among all admissible profiles, the maximum SCF. The technique used here to obtain the SCF for a circular hole with arbitrary irregularities is the boundary perturbation method. Some early works employed other methods to treat holes of arbitrary shapes (Ishida and Tagami, 1959; Averin, 1964); however, those methods are less amenable for the anti-optimization performed as a second step. In Sec. 4.5.2, we find a first-order asymptotic approximation of the SCF at the boundary of a nearly circular hole. The results are compared with the exact solution

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Fig. 4.6

91

Unit vectors on the ideal circular boundary B and on the imperfect boundary B  .

of Tunguskova (1970) for the specific geometry of a hypotrochoid. In Sec. 4.5.3, we solve the anti-optimization problem and find the worst admissible imperfection profile and the corresponding SCF. Some results are presented for typical numerical data. 4.5.2

An asymptotic solution

Consider an infinite isotropic elastic plane with a nearly circular hole. We denote the boundary of the hole B  , as shown in Fig. 4.6, to indicate that it is an -perturbation from an ideal circle denoted by B. The domains outside B and B  are denoted by Ω and Ω , respectively. The boundary of the hole B  is assumed to be traction free. We describe this boundary in polar coordinates by r (θ) = R + h(θ)

(4.69)

where R is the radius of the circle B, r  (θ) is the value of the radial distance r of a point on B  from the center of B,  is a small parameter, and h(θ) is a 2πperiodic function. It is assumed that h(θ) possesses a sufficient number of continuous derivatives, according to the requirements of the asymptotic expansion below, and to the fact that  is small with respect to h/R, h0 /R, etc. Here a prime indicates differentiation with respect to θ. At infinity, a uniform radial tensile stress σ∞ is prescribed. Radially non-uniform loading (e.g., uniaxial tension) may also be treated with exactly the same technique as described below. However, the algebraic manipulations become considerably more complicated in this case. The unit vectors er and eθ for B, and n and t for B  , are defined in Fig. 4.6. The problem to find the stress distribution in Ω is stated as σij,j = 0 in Ω

(4.70a)

σrr = σ∞ , σrθ = 0 at r → ∞

Ti = σij nj = 0 on B



(4.70b) (4.70c)

with a standard notation and summation convention. In Eq. (4.70c), nj is the jth component of the unit vector n normal to B  and pointing outside Ω .

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We shall solve the problem (4.70a)–(4.70c) using a perturbation method. A general treatment of perturbation methods for eigenvalue problems with distorted boundaries has been given by Morse and Feshbach (1953). Van Dyke (1975) used the method to solve Laplace’s equation outside a slightly distorted circle with h(θ) = −R sin2 θ in Eq. (4.69). Parnes and Beltzer (1986) used it to solve elasticity problems in circular domains with eccentric loading and in elliptic domains. We shall use a similar method, but with no restriction on the function h(θ) in Eq. (4.69); namely, we shall find an asymptotic solution valid for any small perturbation from the perfect circular hole. We write the stress σij as a perturbation series: (0)

(1)

(2)

σij = σij + σij + 2 σij + · · ·

(4.71)

Substituting σij in Eq. (4.71) into Eqs. (4.70a) and (4.70b), and setting the coefficients of different powers of  equal to zero, we conclude that (k)

σij,j = 0 in Ω , (k = 0, 1, . . . )

(4.72a)

(0)

(4.72b)

(0) σrr = σ∞ , σrθ = 0 at r → ∞ (k) σrr

=

(k) σrθ

=0

at r → ∞, (k = 1, 2, . . . )

(4.72c)

The derivation of asymptotic boundary conditions on B from Eq. (4.70c) is more complicated. First, the tractions in Eq. (4.70c) are defined with respect to the normal n to B  whereas the asymptotic expressions for the tractions on B have to be given with respect to the normal er , as shown in Fig. 4.6. The components nr and nθ in the directions er and eθ , respectively, are found to be (h0 )2 + O(3 ) (4.73a) 2R2 h0 h0 h −nθ = − + 2 2 + O(3 ) (4.73b) R R where O(n ) denotes the term of order n ; see Givoli and Elishakoff (1992) for details. This agrees with Lomakin’s result (Lomakin, 1968). Now Eq. (4.70c) in polar coordinates reads −nr = 1 − 2

Tr = σrr nr + σrθ nθ = 0 Tθ = σθr nr + σθθ nθ = 0

on B 

(4.74)

Using Eqs. (4.71), (4.73a) and (4.73b) in Eq. (4.74), we obtain two equations in(k) (k) volving σrr (r , θ) and σrθ (r , θ). However, these equations still cannot be equated with respect to powers of  to zero since r  is also a function of . Therefore, we (k) (k) first expand σij (r , θ) in a Taylor series around σij (R, θ): 1 (k) (k) (k) (k) σij (r , θ) = σij (R, θ) + hσij,r (R, θ) + 2 h2 σij,rr (R, θ) + · · · (4.75) 2 After substituting Eq. (4.75) in the two equations obtained by Eq. (4.74), we can equate with respect to powers of  to obtain the desired boundary conditions on the

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circle B. The resulting boundary conditions are written as follows up to the second order: (0) σrr (R, θ) = 0 (4.76a) 0 h (0) (0) (1) (4.76b) σrr (R, θ) = σrθ − hσrr,r R h0 (1) 1 (h0 )2 (0) (2) (1) (0) σrr (R, θ) = σrθ − hσrr,r − h2 σrr,rr + σ R 2 2R2 rr hh0 (0) hh0 (0) + σrθ,r − 2 σrθ (4.76c) R R (0) σrθ (R, θ) = 0 (4.76d) 0 h (0) (0) (1) (4.76e) σrθ (R, θ) = σθθ − hσrθ,r R h0 (1) 1 (h0 )2 (0) (1) (0) 2 σrθ (R, θ) = σθθ − hσrθ,r − h2 σrθ,rr + σ R 2 2R2 rθ 0 0 hh (0) hh (0) + σθθ,r − 2 σθθ (4.76f) R R In Eqs. (4.76a)–(4.76f), all the functions on the right-hand-side are evaluated at r = R. Note that Eqs. (4.76a), (4.76d) and (4.76e) can be used to simplify Eqs. (4.76b), (4.76c) and (4.76f) as (1) (0) σrr (R, θ) = −hσrr,r (4.77a)  0 2 1 h (0) (1) (0) (2) σθθ − hσrr,r − h2 σrr,rr (4.77b) σrr (R, θ) = R 2 h0 (1) 1 hh0 (0) hh0 (0) (1) (0) 2 σrθ (R, θ) = σθθ − hσrθ,r − h2 σrθ,rr + σθθ,r − 2 σθθ (4.77c) R 2 R R Equations (4.72a)–(4.72c) and (4.76a)–(4.76f) define a sequence of problems in domain Ω. In what follows we solve the first two problems in this sequence; namely, we find a first-order asymptotic approximation. The zeroth-order problem is axisymmetric and is easily solved as   R2 (0) σrr = σ∞ 1 − 2 (4.78a) r   R2 (0) (4.78b) σθθ = σ∞ 1 + 2 r (0)

σrθ = 0

(4.78c)

Using this solution as well as Eqs. (4.72a), (4.72c), (4.76b) and (4.76e), we conclude (1) that σij should satisfy (1)

(1) = σrθ = 0 at r → ∞ σrr 2σ∞ (1) h(θ) σrr (R, θ) = − R 2σ∞ 0 (1) σrθ (R, θ) = h (θ) R

(4.79a) (4.79b) (4.79c)

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Optimization and Anti-Optimization of Structures under Uncertainty (1)

The solution for σθθ can be written in the form (1) σθθ

∞ Z 2σ∞ X 2π [−Brn (r, θ, θ0 )h(θ0 ) + Bθn (r, θ, θ0 )h0 (θ0 )]dθ = R n=0 0

(4.80)

where Brn and Bθn are the kernels; see Givoli and Elishakoff (1992) for details. Now Eqs. (4.78c) and (4.80) provide us with an approximate solution on the circular boundary B. We use Eqs. (4.71) and (4.75) again to obtain a solution on the actual boundary B  : (0)

(0)

(1)

σij (r , θ) = σij (R, θ) + [h(θ)σij,r (R, θ) + σij (R, θ)] + O(2 )

(4.81)

which leads to an expression for σθθ on B  . However, the SCF is defined via the circumferential stress σtt on B  rather than via σθθ ; see Fig. 4.6 for the difference between t and eθ . To find σtt , we transform the stresses according to the formula σtt = σrr sin2 φ + σθθ cos2 φ − σrθ sin 2φ

(4.82)

where φ is the angle between er and n, and cos φ = nr , sin φ = −nθ . Then, from Eqs. (4.73a), (4.73b), (4.78c) and (4.80)–(4.82), the stress concentration factor κ along B  is obtained as  ∞ Z 2π X σtt (r , θ)  κ ' =1− h(θ) + [Brn (R, θ, θ0 )h(θ) 2 2σ∞ R 0 n=0 (4.83)  n 0 0 2 − Bθ (R, θ, θ )h (θ)]dθ + O( ) which leads to

"

Z 2π ∞ 1X κ = 2 1 − g(θ) + αn cos n(θ − θ)g(θ)dθ π n=0 0 g(θ) = α0 =

h(θ) R

1 , αn = 2n − 1 for n ≥ 1 2

#

(4.84a) (4.84b) (4.84c)

It is easy to verify that if g(θ) is constant, corresponding to an ideal circle, then κ = 2. Moreover, even when g(θ) = cos θ or g(θ) = sin θ, then Eq. (4.84a) yields the constant value κ = 2. This implies that, for these boundary profiles, the effect of the deviation from the ideal circle on the SCF does not manifest itself in the first-order perturbation term. We now compare our result to that of Tunguskova (1970), who presented an exact expression for the SCF around a hole in the shape of a hypotrochoid (Lockwood, 1961). In cartesian coordinates, this curve is defined by the parametric equations x = R(cos t +  cos kt), y = R(sin t −  sin kt)

(4.85)

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Fig. 4.7

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95

A hypotrochoid with k = 7 and ε = 0.01.

where k is a non-negative integer and 0 < k < 1. The polar coordinates on this curve can easily be written as a function of the parameter t, and be expanded in powers of  as sin(k + 1)t + O(2 ) cos2 t r = R + R cos(k + 1)t + O(2 ) tan θ = tan t − 

(4.86a) (4.86b)

One also can see that cos(k + 1)t = cos(k + 1)θ + O(). Comparing Eq. (4.86b) with Eq. (4.69) and using Eq. (4.84b), we obtain g(θ) =  cos(k + 1)θ

(4.87)

Figure 4.7 describes a hypotrochoid with k = 7 and  = 0.01. Tunguskova’s exact stress concentration factor κ∗ around the hole is κ∗ = with a maximum value of

2(1 − 2 k 2 ) 1 − 2k cos(k + 1)t + 2 k 2 max κ∗ = t

2(1 + k) 1 − k

(4.88)

(4.89)

On the other hand, we obtain κ as follows from our asymptotic solution (4.84a) after substituting the function g(θ) given in Eq. (4.87) and using the orthogonality property of the trigonometric functions: κ = 2 [1 + 2k cos(k + 1)θ]

(4.90)

max κ = 2(1 + 2k)

(4.91)

with a maximum value t

Note that Eqs. (4.90) and (4.91) can be obtained from Eqs. (4.88) and (4.89), respectively, by using Taylor series expansion up to the first order in . In Fig. 4.8, we compare the exact solution (4.89) (solid line) and the asymptotic solution (4.90) (dashed line) for k = 7 and  = 0.01. The agreement between the

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Fig. 4.8 A comparison between Tunguskova’s exact SCF (solid line) and the present asymptotic SCF (dashed line) for the hypotrochoid of Fig. 4.7.

Fig. 4.9

Relative error in the maximal asymptotic SCF as a function of εk.

two solutions is excellent. An easy calculation shows that the relative difference between the maximum values of κ∗ and κ is max κ∗ − max κ 2(k)2 = (4.92) max κ∗ 1 + k This relative error as a function of k is plotted in Fig. 4.9. With the choice of k = 7 and  = 0.01, the relative error amounts to only 0.9%. Increasing  to 0.035 we have an error of about 10%. This may raise the question of whether our first-order asymptotic solution (4.84a) is satisfactory. We claim that in many practical situations, it is satisfactory, since k is very small. We are interested in small irregularities, and an irregularity characterized by  = 0.01 is quite usual in practice. As for the value of k, we confine ourselves to the case where k is of order 10 at the most. If k is large, say 100, we may use Lomakin’s asymptotic solution (Lomakin, 1968), which is valid only for this case of a rapidly oscillating boundary profile. In this sense, our solution and that of Lomakin’s are complementary. For the

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review and numerous references therein on asymptotic methods of the boundary’s form, the reader can consult the review by Guz and Nemish (1987). 4.5.3

A worst-case investigation

Now we shall define a set of admissible functions g(θ), namely, a set of admissible hole profiles. Then we shall find the admissible profile that gives the maximum SCF at a given point on the boundary of the hole. It is first assumed that an admissible profile has only a finite number M of important Fourier components in the angle θ, and that all higher harmonics can be neglected. Thus, g(θ) is written as g(θ) =

M X

(Am cos mθ + Bm sin mθ)

(4.93)

m=0

We also assume that the irregularities are bounded in the L2 sense, namely, Z

2π 0

g 2 (θ)dθ ≤ α

(4.94)

where α is a given constant. The set of admissible profiles g(θ) consists of those functions which are of the form Eq. (4.93) and satisfy Eq. (4.94). We find g(θ) which maximizes κ in Eq. (4.84a) at a given angle θ = θ0 , which will turn out later to be arbitrary. Substituting Eq. (4.93) in Eq. (4.84a) and setting θ = θ0 result in I'

M X κ(θ0 ) =1+ 2(m − 1)(Am cos mθ0 + Bm sin mθ0 ) 2 m=1

(4.95)

In addition, Eqs. (4.93) and (4.94) give the inequality constraint 2πA20 + π

M X

m=1

2 (A2m + Bm )≤α

(4.96)

which is written in the form of an equality constraint J ' 2A20 +

M X

m=1

2 (A2m + Bm ) + p2 −

α =0 π

(4.97)

where p is considered as a new unknown. Now the problem is to find the coefficients Am and Bm that will maximize I subject to the constraint J = 0. This problem is easily solved by the Lagrange multiplier method. The unknowns are A0 , Am , Bm (m = 1, 2, . . . , M ), p and the Lagrange multiplier λ. The necessary

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conditions for extremum are ∂(I + λJ) = 2(m − 1) cos mθ0 + 2Am λ = 0, (m = 1, 2, . . . ) (4.98a) ∂Am ∂(I + λJ) = 2(m − 1) sin mθ0 + 2Bm λ = 0, (m = 1, 2, . . . ) (4.98b) ∂Bm ∂(I + λJ) = 2λp = 0 (4.98c) ∂p ∂(I + λJ) = 4λA0 = 0 (4.98d) ∂A0 From Eqs. (4.98a) and (4.98b), we deduce that λ 6= 0, and therefore from Eq. (4.98c) p = 0; namely, the constraint (4.96) is active, and A0 = 0. Also, from Eqs. (4.98a) and (4.98b), 1 (4.99a) Am = − (m − 1) cos mθ0 , (m = 1, 2, . . . ) λ 1 Bm = − (m − 1) sin mθ0 , (m = 1, 2, . . . ) (4.99b) λ Using these expressions in Eq. (4.97) with p = 0, we obtain B π (4.100) λ2 = M α where M X M (M − 1)(2M − 1)  (4.101) BM = (m − 1)2 = 6 m=2 Then, Eqs. (4.99a) and (4.99b) are reduced to

(m − 1) cos mθ0  π/α)1/2 , (m = 1, 2, . . . ) (BM (m − 1) sin mθ0 = , (m = 1, 2, . . . )  π/α)1/2 (BM

Am =

(4.102a)

Bm

(4.102b)

Note that we chose the negative root of λ2 in Eq. (4.100), because in order to obtain a maximum for I, Am and Bm must have the same signs as cos mθ0 and sin mθ0 , respectively. Equations (4.93), (4.101), (4.102a) and (4.102b) define the admissible worst-case profile. The corresponding maximum stress concentration factor κ is found from Eq. (4.95) to be ! r  α BM (4.103) κ(θ0 ) = 2 1 + 2 π As expected, the maximum κ does not depend on the angle θ0 . The SCF around the hole boundary is given by (cf. Eq. (4.95)) " # M X κ(θ) = 2 1 + 2(m − 1)(Am cos mθ + Bm sin mθ) (4.104) m=2

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Fig. 4.10

The worst-case profile g(θ) for the parameters M = 10 and α = 10 −4 .

Fig. 4.11

The worst-case profile g(θ) for the parameters M = 5 and α = 10 −4 .

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It is interesting to note that the average of κ around the boundary of the hole, R 2π (1/2π) 0 κ(θ)dθ, is always 2, i.e., it is equal to the SCF for a circle. Figure 4.10 is the graph of the profile function g(θ) for the parameters M = 10 and α = 10−4 . The angle θ0 was chosen to be π. Any other choice for θ0 would just shift the plot horizontally. The maximum value of g, namely, the maximum value of imperfection divided by R, is 0.015 at θ0 . Figure 4.11 shows the corresponding distribution of κ around the hole boundary. The maximum SCF is 2.38, namely, 19% higher than the SCF associated with the ideal boundary. Figure 4.12 is the graph of g(θ) for the parameters M = 5 and α = 10−4 . Here, the maximum value of g is 0.010, and the maximum corresponding SCF is now 2.12, or 6% larger than that for a circle. It is instructive to compare the SCF obtained by the worst profile with the one obtained by a hypotrochoid, as found by Tunguskova (1970). The maximum stress

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Fig. 4.12

The distribution of the SCF around the worst-case profile.

concentration factor κmax for a hypotrochoid is given in Eq. (4.91) as κmax = 2+4k. h h To compare this with the worst-case stress concentration factor κmax , we set the maximum number of harmonics to M = k +1 (cf. Eq. (4.87)). Also, the appropriate bound α in Eq. (4.94) is calculated from Eq. (4.87) to be Z 2π α= g 2 (θ)dθ = 2 π (4.105) 0

Then, from Eqs. (4.101) and (4.103), we rdeduce that k(k + 1)(2k + 1) (4.106) κmax = 2 + 4 6 max Comparing Eq. (4.91) with Eq. (4.106), it is easy to verify that always κ ≥ κmax , h and that the equality holds only if k = 1; namely, M = 2. This means that among all the holes whose profiles are described by at most two harmonics with the specified norm α, the hypotrochoid is the least maximum one. Substituting the numerical values  = 0.01 and k = 7, as in Fig. 4.7, we get κmax = 2.30 and κmax = 2.47. In h other words, the hypotrochoid and the worst profile yield SCFs which are larger by 15% and 23.5%, respectively, than that for a circle. One may wonder whether it is also possible to find the best-case profile, for which the stress concentration factor κ would be minimal. Since the average of κ around the nearly circular boundary according to our asymptotic analysis is always 2, there is no profile within the set considered, which gives a SCF smaller than 2 at all the points on the boundary. Therefore, the optimal profile in a general sense is a circle. However, an interesting (although maybe academic) question is: What is the smallest κ that one can achieve within the set of admissible profiles (as defined above)? The answer is obtained by repeating the analysis while choosing the positive root of λ2 in Eq. (4.100) instead of the negative root. The minimum κ is found to be ! r  α BM κ(θ0 ) = 2 1 − 2 (4.107) π

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For M = 10 and α = 10−4 , the minimum SCF is 1.62. The corresponding profile is the same as the one in Fig. 4.10, but with an opposite sign. The maximum κ obtained for this profile is 2.30. Up to this point, the set of admissible imperfection profiles g(θ) was defined via the bound constraint (4.94) and the requirement that g(θ) has a finite number of harmonics. The latter requirement has several deficiencies. First, it is a very strong requirement which causes many otherwise reasonable profiles to be excluded. Second, it may be impractical for the designer to determine how many harmonics are needed in the analysis. Third, and perhaps most important, the SCF result obtained above (see Eqs. (4.101) and (4.103)) grows without limit as the number of harmonics M increases. This is expected since Eq. (4.94) does not pose any restriction on the curvature of the profile, and so the worst-case profile tends to a saw-type function as M tends to infinity. To overcome these deficiencies, we now replace Eq. (4.94) by a constraint on the curvature of the profile, i.e., Z 2π 2 (g 00 (θ)) dθ ≤ γ (4.108) 0

where γ is a given positive number. This enables us to go again through the derivation while totally eliminating the requirement that g(θ) has a finite number of harmonics. The consequence of this derivation for the maximum SCF is found to be (cf. Eq. (4.103)) r   µγ κ=2 1+2 (4.109) π where ∞ X (m − 1)2 = 0.3230 . . . (4.110) µ= m4 m=2 Thus, with the constraint of the form Eq. (4.108), we are able to sum the series in Eq. (4.109), whereas the series in Eq. (4.101), which is based on the constraint (4.94), does not converge as M tends to infinity. Both constraints (4.94) and (4.108) can be viewed as special cases of the general constraint Z 2π

0

[α0 g 2 + α1 (g 0 )2 + α2 (g 00 )2 + · · · + αL (g (L) )2 ]dθ ≤ γ

(4.111)

where αi are the given real constants. The maximum SCF in this general case is again given by Eq. (4.109), where the expression (4.110) for µ is replaced by ∞ X (m − 1)2 µ= (4.112) α + α1 m2 + α2 m4 + · · · + αL m2L m=2 0 The sum in Eq. (4.112) converges if and only if at least one of the coefficients α i for i ≥ 2 is nonzero. Thus, in order that no requirement on the number of harmonics of g(θ) would be needed, the second derivative and/or higher derivatives of g(θ) must be constrained. The generalization of this section to the orthotropic case was given by GanaShvili (1991).

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Fig. 4.13

4.6 4.6.1

A tensegrity structure; thick line: strut, thin line: cable.

Anti-Optimization of Prestresses of Tensegrity Structures Introduction

A tensegrity structure consists of cables and struts that can transmit only tensile and compressive forces, respectively, and it is stiffened by prestresses in the selfequilibrium state. Figure 4.13 illustrates a simple example of a tensegrity structure, where thick and thin lines represent struts and cables, respectively. The term tensegrity was invented by Buckminster Fuller as a contraction of tension and integrity, although the inventor of the structural system is under controversy (Fuller, 1975; Lalvani, 1996). By the rigorous definition, a tensegrity structure is free-standing without any support, and the struts are connected by continuous cables and do not have contact with each other (Motro, 1992). Since the tensegrity structure is unstable in the absence of prestresses, the shape and the stability at the self-equilibrium state strongly depend on the member forces (Zhang and Ohsaki 2006; 2007). Therefore, there exist many difficulties in the design of shape and forces of tensegrity structures, and many methods have been presented for this process. Tensegrity structures usually have several modes of self-equilibrium forces that are governed by a set of linear equations with respect to the member forces, and the member forces are defined as a linear combination of the self-equilibrium force modes. Additional requirements exist for the member forces, because the cables and struts can transmit only tensile and compressive forces, respectively. Furthermore, the upper bounds should be given for the cable forces so that the stress remains within the yield stress divided by the safety factor. A bound for compressive stress should also be given for a strut to prevent buckling. Since these bound constraints are expressed as linear inequalities with respect to the coefficients of the self-equilibrium force modes, the feasible region of the coefficients forms a convex region. The performance of the force design defined by the feasible or admissible set of coefficients should be determined based on the mechanical properties such as the eigenvalues of the tangent stiffness matrix and the displacements against external loads. However, the member forces are very sensitive to the imperfection or variation

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of the geometrical parameters such as the unstressed lengths of members and nodal locations. Therefore, the anti-optimal solution of self-equilibrium forces is defined as the member forces that minimize the performance measures under bound constraints of the forces. If those performance measures have monotonicity and/or convexity properties with respect to the member forces, the anti-optimal solutions can be found by searching the vertices of the feasible region of the member forces. In this section, we present an anti-optimization method of the forces of tensegrity structures (Ohsaki, Zhang and Elishakoff, 2008a). The lowest eigenvalue of the tangent stiffness matrix is considered as the performance measure (Ohsaki, Zhang and Ohishi, 2008b). The properties of the objective functions are investigated, and the anti-optimal solution is found by the enumeration of the vertices of the convex feasible region. 4.6.2

Basic equations

4.6.2.1 Equilibrium equations The members of the tensegrity structures consist of cables and struts connected by pin joints. The self-weight is neglected, and no external load exists at the initial self-equilibrium state. Let m and ns denote the numbers of members and nodes, respectively. The m × ns connectivity matrix is denoted by C s , and the (k, p)-component of C s is s denoted by Ck,p . If member k is connected by nodes i and j (i < j), the kth row of s C is defined as  for p = i 1 s Ck,p = −1 for p = j (4.113)  0 for other cases

The nodes are classified into n free nodes and nf fixed nodes (supports); i.e., n = n + nf . Suppose the nodes are numbered such that the free nodes precede the fixed nodes. Then C s is divided into the m × n matrix C and the m × nf matrix C f corresponding to the free and fixed nodes, respectively, as s

C s = (C, C f )

(4.114)

Let n-vectors x, y, z and nf -vectors xf , y f , z f denote the nodal coordinate vectors of free and fixed nodes in x-, y- and z-directions, respectively. We assume a tensegrity structure in the three-dimensional space for the presentation of general formulations. The coordinate difference vectors of the members are denoted by h x , hy and hz for x-, y- and z-directions, respectively, which are calculated by hx = Cx + C f xf hy = Cy + C f y f z

f f

h = Cz + C z

(4.115)

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and the m × m coordinate difference matrices H x , H y and H z are defined as H x = diag(hx ) H y = diag(hy )

(4.116)

z

z

H = diag(h ) In the following, the components of vectors and matrices are indicated by subscripts x as hx = (hxi ), H x = (Hij ). The length matrix L = (Lij ) is diagonal, and its ith diagonal term is given as q (4.117) Lii = (Hiix )2 + (Hiiy )2 + (Hiiz )2 , (i = 1, . . . , m) Let s = (s1 , . . . , sm )> denote the vector of member forces. In the state of selfequilibrium, the equilibrium equation of a pin-jointed structure in three-dimensional space is written as (Schek, 1974) Ds = 0

(4.118)

where the 3n × m equilibrium matrix D is defined by  C>Hx D = C > H y  L−1 C>H z 

(4.119)

4.6.2.2 Self-equilibrium forces Let r denote the rank of D. Then the equilibrium equation (4.118) has q = m − r set of self-equilibrium force modes, which are found, as follows, by the singular value decomposition of D. The nonzero eigenvalues of D > D are denoted by ωi (i = 1, . . . , r). Then the singular value decomposition of D is written as

where Ω=



D = SΩR>

(4.120)

 diag(ω1 , . . . , ωr ) O O O

(4.121)

and the diagonal terms of the 3n × m rectangular matrix Ω are called singular values of D. The m × m matrix R and the 3n × 3n matrix S satisfy the following orthogonality conditions: R> R = RR> = I m S > S = SS > = I 3n

(4.122)

where I m and I 3n are the m × m and 3n × 3n identity matrices, respectively. By postmultiplying R to Eq. (4.120) and using Eq. (4.122), we obtain DR = SΩ

(4.123)

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The column vectors Ri (i = r +1, . . . , m) of R corresponding to zero singular values satisfy the condition of self-equilibrium force mode as DRi = 0

(4.124)

By denoting g i = Ri+r (i = 1, . . . , q), the self-equilibrium force vector s satisfying Ds = 0 is given as the linear combination of g i as s = α1 g 1 + · · · + αq g q = Gα

(4.125)

where α = (α1 , . . . , αq )> is the coefficient vector, and G = (g 1 , . . . , g q ) is the matrix of the self-equilibrium force modes. Let b> i denote the ith row of G. The components of s are written as si = b > i α, (i = 1, . . . , m)

(4.126)

The self-equilibrium equation with respect to member forces s in Eq. (4.118) can be rewritten in the following form with respect to the nodal coordinates (Zhang and Ohsaki, 2006): Ex + E f xf = 0 Ey + E f y f = 0

(4.127)

f f

Ez + E z = 0 where E is called a force density matrix that is given as E = C > diag(L−1 s)C E f = C > diag(L−1 s)C f

(4.128)

4.6.2.3 Tangent stiffness matrix The tangent stiffness matrix K is expressed as the sum of the linear stiffness matrix K E and the geometrical stiffness matrix K G as K = KE + KG

(4.129)

K E = DKD > , K G = I 3 ⊗ E

(4.130)

with

where I 3 is the 3 × 3 identity matrix, and the ith diagonal component K ii of the diagonal matrix K is the stiffness of the ith member; i.e., K ii = Ai Ei /Lii where Ai , Ei and Lii are the cross-sectional area, Young’s modulus and the length of member i, respectively.

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4.6.2.4 Lowest eigenvalue of tangent stiffness matrix In the following discussions on stiffness, the rigid-body motions are assumed to be constrained, i.e., the structure has 3n-degree-of-freedom with n free nodes. Let λr (λ1 ≤ λ2 ≤ · · · ≤ λ3n ) and Φr denote the rth eigenvalue and eigenvector of K, respectively, which are defined by KΦr = λr Φr , (r = 1, . . . , 3n)

(4.131)

The eigenvector Φr is ortho-normalized by Φ> r Φs = δrs , (r, s = 1, . . . , 3n)

(4.132)

where δrs is the Kronecker delta. The lowest eigenvalue λ1 after constraining the rigid-body motions is minimized as the performance measure in the antioptimization problem defined in Sec. 4.6.3. By differentiation of Eq. (4.131) for r = 1 with respect to αi , and using Eq. (4.132), we obtain ∂K ∂λ1 = Φ> Φ1 (4.133) 1 ∂αi ∂αi The details of eigenvalue sensitivity analysis can be found in Sec. 3.6.3. Incorporation of Eq. (4.129) into Eq. (4.133) leads to ∂(K E + K G ) ∂λ1 = Φ> Φ1 (4.134) 1 ∂αi ∂αi Since K E is independent of αi , we obtain ∂λ1 ∂K G = Φ> Φ1 (4.135) 1 ∂αi ∂αi Furthermore, since K G is a linear function of α, as is seen from Eqs. (4.127) and (4.130), ∂K G /∂αi is a constant matrix. Therefore, if Φ1 is almost constant in the feasible region, then λ1 is a monotonic function of α. Concavity of λ1 with respect to α at a self-equilibrium state can be shown in a similar manner as the proof for the lowest eigenvalue of tangent stiffness matrix; cf. Sec. 5.6. 4.6.2.5 Compliance against external load Suppose small external loads P = (P1 , . . . , P3n )> are applied to the structure. The nodal displacements U = (U1 , . . . , U3n )> are linearly estimated by the tangent stiffness matrix as KU = P

(4.136)

In the field of structural optimization, the external work, which is called compliance is often used as the performance measure. The compliance W , defined as follows, is to be minimized to obtain the stiffest design against the specified loads: W = U >P = U > KU

(4.137)

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which is maximized in the anti-optimization problem in Sec. 4.6.3. By differentiating Eq. (4.136) with respect to αi , the sensitivity coefficient of U with respect to αi is written as ∂U = −K −1 K i U ∂αi

(4.138)

The details of sensitivity analysis of static responses can be found in Sec. 3.6.2. Using the form W = U > KU in Eq. (4.137), the sensitivity coefficient of the compliance is given as ∂W ∂K ∂U = U> U + 2U > K ∂αi ∂αi ∂αi ∂K U = −U > ∂αi Since K E is independent of αi , we obtain ∂K G ∂W = −U > U ∂αi ∂αi

(4.139)

(4.140)

Therefore, if U is almost constant in the feasible region, then W is a monotonic function of α. Convexity of compliance can be proved in several manner (Prager, 1972; Svanberg, 1984, 1994). Stolpe and Svanberg (2001) presented the following simple proof. Let G(α, U ) be defined as

Then, for every α,

  1 G(α, U ) = 2 P > U − U > K(α)U 2

(4.141)

W (α) = max G(α, U )

(4.142)

U

Suppose the extensional stiffness of member i is a concave function (including linear function) of α, which is valid for the tensegrity structures. Then, for each fixed U , G(α, U ) is convex in α. Since a function defined as a pointwise maximum of a collection of convex functions is convex (Boyd and Vandenberghe, 2004), W (α) is a convex function. 4.6.3

Anti-optimization problem

Consider uncertainty in member forces at the self-equilibrium state due to the errors in unstressed (initial) lengths of members or change in member lengths after construction resulting from relaxation of high-tensioned cables. Since the member forces s should satisfy the self-equilibrium equation (4.118), the errors of s cannot be distributed independently. Note that the change in member forces leads to change in nodal locations, which is assumed to be very small in the following; i.e., the equilibrium matrix D is fixed. Therefore, the vectors g i of the self-equilibrium

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force modes are fixed, and the variation of the member forces is investigated in the space of the coefficients α. We first design the nominal value s0 = Gα0 of the self-equilibrium forces. The lower and upper bounds sL and sU , respectively, are appropriately given for s0 to assign the constraints as sL ≤ s 0 ≤ s U (4.143) where the signs of the forces for cables and struts are to be considered; i.e., the lower bounds for the cables and the upper bounds for the struts are 0. The vertices of the feasible region satisfying the constraint (4.143) are enumerated in the space of α. Denote by α0 the mean value of α at all the vertices called the center of the feasible region (Ohsaki, Zhang and Ohishi, 2008b), for brevity, which is conceived as the nominal value of the uncertain coefficient vector α for the self-equilibrium force modes. Note that the nominal value need not be the center of the feasible region, and can be defined arbitrarily, e.g., based on the preference of the designer. Next we assign the range of uncertainty of s as an interval s0 − ∆s ≤ s ≤ s0 + ∆s (4.144) where ∆si is the maximum possible increase or decrease of si . Note that the range is given for the force vector s, although the independent parameters for the forces are α. In the following anti-optimization problem, the deviation of the forces from the desired values is chosen as one of the performance measures. Then the deviation e of α from the desired value α0 is defined as e = (α − α0 )> G> G(α − α0 ) (4.145) which is a convex function of α. We consider the following three performance measures so that the antioptimization problem is formulated as minimization of a concave function, or maximization of a convex function, and the anti-optimal solution exists at a vertex of the feasible region defined by linear inequalities: (1) Lowest eigenvalue λ1 of K after constraining the rigid-body motions, which is a concave function of α, and is to be minimized in the anti-optimization problem. (2) External work called compliance against the specified load after constraining the rigid-body motions, which is a convex function of α, and is to be maximized in the anti-optimization problem. (3) Deviation e of the forces from the desired value, which is a convex function of α, and is to be maximized in the anti-optimization problem. For example, the anti-optimization problem of minimizing λ1 is formulated as minimize λ1 (α) (4.146a) subject to s0 − ∆s ≤ s ≤ s0 + ∆s s = Gα

(4.146b) (4.146c)

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Other problems for maximizing W and e, respectively, are formulated similarly. Then the anti-optimal solution is found by vertex enumeration of the feasible region as follows: Step 1 Specify the geometry, topology, and material property of the tensegrity structure. Step 2 Construct the equilibrium matrix D from Eq. (4.119) and compute the selfequilibrium force modes g 1 , . . . , g q by the singular value decomposition (4.123). Step 3 Assign bound constraints (4.143) for the nominal value of the selfequilibrium forces. Step 4 Generate the list of the vertices of the feasible region of α satisfying (4.143). Step 5 Find the center α0 of the feasible region that is regarded as the nominal value of α. Step 6 Specify ∆s, and generate the list of the vertices of the feasible region of α for the anti-optimization problem satisfying (4.144). Step 7 Assign the support conditions, and compute the tangent stiffness matrix K from Eq. (4.129) at each vertex. Step 8 Compute the lowest eigenvalue λ1 from Eq. (4.131), the compliance W from Eq. (4.137), and the force deviation e from Eq. (4.145) at each vertex of the feasible region satisfying Eq. (4.144), then find the three vertices, each of which corresponds to the anti-optimal solution that minimizes λ1 , maximizes W , or maximizes e. In the following numerical example, merated by cdd+ (Fukuda, 1999; Avis procedure called reverse search. cdd+ ciated active constraints of the region constraints. 4.6.4

the vertices of the feasible region are enuand Fukuda, 1996) based on the efficient can enumerate the vertices and the assodefined by linear inequality and equality

Numerical examples

Consider a tensegrity grid as shown in Fig. 4.14 that consists of the unit cell as shown in Fig. 4.15, where the thick and thin lines are struts and cables, respectively, and the node numbers are shown in Fig. 4.14(a). This grid was first presented by Motro (2003). The tensegrity grid is constructed by consecutively assembling the unit cell in x- and y-directions. Let t and c denote the numbers of rows (x-direction) and columns (y-direction) of the struts, respectively; i.e., there are t + 1 struts in each column and c + 1 struts in each row. Therefore, the structure has 2tc + t + c struts and ns = 2(tc + t + c) nodes. The total number of members is m = 7tc + 5t + 5c − 4. The rank of K E after constraining rigid-body motions is 1; i.e., this kind of structure has only one infinitesimal mechanism irrespective of the values of t and c. The structure in Fig. 4.14 has three and four struts in x- and y-directions,

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22 5

10

400

9

20

18

17 21 3

4 19

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15

300 14

y

7

8 12

100

13 6

1

2

200

0

0

100

200

300

x

11

(a) diagonal view

(b) plan view Fig. 4.14

Fig. 4.15

Tensegrity grid.

Unit cell of the tensegrity grid.

respectively; i.e., t = 3 and c = 2. There are m = 63 members and ns = 22 nodes in total. The x- and y-coordinates (mm) of the nodes are shown in Fig. 4.14, and the height of the grid is 100 mm. The axial forces of the horizontal members are equal to 0 at the self-equilibrium state. Hence, these members are called bars. 2 Young’s modulus is E = 200 × 103 N/mm for all members. The cross-sectional 2 areas are 50 mm for struts and bars, and 5 mm2 for cables. In the following, the units of length and force are mm and N, respectively, which are omitted for brevity. The rank r of the equilibrium matrix D is 59. Therefore, the structure has q = 63 − 59 = 4 self-equilibrium force modes denoted by g 1 , . . . , g 4 with the coefficients α = (α1 , . . . , α4 )> . The lower bound sLi and the upper bound sU i for the axial forces of the struts and cables, respectively, are −10000 and 10000. Therefore, the maximum absolute values of the strains are 0.001 for the struts and 0.01 for the cables in the process of designing the nominal values of the self-equilibrium forces. The vertices of the feasible region have been enumerated by cdd+ (Fukuda, 1999; Avis and Fukuda,

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111

7

Eigenvalue

6.8 6.6 6.4 6.2 6 0

20

40

60

80

100

120

140

Vertex number Fig. 4.16 Lowest eigenvalue of the tangent stiffness matrix of each solution corresponding to the vertex of the feasible region.

5.92

Compliance

5.915 5.91 5.905 5.9 0

20

40

60

80

100

120

140

Vertex number Fig. 4.17 Compliance against the specified load of each solution corresponding to the vertex of the feasible region.

1996) to find 74 vertices. The mean values are calculated for the coefficients α of all the vertices to obtain the nominal value α0 = (−0.4198, −0.5832, −3.2567, 0.4294)>. The maximum and minimum forces of the cables of the nominal self-equilibrium forces are 8507.8 and 1359.6, respectively, whereas those for the struts are −3847.6 and −8290.3, respectively. The value of λ1 at the center is 6.4812. The lowest eigenvalue λ1 of K is positive at all the vertices after constraining the rigid-body motions; i.e., the structure is stable at any set of self-equilibrium forces in the feasible region for designing the nominal values. The worst-case scenario under constraint (4.144) is next investigated. Let ∆si = 500 for all members. Figure 4.16 shows the values of λ1 at the 138 vertices of the

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2600

Force deviation

2400 2200 2000 1800 1600 0

20

40

60

80

100

120

140

Vertex number Fig. 4.18

Force deviation of each solution corresponding to the vertex of the feasible region.

region defined by Eq. (4.144). The worst value of λ1 is 6.0948, which is about 94% of the nominal value 6.4812. The compliance W under static loads is next investigated. The vertical loads of 10 are given asymmetrically in the negative direction of z-axis at nodes 3, 12 and 20 indicated in Fig. 4.14(a). To exclude the rigid-body motions of the structure, the displacements are constrained at nodes 11 and 19 in x-direction, at nodes 6 and 11 in y-direction, and at nodes 6, 10, 11, 18 and 19 in z-direction. Note that the loads are assumed to be sufficiently small to investigate the stiffness against small disturbance by linear analysis. The values of compliances at the vertices of the feasible region are plotted in Fig. 4.17. As is seen, the compliance does not strongly depend on the forces. Finally, the deviation e of forces from the nominal value is plotted in Fig. 4.18. As is seen, the deviation is widely distributed in the region of uncertainty; however, the distribution depends on the definition of ∆s. Although we have considered only three performance measures, the method based on vertex enumeration is very effective for the case where many performance measures are to be evaluated in the process of force design. Here it is worthwhile to mention that some investigators have conducted studies on determining the worst possible stochastic responses due to incompletely known probabilistic characteristics of the input quantities. These works include those of Abbas and Manohar (2005, 2007), Chen, Lian and Yang (2002), Deodatis, Graham and Micaletti (2003a, 2003b), Deodatis and Shinozuka (1989, 1991), Iyengar (1970), Iyengar and Dash (1976), Iyengar and Manohar (1987), Papadopoulos, Deodatis and Papadrakakis (2005), Sarkar and Manohar (1998), Shinozuka (1987), and Wall and Deodatis (1994). The references mentioned in Sec. 6.6.4, pertaining to earthquake engineering, also fall into this category.

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Chapter 5

Anti-Optimization in Buckling

‘Uncertainties and optimization are two major considerations on structural design.’ (Royset, Der Kiureghian and Polak, 2003) ‘Any intelligent fool can make things bigger and more complex... It takes a touch of genius – and a lot of courage – to move in the opposite direction.’ (Albert Einstein)

In this chapter, we first deal with a simple buckling anti-optimization problem. We next treat the anti-optimization problem by minimizing the buckling load. The notion of worst imperfection is introduced for a braced frame with attendant multiple buckling loads, including the problem of mode interaction. The worst-case design and worst imperfection are treated via anti-optimization under stress constraints. We show how to utilize the convexity of the stability region. Finally, we end the discussion with worst imperfection of an arch-type truss associated with the multiple member buckling at limit point. 5.1

Introduction

It is well known that the buckling loads of thin-walled structures such as plates and cylindrical shells are drastically reduced in the presence of initial geometrical imperfection, and the magnitude of reduction of the buckling load strongly depends on the shape of the imperfection. Therefore, extensive research has been carried out for anti-optimization for finding the worst imperfection, which minimizes the buckling load under constraint on the norm of the imperfection. In the general theory of stability of elastic conservative systems, the buckling is characterized as a critical point of equilibrium state, which is classified to limit point and bifurcation point. The details of the classification of critical points may be consulted in Thompson and Hunt (1973, 1984). Worst imperfections have been found for various types of critical points including unstable-symmetric bifurcation point (Ikeda and Murota, 1990), stable-symmetric bifurcation point (Ohsaki, 113

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P

K1 θ

L K2 L2

Fig. 5.1

A bar–spring structure.

2002b), multiple bifurcation points (Ho, 1974; Peek and Triantafyllidis, 1992), and hilltop branching point (Ikeda, Oide and Terada 2002; Ohsaki and Ikeda 2007a, 2007b). Since the buckling loads generally coincide as a result of optimization against buckling (Masur, 1984; Ohsaki, 2000; Olhoff and Rasmussen, 1977), it is very important to investigate worst imperfection for coincident critical points. In the course of finding the worst imperfection, an anti-optimization problem is formulated and solved explicitly or numerically. For the structures where the prebuckling deformation is negligible for the evaluation of the buckling load, antioptimization problems can be formulated in a simple form of linear buckling analysis (de Faria and de Almeida, 2003a). Lindberg (1992a, 1992b) investigated the problem with multimodal dynamic buckling with convex bounds in the Fourier coefficients of the imperfection. Pantelides (1996a) investigated worst imperfection in geometrical and material properties for stiffening elements using the ellipsoidal bounds. Takagi and Ohsaki (2004) defined worst imperfection using the eigenvalues of the tangent stiffness matrix. El Damatty and Nassef (2001) found worst imperfection of a ribbed shell by using a genetic algorithm. 5.2

A Simple Example

Consider a rigid vertical bar with length L as shown in Fig. 5.1 supported by a horizontal spring with extensional stiffness K1 and a rotational spring with stiffness K2 L2 . The bar is subjected to a vertical load P , and the rotation of the bar is denoted by θ. The dotted and solid lines in Fig. 5.1 are the shapes before and after deformation, respectively. The total potential energy Π is written as Π=

1 1 K1 (L sin θ)2 + K2 L2 θ2 − P L(1 − cos θ) 2 2

(5.1)

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The stationary condition of Π with respect to θ leads to the equilibrium equation 1 K1 L2 sin 2θ + K2 L2 θ − P L sin θ = 0 (5.2) 2 which is satisfied along the fundamental path θ = 0. For θ 6= 0, we have from Eq. (5.2) the equation for the bifurcation path θ (5.3) sin θ Further differentiation of Π with respect to θ gives the criticality condition P = K1 L cos θ + K2 L

K1 L2 cos 2θ + K2 L2 − P L cos θ = 0

(5.4)

c

from which the buckling (critical) load P , which is also called the bifurcation load in this case, is obtained as P c = (K1 + K2 )L

(5.5)

The bifurcation load P c can be also obtained by taking the limit θ → 0 for Eq. (5.3) as the intersection between the fundamental path and the bifurcation path. Uncertainty is introduced in the stiffness as K1 = K10 + a1 , K2 = K20 + a2

(5.6)

where K10 and K20 are the nominal values and a1 and a2 are the uncertain parameters. Consider a simple case where the bound of a = (a1 , a2 )> is given by the quadratic constraint as a special case of an ellipsoidal bound as a> a ≤ D

(5.7)

where D is the upper bound for the square of the magnitude of the uncertainty. Then the buckling load is a function of a, and the anti-optimization problem is formulated as minimize

P c (a) >

subject to a a ≤ D

(5.8a) (5.8b)

c

It is seen from Eqs. (5.5) and (5.6) that P is a linear function of a1 and a2 . Therefore, the inequality constraint (5.8b) is satisfied by equality at the anti-optimal solution, and the solution to Problem (5.8) is found from the stationary condition of the Lagrangian as (cf. Sec. 2.4.2) 1 a 1 = a 2 = − L2 2µ r (5.9) 1 2L4 ∗ µ=µ = 2 D where µ is the Lagrange multiplier. Note that µ = −µ∗ corresponds to the optimal solution that maximizes P c (a). For the case where L = 1 and D = 1, the antioptimal solution is computed as a1 = a2 = −0.7071 with µ = µ∗ = 0.7071.

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Consider the case where the bounds are given for a by interval as aL ≤ a ≤ a U

(5.10)

with the lower bound aL and upper bound aU . Since P c is a linear function of a1 and a2 , as observed from Eqs. (5.5) and (5.6), the anti-optimization problem turns out to be a linear programming problem minimize

P c (a) L

(5.11a) U

subject to a ≤ a ≤ a

(5.11b)

The anti-optimal solution exists at a vertex of the feasible region, and the obvious anti-optimal solution is found as (a1 , a2 ) = (aL1 , aL2 ), because P c (a) is a monotonically increasing function of a. 5.3

Buckling Analysis

Consider an elastic conservative structure subjected to quasistatic proportional nodal loads P parameterized by the load factor Λ and the constant vector p of load pattern as P = Λp; i.e., the structure has potential energy and the direction of the dead load P is fixed. Let K denote the tangent stiffness matrix (stability matrix) at an equilibrium state undergoing large deformation. The eigenvalue problem of K is formulated as KΦi = λi Φi , (i = 1, . . . , n)

(5.12)

where λi is the ith eigenvalue numbered in nondecreasing order as λ1 ≤ λ2 ≤ · · · ≤ λn , Φi is the eigenvector associated with λi , and n is the number of degrees of freedom. The eigenvector is ortho-normalized as Φ> i Φj = δij , (i, j = 1, . . . , n)

(5.13)

where δij is the Kronecker delta. The buckling load factor Λc is defined as the smallest positive load factor satisfying the criticality condition λ1 = 0, as KΦc1 = 0 Φc1

(5.14)

where is called the buckling mode. There are many approaches for finding the buckling load of a structure undergoing large deformation (Riks, 1998). However, in this book, the details of buckling analysis are omitted, and we assume that the buckling load can be found numerically within necessary accuracy. Consider a structure for which the deformation before buckling, called the prebuckling deformation, is sufficiently small; e.g., columns under axial compression and building frames under vertical loads. In this case, the axial forces or stresses in each member or element of the structure is almost proportional to Λ, and K is expressed by the sum of the linear stiffness matrix K E that is independent of Λ and the geometrical stiffness matrix K G = ΛK G0 that is proportional to Λ, where K G0

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is a constant matrix defined by p. Hence, the linear buckling load factor Λ b , which is often called simply the linear buckling load, is found as the smallest positive value of Λ satisfying (K E + Λb K G0 )Φb = 0

(5.15)

where Φb is the linear buckling mode. Note that the linear eigenvalue problem (5.15) generally has negative buckling load factors, because the structure may buckle against the loads applied to the reverse direction and the symmetric matrix K G0 is indefinite. However, the direction of the load is generally fixed, and we can assume that the buckling load factor takes a positive value. The ith positive smallest value of Λb is called the ith linear buckling load factor; however, we are interested in the lowest positive buckling load only. 5.4

Anti-Optimization Problem

The buckling load factor represents the safety against static loads when plastic failure is not considered. Therefore, in the following, the linear buckling load factor is minimized in an anti-optimization problem. As demonstrated with the simple example in Sec. 5.2, the anti-optimization problem can be formulated with either the ellipsoidal or linear bound of the uncertain parameter vector a representing variability of load, shape, material property, etc. For the ellipsoidal bound with weight matrix W , the anti-optimization problem is formulated as minimize

Λb (a) >

subject to a W a ≤ 1

(5.16a) (5.16b)

The details of the ellipsoidal model can be found in Sec. 3.4. The anti-optimal solution to Problem (5.16) can be obtained explicitly if the buckling load is linearized using sensitivity coefficients. Sensitivity analysis of the eigenvalue of vibration has been presented in Sec. 3.6.3 as the simple case of the eigenvalue sensitivity. For linear buckling load, differentiation of Eq. (5.15) with respect to ai leads to ∂Φb ∂Λb ∂K G0 b ∂Φb ∂K E b Φ + KE + K G0 Φb + Λb Φ + Λb K G0 =0 ∂ai ∂ai ∂ai ∂ai ∂ai

(5.17)

Since K E is positive definite and the buckling load Λb is positive, Φb is normalized in view of Eq. (5.15) as Φb> K G0 Φb = −1

(5.18)

Premultiplying Φb> to Eq. (5.17) and using Eqs. (5.15) and (5.18), we obtain the sensitivity coefficient   ∂Λb ∂K E ∂K G0 = Φb> + Λb Φb (5.19) ∂ai ∂ai ∂ai

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It is seen by comparing Eq. (5.19) with Eq. (3.96) in Sec. 3.6.3 that the sensitivity coefficients of the buckling loads can be computed in a similar manner as those of the eigenvalues of free vibration. However, the geometrical stiffness matrix K G0 for the unit load factor corresponding to the load pattern vector p implicitly depends on a i through the axial forces or stresses against p. Therefore, sensitivity analysis of the responses against the static load p is needed to compute the sensitivity coefficients of the linear buckling load (see Sec. 3.6.2 for sensitivity analysis of static responses). When the bounds of the uncertain parameters are given by interval, and Λb (a) is linearized with respect to a, the anti-optimization problem turns out to be a linear programming problem. For the case without linearization of Λb (a) and with linear constraints for a, the anti-optimal solution can be found using vertex enumeration of the feasible region as shown in Sec. 5.6.

5.5

5.5.1

Worst Imperfection of Braced Frame with Multiple Buckling Loads Definition of frame model

Imperfection sensitivity of multiple buckling loads has been extensively investigated to demonstrate that imperfection sensitivity increases as the result of the coincidence of buckling loads (Thompson and Hunt, 1984). However, Ohsaki (2002a) showed that optimization of a braced frame that results in multiple buckling loads does not lead to any enhancement of imperfection sensitivity. In this section, the worst imperfection is investigated for a braced frame with multiple (coincident) linear buckling loads. Consider a 3-story 3-span braced plane frame as shown in Fig. 5.2 subjected to nodal proportional loads ΛP in vertical direction at all the nodes connecting the beams and columns, where P = 1.0 × 104 N. In the following, the units mm and N of the length and force, respectively, are omitted for brevity. The braces are modeled by the truss elements. The intersecting pair of braces are not connected at the centers, and Euler buckling of the braces is ignored. Sandwich cross-sections are assumed for columns and beams; i.e., the second moment of inertia I and the section modulus Z are functions of the cross-sectional area A as I = r2 A, Z = rA

(5.20)

where r is the distance between the flange and the center axis of the member. In the following examples, the value of r is 80.0 for the interior columns in the first story, and 60.0 for the remaining beams and columns. Young’s modulus is 2.0 × 105 for all members. The standard assumption of rigid floor is used, i.e., the nodes in the same floor has the same horizontal displacement. Each column is divided into two beamcolumn elements, and the Green strain is used for the strain-displacement relation. Since the prebuckling deformation of a frame subjected to vertical loads is very

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y

ΛP

ΛP

ΛP

ΛP

5000

9

10

11

12

5000

5

6

7

8

6000

1

2

3

4

x

4000

Fig. 5.2

4000

4000

A 3-story 3-span braced plane frame.

small, the buckling load is evaluated by linear buckling analysis, whereas the buckling load of an imperfect structure is evaluated by nonlinear path-tracing analysis, for which the displacement increment approach is used. The effect of yielding is not considered here to investigate the elastic properties of the frame, thereby simplifying the anti-optimization process. Let σi (Λ) denote the maximum absolute value of the stress of the flange at the two ends of the ith beam-column element. The maximum value of σi (Λ) throughout the frame is denoted by σ m (Λ) as σ m (Λ) = max σi (Λ) i

(5.21)

The maximum load factor Λm (Λ) is defined, as follows, so that σ m (Λ) reaches the specified upper bound σ m : Λm = max{Λ | σ m (Λ) ≤ σ m }

(5.22)

The purpose of this section is summarized as follows: (i) The worst imperfection is investigated for a braced frame, where the norm of the imperfection is defined by the absolute value of the inclination of each story and the out-of-straightness of each column; i.e., these geometrical imperfections are regarded as the uncertain parameters for formulating the anti-optimization problem. The maximum load Λm is defined by the stresses as Eqs. (5.21) and (5.22) at the two ends of the elements. (ii) The worst imperfection mode of the nodal locations found by geometrically nonlinear path-tracing analysis is compared with the buckling mode, and the effect of modal interaction on imperfection sensitivity is evaluated using the

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Cross-sectional areas of the optimal solution.

External column Internal column External beam Internal beam External brace Internal brace

1st story

2nd story

3rd story

5838.64 6139.51 1500.00 1500.00 44.42 22.56

4886.90 4930.91 1500.00 3102.45 0.0 76.69

1999.45 2060.32 1500.00 1500.00 8.59 5.30

third- and fourth-order differential coefficients of the total potential energy with respect to the generalized displacements in the directions of buckling modes. (iii) The worst-case design and the corresponding worst imperfection that maximize the increase of the maximum stress are found by linear analysis under specified loads, and compared with the worst imperfection obtained by the nonlinear path-tracing analysis.

5.5.2

Worst imperfection of optimized frame

The optimal braced frame for minimizing the total structural volume is first found under constraint on the linear buckling load factor with the lower bound Λ = 100, where the optimization library DOT Ver. 5 (VR&D, 1999) is used. Note that the purpose here is to obtain a frame with multiple linear buckling loads as a result of optimization. Therefore, the properties of the optimal braced frame are not discussed. Using the symmetry condition of the frame with respect to the y-axis, the members are classified to those in internal and external spans in each story. Hence, the number of design variables is 18, including beams, columns, and braces. Table 5.1 shows the optimal cross-sectional areas. Note that the beams in the 1st, 2nd, and 3rd stories refer to those in the 2nd floor, 3rd floor, and roof, respectively. The four lowest linear buckling loads are 99.99, 100.00, 100.01, and 111.88, i.e., the three buckling loads are almost equal to Λ at the optimal solution. The coincident buckling modes of the three multiple buckling loads are shown in Fig. 5.3(a)–(c); i.e., the buckling loads corresponding to two sway buckling modes (Modes 1 and 3) and one non-sway buckling mode (Mode 2). The imperfections are given for the x-coordinates of the nodes including those at the centers of the columns. Let R denote the upper bound of the inclination Ri (i = 1, . . . , f ) of the ith story, as illustrated in Fig. 5.4, where f is the number of stories that is equal to 3 for this example. The local imperfection (out-of-straightness of the column) is defined by the additional inclination θi of the elements in the ith column as shown in Fig. 5.4, and its upper bound is denoted by θ. Note again that each column is divided into two elements. The number of columns, which is 12 for this example, is denoted by mc . Since the locations of all the nodes, including the centers of columns, are defined by

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(a) Mode 1 Fig. 5.3

121

(b) Mode 2

(c) Mode 3

Buckling modes of optimal solution under linear buckling constraint.

Ri θi

Fig. 5.4

Definition of imperfection parameters Ri and θi .

their original locations and the imperfection vectors R = (R1 , . . . , Rf )> and θ = (θ1 , . . . , θmc )> , R and θ are regarded as the uncertain parameters in the antioptimization problem. The maximum load Λm of the imperfect frame in Table 5.1 is defined by the stress constraints (5.21) and (5.22), as the load factor when the maximum stress σ m computed by path-tracing analysis for specified R and θ reaches the upper bound. Hence, the anti-optimization problem is stated as minimize

Λm (R, θ)

(5.23a)

subject to |Ri | ≤ R, (i = 1, . . . , f ) c

|θi | ≤ θ, (i = 1, . . . , m )

(5.23b) (5.23c)

The maximum stress σ m at buckling of the optimal perfect frame is 511.37. Therefore, the upper-bound stress of imperfect frames has been set as a slightly smaller value σ m = 510.0, because the maximum stress of an imperfect frame may rapidly increase at the load level slightly below the buckling load of the perfect frame. The upper bounds for the story inclination and the out-of-straightness of the column are given as R = θ = 1/500, which is a practically acceptable value (Kim and Chen, 1996).

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Fig. 5.5

Worst imperfection of the optimal frame.

Since the maximum loads of imperfect systems are highly nonlinear functions of the imperfection parameters R and θ, it is very difficult to obtain the antioptimal solution by a gradient-based nonlinear programming approach. Therefore, the worst imperfection is searched for at the upper and lower bounds of the column out-of-straightness θi = ±1/500, whereas the three cases Ri = −1/500, 0, 1/500 are searched for story inclination. Hence, the anti-optimization problem is formulated as a mixed integer nonlinear programming problem. From f = 3, mc = 12, and symmetry of the frame, the total number of solutions is 33 × 212 /2 = 55296. Therefore, it is possible to find the anti-optimal solution by the enumeration process. Incremental path-tracing analysis has been carried out for each set of imperfection parameters to find the maximum load. The worst imperfection found by this enumeration is shown in Fig. 5.5 and in the first row of Table 5.2. In solution of the table, the first three columns give the story inclination R1 , R2 , R3 , and the remaining show the out-of-straightness of the columns θ1 , . . . , θ12 , where the column numbers are given in Fig. 5.2. The signs + and − correspond to 1/500 and −1/500, respectively. As is seen from Fig. 5.5, the worst imperfection is similar to a sway mode (Mode 1 in Fig. 5.3) and is asymmetric with respect to the y-axis. The maximum load of the imperfect system with worst imperfection is 76.067, which is about 76% of the maximum load 100 of the perfect system. The relations between σ m and the load factor obtained by path-tracing analysis for the perfect frame and the frame with worst imperfection are plotted in Fig. 5.6. Note that no drastic decrease due to geometrical nonlinearity is observed in the imperfect system. For a larger frame with more members, it is very difficult to find the anti-optimal solution by enumeration. Therefore, in order to present a general method, we next find approximate anti-optimal solutions by simulated annealing (see Sec. 2.6.3 for details of the method). The initial temperature parameter is 0.02, and the temperature is multiplied by 0.99 at each iterative step. The iteration is terminated after 500 steps. The results from the randomly generated ten different initial solutions are listed as SA-1, . . . , SA-10 in Table 5.2. It is seen that a good approximate

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123

Optimization results by enumeration and simulated annealing. Maximum load

Enumeration SA-1 SA-2 SA-3 SA-4 SA-5 SA-6 SA-7 SA-8 SA-9 SA-10

Solution R1 R2 R3 θ1 θ2 θ3 θ4 θ5 θ6 θ7 θ8 θ9 θ10 θ11 θ12

76.067

+ + + − − + − + + − + + −

76.306 76.216 76.154 76.306 76.454 76.638 77.397 76.995 76.398 76.763

− + + − − − − + + −

− + + − − − − + + −

0 + + 0 0 − − 0 + −

+ − − + + + + − − +

− − − − − − − − − +

+ + + + + + − + − −

+ − − + + + + − − +

− + + − − − − − + −

+ − + + + − + + − +

− − − − + + + − − −

− + + − − − + + + +

+ + − + + + − + + +

− + + − − − − + + −

+

−

+ + + + − − − + + −

− − − − − − − + − −

100 Perfect Load factor

80 Imperfect

60 40 20 0 0

0.5

1

1.5

Maximum stress Fig. 5.6 Relations between maximum stress σ m and load factor of the perfect and imperfect systems.

solution with Λm = 76.154 has been found by the SA with random start, where the total number of analysis is 500 × 10 = 5000, which is less than 1/10 of the enumeration. 5.5.3

Mode interaction

It is well known that the maximum load of a structure with coincident buckling loads may be drastically reduced in the presence of small imperfection due to the interaction of buckling modes (Thompson and Hunt, 1974; Thompson and Supple, 1973). In order to investigate the effect of mode interaction, the imperfection in the

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Optimization and Anti-Optimization of Structures under Uncertainty Table 5.3 Relation between coefficients (c1 , c2 , c3 ) and the maximum load Λm . (c1 , c2 , c3 )

Λm

(c1 , c2 , c3 )

Λm

(c1 , c2 , c3 )

Λm

(1, 0, 0) (0, −1, 0) (1, 0, 1) (0, −1, 1)

80.44 79.42 79.84 78.71

(0, 1, 0) (1, 1, 0) (1, 0, −1)

79.37 79.65 79.20

(0, 0, 1) (1, −1, 0) (0, 1, 1)

78.93 79.89 78.73

linear combination of the eigenmodes corresponding to the zero eigenvalues of the tangent stiffness matrix at the critical point of the perfect system should be considered. However, these eigenmodes are almost the same as those of the linear buckling modes in Fig. 5.3(a)–(c), because the prebuckling deformation of the perfect system is negligibly small. Therefore, we use the linear buckling modes for investigation of the effect of mode interaction. Let c1 , c2 and c3 denote the coefficients for the three lowest linear buckling modes Φc1 , Φc2 and Φc3 , respectively, and their linear combination is defined as Φc = c1 Φc1 + c2 Φc2 + c3 Φc3

(5.24)

c where Φci (i = 1, 2, 3) are normalized by Φc> i Φi = 1. Denote by ψi the inclination of the ith column element corresponding to Φc . The inclination of the column element from the vertical axis is then defined such that 1/500 for ψi > 0 and −1/500 for ψi ≤ 0, and the nodal locations are computed from the base to the top by summing up the inclinations of column elements multiplied by the element lengths. This way, the imperfection that has θi = 1/500, 0, or −1/500 is generated by assigning the set (c1 , c2 , c3 ). Note that the inclination may be different for the columns in the same story, for simplicity. The maximum load Λm computed by path-tracing analysis for various values of (c1 , c2 , c3 ) are listed in Table 5.3. Note again that the buckling load of the perfect system is 100. As is seen, the maximum load does not strongly depend on the imperfection pattern, and no mode interaction is observed; e.g., Λm = 79.65 for (c1 , c2 , c3 ) = (1, 1, 0) corresponding to the imperfection in the intermediate direction between Modes 1 and 2 is even greater than Λm = 79.37 for the imperfection (c1 , c2 , c3 ) = (0, 1, 0) in the direction of Mode 2. The effect of mode interaction is investigated more accurately below using the differential coefficients of the total potential energy (Thompson and Hunt, 1984; Ohsaki, 2002a). It is known that the interaction in the third-order differential coefficients of the total potential energy, called third-order interaction, may exist in a semi-symmetric system (Huseyin, 1975; Ohsaki and Ikeda, 2007a), where a symmetric and an asymmetric bifurcation points coincide as is the case with this braced plane frame. Note that the bifurcation corresponding to Mode 2 in Fig. 5.3 is asymmetric, although the mode shape is symmetric with respect to the y-axis. The asymmetry

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exists in the potential energy Π. Let U c denote the value of the displacement vector at the critical point of the perfect system. For Mode 2, the deformation corresponding to the displacement vector U = U c + Φc2 and U = U c − Φc2 have different values of Π. However, the asymmetry is very small, because the maximum load for the imperfections (c1 , c2 , c3 ) = (0, 1, 0) and (0, −1, 0) are almost the same as those shown in Table 5.3. Let Π,i denote the differential coefficient of Π with respect to the generalized displacement in the direction of Φci . The third- and fourth-order differential coefficients are computed as Π,222 = −8.7 × 10−4 , Π,112 = −1.3 × 10−4 , Π,233 = 6.6 × 10−4 , Π,1111 = 3.9 × 10−2 , Π,2222 = 5.1 × 10−2 , Π,3333 = 8.5 × 10−1 , Π,1122 = 2.9 × 10−2 , Π,1133 = 5.7 × 10−2 , Π,2233 = 3.3 × 10−2

(5.25)

where the vanishing terms are not shown, e.g., Π111 = Π333 = 0 due to symmetry (see Ohsaki and Ikeda (2007a) for details of computation of the higher-order differential coefficients, where the symbolic computation software package Maple 9 (Maplesoft, 2004) has been extensively used). To investigate the effect of third-order interaction, let q denote the generalized coordinate in the direction of the asymmetric bifurcation mode Φc2 : U = U c + qΦc2

(5.26)

Then Π can be expanded as 1 1 1 (5.27) Π = Πc + Π,2 q + Π,22 q 2 + Π,222 q 3 + Π,2222 q 4 2 6 24 where Πc is the value of Π at the critical point. Since Π,2 = 0 and Π,22 = 0 are satisfied from the equilibrium and the criticality conditions, the leading term of Π governing the symmetry is Π,222 . Suppose q = 100 corresponding to maximum out-of-straightness of the column about 1/100 in the direction of Φc2 , which is larger than the expected value in design practice. In this case, the orders of the third- and fourth-order differential coefficients Π,222 and Π,2222 , respectively, are 102 and 106 . Therefore, the third-order term is negligibly small compared to the fourth-order term, and the asymmetry of Φc2 is very small; hence, the third-order interaction can be neglected. The fourth-order interaction should also be investigated for this almost symmetric system. Since the fourth-order interaction terms Π,1122 , Π,1133 and Π,2233 among the three modes are all positive and their absolute values are in the same order as those of Π1111 , Π,2222 and Π,3333 , no increase of imperfection sensitivity will be observed for this frame (Thompson and Hunt, 1984). In order to further investigate the effect of mode interaction, maximum loads of imperfect systems of non-optimal frames are computed. If the cross-sectional areas of the braces of the optimal frame in Table 5.1 are replaced by 0, i.e., if we remove the braces, the lowest buckling mode is a sway mode, and the three lowest buckling loads are 22.53, 35.89 and 53.43, respectively, which do not coincide. The maximum

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stress at buckling is 115.76. The worst imperfection is computed with σ m = 115.0, θi = ±1/500, and Ri = −1/500, 0, 1/500, by enumeration to find the corresponding maximum load is 15.38. Therefore, the reduction of the maximum load due to the worst imperfection is about 32%, which is larger than the 24% of the optimal frame with coincident buckling. Note that the worst imperfection in this case is similar to the sway mode (Mode 1 in Fig. 5.3). In conclusion, the coincidence of buckling loads does not enhance imperfection sensitivity of a braced frame. 5.5.4

Worst-case design and worst imperfection under stress constraints

Since the effect of geometrical nonlinearity on the maximum load defined by stress constraint of an imperfect frame with practically acceptable proportion is rather small as demonstrated in Sec. 5.5.3, the worst imperfection may be successfully found by evaluating the maximum load under stress constraints by linear analysis. In order to investigate the effect of imperfection on the maximum stress, consider a pin-supported column with cross-sectional area A under axial load N . Suppose an imperfection (out-of-straightness) δ is given at the center of the column. The normal stress σ0 under axial load N is computed as N (5.28) A Since the bending moment at the center is N δ, the maximum stress due to bending is Nδ σ1 = (5.29) Z where Z is the section modulus. Suppose the column consists of a sandwich cross-section, and denote by r the distance between a flange and the center axis of the column. Then Z is given as Z = Ar, and the ratio of bending stress to the axial stress is obtained as σ0 =

δ σ1 = σ0 r

(5.30)

i.e., increase ratio of stress due to imperfection is δ/r. Let θ and κ denote the angle of out-of-straightness of the column due to the imperfection δ at the center and the slenderness ratio of the column, respectively. Then the following relation holds: σ1 θκ = σ0 2

(5.31)

For example, for θ = 1/500 and κ = 100, the ratio is obtained as σ1 /σ0 = 0.1. Since both ends of the column are supported by beams when we consider buckling of a frame, and κ is usually less than 100, the increase ratio due to imperfection will be less than 10%. Therefore, we can conclude that the effect of geometrical imperfection on the maximum stress is not very large.

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Table 5.4 Cross-sectional areas of the worst-case design for maximizing the stress ratio σ I /σ P evaluated by linear analysis.

External column Internal column External beam Internal beam External brace Internal brace

Fig. 5.7

1st story

2nd story

3rd story

5850.58 6152.71 1504.06 3106.50 46.04 13.14

4896.46 4941.28 1503.20 1501.88 8.32 91.57

2009.12 2070.51 1502.42 1500.96 19.77 15.52

Worst imperfection for maximizing the stress ratio σ I /σ P by linear analysis.

We next consider the worst-case design of the 3-story 3-span frame. Let σ P and σ denote the maximum absolute value of the stresses of the perfect and imperfect systems, respectively, obtained by linear analysis, where in general, σ I > σ P holds. The worst imperfection is defined so as to maximize the ratio σ I /σ P due to the imperfection defined by the vector θ consisting of the out-of-straightness of each column element, which is assumed to be a continuous variable with an upper bound of 1/500 and a lower bound of −1/500. The nodal locations are computed from the base to the top by summing up the inclinations of column elements multiplied by the element lengths. The vector A of the cross-sectional areas is also considered as design variable vector to find the worst-case design, where the symmetry of the frame is incorporated. The constraint is given for the linear buckling load as Λb ≥ 100. Anti-optimization to find the worst-case design is carried out by the optimization software package IDESIGN Ver. 3.5 (Arora and Tseng, 1987), where sequential quadratic programming is used. The cross-sectional areas of the worst-case design are listed in Table 5.4, and the corresponding worst imperfection is shown in Fig. 5.7. The value of the objective function σ I /σ P at λ = 100 is 1.0912; i.e., the maximum stress increases about 9.1% due to the worst imperfection. Therefore, if the maxiI

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100

Load factor

80

Perfect

60

Imperfect

40 20 0 0

0.5

1

1.5

Maximum stress Fig. 5.8 Relations between maximum stress ratio σ m /σ P and load factor by path-tracing analysis of the worst-case design.

mum stress is specified, the ratio of the maximum load of the imperfect system to that of the perfect system is 1/1.0912 = 0.91642 due to the worst imperfection. Geometrically nonlinear analysis has been carried out to verify the validity of using linear analysis for anti-optimization. The maximum loads of the imperfect system defined by the upper-bound stress σ m = σ P at Λ = 100 found by geometrically nonlinear analysis is Λ = 81.189. Therefore, the reduction of the maximum load is 19%, and the reduction ratio is magnified twice due to geometrical nonlinearity. Figure 5.8 shows the relation between the maximum stress ratio σ m /σ P and the load factor of the perfect and imperfect systems. The initial stiffnesses are almost the same, and a slightly nonlinear behavior is observed near the maximum load factor.

5.6

Anti-Optimization Based on Convexity of Stability Region

In this section, we briefly summarize the anti-optimization method for linear buckling loads of multi-parameter systems utilizing the convexity of the stability region that is derived from the convexity of the reciprocal value of the linear buckling load. Consider the case where uncertainty exists in the load vector P , which is parameterized by N X P = a i pi (5.32) i=1

where a = (a1 , . . . , aN )> is the vector of uncertain parameters, which take the role of coefficients for the load pattern vectors pi , and N is the number of loading patterns. Consider the situation where the total load vector P is multiplied by the load factor Λ, which is increased to find the buckling (critical) load factor Λ c for each

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specified parameter vector a. For this multi-parameter loading condition, the set of critical points in the space of Λa forms a surface that is called the stability boundary (Huseyin, 1975). The boundary degenerates to a point for the conventional case of proportional loading with single load pattern. If the prebuckling deformation is negligibly small, and the buckling load is estimated by linear buckling analysis, the anti-optimization problem can be solved efficiently, as follows, utilizing the convexity of the stability region (de Faria, 2000; de Faria and Hansen, 2001a, 2001b). De Faria and de Almeida (2003a) presented an anti-optimization method of a plate with variable thickness. Although they presented detailed investigation for general cases, we briefly present the results below based on the concavity of the reciprocal value of the buckling load. We first derive the concavity property of the lowest eigenvalue of the tangent stiffness matrix at an equilibrium state, which is assumed to be a linear function of a. Let ζ1I and ζ1II denote the lowest eigenvalues of the tangent stiffness matrices K(aI ) and K(aII ), respectively, for two sets of parameter values aI and aII . The interpolation between K(aI ) and K(aII ) is given by the parameter a∗ = αaI + (1 − α)aII (0 ≤ α ≤ 1). The eigenvector associated with the lowest eigenvalue of K(a ∗ ) is denoted by Φ∗ . Based on the Rayleigh principle, the following relations hold: Φ∗> K II Φ∗ Φ∗> K I Φ∗ ≥ ζ1I , ≥ ζ1II ∗> ∗ Φ Φ Φ∗> Φ∗

(5.33)

Therefore, the following inequality is derived for the lowest eigenvalue ζ1∗ of K(a∗ ): Φ∗> [αK I + (1 − α)K II ]Φ∗ Φ∗> Φ∗ Φ∗> K II Φ∗ Φ∗> K I Φ∗ + (1 − α) =α ∗> ∗ Φ Φ Φ∗> Φ∗ ≥ αζ1I + (1 − α)ζ1II

ζ1∗ =

(5.34)

It is seen from Eq. (5.34) that the lowest eigenvalue ζ1 (a) of K(a) is a concave function of the load parameters a. Hence, the feasible (stable) region S in the space of a satisfying ζ1 ≥ 0 is convex. Consider next the linear buckling problem, where the tangent stiffness matrix K(a) is divided into the linear stiffness matrix K E and the geometrical stiffness matrix K G = ΛK G0 . Note that K E is independent of P , and K G is a linear function of P , hence, K G is a linear function of a. Then the problem of finding the linear buckling load factor Λb is formulated as (K E + Λb K G0 )Φ = 0

(5.35)

Suppose we are interested in the positive smallest buckling load factor and define η b as ηb =

1 Λb

(5.36)

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Then η b is the largest eigenvalue of the following problem, which is denoted by η max : −K G0 Φ = η b K E Φ

max

(5.37)

Convexity of η can also be proved based on the Rayleigh principle. For the two vectors of uncertain parameters aI and aII , and the interpolation parameter α (0 ≤ α ≤ 1), define a∗ = αaI + (1 − α)aII and denote by Φ∗ the linear buckling mode for a = a∗ . Then the following relations hold from the Rayleigh principle and positive definiteness of K E : −

∗ Φ∗> K II Φ∗> K IG0 Φ∗ GΦ maxI ≤ η , − ≤ η maxII Φ∗> K E Φ∗ Φ∗> K E Φ∗

(5.38)

where η maxI and η maxII are the values of η max corresponding to a = aI and aII , respectively. Therefore, the following inequality is derived for η max∗ corresponding to a = a∗ : ∗ Φ∗> [αK IG + (1 − α)K II G ]Φ Φ∗> K E Φ∗ Φ∗> K II Φ∗ Φ∗> K I Φ∗ = −α ∗> G ∗ − (1 − α) ∗> G ∗ Φ KEΦ Φ K EΦ ≤ αη maxI + (1 − α)η maxII

η max∗ = −

(5.39)

It is seen from Eq. (5.39) that the largest eigenvalue η max (a) of Eq. (5.37) is a convex function of a for Λb > 0. On the other hand, for Λb < 0, the buckling load with the smallest absolute value corresponds to the smallest eigenvalue η min of Eq. (5.37), which can be shown to be a concave function of a in a similar manner as the case of Λb > 0, using the Rayleigh principle. If we define the linear buckling load as the buckling load with the smallest absolute value, then its reciprocal value η b is found from η b = max{η max , −η min}

(5.40)

Since η min is concave, −η min is convex, and η b is defined as the maximum value of the two convex functions, which is also convex. Therefore, the reciprocal value of the linear buckling load factor is a convex function of the load pattern vector a irrespective of the sign of Λb . Suppose ai ≥ 0 (i = 1, . . . , N ), for simplicity, and consider the problem of finding the worst-load scenario defined by the parameters a under the constraints N X i=1

ai ≤ D,

ai ≥ 0, (i = 1, . . . , N )

(5.41a) (5.41b)

where D is a specified positive value. The objective of the anti-optimization here is to find the minimum buckling load factor Λb associated with the set of load parameters satisfying Eq. (5.41). Therefore, the problem is converted to a maximization

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P2 P1

H x

H

H

y

Fig. 5.9

A 2-bar truss.

2

a2

1.5

1

0.5

0 0

0.2

0.4

0.6

0.8

1

1.2

a1 Fig. 5.10

Stability boundary (solid line) of the 2-bar truss; •: anti-optimal solution.

problem of η b (a). Since the feasible region defined by Eq. (5.41) is convex, the anti-optimal solution that maximizes the convex function η b (a) exists at a vertex of the convex polyhedron (5.41). Let V denote the set of vertices of the region (5.41). The anti-optimal solution is found by solving the following problem: maximize η b (a)

(5.42a)

subject to a ∈ V

(5.42b)

The solution of Problem (5.42) can be found by vertex enumeration, if the number of uncertain load parameters is moderately small. For example, consider a 2-bar truss as shown in Fig. 5.9, where H = 1000 mm. The two members consist of the same cross-section, where the cross-sectional area is 300 mm2 , and the moment of inertia is 7.5 × 105 mm4 . Young’s modulus is 2 200 kN/mm . Each member is divided into two elements to consider member buckling. In the following, the units of length and force are mm and kN, respectively.

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0.03 0.025 0.02 0.015 0.01 0.005 0 1 0.8 0.6

0 0.2

0.4

a2

0.4

a1

0.6

0.2 0.8 10

Fig. 5.11

Reciprocal value of linear buckling load η b = 1/Λb .

The horizontal load P1 and the vertical load P2 are parameterized as P1 = 1000a1, P2 = 1000a2

(5.43)

The linear buckling load factor Λb is computed for various sets of (P1 , P2 ), and the values of Λb P that define the stability boundary are plotted as shown in the solid line in Fig. 5.10. Suppose the constraint is given as a1 + a2 ≤ 1.0, a1 ≥ 0, a2 ≥ 0

(5.44)

Then the maximum load factor is attained at P = (0, 1.0587)> indicated by • in Fig. 5.10, which is located at a vertex of the stability boundary. Consider next the braced frame shown in Fig. 5.2 in Sec. 5.5. The cross-sectional areas are 1500.0 for beams and columns, and 100.0 for braces. Other parameters are the same as those in Sec. 5.5. Let P1 and P2 denote the vertical loads applied at the exterior and interior nodes, respectively, which are parametrized by a 1 and

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1

0.8

0.028 0.026 0.024 0.022 0.02 0.018 0.016 0.014 0.012 0.01 0.008 0.006 0.004 0.002

a2

0.6

0.4

0.2

0 0

0.2

0.4

0.6

0.8

1

a1 Fig. 5.12

Contour lines of the reciprocal value of linear buckling load η b = 1/Λb .

a2 as P1 = 10000a1, P2 = 10000a2

(5.45)

The reciprocal value η b = 1/Λb of the linear buckling load is computed for each set of (P1 , P2 ) for 0 ≤ a1 ≤ 1 and 0 ≤ a2 ≤ 1 to obtain the plot in Fig. 5.11, where the contour lines are also plotted. It is confirmed from Fig. 5.11 that η b is a convex function of (a1 , a2 ). Suppose the constraint is given as a1 + a2 ≤ 0.8, a1 ≥ 0, a2 ≥ 0

(5.46)

Then the feasible region is a triangle as shown in the thick straight lines in Fig. 5.12, and the worst value of η b is about 0.0187, which is attained at the vertex (a1 , a2 ) = (0.8, 0). 5.7

5.7.1

Worst Imperfection of an Arch-type Truss with Multiple Member Buckling at Limit Point Introduction

It has been pointed out by many researchers that imperfection sensitivity generally increases as a result of coincidence of bifurcation points. Thompson and Hunt

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(1973) showed that the critical load of a latticed column drastically decreases due to interaction of the critical loads associated with local and global buckling modes. However, very large imperfection beyond the practically acceptable level for civil engineering was considered therein. Ohsaki (2002a, 2002c) demonstrated that a building frame with practical proportions does not have severe reduction of buckling loads due to mode interaction (see Sec. 5.5 for details). Thompson and Schorrock (1975) derived imperfection sensitivity laws for the hilltop branching point, often abbreviated as hilltop point, that has a bifurcation point at a limit point. The critical loads of imperfect systems are expressed as a piecewise linear function of the imperfection parameter. Therefore, imperfection sensitivity is even decreased by the coincidence of critical points from the imperfection-sensitive structure with a simple bifurcation point. Ohsaki (2000) demonstrated that optimization of shallow shell-type truss against buckling often leads to a hilltop point with multiple bifurcation. Ikeda, Oide and Terada (2002) verified the imperfection sensitivity laws using random imperfections. Ohsaki (2003b) derived imperfection sensitivity law for the case where an asymmetric bifurcation point exists at a limit point. Ikeda, Ohsaki and Kanno (2005) derived imperfection sensitivity laws for hilltop points of symmetric systems based on group theoretic approach. Ohsaki and Ikeda (2006) derived a piecewise linear imperfection sensitivity law for the hilltop point with many symmetric bifurcation points. Imperfection sensitivity of degenerate hilltop points was investigated by Ohsaki and Ikeda (2009). The worst imperfection for hilltop points is presented in Ikeda, Oide and Terada (2002) and Murota and Ikeda (1991). In this section, we verify the results in Ohsaki and Ikeda (2006) using random imperfections. Interested readers may consult with Ohsaki and Ikeda (2007a, Chap. 11) for details on random imperfection. Antioptimization problems are formulated to derive the worst imperfection. Numerical examples are given for an arch-type truss. 5.7.2

Hilltop branching point of perfect system

Consider a finite-dimensional structure subjected to quasistatic proportional loads P , which is defined by the load factor Λ and the vector p of load pattern as P = Λp. Let n denote the number of degrees of freedom, and define the nodal displacement vector as U = (U1 , . . . , Un )> . We consider an conservative linear elastic system that has the total potential energy denoted by Ve (U , Λ). The stability matrix (tangent stiffness matrix) K is given as ! ∂ 2 Ve (5.47) K= ∂Ui ∂Uj The ith eigenvalue ζi (ζi ≤ ζi+1 ) and eigenvector Φi are defined by KΦi = ζi Φi , (i = 1, . . . , n)

(5.48)

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where Φi is normalized as Φ> i Φi = 1, (i = 1, . . . , n)

(5.49)

Suppose ζ1 > 0 is satisfied at the initial state Λ = 0. The first critical point as Λ is increased from 0 is defined by ζ1 = 0 with the critical load factor Λ = Λc . Consider the hilltop point where m − 1 bifurcation points exist at a limit point; i.e., m eigenvalues ζ1 , . . . , ζm vanish at the critical point. Let qi denote the generalized displacement in the direction of Φci as c

U =U +

n X

qi Φci

(5.50)

i=1

where U c is the displacement vector at the critical point. The coordinates defined by q1 , . . . , qm are called active coordinates, whereas those by qm+1 , . . . , qn are called passive coordinates. The increment of Λ from Λc is denoted by λ as λ = Λ − Λc

(5.51) >

Then the total potential energy is written in terms of q = (q1 , . . . , qn ) and λ as V (q, λ). In the following, differentiation of V with respect to qi is denoted by the subscript ( · ),i , and all variables are evaluated at the critical point without explicit notation of the superscript ( · )c . Differentiation of V with respect to Λ is denoted by the subscript ( · ),Λ . 5.7.3

Imperfection sensitivity of hilltop branching point

Let a = (a1 , . . . , aN )> denote the imperfection vector representing errors in nodal locations, cross-sectional areas, etc. The vector a is defined by the value of the perfect system (nominal value) a0 , imperfection parameter ξ, and the vector d of imperfection pattern as a(ξ) = a0 + ξd

(5.52)

Differentiation of the total potential energy with respect to ξ is denoted by the subscript V,ξ as V,ξ =

N X ∂V di ∂ai i=1

(5.53)

The load increment λ for an imperfect system is defined by λ = Λ − Λc0

(5.54)

where Λc0 is the critical load factor of the perfect system. Hence, the total potential energy of an imperfect system is written as V (q, λ, ξ), as a function of n + 2 variables λ, q1 , . . . , qn and ξ. Suppose Φ1 , . . . , Φm−1 correspond to bifurcation modes, and Φm to limit point mode. The symmetry of the

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total potential energy is characterized by the third-order differential coefficients with respect to qi . Consider the case, where m − 1 bifurcation points are individually symmetric (Thompson and Hunt, 1984) as V,ijk = 0, (i, j, k = 1, . . . , m − 1) (5.55) which is satisfied in the following numerical examples where Φ1 , . . . , Φm−1 are the member buckling modes of a truss. Note that the orthogonality conditions Φ> i Φj = 0 (i 6= j) for Φ1 , . . . , Φm−1 are relaxed to satisfy Eq. (5.55). Since V is symmetric in the direction of bifurcation modes, and asymmetric to the limit point mode, we have V,ijm 6= 0, (i, j = 1, . . . , m − 1; i 6= j) (5.56a)

V,imm = 0, (i = 1, . . . , m − 1) (5.56b) Furthermore, from the definition of limit point, V,mmm 6= 0 (5.57) is satisfied (see Ohsaki and Ikeda (2007a) for details of stability analysis of elastic conservative system exhibiting hilltop branching point with multiple bifurcation points). The incremental load factor λc at the critical point of an imperfect system is obtained as (Ohsaki and Ikeda, 2006) V,mξ 1 p V,mmm Cm |ξ| (5.58) λc = − ξ− V,mΛ V,mΛ where m−1 X m−1 X Cm = V,ijm qei qej (5.59) i=1 j=1

and qei is obtained by solving the following m − 1 simultaneous linear equations: m−1 X V,ijm qej + V,iξ = 0, (i = 1, . . . , m − 1) (5.60) j=1

Note that the sign of the limit point mode Φcm is defined so that V,mΛ > 0 is satisfied, and V,mmm Cm > 0 holds. As is seen from Eq. (5.58), the critical load of an imperfect system is a piecewise linear function of the imperfection parameter ξ, which means that the structure is not imperfection-sensitive even though multiple bifurcation points exist at the critical point of the perfect system. 5.7.4

Worst imperfection

The worst imperfection pattern that minimizes the critical load for a specified norm of imperfection is derived below. Let B denote the matrix for which (j, i)-component is given as ∂ 2 V /∂qj ∂ai . Define hk = V,kξ for brevity. Then we obtain from Eq. (5.53) ! n N X X ∂ ∂V hk = di Φckj ∂q ∂a j i (5.61) j=1 i=1 = Φc> k Bd, (k = 1, . . . , m)

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where Φckj is the jth component of Φck . In the following, the component of a vector e = (e is denoted by a subscript as h = (hi ), q qi ). The vector h is divided into the components hB = (h1 , . . . , hm−1 )> corresponde ing to the bifurcation modes and hL = hm to the limit point mode. The vector q eB = (e is similarly divided into q q1 , . . . , qem−1 )> and qeL = qem . Let F B denote the (m − 1) × n matrix for which the ith row is equal to Φc> i . Then, from Eq. (5.61), we obtain hB = F B Bd

(5.62)

The (m − 1) × (m − 1) matrix G is defined so that the (i, j)-component is equal to V,ijm . Then, from Eq. (5.60), eB = −G−1 F B Bd q

(5.63)

is derived. Hence, from Eq. (5.59), we have

Cm = d> B > F B> (G−1 )> F B Bd

(5.64)

e = V,mmm B > F B> (G−1 )> F B B G

(5.65)

For simplicity, we write

Then Eq. (5.58) is written as

λc (d) = −

1 p >e Φc> m Bd d Gd |ξ| ξ− V,mΛ V,mΛ

(5.66)

Finally, the anti-optimization problem for finding the worst imperfection pattern that minimizes λc is formulated as λc (d)

minimize subject to

N X

d2i = 1

(5.67a) (5.67b)

i=1

where Eq. (5.67b) is the norm constraint for d. We next investigate the imperfection in the direction of active coordinates; i.e., the imperfection pattern vector d is defined as a linear combination of the critical eigenmodes Φci (i = 1, . . . , m) of the perfect system as d=

m X

ci Φci

(5.68)

i=1

where ci is the coefficient, and the following orthogonality conditions are satisfied between a bifurcation mode and the limit point mode: c Φc> i BΦm = 0, (i = 1, . . . , m − 1)

(5.69)

The accuracy of this assumption is investigated in the numerical examples. Note that N = n holds in this case.

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Let D B denote the n × (m − 1) matrix for which the ith column is Φci , and divide c into the components cB = (c1 , . . . , cm−1 )> and cL = cm , corresponding to the bifurcation modes and the limit point mode, respectively. By defining >

e DB G∗ = D B> G

em =

c Φc> m BΦm

(5.70a) (5.70b)

and using Eqs. (5.68) and (5.69), Eq. (5.66) is rewritten as λc (c) = −

1 √ B> ∗ B em cL ξ− c G c |ξ| V,mΛ V,mΛ

(5.71)

Therefore, λc defined in Eq. (5.58) is conceived as a function of c, and the anti-optimization problem for minimizing λc is formulated as λc (c) m X subject to c2i = 1 minimize

(5.72a) (5.72b)

i=1

where Eq. (5.72b) is the norm constraint for c. In the process of minimizing λc (c), the right-hand-side of Eq. (5.71) is divided into terms corresponding to cL and cB , and we first consider the maximization problem of cB> G∗ cB under the constraint cB> cB = κ, (0 ≤ κ ≤ 1)

(5.73)

Since G∗ is a real symmetric matrix, the minimum of the second term in Eq. (5.71) √ is obtained as −(1/V,mΛ ) κτ |ξ|, where τ is the maximum eigenvalue of G∗ . Hence, using Eq. (5.72b), κ is obtained by minimizing λc (κ) as λc (κ) = −

em √ 1 √ 1−κξ− κτ |ξ| V,mΛ V,mΛ

(5.74)

By differentiating Eq. (5.74) with respect to κ, the value of κ corresponding to the worst imperfection is obtained as τ κ= p 2 em + τ

Then the anti-optimal solution of Problem (5.72) is given as r τ e2 g cL = 2 m , c B = 2 em + τ em + τ

(5.75)

(5.76)

where g is the eigenmode corresponding to the maximum eigenvalue of G∗ , which is normalized by g > g = 1.

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Λp 3 4

4

4

2

1

L

2

H

3 4

4

4

4

4 1

L

L

L

L

L

L

x L

y Fig. 5.13

5.7.5

An arch-type truss.

Worst imperfection of an arch-type truss

Imperfection sensitivity analysis is carried out for an arch-type truss as shown in Fig. 5.13, where L = 1000, H = 400, and the height of the center node is 200. A vertical load Λp is applied at the center node. Young’s modulus is 1.0, for simplicity, and p = 0.001. In the following, the units of length and force are omitted. The numbers in Fig. 5.13 indicate group numbers; i.e., the members are classified into four groups. Let Ai and Ii denote the cross-sectional area and the second moment of inertia of the members in group i, which are related as Ii = γi2 Ai

(5.77)

where γi is a parameter that is independent of Ai . Each member is divided into four beam elements. Note that the members are pin-jointed at the nodes. The shape of the member is approximated as piecewise linear when considering the imperfection in the direction of eigenmodes. The Green strain is used, and the equilibrium paths are traced by the displacement increment method. A symbolic computation software package Maple 9 (Maplesoft, 2004) is used for differentiation of equations. The relation between Λ and the vertical displacement v at the center node is shown in Fig. 5.14 for the design (A1 , A2 , A3 , A4 ) = (100, 100, 1000, 300) and (γ1 , γ2 , γ3 , γ4 ) = (18.15, 18.05, 30.0, 100.0). In this case, a limit point is reached at Λ = 4.7681. The five lowest eigenvalues are plotted against v in Fig. 5.15, where the solid line and other lines correspond to the limit point mode and bifurcation modes, respectively. Note that the chained line has duplicate eigenvalues. The five modes Φc1 , . . . , Φc5 are shown in Fig. 5.16, where Φc1 , . . . , Φc4 correspond to member buckling modes, and Φc5 to limit point mode. All the modes are normalized by Eq. (5.49). If we consider the imperfection in the direction of the sum of the five eigenmodes P5 as d = i=1 Φci , the relation between ξ and λc is written as λc = −1.8393 × 10−2 ξ − 0.22176|ξ|

(5.78)

and Λc = λc + Λc0 is plotted in Fig. 5.17. Note that ‘+’ in Fig. 5.17 indicates the maximum load obtained by the path-tracing analysis. The maximum value of the

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5

Load factor

4 3 2 1 0 0

20

40

60

80

100

Displacement Fig. 5.14

Relation between load factor and vertical displacement at the center node.

0.00015

Eigenvalues

duplicate 0.0001

0.00005 limit point 0 0

20

40

60

80

100

Displacement Fig. 5.15

Relation between eigenvalues and vertical displacement at the center node.

nodal imperfection corresponding to ξ = 1 is 0.5, which is very small compared with the dimension of the arch. Therefore, from Fig. 5.17, the piecewise linear law Eq. (5.78) has a good accuracy if the imperfection parameter is sufficiently small. Next, we carry out path-tracing analysis to compute the critical loads corresponding to random imperfection of nodal coordinates. The imperfection subject to normal distribution N(0, δ) is given for the (x, y)-coordinates of the nodes. The similar imperfection with normal distribution is given for the initial deflection (outof-straightness) of each member as shown in Fig. 5.18. The standard deviation √ of the imperfection at the center node is δ, whereas the standard deviation is δ/ 2 for the interior nodes that are 1/4 of the member length form the two nodes, respectively.

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141

Mode 1 Φc1 Mode 2 Φc2 Mode 3 Φc3 Mode 4 Φc4 Mode 5 Φc5 (Limit point mode) Fig. 5.16

Four member buckling modes Φc1 , . . . , Φc4 and limit point mode Φc5 .

Maximum load

4.775 4.770 4.765 4.760 4.755 −0.02

−0.04

0

0.02

0.04

Imperfection parameter Fig. 5.17 Relation between Λc and ξ for imperfection in the direction of sum of five modes; solid line: linear estimation using sensitivity coefficient, ‘+’: critical load found by path-tracing analysis.

The worst imperfection in the space of buckling modes, obtained by solving Problem (5.72), is c1 = 0.0, c2 = −0.93362, c3 = 0.0

c4 = 0.32002, c5 = 0.16108

(5.79)

which corresponds to the shape in Fig. 5.19. Since strict orthogonality is not satP5 isfied in the modes Φci (i = 1, . . . , 5), the norm of d = i=1 ci Φci is 1.0478, which c is not equal to 1. However, the absolute value of Φc> i BΦm (i = 1, . . . , m − 1) c> c is less than 1/100000 of the absolute value of Φm BΦm ; i.e., the assumption of orthogonality in Eq. (5.69) is satisfied.

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δ/ 2

δ

δ/ 2

Fig. 5.18

Definition of column imperfection.

Fig. 5.19 Worst imperfection in the space of buckling modes as the anti-optimal solution of Problem (5.72).

Maximum load

4.8 4.75 4.7 4.65 4.6 4.55 0

0.5

1

1.5

2

Norm of imperfection Fig. 5.20 Relation between Λc and imperfection norm for small random imperfection δ = 0.1; solid and dotted lines: linear estimation using Problem (5.67) and Problem (5.72), respectively, ‘+’: critical load found by path-tracing analysis.

The worst objective value of Problem (5.72) is −0.11231, which is divided by the norm 1.0478 to obtain the sensitivity −0.10719 with respect to the imperfection parameter ξ. The estimation by linear approximation using this sensitivity coefficient is plotted by dotted line in Fig. 5.20, where ‘◦’ is the result of path-tracing analysis for various norms of imperfections in the direction of the worst imperfection. The ‘+’ mark shows the value of Λc for small random imperfection corresponding to δ = 0.1. Similar results for moderately large imperfection δ = 1.0 are shown in dotted line in Fig. 5.20. As is seen, the critical loads of imperfect systems can be linearly approximated by sensitivity coefficients with good accuracy in the range of

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Maximum load

4.8 4.6 4.4 4.2 4 0

5

10

15

20

Norm of imperfection Fig. 5.21 Relation between Λc and imperfection norm for moderately large random imperfection δ = 1.0; solid and dotted lines: linear estimation using Problem (5.67) and Problem (5.72), respectively, ‘+’: critical load found by path-tracing analysis.

Fig. 5.22 Worst imperfection in the general direction as the anti-optimal solution of Problem (5.67).

small imperfection. The solution of Problem (5.72) gives a good lower bound for imperfections defined by a moderately smooth shape as shown in Fig. 5.18. On the other hand, the worst imperfection mode in the general direction obtained by solving Problem (5.67) is as shown in Fig. 5.22, which is more irregular than the mode in Fig. 5.19. Therefore, in order to incorporate it into practical situations, the imperfection mode should be smoothed by restricting the shape as a combination of possible imperfection shapes during fabrication and manufacturing. The sensitivity coefficient with respect to the imperfection parameter ξ corresponding to this worst imperfection is −0.25822. The solid lines in Figs. 5.20 and 5.21 are the critical loads by linear approximation using the sensitivity coefficient. The result of path-tracing analysis is indicated by ‘•’. As is seen, the worst imperfection defines a conservative lower bound of the critical load for the specified norm of smooth imperfection shown in Fig. 5.18. If we express the worst imperfection mode of Problem (5.67) by Φci (i = 1, . . . , 5), the coefficients are obtained as 0.0, 0.41847, 0.0, 0.10663, −0.071310 for which the norm is 0.43769, far less than 1. Therefore, the worst mode has components of the passive coordinates, and we can conclude that the worst imperfection cannot be obtained by considering the active coordinates only.

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Some Further References

For other pertinent references on treatment of uncertainty in buckling, antioptimization in particular, the reader may consult with the monographs by Elishakoff, Li and Starnes (2001) and by Qiu and Wang (2008) as well as papers by Adali (1992), Adali, Elishakoff, Richter and Verijenko (1994a), Adali, Lene, Duvaut and Chiaruttini (2003), Adali, Richter and Verijenko (1995b, 1997), Baitsch and Hartmann (2006), Ben-Haim (1993a, 1993b, 1995), Ben-Haim and Elishakoff (1989b, 1990), Cho and Rhee (2003, 2004), Chryssanthopoulos and Poggi (1995), Conrado, de Faria and de Almeida (2005a, 2005b), de Faria (2000, 2002, 2004, 2007), de Faria and de Almeida (2003a, 2003b, 2004), de Faria and Hansen (2001a, 2001b), Deml and Wunderlich (1997), Elseifi (1998), Elseifi, G¨ urdal and Nikolaidis (1999), Elishakoff (1998a, 2000b), Elishakoff and Ben-Haim (1990a), Elishakoff, Cai and Starnes (1994a), Elishakoff, Li and Starnes (1994d, 1996), Elishakoff, van Manen, Vermeulen and Arbocz (1987), Kim and Sin (2001), Kogiso, Shao and Murotsu (1996), Lindberg (1992a,1992b), Pantelides (1996a, 1996b), Qiu, Elishakoff, and Starnes (1996d), Qiu (2005), Qiu, Ma and Wang (2006a), Qiu and Wang (2005), Reitinger, Bletzinger and Ramm (1994), Zharii, Jih-Hua, Chi-Ti, Li-Wei and Pantelides (1996), and Wang, Elishakoff, Qiu and Ma (2009).

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Chapter 6

Anti-Optimization in Vibration

‘It was the best of times, it was the worst of times...’ (Charles Dickens, A Tale of Two Cities) ‘... singular praise it is to have done the best things in the worst times...’ (Inscription in Harold Church, Staunton, for Sir Robert Shirley, Baronet)

In this chapter, we start, as in previous chapters, with a simple example. Then we review Bulgakov’s problem, which apparently was the first problem in the literature on anti-optimization in vibration. Then we deal with the effect of scatter in material properties on the dynamic response of viscoelastic beams by treating the aeroelastic optimization and anti-optimization. 6.1

Introduction

The eigenvalues of vibration and responses to dynamic loads play important roles in the design process of various fields of engineering. For eigenvalue problems, the uncertainty in structural parameters can be introduced in a similar manner as the static problems presented in Chap. 4. However, for the responses to dynamic loads such as seismic excitations, the uncertainty of loads in time domain should be considered. Thus, the anti-optimization problem becomes very difficult to formulate. Chen and Wu (2004a) applied interval parameters to anti-optimization considering dynamic responses, where the nonlinear responses are approximated by sensitivity analysis. The beams under uncertain excitation was studied by Sadek, Sloss, Adali, and Bruch (1993). Anti-optimization for earthquake excitation was studied by Baratta, Elishakoff, Zuccaro and Shinozuka (1998) and Zuccaro, Elishakoff and Baratta (1998). Modares, Mullen and Muhanna (2006) presented an interval formulation for the anti-optimization of eigenvalues of vibration, where linear perturbation by positive semidefinite matrix representing the uncertainty in elastic modulus is considered and 145

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1

K1 Fig. 6.1

M0

2

M0

K2

A 2-bar lumped mass structure.

the monotonicity of the eigenvalue is used. Interval matrix problem for eigenvalue analysis was presented by Qiu, Chen and Elishakoff (1995a, 1996b) (see Chap. 7 for basic properties of interval matrices). 6.2

A Simple Example of Anti-Optimization for Eigenvalue of Vibration

Let K and M denote the stiffness matrix and mass matrix, respectively, of a finite-dimensional structure. The eigenvalue problem of undamped free vibration is formulated as KΦi = Ωi M Φi , (i = 1, . . . , n)

(6.1)

where Ωi and Φi are the ith eigenvalue and eigenvector, respectively, and n is the number of degrees of freedom. Φi is orthonormalized as Φ> i M Φj = δij , (i, j = 1, . . . , n)

(6.2)

where δij is the Kronecker delta. An anti-optimization problem of free vibration is formulated to find the worst-case scenario for minimizing the lowest eigenvalue. Consider a 2-bar structure as shown in Fig. 6.1 that has lumped mass M0 at nodes 1 and 2. The extensional stiffnesses of members 1 and 2 are denoted by K1 and K2 , respectively. The stiffness matrix K and mass matrix M are defined as     K1 + K2 −K2 M0 0 K= , M= (6.3) −K2 K2 0 M0 The lowest eigenvalue and eigenvector are obtained as   q 1 K1 + 2K2 − K12 + 4K22 Ω1 = 2M0 p   M0 −K1 + K12 + 4K22 Φ1 = 2K2 τ where τ 2 = 2K12 + 8K22 − 2K1

q K12 + 4K22

(6.4)

(6.5)

The sensitivity coefficients of the eigenvalue can be computed generally using the formulations in Sec. 3.6.3. However, for this simple example, the sensitivity coefficients c1 and c2 are obtained by directly differentiating Ω1 with respect to K1 and

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K2 as

147

! K1 1 − p >0 2 2 K12 + 4K22 ! 2K2 >0 1− p 2 K1 + 4K22

1 ∂Ω1 = c1 = ∂K1 M0 ∂Ω1 1 c2 = = ∂K2 M0

(6.6)

Therefore, as expected, Ω1 is a monotonically increasing function of K1 and K2 . Consider uncertainty in the stiffnesses as K1 = K10 + a1 , K2 = K20 + a2

(6.7)

where K10 and K20 are the nominal values and a1 and a2 are the uncertain parameters. If the bound of a = (a1 , a2 )> is given as follows by the quadratic inequality constraint a> a ≤ D

(6.8)

which is a simple form of an ellipsoidal bound, then the anti-optimization problem is formulated as minimize

Ω1 (a)

(6.9a)

>

subject to a a ≤ D

(6.9b)

Since Ω1 is a nonlinear function of a1 and a2 , the anti-optimal solution cannot be obtained explicitly by applying the Lagrange multiplier approach, as was the case for the static response with load uncertainty in Sec. 4.1. Let c = (c1 , c2 )> , and denote by c0 the value of c at (K1 , K2 ) = (K10 , K20 ). Then Ω1 (a) is linearized at a = 0; i.e., at (K1 , K2 ) = (K10 , K20 ), to reformulate the problem to a linearized problem: minimize

Ω1 (0) + c0> a

(6.10a)

>

subject to a a ≤ D

(6.10b)

In a similar manner as the simple example of static anti-optimization problem in Sec. 4.1, the anti-optimal solution can be obtained as r 1 1 c> ∗ 2 c2 a = − c2 , µ = µ = (6.11) 2µ 2 D Note that µ = −µ∗ corresponds to the optimal solution that maximizes Ω1 . Consider the case, for simplicity, where M0 = 1, K10 = 2, K20 = 1, and D = 1. Then Problem (6.10) is explicitly written as minimize

0.5858 + 0.1464a1 + 0.2928a2

subject to

a21

+

a22

(6.12a)

≤1

and the anti-optimal solution is found to be a = (−0.4472, −0.8944) µ∗ = 0.1637.

(6.12b) >

with µ =

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If Ω1 (a) is not linearized, and the bounds are given for a by the interval aL ≤ a ≤ a U

(6.13)

with the lower bound aL and upper bound aU , then the anti-optimization problem has a nonlinear objective function and the linear bound constraints as minimize

Ω1 (a) L

(6.14a) U

subject to a ≤ a ≤ a

(6.14b)

Since Ω1 is a monotonically increasing nonlinear function of a1 and a2 , the antioptimal solution for Problem (6.14) exists at a vertex of the feasible region, and the obvious anti-optimal solution is found as a = aL . 6.2.1

Anti-optimization for forced vibration

For forced vibration of a linear elastic system, the anti-optimization problem is formulated against the time-history responses (Elishakoff and Pletner, 1991). Consider, for simplicity, a single-degree-of-freedom system subjected to forced base acceleration u ¨b as Mu ¨ + C u˙ + Ku = −M u ¨b

(6.15)

where u is the relative displacement from the base, M is the mass, C is the damping coefficient, K is the stiffness, and ( ·˙ ) is the time derivative. Suppose fb = −M u ¨b is expanded in Fourier series in time domain with N modes as fb (t) =

N X

ai sin ωi t

(6.16)

i=1

where ai is the Fourier coefficient for the vibration mode with circular frequency ωi . Let θi (t) denote the response corresponding to the ith mode that is an explicit function of t. Then the response is written as a linear function of ai as u(t) =

N X

ai θi (t)

(6.17)

i=1

Consider the case where the displacement, velocity, and acceleration of the base excitation is linearly bounded with the specified upper bound denoted by ( · )U as U |¨ ub | ≤ u ¨U ˙ b | ≤ u˙ U b , |u b , |ub | ≤ ub

(6.18)

Then the constraints are written explicitly with respect to a, and the antioptimization problem for finding the worst excitation for maximizing |u(t)| at a discretized time step, or maximizing the maximum value of |u(t)|, |u(t)| ˙ or |¨ u(t)| during the specified time period, is formulated as a linear programming problem. If the ellipsoidal bound is given for a = (a1 , . . . , aN )> , the problem turns out to be a quadratic programming problem.

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The bounds in Eq. (6.18) can be more generally written as time-varying functions: U |¨ ub | ≤ u ¨U ˙ b | ≤ u˙ U b (t), |u b (t), |ub | ≤ ub (t)

(6.19)

Michalopoulos and Riley (1972) investigated a two-degree-of-freedom system and found that the worst excitation tends to be a bang-bang type function in control theory. A global measure can also be used for defining the bound of the input force (Tzan and Pantelides, 1996a). For example, an energy bound can be given as Z T [fb (t)]2 dt ≤ E U (6.20) 0

where E U is the specified upper bound, and T is the duration of the motion.

6.3

Bulgakov’s Problem

In his pioneering paper Bulgakov (1946) introduces his problem as follows: ‘Damped linear systems with constant parameters cannot develop unlimited forced oscillations if the exciting actions remain finite. Under unfavorable circumstances, however, the disturbances become considerable. The purpose ... is to calculate the upper bounds of the deviations when the disturbing actions yρ (t) (ρ = 1, . . . , r) satisfy the conditions |yρ (t)| ≤ lp (lp constant) but are otherwise arbitrary one-valued continuous functions of the time t possessing as many derivatives as necessary; points of discontinuity of the first kind are admitted, but their number in any finite interval must also be finite.’ Bulgakov (1946) derives the solution of some set of dynamic equations of motion in the form  Z tX r r  X d σjρ (τ ) yρ (t − τ )dτ (6.21) xj (t) = [Sjρ yρ (t) − δjρ yρ (0)] + 0 ρ=1 dτ ρ=1

where yρ (t) (ρ = 1, . . . , r) are excitations, and the expressions for Sjρ and σjρ (τ ) depend on the particular system under consideration. We refrain from reproducing the expressions for σjρ (t), Sjρ and δjρ . According to the author, the expression under the sum sign ‘attains its maximum value mjρ (t) if the absolute magnitude of yρ (t − τ ) retains the value lρ , whereas its sign changes so as to make all elements of the integral and the two other terms positive. In other words, the sign of yρ (t−τ ) must be altered for τ = t1 , t2 , . . . and possibly for τ = 0, t, the ti ’s being the successive positive roots of the equation βjρ (t) = 0.’

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This equation is not reproduced here. Bulgakov (1946) writes the maximum maximorum as the sum of the convergent series:     Z t1  Z t2  d d σjρ (τ ) (−lp )dτ + σjρ (τ ) (+lp )dτ + · · · mjρ (∞) = Sjρ (∓lp ) ± dτ dτ t1 0 (6.22) The signs must be chosen so that both terms will be positive. Bulgakov (1946) expresses mjρ (∞) as mjρ (∞) = Ajρ lρ , with   Ajρ = ∓Sjρ ± σjρ (0) − 2σjρ (t1 ) + 2σjρ (t2 ) − · · · (6.23)

where Ajρ has a physical sense of influence coefficients. Finally, the maximum deviation in the response xj (t) caused by all disturbing actions combined is expressed in the following form: Mj (∞) =

r X

Ajρ lρ

(6.24)

ρ=1

Special cases of this problem have been discussed in his book (Bulgakov, 1939) and paper (Bulgakov, 1940). Bulgakov’s solution is referred to in the Russian literature as the method of accumulation of disturbances. Bulgakov’s approach was extended by Balandin (1993), Bulgakov and Kozovkov (1950), Gnoenskii (1961), Korneev and Svetlitskii (1995), Krichevskii and Ulanov (1973), Mironovkii and Slaev (2002), Naishul and Svetlitskii (1956), Nguyen (1962, 1965), Roitenberg (1958), Svetlitskii (1958, 1967, 2003), Svetlitskii and Prokhorov (1970), and Svetlitskii and Stasenko (1967). Analogous solutions were also developed in the West, especially in the theory of control (see, e.g., Michalopoulos and Riley (1972), Sage (1968)) and is referred to as the bang-bang optimal control problem. Later works, in the context of free and forced vibrations, are also associated with evaluation of bounds albeit without uncertainty aspects. The reader may like to consult, for example, works by Hu and Schiehlen (1996), Sharuz (1996), Sharuz and Krishna (1996), and Schiehlen and Hu (1997). 6.4

6.4.1

Non-probabilistic, Convex-Theoretic Modeling of Scatter in Material Properties Introduction

Behavior, vibration, and stability of viscoelastic structures have been dealt with in a number of monographs (Bland, 1960; Christensen, 1982; Fl¨ ugge, 1975). In these studies, material properties of the structure have been fixed at some deterministic parameters. However, it is well known that the viscoelastic properties of structures exhibit a large scatter (Koltunov, 1976). This scatter is usually accounted for by considering the material properties as random variables. Cozzarelli and Huang (1971) and Huang and Cozzarelli (1972) were apparently the first investigators

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151

to include material uncertainty in their analyses. Hilton, Hsu and Kirby (1991) extended the elastic-viscoelastic analogies to the stochastic case due to random linear viscoelastic material properties. Both Gaussian and beta distributions were considered for modeling the uncertainty in the data. In probabilistic analyses, the needed probabilistic information for analysis was postulated as given. For example, Huang and Cozzarelli (1973) utilized a log-normal density or a truncated log-normal density. However, extensive experimental data are needed to substantiate the probability densities with regards to the data. More often, the necessary data are simply lacking, or only partial information is available about the parameters. In these circumstances, the usefulness of the results of probabilistic modeling based on incomplete data may be questionable. Shinozuka (1987) studied the response variability in the stochastic context. Upper bound results were derived in two cases in which the spectral density function of the stochastic field has assumed limiting shapes. The importance of such response variability studies is immediately understood if one recognizes that it is rather difficult to estimate experimentally the autocorrelation function or, equivalently, the spectral density function of the stochastic variation of material properties. In this section, we present the non-probabilistic, convex modeling (Ben-Haim and Elishakoff, 1990; Elishakoff and Ben-Haim, 1990a) for dealing with material uncertainty and attendant response variability for viscoelastic structures. In particular, the problem of forced vibrations of viscoelastic beams is studied. First, the analytical solution by Inman (1989) is generalized for a deterministic set of variables, describing material properties. Next, these variables are treated as varying in a solid ball in the four-dimensional space, thus modeling the scatter in material properties. The response uncertainty is directly related to the material uncertainty. The least favorable response needed for the design of the structure is determined. 6.4.2

Basic equations for vibrating viscoelastic beam

This section follows closely the study by Elishakoff, Eliseeff and Glegg (1994b). Transverse vibration of a viscoelastic beam is governed by the following differential equation: Z t ∂ 4 u(x, τ ) ∂ 4 u(x, t) + I g(t − τ ) dτ E0 I ∂x4 ∂x4 0 (6.25) ∂u(x, t) ∂ 2 u(x, t) +c +ρ = f (x, t) ∂t ∂t2 where E0 is Young’s modulus, c is the viscous damping coefficient, I is the moment of inertia, u(x, t) is the transverse displacement, which is a function of axial coordinate x and time t, g(t − τ ) is the viscoelastic kernel, ρ is the mass per unit length of the beam, and f (x, t) is the transverse excitation. Equation (6.25) is supplemented by appropriate boundary and initial conditions. We shall seek the solution of the problem through separation of variables. Normal

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modes φn of the beam with constant Young’s modulus E0 , uniform cross-section, and uniform mass density satisfy the equation ∂ 2 φn ∂ 4 φn + ρ =0 (6.26) ∂x4 ∂t2 Letting u = φn (x) exp(iωn t) with the circular frequency ωn = βn2 , the equation for φn reads E0 I

d4 φn (x) = ρβn4 φn dx4 The normal modes satisfy the orthogonality condition Z l φn (x)φm (x)dx = δmn , E0 I

0

E0 I

Z

l

0

d4 φn (x) φm (x)dx = ρβn4 δmn dx4

(6.27)

(6.28)

where δmn is the Kronecker delta, and l is the length of the beam. The solution of Eq. (6.25) is written in the following form: u(x, t) =

∞ X

an (t)φn (x)

(6.29)

n=1

where an (t) is the modal response, which is a function of time t. We substitute Eq. (6.29) into Eq. (6.25), multiply by φm , and integrate with respect to x over the interval (0, l) to yield ∞ X d2 a n dan ρ 2 (φn , φm ) + c (φn , φm ) + E0 Ian (t)(φIV n , φm ) dt dt n=1 Z t +I g(t − τ )an (τ )(φIV n , φm )dτ = (f, φm )

(6.30)

0

φIV n

where is the fourth-order derivative of φn with respect to x, and (φ, ψ) is the inner product defined by Z l (φ, ψ) = φ(x)ψ(x)dx (6.31) 0

Because of the orthogonality property (6.28), we are left with Z t d2 a m dam (φIV (φIV , φm ) m , φm ) ρ 2 +c + E0 I am + I g(t − τ )am (τ ) m am dτ dt dt (φm , φm ) (φm , φm ) 0 (f, φm ) = , (m = 1, 2, . . . ) (6.32) (φm , φm ) or, more specifically, c dam d2 a m 4 4 + + βm am (t) + βm 2 dt ρ dt

Z

t 0

1 1 g(t − τ )am (τ )dτ = fm (t), E0 ρ

g(t) = 0 for t < 0

(6.33)

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where (f, φm ) (6.34) (φm , φm ) We assume zero initial conditions. Taking the Laplace transform of Eq. (6.33) yields   β4 1 c 4 Am (s) + m G(s)Am (s) = Fm (s) (6.35) s2 + s + β m ρ E0 ρ We now follow Golla and Hughes (1985) and Inman (1989) and take the following analytical form for G(s): αs2 + γs G(s) = 2 (6.36) s + βs + δ Some restrictions will be imposed later on the values of the parameters α, β, γ and δ. Now, G(s) can be represented as   γ  s δ G(s) = α 1 + −β 2 − 2 (6.37) α s + βs + δ s + βs + δ so that the positive kernel function g(t) is given as    aeat − bebt γ √ δ −β − √ e−(1/2)βt sinh Dt g(t) = α δ(t) + (6.38) α a−b D where δ(t) is the Dirac delta function, and β √ β √ β2 −δ (6.39) a = − + D, b = − − D, D = 2 2 4 In addition, we assume that the discriminant D is nonnegative. Substitution of Eq. (6.36) into Eq. (6.35) results in   4 c αs2 + γs 1 βm 2 4 s + s + βm + Am (s) = Fm (s) (6.40) 2 ρ E0 s + βs + δ ρ Equation (6.40) corresponds to an equation with the fourth degree of s; hence, it can be written as two second-degree equations of the form           1 F (s) m1 m2 2 c1 c2 k1 k2 Am (s) m  (6.41) s + s+ = ρ m2 m3 c2 c3 k2 k3 Zm (s) 0 fm (t) =

where Zm (s) is an artificial variable; for the general issue of artificial variables, one may consult studies by Ahrens (1990), McTavish (1988), McTavish, Hughes, Soucy and Graham (1992), and Ottl (1987). The matrices in Eq. (6.41) are supposed to be symmetric to simplify the analysis. The artificial variable Zm (s) is chosen in such a way that the transfer functions in Eqs. (6.40) and (6.41) are coincident. Equation (6.40) reads "       4 4 c cβ βm cδ βm γ 4 3 4 2 4 s + + β s + βm + +δ+α s + + ββm + s ρ ρ E0 ρ E0 # (6.42) 1 2 4 + δβm am (s) = Fm (s)(s + βs + δ) ρ

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Equation (6.41) becomes Fm (s) ρ =0

(m1 s2 + c1 s + k1 )Am + (m2 s2 + c2 s + k2 )Zm =

(6.43a)

(m2 s2 + c2 s + k2 )Am + (m3 s2 + c3 s + k3 )Zm

(6.43b)

To eliminate Zm (s) from Eqs. (6.43a) and (6.43b), we first obtain from Eq. (6.43b) m2 s 2 + c 2 s + k 2 Am (s) m3 s 2 + c 3 s + k 3 We substitute Eq. (6.44) into Eq. (6.43a) to yield Zm (s) =

(6.44)

[(m1 m3 − m22 )s4 + (m3 c1 + m1 c3 − 2m2 c2 )s3

+ (m1 k3 + m3 k1 + c1 c3 − c22 − 2k2 m2 )s2 + (c1 k3 + c3 k1 − 2k2 c2 )s 1 + (k1 k3 − k22 )]Am (s) = (m3 s2 + c3 s + k3 )Fm (s) ρ

(6.45)

Comparison of Eqs. (6.42) and (6.45) shows that it is sufficient to choose parameters as m1 = m3 = 1, m2 = 0, c3 = β, k3 = δ

(6.46)

In addition 4 c αβm 4 , k1 = c22 + βm + ρ E0 We are left with two additional equations

c1 =

(6.47)

4 4 βm αδβm , δc22 − k22 = − (6.48) E0 E0 We consider the specific case in which the parameters α, β and γ are dependent. In particular, we assume that αβ = γ, yielding βc2 (6.49) k2 = 2 Then Eq. (6.48) takes the form  2  4 β αδβm − δ c22 = (6.50) 4 E0

βc22 − 2k2 c2 = (γ − αβ)

which suggests that we should require that β 2 > 4δ for c2 to be real. This, however, is implied by Eq. (6.39). Hence, we put, as is done in Inman (1989), β 2 = 8δ which is in the allowable range of variation. Equation (6.50) then leads to r α 2 c2 = β E0 m Bearing in mind Eq. (6.49), we obtain r   √ 2αδ 2 2α 4 k2 = βm , k 1 = 1 + βm , c3 = β = 2 2δ E0 E0

(6.51)

(6.52)

(6.53)

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The function G(s) in Eq. (6.36) becomes

√ α(s2 + 2s 2δ) √ (6.54) s2 + 2s 2δ + δ Effectively we are left with two parameters, α and δ. Substitution of k1 , k2 , k3 , c1 , c2 , and c3 into Eq. (6.41) results in r   α 2 c    βm     a˙ m ρ E0 1 0 a ¨m  r +   z˙m √ 0 1 z¨m α 2 βm 2 2δ E0 r (6.55)     2αδ 2 α 4   1 β (1 + 2 E0 )βm E0 m   am =  ρ fm (t) + r   zm 2αδ 2 0 βm δ E0 which should be solved for the specific excitation. G(s) =

6.4.3

Application to a simply supported beam

Consider now the beam that is simply supported at its ends. The mode shapes and the eigenvalues are r  mπ 4 E I 2 mπx 0 4 sin , βm (6.56) = φm = l l l ρ Consider first an excitation that is spacewise uniform and timewise harmonic as f (x, t) = q0 exp(iωt)

(6.57)

We will confine ourselves to the steady-state solution. Via Eq. (6.34) we obtain fm (t) = Rm exp(iωt)

(6.58)

where

√ 2lq0 [1 − (−1)m ] Rm = mπ A steady-state solution of Eq. (6.55) is sought in the form

(6.59)

am (t) = am exp(iωt), zm (t) = z m exp(iωt)

(6.60)

Substituting Eq. (6.60) into Eq. (6.55) yields "   α c 4 2 βm am = − ω + iω + 1 + 2 ρ E0 ! #−1 r r α 2 2αδ 2 1 + β iω + β Pm Rm E0 m E0 m ρ z m = P m am Pm =

1 √ + ω 2 − 2 2δiω − δ

r

α 2 β iω + E0 m

r

2αδ 2 β E0 m

(6.61a) !

(6.61b) (6.61c)

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The steady-state response in the middle cross-section is ) ( ∞ X am m iωt [1 − (−1) ]e U (l/2, t) = Re 2 m=1

(6.62)

where Re denotes the real part. Consider now a case of a general nonharmonic excitation. We seek the complementary solution of Eq. (6.55) in the following form: am (t) = E1 ert , zm (t) = E2 ert

(6.63)

Substituting Eq. (6.63) into Eq. (6.55) and requiring that E1 and E2 do not vanish simultaneously, we arrive at the following equation:       √ α 2c √ c 4 3 4 r + 2 2δ + r + δ+ 1+ βm + 2δ r2 ρ E0 ρ     (6.64) √ α cδ 4 4 βm r + β m δ = 0 + 2 2δ 1 + + ρ E0 We first study the behavior of the roots. It can be shown, through use of the Routh–Hurwitz criterion, that the equation r 4 + λ3 r 3 + λ2 r 2 + λ1 r + λ 0 = 0

(6.65)

has roots with negative real parts, if and only if λ1 λ2 λ3 > λ21 + λ0 λ23

(6.66)

It can be verified that this condition is satisfied. It should be noted that inequality (6.66) is also satisfied for the vanishing viscous damping. This implies that the viscoelastic behavior is introducing an additional damping to the structure. Since the roots of the characteristic equation have negative real parts, the complementary solution vanishes when t → ∞. We are interested in the behavior of the displacement after sufficient time has elapsed from the start. In this case, the complementary solution can be discarded. For the excitation f (t) = bt

(6.67)

Eq. (6.34) yields

√ b 2I fm (t) = hm t, hm = [1 − (−1)m ] ρmπ The particular solution of Eq. (6.55) is given in the following form: am (t) = S1 t + S2 , zm (t) = G1 t + G2 Substitution of am (t) and zm (t) in Eq. (6.69) into Eq. (6.55) results in ! r 2 α0 c hm + 8 hm , S1 = 4 , S2 = − 2f 4 βm δ E 0 βm ρβm v u √ ! r 2α hm α 4 α 2c hm u 3 t √ +√ G1 = − , G2 = 2 + 4 2 δE0 βm βm E 0 δ ρβm δ δ E0

(6.68)

(6.69)

(6.70)

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The solution for am (t) can be written as   1 hm am (t) = 4 t − 4 ceq βm ρβm

(6.71)

where ceq is an equivalent viscous damping, given as follows, representing the combined effect of both genuine viscous damping and additional damping introduced by the viscoelastic behavior: r 4 2 αβm ceq = c + e c, e c = 2ρ (6.72) δ E0 Note that, as expected, for α = δ = χ and χ tending to zero, the equivalent viscous damping ceq reduces to the genuine viscous damping c, since the contribution due to viscoelasticity vanishes. The non-dimensional response reads 1 e am (t) = am (t) = K1 (K2 T − 1) (6.73) re p where re = I/A is radius of inertia, T = ωm t is non-dimensional time, and K1 =

Finally, we obtain

ρβ 2 hm ceq , K2 = m 8 ρβm re ceq

U (l/2, t) =

∞ X

(6.74)

am (t) sin(mπ)

(6.75)

m=1

6.4.4

Least and most favorable responses

Hereinafter, we will apply the methodology of anti-optimization for modeling uncertainty in material properties of viscoelastic structure. We discuss the realistic situation, assuming that we possess only some limited information on material parameters α, β, γ and δ; namely, that they belong to some convex sets. The displacement in either Eq. (6.62) or Eq. (6.75) is denoted as U (l/2, t) = f (α, β, γ, δ)

(6.76)

We expand function f (α, β, γ, δ) in Taylor series and take into account only the linear terms as f (α, β, γ, δ) ' f (α0 , β0 , γ0 , δ0 ) + Φ> ζ where Φ= ζ=





α1

∂f ∂f ∂f ∂f , β1 , γ1 , δ1 ∂α ∂β ∂γ ∂δ

>

α − α 0 β − β0 γ − γ 0 δ − δ 0 , , , α1 β1 γ1 δ1

(6.77)

at (α, β, γ, δ) = (α0 , β0 , γ0 , δ0 ) >

(6.78) = (ζ1 , ζ2 , ζ3 , ζ4 )

>

and α0 , β0 , γ0 and δ0 are the average values of parameters. In addition, parameters α1 , β1 , γ1 and δ1 are introduced for dimensionless formulation. In particular, the

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following values have been taken: α1 = 1 N/m , β1 = 1 s−1 , γ1 = 1 N/m s, δ1 = 1 s−2 . The scatter is modeled as the vector ζ belonging to some set Z(P ) taken as a four-dimensional solid ball given by Z(P ) = {ζ|ζ > ζ ≤ P 2 }

(6.79)

where P is the radius of the sphere. It should be noted that the derivation of radius P assumes that the parameters α, β, γ and δ are of similar order of magnitude and that the variations of these values are also of similar magnitude. On the other hand, McTavish (1988) gives a numerical example (taken from Bagley and Torvik (1983)) that demonstrates that these parameters have multiple orders of magnitude difference. However, this may be simply remedied by choosing α1 , β1 , γ1 and δ1 such that an approximate distribution may be represented by a sphere in the appropriately transformed coordinates. We are interested in the maximum possible value the function f (α, β, γ, δ) may take in this solid ball. This will result in the least favorable response the system may experience when the parameters vary in the solid ball defined by Eq. (6.79). We will also be interested in the minimum value of f (α, β, γ, δ). This will be identified with the best possible response. Let µ1 and µ2 denote the maximum and minimum of the function f (α, β, γ, δ) within the solid ball, given in Eq. (6.79): µ1 (P ) = max f (α, β, γ, δ), µ2 (P ) = min f (α, β, γ, δ) ζ∈Z

ζ∈Z

(6.80)

Since Z(P ) is convex, the linear function f (α, β, γ, δ) reaches its maximum and minimum values on the boundary; i.e., on the set B(P ) = {ζ|ζ > ζ = P 2 }

(6.81)

We define the Lagrangian as follows, in view of Eq. (6.77): L(ζ, λ) = f (α0 , β0 , γ0 , δ0 ) + Φ> ζ + λ(ζ > ζ − P 2 )

(6.82)

and demand that ∂L = Φ + 2λζ = 0 ∂ζ

(6.83)

yielding ζ =−

1 Φ 2λ

However, in view of the equality ζ > ζ = P 2 , we arrive at p PΦ Φ> Φ , ζ = ±p λ=± 2P Φ> Φ

(6.84)

(6.85)

where the plus sign is associated with the maximum and the minus sign with the minimum.

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Therefore, the final results for the least favorable response (denoted as LF R) and for the most favorable response (indicated as M F R) are p (6.86a) LF R = µ1 (P ) = f0 + P Φ> Φ p M F R = µ2 (P ) = f0 − P Φ> Φ (6.86b) where f0 = f (α0 , β0 , γ0 , δ0 ). Equation (6.76) can be put in the following form, in conjunction with Eq. (6.58): 1 (6.87) f (α, β, γ, δ) = (Ψeiωt + Ψ∗ e−iωt ) 2 where asterisk denotes complex conjugate. The function Ψ, with term m = 1, retained is √ B1 2q0 2l α1 = (δ − ω 2 + iβω) (6.88) , B1 = C1 πρ where   (E0 − α)I  π 4 cβ 4 + ω2 C1 = ω − δ + ρ L ρ     (6.89) cδ (βE0 − γ)I  π 4 c δE0 I  π 4 2 + iω + − +β ω + ρ L ρ ρ L ρ In the following, all the functions and their derivatives are evaluated at (α, β, γ, δ) = (α0 , β0 , γ0 , δ0 ). The derivatives of Ψ are calculated as      ∂Ψ 1 4q0 I π 4 2 2 = 2 (δ − ω + iβω) ω (6.90a) ∂α C1 πρ ρ l    4q0 iω E0 I  π 4 iωc ∂Ψ 2 2 = C − (δ − ω + iβω) −ω + + (6.90b) 1 ∂β πC12 ρ ρ l ρ   ∂Ψ 1 4q0 iω  π 4 = 2 (δ − ω 2 + iβω) I (6.90c) ∂γ C1 πρ ρ l    4q0 E0 I  π 4 iωc ∂Ψ = C1 − (δ − ω 2 + iβω) −ω 2 + + (6.90d) 2 ∂δ πC1 ρ ρ l ρ Equations (6.86a) and (6.86b) become !1/2 ∂Ψ0 2 ∂Ψ 2 ∂Ψ 2 ∂Ψ 2 LF R = f0 − P + + + (6.91a) ∂α ∂β ∂γ ∂δ M F R = f0 + P

!1/2 ∂Ψ 2 ∂Ψ 2 ∂Ψ 2 ∂Ψ 2 + + + ∂α ∂β ∂γ ∂δ

(6.91b)

In the case of the timewise linearly increasing excitation given in Eq. (6.67), the least favorable response as a function of non-dimensional time T reads, in view of Eq. (6.73), as !1/2 ∂am 2 ∂am 2 + (6.92a) LF R(T ) = am (T ) + P ∂β ∂α !1/2 ∂am 2 ∂am 2 + M F R(T ) = am (T ) − P (6.92b) ∂β ∂α

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LFR

LFR

MFR

MFR

(a)

(c)

LFR

LFR

MFR

MFR

(b)

(d)

Fig. 6.2 Dependence of the least favorable response (LF R) and most favorable response (M F R) (a) (a) (b) (a) upon uncertainty radius P : (a) a0 = 34.35×109 N/m2 , δ0 = 7.8125×105 s−2 ; (b) a0 = 2a0 , (b)

δ0

(a)

(c)

= δ0 ; (c) α0

(a)

(c)

= α0 , δ 0

(a)

(d)

= 2δ0 ; and (d) α0

(a)

(d)

= 2α0 , δ0

(a)

= 2δ0 .

For numerical values hm = βm , we have √ √ 2 αωm ∂am 2 2 ωm ∂am =− , =− 2 ∂α δ E0 ∂δ δ E0

6.4.5

(6.93)

Numerical examples and discussion

Consider first the numerical results for the case of the harmonic excitation. Figure 6.2(a) portrays the dependence of the least and most favorable responses on the uncertainty radius P . The parameters are fixed at the values defined by the superscript ( · )(a) as (a)

2

(a)

a0 = 34.35 × 109 N/m , δ0 (a)

E0

= 7.8125 × 105 c−2 , m = 1, (a)

= 68.9 × 109 N/m2 , ρ(a) = 27 kg/m, (β1 )4 = 3.0688 × 105 ,

I (a) = 10−4 m4 , l(a) = 3 m

(6.94)

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161

norm of ∇f

δ √

Fig. 6.3 Absolute value φ = Φ> Φ of the norm of gradient of the function f (α, β, γ, δ) as a function of parameters α and δ.

The value of the viscous damping is fixed at c(a) = 5983.2 kg/(m · s), corresponding to the coefficient ζ = c/(2ρω1 ) = 0.2. As is seen in Fig. 6.2(a), the amplitude of vibration for the nominal structure is 1.424 × 10−3 . For the beam without viscoelastic effects, the response of the structure evaluated through the use of the expression 4q0 Y2 = (6.95) π|EI(π/l)4 − ω 2 ρ + iωc|

is 1.979×10−3 m3/2 . This is 28% more than the nominal response of the viscoelastic beam. The least favorable response linearly increases with P , whereas the most favorable response linearly decreases with P . At P = 1000, for example, LF R = 1.293 × 10−3 m3/2 ; i.e., about 9% larger than the nominal response. Figure 6.2(b) (b) (a) is associated with the same data as in Fig. 6.2(a) except that now a0 = 2α0 . (c) (a) In Fig. 6.2(c), the value α0 = α0 , but the nominal value of δ is twice as much (c) (a) as in Fig. 6.2(a), namely, δ0 = 2δ0 . In Fig. 6.2(d), the data are the same as in (d) (a) (d) (a) Eq. (6.94) except α0 = 2α0 and δ0 = 2δ0 . Figure 6.3 depicts the behavior p of the absolute value φ of the gradient of the function f (α, β, γ, δ), namely, φ = Φ> Φ defining the least and most favorable responses in Eqs. (6.86a) and (6.86b). As is seen for large values of α and small values of δ, the values of φ can be significant, contributing to larger values of the least favorable response. On the other hand, for materials with small values of α and relatively large values of δ, the effect of uncertainty will be less pronounced. Figure 6.4 is associated with the response to timewise linearly increasing excitation. It should be noted that the scatter in the responses stems from the constant term in Eq. (6.71). Remarkably, the viscoelastic beam and its elastic counterpart share the same slope in the asymptotic response, when t → ∞. In these circum-

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LFR MFR

Fig. 6.4 Response variability region in the beam subjected to timewise linearly increasing exci(3) (3) tation (α0 = E0 , δ0 = 10 s−2 ); response variability region is hatched.

LFR MFR

Fig. 6.5 Response variability region in the beam subjected to timewise linearly increasing exci(4) (3) (4) (3) tation (α0 = 2α0 , δ0 = 2δ0 ); response variability region is hatched.

stances, however, use of the nonlinear governing equations is necessary, since the response becomes too large to justify employing the linearized analysis. The data (3) (3) in Fig. 6.4 are the same as in Fig. 6.2(a) except α0 = E0 and δ0 = 10 s−2 ; in addition, the uncertainty radius is fixed at P = 2. The upper curve represents the LF R, whereas the lower curve represents the M F R. The hatched area is a region of the response variability (or uncertainty). Equation (6.71) in conjunction with Eqs. (6.77) and (6.88) was utilized for obtaining the M F R and the LF R. It is noteworthy that the contribution of the viscoelastic effects to the equivalent viscous damping ceq in Eq. (6.72) through e c is much more important than that of the purely viscous damping for the numerical data chosen in Fig. 6.4. Therefore, the time shift predicted by the second term in Eq. (6.71) is non-negligible. In Fig. 6.5, the data (4) (3) (4) (3) were changed to α0 = 2α0 and δ0 = 2δ0 with the same value of uncertainty radius. As is seen, the response variability is reduced in comparison with Fig. 6.3.

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LFR

MFR

Fig. 6.6 Response variability region in the beam subjected to timewise linearly increasing exci(5) (3) (5) (3) tation (α0 = α0 /2, δ0 = δ0 /2); response variability region is hatched.

(5)

(3)

In Fig. 6.6, the data are the same as in Fig. 6.2(a) except that now α0 = α0 /2 (5) (3) and δ0 = δ0 /2. The response uncertainty region is now increased. This is understandable since the relative measure of uncertainty (RM U ) could conveniently be defined as P (6.96) RM U = Q where P is the radius of input uncertainty, and Q is the measure of the nominal location of the uncertain vector as s  2  2  2  2 β0 γ0 δ0 α0 Q= + + + (6.97) α1 β1 γ1 δ1 where α1 , β1 , γ1 and δ1 are defined in Eq. (6.78). With increasing RM U , one would anticipate the increase in the response uncertainty. This qualitative observation is in agreement with the obtained results. Moreover, the method presented in this section provides a rigorous way to quantify the response variability, if scant information on variability is available. 6.5 6.5.1

Anti-Optimization of Earthquake Excitation and Response Introduction

There is a variety of literatures on modeling earthquake excitations. Modern analysis is based on the recognition that this excitation is an uncertain process. In an overwhelming majority of the studies, the uncertainty is modeled as a random process, either stationary or nonstationary with various approximations and attendant models. The works by Drenick (1970, 1977b), Drenick, Novomestky and Bagchi (1989), Drenick and Yun (1979), Elishakoff and Pletner (1991), and Shinozuka (1970) have been the exceptions in the literature dedicated to uncertainty modeling

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the earthquakes, in the sense that they utilized an alternative, non-probabilistic avenue. Drenick (1970, 1977b) used a constraint on the total energy, which the earthquake is likely to develop at the certain site, as a description of uncertainty. He used the Cauchy–Schwarz inequality to determine the maximum response of the system to such an excitation. In the opinion of several investigators such a bound was too conservative (Drenick, 1993). Shinozuka (1970) has suggested characterizing the earthquake uncertainty by specifying an envelope of the Fourier amplitude spectrum. Numerical calculations have demonstrated that the maximum response of the structure predicted by this method is less than that predicted in Drenick (1970, 1977b), Drenick, Novomestky and Bagchi (1989) and Drenick and Yun (1979). Elishakoff and Pletner (1991) investigated the modification of the response prediction when the global information on the excitation is increased. In particular, the maximum possible response, which the structure may develop, was evaluated under the assumption that only the bound on base acceleration is known; then the maximum response was modified under the assumption that, in addition to the base acceleration bound, the bounds on base velocity and/or displacement were specified. In this section, we resort to ellipsoidal modeling of earthquake excitation. 6.5.2

Formulation of the problem

Consider a single-degree-of-freedom linear system subjected to earthquake excitation. The motion is governed by m¨ x + cx˙ + kx = −m¨ xb

(6.98)

where x(t) is the relative displacement of the structure from the base, c is the damping coefficient, k is the stiffness, m is the mass, and x ¨b (t) is the acceleration of the earthquake excitation. If the excitation force −m¨ xb (t) is known, the response is given by the Duhamel integral Z t x(t) = [−m¨ xb (τ )]h(t − τ )dτ (6.99) 0

Here h(t) is the impulse response function given as 1 exp(−µω0 t) sin ωd t (6.100a) h(t) = mωd p ωd = ω 0 1 − µ 2 (6.100b) p p where ω0 = k/m is the natural circular frequency, and µ = c/(2 k/m) is the damping ratio. Let us represent the excitation as a series in terms of complete and orthogonal deterministic set of functions {φi (t)}: ∞ X x ¨b (t) = Ai φi (t) (6.101) i=1

where Ai is the coefficient, and {φi (t)} satisfies the orthogonality condition Z T φi (t)φj (t)dt = 0 for i 6= j (6.102) 0

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in interval (0, T ), where T is the duration of the earthquake. If the excitation is known, the coefficients Ai are readily obtainable as follows, by multiplying both sides of Eq. (6.101) by φj (t), integrating the result in the interval (0, T ), and using the orthogonality property (6.102): Z T 1 x ¨b (t)φi (t)dt (6.103) Ai = 2 Vi 0 where Z T Vi2 = [φi (t)]2 dt (6.104) 0

The response, given in Eq. (6.99), is rewritten as ∞ X x(t) = Ai ψi (t)

(6.105)

i=1

where

ψi (t) = −m

Z

T

0

φi (t)h(t − τ )dτ

(6.106)

Let us suppose now that we possess some fragmentary information at a given site, namely, we assume that the historical data is available on the accelerograms (1) (m) x ¨b , . . . , x ¨b , where the superscript denotes the serial number of the earthquake, and m is the total number of accelerograms. Then, using the decomposition of type (6.101) for each of the earthquake realization ∞ X (k) (k) x ¨b (t) = Ai φi (t), (k = 1, . . . , m) (6.107) i=1

we arrive at m vectors of the excitation parameters (1)

(1)

(2)

(2)

A(1) = (A1 , . . . , AN )> , A(2) = (A1 , . . . , AN )> , .. . (m)

A(m) = (A1

(6.108)

(m)

, . . . , AN )>

where N is the number of most significant terms in the series (6.101) and (6.105), which are rewritten as N N X X x ¨b (t) = Ai φi (t), x(t) = Ai ψi (t) (6.109) i=1

i=1

Thus we replace m accelerograms by m points in an N -dimensional space. To this end, to use the ellipsoidal modeling of these point-images of original accelerograms, we model these points as belonging to an N -dimensional ellipsoid A> W A ≤ θ 2

(6.110)

where W is a symmetric positive-definite matrix defining the shape of the ellipsoid, and θ is a positive constant defining the size of the ellipsoid. The way of obtaining the matrix W and constant θ will be discussed in Sec. 6.5.4.

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Maximum structural response

We are interested in determining the maximum response of the structure at time instant t. Mathematically the problem reads as follows: N X maximize Ai ψi (t) = A> ψ(t) (6.111a) i=1

subject to A> W A ≤ θ2

(6.111b)

We define the Lagrangian

L(t, λ) = A> ψ(t) + λ(θ2 − A> W A)

(6.112)

From the stationary conditions of L with respect to A and λ, we obtain q −1 ψ > W −1 ψ W ψ Aworst = θ q , λ= 2θ ψ > W −1 ψ

(6.113)

The worst response is obtained by substituting Eq. (6.113) into Eq. (6.101): q xworst (t) = A> ψ = θ ψ > W −1 ψ (6.114) worst

Once the basic excitation functions φi (t) are chosen, the basic response function ψi (t) are readily obtained from Eq. (6.106). Then, having the information on matrix W and constant θ from the accelerograms, we find the maximum response by employing Eq. (6.114). 6.5.4

Ellipsoidal modeling of data

6.5.4.1 Basic ideas Consider any collection of data to be processed by the procedure in Sec. 6.5.3. Assume that one record consists of N numerical parameters, so that any observation j of the phenomena is fully described by a point Pj in N -dimensional Euclidean vector space EN . The problem treated in this section is to find the smallest (in some sense) ellipsoid containing all the observed data; i.e., to set the matrix W in Eq. (6.110), and the shift of the origin to the center of the ellipsoid. This problem is rather hard to solve, if the ellipsoid width defining the minimum volume is required, the approach leads to cumbersome optimizations that may be impractical. So the problem is treated aiming at a simple, as far as possible, solution. In this view, and for simplicity of graphic representation only, let us consider the three-dimensional case as shown in Fig. 6.7; i.e., a collection of m points P1 , . . . , Pm that are the recorded observations of the phenomenon to be included in the ellipsoid, and let       x11 x21 xm1       x1 =  ...  , x2 =  ...  , . . . , xm =  ...  (6.115) x1N

x2N

xmN

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π'

π''

Fig. 6.7 Collection of m = 7 points representing the recorded observations, to be included in the N = 3 dimensional ellipsoid.

be the coordinate vectors of the points, collecting any relevant parameters to identify the characters of the phenomenon at the observation Pi in the reference frame with origin O, which is given, e.g., as follows for N = 3:       0 0 1 (6.116) (e1 , e2 , e3 ) with e1 = 0 , e2 = 1 , e3 = 0 1 0 0 The proposed procedure is conducted basically in two phases:

(A) Find the smallest parallelepiped P in EN containing all homothetic points Pi (i = 1, . . . , m). (B) Find the smallest ellipsoid in EN , also containing all points Pi (i = 1, . . . , m), searching in the family of ellipsoids Eλ that are homothetic to the ellipsoid E1 inscribed in P having the same center as P and a principal axis parallel to the sides of P. The second phase (B) is straightforward, as it will be shown later. Phase (A) is founded on the solution of two auxiliary problems as follows. 6.5.4.2 Preliminary statements Consider the following auxiliary problems: Problem I : Given the set of m points P1 , . . . , Pm in EN , find the couple of parallel hyperplanes, as shown in Fig. 6.8 for three-dimensional case, containing all the points and having the minimum distance from each other. Let π 0 and π00 be any two parallel hyperplanes in EN with α being the unit vector orthogonal to both; let δ 0 and −δ 00 be the orthogonal distances of π 0 and π 00 , respectively, from the point O that is assumed to be the origin of the reference frame as shown in Fig. 6.8 when δ 0 > 0 and δ 00 < 0. Any point Pi is internal to the strip included by π 0 and π 00 if

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π' i

π''

Fig. 6.8

Two parallel planes π 0 and π 00 ; α is the unit vector orthogonal to both.

and only if 0 > 00 x> i α ≤ δ , xi α ≥ δ

(6.117)

X > α ≤ δ 0 1, X > α ≥ δ 00 1

(6.118)

 >   x1 1  ..   ..  1 = . , X =  . 

(6.119)

are satisfied simultaneously. By applying Eq. (6.117) for i = 1, . . . , m, one obtains that the strip includes all points if and only if where

x> m

1

The total width of the strip is given by δ 0 − δ 00 . Therefore, the first problem denoted by Problem I turns out to be find

α ∈ SN 1 , δ 0 and δ 00 that minimize δ 0 − δ 00

subject to X > α ≤ δ 0 1 >

(6.120a) (6.120b)

00

X α≥δ 1

(6.120c)

with SN 1 being the unit sphere in EN . This problem is approached in two steps: Step I.1: For the given α ∈ SN 1

(δ 0α , δ 00α ) that minimize δ 0α − δ 00α

find

>

0α

subject to X α ≤ δ 1

(6.121b)

X > α ≥ δ 00α 1

It is trivially given as

(6.121c)

> 00α > = min x> δ 0α = max x> i α = xs α i α = xr α, δ i

(6.121a)

i

(6.122)

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Step I.2: In order to find the smallest strip containing all points, consider the following problem: α1 that minimizes δ 0α1 − δ 00α1

find

>

0α1

>

00α1

subject to X α1 ≤ δ X α1 ≥ δ

α> 1 α1

1 1

=1

(6.123a) (6.123b) (6.123c) (6.123d)

where δ 0α1 and δ 00α1 are implicitly related to α1 by Eq. (6.122). The above problem is a nonlinear programming problem, mainly because of the constraint α> 1 α1 = 1. Any constrained search procedure can be applied to find the optimal vector α1 . The simplest, and also effective, as tested in this investigation, is a random search. The only difficulty lies in generating unit random vectors α1 ’s satisfying the constraint (6.123d). The following procedure is suggested. Let β be a generic unit random vector and α1 the unit vector that minimizes δ 0α1 − δ 00α1 . Let ni and si (i = 1, . . . , N ) be any random numbers in [0, 1], θi = 2πni the random angle in [0, 2π] of β and SGi the sign of the ith component of β with  SGi = +1 if si ≤ 0.5 (6.124) SGi = −1 if si > 0.5 The components of β are chosen through the following steps: (a) RN = 1, (b) RN −i = sin2 (θp i )RN −i+1 (i = 1, . . . , N − 1) and θi = 2πni , (c) βi+1 = SGi+1 Ri+1 − Ri (i = N − 1, N − 2, . . . , 1).

Problem I is thus solved after Steps I.1 and I.2.

Problem II : Suppose α1 is found, and the first minimal strip is obtained, with width δ10 − δ100 = d1 . Let α1 , . . . , αk (k < N ) be given unit vectors, mutually orthogonal, i.e., α> i αj = δij , and consider the following Problem II: find

αk+1 that minimize δ 0αk+1 − δ 00αk+1 >

subject to X αk+1 ≤ δ

0αk+1

(6.125a)

1

(6.125b)

X > αk+1 ≥ δ 00αk+1 1

(6.125c)

α> k+1 αk+1 = 1 > αi αk+1 = 0,

(i = 1, . . . , k)

(6.125d) (6.125e)

Problem II has k constraints in addition to those in Problem I, and it can be regarded as a generalization of Problem I, by making reference to SN −k , which is the vector subspace of EN orthogonal to α1 , . . . , αk . To generate SN −k , one has to infer an orthogonal unit minimal base, say (β k+1 , . . . , β k+1 1 N −k ), of SN −k . To this aim one can take recourse to the Gram–Schmidt orthogonalization procedure as follows.

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Let β k1 , . . . , β kN −k+1 be the minimal orthogonal base of SN −k+1 so that both sets of unit vectors     α> α> 1 1     .. ..     . .      α>   α>     k−1 k−1      k k > k+1 > B =  (β 1 ) =  αk (6.126) , B      k+1 > ..    (β1 )      . ..      (βkN −k )>    . > (β k+1 ) (β kN −k+1 )> N −k

are orthogonal unit bases for the entire EN , where the first is a known base and the second is known up to αk . Note that  > e1  ..  1 (6.127) B = .  e> N

One has to infer B k+1 from B k . Let γ 0 = αk , and in accordance with the Gram– Schmidt orthogonalization method, one sets γ 1 = λ11 αk + βk2 γ 2 = λ12 αk + λ22 γ 1 + β k3 γ 3 = λ13 αk + λ23 γ 1 + λ33 γ 2 + β k4 .. .

(6.128)

γ N −k = λ1N −k αk + λ2N −k γ 1 + λ3N −k γ 2 + · · · + βkN −k+1 where, by orthogonalization λ11 = −(βk2 )> αk (βk )> γ

λ12 = −(βk3 )> αk ,

λ22 = − (γ3 )> γ 1

λ13 = −(βk4 )> αk ,

> (βk 4 ) γ1 (γ 1 )> γ 1

1

λ23 = −

1

(β k )> γ

λ33 = − (γ 4 )> γ 2

,

2

.. . λ1N −k = −(β kN −k+1 )> αk , λ2N −k = − ···

2

.. . > (βk N −k+1 ) γ 1 , (γ 1 )> γ 1

(βk

> (βk N −k+1 ) γ 2 , (γ 2 )> γ 2

)> γ

N −k N −k+1 N −k+1 λN −k = − (γ )> γ N −k+1

λ3N −k = −

N −k+1

(6.129) After the orthogonal vectors γ i have been found, one can normalize β k+1 as i γi k+1 , (i = 1, . . . , N − k) (6.130) βi = |γ i |

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Note that each γ i for i > 0, and consequently each β k+1 , is orthogonal to αj i (i = 1, . . . , k). In fact, every γ i turns out to be a linear combination of vector αk and vectors β ki . All these β vectors are assumed to have been built up orthogonal to α1 , . . . , αk in the previous step. Hence, one can write for suitable aij γ i = ai0 αk +

NX −k+1

air βkr

(6.131)

air βkr αj = 0

(6.132)

r=1

and > γ> i αj = ai0 αk αj +

NX −k+1 r=1

Therefore, the problem (6.125) is reduced to find

αk+1 ∈ SN −k that minimizes δ 0αk+1 − δ 00αk+1 >

subject to X αk+1 ≤ δ

0αk+1

1

(6.133b)

>

X α1 ≥ δ 00αk+1 1

α> k+1 αk+1

(6.133c)

=1

where αk+1 ∈ SN −k means αk+1 =

(6.133a)

(6.133d)

N −k X

ηik β k+1 i

(6.134)

i=1

with (αk+1 )> αk+1 =

N −k N −k X X

ηik ηrk (β k+1 )> β k+1 r i

(6.135)

i=1 r=1

where, by orthonormality of β’s,

(αk+1 )> αk+1 =

N −k X

(ηik )2

(6.136)

i=1

With the above relations, Problem II is reformulated as find

η k ∈ SN −k that minimizes δ 0αk+1 − δ 00αk+1 N −k X

(6.137a)

ηik β k+1 i

(6.137b)

X > αk+1 ≤ δ 0αk+1 1

(6.137c)

subject to αk+1 =

i=1

>

X α1 ≥ δ

η

k> k

η =1

00αk+1

1

(6.137d) (6.137e)

Thus, one recognizes that Problem II is reduced to Problem I in the vector space EN −k . Then phase (A) of the main problem is articulated in N steps as follows:

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Step A.1: Solve Problem I, and find α1 , δ 0α1 , δ 00α1 . After this step a first strip between hyperplanes π 0 and π 00 is determined. Step A.2: Solve Problem II with k = 2 in the form (6.137), and find αk+1 , δ 0αk+1 , δ 00αk+1 . Step A.3,. . . , A.N : Iterate step A.2 for k = 3, . . . , N . After Step A.N, N couples of hyperplanes are individuated; each couple defines a minimal strip in EN containing all m points. By iterating these N steps, the minimal parallelepiped is found. 6.5.4.3 Search of the smallest ellipsoid containing all points The smallest ellipsoid is assumed to have the same centroid as P and principal axis parallel to the edges of P which are identified in the unit vectors α1 , . . . , αN . The diametral lengths are assumed to be proportional, respectively, to d1 = δ 0α1 − δ 00α1 , . . . , dN = δ 0αN − δ 00αN . Let x0 be the position vector of the centroid of P, and shift the origin of the reference system to x0 . Thus, the data points are identified in the new reference frame by Ai = xi − x0 , (i = 1, . . . , m)

(6.138)

so that the ellipsoid is identified by A> W A ≤ θ 2 In order to identify matrix W , let R be the rotation matrix yielding  > αi  ..  RB 1 =  .  = B N

(6.139)

(6.140)

α> N

In the frame of reference B N , the equation of the ellipsoid is given by m X zi2 > 2 2 = Z QZ ≤ θ d i=1 i

(6.141)

where zi are the coordinates of any point with respect of B N and Q = diag(1/d21 , . . . , 1/d2N )

(6.142)

In the reference frame B 1 , one has the coordinates A of any point P related to the corresponding coordinates in B N > > > P = B> 1 A = BN Z = B1 R Z

(6.143)

A = R> Z, Z = RA

(6.144)

whence

Therefore, the equation of the ellipsoid is Z > QZ = A> R> QRA ≤ θ2

(6.145)

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173

Coordinates of points Pi (i = 1, . . . , 14) to be enclosed by a three-dimensional P2

P3

P4

P5

P6

P7

P8

P9 P10 P11

P12

P13

P14

x1 100 −300 700 400 −200 400 0 100 −300 700 400 −200 −750 1380 x2 500 550 450 470 500 500 500 −200 500 800 600 800 1370 −250 x3 800 800 600 600 400 400 200 200 100 100 900 900 1600 −260

whence, in B 1 (having again shifted back to x0 ) A> W A ≤ θ 2

(6.146)

θ2 = max A> i W Ai

(6.147)

where W = R> QR. Thus, the matrix W has been produced. Now the minimal value for θ 2 can be found by the sequence of unidimensional search i=1,...,m

Note that the above minimal ellipsoid may not be the absolute minimum, since it is minimal under the assumption that it is related to the minimal parallelepiped. The ellipsoid so found is in general included between the ellipsoid contained in P and the one containing P. 6.5.4.4 Numerical application A numerical application of the procedure proposed above is performed. In order to show the results graphically, the example is carried out for N = 3. Let us consider a set of points Pi (i = 1, . . . , m = 14) with the coordinates given in Table 6.1. The unit vectors α1 , α2 , α3 in E3 , orthogonal to the couples of planes that define the minimal strips containing all points P , are       0.527 −0.824 0.205 (6.148) α1 = −0.670 , α2 = −0.510 , α3 = −0.538 −0.657 −0.242 0.713 The minimal distances between the planes are

d1 = 647, d2 = 981, d3 = 3240

(6.149)

Figure 6.9 shows the minimal parallelepiped P, the ellipsoid E1 contained in P, and the principal planes of the ellipsoid. Figure 6.10 shows the sectional view of the three principal planes. Figure 6.11 shows the ellipsoid Eλ found by enlarging homothetically E1 in order to include all points. In conclusion, we would like to note that, Elishakoff, Wang and Qiu (2008) and Wang, Elishakoff and Qiu (2008a) recently addressed the following question: Which analysis is preferable, ellipsoidal modeling or interval analysis? They showed that the type of analytical treatment that should be adopted for non-probabilistic analysis of uncertainty depends on the available experimental data. The main idea is

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Fig. 6.9

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Minimal parallelepiped, ellipsoid contained in it, and principal planes of the ellipsoid.

Fig. 6.10

Sectional views of three principal planes of the ellipsoid.

based on the consideration that the maximum structural response predicted by the preferred method ought to be minimal. Prior to analytical treatment, the existing data ought to be enclosed by the minimum-volume hyper-cube of volume V1 that contains all experimental data. The available data also have to be enclosed by the minimum-volume ellipsoid V2 . If the response R(V1 ) based on V1 is smaller than the response R(V2 ) based on V2 , then one has to prefer interval analysis. If, however, R(V1 ) is in excess of R(V2 ), one is recommended to utilize ellipsoidal analysis. If R(V1 ) and R(V2 ) equal each other, or are in close vicinity, then two approaches are equally valid. For further details and examples of data for 7-bar planar truss

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Fig. 6.11 Ellipsoid ελ obtained by homothetic enhancement of ellipsoid ε1 to include all points of observation.

structure, or for 60-bar space truss structure, one is referred to the above papers.

6.6

6.6.1

A Generalization of the Drenick–Shinozuka Model for Bounds on the Seismic Response Preliminary comments

This section closely follows the article by Baratta, Elishakoff, Zuccaro and Shinozuka (1998). It aims at establishing basic criteria for treatment of uncertainty in seismic excitation through anti-optimization, with the purpose of setting up convex models for forecasting the maximum possible response parameters of an SDOF structure in the seismic environment. Consider the simple linear SDOF shear frame under the action of a ground acceleration a(t). The equation of motion reads u ¨ + 2µω0 u˙ + ω02 u = −a(t)

(6.150)

u(0) = 0, u(0) ˙ =0

(6.151)

where u(t) is the relative displacement from the ground, a(t) is the ground acceleration, ω0 is the natural circular frequency of the undamped system, and µ is the damping ratio. The initial conditions read The solution, as well known, is given by Z t u(t) = − a(τ )h(t − τ )dτ 0

(6.152)

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where h(t) is the impulse response function defined as   1 exp(−µω0 t) sin(ωd ) for t ≥ 0 (6.153) h(t) = ωd  0 for t < 0 p where ωd = ω0 1 − µ2 . The basic idea, first suggested by Drenick (1970), is to find the base accelerogram function yielding the maximum possible value of the response u(t). Following Drenick, assume that a(t) has a finite duration, say T , and a bounded energy E0 as Z T a2 (t)dt = kak2 = E02 (6.154) 0

Let P(E0 ) be the set of functions defined in interval (0, T ) and possessing the given energy E02 . Then the problem reduces to maximize u(t) = kukT = E0 HT , t ∈ (0, T )

subject to a ∈ P(E0 )

The following relation was proved by Drenick: Z T Z T HT2 = max h2 (t − τ )dτ = h2 (t0 − τ )dτ, t ∈ (0, T ) 0

(6.155a) (6.155b)

(6.156)

0

As T → ∞, the base accelerogram ac (t) yielding the maximum peak response, and coined by Drenick (1973) in a later paper as the critical excitation, has a shape coincident with the impulse response function reversed with respect to time, and is given by ac (t) = ±

E0 h(−t) H∞

(6.157)

The maximum peak response is given by kuk = uc (0) = E0 H∞

(6.158)

where uc (t) denotes the response due to acceleration ac (t). Some criticism was expressed by Drenick (1973) himself, mainly about the circumstance that aseismic design based on critical excitation could be ‘... far too pessimistic to be practical.’ Yet Drenick (1973) investigated the ratio of the maximum response due to critical excitation, to the maximum response of actually recorded seismic accelerograms. Response spectra were derived on the basis of both the critical excitation and a number of accelerograms from the set of recorded earthquakes, with attendant comparison of results. It was shown that for the range of structures with natural period from 0.5 to 1.2 sec and damping ratio in the range of 0.02 and 0.10, the ratio of the critical to the experimental response was almost invariably in the neighborhood of 2. By extending the class of possible accelerograms from only the recorded ones to linear combinations of these, the above ratio decreases to the range of values between 1.3 and 1.6.

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In later papers, Baratta (1980, 1981) attempted to develop a modified procedure to obtain the least favorable excitation, searching in the set produced by a generator of artificial earthquake accelerograms. The model chosen for such a generator followed the procedure of Ruiz and Penzien (1969). The class of admissible seismograms was constrained by a specified value of the maximum peak ground acceleration (PGA). The generator’s parameters were calibrated on the basis of four available accelerograms recorded at Tolmezzo (northeastern Italy) during the earthquake of May 1976 (namely the earthquakes of 6–9 May 1976), so that the class of admissible accelerograms was compatible with these samples. The comparison of the least-favorable responses of the undamped structure with the envelope of responses obtained through the processing of four recorded accelerograms, also yielded over-shooting values of about 2 in the range of vibrational period of about 0.5 sec of the structure, to the value of 7 for more deformable structures. The result appears to confirm that at least Drenick’s critical response, if not the critical excitation, should be in the neighborhood of some realizable excitation during an earthquake, if the behavior of the structure is in the elastic range. Analogous results were confirmed by Baratta and Zuccaro (1992) by re-evaluation of the same seismic model. In this case, the analysis was carried out for 5% damping, and the overshooting ratio turned out to be around 2 or 2.5, in a much wider range around T = 0.5 sec. The extension of this range was probably due to the influence of damping. Again, Drenick’s results may well be confirmed: ‘Earthquake-type functions of a given site may give responses that are comparable to those by critical excitation.’ It must also be stressed that, by introducing the least-favorable earthquake in a full probabilistic analysis of seismic hazard, the severity of the approach is quite mitigated by the combination with uncertainties deriving from regional seismicity (Baratta and Zuccaro, 1992). It is possible that inelastic excursions of the structure can further reduce the gap between worst excitations and recorded ones. From the above considerations, one can conclude that the original idea of critical excitation is worth pursuing further. This is why the study in this section is presented to set up a convex model to predict the worst possible shapes of accelerograms acting on linear structures, following the early idea by Shinozuka (1970), that some bound on spectral ordinates can be used to improve a priori knowledge of the excitation. In this section, we modify and combine the approaches of Drenick (1970) and Shinozuka (1970). We are interested in the maximum stress the structure can attain by earthquake action at a given site, and assume that the structural response does not exceed the appropriate linear threshold. Finally, we assume that the base accelerogram belongs to a set of possible accelerograms, whose basis is composed of accelerograms ai (t) (i = 1, 2, . . . ), so that any possible accelerogram a(t) can be expanded in the linear series of the base accelerograms as ∞ X a(t) = ci ai (t) (6.159) i=1

Due to practical numerical reasons, this series should be truncated to N , where N

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is the number of retained terms. One simple example of such a space is the Fourier expansion. As a result, every accelerogram can be represented by a linear combination of harmonic functions. Let us assume that the duration of the earthquake is known, and that it is possible (from geophysics, for instance) to find a bound for the energy of the excitation, as in Drenick’s problem Z T a2 (t)dt ≤ E02 (6.160) 0

If the base functions ai (t) are orthogonal and all scaled to the same energy, every function of the type (6.159) satisfying the constraint (6.160) will possess coefficients ci satisfying the following inequality: N X i=1

c2i ≤ 1

(6.161)

Any possible function that satisfies Eq. (6.160) is referred to as an admissible accelerogram. Thus, admissible functions can be mapped in the Euclidean space of N -dimensional vector c = (c1 , . . . , cN )> . Admissible accelerograms are contained in the unit sphere S0 centered at the origin. The values of the maximum and minimum displacements of the structure are defined for a(t) varying within S0 . Let ui (t) denote the response function under ai (t). The response under any a(t) expressed by Eq. (6.159) are given as follows due to the linearity of system: u(t) =

N X

ci ui (t)

(6.162)

i=1

Thus, with matrix notation, the problem is set as follows for any fixed instant t in (0, T ): find

max c> u(t) c >

subject to c c ≤ 1

(6.163a) (6.163b)

>

where u(t) = (u1 (t), . . . , uN (t)) . Since Problem (6.163) is to maximize a linear function within a convex domain, the anti-optimal solution exists on the boundary of S0 , and the problem is reformulated as find

max c> u(t) c >

subject to c c = 1

(6.164a) (6.164b)

which differs from Problem (6.163) in that the inequality in Problem (6.163) is replaced by an equality. Let us now introduce notations r(c) = c> u, f (c) = c> c − 1

(6.165)

At the solution point, with λ ≥ 0,

∇f = λ∇r for c> u → max.

(6.166)

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u

N=200 N=100 N=50 N=25 t

Fig. 6.12

Influence of N and the number of harmonics on the bound (6.170).

and ∇f = −λ∇r for c> u → min.

(6.167)

c(t) = ±η(t)u(t)

(6.168)

where λ is the Lagrange multiplier (see Sec. 2.4 for details). Hence, with Eq. (6.164b), the optimal and anti-optimal solutions for the fixed time instance t is given as with

η 2 (t) =

1 u(t)> u(t)

The maximum displacement at any instant t in (0, T ), is finally given by q umax (t) = −umin (t) = u(t)> u(t)

(6.169)

(6.170)

A numerical example has been carried out by assuming that admissible functions are continuous. All ai (t)’s were taken as harmonic functions, as in the Fourier expansion:  A0 sin(φi t) for i ≤ m ai (t) = A0 cos(φi t) for m < i ≤ N (6.171) 2πi 2 2 2 φi = , A0 = E0 T T with N even, and m = N/2. In Fig. 6.12, the bound (6.170) is plotted for different values of m = 25, 50, 100, 200 for t ranging from 0 to 4 sec. The figure shows how the refinement of the functional space of the excitation makes the bounds increase progressively, and a convergence is attained for m = 100. Figure 6.13 depicts a sample maximizing function for T = 30 and m = 100. These bounds are of the type of Drenick’s critical excitation. It appears to be highly questionable that such functions can be accepted as credible accelerograms.

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u

ENVELOPES OF RESPONSE IN (0, 38)

a)

b) t

a t

MAXIMAL ACCELEROGRAM AT T = 15 Fig. 6.13 Upper and lower bounds and sample maximal accelerogram for T = 30 sec: (a) bounds (6.170); (b) umax of any harmonic component.

6.6.2

Credible accelerograms

It is possible to investigate how the above bound is affected by the presence of additional information. Assume that earthquakes, in general, exhibit a nominal power spectrum whose shape is constructed to fit the following expression: f (φ; σ1 , α1 , σ2 , α2 ) = K[g(φ; σ1 , α1 ) + g(φ; σ2 , α2 )]

(6.172)

with g(φ; σ, α) =

(σ 2

σ 2 + 4α2 φ2 − φ2 )2 + 4α2 φ2

(6.173)

where σ and α are parameters governing the shape of the spectrum, and K is a normalizing factor. These parameters are assumed to have assigned values for any given site under examination. Note that Eq. (6.172) is nothing else but the sum of two functions of the Kanai–Tajimi type; the superposition is introduced in order to have a better approximation for the spectra of recorded earthquakes exhibiting more than one dominant frequency. Once this shape is assumed as the central spectrum, it is to be expected that accelerograms should not differ from the central spectrum by more than a given amount. Let c0 be the vector of combination coefficients fitting the central spectrum; i.e., with reference to a basis of the type (6.171): c20,i = c20,i+m = f (φ; σ1 , α1 , σ2 , α2 ), (i ≤ m)

(6.174)

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We assume that through the analysis of the previous recorded earthquakes, it is possible to qualify earthquakes and their variability, as accelerograms with given norm of Eq. (6.160), but with distance from c0 not larger than a given amount, say θ. Thus, the constraints on the coefficients are of the type N X c2i = 1 (6.175a) i=1

N X i=1

(ci − c0,i )2 ≤ θ2

which should be considered in conjunction with the condition N X c20,i = 1

(6.175b)

(6.176)

i=1

expressing the fact that c0 yields an accelerogram of a given norm, realized by assigning E0 a proper value. The introduction of this additional information implies that one intends to deal with functions that are not far from earthquake-type accelerograms. The expected result is the maximum stress in a linear SDOF structure subjected to spectrum-compatible earthquakes. The term spectrum-compatible indicates that at the given site the seismic excitation is qualified by its distance (6.175b) from the central spectrum. The problem is similar to the one dealt with in Sec. 6.5, except the fact that now the admissible set of excitations is given by the intersection of the unit sphere S0 in Eq. (6.175a), representing the energy constraint, and the side sphere Sd in Eq. (6.175b), namely, the spectrum-compatible sphere as shown in Fig. 6.14. The problem of finding the maximum displacement at a certain time instance t reads: find max c> u(t) (6.177a) c

subject to (c − c0 )> (c − c0 ) ≤ θ2

(6.177b)

>

c c≤1 (6.177c) where θ < 4, otherwise Sd includes S0 , and the problem reduces to the formulation given in Problem (6.164). Problem (6.177) can be solved in two steps: 2

Step 1: Solve the simpler Problem (6.164) and find the anti-optimal solution c1 . If c1 is contained in Sd , then it represents also the solution of Problem (6.177). If c1 is on S0 but outside Sd , then no internal point to Sd exists where the antioptimality (optimality) condition (6.166) (respectively, condition (6.167)) holds, and the solution must be searched for on the boundary of the intersection of S0 and Sd , as described in Step 2. Step 2: Now the problem transforms into the following one: find max c> u(t) (6.178a) c

subject to (c − c0 )> (c − c0 ) = θ2 >

c c≤1

(6.178b) (6.178c)

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c2 Sd c0 c1 So

Fig. 6.14

Set of admissible coefficients.

Note that the constraint concerning Sd can be written as N X i=1

(ci − c0,i )2 =

N X i=1

(c2i − 2ci c0,i + c20,i ) = θ2

(6.179)

Taking into account Eq. (6.176), we can state that N X i=1

with

ci c0,i = 1 −

θ2 =a 2

|a| < 1

(6.180)

(6.181)

As a consequence, the problem can be formulated as follows: find

max c> u(t) c >

subject to c c = 1, c> 0 c=a

(6.182a) (6.182b)

where all constraints appear in the form of equalities. In order to solve Problem (6.182), we consider L(c, r1 , r2 ; t) = c> u(t) + r1 (c> c − 1) + r2 (c> 0 c − a)

(6.183)

where r1 and r2 are Lagrange multipliers. We require that the gradient of the Lagrangian with respect to c vanishes as ∇L = u(t) + 2r1 c + r2 c0 = 0

(6.184)

1 (r2 c0 + u(t)) 2r1

(6.185)

Hence, c = c(t) = −

Substituting Eq. (6.185) into the second constraint c> 0 c = a in (6.182b) for S0 , we obtain 1 > c> c = 2 (u(t)> u(t) + r22 c> (6.186) 0 c0 + 2r2 c0 u(t)) = 1 4r1

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where one puts > 2 c> 0 u(t) = b, u(t) u(t) = 4U (t)

(6.187) >

Bearing in mind that we must have the solution satisfying c c = arrive at 4r12 − r22 − 2r2 b − 4U 2 (t) = 0 Substituting now Eq. (6.185) into the second constraint with equality, we obtain 1 (r2 + b) = a c> 0c=− 2r1 Hence r2 = −(2r1 a + b)

c> 0c

c> 0 c0

= 1, we (6.188)

= a in (6.182b) for Sd (6.189) (6.190)

Substituting the latter expression into Eq. (6.188), the following equation for r 1 is obtained: 4(1 − a2 )r1 + b2 − 4U 2 = 0 (6.191)

Therefore,

r 1 4U 2 − b2 (6.192) r1 = ± 2 1 − a2 Note that the expression under the root is positive. Indeed, 0 < a2 < 1 from Eq. (6.181), and by elementary use of the Cauchy–Schwarz inequality, we establish 2 > > > 2 b2 = (c> (6.193) 0 u(t)) ≤ (c0 c0 )(u(t) u(t)) = u(t) u(t) = 4U After r1 has been calculated, r2 is given by Eq. (6.190). Consider now the elements of the Hessian  ∂ 2 L(c, r1 , r2 ; t) 2r1 for i = j (6.194) = 0 for i 6= j ∂ci ∂cj Thus, we verify that the matrix of second-order derivatives of the Lagrangian with respect to the basic variables ci is diagonal. Moreover, it is positive definite if r1 is positive; then the solution corresponds to the local minimum of the objective function c> u. Analogously, if r1 < 0, the matrix of second-order derivatives of the Lagrangian is negative definite, and the solution corresponds to the local maximum of the objective function. In conclusion, the solution for the problem of the maximum response is given by r 1 4U 2 − b2 r1 = , 2 1 − a2 (6.195) r2 = −(2r1 a + b), 1 (r2 c0 + u(t)) 2r1 Likewise, the solution for the minimum response is given by changing the sign of r1 as r 1 4U 2 − b2 r1 = − (6.196) 2 1 − a2 with the same definition of r2 and c as in Eq. (6.195). c = c(t) = −

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Optimization and Anti-Optimization of Structures under Uncertainty Table 6.2 Values of epicentral distance, local intensity, PGA, duration, and norm at various sites.

6.6.3

Site

Epicentral distance (km)

Local intensity (MCS)

PGA (g/10)

Duration (sec)

Norm (cm · sec −3/2 )

Torre del Greco Brienza Sturno Bagnoli Irpino

80.1 41.3 34.8 22.3

7.0 7.0 6.0 6.0

0.59 2.24 2.25 1.31

52.9 78.7 70.7 79.1

79.46 185.19 284.53 152.29

Application

In order to have a quantitative estimate of the proposed procedure, consider a particular area, namely, the Campania region in southern Italy, where a number of accelerograms have been recorded during the Campania-Lucania earthquake which took place 23 November, 1980. The magnitude of the event was estimated to be 6.5, the epicentral intensity was set at 7.5 degree the Mercalli–Cancani–Sieberg (MCS) intensity scale. The event was rather non-typical, mainly because of its duration that was up to almost 2 min at some sites. It should be noted that only the duration of the strong motion will be considered here, which varies from about 52 sec to more than 86 sec. Information on the local characters of ground motion was derived from the direct inspection of the recorded accelerograms, limiting the analysis, for simplicity, to only the NS component. The considered records are summarized in Table 6.2. Recorded ordinates of all accelerograms are converted to cm·sec−2 . All earthquakes are preliminarily reduced to the same energy norm, given in Eq. (6.154) as the one recorded in Torre del Greco, the closest site to Naples, the town with the greatest intensity in the region, so that every record possesses energy E0 = 79.46 cm · sec−3/2 . The analysis was carried out including m = 400 sine and cosine waves for Torre del Greco and Sturno, while m is increased to 800 for the accelerograms recorded in Brienza and Bagnoli Irpino that exhibit power spectra scattered on a wider range of frequencies. A plot of a typical accelerogram with the approximation resulting from the associated Fourier expansion and its power spectrum is given in Fig. 6.15. Note that the plot of Fourier approximation in Fig. 6.15(b) looks almost the same as the original accelerogram Fig. 6.15(a). Let fij be the ith coefficient of the Fourier expansion of the jth accelerogram (i = 1, . . . , m; j = 1, . . . , 4). The central distribution of coefficients, collected in the vector c0 , were obtained for each of the four analyzed sites, through the following steps: Step 1: Normalize fij , for every j, to unit modulus through division by a factor Fj = |f j |, with f j = (f1j , . . . , fmj )> , and let cij = fij /Fj . Step 2: Determine, for every site, the best values of the parameters σj and αj that

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PLOT OF RECORDED ACCELEROGRAM - TORRE DEL GRECO-NS a POWER SPECTRUM S

t

(a) a*

ω (c)

t

(b) Fig. 6.15 Torre Del Greco-Campania Earthquake of Nov. 23, 1980; (a) recorded accelerogram; (b) Fourier approximation; (c) power spectrum.

give the optimal fit of the ordinates of the Kanai–Tajimi-type spectrum given in Eq. (6.172) with the values of the spectrum corresponding to the normalized coefficients cij from the recorded accelerograms on the corresponding sites. Step 3: Calculate, for every value of φ, the target spectrum s0,ij , e.g., the spectrum in Eq. (6.172), as follows: s20,ij = f (φ; σ1 , α1 , σ2 , α2 ),

m X

s20,ij = 1

(6.197)

i=1

Step 4: Let βij =

ci+m,j , (i = 1, . . . , m) cij

(6.198)

and, finally, calculate the central target spectra at the considered sites as follows: s0,ij sign fij (6.199) c0,ij = q 2 1 + βji

Step 5: Calculate, for every site, the quadratic scatter of the coefficients obtained by directly processing the accelerogram and those calculated by the target spectrum m X θj2 = (cij − c0ij )2 (6.200) i=1

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RECORDED SPECTRUM TORRE DEL GRECO-NS OMB=9.920058 CSIB=.109198 OMC=14.54868 CSIC=.1705714 R2=.2608822 S

ω

Fig. 6.16 Processed power spectrum from recorded accelerograms and fitted central spectrum, in the location of Torre del Greco.

Table 6.3

Values of θj2 , σ1 , α1 , σ2 , and α2 at various sites.

Site

θj2

σ1 (sec−1 )

α1

σ2 (sec−1 )

α2

Torre del Greco Brienza Sturno Bagnoli Irpino

0.261 0.219 0.186 0.257

9.92 32.98 17.94 5.15

0.109 0.478 0.347 0.142

14.55 37.67 3.77 28.64

0.171 0.403 0399 0.762

In Fig. 6.16, the processed spectrum at Torre del Greco and the central spectrum are plotted. In Table 6.3, the final values of θj2 are listed illustrating that the inequality θj2 ≤ 4 holds. Assuming that the central spectrum is site-dependent, but that independence holds for the quadratic scatter, it is possible to infer from the data that θj2 ' 0.26 for the area under examination. After introducing this value in the results discussed in Sec. 6.6.2, one obtains the bounds plotted in Figs. 6.17 and 6.18 for ω0 = 10 ∼ 100 sec−1 . It is also possible to perform a comparison with the previous, spectrum-free bound, showing that the new bound is approximately half of the one constrained only by the upper bound by Drenick. The response displacement spectrum by the present procedure is finally calculated, for a value of the damping coefficient µ = 0.05. The comparison with the envelope of the same spectra calculated with reference to the processed accelerograms shows that the present results, although better founded and less sensitive to uncertain parameters than recorded accelerograms, are not too conservative, but

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Sn

187

(cm) PRESENT BOUNDS SPECTRA OF RECORDED EARTHQUAKES: TORRE DEL GRECO BRIENZA STURNO CALITRI BAGNOLI IRINO

5 4 3 2 1

20

40

60

80

100

ω0(sec-1)

Fig. 6.17 Response spectra of displacement: Comparison of present bounds with the envelope of spectra from recorded accelerograms normalized to the same norm as Torre del Greco.

Sn

(cm)

SHINOZUKA’S BOUND DRENICK’S BOUND

5

PRESENT BOUNDS ENVELOPE OF RECORDED SPECTRA

4 3 2 1

20

40

60

80

100

ω0(sec-1)

XMMAX=5.62035 YMMAX=4.21342 YMMIN.02036

Fig. 6.18 Response spectra of displacement: Comparison of present bounds with Drenick’s and Shinozuka’s bounds.

yield a magnification by a factor of about two in the design forces, in the whole range of natural periods of the structures; this result can be directly incorporated into the safety factor. Following Shinozuka’s (1970) model, and the assumption given in Eq. (6.169),

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with the parameters presented in Table 6.3, one obtains the following bound for the maximum displacement over the whole duration of the excitation: Z ∞ 1 |H(φ)|X(φ)dφ (6.201) Iei = 2π −∞ where H 2 (φ) =

1 (ω02 − φ2 )2 + 4µ2 ω02 φ2

(6.202)

and X(φ) = K[g(φ|σ1 , α1 ) + g(φ|σ2 , α2 )] with K chosen so that 1 2π

Z

(6.203)

∞ −∞

X 2 (φ)dφ = E02

(6.204)

From inspection of Figs. 6.17 and 6.18, it is possible to infer that Problem (6.164) yields the bound in the norm substantially close to that obtained via the Drenick’s original approach, apart from the range of higher frequencies, due to the truncation of the expansion (6.171) by a finite number of harmonics (say, for φi > 50 sec−1 ). Moreover, the spectrum-compatible bound in Problem (6.177) is not significantly different from Shinozuka’s bound, although it yields slightly closer bounds in the range of low frequencies. The opposite happens in the range of higher frequencies. The ratio of the ordinates of the spectrum-compatible bound to the corresponding ordinates of the envelope of spectra related to actual seismic records, normalized to the same norm as Torre del Greco, vary in the range 2.0 ∼ 2.5. 6.6.4

Discussion and conclusion

In this section, an approach has been presented for response analysis of a linear structure subjected to seismic loads, and for calculation of an upper bound of the response based on the solution of a convex anti-optimization problem. Starting with a specialized function space with a given basis, the anti-optimal combination of the coefficients can be found analytically. The data required for the analysis in this section are: (i) the estimated energy of the earthquake, (ii) an estimate of the shape of the central power spectrum at the site, (iii) an upper bound on the maximum distance of the power spectrum of realizable earthquakes at the site from the central one. A direct comparison has been performed with Shinozuka’s (1970) approach, yielding at some cases very close results. It should be noted that in Shinozuka’s approach, however, the admissible spectra are bounded from above by the central spectrum, while the approach in this section includes earthquakes whose spectra are allowed to exceed the central spectrum; note, moreover, that in Shinozuka’s approach the bound involves only the steady-state part of the response, while in

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the approach in this section, as in Drenick’s (1970) one, the bound involves also the transient response starting with homogeneous initial conditions. It should be emphasized that the present convex modeling combines positive features of the approaches developed by Drenick (1970) and Shinozuka (1970), and considerably reduces the estimates of the maximum possible response. One should stress that the real structures are generally not SDOF-systems where the maximum stresses are governed by the maximum displacement of a single mode. The generalization of this method to multi-degree-of-freedom realistic structures is possible through casting it in a state-space form as a vector differential equation. One should also note that the maximum stress at two distinct positions of the structure will be produced by the different worst-case combinations of excitations. Obviously, one is interested in the worst response at the critical location of the structure. Determining such a location may be a non-trivial task. Then, if the structure is represented as the multi-degree-of-freedom system, one must explore the maximum responses of each of the masses, and the maximum of these maxima will constitute a critical response. Analogously, a distributed system must be discretized through a fine mesh and the maximum responses at each nodal point must be determined. Then the location with maximum response amongst critical responses at each node will constitute the globally critical response. The attractive feature in the present method is the fact that one does not guess the critical location, but rather determine it through a discretization scheme combined with convex optimization procedure for each nodal point of the mesh. Note that the present approach is restricted to linear systems. The generalization to realistic nonlinear systems appears to be necessary for practical applicability of this method. (This appears to be doable if and when the funding agencies are more receptive to high-risk high-payoff methodologies.) In this context, convex models have been applied to the nonlinear structures through treating the nonlinear transfer function between the uncertain quantities and the output of the nonlinear transformation, as a numerical code utilized by Ben-Haim and Elishakoff (1989b) and the automatic differentiation procedure. Another alternative is the use of the nonlinear programming schemes. For the application of the latter method the reader may consult the paper by Li, Elishakoff, Starnes and Shinozuka (1996). To sum up, it appears that the time is ripe to critically revisit existing stochasticity concept-based approaches in earthquake engineering and examine the possibility of utilizing alternative approaches. One such alternative approach was attempted in this section, although to an SDOF system in the linear setting. Related subjects, although in different settings were pursued by Abbas and Manohar (2005, 2007), Ben-Haim, Chen and Soong (1996), Elishakoff (1991b), Elishakoff and Pletner (1991), Ganzerli, Pantelides and Reaveley (2000), He and Zhang (1997), Iyengar and Manohar (1987), Lignon and J´ez´equel (2006), Ma, Leng, Meng and Fang (2004), Manohar and Sarkar (1995), Moustafa (2009), Pantelides and Tzan (1996), Philap-

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pacopoulos (1980), Schmitendorf, Jabbari and Yang (1994), Srinivasan, Corotis and Ellingwood (1992), Srinivasan, Ellingwood and Corotis (1991), Takewaki (2002a, 2002b, 2006b), Tzan and Pantelides (1996b), Wu and Soong (1996), Yoshikawa (2002, 2003), Zhai and Xie (2007), and possibly others. Finally, we mention a definitive monograph by Takewaki (2006a) and references therein. It generalizes earlier investigations on convex modeling of earthquakes. Shinozuka (1970) pioneered the determination of the maximum response in stochastic settings. Further stochastic anti-optimization-based studies include those by Deodatis, Graham and Micaletti (2003a, 2003b), Deodatis and Shinozuka (1989, 1991), Manohar and Sarkar (1995), Sarkar and Manohar (1998), Shinozuka (1987), Shinozuka and Deodatis (1988), and others. 6.7 6.7.1

Aeroelastic Optimization and Anti-Optimization Introduction

In this section, we apply the hybrid optimization and anti-optimization to aeroelastic problems, closely following the article by Zingales and Elishakoff (2001). Optimization of structures with aeroelastic constraints has been dealt with by several authors. A partial list of works includes those by Ashley (1982), Bishop, Eastep, Striz and Venkayya (1998), Librescu and Bainer (1983), Livne and Mineau (1997), Plaut (1971), Pierson and Genalo (1977), Ringertz (1994), Shirk, Hertz and Weisshaar (1986), and Turner (1982). However, the uncertainty in elastic moduli or in the material properties in conjunction with the aeroelastic optimization is a relatively new topic. It has been addressed in the probabilistic setting by Kuttekeuler and Ringertz (1998) and Allen and Maute (2004). The stochastic finite element method for reliability of plates in supersonic flow was introduced by Liaw and Yang (1990). In the future one would anticipate an increased utilization of the stochastic finite element method in conjunction with aeroelastic phenomena. On the other hand, it is instructive to quote Shinozuka (1987): ‘... It is recognized that it is rather difficult to estimate experimentally the auto-correlation function, or in the case of weak homogeneity, the spectral density function of the stochastic variation of material properties. In view of this, the upper bound results are particularly important, since the bounds derived ... do not require knowledge of the auto-correlation function.’ This observation may limit the applicability of probabilistic analysis to uncertainty. In these circumstances one should look for alternatives to the notion of stochasticity. As Livne (1997) mentions in his careful overview of the subject of aeroelastic optimization, ‘... approaches for addressing uncertainty in design optimization of

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structural systems are still in various stages of study and evaluation ... Ranging from statistical methods in which the statistical characteristics of uncertainties are known or assumed or methods based on fuzzy logic, as well as methods which use bounds on the system uncertainties to optimize for a worst case combination of pre-assigned parameter and modeling errors – all these methods have not been compared yet in the context of aeroelasticity and aeroservoelasticity of fixed wing airplanes.’ In this section, we adopt an anti-optimization method that utilizes information that is easier to obtain (than either a stochasticity or fuzzy-sets based approach); namely, the region of variation of the elastic moduli. Because of this partial information it is sensible to require only the fragmentary characterization of the response; namely, the determination of the maximum and minimum values of the critical velocity. Once the minimum of the velocity is determined, the structural parameters are chosen so as to exceed some preselected velocity. Numerous examples are elucidated to gain physical insights into the problem.

6.7.2

Deterministic theoretical analysis

Let us consider a Bernoulli–Euler beam with variable cross-section and nonhomogeneous elastic properties. The beam is subjected to a supersonic stream flow in the direction of beam axis (x-direction). Young’s modulus E(x) of the beam is represented by the following expression: E(x) = φ1 E1 + φ2 E2 (6.205) where E1 and E2 can be characterized as amplitudes, and φ1 (x) and φ2 (x) are continuous non-dimensional functions governing the variation of the modulus E(x). In this section, we abstain from specifying particular forms of the functions φ1 (x) and φ2 (x), except noting that they should be chosen to satisfy the obvious physical requirement E(x) > 0. The width b(x) of the beam and its depth h(x) are considered as varying with respect to x: b(x) = b0 χb (x), h(x) = h0 χh (x) (6.206) where b0 and h0 are positive constants, and χb (x) and χh (x) are shape functions governing the variation of b(x) and h(x), respectively. Moreover, χb (x) and χh (x) equal unity at the coordinate origin; i.e., χb (0) = χh (0) = 1. Therefore, b0 and h0 represent the width and the depth of the cross-section at x = 0. The description of the aeroelastic interaction between the transverse deflection w(x, t) of the beam and the air pressure load will be performed by means of the piston theory (Ashley and Zartarian, 1956; Il’yushin, 1956; Bolotin, 1963). The effect of the structural and aerodynamic damping on w(x, t) will be excluded for the sake of simplicity. The equation reads   of motion   ∂ 2 w(x, t) ∂ 2 w(x, t) kp∞ ∂w ∂2 E(x)I(x) + ρA(x) + U =0 (6.207) ∂x2 ∂x2 ∂t2 c∞ ∂x

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where A(x) is the cross-sectional area, I(x) is the moment of inertia, and ρ is the mass per unit length of the beam. The gas-flow interaction is represented by the last term in Eq. (6.207), where k is the exponent of the polytropic thermodynamic transformation, p∞ is the pressure of the nondisturbed gas flow, c∞ is the speed of the shock waves in the nonperturbed airstream, and U is the relative velocity between the gas flow and the system. The boundary conditions associated with the simply supported beam read ∂ 2 w(x, t) = 0 at x = 0 and L (6.208) ∂x2 where L is the length of the beam. The initial conditions are ∂w(x, t) = 0 at t = 0 (6.209) w(x, t) = ∂t The governing equation for the deflection w(x, t) is not solvable exactly for arbitrary choices of the functions φ1 (x), φ2 (x), χb (x), and χh (x). Hereinafter, the Bubnov– Galerkin approach will be utilized. We approximate the solution as w(x, t) =

N X

w(x, t) =

ψi (x)fi (t)

(6.210)

i=1

with N denoting the number of retained terms in the expansion, and ψi (x) designating known comparison functions satisfying all boundary conditions in Eq. (6.208). We substitute Eq. (6.210) into Eq. (6.207). Naturally, the expression Eq. (6.210) does not satisfy Eq. (6.207). The result of the above substitution, therefore, differs from zero. We denote it by ε. We require that the inner product vanishes: (ε, ψj ) = 0, (j = 1, . . . , N ) where (ε, ψj ) =

Z

(6.211)

L

ε(x)φj (x)dx

(6.212)

0

This procedure leaves us with a set of ordinary differential equations: N X

α=1

Kαβ fα (t) +

N X

kp∞ Mαβ f¨α (t) + U c∞ α=1

N X

Nαβ fα (t) = 0, (β = 1, . . . , N ) (6.213)

α=1

The coefficients appearing in Eq. (6.213) are given as follows:   Z L d2 d2 ψα (x) ψβ (x) 2 E(x)I(x) dx, Kαβ = dx dx2 0 Z L Mαβ = ρA(x)ψα (x)ψβ (x)dx, Nαβ =

0 Z L 0

(6.214)

dψα (x) ψβ dx dx

The solution of the system in Eq. (6.213) is furnished in the form fα (t) = Aα eiωt , (α = 1, . . . , N )

(6.215)

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where ω represents the eigenfrequency. We arrive at an algebraic homogeneous system of equations for Aα : N X

α=1

Kαβ Aα −

N X

ω 2 Mαβ Aα + U

α=1

N X

Nαβ Aα = 0, (β = 1, . . . , N )

(6.216)

α=1

To obtain a the nontrivial solution; namely, N X j=1

A2j 6= 0

(6.217)

one requires the determinant of the coefficient matrix of the system in Eq. (6.216) to vanish: det[−ω 2 Mαβ + U Nαβ + Kαβ ] = 0

(6.218)

leading to the approximate eigenfrequency equation for ω. 6.7.3

Stability analysis within two-term approximation

Let us confine our analysis to the case where only two terms are retained in Eq. (6.210) A1 (K11 − ω 2 M11 + U N11 ) + A2 (K12 − ω 2 M12 + U N12 ) = 0, A1 (K21 − ω 2 M21 + U N21 ) + A2 (K22 − ω 2 M22 + U N22 ) = 0

(6.219)

It is instructive to write Eq. (6.219) in terms of the circular natural frequencies of the system ω 1 and ω 2 . First, as an auxiliary problem, consider the free-vibration case det[−ω 2 Mαβ + Kαβ ] = 0, (α, β = 1, 2)

(6.220)

resulting in the biquadratic equation ω 4 − c1 ω 2 + c2 = 0

(6.221)

K11 M22 + K22 M11 − K12 M21 − K21 M12 , det[Kαβ ] det[Kαβ ] c2 = det[Mαβ ]

(6.222)

with the coefficients given by c1 =

The solution is given as (Panovko and Gubanova, 1965) q q 1 1 ω 21 = (c1 − c21 − 4c2 ), ω 21 = (c1 + c21 − 4c2 ) 2 2

(6.223)

Because of Vieta’s theorem, c1 = ω 21 + ω22 and c2 = ω 1 ω 2 . Hence, Eq. (6.218) can be rewritten in the following form: ω 4 + [(ω 21 + ω22 ) + d1 U ]ω 2 + ω 1 ω2 + d2 U + d3 U 2 = 0

(6.224)

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with terms di defined as N11 M22 + N22 M11 − N12 M21 − N21 M12 , det[Mαβ ] K11 N22 + K22 N11 − K12 N21 − K21 N12 d2 = , det[Mαβ ] det[Nαβ ] d3 = det[Mαβ ]

d1 =

(6.225)

Equation (6.215) suggests that once the frequency ω takes a complex value, the dynamic instability called flutter occurs. The critical condition appears when the discriminant D = [U d1 + (ω 21 + ω 22 )]2 − 4[ω21 ω22 + d2 U + d3 U 2 ]

(6.226)

of the biquadratic equation (6.224) vanishes, yielding the flutter velocity Ucr

 1 d2 (ω2 + ω 22 ) + 2d2 = 2 d1 − 4d3 1 1  q 2 2 2 2 2 2 2 2 + [d1 (ω 1 + ω 2 ) + 2d2 ] − (ω1 + ω 2 ) (d1 − 4d3 )

(6.227)

For values larger than Ucr , the discriminant takes a negative sign, and the roots are complex conjugate. For the case of the beam simply supported at both ends, the following set can be utilized for the comparison functions: φα (x) = sin

 απx  L

, (α = 1, 2, . . . )

(6.228)

which represent the exact mode shapes of the associated uniform beam. Note that in these circumstances, N11 = N22 = 0, N12 = −N21

(6.229)

yielding Ucr

p (ω 22 − ω22 ) det[Mαβ ] ω 22 − ω22 = = √ 2N12 2 d3

(6.230)

Observe that Eq. (6.230) formally coincides with expression (4.104) in the monograph by Bolotin (1963) for the uniform beam. The difference lies in the definition of the circular natural frequencies ω α . Recall that E1 and E2 are uncertain variables, leading to variability of the flutter velocity. The following question arises: How should one take into account the variability of the elastic modulus on Ucr ?

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6.7.4

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Convex modeling of uncertain moduli

In this section, the uncertainty in the elastic modulus E(x), represented in Eq. (6.205), will be specified. In many recent investigations, the elastic modulus is treated as a random variable and the problems are solved by the finite element method necessitating the knowledge of the autocorrelation function of the elastic modulus. However, such information is often if not always unavailable. In these circumstances, one can act in one of the following ways: Either postulate that such an autocorrelation function is known, or pick up (as is usually done) any allowable autocorrelation function based on the assumption that once the autocorrelation function becomes available it can be introduced into the analysis. Another way will be to abandon this excessively demanding analysis and look for an approach that does not postulate that uncertainty and probability are synonymous. In this section, we use a non-probabilistic model of uncertainty. The two amplitude parameters E1 and E2 are treated as uncertain variables, varying in a convex domain C. The admissible uncertainty region of variation C as shown in Fig. 6.19 is then described by an inequality: (E1 , E2 ) ∈ C

(6.231)

The domain C for E1 and E2 will be assumed to lie entirely in the positive quadrant of the plane (E1 , E2 ). Because of the variability of Ucr , it appears natural to evaluate the worst critical velocity Ucr,worst , when E1 and E2 vary in C. Thus we are looking for Ucr,worst =

min

(E1 ,E2 )∈C

Ucr (E1 , E2 )

(6.232)

Note that the coefficient c1 in Eq. (6.222) is a linear function of Ej (j = 1, 2), whereas the coefficient c2 is a quadratic function of them, that is, c1 = δ1 E1 + δ2 E2 , c2 = ε11 E12 + ε12 E1 E2 + ε22 E22

(6.233)

where δ1 = M22 K111 + M11 K221 − M12 K211 − M21 K121 ,

δ2 = M22 K112 + M11 K222 − M12 K212 − M21 K122 ,

ε11 = K111 K221 − K121 K221 ,

(6.234)

ε22 = K112 K222 − K122 K212 ,

ε12 = K111 K222 + K112 K221 − K121 K212 − K122 K211

The expression for Kijl reads   Z L d2 φi (x) d2 dx Kijl = ψj (x) 2 I(x)φl (x) dx dx2 0

(6.235)

Bearing in mind Eqs. (6.223) and (6.233), we can rewrite Eq. (6.230) for the square of the critical velocity as follows: 2 Ucr (E1 , E2 ) = d11 E12 + 2d12 E1 E2 + d22 E22

(6.236)

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Shape and location of the admissible uncertainty region C and of the η 2 ellipses.

where δ12 − 4ε11 det[Mαβ ] , 2 4N12 2δ1 δ2 − 4ε12 det[Mαβ ] , = 2 8N12 δ 2 − 4ε22 det[Mαβ ] = 2 2 4N12

d11 = d12 d22

(6.237)

One recognizes that the expression of the square of critical velocity is a quadratic form in variables E1 and E2 . In matrix form, Eq. (6.236) can be rearranged as 2 Ucr = E > DE

with E=



   E1 d11 d12 , D= E2 d21 d22

(6.238)

(6.239)

Accounting for the physical meaning of the right-hand-side in Eq. (6.235), we can conclude that the matrix D is positive definite. Therefore, Eq. (6.236), conveniently rewritten in the form 2 E > DE − Ucr =0

(6.240)

represents an ellipse in the plane (E1 , E2 ) with the center coinciding with the origin of the coordinate system. We thus obtain a family of homothetic ellipses correspond2 ing to different values of Ucr if the latter is treated as an independent parameter. 2 Naturally, the larger values of Ucr correspond to larger semi-axes of the ellipse in the 2 plane. For small values of Ucr the ellipse in Eq. (6.236) and the region in Eq. (6.231) are disjoint. For large values of it they have a common area. Therefore, there exist 2 2 2 2 2 two values of Ucr , denoted by Ucr,worst and Ucr,best with Ucr,worst < Ucr,best , for

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which the two convex domains in Eqs. (6.231) and (6.236) share a single point. One of the values corresponds to the case when one ellipse contains the other one entirely, whereas the other point is associated with the case when they are on different sides of the common tangent. 2 2 2 Values of Ucr falling outside of the interval [Ucr,worst , Ucr,best ] do not represent feasible critical velocities, because the set (E1 , E2 ) will not satisfy Eq. (6.231). Note 2 2 that the two values of velocities, Ucr,worst and Ucr,best , respectively, represent the smallest and the largest solutions allowed by Eq. (6.231). To determine these values, we need to examine closely the relative spacing of the critical velocity ellipse and the region of uncertain variation of the elastic modulus. According to the interpretation of the parameters E1 and E2 as some effective elastic modulus, we shall confine our analytical derivation to the arc of the ellipse in Eq. (6.236) contained in the first quadrant of the plane (E1 , E2 ). It can be shown (see Zingales and Elishakoff (2001)) that, under some assumptions about the functions φi (x), the critical velocity ellipse in Eq. (6.240) is always oriented with its major axis in the second and fourth quadrant of the plane (E1 , E2 ), as shown in Fig. 6.19. 2 Once the orientation of the ellipse, representing the critical velocity Ucr with respect to the region C of the uncertain parameters E1 and E2 has been identified, one can solve analytically the anti-optimization problem. The following sections will be concerned with the solution of Problem (6.232) for various practical shapes of the uncertainty region. Before proceeding further, let us pose the following question: Why do we need to consider various shapes of the uncertainty region? To answer this question, we visualize that the results of the experimental measurements yield an ensemble of functions Ei (x). These functions are then decomposed to sets of pairs (E1i , E2i ). Each of these pairs forms a point in the plane (E1 , E2 ). We then are interested in determining the region that contains all of these points. A convex hull of these points is naturally such a set, forming a polygonal uncertainty region. Also, it is easily visualized that different researchers and engineers may approximate the uncertainty region via differing means. It makes sense, therefore, to treat different possibilities. 6.7.5

Anti-optimization problem: polygonal region of uncertainty

In this section, we consider an uncertainty region C possessing a polygonal shape with m vertices. The vertices Pi (i = 1, . . . , m) of the polygon have coordinates Pi = (E1i , E2i ), respectively. We denote the vertex in the lowest left corner by P1 , the others are numbered counterclockwise as P2 , P3 , etc. as shown in Fig. 6.19. Hereinafter we form a vector V of distances di of the vertices from the origin O: q 2 + E2 (6.241) V = (d1 , . . . , dm )> , di = E1,i 2,i

The result of anti-optimization given by the joint points of the homothetically inflated ellipse in Eq. (6.237) depends on the values of the design variables b0 and

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h0 appearing in Eq. (6.206), in addition to uncertain variables E1 and E2 . This dependence can be highlighted by rewriting Eq. (6.236) in the form 2 Ucr = b20 h60 η 2

(6.242)

η 2 = d011 E12 + 2d012 E1 E2 + d022 E22

(6.243)

with the positive coefficient

d0ij

The coefficients (i, j = 1, 2) in Eq. (6.243) are obtained from Eq. (6.236) by formally letting b0 = h0 = 1. Because the critical velocity depends on the uncertain parameters of elasticity Ei through the coefficient η 2 , the anti-optimization problem will focus on the determination of the extremal of the coefficient η 2 , provided that E1 and E2 vary in region C. Equation (6.243) represents a family of concentric ellipses in the plane E1 and E2 obtained by varying the value of the coefficient η 2 . The major axis passes through the second and fourth quadrants of the plane (E1 , E2 ). As shown in Sec. 6.7.4 for the critical velocity (and hence η 2 ), the ellipse in Eq. (6.243) and the region of uncertainty C must share a common point. There are two possibilities for such a point to exist. One, Pbest corresponds to the maximum of η 2 , whereas the minimum is associated with Pworst . 2 2 The extreme values ηmin and ηmax of the coefficient η 2 are found by substituting the coordinates (E1,worst , E2,worst ) and (E1,best , E2,best ) of the points Pworst and Pbest into Eq. (6.243). Bearing in mind Eq. (6.243), the expressions for the 2 2 maximum and minimum values of the critical flutter velocity Ucr,best and Ucr,worst become 2 2 Ucr,best = b20 h60 ηmax

= 2 Ucr,worst

= =

2 2 b20 h60 (d011 E1,best + 2d012 E1,best E2,best + d022 E2,best ) 2 6 2 b0 h0 ηmin 2 2 b20 h60 (d011 E1,worst + 2d012 E1,worst E2,worst + d022 E2,worst )

(6.244a) (6.244b)

The expressions of the coordinates corresponding to the points Pworst and Pbest are given in Zingales and Elishakoff (2001) for all the cases reported in Figs. 6.20(a)–(c). Comparing the ellipses in Fig. 6.19 with those in Fig. 6.20, we observe that they possess different ratios of their respective semi-axes e = λ1 /λ2 , referred to as eccentricity e. This is because the functions φ1 and φ2 chosen in Fig. 6.19 are φ1 = 1 + 5.3(x/L)2 + 0.5(x/L)3 and φ2 = 1 + 2(x/L), whereas in Fig. 6.20 they read φ1 = 1 − 3(x/L)2 and φ2 = 0.001 + 0.3(x/L). The shape functions φj (x) for Fig. 6.19 are positive in the interval √ [0, L]. The function φ1 (x) for Fig. 6.20 takes negative values in the interval [L/ 3, L]. Still, the modulus of elasticity is a positive quantity. Despite the function φ1 (x) not being positive in the whole entire interval [0, L], the major axis of the η 2 ellipse lies in the second and fourth quadrants of the plane (E1 ,E2 ). Numerical calculations show that the positiveness requirement of functions φ1 (x) and φ2 (x) in the interval [0, L] leads to a large eccentricity e of the η 2 ellipse. The weaker condition that E(x) is positive but φj (x) may not be

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199

(b)

(c) 2 2 Fig. 6.20 Regions of uncertainty: (a) rectangular region of uncertainty, η max and ηmin ellipses, (b) polygonal region of uncertainty with particular location of vertices P 3 and P6 , (c) singular location for rectangular region of uncertainty; extremal that does not coincide with one of the vertices.

results in lower values of e. The particular choice of φ1 (x) and φ2 (x) in Fig. 6.20 has been made to enable an easier visualization of the ellipses corresponding to 2 2 extremal values ηmin and ηmax . The objective of the next section is to derive approximations of the admissible region for E1 and E2 by a continuous smooth curve for analytical purposes.

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(a)

Eq.(6.244),

(b) Fig. 6.21 Ellipsoidal region of uncertainty: (a) ellipse, (b) ellipse corresponding to the extremal values of the coefficient η 2 .

6.7.6

Anti-optimization problem: ellipsoidal region of uncertainty

Let us assume that the region C is represented by an ellipse with the center E0 = (E10 , E20 ): (E2 − E20 )2 (E1 − E10 )2 + ≤1 (6.245) 2 a b2 where a and b represent the semi-axes of the ellipse as shown in Figs. 6.21(a) and (b). We are looking for the points Pworst = (E1,worst , E2,worst ) and Pbest =

optimization

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(E1,best , E2,best ) that satisfy Eqs. (6.244a) and (6.244b) and correspond, respectively, to the worst and the best critical velocities Ucr,worst and Ucr,best. In the following, we denote the coordinates of the points Pworst and Pbest , respectively, as 2 2 (e1,worst , e2,worst ) and (e1,best , e2,best ). Then Ucr,worst , and Ucr,best are given by 2 Ucr,worst = (d011 e21,worst + 2d012 e1,worst e2,worst + d022 e22,worst )b20 h60 , 2 Ucr,best = (d011 e21,best + 2d012 e1,best e2,best + d022 e22,best )b20 h60

(6.246)

2 The obtained solution for the anti-optimized critical flutter velocity Ucr,worst shall be utilized in the next section to obtain the best possible value of a certain objective function involving it.

6.7.7

Minimum weight design

The optimal design process involves the weight W of the beam Z L W = ρb0 h0 χb (x)χh (x)dx

(6.247)

0

which is a monotonically increasing function of the design variables; namely, the width b0 and the depth h0 at x = 0. The functions χb (x) and χh (x) governing the shape of the cross-section of the beam have been introduced in Eq. (6.206). We are interested in designing the system, i.e., determining the values of b0 and h0 , that minimizes the weight W under constraint on the critical flutter velocity 2 Ucr whose value must be greater than the specified velocity U0 : Ucr (b0 , h0 ) ≥ U0

(6.248)

Because Ucr is an interval variable [Ucr,worst , Ucr,best], fulfilment of the inequality Ucr,worst (b0 , h0 ) ≥ U0

(6.249)

automatically leads Eq. (6.248) to be satisfied. Hereinafter we will deal with Eq. (6.249). It is rewritten in a form that is more appropriate to the design by utilizing the representation in Eq. (6.243). This yields Ucr,worst = ηmin b0 h30 ≥ U0

(6.250)

where ηmin is defined in Eq. (6.244b). Let the manufacturing requirements demand that the values b0 and h0 vary in a box Γ: b0L ≤ b0 ≤ b0U , h0L ≤ h0 ≤ h0U

(6.251)

where b0L , b0U and h0L , h0U denote the bounds for the width and the thickness, respectively. An additional constraint involving the design parameters b0 and h0 is concerned with the range of validity of the utilized mechanical theory. Because we apply the Bernoulli–Euler theory, we specify an additional constraint over the design variables as b0 ≤ zh0

(6.252)

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Fig. 6.22

Alternative location of the design box with respect to the constraint g 2 ≥ 0.

Specifically, for the sake of determinacy, z is fixed at 10. The optimization problem is then stated as minimize

W (b0 , h0 )

subject to g1 (b0 , h0 ) =

(6.253a) ηmin b0 h30

− U0 ≥ 0

g2 (b0 , h0 ) = 10h0 − b0 ≥ 0

(b0 , h0 ) ∈ Γ

(6.253b) (6.253c) (6.253d)

The optimization problem (6.253) will be solved by employing geometrical constraints. Note that if in Eq. (6.253c) the entire region of variation is on the right side of the line g2 (b0 , h0 ) = 0, then no acceptable solution exists. This case is represented by the design region Γ1 drawn with a dotted line in Fig. 6.22. If, however, the region Γ is located on the left side of the line g2 (b0 , h0 ) = 0, the entire region is feasible except for the additional constraint Eq. (6.253b) (see domain Γ3 in Fig. 6.22). If the line g2 (b0 , h0 ) = 0 passes through the region, only the upper part (see the filled part Γ of the region Γ2 bounded by solid lines in Fig. 6.22) is feasible. We denoted by Q1 and Q2 , respectively, the intersection points of the line g2 (b0 , h0 ) = 0 with the region Γ2 that are closest and farthest from the origin O. Depending on the location of the region Γ2 with respect to the origin, several possibilities arise. If the line g2 (b0 , h0 ) = 0 crosses the edges AD and CD, the intersection points have the coordinates Q1 = (b0L , 1/10b0L), Q2 = (h0U , 1/10h0U)

(6.254)

as shown in Fig. 6.22. Other possibilities of crossing the region by the line g2 = 0 are elucidated in Figs. 6.23(a)–(c), where the coordinates of the intersection points are also indicated. To obtain the solution of the optimization problem in Eq. (6.253), let us examine closely the critical velocity constraint g1 (b0 , h0 ) ≥ 0. If the curve

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(a)

(b)

(c) e for specified velocity U0 , (c) Fig. 6.23 Feasible regions: (a) and (b) feasible design region Γ alternative location of the design box with respect to constraints g 1 ≥ 0 and g2 ≥ 0 for specified velocity U0 .

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(a)

(b) e (b) in correFig. 6.24 Solution of the optimization problem: (a) on the vertex of the region Γ, spondence of points Q1 and Q2 in Eq. (6.254).

defined by g1 (b0 , h0 ) = ηmin b0 h30 − U0 = 0

(6.255)

lies below the feasible region Γ, then the constraints (6.253b) and (6.253c) are not active for the specified value of the velocity U0 . In this case, the solution of the optimization problem coincides with the closest vertex of the hatched region Γ to the origin O. This vertex, denoted hereinafter by Qopt , is either the point Q1 , as shown in Figs. 6.22 and 6.23(b), or point A, as shown in Figs. 6.23(a) and (c), respectively, if the straight line g2 (b0 , h0 ) = 0 intersects the edge AD or AB. However, if the line in Eq. (6.255) is entirely above the hatched region Γ, then no solution of the optimization problem exists for the specified value of the velocity U0 . Thus, we identify an interval [U0,min , U0,max ], where the specified velocity U0 must lie, so that the inequality Ucr,worst ≥ U0 may hold. The extreme values U0,min and U0,max of

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the admissible range of U0 are given by U0,min = ηmin b01 h301

(6.256a)

ηmax b02 h302

(6.256b)

U0,max =

where the values b0i and h0i (i = 1, 2) denote, respectively, the points in the feasible domain Γ with smallest and largest distances from the origin O, as represented in Figs. 6.24(a) and (b). In Fig. 6.24(a) such extremal points are A and C, corresponding to U0,min = 1.75 × 107 and U0,max = 3.65 × 108 , respectively. Hereinafter we assume that the specified velocity belongs to the region U0 ∈ [U0,min , U0,max ]. e ⊆ Γ a region that is both feasible and satisfies the constraint on the Denote by Γ critical velocity Ucr,worst in Eq. (6.253b). The weight in Eq. (6.247) is a monotonically increasing function of the variables b0 and h0 . Therefore, the solution Qopt = (b0,opt , h0,opt ) of the optimization problem lies on the boundary of the ree closest to origin O. gion Γ To obtain the point Qopt on the boundary with minimum distance OQ, we again resort to a geometric argument. Note that there exists a specific value of the distance OQ such that the family of circles (denoted by the dotted line in Figs. 6.24(a) and (b)) b20 + h20 = OQ

2

(6.257)

and the curve g1 (b0 , h0 ) = 0 share a single point; that is, they have a common tangent in point Q(b0 , h0 ) as shown in Figs. 6.24(a) and (b). The tangent line to the curve g1 (b0 , h0 ) = 0 in a plane Ob0 h0 passing through the point Q = (b0 , h0 ) reads 4 dh0 ηmin h0 h0 − h 0 = (6.258) (b0 − b0 ) = − (b0 − b0 ) db0 h0 =h0 ,b0 =b0 3U0

where

dh0 = db0



db0 dh0

−1

"

d = dh0

U0 3

ηmin h0

!#−1

=

−

3U0 4

ηmin h0

!−1

(6.259)

The tangent to the circle in Eq. (6.257) is represented by 2

b0 b0 + h0 h0 − OQ = 0

(6.260)

At point Q the two straight lines in Eqs. (6.258) and (6.259) coincide. Therefore, the coefficients of the variables b0 and h0 in Eqs. (6.258) and (6.259) must be proportional, with proportionality constant denoted by θ: 4

b0 = θηmin h0

(6.261a)

h0 = 3θU0

(6.261b)

The solution reads 5

θ=

h0 h ηmin , b0 = 0 3ηmin 3U0

(6.262)

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We substitute b0 into Eq. (6.256a) to obtain h0 : ! 41 √ 3U0 h0 = ηmin

(6.263)

The value of b0 is found by substituting Eq. (6.263) and the expression for θ in Eq. (6.262) into Eq. (6.261a) as 1  3U0 4 b0 = (6.264) ηmin We denote by R1 and R2 the intersection points of the curve g1 (b0 , h0 ) = 0 with e If the point Q with coordinates (b0 , h0 ), defined the boundaries of the region Γ. in Eqs. (6.263) and (6.264), lies on the boundary of the feasible region as shown in Fig. 6.24(b), i.e., is located on the arc R1 R2 of the curve g1 (b0 , h0 ) = 0, then the values of b0 and h0 so obtained represent the solution of the optimization problem (6.253). If, however, the point Q is outside the admissible region as shown in Fig. 6.24(a), then the solution of the optimization problem is given by the vertex e closest to the origin O. of the region Γ 6.7.8

A numerical example

In this section, we apply the present hybrid optimization and anti-optimization procedure to the minimum weight design of a simply supported beam with length L = 15 m. The functions φi (x) in Eq. (6.205) governing the variation of the elastic modulus E(x) along the axis of the beam are chosen in the form  x 4  x 5  x 2 + σ3 , φ2 (x) = σ2 (6.265) φ1 (x) = 1 + σ1 L L L where σi (i = 1, 2, 3) are the coefficients given as σ1 = 1.1, σ2 = 5.4, σ3 = 2.2. The shape functions χb (x) and χh (x) in Eq. (6.206) governing, respectively, the geometric shape of the width b(x) and of the height h(x) are  x 2 x (6.266) χb (x) = 1 + ε1 , χh (x) = 1 + ε2 L L where εi (i = 1, 2) are the coefficients, which are given as ε1 = −0.7, ε2 = −0.73. The uncertain coefficients E1 and E2 in Eq. (6.205) are assumed to vary in an ellipsoidal region. The coordinates E10 and E20 of the center and the semi-axes a and b of the ellipse are listed in Fig. 6.25(a). The mass density is set at ρ = 2.7 × 10−6 kg · s2 /cm4 , whereas the aeroelastic parameters in Eq. (6.213) are c∞ = 3.3 × 104 m/s and k = 1.66. The values of the coefficients d0ij are d011 = 3.4 × 10−5 , d012 = 3.91 × 10−6 , and d022 = 5.5 × 10−6 , which yield a family of concentric ellipses after substituting the values of the coefficients d0ij into Eq. (6.243). The generic ellipse of the family is represented by the dotted line in Fig. 6.25(a). Equation (6.247), used to determine the coordinates e1 and e2 , becomes γ 4 − 7.81 × 106 γ 3 − 6.32 × 1013 γ 2 + 1.11 × 1011 γ − 5.22 × 1023 = 0

(6.267)

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(a)

(b) Fig. 6.25

Application of the hybrid optimization and anti-optimization.

whose smallest solution is γ = −5.07 × 106 . The values of e1 and e2 are then e1 = 1.06 × 105 and e2 = 2.98 × 105 . In Fig. 6.25(a), the ellipse corresponding to the minimum η 2 and tangent to the region of uncertainty in Pworst = (e1 , e2 ) is depicted with a solid line. We assume that the design variables b0 and h0 are in the region 6 cm ≤ b0 ≤ 30 cm, 0.5 cm ≤ h0 ≤ 2.5 cm

(6.268)

represented in Fig. 6.25(b). The lines corresponding to the extremal of the interval [U0,min, U0,max ] are depicted, respectively, with dashed and dot-dashed lines. The interval where the constraint g1 (b0 , h0 ) ≥ 0 is active is represented by [1.9 × 103 , 1.02 × 106 ]. We set U0 = 900 m/s. e of variation of the design variables b0 and h0 is depicted as a The region Γ

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rectangle in Fig. 6.25(b). The values of b0 and h0 corresponding to Eqs. (6.263) and (6.264) are listed in Fig. 6.25(b). It is seen that they do not belong to the e as shown in Fig. 6.25(b). Therefore, the combination of boundary of the region Γ the design values b0 and h0 corresponding to the minimum of the weight W (b0 , h0 ) is found at the point Qopt = (6.0 cm, 1.79 cm). With the obtained values, the minimum weight in this simple model is W = 38.7 kg, which is the smallest value satisfying all the constraints imposed on the system. 6.7.9

Conclusion

We discussed the flutter of a beam with uncertain elastic modulus that is simply supported at both ends in a stream of gas flow. The structural model has a variable cross-section and variable elastic modulus along the axis of the beam to reflect the realistic situation. We addressed the following question: How can the optimization methods in the presence of the bounded uncertainty, which is neither probabilistic nor fuzzy, be utilized? The parameters describing the elastic modulus were modeled as variables belonging, respectively, to a polygonal hull of experimental points, or to an ellipsoidal region. Hybrid optimization and anti-optimization proved to be an effective tool for dealing with optimum design in the presence of uncertainty. We first solved the problem to find an expression for the worst possible combination of the uncertain parameters involved in one of the constraints of the design variables. As a second step we optimized the design variables under the anti-optimized version of one of the constraints. Note that the further studies on flutter problems in nonprobabilistic setting were conducted by Wang and Qiu (2009) and Khodaparasi, Marques, Badcock and Mottershead (2009). 6.8

Some Further References

For additional references to anti-optimization in the vibration context, the reader may consult, for example, the works by Ben-Haim (1994, 1998a, 1998b), Ben-Haim, Fr´ yba and Yoshikawa (1999), Ben-Haim and Natke (1992), Capiez-Lernout and Soize (2008), Chen, Chu, Yan and Wang (2007), Chen and Lian (2002), Chen, Qiu and Liu (1994), Chen, Qiu and Song (1995), Chen and Wu (2004a), Chen, Wu and Yang (2006), Chen, Yang and Lian (2000), Craig, Stander and Balasubramanyam (2003), Drenick (1968, 1970, 1973, 1977a, 1977b, 1993), Drenick, Novomestky and Bagchi (1989), Drenick and Park (1975), Drenick, Wang, Yun and Philappacopoulos (1980), Drenick and Yun (1979), Dyne, Hammond and Davies (1988), Elishakoff (1991a), Elishakoff and Ben-Haim (1990b), Elishakoff and Duan (1994), Fr´ yba, Nakagiri and Yoshikawa (1998), Fr´ yba and Yoshikawa (1997, 1998) Gao, Zhang, Ma and Wang (2008), Hlav´ aˇcek (2002b), Hlav´ aˇcek, Pleˇsek and Gabriel (2006), Leng and He (2007), Li, Elishakoff, Starnes, and Shinozuka (1996), Lignon and J´ez´equel (2006), Modares, Mullen and Muhanna (2006), Mullen and Muhanna (1999), Nakagiri and

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Suzuki (1997), Qiu, Chen and Elishakoff (1995a, 1996a, 1996b) Qiu, Chen and Jia (1995b), Qiu, Elishakoff and Starnes (1996d), Qiu and Hu (2008), Qiu, Hu, Yang and Lu (2008a), Qiu, Ma and Wang (2004c), Qiu, M¨ uller and Frommer (2004e), Qiu and Wang (2003, 2005, 2006), Sadek, Sloss, Adali, and Bruch (1993), Srinivasan, Corotis and Ellingwood (1992), Srinivasan, Ellingwood and Corotis (1991), Venini (1998), Wang and Qiu (2009), Yamaguchi, Kogiso and Yamakawa (2007), and Yoshikawa and Fr´ yba (1997).

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Chapter 7

Anti-Optimization via FEM-based Interval Analysis

‘Make the best of things.’ (Russian proverb) ‘... approaches based on the concept of “critical excitation” or “worst-case input” are promising and seem to lead to rational design concept.’ (Takewaki, 2002b)

In this chapter, we deal with anti-optimization in the context of the finite element method, which is apparently the most universal method currently for structural analysis. We first discuss the interval analysis of multi-degree-of-freedom systems. Then we deal with interval finite element analysis of shear frame. 7.1

Introduction

In many real-life problems of structural engineering, neither the initial conditions, external forces, nor the parameters of the constitutive relations used in procedures to calculate response quantities can be perfectly described, because of lack of detailed knowledge and also due to the inherent uncertainties. Therefore, there have been many approaches presented for numerical anti-optimization based on the general finite element method (FEM). Pantelides and Booth (2000) developed a numerical optimization algorithm with ellipsoidal bounds of uncertainty, and applied it to reinforced concrete structures and trusses. Muhanna, Mullen and Zhang (2005) presented a penalty-based approach. Lanzi and Giavotto (2006) used a commercial finite element analysis package called ABAQUS for buckling optimization. General methodologies of anti-optimization in parallel computing environment is discussed in Gurav, Goosen and van Keulen (2005a). Presently, there are three paradigms to describe uncertainty: (a) theory of probability and random processes, (b) fuzzy sets, and (c) convex set-theoretical analysis. These three corners of the uncertainty triangle (Elishakoff, 1990), as mentioned in the preface of this book, can be combined in various manners. The hybrid stochastic-fuzzy approach was developed by Hoh and Hagaki (1989). Convex mod211

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eling of uncertainty was studied in the book by Ben-Haim and Elishakoff (1990), in which a complete literature survey for continuous systems and several illustrations for the applications in mechanics are presented. Elishakoff and Colombi (1993) developed a hybrid stochastic-convex approach to solve some actual space-shuttle problems. Interval analysis is one of the tools for incorporating uncertainty in structural analysis. We presented in Sec. 3.3 the basic properties of interval analysis using simple mathematical and structural models. However, for practical design process, response quantities, in the form of internal forces, stresses, displacements and strains, are needed to evaluate the safety and serviceability of structures with many degrees of freedom. Interval analysis for multidimensional structures is studied in Muhanna and Mullen (2001) and Muhanna, Zhang and Mullen (2007). Moens and Vandepitte (2005) summarized the interval finite element methods for static and eigenvalue analyses. In this chapter, we present some results of FEM-based interval analysis of static responses.

7.2

Interval Analysis of MDOF Systems

In this section, interval analysis of linear equations is summarized following the notations of Muhanna and Mullen (2001) and Muhanna, Zhang and Mullen (2007). Consider a set of linear equations formulated as Ax = b

(7.1) >

where A = (Aij ) is a constant n × n matrix, b = (b1 , . . . , bn ) is a constant vector, and x = (x1 , . . . , xn )> is the variable vector. Suppose the components of A and b are bounded by intervals as ALij ≤ Aij ≤ AU ij , (i, j = 1, . . . , n) bLi ≤ L

bi ≤

bU i , U

(i = 1, . . . , n)

(7.2a) (7.2b)

where the superscripts ( · ) and ( · ) indicate specified lower and upper bounds, respectively. The set of A and b satisfying the conditions (7.2a) and (7.2b), respectively, are denoted by the interval matrix A and interval vector B. The interval values are denoted by calligraphic letters throughout this chapter. Then the family of equations (7.1) satisfying (7.2a) and (7.2b) are given as Ax = b, (A ∈ A, b ∈ B)

(7.3)

AH B ≡ ♦S(A, B)

(7.4)

The solution set of Eq. (7.3) is denoted by S(A, B) that is not generally an interval vector. In the following, we assume that all the matrices A satisfying the condition (7.2a) are regular (nonsingular) so that S(A, B) is bounded. Let ♦S(A, B) denote the smallest interval components called hull of S(A, B). The hull inverse AH of the matrix A is defined by

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i.e., AH B = ♦{A−1 b|A ∈ A, b ∈ B} (7.5) On the other hand, the matrix inverse A−1 of a regular interval square matrix A is defined as A−1 ≡ ♦{A−1 |A ∈ A} (7.6) Then the following relation holds: AH B ⊆ A−1 B (7.7) where the equality holds if A is diagonal. The problem of finding ♦S(A, B) is a combinatorial problem that cannot be solved in a practically acceptable computational time for a large-scale problem. Therefore, our purpose here is to find the narrowest possible interval vector X satisfying AH B = ♦S(A, B) ⊆ X (7.8) for which many algorithms exist such as interval Gaussian elimination, that may break down for a large-scale problem (Nickel, 1977), Gauss–Seidel iteration (Gay, 1982), and its generalization (Neumaier, 1982; Rump, 1992). The typical approach utilizes a regular n × n transfer matrix R for converting the problem Ax = b to a problem of finding the fixed point g(x) = x of an operator g(x): g(x) = x − R(Ax − b) = Rb + (I − RA)x (7.9) where I is the n × n identity matrix (see Muhanna, Zhang and Mullen (2007) for details of this approach). As an example, consider the case when uncertainty exists only in some components of A as     2 [−1, 0] 1.2 A= , B= (7.10) [−1, 0] 2 −1.2 By computing x for the four cases (A12 , A21 ) = (−1, −1), (−1, 0), (0, −1), and (0, 0), we have         0.4 0.3 0.6 0.6 x = A−1 b = , , , (7.11) 0.4 −0.6 −0.3 −0.6 Therefore, AH B is obtained as   [0.3, 0.6] H A B= (7.12) [−0.6, −0.3] On the other hand, we can compute A−1 for the four cases (A12 , A21 ) = (−1, −1), (−1, 0), (0, −1), and (0, 0) as         2/3 1/3 1/2 1/4 1/2 0 1/2 0 −1 A = , , , (7.13) 1/3 2/3 0 1/2 1/4 1/2 0 1/2 Therefore, the interval of A−1 is obtained as   [1/2, 2/3] [0, 1/3] −1 A = (7.14) [0, 1/3] [1/2, 2/3] and A−1 B results in   [0.2, 0.8] −1 A B= ⊃ AH B (7.15) [−0.8, −0.2] Hence, the relation (7.7) is satisfied.

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U1

U2 P1 K2

K1

Fig. 7.1

7.3

optimization

P2

A 2-bar structure.

Interval Finite Element Analysis for Linear Static Problem

As an illustrative example, consider a simple 2-bar structure as shown in Fig. 7.1 subjected to the loads P = (P1 , P2 )> . The extensional stiffnesses of bars 1 and 2 are denoted by K1 and K2 , respectively. The stiffness matrix K is given as   K1 + K2 −K2 K= (7.16) −K2 K2 The displacement vector U = (U1 , U2 )> is found from      K1 + K2 −K2 U1 P1 = −K2 K2 U2 P2 The inverse of K is obtained as K −1



(7.17)

 1 1  K1 K1   =  1 K1 + K 2  K1 K1 K2

(7.18)

K1 + K 2 1 , U2 = K1 K1 K2

(7.19)

Consider the case where the loads have deterministic values P1 = 0, P2 = 1. The displacements U = (U1 , U2 )> are explicitly obtained as U1 =

Suppose K1 and K2 are interval values K1 = [1.9, 2.1] and K2 = [0.9, 1.1]. Then U2 =

[2.8, 3.2] [1.9, 2.1] + [0.9, 1.1] = = [1.21, 1.87] [1.9, 2.1] × [0.9, 1.1] [1.71, 2.31]

(7.20)

for which the diameter is 0.66. However, this expression gives an overestimation of the interval of U2 due to the multiple appearance of K1 and K2 in a component of K −1 as discussed in Sec. 3.3. The exact interval can be found by rewriting U2 as U2 =

1 1 + K1 K2

from which we obtain 1 1 U2 = + = [0.48, 0.53] + [0.91, 1.11] = [1.39, 1.64] [1.9, 2.1] [0.9, 1.1] and the diameter is reduced to 0.25.

(7.21)

(7.22)

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For a general finite element analysis problem, the explicit form of the displacement vector cannot be obtained. Muhanna, Mullen and Zhang (2005) noted that the accuracy of the interval vector of the displacements depends on the order of the sequence of arithmetic operations in the solution process by finite element analysis. They presented an element-by-element approach to delay the operation involving the coupling of the interval variables, where the penalty terms are used to ensure the compatibility among the element displacements as described below. In the standard procedure of finite element analysis, the element stiffness matrices are assembled to formulate the global stiffness matrix. If we write the stiffness matrix of member i of the 2-bar structure as   Ki −Ki Ki = (7.23) −Ki Ki then the four components of the matrix are treated as independent variables in the interval analysis, although they depend on Ki only. If we write K i as    Ki 0 1 −1 Ki = (7.24) 0 Ki −1 1 then the number of independent variables in interval analysis is reduced to two, and the variables exist in a diagonal matrix. Another source of overestimation of the intervals of displacements is the coupling of elements through the common nodes; e.g., the term K1 + K2 appears due to the coupling of the bars at the center node 1 of the 2-bar structure in Fig. 7.1. The Lagrange multiplier approach can be used to delay the coupling in finite element analysis (Muhanna and Mullen, 2001). The analysis problem is formulated as a minimization problem of the total potential energy 1 (7.25) Π = U > KU − U > P 2 under constraints CU = V

(7.26)

where C and V are the specified matrix and vector, respectively, to represent the compatibility conditions between the elements. Note that the structure of K in Eq. (7.25) is different from that of the conventional global stiffness matrix; it has a block-diagonal form of the element stiffness matrices. Accordingly, the nodal displacements of different elements at the same node are regarded as independent variables. Although the definitions and sizes of K, U , and P are different from those for conventional analysis, the same notations are used for simplicity. Let n e and m denote the number of degrees of freedom of each element and the number of elements, respectively. Then the size of K is ng × ng with ng = ne m, and U and P also have ng components. For example, for the 2-bar structure in Fig. 7.1, U is written as (1)

(1)

(2)

(2)

U = (U0 , U1 , U1 , U2 )>

(7.27)

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where the superscript ( · )(i) denotes the value in the ith bar, and the fixed support is denoted by node 0. The compatibility conditions are given as  (1)  U    0(1)    0 1 0 0 0 U 1  (7.28)  (2)  = 0 0 1 −1 0 U1  (2)

U2

The stationary conditions for Π with incorporated constraints (7.26) using the vector Q of Lagrange multipliers lead to the following system of equations:      U P K C> = (7.29) C 0 Q V

The reader may consult Sec. 2.4 for details of the Lagrange multiplier approach. Suppose the interval form K of K is expressed as K = DS, where D is the uncertain diagonal matrix, and S is a deterministic matrix as demonstrated in Eq. (7.24) for an element stiffness matrix. Then the first equation of Eq. (7.29) is written in an interval form DSU = P − C > Q

(7.30)

CU = 0

(7.31)

where P is assumed to be deterministic, for simplicity. Using V = 0 for general compatibility conditions without forced displacements, the interval form of the second equation in Eq. (7.29) is written as which is premultiplied by C

>

and added to Eq. (7.30) to obtain

D(S + C > C)U = P − C > Q

(7.32)

Eq. (7.32) can be rewritten as >

DHU = P − C > Q

(7.33)

U = H −1 D−1 (P − C > Q)

(7.34)

with H = S + C C. Hence, U is obtained as −1

where D is the exact inverse of the diagonal interval matrix D. Note that Q has the physical meaning as internal force vector. Therefore, for a statically determinate structure, for which the internal loads do not depend on the stiffnesses of the elements, the vector P − C > Q is deterministic, corresponding to the deterministic (nominal) value of Q. For the interval of displacement vector, suppose D is formulated so that each interval stiffness parameter appears only once. In this case, the exact interval U can be successfully computed from Eq. (7.34), if the structure is statically determinate. However, for a statically indeterminate structure, the use of deterministic (nominal) value of Q in Eq. (7.34) leads to underestimation of the interval U. By contrast, the use of Eqs. (7.31) and (7.32) for computing the interval Q results in overestimation of U. Muhanna and Mullen (2001) presented an iterative approach to obtain a sharp interval hull for U.

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5

K5 /2

K5 /2

4

K4 /2

K4 /2

3

K3 /2

K3 /2

2

K2 /2

K2 /2

1

K1 /2

Fig. 7.2

7.4 7.4.1

K1 /2

A 5-story shear frame.

Interval Finite Element Analysis of Shear Frame Basic equations

A multi-degree-of-freedom shear frame, e.g., the 5-story shear frame as shown in Fig. 7.2, is considered. The bending stiffness of the storey columns are taken as uncertain variables. The bending stiffnesses are modeled using a set-theoretical interval approach, where the uncertain variable is assumed to be strictly bounded from below and above. This section summarizes the results of K¨ oyl¨ uoˇ glu and Elishakoff (1998) with modification of notations to be consistent with other parts of this book. Finite element models with cubic deterministic shape functions are applied to discretize the uncertain displacement fields. This yields the 2 × 2 element stiffness matrix K e of the shear beam element. When the interval approach is applied, the 2×2 interval stiffness matrix Ke of the eth element is obtained (K¨ oyl¨ uoˇ glu, C ¸ akmak and Nielsen, 1995). Note that, in the following, the stiffness of each story is modeled by a shear beam element representing the sum of the stiffnesses of the two columns. All components Ke,ij of Ke are bounded from below and above. The interval matrix ce with a radius matrix ∆K e , Ke is located symmetrically around a central matrix K i.e., ce − ∆K e , K ce + ∆K e ] Ke = [K

(7.35)

The element stiffness matrices of the shear frame appear in the uncertain set of linear equations, which is solved using Fuchs’ method (Fuchs, 1991) to obtain the nodal displacements in closed form with respect to uncertain story stiffnesses, where the element stiffness matrix is first diagonalized and then assembled accordingly. Hence, the lower and upper bounds for the nodal displacements are functionally related to

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those of the interval element stiffness matrices. Elishakoff, Ren and Shinozuka (1997) applied Fuchs’ method to statically determinate and indeterminate beams with random stiffness. According to the Bernoulli–Euler beam theory, the displacement field v(x) in the transverse direction of the eth element is related to the loading field p(x) by [(EI)e (x)v(x),xx ],xx = p(x)

(7.36)

where x is the coordinate along the beam axis, (EI)e (x) is the bending stiffness field, and ( · ),x denotes differentiation with respect to x. The assumptions and the differential relationship of the Bernoulli–Euler beam theory are assumed to be valid even when (EI)e (x) possesses some uncertainties. This means that (EI)e (x) is such that these assumptions are not violated. In what follows, the uncertain field (EI)e (x) is modeled using set-theoretical interval approach, where (EI)e (x) is assumed to be bounded from below and above with constant envelope as (EI)Le ≤ (EI)e (x) ≤ (EI)U e , (0 ≤ x ≤ Le )

(7.37)

where (EI)Le and (EI)U e are the lower and upper bounds for (EI)e (x), and Le is the length of element e. The central value of the bending stiffness becomes (EI)0e = [(EI)Le + (EI)U e ]/2. For a shear frame in Fig. 7.2, the rotation at each end of the column element is fixed. Therefore, the uncertain displacement field V(x) of an element is approximated as a linear combination of the uncertain vector of two-degree-of-freedom transverse nodal displacements Ve multiplied by the independent vector of cubic interpolation functions N (x) = (N1 (x), N2 (x))> : V(x) = N > (x)Ve

(7.38)

where N1 = 1 −

3 2 2 3 2 x + 2 x3 , N 2 = 3 x2 − 2 x3 3 Le Le Le Le

(7.39)

Following the standard finite element methodology, the interval stiffness matrix Ke of the beam element with (i, j)-component Ke,ij is evaluated as Z Le d2 Ni (x) d2 Nj (x) dx (7.40) Ke,ij = (EI)e (x) dx2 dx2 0 where (EI)e (x) is the interval bending stiffness field. Owing to the uncertainty in the bending stiffness, all components of Ke are uncertain, yet bounded from above and below, as follows, as a result of the interval model: U L ≤ Ke,ij ≤ Ke,ij Ke,ij

(7.41)

Let Re be defined as Re =

12(EI)e L3e

(7.42)

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with its lower and upper bounds ReL and ReU , respectively. The lower bound K Le and the upper bound K U e of K e are found, as follows, from minimization/maximization of the integral in Eq. (7.40) and j independently using  for each i    Eq. (7.37): ReL −ReU ReU −ReL  , KU   K Le =  (7.43) e = U L L U −Re Re −Re Re We consider the eigenvalues λ1 and λ2 (λ1 < λ2 ) of K e . The independent computation for each Ke,ij in Eq. (7.43) gives a four-dimensional rectangular prism L U with sides defined by the coordinates Ke,ij and Ke,ij , within which K e lies for any realization of (EI)e (x), and leads to conservative intervals compared to the intervals where the interdependency is taken into account. For that reason, working with Eq. (7.43) to obtain the bounds for the eigenvalues of the element interval matrix leads to very conservative results. Moreover, the smallest eigenvalue that is obviously equal to zero for any realization of a beam element without constraining rigid-body motions would not become zero, but become an interval value containing zero. Thus, the eigenvalues are first computed from the integral of Eq. (7.40) directly, which leads to Z Le 72 dx (7.44) (EI)e (x) 6 2 λ1 = 0, λ2 = 2 L (L − 4L e x + 4Le x ) 0 e e Hence, the lower and upper bounds λL2 and λU 2 , respectively, on the second eigenvalue λ2 can be obtained from minimization and maximization of λ2 in Eq. (7.44), respectively, as λL2 = 14ReL − 12ReU, (7.45a)

L U λU (7.45b) 2 = −12Re + 14Re 0 Then, the central value λ2 and the radius ∆λ2 of λ2 are computed as λL + λ U 2 = ReL + ReU λ02 = 2 2 (7.46) U L ∆λ2 = 13(Re − Re ) We next evaluate the displacements of a general multi-story shear frame subjected to static loads. Let K and P denote the global stiffness matrix and the applied nodal loads, respectively. We assume that K is an interval matrix, but P is a deterministic vector, for simplicity. Then the nodal displacement vector U is also an interval vector denoted by U. One can write down the interval nodal displacements as a function of interval eigenvalues as U = K−1 P = (Q−1 )> J −1 Q−1 P (7.47) where J −1 is a diagonal matrix with 2/λ for all the diagonal components, and 2   −1 1   −1 1     · ·   (7.48) Q=    · ·    −1 1  −1

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For further details the reader may consult with K¨ oyl¨ uoˇ glu and Elishakoff (1998). Then, the lower and upper bounds U L and U U of U can be computed from minimization/maximization of U in Eq. (7.47) as U L = min[(Q−1 )> J −1 Q−1 P ]

U

U

= max[(Q

−1 >

) J

−1

Q

−1

P]

(7.49a) (7.49b)

In the process of minimization/maximization of Eqs. (7.49a) and (7.49b), subtraction and addition of intervals appear. It should be noted that when two intervals are added or subtracted, the resulting interval has a central value which is the sum or subtraction of the central values of the two intervals, and has a radius which is the sum of the radii of these two intervals (see Sec. 3.3 for the basic properties of interval arithmetic). Furthermore, interval arithmetic is basically used to find bounds for the discrete values of the displacements at the nodes, and these bounds for the shape functions could only become approximate bounds for the uncertain displacement field. In order to find bounds for the uncertain displacement field, one needs to consider an interval validation methodology for the initial differential equation (7.36) (Alefeld and Herzberger, 1983; Neumaier, 1990). 7.4.2

A numerical example

Consider a 5-story shear frame as shown in Fig. 7.2. Let the height of each story be 1 and (EI)0e = 1/12 so that each story with deterministic properties has a stiffness Re = 12(EI)0e /L3e = 1, and, from Eq. (7.46), the central value λ02 of the eigenvalue is equal to 2Re = 2. Moreover, for the 5-story shear frame, Q−1 can be obtained in closed form as   −1 −1 −1 −1 −1  −1 1 −1 −1   −1  (7.50) Q = −1 −1 −1   −1 1  −1

Four different loading scenarios are considered. The global load vector is chosen from P 1 = (1, 1, 1, 1, 1)>, P 2 = (1, 2, 3, 4, 5)>, P 3 = (1, 1, 1, −1, −1)>, and P 4 = (−2, 1, 2, 3, −5)>, which are called Loadings 1, 2, 3, and 4, respectively. Under these loads, the nodal displacements of the shear frame with deterministic properties are computed, and the results are listed in Table 7.1. Suppose the lower and upper bounds for the bending stiffness field are taken 3 0 such that ReL = 12(EI)Le /L3e = 0.97, ReU = 12(EI)U e /Le = 1.05. Then Re = 12(EI)0e /L3e = 1.01. From Eq. (7.46), we obtain λL2 = 0.98 and λU 2 = 3.06. The bounds for the global nodal displacements are given in Table 7.2, which covers all possible displacements for realizations of the shear frame within the bounds on Re . In addition to such a relation, non-dimensional quantities wU1 , . . . , wU5 , which are called coefficients of interval uncertainty and defined as the ratio of the radius to the

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Table 7.1 Global nodal displacements of the deterministic shear frame.

U1 U2 U3 U4 U5

Loading 1

Loading 2

Loading 3

Loading 4

5 9 12 14 15

15 29 41 50 55

1 1 0 −2 −3

−1 0 0 −2 −7

Table 7.2 Lower and upper bounds for the global nodal displacements of the shear frame with interval stiffness properties. Loading 1 U1 U2 U3 U4 U5

[3.27, [5.88, [7.84, [9.15, [9,80,

10.20] 18.37] 24.49] 28.57] 30.61]

Loading 2

Loading 3

Loading 4

[9,80, 30.61] [18.95, 59.18] [26.80, 83.67] [32.68, 102.04] [35.95, 112.24]

[0.65, 2.04] [0,65, 2.04] [−1.39, 1.39] [−5.47, 0,08] [-7.51, -0.57]

[−2.04, −0,65] [−1.39, 1.39] [−1.39, 1.39] [-5.47, 0.08] [−15.67, -3.19]

Table 7.3 Coefficients of interval dimensions of the global nodal displacements of the shear frame with interval stiffness properties.

w U1 w U2 w U3 w U4 w U5

Loading 1

Loading 2

Loading 3

Loading 4

0.5149 0.5149 0.5149 0.5149 0.5149

0.5149 0.5149 0.5149 0.5149 0.5149

0.5149 0.5149 — 1.0297 0.8581

0.5149 — — 1.0297 0.6620

absolute value of the central value, are obtained as shown in Table 7.3 for U1 , . . . , U5 . Note that ‘–’ in Table 7.3 means that the coefficient cannot be computed, because the central value vanishes. 7.5

Interval Analysis for Pattern Loading

In this section, we follow the basic ideas of interval analysis of beams and frames subjected to pattern loads (K¨ oyl¨ uoˇ glu, C ¸ akmak and Nielsen, 1995); however, the notations are modified to be consistent with other parts of the book. Furthermore, the concepts are presented in a simpler form, and only new numerical examples are given. Consider a Bernoulli–Euler beam subjected to a uniformly distributed load p, which exists in the interval [p0 − ∆p, p0 + ∆p]. Based on the conventional finite

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element formulation with cubic displacement interpolation function, the element stiffness matrix is derived in a standard manner, and the nodal load vector p of an element corresponding to the uniformly distributed load p is written as   p/2  pL/12   p= (7.51)  p/2  −pL/12

The element stiffness matrices and the element load vectors are assembled to form the n × n global stiffness matrix K and the global nodal load vector P , respectively, which are defined as interval matrix and vector: K = [K 0 − ∆K, K 0 + ∆K], P = [P 0 − ∆P , P 0 + ∆P ]

(7.52)

Bounds for the displacement vector U = (U1 , . . . , Un )> can be found, as follows, by using the Oettli–Prager lemma (Neumaier, 1990), which states that |K 0 U − P 0 | ≤ ∆K|U | + ∆P

(7.53)

where, e.g., |U | denotes the vector consisting of the absolute values |Ui | of the components of U . The sign of Ui , denoted by sign Ui , is defined as   1 for Ui > 0 sign Ui = 0 for Ui = 0 (7.54)  −1 for Ui < 0

Suppose the sign of each component of Ui is known. In most of the engineering problems, the direction of the load vector is fixed, therefore, this is a very reasonable assumption. Let S be a diagonal matrix with the signs of Ui : S = diag(sign Ui )

(7.55)

Then the following relations hold for the absolute values of Ui : (K 0 S − ∆K)|U | − P 0 − ∆P ≤ 0 0

0

(−K S − ∆K)|U | + P − ∆P ≤ 0

(7.56a) (7.56b)

Therefore, |Ui | can be conceived as independent variables, and the conservative upper bounds (lower bounds) for |Ui | can be found by solving a linear programming problem for maximizing (minimizing) |Ui | under constraints (7.56) and |Ui | ≥ 0 (i = 1, . . . , n). Note that we must solve linear programming problems 2n times to obtain the upper and lower bounds of the absolute values of all displacement components. Consider next a pattern loading problem without uncertainty in the stiffness matrix. The upper and lower bounds for the uniformly distributed load pe on L element e are denoted by pU e and pe , respectively. The ith component of the global

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223

2.2

U2

2.1

2.0

1.9

1.8 0.9

Fig. 7.3

1.0 U1

1.1

Bounds of nodal displacements of the 2-bar structure in Fig. 7.1.

displacement vector against the uniformly distributed unit load on element e is denoted by Die . Then, the maximum and minimum values UiU and UiL of Ui are obtained from m X U e L Ui = max{Die pU (7.57a) i , Di pi } e=1

UiL =

m X e=1

e L min{Die pU i , Di pi }

(7.57b)

where m is the number of elements. On the other hand, an interval can be defined for the nodal load vector P so that any realization of the set of distributed loads on m elements are contained in the interval [P L , P U ] = [P 0 − ∆P , P 0 + ∆P ], where P 0 is the nominal value and ∆P is the smallest radius. Then, the extremum values of Ui can be computed by simple addition of extremum of the two terms as n X −1 −1 UiU = max{Kij (Pj0 + ∆Pj ), Kij (Pj0 − ∆Pj )} (7.58a) j=1

UiL =

n X j=1

−1 −1 min{Kij (Pj0 + ∆Pj ), Kij (Pj0 − ∆Pj )}

(7.58b)

For a frame structure, the total number of degrees of freedom is generally larger than the number of members; i.e., n > m. The number of independent load parameters is m for Eq. (7.57), and n for Eq. (7.58). Therefore, Eq. (7.58) leads to a more conservative (wider) interval than Eq. (7.57), as illustrated in the following simple example. We first demonstrate a simple interval algebra using the 2-bar structure in Fig. 7.1, where the stiffnesses are fixed at K1 = 2, K2 = 1; i.e.,   3 −1 K= (7.59) −1 1

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p1

1

p2

2

(1) 2

4 1

Fig. 7.4

(2)

3

A cantilever beam consisting of two elements subjected to distributed loads.

The nominal values of the loads are P10 = P20 = 1, and the load uncertainty is given by ∆P1 = ∆P2 = 0.1, which are independent of each other. Using Eq. (7.58), the intervals of U1 and U2 are obtained as [0.9,1.1] and [1.8,2.2], respectively. The displacements for the loads at the vertices of its feasible region (0.9,0.9), (0.9,1.1), (1.1,0.9), (1.1,1.1) are obtained as (0.9,1.8), (1.0,2.1), (1.0,1.9), (1.1,2.2), respectively. Since the nodal displacements are linear functions of the nodal loads, the existable region of the nodal displacements (U1 , U2 ) is a convex region as indicated in the gray area of Fig. 7.3, which verifies that the narrowest intervals of U 1 and U2 are [0.9,1.1], and [1.8,2.2], respectively. Consider next the cantilever beam as shown in Fig. 7.4 consisting of two beam elements, where the displacement numbers are defined in the lower figure. The beam is subjected to uniformly distributed loads p1 and p2 on members 1 and 2, respectively. Suppose the length L and the bending stiffness EI are 1, for simplicity, for the two members. Then the inverse of the stiffness matrix is obtained as   2 3 5 3 1 3 6 9 6   (7.60) K −1 =  6 5 9 16 12 3 6 12 12

The load vectors corresponding to the nominal values p01 = p02 = 1 of distributed loads are given as     6 6  1 1  1 −1 0 0   , p2 =   p1 = (7.61) 12  0  12  6  0 −1 When the radii ∆p1 and ∆p2 of uncertainty are equal to 0.1, the upper bound P U = P 0 + ∆P and the lower bound P L = P 0 − ∆P are obtained as     1.1000 0.9000  0.0167  −0.0167 L    PU =  (7.62)  0.5500  , P =  0.4500  −0.0750

−0.0917

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225

Since all components of K −1 are positive, the upper and lower bounds U U and U L are easily computed from Eq. (7.58) as     0.7959 0.6208 1.3167 1.0166 L    UU =  (7.63) 2.2584 , U = 1.7416 1.5167 1.1499

We next use Eq. (7.57) for the same nominal values and radii of p1 and p2 . Then from the displacements corresponding to the loads (p1 , p2 ) = (1.1, 1.1), (1.1,0.9), (0.9,1.1), and (0.9,0.9), the following upper and lowers bounds are obtained:     0.7792 0.6375 1.2833 1.0500 L    UU =  (7.64) 2.2000 , U = 1.8000 1.4667 1.2000

Therefore, narrower bounds are obtained by evaluating the uncertainty in the distributed loads; i.e., evaluation after converting to nodal loads leads to a conservative bound, because the number of independent variables increases from two to four as a result of conversion. 7.6

Some Further References

For additional references to anti-optimization in the finite element context, the reader may consult, for example, the works by Chen and Yang (2000), Dessombz, Thouverez, Lain´e and J´ez´equel (2001), Elishakoff (1998c), Guo and L¨ u (2001), K¨ oyl¨ uoˇ glu and Elishakoff (1998), Matsumoto and Iwaya (2001), Muhanna and Mullen (2002, 2005), Muhanna, Kreinovich, Sˇ olin, Chessa, Araiza and Xiang (2006), Muhanna, Mullen and Zhang (2005), Muhanna, Zhang and Mullen (2007), Mullen and Muhanna (2002), Nakagiri (1995, 1996), Nakagiri and Suzuki (1996, 1997, 1999), Nakagiri, Hoshi and Yamada (2003), Nakagiri and Yoshikawa (1996), Rao, Chen and Mulkay (1998), Saxena (2003), Suzuki and Otsubo (1995), Vitali, Blanc, Larque, Duplay, Morvan, Nakagiri and Suzuki (1999), and Zhang (2005).

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Chapter 8

Anti-Optimization and Probabilistic Design

‘The worst-case and probabilistic world should not be opposed, but considered together.’ (Ben-Tal, El Ghaoui and Nemirovski, 2006) ‘Make the best out of the worst.’ (English proverb)

This chapter is devoted to systematic comparison of probabilistic and antioptimization approaches in the dynamic environment. We will consider impact buckling of uniform columns possessing uncertain initial imperfections. A single imperfection parameter is considered first, and then the methodologies and results are extended to the case of multiple imperfection parameters. 8.1

Introduction

In Chapter 1, we have already found, albeit on a simple example of a bar subjected to a bounded random load, that the worst-case scenario and probabilistic methods yield coincident or close results. In this chapter, less transparent problems are considered. The probabilistic modeling of various kinds of uncertainties in the dynamic buckling setting was dealt with by Goncharenko (1962), Budiansky and Hutchin´ son (1964), Lindberg (1965), Ariaratnam (1967), Kildibekov (1971), Amazigo and Frank (1973), Amazigo (1974), Lockhart and Amazigo (1975), Maymon and Libai (1977), Elishakoff (1978b), Bogdanovich (1987), and others. Convex modeling of the dynamic buckling problems was facilitated by Ben-Haim and Elishakoff (1989a), Elishakoff and Ben-Haim (1990a), Lindberg (1992a, 1992b), and Ben-Haim (1993a). These two avenues of thoughts on uncertainty modeling, namely, the probabilistic and anti-optimization ones, have not been compared in either of the above studies. This will be the principal objective of the present chapter. The direct comparison of designs yielded by two alternative approaches sheds light on the possible compatibility or incompatibility of these two alternative approaches. According to Einstein, ‘As far as the laws of mathematics refer to reality, they 227

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are not certain, and as far as they are certain, they do not refer to reality.’ How much more does this statement appear to apply to engineering, where numerous uncertain factors are encountered almost in every problem? The pertinent question arises of how to model this uncertainty. It is nowadays widely accepted that there are three competing methods of describing uncertainty, as discussed in the preface of this book; namely, the probabilistic method, the fuzzy-sets based approach, and the method which is referred to as anti-optimization (Elishakoff and Ben-Haim, 1990a) or as the guaranteed approach by Kurzhanski (1977) and by Chernousko (1994). Each of these approaches has generated numerous research papers and monographs. Yet the systematic comparisons among these three approaches are still missing. Naturally, there may be a strong argument put forward against such a comparison, in the first place. Indeed, these three approaches provide judgments of different kinds: The probabilistic approach furnishes the reliability of the structure, namely, the probability that the stresses, strains or displacements of the structure will not take values over some safe threshold. The fuzzy-sets based approach yields the safety margin via the concept of a membership function. Anti-optimization (or the guaranteed approach) provides the least favorable, maximum displacements which can be contrasted with the threshold values. If the maximum response does not cross the threshold, we can guarantee that the failure will not occur. This corresponds to the statement that if the least favorable, or an anti-optimized, response is below the failure boundary, a successful performance will be secured. This particular thought, namely, that three approaches provide different types of answers, precluded the very attempt to compare them for a long time. The first numerical comparison has been performed independently by Elishakoff, Cai and Starnes (1994a) and Kim, Ovseyevich and Reshetnyak (1993) in different but complementary contexts: The former study dealt with nonlinear static buckling, whereas the latter investigated linear dynamic response. In the paper by Elishakoff, Cai and Starnes (1994a), for certain regions of variations of the parameters, the design values of the buckling loads obtained via applications of two alternative approaches were found to be extremely close to each other. In the paper by Kim, Ovseyevich and Reshetnyak (1993), the design value corresponding to the mean plus three times standard deviation, was adopted. Elishakoff and Li (1999) studied a benchmark static problem, leading to direct contrast of the probabilistic and non-probabilistic analyses of uncertainty. General understanding on the limitation of probabilistic methods is summarized in Elishakoff (2000a). Langley (2000) presented a unified framework of combined probabilistic and possibilistic (deterministic) approaches. In Secs. 8.2 and 8.3, probabilistic and anti-optimization approaches are compared for impact buckling of uniform columns possessing single and multiple uncertain initial imperfections following the studies by Elishakoff and Zingales (2000) and Zingales and Elishakoff (2000), respectively.

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229

Initial imperfection amplitude Additional displacement amplitude

P(t)

P(t) h w Total displacement

L/2 L Fig. 8.1

8.2 8.2.1

Description of the structural model.

Contrasting Probabilistic and Anti-Optimization Approaches Problem formulation

Let hti◦ denote a singularity function, namely, a unit step function defined as  0, t < 0 (8.1) hti◦ = 1, t ≥ 0 where t is the time. The differential equation for the uniform column as shown in Fig. 8.1 under axial impact load P (t) = P hti◦

(8.2)

reads ∂4w ∂2w ∂2w ∂ 2 w0 + P (t) + mJ = −P (t) (8.3) ∂x4 ∂x2 ∂t2 ∂x2 where x is the axial coordinate, E is Young’s modulus, I is the moment of inertia, m is the mass density, J is the cross-sectional area, w0 (x) is the initial imperfection constituting a small deviation of the initial shape of the unstressed column, and w(x) is the additional transverse deflection of the axis of the column, so that EI

wT (x, t) = w0 (x) + w(x, t)

(8.4)

represents the total displacement. The boundary conditions for the column that is simply supported at both ends read w(x, t) = 0

at x = 0 and x = L

(8.5a)

2

∂ w(x, t) = 0 at x = 0 and x = L ∂x2 where L is the length of the column. The initial conditions are w(x, t) = 0 and

∂w(x, t) = 0 at t = 0 ∂t

(8.5b)

(8.6)

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We confine ourselves to the case where the initial imperfection w0 (x) has a multiplicative representation  πx  w0 (x) = h sin (8.7) L where h is the magnitude of initial imperfection. Following Elishakoff (1978a) we will use the non-dimensional quantities ξ=

w0 w P x , λ = ω1 t, u0 = , u= , γ= L ∆ ∆ Pcl

(8.8)

where ξ is the non-dimensional axial coordinate, λ is the non-dimensional time, u0 (ξ) is the non-dimensional initial displacement, u(ξ, t) is the non-dimensional additional displacement, and γ is the non-dimensional axial load. In Eq. (8.8) r  π 2 r EI I πEI 2 ω1 = , ∆= , Pcl = (8.9) L mJ J L2 where ω1 is the fundamental circular natural frequency of the ideal column, i.e., of a column with neither initial imperfections nor axial load, ∆ is the radius of inertia, and Pcl is the classical Euler buckling load. The differential equation (8.3) reduces to the following non-dimensional one: 2 2 2 ∂4u 2 ∂ u 4∂ u 2 ∂ u0 + π γ + π = −π γ ∂ξ 4 ∂ξ 2 ∂λ2 ∂ξ 2

(8.10)

We introduce the non-dimensional initial imperfection g as h (8.11) ∆ The additional displacements are expressed in a separable form analogous to Eq. (8.7), namely, g=

u(ξ, λ) = e(λ) sin(πξ)

(8.12)

where e(λ) is a time-dependent function. Substituting Eqs. (8.7) and (8.12) in conjunction with Eq. (8.11) into Eq. (8.10) yields the following ordinary differential equation with respect to the function e(λ): d2 e(λ) + (1 − γ)e(λ) = γg dλ2 The solution satisfying the initial conditions (8.6) is obtained as  γg  (cosh(rλ) − 1), γ > 1    1−γ 1 2 e1 (λ) = gλ , γ=1  2   γg   (1 − cos(rλ)), γ<1 1−γ

where

r=

p

|γ − 1|

(8.13)

(8.14)

(8.15)

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231

The non-dimensional total displacement z(ξ, λ) equals 1 wT (x, t) = ν(λ) sin(πξ) ∆ with ν(λ) being a function of a non-dimensional time λ alone   γg 1 ν(λ) = cosh(rλ) − , γ > 1, 1−γ γ g(λ2 + 2) , γ = 1, ν(λ) = 2  γg 1 − cos(rλ) , γ<1 ν(λ) = 1−γ γ z(ξ, λ) =

(8.16)

(8.17a) (8.17b) (8.17c)

The next sections will deal with various uncertainty analyses. 8.2.2

Probabilistic analysis

We treat the uncertain initial imperfection magnitude as a random variable with specified probability distribution function FG (g) = Prob(G ≤ g)

(8.18)

We are interested in finding the survival reliability of the structure, which is defined in general form as the probability that the structure will perform its specified mission. Herewith we adopt Hoff’s criterion (Hoff, 1965): ‘A structure is in a stable state if admissible finite disturbances of initial state of static or dynamic equilibrium are followed by displacements whose magnitude remains within allowable bounds during the required lifetime of the structure.’ Within this definition, we introduce the admissible state as −d ≤ z(ξ, λ) ≤ c

(8.19)

where c is the non-dimensional upper bound, and −d is the non-dimensional lower bound of the safety region. On the other hand, the inequalities max z(ξ, λ) > c or min z(ξ, λ) < −d ξ

ξ

(8.20)

mean unsafe regions, or, as defined by Hoff, as regions of impact buckling. Hereinafter we denote random variables with capital letters, whereas the possible values they take on are designated with lower-case notation. Reliability R(λ) at a preselected non-dimensional time λ is defined as a probability that the event described in Eq. (8.19) takes place: R(λ) = Prob(−d ≤ Z(ξ, λ) ≤ c)

(8.21)

V (λ) = Z(1/2, λ)

(8.22)

Define V (λ) as

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Then the reliability becomes R(λ) = Prob(−d ≤ V (λ) ≤ c)

(8.23)

Note that V (λ) is a random function of λ, and is represented as a product V (λ) = Ga(λ) where G is a random variable and a(λ) is a deterministic function as   1 γ cosh(rλ) − , γ > 1, a(λ) = 1−γ γ λ2 + 2 , γ = 1, a(λ) = 2   1 γ − cos(rλ) , γ < 1 a(λ) = 1−γ γ

(8.24)

(8.25a) (8.25b) (8.25c)

Reliability, as per in Eq. (8.21), becomes R(λ) = Prob(V (λ) ≤ c) − Prob(V (λ) ≤ −d)     d c − Prob G ≤ − = Prob G ≤ a(λ) a(λ)

(8.26)

since a(λ) is a positive-valued function. In terms of probability distribution function FG (g), Eq. (8.26) is rewritten as     c d R(λ) = FG − FG − (8.27) a(λ) a(λ) Equation (8.27) depends, in general, on the specific form of the probability distribution function FG . The influences of the specified probability distribution function over the design of the geometric characteristic of the column will be the objective of the next sections. 8.2.3

Uniformly distributed random initial imperfections

Consider first the case when the initial imperfection amplitude G for the sinusoidal imperfection in Fig. 8.1 possesses a uniform probability density function over the interval [α1 , α2 ] and vanishes elsewhere. The probability distribution function FG (g) then reads FG (g) =

1 (hg − α1 i1 − hg − α2 i1 ) α2 − α 1

where hg − αi1 is a singularity function as  0, g<α 1 hg − αi = g − α, g ≥ α

(8.28)

(8.29)

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The reliability of the column becomes  1  1 ! c 1 c R(λ) = − α1 − − α2 α2 − α 1 a(λ) a(λ)  1  1 ! d d 1 − − α1 − − − α2 − α2 − α 1 a(λ) a(λ)

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233

(8.30)

Hereinafter, for the sake of simplicity, we take α1 ≥ 0, α2 ≥ 0. Since the arguments in the third and fourth terms in Eq. (8.30) are negative, they vanish. Therefore, we have the following expression: 1  1 !  1 c c − α1 − − α2 (8.31) R(λ) = α2 − α 1 a(λ) a(λ) We are interested in designing the system; namely, choosing the radius ρ of a circular solid cross-section such that the reliability R(λ) will be no less than some codified reliability rc as R(λ) ≥ rc

(8.32)

Consider first a sub-case when the applied axial load exceeds the classical buckling load, so that γ=

P P L2 M 4P L2 = 2 = 4 > 1, M = 3 Pcl π EI ρ π E

(8.33)

where M is a load-dependent parameter. By virtue of Eq. (8.25), the reliability reads  1 c(1 − ρ4 /M ) 1 − α1 R(λ) = α2 − α1 cosh(rλ) − ρ4 /M (8.34)  1 c(1 − ρ4 /M ) 1 − − α2 α2 − α1 cosh(rλ) − ρ4 /M

Let us closely investigate this expression. If, for the pre-selected non-dimensional time instant λ, the argument of the singularity functions in Eq. (8.34) satisfies the inequality c(1 − ρ4 /M ) < α1 cosh(rλ) − ρ4 /M

(8.35)

c(1 − ρ4 /M ) > α2 cosh(rλ) − ρ4 /M

(8.36)

then the reliability vanishes identically. On the other hand, if

then the reliability equals unity. In the intermediate range, i.e., for α1 ≤

c(1 − ρ4 /M ) ≤ α2 cosh(rλ) − ρ4 /M

(8.37)

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the reliability varies between zero and unity. Generally, in engineering practice, the codified reliability rc is not taken to be unity. Rather, it is a pre-selected quantity in extremely close vicinity of unity. For rc tending to unity from below, the inequality in Eq. (8.37) is valid. Hence the argument in the second term in Eq. (8.34) is negative. Therefore, it vanishes, and we are left with 1  1 c(1 − ρ4 /M ) − α (8.38) R(λ) = 1 α2 − α1 cosh(rλ) − ρ4 /M In order for the reliability to tend to unity from below, it is necessary and sufficient that lim

ρ→ρd

c(1 − ρ4 /M ) = α2 cosh(rλ) − ρ4 /M

(8.39)

where ρd is the design value of the cross-sectional radius. For this we obtain the following equation: c(1 − ρ4d /M ) p = α2 cosh(λ M/ρ4d − 1) − ρ4d /M

(8.40)

It should be noted that the non-dimensional time parameter λ is proportional to the radius, through Eq. (8.9); i.e., λ is expressed using t = λ/ω1 : " r # 1  π 2 E λ= tρd (8.41) 2 L m Equation (8.40) is a transcendental equation in terms of ρd , once the time instant t at which reliability rc is desired, and the failure boundary c and other parameters are fixed. Design process requires us to obtain the cross-sectional radius ρd as a function of these quantities, as ρd = ρd (t, c, E, m, L) either analytically or numerically. It is advantageous, however, to view Eq. (8.40) as a function of the failure boundary c depending upon the radius ρd and the remaining parameters: p α2 [cosh(λ M/ρ4d − 1) − ρ4d /M ] (8.42) c(ρd , t, E, m, L) = 1 − ρ4d /M This relationship enables the design of the column graphically. One obtains a curve c versus ρd with other parameters fixed. Once such a curve is drawn, the determination of the required cross-sectional radius ρd becomes a straightforward task: One specifies the failure boundary c and reaches the corresponding radius in the graph. Two other sub-cases, associated with P = Pcl or P < Pcl , respectively, are dealt with in a perfect analogy. For P =√Pcl , the design value of the cross-sectional radius is immediately expressed as ρd = 4 M . Then   α2  π 4 E 2 √ c(ρd , t, E, m, L) = t M +2 (8.43) 2 L m

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235

1 c = 0.5 α1 = 0.1

0.8

α2 = 0.6

R(λ)

0.6

0.4

0.2

R ∞ = 0.133

0

1

2

3

4

5

6

λ

Fig. 8.2 Reliability versus non-dimensional time; initial imperfections with uniform probability density (P = 0.5Pcl ), Eqs. (8.25c) and (8.30).

1 c = 0.5 α1 = 0.1

0.8

α2 = 0.6

R(λ)

0.6

0.4

0.2

0

0.5

1

1.5

2

2.5

3

3.5

λ

Fig. 8.3 Reliability versus non-dimensional time; initial imperfections with uniform probability density (P = Pcl ), Eqs. (8.25b) and (8.30).

whereas for P < Pcl this equation takes the following form: p α2 [ρ4d /M − cos(λ M/ρ4d − 1)] c(ρd , t, E, m, L) = (8.44) 1 − ρ4d /M The results of sample calculations are presented in Figs. 8.2–8.4 portraying reliability R(λ) versus non-dimensional time λ for P < Pcl , P = Pcl , and P > Pcl , respectively, for c = 0.5, α1 = 0.1, α2 = 0.6. In the following, we assume the mechanical

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1 c = 0.5 α1 = 0.1

0.8

α2 = 0.6

R(λ)

0.6

0.4

0.2

0

0.5

1

1.5

2

λ

Fig. 8.4 Reliability versus non-dimensional time; initial imperfections with uniform probability density (P = 2Pcl ), Eqs. (8.25a) and (8.30).

1 3.5 Probability of initial failure

3 2.5 fG (g)

2 1.5 1 0.5

0

0.2

0.4

0.6

0.8

g

Fig. 8.5

Geometrical representation of the reliability at the initial time instant.

and geometrical parameters of the column as L = 10.0 m, E = 2.1 × 106 kg/cm2 , m = 2.1 × 10−3 /9.81 kg · s2 /cm4 . The axial load P is set at P = 3000 kg. As it is seen from Fig. 8.2, R(λ) does not assume a value of zero or unity. At time instant zero, the reliability equals 0.8. This value is deducible directly; indeed, probability density of the initial imperfections for g ∈ [0.1, 0.6] is 1 fG (g) = =2 (8.45) 0.6 − 0.1 as shown in Fig. 8.5. Hence, the area under the probability density curve between

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5

Boundary c (cm)

4

II

3

2

1

I

c = 0.6 4

√M = 3.86 0

2

4 6 Radius ρ (cm)

8

10

Fig. 8.6 Design curve c = c(ρd ), t = 0.2 with G having a uniform probability density and unity reliability requirement, Eq. (8.44).

the values c = 0.5 and the uppermost value that G may take, namely, 0.6, equals 2(0.6 − 0.5) = 0.2, which constitutes the probability of failure Pf (λ) at the initial time λ = 0 denoted by Pf (0) as shown in Fig. 8.5. This means that the reliability curves in Figs. 8.2–8.4 have an initial value equal to R(0) = 1 − Pf (0) = 0.8. This probability is equivalent to the failure probability of the item at delivery. For the actual load less than the classical buckling load, the variation of the total displacement with time is harmonic. When λ approaches value π/r, the absolute value of the total displacement attains its maximum. At this time, the reliability of the system reaches its minimum. Values of λ larger than π/r yield a constant value of reliability achieved at the latter time instant R(λ) = R(π/r) for λ ≥ π/r

(8.46)

In Fig. 8.2 the value π/r equals 4.1. Therefore, the reliability in Fig. 8.2 remains constant and equals 0.15. Figure 8.6 shows the relation between the failure boundary and the cross-sectional radius ρd with other parameters fixed. This curve enables us, by specifying the failure boundary c, to obtain the smallest required cross-sectional radius yielding the prescribed reliability for the prescribed external load. One must observe that with values of c less than α2 = 0.6, it is not possible to design the column in a manner so as to attain a unity or near-unity reliability. In fact, if ρ approaches infinity, the column gains an infinite frequency of vibration and the displacement at the center remains fixed at the initial imperfection value. Naturally, if the failure boundary happens to be less than or equal to α1 , i.e., the minimum value that the initial imperfection can assume, there is no finite value of ρd that will ensure extremely high reliability, since the structure buckles at the outset of the experiment. Likewise, if c ∈ [α1 , α2 ) the reliability that is arbitrarily close to

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8 6 Boundary c

5

4

4

2 3 2

2

Time t (sec)

4 Radius ρ (cm)

1

6 8 10

Fig. 8.7 Design surface c = c(ρd ), with G having a uniform probability density and unity reliability requirement, Eq. (8.44).

unity cannot be obtained, since the non-zero probability of failure occurs. On the other hand, when c ≥ α2 , the initial probability of collapse is absent, and the design with near-unity reliability is feasible. Figure 8.7 depicts the surface ρ = ρ(c, t) for a prescribed value of the external load and unity reliability. With increase of the time interval [0, t], where safe performance is required, the value of the failure boundary allowing unity reliability should be greater. Naturally, values of ρd are larger than the ones obtained from such a surface for the pre-selected time instant t corresponding to unity reliability. This behavior is elucidated in Fig. 8.8, where the design curves are presented for various values of the design times t1 = 0.3 sec, t2 = 0.8 sec, and t3 = 2.5 sec. Figure 8.9 represents the cross-section of the surface in Fig. 8.7 with the plane t = 0.5 sec (solid line), contrasted with other design curves, which are obtained with smaller values of the required reliability, namely, rc = 0.9 or 0.8, at the same time instant. It is seen that, by requiring less reliability, the design value of the cross-sectional radius decreases, as expected, for the same pre-selected value of the safety bound c. We note that in Figs. 8.6 and 8.8 the abscissa axes are divided into two √ subregions corresponding to different behaviors of the system. We denote the value 4 M by ρdc . If the design radius ρd satisfies ρd ≤ ρdc , then the load ratio P/Pcl satisfies P/Pcl ≥ 1, corresponding to region I. In this region, the displacement maxξ z(ξ, λ) is an exponential function of the time t. The design relationships, conveniently expressed as the failure boundary function depending on the variables ρd , t, E, m and L as c(ρd , t, E, m, L), takes the form of either Eq. (8.42) or Eq. (8.43). The

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7

Boundary c (cm)

6

Solid line t =0.3 sec Dashed line t 0.8 sec

5

Dot-Dashed line t = 2.5 sec 4 3 II

2 I

1

0

2

4 6 Radius ρ (cm)

8

10

Fig. 8.8 Design curves c = c(ρd ), with G having uniform probability density and unity reliability requirement for different safe time instants, Eq. (8.44).

5

Boundary c (cm)

Solid line rc = 1 Dashed line rc = 0.83

4

Dot-Dashed line rc = 0.75 3 II

2

I

1

0

2

4 6 Radius γ (cm)

8

10

Fig. 8.9 Comparison of design curve, with G having uniform probability density and different codified reliabilities, t = 0.5, Eq. (8.44).

argument of the hyperbolic function in Eq. (8.42), with the aid of Eq. (8.41), takes the form s r p (ρd )4 − M  π 2 E M =t (8.47) λ 1− (ρd )4 ρd L m

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By inspecting Eqs. (8.42) and (8.47), we conclude that the failure boundary function in Eq. (8.42) is a monotonically decreasing function of the cross-sectional radius ρ d in region I as shown in Figs. 8.6–8.8. On the other hand, if ρd > ρdc ; i.e., region II in Figs. 8.6–8.8, then the displacement function is represented by a non-monotonic trigonometric function Eq. (8.17c). In this case, the function c(ρd , t, E, m, L) is given in Eq. (8.44) with the attendant argument of the cosine function in Eq. (8.47). Inspecting the expression of the function c(ρd , t, E, m, L), we note that its maximum value is reached for the first time when the expression in Eq. (8.47) reaches π. If, for a combinations of design radius ρd and time t, the expression in Eq. (8.47) has a value larger than π, then the displacement in Eq. (8.17c) has already reached its maximum value at a time t < t. Therefore, in order for the column to perform satisfactorily with the required reliability until the specified time t, the trigonometric function in Eq. (8.44) must acquire the value cos(π) = −1, and Eq. (8.44) becomes c = α2

ρd /M + 1 ρd /M − 1

(8.48)

Interestingly, an asymptotic behavior of the design curves may be observed from Figs. 8.6–8.8. The asymptotic value of the cross-sectional radius ρd corresponds to an infinite value of the first natural circular frequency of vibration ω1 in Eq. (8.9). In these circumstances the column behaves statically at every time instant t. Unity reliability R(λ) is achieved by setting the failure boundary c equal to the maximum value of the initial imperfection g. The above discussion is confirmed analytically by calculating the limit of the expression as lim c = (α2 − α1 )rc + α1

ρd →∞

(8.49)

which, in case the specified reliability rc = 1− , will yield α2 as the final result. 8.2.4

Random initial imperfections with truncated exponential distribution

Assume that the probability density function differs from the uniform one, although initial imperfection amplitude G varies in the interval [α1 , α2 ], namely, G is supposed to have a truncated exponential probability density function fG = A exp(−qg)[hg − α1 i◦ − hg − α2 i◦ ]

(8.50)

where A=

1 exp(−qα1 ) − exp(−qα2 )

(8.51)

is the normalization coefficient, and h · i◦ is the singularity function defined in Eq. (8.1). The parameter q is a deterministic positive number characterizing the probability distribution. For α1 = 0 and α2 tending to infinity, q is the reciprocal to the

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mathematical expectation of the random variable G. The probability distribution function, obtained by integrating Eq. (8.50), reads FG (g) = A[exp(−qα1 ) − exp(−qg)]hg − α1 i◦

− A[exp(−qα2 ) − exp(−qg)]hg − α2 i◦

(8.52)

In this new circumstance, Eq. (8.27) involves the probability distribution function in Eq. (8.52). Hence,   c R(λ) = FG a(λ)  ◦ c = A[exp(−qα1 ) − exp(−qc/a(λ))] − α1 (8.53) a(λ)  ◦ c − A[exp(−qα2 ) − exp(−qc/a(λ))] − α2 a(λ) Note that a(λ) is given in Eq. (8.25); it depends upon the ratio of the external load P to the classical buckling load Pcl . Figure 8.10 depicts the reliability function for the case P/Pcl = 0.5 versus the non-dimensional time λ. It is seen that, as λ approaches infinity, the reliability R(λ) attains a constant non-vanishing value. In Figs. 8.11 and 8.12 the reliability R(λ) is portrayed for the cases P/Pcl = 1 and P/Pcl = 2, respectively. In these cases, the reliability tends to zero when λ → ∞. This behavior is explained through closer examination of Eqs. (8.25) and (8.53). Inspecting Eq. (8.53), we find that a vanishing value of the reliability R(λ) can be obtained only if c c − α1 ≤ 0 and − α2 ≤ 0 (8.54) a(λ) a(λ) In the case γ = P/Pcl = 0.5, as shown in Fig. 8.10, the function a(λ) assumes the expression in Eq. (8.25c), i.e., a harmonic dependence on the non-dimensional time λ. We denote by λm = π/r, which is the value of the non-dimensional time corresponding to the minimum of the term c/a(λ). We conclude that if both conditions hold in Eq. (8.54) for λ ≤ λm , then the reliability will vanish. If, instead, the conditions in Eq. (8.54) are not fulfilled for the non-dimensional time in the range 0 ≤ λ ≤ λm , the reliability R(λ) never attains zero. For the non-dimensional time λ > λm , the reliability R(λ) maintains its minimum value achieved at λ = λm . The minimum non-vanishing value of the reliability can be analytically evaluated by calculating      c γ+1 lim R(λ) = lim FG = A exp(−qα1 ) − exp −qc (8.55) λ→∞ λ→∞ a(λ) 1−γ

For α1 = 0.1, α2 = 0.6, c = 0.5 and q = 0.1, this value is 0.233 as shown in Fig. 8.9. In the cases γ = P/Pcl = 1 and γ = P/Pcl = 2 of Figs. 8.10 and 8.11, respectively, a(λ) is a monotonically increasing function with respect to the non-dimensional time. Hence there always exists a value λc such that both of the conditions in Eq. (8.54) are satisfied for λ ≥ λc , resulting in a vanishing limiting

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1 c = 0.5 α1 = 0.1

0.8

α2 = 0.6

R(λ)

0.6

0.4 R∞ = 0.233 0.2

0

1

2

3

4

5

6

λ

Fig. 8.10 Reliability versus non-dimensional time, with G having truncated exponential probability distribution (P = 0.5Pcl ), Eqs. (8.25c) and (8.53).

2 c = 0.5 α1 = 0.2

0.8

α2 = 0.6

R(λ)

0.6

0.4

0.2

0

0.5

1

1.5

2

2.5

3

3.5

λ

Fig. 8.11 Reliability versus non-dimensional time, with G having truncated exponential probability distribution (P = Pcl ), Eqs. (8.25b) and (8.53).

value of reliability R(λ). We are interested in the design of a column with nearunity reliability. It may be found from Eq. (8.53) that the necessary and sufficient condition to achieve this value reads c → α2 (8.56) a(λ) where the function a(λ) assumes one of the three expressions (a), (b) and (c) in

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1 c = 0.5 α1 = 0.1

0.8

α2 = 0.6

R(λ)

0.6

0.4

0.2

0

0.5

1

1.5

2

λ

Fig. 8.12 Reliability versus non-dimensional time, with G having truncated exponential probability distribution (P = 2Pcl ), Eqs. (8.25a) and (8.53).

7 Solid line r0 = 1

6

Dashed line r0 = 0.83

Boundary c (cm)

5

Dot-Dashed line r0 = 0.75

4 II

3 2 I 1

0

2

4 6 Radius ρ (cm)

8

10

Fig. 8.13 Comparison of design curves for different codified reliabilities, t = 0.5 with imperfections possessing truncated exponential distribution, Eq. (8.44).

Eq. (8.25) depending upon the value of the external load P . Specifying the appropriate expression for a(λ) and choosing the time interval [0, t] required for successful performance, Eq. (8.56) reduces to either Eq. (8.42), Eq. (8.43) or Eq. (8.44), respectively, for P > Pcl , P = Pcl or P < Pcl . Thus we conclude that, although the imperfections possess an exponential distribution curve, the curve in Fig. 8.6 obtained for uniformly distributed imperfections can be used for the column design

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with near-unity reliability. Several design curves corresponding to different values of reliability are portrayed in Fig. 8.12, all corresponding to the same pre-selected time instant t. An inspection of Fig. 8.13 reveals that lower reliability demand corresponds to smaller values of the cross-sectional radius ρd , for a given value of the failure boundary c. A natural question arises: Is it possible that, for a near-unity reliability requirement, the same design curve will be obtained for initial imperfections distributed with a completely generic, but bounded, probability density function? The answer to this question will be provided in the following section. 8.2.5

Random initial imperfection with generic truncated distribution

Let us assume that the initial imperfection amplitude is a random variable with a generic probability density function fG (g) defined as follows in the interval [α1 , α2 ] with α1 and α2 both positive:  ∗ AfG (g) for g ∈ [α1 , α2 ] (8.57) fG (g) = 0 elsewhere where A is a normalization coefficient such that Z ∞ fG (g)dg = 1

(8.58)

−∞

∗ and fG (g) is the generic probability density function defined in (−∞, ∞). With the aid of singularity functions, Eq. (8.57) can be written as ∗ fG (g) = AfG (g)(hg − α1 i◦ − hg − α2 i◦ )

(8.59)

The probability distribution function FG (g) is obtained through formal integration of Eq. (8.59), reading Z g Z g ∗ FG (g) = fG (y)dy = AfG (y)dy (8.60) −∞

−∞

The reliability R(λ) of the column is given through Eq. (8.27) with FG (c/a(λ)) represented by Eq. (8.60). In order to design a column with an extremely high reliability requirement, it is sufficient that the argument c/a(λ) → α2 , resulting in   Z c/a(λ) c ∗ lim FG = lim AfG (g)[hg − α1 i◦ − hg − α2 i◦ ]dg a(λ) c/a(λ)→α2 c/a(λ)→α2 −∞ Z c/a(λ) ∗ AfG (g)[hg − α1 i◦ − hg − α2 i◦ ]dg = lim =

c/a(λ)→α2 α1 Z α2 ∗ AfG (g)dg α1

=1

(8.61)

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e that is less Necessity could be demonstrated as follows: If c/a(λ) equals some α than α2 , then the reliability equals F (e α) that is smaller than unity. Hence it is necessary that the argument c/a(λ) approaches α2 to obtain unity reliability. To sum up, in this section, we established that, for the imperfection amplitude having either uniform distribution or truncated exponential distribution, the nearunity reliability design requires that c/a(λ) tends to α2 . Now, we observe that the analogous conclusion holds without specifying the type of probability distribution, except that its density is bounded between α1 and α2 . Thus, irrespective of the specified character of the bounded density, the same design value of the cross-sectional radius ρd is achieved. This independence of the design from the probabilistic contents appears to be remarkable. It leads to the following question: Since the obtained designs are independent of the particular form of the probability density, may it be that the same result is obtainable without resorting to the probability notion at all? The reply to this question is affirmative as will be shown below. 8.2.6

Buckling under impact load: Anti-optimization by interval analysis

Let us postulate that the initial imperfections constitute an uncertain variable, although not a random one. Specifically, we assume that imperfections constitute an interval variable g = [α1 , α2 ] (8.62) with α1 and α2 both positive. The interval analysis is a mathematical tool dealing with intervals rather than with numbers (see Sec. 3.3 for details of interval analysis). In the present context, the analysis is straightforward, since the response function ν(λ), as seen in Eq. (8.17), is a linear function of the initial imperfection amplitude g. We define the functions ν1 (λ) and ν2 (λ) as the lower and upper envelopes of the response, respectively. These constitute the interval responsefunction [ν(λ)] as   γ 1   [α1 , α2 ] cosh(rλ) − , γ>1    1 − γ γ    λ2 + 2 (8.63) [ν(λ)] = [ν1 (λ), ν2 (λ)] = [α1 , α2 ] , γ=1  2       γ 1   − cos(rλ) , γ<1  [α1 , α2 ] 1−γ γ Inspection of Eq. (8.63) reveals that the interval displacement function of nondimensional time λ is represented as a product of an interval variable [α1 , α2 ] and a deterministic function a(λ) defined in Eq. (8.25). In order to design the column, i.e., to guarantee its successful performance until the pre-selected time t, or its non-dimensional counterpart λ, it is necessary and sufficient to bound the largest possible displacement to satisfy the inequality ν2 (λ) ≤ c (8.64)

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where c is the failure boundary. Bearing in mind Eq. (8.63) in conjunction with Eq. (8.25), we obtain a(λ)α2 ≤ c

(8.65)

We note that the design condition in Eq. (8.65) coincides formally with Eq. (8.56), when the required reliability approaches unity from below. This proves the equivalence between the results furnished by interval analysis and probabilistic one, when rc attains values extremely close to unity. 8.2.7

Conclusion

A one-dimensional impact-buckling problem has been studied in this section in order to contrast the probabilistic and non-probabilistic analyses of uncertainty. Simplicity of the example allows us not to lose sight of the forest for the trees of analytical and numerical derivations. In particular, this example illustrates that probabilistic analysis and non-probabilistic interval analyses are compatible with each other. In probabilistic analysis, one postulates the knowledge of the probability density function. However, such information is often unavailable. To model this situation, we constructed two alternative models of probability density utilizing a random variable with either uniform or non-uniform truncated density. Both yielded the same value for the design variable, if the required reliability is extremely close to unity. This result leads to the general conclusion that irrespective of the density involved, if it extends between the lower possible value α1 and the upper possible one α2 , the same designs will follow. This could be interpreted as good news for the probabilists. Researchers in the field of stochasticity could argue that the unavailability of the probability density, except the information on its boundedness, may be immaterial. Thus any assumption on the probability density will be equally efficient. If, however, the experimental data is available, then an appropriate density should be utilized. We maintain that a design point at the bound of the distribution provides a near-unity probability. This seems blatantly transparent; yet this fact appears not to be emphasized by the probabilistic analysts. Even though the knowledge of the probability density function is not needed in interval analysis, one must face the issue on how to properly define bounds. Except in special cases where there exist well defined bounds, the selection of the bounds will likely rely on a decision, perhaps based on data analysis or physical arguments. Remarkably, the same designs furnished by the probabilistic analysis are also obtained through the use of the interval analysis. This suggests the following idea: Since the initial imperfection is an uncertain value, and often no data is available to justify the particular choice of the density, why not employ the simplest possible approach? Interval algebra is indeed such a method. Although in this particular case interval analysis may appear simple, it provides the same result as derived by more sophisticated and algebraically lengthier probabilistic analysis. It is concluded, therefore, due to engineering pragmatism, that in some circumstances the non-

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probabilistic analysis of uncertainty may be preferable to the probabilistic one, since it leads to the same results in a much simpler manner.

8.3 8.3.1

Anti-Optimization Versus Probability: Vector Uncertainty Introduction

In Sec. 8.2, a direct comparison was performed between the widely popular probabilistic methods and interval analysis, which represents a simplest version of the anti-optimization technique. Immediately a natural question arises: How can we compare these two competing approaches in the case of the vector uncertainty? This problem is investigated in this section. Whereas in a single-dimensional case uncertainty can be identified as a variable belonging to an interval, the two-or-more-dimensional case presents more possibilities. For example, in a two-dimensional case, the uncertain vector is identified with two coordinates varying in a rectangular region, which enables one to use interval analysis in a vector setting. It is intuitively understood that the narrowest interval shall be chosen to characterize the one-dimensional uncertainty; for the two-dimensional case one should seek a rectangle of minimum area so that the further evolution of the system will be more closely bracketed. Yet, this may not be a best representation of the available data whose scatter must be modeled. Indeed, in some cases, the use of regions other than the rectangle may result in an even smaller area enclosing all available data. The possibility arises, for example, of enclosing the data by the minimum area ellipse, whose area may turn out to be smaller than that of the minimum-area rectangle. Along these thoughts, in addition to interval analysis, ellipsoidal modeling was developed (Schweppe, 1973; Chernousko, 1994; Kurzhanski and Valyi, 1992) for uncertainty analysis (see Sec. 3.4 for details of ellipsoidal model). Interestingly, these two lines of thoughts (on intervals and ellipsoids) intersected in extremely few works, and essentially have been developed in parallel, mostly without knowledge about the developments in the other field. It appears that the ellipsoidal framework has some advantages over interval analysis in the sense that it deals with a smooth, convex boundary of the enclosed data with associated straightforward analytical or numerical treatment. Yet the ellipsoidal model may suggest that the components are functionally dependent. This tacit assumption may be unjustified in some circumstances. Hence the independent data may be better justified as enclosed by a rectangular region. Interval and ellipsoidal modeling are particular cases of convex modeling (Ben-Haim and Elishakoff, 1990; Ben-Haim, 1996). In fact, convex description of uncertainty is richer than the ellipsoidal one: In addition to the ellipsoids per se, it includes sets with functions with envelope bounds, or those with bounded integral rectangles, or those with bounded integral rectangles of its derivatives and so on. The case where the uncertain variables do not belong to a convex set may be

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a=5 P(t)

P(t) h2 h 1 Initial imperfections

L/2 L Fig. 8.14

Description of the structural model.

dealt with using the methods of nonlinear programming (Li, Elishakoff, Starnes and Shinozuka, 1996; Luenberger, 2003) (see Sec. 2.4.2 for details of nonlinear programming). All these analyses share the main ideas of anti-optimization, namely, of the desire to determine least favorable responses, in order to guarantee the successful performance despite the presence of uncertainty. In this section, the two-dimensional uncertain imperfections are considered in the context of the column impact problem (Zingales and Elishakoff, 2000), whose one-dimensional counterpart was discussed in Sec. 8.2. 8.3.2

Deterministic analysis

The differential equation describing the problem of the simply supported column, as shown in Fig. 8.14, under axial impact load with initial imperfections is given in Eq. (8.3) in Sec. 8.2. The same boundary and initial conditions are set in this section, so that Eqs. (8.5) and (8.6) in Sec. 8.2 remain unchanged. Let the initial imperfections be given as a sum of two sinusoidal terms  aπx   πx  + h2 sin (8.66) w0 (x) = f1 (x) + f2 (x) = h1 sin L L where h1 and h2 are the amplitudes of f1 (x) and f2 (x), respectively. In Eq. (8.66), the integer a > 1 represents the wave number in the variation of f2 (x). The same non-dimensional parameters as adopted in Eqs. (8.8) and (8.9) in Sec. 8.2 are used hereafter. The non-dimensional amplitudes of initial imperfections are given as h1 h2 , g2 = (8.67) ∆ ∆ The non-dimensional differential equation governing the problem is formulated in a similar manner as Eq. (8.10) in Sec. 8.2. The additional displacement is sought as a superposition g1 =

u(ξ, λ) = u1 (ξ, λ) + u2 (ξ, λ)

(8.68)

where the functions uj (ξ, λ) (j = 1, 2) are the solutions of the following equations: (j)

∂ 2 uj ∂ 2 uj d2 u 0 ∂ 4 uj + π2 γ + π4 = −π 2 γ , (j = 1, 2) 4 2 2 ∂ξ ∂ξ ∂λ dξ 2

(8.69)

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with the initial imperfections expressed as (1)

(2)

u0 (ξ) = g1 sin(πξ), u0 (ξ) = g2 sin(aπξ)

(8.70)

The functions uj (ξ, λ) are represented in the separable forms as the initial imperfections in Eq. (8.70), namely, u1 (ξ, λ) = e1 (λ) sin(πξ), u2 (ξ, λ) = e2 (λ) sin(aπξ)

(8.71)

where e1 (λ) and e2 (λ) are time-dependent functions. Substitution of Eqs. (8.70) and (8.71) into Eq. (8.69) yields ordinary differential equations for e1 (λ) and e2 (λ) d2 e1 (λ) + (1 − γ)e1 (λ) = γg1 (8.72a) dλ2 d2 e2 (λ) + a2 (a2 − γ)e2 (λ) = γa2 g2 (8.72b) dλ2 Satisfying the initial conditions ej = 0 and dej /dt = 0 at λ = 0 (j = 1, 2) yields  γg1 (cosh(rλ) − 1), γ > 1    γ−1    g 1 λ2 e1 (λ) = (8.73) , γ=1  2      γg1 (1 − cos(rλ)), γ<1 1−γ and  γg2  (cosh(aqλ) − 1), γ > a2   γ − a2   1 γa2 g2 λ2 , γ = a2 e2 (λ) = (8.74) 2    γg2   (1 − cos(aqλ)), γ < a2 a2 − γ where p p (8.75) r = |γ − 1|, q = |γ − a2 | The non-dimensional modal displacements zi (ξ, λ) (i = 1, 2) are introduced as (1)

z1 (ξ, λ) = u0 (ξ) + u1 (ξ, λ) = [g1 + e1 (λ)] sin(πξ), (2)

z2 (ξ, λ) = u0 (ξ) + u2 (ξ, λ) = [g2 + e2 (λ)] sin(aπξ)

(8.76)

With the aid of Eqs. (8.71) and (8.72), we form the total displacement wT (x, t) or its non-dimensional counterpart z(ξ, λ) as 1 z(ξ, λ) = wT (x, t) = z1 (ξ, λ) + z2 (ξ, λ) (8.77) ∆ (j)

(j)

By defining the non-dimensional displacement wT (x, t) = w0 (x) + wj (x, t) (j = 1, 2), we have 1 (1) z1 (ξ, λ) = wT (x, t) = ν1 λ sin(πξ), ∆ (8.78) 1 (2) z2 (ξ, λ) = wT (x, t) = ν2 λ sin(aπξ) ∆

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where the functions νj (λ) (j = 1, 2) of the non-dimensional time λ alone read   1 γg1 (1) cosh(rλ) − = g1 a1 (λ), γ>1 (8.79a) ν1 (λ) = γ−1 γ g1 (λ2 + 2) (2) = g1 a1 (λ), γ=1 (8.79b) ν1 (λ) = 2  γg1 1 (3) ν1 (λ) = − cos(rλ) = g1 a1 (λ), γ<1 (8.79c) 1−γ γ   a2 γg2 (1) cosh(aqλ) − = g2 a2 (λ), γ > a2 (8.79d) ν2 (λ) = γ − a2 γ g2 (γa2 λ2 + 2) (2) = g2 a2 (λ), γ = a2 (8.79e) ν2 (λ) = 2  γg2 a2 (3) ν2 (λ) = 2 − cos(aqλ) = g2 a2 (λ), γ < a2 (8.79f) a −γ γ Since there exist different analytical expressions for νj (λ) depending on the value of γ, the total displacement possesses various analytical representations. In particular, five different cases occur as follows: (3)

(3)

(8.80a)

(2)

(3)

(8.80b)

z(ξ, λ) = a1 (λ)g1 sin(πξ) + a2 (λ)g2 sin(aπξ), γ < 1 z(ξ, λ) = a1 (λ)g1 sin(πξ) + a2 (λ)g2 sin(aπξ), γ = 1 z(ξ, λ) =

(1) a1 (λ)g1 (1) a1 (λ)g1

z(ξ, λ) =

(1) a1 (λ)g1

z(ξ, λ) =

sin(πξ) +

(3) a2 (λ)g2 (2) a2 (λ)g2

sin(aπξ), γ = a

2

(8.80d)

sin(πξ) +

(1) a2 (λ)g2

sin(aπξ), γ > a2

(8.80e)

sin(πξ) +

sin(aπξ), 1 < γ < a2

(8.80c)

(i)

Note that, although there are five different cases, each of the parameters a 1 and has three separate expressions. If the initial imperfection amplitudes g1 and g2 are given deterministically, the design is performed so that the total displacement does not exceed the threshold value c. Hereafter the cross-section of the column is assumed to have a circular solid shape. The main thrust of this section is an investigation of the effect of uncertainty in initial imperfections on the resulting design values of the cross-sectional radius. If g1 and g2 constitute uncertain variables, the output z(ξ, λ) will likewise represent the uncertain function. Properties of the function z(ξ, λ) depend on the information provided about the uncertain variables g1 and g2 . We will investigate several alternative avenues of describing this vector uncertainty, either probabilistically or without recourse to the stochasticity concept. (j) a2

8.3.3

Probabilistic analysis

Let the amplitudes of the initial imperfections g1 and g2 constitute a random vector with specified joint probability density function fG1 G2 (g1 , g2 ). Capital letters denote the random variables, whereas lower-case notation is reserved for the possible values.

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251

We are interested in finding the reliability of the system; namely, the probability that the total displacement will remain in a safe region, in accordance with Hoff’s criterion given by Eq. (8.19) in Sec. 8.2. The reliability is defined as the probability that the maximum displacement Y = maxξ Z(ξ, λ) remains in a safe region. For simplicity we assume that a is an odd number. Hence Y ≡ max Z(ξ, λ) = Z(1/2, λ) = V1 (λ) + V2 (λ) ξ

(8.81)

Functions V1 (λ) and V2 (λ) are multiplicative random processes depending upon the non-dimensional parameter λ (i)

(j)

V1 (λ) = G1 a1 (λ), V2 (λ) = G2 a2 (λ), (i, j = 1, 2, 3)

(8.82)

and the reliability is given as R(λ) = Prob(−d ≤ Y = V1 (λ) + V2 (λ) ≤ c)

(8.83)

In view of Eq. (8.81), the reliability in Eq. (8.83) becomes R(λ) = Prob(Y ≤ c) − Prob(Y ≤ −d)

(8.84)

Equation (8.84) can be written as R(λ) = FY,λ (c, λ) − FY,λ (−d, λ)

(8.85)

where FY,λ is the probability distribution functions of random process Y : (i)

(j)

Y (λ) = G1 a1 (λ) + G2 a2 (λ), (i, j) ∈ {(1, 1), (1, 2), (1, 3), (2, 3), (3, 3)}

(8.86)

Note that Y (λ) represents a linear combination of two random variables with deterministic real-valued positive functions of non-dimensional time λ as coefficients. The evaluation of the reliability, as stated in Eq. (8.84), needs the particularization of the joint probability density function of the random variables G1 and G2 . We will consider the cases in which G1 and G2 are either statistically independent or dependent. 8.3.4

Initial imperfections with uniform probability density: Rectangular domain

Let the initial imperfections be independent random variables with a uniform probability density function in a rectangular domain as shown in Fig. 8.15, which is defined as 1 1 [hg1 − α1 i◦ − hg1 − α2 i◦ ] fG1 G2 (g1 , g2 ) = α 2 − α 1 β2 − β 1 (8.87) × [hg2 − β1 i◦ − hg2 − β2 i◦ ]

where h · i◦ is the singularity function defined in Eq. (8.1) in Sec. 8.2. The variations of G1 and G2 are confined to the intervals G1 ∈ [α1 , α2 ] and G2 ∈ [β1 , β2 ],

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3 f G 1 G2

2

2

1 1.5 g2

0 1 g1 Fig. 8.15

1

1.5 2

Uniform probability density function over a rectangular domain.

respectively, as shown in Fig. 8.15. Moreover, we assume, for the sake of simplicity, that αj > 0, βj > 0 (j = 1, 2). Marginal probability densities read 1 [hg1 − α1 i◦ − hg1 − α2 i◦ ], α2 − α 1 1 [hg2 − β2 i◦ − hg2 − β2 i◦ ] fG2 (g2 ) = β2 − β 1

(8.88)

fG1 G2 (g1 , g2 ) = fG1 (g1 )fG2 (g2 )

(8.89)

fG1 (g1 ) =

with

In order to perform the reliability analysis of the column, it is necessary to find the probability distribution function FY,λ (y, λ) of the random variable Y (λ) in Eq. (8.85). With the second term in Eq. (8.85) vanishing identically, we obtain

R(λ) = FY,λ (c, λ) =

Z

∞ −∞

Z

(j) c−a (λ)g2 2 (i) a (λ) 1

fG1 G2 (g1 , g2 )dg1 dg2

(8.90)

−∞

By substituting Eqs. (8.88) and (8.89) into Eq. (8.90) and performing integration,

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Anti-Optimization and Probabilistic Design

the reliability function is obtained as 1 1 R(λ) = α 2 − α 1 β2 − β 1 1 1 − α 2 − α 1 β2 − β 1 1 1 − α 2 − α 1 β2 − β 1 1 1 − α 2 − α 1 β2 − β 1 where

" " " "

1 (j) (i)

a2 a1 1

(j) (i)

a2 a1 1

(j) (i)

a2 a1 1

(j) (i)

a2 a1

hc −

(i) a 1 α1

hc −

(i) a 1 α1

hc −

(i) a 1 α2

hc −

(i) a 1 α2

253

−

(j) a 2 β1 i 2

−

(j) a 2 β2 i 2

−

(j) a 2 β1 i 2

−

(j) a 2 β2 i 2

  x2 , x≥0 2 hxi =  0,2 x<0

# # #

(8.91)

#

(8.92)

Equation (8.91) suggests several useful conclusions regarding the characterization (i) of the reliability. At the initial time instance t = 0 (or λ = 0), we have a1 (0) = (j) a2 (0) = 1. We conclude that if the failure boundary c is set as c ≤ α 1 + β1

(8.93)

then the reliability vanishes, and no possibility of design associated with nonvanishing reliability exists for the column subjected to non-zero values of the applied load P . Figures 8.16–8.19 portray some interesting aspects that may be deduced by investigation of the various terms in Eq. (8.91). We first observe that each one of the four terms represents the area of the region created by the straight line (i)

(j)

c = a 1 g1 + a 2 g2

(8.94)

and an appropriate boundary of the rectangular domain of the initial imperfection amplitudes g1 and g2 . Let us consider various straight lines passing through either of the four corners of the boundary, and parallel to the line given in Eq. (8.94). The plane (g1 , g2 ) is subdivided into five regions denoted by Γj (j = 1, . . . , 5) as shown in Fig. 8.16. The broken line in Fig. 8.16 represents the line in Eq. (8.94). Four dotted lines are parallel to it, and pass through either corner A, B, C, or D. In the case where the line in Eq. (8.94) belongs to region Γ2 , the coordinates of the intersection points A∗ and B ∗ with the edges AD and AB, respectively, are given by ! ! (j) (i) c − a 2 β1 c − a 1 α1 ∗ ∗ A = , β1 , B = α1 , (8.95) (i) (j) a1 a2 The distances AA∗ and AB ∗ are (j)

AA∗ =

c − a 2 β1 (i)

a1

(i)

− α1 , AB ∗ =

c − a 1 α1 (j)

a2

− β1

(8.96)

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2.75 Γ4

Γ3

2.25

Γ5

g2

B

C

1.75 B*

.75 0.75

A*

A

1.25

D Γ2

Γ1

1

1.25

1.5

1.75

2

2.25

2.5

g1 Fig. 8.16

Geometrical representation of the first term in reliability expression, Eq. (8.97).

2.75

g2

Γ4

Γ3

2.25

B* l B

Γ5

C*

C

1.75

.75 0.75

A*

A

1.25

1

1.25

D

Γ2

Γ1

1.5

1.75

2

2.25

2.5

g1 Fig. 8.17

Geometrical representation of the second term in reliability expression, Eq. (8.100).

The area SAB ∗ A∗ under the triangle AB ∗ A∗ , denoted as a hatched region in Fig. 8.16, equals (i)

SAB ∗ A∗ =

(j)

(c − a1 α1 − a2 β1 )2 1 (AA∗ )(AB ∗ ) = (i) (j) 2 2a a 1

(8.97)

2

The reliability of the system is determined by multiplying the area in Eq. (8.97) by the constant value of the probability density function (α2 − α1 )−1 (β2 − β1 )−1 . In this case, only the first term in the bracketed expression in Eq. (8.91) is nonvanishing. By inspection of Eq. (8.91), the coincidence between this first term and the area SAB ∗ A∗ is recognized. Figure 8.17 deals with the case in which the straight line in Eq. (8.94) belongs to

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Anti-Optimization and Probabilistic Design

3.25

255

B*

2.25 Γ4

Γ3

2.75

Γ5

g2

B

C*

1.75 A

1.25

Γ1

0.75 0.75

1

C D* D

A*

2

2.25

Γ2

1.25

1.5

1.75

2.5

g1 Fig. 8.18

Geometrical representation of the third term in reliability expression, Eq. (8.103).

3.5 B* 3 2.5 Γ4

g2

Γ3

Γ5

D* C C*

B

2 1.5

A*

A

D

Γ1

1

Γ2

1.25

1.5

1.75

2

2.25

2.5

g1 Fig. 8.19

Geometrical representation of the fourth term in reliability expression, Eq. (8.104).

the region Γ3 . The shaded area ABC ∗ A∗ , multiplied by the above constant value of the probability density function, represents the reliability of the system. This area is obtained as a difference between the area of the triangle AB ∗ A∗ and the area of the triangle BB ∗ C ∗ . The coordinates of the intersection point C ∗ are ! (j) c − a β 2 2 , β2 (8.98) C∗ = (i) a1 whereas the expression of the coordinates of B ∗ coincides with Eq. (8.95). The side lengths for the triangle AB ∗ A∗ are then given by Eq. (8.96) and the ones of the

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triangle BB ∗ C ∗ read (i)

BB ∗ =

c − a 1 α1 (j)

a2

(j)

− β2 , BC ∗ =

c − a 2 β2 (i)

a1

− α2

(8.99)

The area SBB ∗ C ∗ equals (i)

SBEC ∗ =

(j)

1 (c − a1 α1 − a2 β2 )2 (BB ∗ )(BC ∗ ) = (i) (j) 2 2a a 1

(8.100)

2

and is recognized as the second term in Eq. (8.91). Note that the area of the triangle AB ∗ A∗ coincides with the first term of Eq. (8.91). Analogously, the geometrical meanings of the third and fourth terms involved in Eq. (8.91) may be deduced by examining Figs. 8.18 and 8.19, respectively. In fact, Fig. 8.18 deals with the case that the broken line belongs to region Γ 4 . In this case, the reliability is given by the area ABC ∗ D∗ D, and is obtained as the algebraic sum SBC ∗ D∗ DA = SAB ∗ A∗ − SBEC ∗ − SDD∗ A∗

(8.101)

The first two terms of this expression have been already identified. In order to obtain the last term in Eq. (8.101), we determine the coordinates of point D ∗ as ! (i) c − a α 2 1 (8.102) D ∗ = α2 , (j) a2 Note that the coordinates of the point A∗ are given in Eq. (8.95). The area of the triangle DD∗ A∗ is so obtained as (i)

SDD∗ A∗ =

(j)

1 (c − a1 α2 − a2 β1 )2 (DD∗ )(DA∗ ) = (i) (j) 2 2a a 1

(8.103)

2

which coincides with the third term in Eq. (8.91). To determine the geometric interpretation of the fourth term in Eq. (8.91), let us consider the case when the broken line belongs to region Γ5 as shown in Fig. 8.19. The reliability is given as the product of the area of the rectangle ABCD and the density (α2 − α1 )−1 (β2 − β1 )−1 . The area itself equals (α2 − α1 )(β2 − β1 ). Hence, the reliability is unity as expected. On the other hand, in order to identify the fourth term in Eq. (8.91), we represent the area ABCD as the following algebraic sum: SABCD = SAB ∗ A∗ − SBB ∗ C ∗ − SDD∗ A∗ + SCC ∗ D∗

(8.104)

The first three terms in Eq. (8.104) are given by their respective counterparts in Eq. (8.91). The fourth term in Eq. (8.104) may be easily obtained by inspection of Eqs. (8.98) and (8.102) as (i)

SCC ∗ D∗ =

(j)

1 (c − a1 α2 − a2 β2 )2 CC ∗ )(CD∗ ) = (i) (j) 2 2a a 1

(8.105)

2

As is seen, each term in the reliability expression has an appropriate geometrical meaning. Naturally, if the broken line lies in region Γ1 , then one immediately

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Anti-Optimization and Probabilistic Design

c = 1.2(α2+β2) α = 0.5

1.2

257

β1=2.39 β2=2.0

1 0.8

Γ3

Γ4

R

Γ5

0.6 0.4 0.2 β4 0

1

β3

2

3

4

5

β Fig. 8.20 Reliability versus non-dimensional time, initial imperfections with uniform probability density (D = [1.2, 2] × [1.4, 2]), Eqs. (8.80a) and (8.91).

c = 1.2(α2+β2) γ= 1

1.2

β1=1.640 β2=1.327 β3=1.133 β4=0.796

1 0.8

Γ5

Γ3

Γ2

Γ1

R

Γ4

0.6 0.4 0.2 β4 0

0.5

β3

β2

1

1.5

β1 2

2.5

β Fig. 8.21 Reliability versus non-dimensional time, initial imperfections with uniform probability density (D = [1.2, 2] × [1.4, 2]), Eqs. (8.80b) and (8.91).

deduces that the reliability vanishes. Likewise, if the broken line passes through region Γ5 , the corresponding reliability is unity. The results of sample calculations are portrayed in Figs. 8.20–8.24, with the same mechanical and geometrical parameters used in Sec. 8.2, and the second term of imperfection in Eq. (8.66) is given by a = 5, which is an odd number. The cases (i) (j) correspond to the five independent expressions for the coefficients a1 and a2 in Eq. (8.80), which correspond, respectively, to γ < 1, γ = 1, 1 < γ < 25, γ = 25, γ > 25. For a specified value of the failure boundary c and the different values of the ratio P/Pcl , the reliability functions are depicted as functions of a non-dimensional time λ. For the case γ < 1 in Fig. 8.20, buckling occurrence is not a certain event: The structure may or may not buckle depending upon the parameters of the system. Hence, the reliability does not necessarily tend to zero with the increase of time λ.

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c = 1.2(ρ2+β2) γ= 2

1.2 1 0.8

Γ5

Γ4

Γ2

Γ1

R

Γ3

0.6 λ1=0.471 λ2=0.743 λ3=0.982 λ4=1.203

0.4 0.2

λ4 0

0.2

0.4

λ2

λ3 0.6

0.8

λ1

1

1.2

1.4

λ Fig. 8.22 Reliability versus non-dimensional time, initial imperfections with uniform probability density (D = [1.2, 2] × [1.4, 2]), Eqs. (8.80c) and (8.91).

λ1=0.0697 λ2=0.0550 λ3=0.0500 λ4=1.0351

c = 1.2(α2+β2) γ = 25

1.2 1 0.8

Γ5

Γ4

Γ2

Γ1

R

Γ3

0.6 0.4 0.2 λ4 0

0.02

λ3 0.04

λ2

λ1

0.06

0.08

0.1

λ Fig. 8.23 Reliability versus non-dimensional time, initial imperfections with uniform probability density (D = [1.2, 2] × [1.4, 2]), Eqs. (8.80d) and (8.91).

Instead, it gains a minimum value given by R(λ) = R(π/r) for λ ≥ π/r

(8.106)

depending on the value c that delimits stable and unstable states. When the nondimensional time reaches the value π/r, the displacement at the center of the column will achieve its maximum value, corresponding to the least possible reliability. In these circumstances, the value R(π/r) represents the guaranteed minimum reliability that the column may possess. Our main objective is to design the system, i.e., to obtain the radius ρd of the circular solid cross-section of the column so as to maintain a prescribed reliability rc , up to a pre-selected time instant t, for a specified value of the failure boundary c.

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Anti-Optimization and Probabilistic Design

λ1=0.0680 λ2=0.0538 λ3=0.0489 λ4=1.0343

c = 1.2(∆2+β2) γ = 26

1.2 1 0.8

259

Γ5

Γ4

Γ2

Γ1

R

Γ3

0.6 0.4 0.2 λ4 0

λ3

0.02

0.04

λ2

λ1

0.06

0.08

0.1

λ Fig. 8.24 Reliability versus non-dimensional time, initial imperfections with uniform probability density (D = [1.2, 2] × [1.4, 2]), Eqs. (8.80e) and (8.91).

10

√ Ma = 1.65 2

8

4

Boundary c

√ M = 3.68 6 4 I 2

II

√ Ma

2

0

0

2

III

4

√M 4

6 Radius ρd (cm)

8

10

12

Fig. 8.25 Design curve c = c(ρd ) at t = 0.2 for uniform probability density function and unity reliability requirement (P = 3000 kg, D = [1.2, 2] × [1.4, 2]), Eq. (8.91).

It is advantageous, however, to view c as a function of the remaining parameters, as c(ρd , t, E, L, m), which yields the failure boundary, for various combinations of its arguments, corresponding to satisfactory performance with the specified reliability rc . Thus, specifying the mechanical and the geometrical characteristics of the column and the external load allows one to obtain design curve c(ρd ) and surface c(ρd , t) shown in Figs. 8.25–8.27. The value of the cross-sectional radius ρd necessary for a successful performance is obtained by means of these curves, specifying the value of the failure boundary c. Note that the ratio γ = P/Pcl depends upon

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6

3

4

Boundary c

4

2

0 Time t (sec) 1

2 4 Radius ρd (cm)

6

0

Fig. 8.26 Design surface c = c(ρd , t) for uniform probability density function and unity reliability requirement (P = 3000 kg, D = [1.2, 2] × [1.4, 2]), Eq. (8.91).

10 Solid Line rc = 1 Dashed Line rc = 0.9 Dotted Line rc = 0.8

Boundary c

8 6 4 I

II

III

2 0

0

2

4

6 Radius ρd (cm)

8

10

12

Fig. 8.27 Comparison of design curves; uniform probability density function and different codified reliabilities at t = 0.5 (P = 3000 kg, D = [1.2, 2] × [1.4, 2]), Eq. (8.91).

the cross-sectional radius ρ as γ=

P P L2 M 4P L2 = 2 = 4, M = 3 Pcl π EI ρ π E

(8.107)

It is immediately observed, in view of Eq. (8.91), that when (i)

(j)

lim a1 (ρ, t)α2 + a2 (ρ, t)β2 = c

ρ→ρd

the reliability tends to unity from below.

(8.108)

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261

Figure 8.25 depicts the failure boundary c versus the radius ρd for the prescribed time t = 0.5 and unity reliability. As the dependence c = c(ρd ) has an asymptote at c = 4, when α2 = 2 and β2 = 2, this numerical result can be derived from analytical considerations. In fact, when ρd tends to infinity, (3)

lim aj (ρd , t) = 1, (j = 1, 2)

ρd →∞

(8.109)

and Eq. (8.108) takes the following form: c = α 2 + β2

(8.110)

Since α2 + β2 = 4 in Fig. 8.25, the asymptote ρd → ∞ is represented by c = 4. This result is straightforward: an infinite value of the cross-sectional radius ρd yields an infinite frequency of vibration of the column with J → ∞ as observed in Eq. (8.9) in Sec. 8.2. In these circumstances, the system will remain in the same position represented by its initial imperfection alone. Figure 8.26 portrays a design surface c = (ρd , t) allowing one to obtain, for a specified time t and failure boundary c, the required radius ρd of the cross-sectional area so that the reliability is rc . In Fig. 8.26, rc is taken to be unity. Figure 8.27 depicts design curves for various codified reliabilities rc = 0.8, 0.9, and 1.0. These figures show that smaller values of design cross-sectional radius are needed when less stringent reliabilities are imposed for a specified failure boundary, as it should be. Let us examine the dependence c = c(ρd ) in Figs. 8.25 and 8.27. Each figure is composed of three different sub-regions. Values of the cross-sectional radius (2) ρd ≤ ρdc = (M/a2 )1/4 (Region I) lead to a load ratio γ ≥ a2 . Therefore, the (i) (j) functions a1 (ρd , t) and a2 (ρd , t) in the expression of the failure boundary c are represented by Eqs. (8.79a) and (8.79d), respectively. In this case, the displacement function maxξ Z(ρd , t) is a monotonically increasing function with time t. A large failure boundary c, combined with small values of the time interval t, where the successful probabilistic performance is required, is necessary for high prescribed reliabilities rc . √ (2) (1) In the case where ρdc < ρd < ρdc = M , the load ratio γ belongs to the open interval (1, a2 ). The expression of the failure boundary c contains two functions: A (3) (1) trigonometric function for a2 (ρd , t) and an expression a1 (ρd , t) that is monotonically increasing. Also for this case, large values of c along with small performance times would guarantee the high reliability requirement. To obtain more insights on the dependence upon time, we express the argument ε1 of the cosine function in Eq. (8.79f), in view of Eq. (8.107), as r p a2 ρ4d − M  π  E ε1 = t (8.111) ρd L m allowing one to have a better look at the influence of the pre-selected time t. In (3) fact, for the function a2 (ρd , t) in Eq. (8.79f) that attains its maximum for values of ρd and t making ε1 = π, reliability reaches its minimum level. For values of

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parameters leading to ε1 > π, the previously achieved minimum reliability is valid. (1) Values of ρd ≥ ρdc = (M )1/4 lead to a load ratio γ < 1; hence, the expression of (j) the failure boundary c(ρd , t) involves Eq. (8.79a) for a1 (ρd , t). Interestingly, if the (3) argument of the trigonometric function in the expression of a1 (ρd , t), bearing in mind Eq. (8.107), r p ρ4d − M  π  E (8.112) ε2 = t ρd L m

is larger than π for a specified time t and ρd , the reliability achieved for ε2 = π is (3) retained. Hence, the cosine function in the expression of a1 (ρd , t) must be replaced (3) by −1, and the function a1 (ρd , t) becomes (3)

a1 (ρd , t) = −

1 + ρ4d /M 1 − ρ4d /M

(8.113)

As may be observed from Fig. 8.27, different values of the required reliability rc lead to appropriate asymptotic values. In fact, let us design the system for high reliability requirement as shown in region Γ4 in Fig. 8.18. The expression of the failure boundary as a function of reliability, with the aid of Eq. (8.82), is given by q (i) (j) (i) (j) (8.114) c = α2 a1 + β2 a2 − 2a1 a2 (1 − rc )(α2 − α1 )(β2 − β1 )

Evaluation of the limit value of Eq. (8.115) for an infinite value of ρd reads p lim = α2 + β2 − (1 − rc )(α2 − α1 )(β2 − β1 ) (8.115) ρd →∞

Equation (8.115) allows us to obtain the asymptotic value of the failure boundary function c(ρd , t) by specifying the reliability value. The results of this section clearly demonstrate that for high values of required reliability, the design radii ρd of the cross-section are extremely close to the values obtained by non-probabilistic analysis. The following question can be asked: Do the design values of the cross-sectional radius ρd depend upon the particular form of the probability density function fG1 G2 (g1 , g2 )? One can anticipate, generally speaking, that the designs depend on the probabilistic inputs. Yet, as it will be elucidated, the design in the extremely high reliability region is practically independent of the input probabilistic information. 8.3.5

Initial imperfections with general probability density: Rectangular domain

Let the initial imperfections possess a general probability density function in a rectangular domain ∗ fG1 G2 (g1 , g2 ) = AfG (g1 , g2 )[hg1 − α1 i◦ − hg1 − α2 i◦ ] 1 G2

× [hg2 − β1 i◦ − hg2 − β2 i◦ ]

(8.116)

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located in the first quadrant of the plane (g1 , g2 ), where A represents a normalization coefficient depending upon the specific expression of the probability density function ∗ fG (g1 , g2 ) that extends over the entire plane. 1 G2 The reliability expression with z = c substituted into Eq. (8.90), and taking into account Eq. (8.116), read, Z

R(λ) =

∞

dg2

(j) c−a g2 2 (i) a1

Z

−∞ β 1 i◦

−∞

∗ AfG (g1 , g2 )[hg1 − α1 i◦ − hg1 − α2 i◦ ] 1 G2

× [hg2 − − hg2 − β2 i◦ ]dg1 Z ∞ = [hg2 − β1 i◦ − hg2 − β2 i◦ ]dg2

(8.117)

−∞

×

Z

(j) c−a2 g2 (i) a1

−∞

∗ AfG (g1 , g2 )[hg1 − α1 i◦ − hg1 − α2 i◦ ]dg1 1 G2

Denoting the inner integral in Eq. (8.117) by I(g2 ), Eq. (8.117) can be rewritten as Z ∞ R(λ) = I(g2 )[hg2 − β1 i◦ − hg2 − β2 i◦ ]dg2 =

=

Z Z

−∞ β2

I(g2 )dg2

(8.118)

β1

Z

β2

dg2 β1

(j) c−a2 g2 (i) a1

−∞

∗ AfG (g1 , g2 )[hg1 − α1 i◦ − hg1 − α2 i◦ ]dg1 1 G2

In a high reliability range as shown in region Γ4 in Fig. 8.18, c lies in the vicinity of a critical value c∗ given as (i)

(j)

c ∗ = a 1 α 2 + a 2 β2

(8.119)

The highest reliability requirement will be obtained when the line in Eq. (8.94) tends to pass through the point B. This corresponds to value c = c∗ substituted in Eq. (8.94). Thus we calculate the limit lim R(λ) = lim∗

c→c∗

c→c

= =

Z

Z

dg2 β1

β2

dg2

β1 Z β2

dg2

β1

+

β2

Z

(j) c−a2 g2 (i) a1

−∞

Z

(j) c∗ −a2 g2 (i) a 1

−∞ Z α2 −∞

β2

dg2 β1

Z

Z

∗ AfG (g1 , g2 )[hg1 − α1 i◦ − hg1 − α2 i◦ ]dg1 1 G2

∗ AfG (g1 , g2 )[hg1 − α1 i◦ − hg1 − α2 i◦ ]dg1 1 G2

α2 + α2

∗ AfG (g1 , g2 )[hg1 − α1 i◦ − hg1 − α2 i◦ ]dg1 1 G2

a

(j) 2 (i) 1

a

(β2 −g2 )

∗ AfG (g1 , g2 )[hg1 − α1 i◦ − hg1 − α2 i◦ ]dg1 1 G2

(8.120)

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3 2.5 A2 2

A1

g2

∆Φ G0

1.5

K

D

1 0.5

0

1

2 g1

3

4

Fig. 8.28 Domain of integration of the probability density function (g 10 = 2, g20 = 1.5, K = 1.0), Eq. (8.122).

The inner integral vanishes, because integration is carried out in the interval (j) (i) [α2 , α2 + (a2 /a1 )(β2 − g2 )]. Since β1 ≤ g2 ≤ β2 , the upper limit of integration is greater than α2 . However, in the interval beyond α2 , the integrand in the square parenthesis vanishes. Reliability is then given only by the first term in Eq. (8.120), yielding Z α2 Z β2 ∗ AfG (g1 , g2 )dg1 = 1 (8.121) dg2 lim∗ R(λ) = lim∗ 1 G2 c→c

c→c

β1

α1

which represents the integral of the probability density function in its entire domain. In this case of unitary reliability, the design value of the cross-sectional radius ρ d is found from Eq. (8.119). 8.3.6

Initial imperfection with uniform probability density function: Circular domain

This section deals with a different shape of the variable domain. The magnitudes of the initial imperfections G1 and G2 are modeled as random variables belonging to a circular domain on the plane (g1 , g2 ). The joint probability density function is assumed to be uniform as ( 1 , for (g1 − g10 )2 + (g2 − g20 )2 ≤ K 2 fG1 G2 (g1 , g2 ) = (8.122) πK 2 0, elsewhere where (g10 , g20 ) is the center of the circle, and K the radius. As may be seen in Fig. 8.28, we assume, without loss of generality, that the circle is placed in the first

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quadrant of the plane (g1 , g2 ). Also we assume that the straight line (j)

(i)

c = a1 (λ)g1 + a2 (λ)g2

(8.123) (i) a1 (λ)

is drawn in a generic position depending on the values of the function and (j) a2 (λ). According to Eq. (8.85), in order to find the reliability, it is necessary to integrate the probability density function below the straight line in Eq. (8.122), or over the hatched area in Fig. 8.28 as R(λ) =

ZZ

fG1 G2 (g1 , g2 )dg1 dg2 = D

Z

∞ −∞

Z

(j) c−a2 g2 (i) a1

−∞

1 dg1 dg2 πK 2

(8.124)

The double integral in the latter equation will be calculated with the aid of the following polar coordinate system whose origin is at the center of the circle: g1 = g10 + δ cos φ, g2 = g20 + δ sin φ

(8.125)

where δ is the polar distance and φ is the polar angle. In this coordinate system, the reliability becomes ZZ δ dδdφ (8.126) R(λ) = 1 − 2 Dc πK where Dc is the non-hatched area in Fig. 8.28. Eq. (8.126) yields 1 [∆φ − sin(∆φ)] (8.127) 2π where ∆φ = φ1 − φ2 is the angular difference of the phase angles φ1 and φ2 between the polar axis G0 D and the polar directions G0 A1 and G0 A2 , respectively. The polar phase angles φ1 and φ2 may be obtained by solving the following set of equations: R(λ) = 1 −

(i)

(j)

a1 cos φ + a2 sin φ =

1 (i) (j) (c − a1 g10 − a2 g20 ) δ

δ 2 (sin2 φ + cos2 φ) = K 2

(8.128a) (8.128b)

where the first equation represents a straight line in the polar coordinate system, and the latter is the equation of a circle. Equations (8.128a) and (8.128b) have the solution q   (i) 2 (j) 2 (i) (j) 2 (a ) + (a ) − f f a + a 1 2 1 2  φ1 = cos−1  (8.129a) (i) (j) (a1 )2 + (a2 )2 q   (i) 2 (j) 2 (j) (j) 2 (a ) + (a ) − f f a + a 1 2 1 2  (8.129b) φ2 = cos−1  (i) 2 (j) 2 (a1 ) + (a2 )

with

f=

1 (i) (j) (c − a1 g10 − a2 g20 ) K

(8.130)

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In case of R(λ) tending to unity from below, we see from Eq. (8.127) ∆φ → 0 or φ1 → φ2

(8.131)

Hence, the square-root terms in Eq. (8.129) must gain (i)

Thus

(j)

(a1 )2 + (a2 )2 − f 2 → 0+

(8.132)

q (i) (j) f → (a1 )2 + (a2 )2

(8.133)

The geometrical meaning of the condition stated in Eq. (8.133) can be shown, expressing the system in Eqs. (8.127a) and (8.127b) back in Cartesian coordinates. Thus, bearing in mind Eqs. (8.125) and (8.133), we write Eq. (8.127) in the form q (i) (j) (i) (j) (i) a1 g10 + a2 g20 − a1 g1 + K (a1 )2 + (a2 )2 (8.134a) g2 = (i) a1 (g1 − g10 )2 + (g2 − g20 )2 = K 2

(8.134b)

Substituting Eq. (8.134a) into Eq. (8.134b), we obtain a quadratic equation for the unknown g1 q0 (g1 )2 − 2g1 q1 + q2 = 0

(8.135)

with coefficients  !  (i) 2 a 1  q0 =  1 + (j) a2 q   (i) 2 (j) 2 (i) (i) 2 (a ) + (a ) 2a K 2(a ) g 1 2 1 10 1  + q1 = g10 + (j) 2 (j) 2 (a2 ) (a2 ) q   ! ! (i) 2 (j) 2 (i) (i) 2 (i) 2 (a ) + (a ) 2a a a 1 2 1 1 1 q2 = (g10 )2 + (g10 )2 + K2 + g10 K  (j) (j) (j) a2 a2 (a2 )2

(8.136)

of which the discriminant q12 − 4q0 q1 is identically zero. Hence, the two coinciding solutions of Eq. (8.135) read (1) (2) g1 = g1 = g10 + q

(i)

a1 K (i)

(j)

(8.137)

(a1 )2 + (a2 )2

This implies that the straight line in Eq. (8.134a) and the circle in Eq. (8.134b) share one common point. Thus, we conclude that the straight line is tangent to the circular domain of the initial imperfections. Eq. (8.133) can then be interpreted as the condition for this situation.

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∆Φ1 = 1.628 ∆Φ2 = 2.490

1 rc = 0.9

0.8

rc = 0.7

rc

0.6 0.4 0.2

∆Φ2

∆Φ1 0

1

2

3

4

5

6

∆Φ

Fig. 8.29

Reliability versus angular difference, Eq. (8.127).

Boundary c (cm)

10 Solid Line rc = 1 Dashed Line rc = 0.95 Dotted Line rc = 0.9

8

6

4

2

I

0

II

2

III

4

6 8 10 Radius ρd (cm)

12

14

Fig. 8.30 Comparison of design curves for uniform probability density function over a circular domain and different required reliabilities (t = 0.5, P = 3000 kg, g10 = 2, g20 = 1.5, K = 1.0), Eq. (8.139).

In order to design the system with a prescribed required reliability rc , we must solve Eq. (8.127) for the angular difference ∆φ, with respect to rc substituted for R. Figure 8.29 depicts the curve R = R(∆φ), allowing one to obtain the angular difference ∆φ given the value of the design reliability rc . For example, for rc = 0.9, ∆φ = 1.628. Having found ∆φ that corresponds to rc , we proceed to design the system. In order to obtain the design curves c = c(ρd ) for specified time and pre-selected

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7 6 Boundary c

5

5

4

4 3

3 2

2

t (sec)

4 1

6

Radius ρd (cm)

8 10

Fig. 8.31 Design surface c = c(ρd , t), uniform probability density function over a circular domain and required unity reliability (P = 3000 kg, g10 = 2, g20 = 1.5, K = 1.0), Eq. (8.139).

reliability rc , one subtracts Eq. (8.128) from Eq. (8.129), leading to ∆φ = φ2 − φ1 

(i)

(j)

f a1 + a2

q

(i)

(j)

(a1 )2 + (a2 )2 − f 2



 (i) (j) (a1 )2 + (a2 )2 q   (i) 2 (j) 2 (j) (j) 2 (a ) + (a ) − f f a + a 1 2 1 2  − cos−1  (i) (j) (a1 )2 + (a2 )2 −1

= cos



(8.138)

Once ∆φ is determined from Fig. 8.29, one can calculate the value from Eq. (8.138). The solution of Eq. (8.138) for f , in conjunction with Eq. (8.130), allows one to obtain the required expression for the design curve c = c(ρd ), which reads (i)

(j)

c(ρd ) = a1 (ρd )g10 + a2 (ρd )g20 q K (i) (j) (a1 (ρd ))2 (1 + cos(∆φ)) + (a2 (ρd ))2 +√ 2

(8.139)

and is portrayed in Fig. 8.30 for specified time t = 0.5 and different values of rc . Figure 8.31 represents the design surface for unity reliability as a function c = c(ρd , t), allowing one to find the cross-sectional radius ρd for specified exploitation time t and the failure boundary c.

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8.3.7

269

Initial imperfections as interval variables: Interval analysis

Let the initial imperfection amplitude be represented by a vector interval variable g1 = [α1 , α2 ], g2 = [β1 , β2 ]

(8.140)

so that the additional displacement function at the center of the column is an interval variable as well: (i)

(j)

z = [ζ1 , ζ2 ] = [α1 , α2 ]a1 (λ) + [β1 , β2 ]a2 (λ)

(8.141)

where ζ1 and ζ2 are the lower and upper bounds of z, respectively. Since g1 and g2 are positive, so are the additional displacements. Hence, in the safety requirement condition −d ≤ z ≤ c, only the condition z ≤ c is operative. The column design via the anti-optimization technique identifies the worst possible reachable condition for the uppermost bound ζ2 of the interval variable z. If ζ2 ≤ c, the system will remain in the safe domain, otherwise it will fail. Expressing the argument of the functions (i) (j) a1 (λ) and a2 (λ) in terms of the cross-sectional radius ρd and the pre-selected time t by means of Eq. (8.108) (see Sec. 8.2), we obtain the formal design relation (i)

(j)

ζ2 = α2 a1 (ρd , t) + β2 a2 (ρd , t) = c

(8.142)

Comparing Eq. (8.142) with Eq. (8.108), we conclude that the former matches the design condition in case of unity reliability requirement given in Eq. (8.108). If the uncertainty region coincides with the domain represented in Fig. 8.15, then the design surface c(ρd , t) represented by Eq. (8.142) coincides with Eq. (8.77). Thus, the anti-optimization method and the probabilistic one with the highest reliability requirement lead to the same design values for the cross-sectional radius ρd . 8.3.8

Initial imperfections as convex variables: Circular domain

Let us consider the case when the information about the uncertain imperfection magnitude is given as (g1 − g10 )2 + (g2 − g20 )2 ≤ K 2

(8.143)

where (g10 , g20 ) is the center of the circle, and K is the radius as shown in Fig. 8.32. The displacement at the center of column reads (i)

(j)

z = g1 a1 (λ) + g2 a2 (λ)

(8.144)

where g1 and g2 are uncertain variables belonging to the set defined by Eq. (8.143), which is convex. The Lagrangian for the problem of maximizing z under constraint (8.143) is given as (i)

(j)

Π = a1 g1 + a2 g2 + η[(g1 − g10 )2 + (g2 − g20 )2 − K 2 ]

(8.145)

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3 S 2.5 A

g2

2 G0

1.5

d K

B

1 0.5

0

1

2 g1

3

4

Fig. 8.32 Initial imperfection amplitudes modeled by convex variables: anti-optimization design (g10 = 2, g20 = 1.5, K = 1.0), Eq. (8.152).

where η is the Lagrange multiplier (see Sec. 4.1 for details of the Lagrange multiplier approach). Necessary conditions for the extremal read ∂Π (i) = a1 + 2η(g1 − g10 ) = 0 ∂g1 ∂Π (j) = a2 + 2η(g2 − g20 ) = 0 ∂g2 ∂Π = (g1 − g10 )2 + (g2 − g20 )2 − K 2 = 0 ∂η

(8.146a) (8.146b) (8.146c)

The solutions of Eqs. (8.146a)–(8.146c), denoted by g 1 and g2 , corresponding to maximum z read (i)

Ka1

g 1 = g10 + q

(a1 )2 + (a2 )2

g 2 = g20 + q

(a1 )2 + (a2 )2

(i)

(j)

(8.147a)

(j)

Ka2 (i)

(j)

(8.147b)

Note that Eq. (8.147a) coincides with Eq. (8.137) which represents the abscissa of the common point between the circular domain of the random variables in Eq. (8.122) and the straight line in Eq. (8.123). Let us investigate the geometrical meaning of Eqs. (8.146a)–(8.146c) and the results in Eqs. (8.147a) and (8.147b). Eliminating parameter η from Eqs. (8.146a)

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and (8.146b) leads to (j)

g2 =

a2

g (i) 1 a1

+

1 (i) a1

(i)

(j)

(a1 g20 − a2 g10 ) (j)

(8.148) (i)

This expression represents a straight line with slope a2 /a1 passing through the center of the circle, which is identified in Fig. 8.32 with label S. Eq. (8.148) must be solved in conjunction with Eq. (8.146c). Thus the solution of the anti-optimization problem is found as the intersection point of the straight line S and the circle represented by Eq. (8.146c). There are two intersection points in Fig. 8.32 denoted by A and B, respectively, corresponding to the maximum and minimum of the displacement function. In order to deal with the anti-optimization technique, one must look for the maximum of the displacement function, namely, the point A in Fig. 8.32, which is obtained in Eq. (8.147). The design problem can be stated differently. Let us consider the expression (j)

g2 =

a2

g (i) 1

a1

+



(i) a1 g10

−

(j) a2 g20

 q (i) 2 (j) 2 + K (a1 ) + (a2 )

(8.149)

representing a line orthogonal to S and tangent to the circle at point A. One may obtain the solution of the system in Eq. (8.146) by solving (i)

(j)

a1 g1 + a 2 g2 − T = 0 2

(8.150a) 2

(g1 − g20 ) + (g2 − g20 ) = K

2

(8.150b)

where the line in Eq. (8.150a) is perpendicular to the line in Eq. (8.149). The parameter T is a free parameter that must be chosen so that the straight line in Eq. (8.150a) constitutes a tangent to the circle in Eq. (8.150b). It may be deduced by inspection of Eq. (8.147a) that the parameter T coincides with the failure boundary value c. The value of T is obtainable as follows by solving Eqs. (8.150a) and (8.150b): q (j) (i) (j) (i) T = c = a1 (ρd , t)g10 + a2 (ρd , t)g20 + K (a1 (ρd , t))2 + (a2 (ρd , t))2 (8.151)

Rewriting Eq. (8.139) in Cartesian coordinates for unity required reliability r c as q (j) (i) (j) (i) (8.152) c = a1 (ρd , t)g10 + a2 (ρd , t)g20 + K (a1 (ρd , t))2 + (a2 (ρd , t))2 one immediately recognizes that the different models of uncertainty treated in Sec. 8.3.6 and in the present one lead to the same analytical expression of the failure boundary surface. Then, with the initial imperfection amplitude given in the same region of variation, these different approaches result in the same design surface.

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Concluding remarks

As shown in this section, the two-dimensional interval analysis provides a design that is a limiting case of the probabilistic design when the required reliability tends to unity from below. Likewise, convex modeling yields a design to which the probabilistic calculations tend, when the variation of uncertain variables is limited to the circular domain. The natural question arises: Can the non-probabilistic method be preferred to the probabilistic approach? In general, should the simpler method be preferred to the more complex one? It appears instructive to reproduce here three relevant quotations: (i) ‘The aim of science is to seek the simplest explanations of complex functions. We are apt to fall into the error of thinking that the facts are simple because simplicity is the goal of our quest. The guiding motto in the life of every natural philosopher should be, “Seek simplicity and distrust it”.’ (Alfred North Whitehead). (ii) ‘Everything should be made as simple as possible, but not one bit simpler.’ (Albert Einstein) (iii) ‘Never use a long word where a short one will do. If it is possible to cut a word, always cut it out.’ (George Orwell) If one follows these recommendations, one concludes that interval analysis is the simplest yet tenable method of describing uncertainty. It leads to the same answer as the theory of probability, but without using long words, as Orwell put it; it also appears that uncertainty modeling cannot be made one bit simpler, to borrow Einstein’s words, than the interval analysis. Yet, in accordance with Whitehead, we do not discourage the distrust. Indeed, the following challenging question appears to be in order: Can interval or ellipsoidal analyses describe first-excursion failure, fatigue problems, wind loads, earthquake engineering and a long list of other issues dealt with to some degree of success, recognized at least by some researchers, by the theory of probability and random processes? This question remains yet to be answered. The establishment of identity of designs furnished by the probabilistic and non-probabilistic analyses shows that all the ‘analytical roads’ lead to the same results for the problem in question. Note that hybrid probabilistic and non-probabilistic analysis of structures apparently was initiated by Elishakoff and Colombi (1993) and Elishakoff, Lin and Zhu (1994e) in the context of space shuttle application. The later works include those by Elishakoff and Li (1999), Kreinovich, Xiang, Starks, Longpr´e, Ceberio, Araiza, Beck, Kandathi, Nayak, Torres and Hajagos (2006), Adduri and Penmetsa (2007), Qiu, Yang and Elishakoff (2008b, 2008c), and possibly others.

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Chapter 9

Hybrid Optimization with Anti-Optimization under Uncertainty or Making the Best out of the Worst ‘Make the best of a bad bargain.’ (Japanese proverb) ‘Things turn out the best for people who make the best of the way things turn out.’ (John Wooden) ‘A given form will be optimum if all failure modes which can possibly intersect occur simultaneously under the action of the load environment.’ (Spunt, 1971) In this chapter, as the main purpose of this book, hybrid approaches of optimization with anti-optimization are presented for static and buckling problems. We first present the framework of the problem and the methodologies for solving the twolevel optimization problem with anti-optimization. Several applications are shown for buckling optimization, static problems considering stress, displacements and compliance. The design methods for flexible structures are also shown, and finally, an example of force identification of a tensegrity structure is presented. 9.1

Introduction

Structural design problems are usually formulated, or implicitly formulated, to minimize the structural cost under constraints on structural performance, defined by the responses to design loads. In the practical design process, the design loads and the bounds on the responses are considered to be deterministic, where the bounds are decreased, or the design loads are increased by incorporating the safety factor. However, it is desirable to explicitly incorporate uncertainties in the formulation of the design problem. Hence, the responses or the design loads are obtained by solving anti-optimization problems, and the design problem turns out to be a hybrid and a two-stage optimization–anti-optimization problem. Elishakoff, Haftka and Fang (1994c) presented a basic concept of optimization with anti-optimization. Kwak and Haug (1976) formulated a worst-case design problem and developed a parametric programming approach. They applied it to 273

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P2 , U2 P1 , U1

H x y

W1

Fig. 9.1

W2

An asymmetric 2-bar truss.

problems with loading uncertainty and a vibration isolator design with a specified band of excitation frequency. Lanzi and Giavotto (2006) used a multiobjective genetic algorithm for buckling optimization with uncertainty. El Damatty and Nassef (2001) optimized a ribbed shell by using a genetic algorithm. Structural optimization considering the worst-case scenario is closely related to the so-called robust design. The purpose of robust design is to achieve the design that is least sensitive to the uncertain parameters (Ben-Haim, Fr´ yba and Yoshikawa, 1999; Au, Cheng, Tham and Zheng, 2003). Various methodologies have been developed based on the intuitively defined robustness function. Structural optimization under uncertainty has been extensively studied in the framework of reliability-based design (Palassopoulos, 1991; Kogiso, Shao, Miki and Murotsu, 1997; Hajela and Vittal, 2006). There are many methods for reliabilitybased structural optimization; see, for instance, Youn, Choi and Park (2003), Kharmanda, Olhoff and El-Hami (2004), Cheng, Xu and Jiang (2006), and Kim and Choi (2008). Optimization considering uncertainty can also be formulated by the fuzzy set theory. Pantelides and Ganzerli (2001) showed that similar designs can be found by fuzzy set and convex model (see Sec. 2.9 for fuzzy-set based optimization). 9.2

A Simple Example

Consider again the asymmetric 2-bar truss as shown in Fig. 9.1, which has been √ used in Sec. 4.1, subjected to static loads P1 and P2 , where H = 1, W1 = 3, and W2 = 1. Let Ai and Li denote the cross-sectional area and the length of the ith member, respectively. Young’s modulus is denoted by E. The horizontal and vertical displacements are denoted by U1 and U2 , respectively. We consider a simple optimization problem to find the design variable vector A = (A1 , A2 )> for minimizing the total structural volume V = A 1 L1 + A 2 L2

(9.1)

U2 ≤ U2U

(9.2)

under displacement constraint

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where U2U is the specified upper bound for U2 , which is assumed to be positive. The relation between the load vector P = (P1 , P2 )> and the displacement vector U = (U1 , U2 )> is written as √ √ √ √ 2 √ [(A1 + 2 2A2 )P1 − ( 3A1 − 2 2A2 )P2 ] U1 = A1 A2 E(2 + 3) (9.3) √ √ √ √ 2 √ [( 3A1 − 2 2A2 )P1 + (3A1 + 2 2A2 )P2 ] U2 = A1 A2 E(2 + 3) Uncertainty is introduced for P as P = P0 + a

(9.4)

0

where P = (P10 , P20 )> is the vector of nominal values, and a = (a1 , a2 )> is the uncertain parameter vector. Since U2 depends on A and a, it is written as U2 (a, A). We are interested in designing the truss, i.e., selecting the values of A1 and A2 , so that the displacement U2 under the worst possible load scenario satisfies the constraint (9.2). b (A) denote the worst load scenario for maximizing U2 . Since the worst load Let a scenario depends on the design variable vector A, the value of U2 corresponding b2 (A) = U2 (b b is denoted as U to a a(A), A). The lower and upper bounds for Ai are denoted by ALi and AU , respectively. Then the optimization problem is formulated i as minimize

V = A 1 L1 + A 2 L2

b2 (A) ≤ subject to U ALi

U2U

≤ Ai ≤

(9.5a) (9.5b)

AU i ,

(i = 1, 2)

(9.5c)

Consider the case where the bound of a is given by the quadratic constraint (ellipsoidal bound) as a> a ≤ D

(9.6)

where D is the specified upper bound. Then the lower-level anti-optimization probb2 (A) is formulated as lem for finding U b2 (A) = max U2 (a, A) find U (9.7a) a

subject to a> a ≤ D

(9.7b)

Here, A is regarded as a parameter vector. Thus, optimization with the worst load scenario is formulated as a two-level optimization–anti-optimization problem. In this example, we can obtain the anti-optimal solution explicitly, as follows, in the same manner as Sec. 4.1 by using the Lagrange multiplier approach: 1 b (A) = c(A) (9.8) a 2µ where µ ≥ 0 is the Lagrange multiplier, and c = (c1 , c2 )> is given as √ √ 3 2A1 + 4A2 6A1 − 4A2 √ , c2 (A) = √ c1 (A) = (9.9) A1 A2 E(2 + 3) A1 A2 E(2 + 3)

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Since the constraint (9.7b) is satisfied with equality at the anti-optimal solution, we obtain r 1 c> c (9.10) µ= 2 D Hence, by incorporating Eqs. (9.8)–(9.10) into Eqs. (9.3) and (9.4), the load b2 (A) corresponding to the worst-case b(A) and the displacement U vector P = P 0 + a scenario are written as explicit functions of A, and the two-level optimization–antioptimization problem can be reformulated as a single-level optimization problem. However, in general, such an explicit reformulation is not possible. Therefore, many algorithms for solving the two-level problems have been developed.

9.3

Formulation of the Two-Level Optimization–Anti-Optimization Problem

The general formulations of the two-level optimization–anti-optimization problem are presented below (Lombardi and Haftka, 1998; Lombardi, 1998). Let x = (x1 , . . . , xm )> denote the vector of m design variables. The objective function, such as the total structural volume, to be minimized is denoted by F (x). Constraints are given as Hj (x) ≤ 0, (j = 1, . . . , N I ) (9.11) I where N is the number of inequality constraints, which include the side (bound) constraints of the design variables for a simple formulation of the problem. Note that equality constraints are omitted also, for simplicity, because the design requirements in practice are usually given in inequalities. The structural optimization problem is stated as minimize F (x) (9.12a) subject to Hj (x) ≤ 0, (j = 1, . . . , N I ) (9.12b) The various definitions of the problem can be found in Chap. 2. Consider the case where uncertainty exists in the vector of N parameters a = (a1 , . . . , aN )> defining the loads, geometry, etc., and Hj (x) is regarded as a function of a and x as Hj (a, x). Then the constraints should be given for the b j (x) of Hj (a, x). Then the upperworst-case scenario (maximum possible value) H b j (x) level optimum design problem is to be reformulated with the maximum value H as minimize F (x) (9.13a) b j (x) ≤ 0, (j = 1, . . . , N I ) subject to H

(9.13b) b where Hj (x) is obtained by solving the following lower-level anti-optimization problem: b j (x) = max Hj (a, x) find H (9.14a) a

subject to Ck (a) ≤ 0, (k = 1, . . . , N C )

(9.14b)

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Here, the feasible domain C for a is defined by the inequality constraints Ck (a) ≤ 0 (k = 1, . . . , N C ). Note that if a simple approach is used, Problem (9.14) should be solved N I times at each step of modification of x in the upper-level problem (9.13). Alternatively, the two-level problem can be formally written as minimize

F (x)

(9.15a)

b j (x) = max Hj (a, x) ≤ 0, (j = 1, . . . , N I ) subject to H a∈C

9.4 9.4.1

(9.15b)

Algorithms for Two-Level Optimization–Anti-Optimization Cycle-based method

As can be seen from the formulations of the upper-level problem (9.13) and the lower-level problem (9.14), we have to solve a nested optimization problem. The simplest method for this purpose is the cycle-based method that is summarized as follows: (0)

Step 1: Assign an initial value aj for a corresponding to each of the inequality constraints Hj (a, A) ≤ 0 (j = 1, . . . , N I ), and set the iteration counter k = 0. (k) Step 2: Solve Problem (9.13) for fixed a = aj , for the jth constraint to obtain the optimal solution x(k) . (k) Step 3: Fix x at x(k) and find anti-optimal solution aj of Problem (9.14) for each response function Hj (a, x(k) ) (j = 1, . . . , N I ), and set k ← k + 1. Step 4: Go to Step 2, if the stopping criteria for the cycle are not satisfied. The readers can consult with Sec. 6 of Gurav (2005), Lombardi and Haftka (1998), etc., for the details. In this method, the lower-level anti-optimization problem should be solved many times at each iterative step of design optimization, if constraints are given for many response quantities as Hj (a, x) ≤ 0 (j = 1, . . . , N I ). Since the anti-optimization problem for finding the worst value of Hj (a, x) can be solved simultaneously, the use of parallel computation will be very effective for reducing the computational cost (Gurav, Goosen and van Keulen, 2005a). Although this cycle-based approach is simple and easy to implement, this approach does not rapidly converge if the anti-optimal solution strongly or nonlinearly depends on x. Lombardi and Haftka (1998) applied this algorithm to the minimum weight design of a composite laminated structure with displacement constraints, where the thickness and angle of each ply are chosen as design variables, and the uncertainty is introduced for the load components, which are expressed by the multipliers ai (i = 1, . . . , N ) for the load pattern vectors p1 , . . . , pN . The displacement vector against pi is denoted by U i = (U1i , . . . , Uni )> , where n is the number of degrees of freedom. Then the total displacement vector, considering the uncertainty in the

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load vector, is given as a linear function of a as U=

N X

ai U i

(9.16)

i=1

The bounds for a are defined by the linear inequalities bL ≤ Ba ≤ bU

(9.17)

where B is a constant matrix, and bL and bU are constant vectors of the lower and upper bounds. Then the anti-optimization problem for finding the worst value of the displacement component Uj is formulated as find

bj (x) = max U a

subject to bL ≤ Ba ≤ b

N X

ai Uji

i=1 U

(9.18a) (9.18b)

where Uj has been assumed to be positive for simplicity. Then the cycle-based method is applied to find the optimal distributions of the thickness and ply-angle. It has been observed from their numerical experiments that the convergence of the cycle-based method is very good if a is bounded with linear inequalities as in Eq. (9.18b). 9.4.2

Methods based on monotonicity or convexity/concavity

Consider the case where the bounds for a are given by the interval aLi ≤ ai ≤ aU i , (i = 1, . . . , N )

(9.19)

and no other constraint is given. If the objective function Hj (a, x) that is to be maximized in the lower-level anti-optimization problem (9.14) is a continuous and concave function of a, then the anti-optimal solution is the global maximum of a nonlinear programming problem, which can be easily found using a gradient-based approach. If Hj (a, x) is a continuous and monotonic function of a, the global antioptimal solution exists at a vertex of the feasible region defined by Eq. (9.19), which can be found easily from the signs of differential coefficients of Hj (a, x) with respect to a. If Hj (a, x) is a convex function of a, the anti-optimal solution also exists at the vertex of the feasible region. In this case, the solution can be found by enumerating the values of Hj (a, x) at the vertices of the feasible region. Note that the solution can exist on an edge or on a hyper-plane for a non-generic case where the differential coefficient of Hj (a, x) with respect to a parameter ai vanishes at the anti-optimal solution. Let V denote the set of 2N vertices of the feasible region; i.e., ai (i = 1, . . . , N ) is equal to either aLi or aU i for the parameter set in V. Then the two-level optimization

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problems (9.13) and (9.14) are combined into a single-level minimization problem as minimize

F (x)

(9.20a) I

subject to Hj (a, x) ≤ 0, (j = 1, . . . , N ; a ∈ V)

(9.20b)

Although this problem formulation is simple, the total number of constraints (9.20b) is 2N × N I ; hence, the computational cost increases as an exponential function of the number N of the uncertain parameters, if all the vertices of the feasible region must be enumerated. There are several studies based on this enumeration approach (Ganzerli and Pantelides, 2000) (see Sec. 4.6 for anti-optimization of a tensegrity structure by enumeration, and Sec. 9.6 for hybrid optimization–anti-optimization of a truss under stress and displacement constraints). 9.4.3

Other methods

9.4.3.1 Approximation methods To reduce the computational cost of solving the lower-level anti-optimization problem with large numbers of design variables and uncertain parameters, approximation methods such as response surface method (Myers and Montgomery, 1995) can be used (see, e.g., Gurav, Goosen and van Keulen (2005a) and Secs. 3 and 7 of Gurav (2005) for details of this approach). In this method, the responses are approximated by a polynomial of the parameters as described in Sec. 2.5. Then the lower-level anti-optimization problem turns out to be a small optimization problem that is easily solved by a gradient-based nonlinear programming for maximizing a polynomial objective function in a convex feasible region for a. Lanzi and Giavotto (2006) solved a multiobjective optimization problem for simultaneously minimizing the total structural volume and maximizing the collapse load in the postbuckling range, using a genetic algorithm with approximation by neural networks, response surface, radial basis function (Nakayama and Hattori, 2003) or kriging method (Sakata, Ashida and Zako, 2003). 9.4.3.2 Parametric programming approach In the process of solving the upper-level design optimization problem (9.13), it is b j (x) for the jth effective to incorporate the sensitivity of the anti-optimal solution a b j (x) at each constraint with respect to the design variables x, rather than fixing a step of optimizing x. Therefore, the parametric programming approach presented in Sec. 2.4.3 can be used to accelerate the convergence of the cyclic optimization process. This idea has been mentioned in Kwak and Haug (1976) and Gurav, Goosen and van Keulen (2005a), but has not been actually implemented. Baitsch and Hartmann

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(2006) used this approach for optimizing the truss under constraint on strain energy (compliance) considering uncertain geometrical parameters.

9.5 9.5.1

Optimization against Nonlinear Buckling Introduction

There has been extensive controversy over the effectiveness of optimization under nonlinear buckling constraints. Thompson and Supple (1973) cautioned the danger of optimization due to interaction of critical modes at buckling: ‘If there is severe imperfection-sensitivity associated with the coupling then the optimization cast in this form may be invalid. We conclude, therefore, that any optimization scheme which is to be applied to a structural system displaying multi-mode buckling must be so defined as to take into account the complete buckling characteristics including the effects of unavoidable manufacturing imperfections.’ As one can see by careful reading of their comments, they did not unconditionally reject the idea of optimization against buckling. However, the terms if and so defined have been omitted for some reason by some researchers to state that optimization against buckling is unconditionally meaningless. The safety against buckling of structures can be effectively improved by optimization if uncertainties in geometry, material, etc., are appropriately incorporated. Furthermore, optimization does not always enhance imperfection sensitivity, and for some structures, sensitivity is even decreased by optimization. For plates and frames, for which the prebuckling deformation is negligibly small, the buckling loads can be accurately estimated by linear buckling analysis. Angularply laminas are extensively studied for optimization under buckling constraints with uncertain initial imperfection (Adali, Richter and Verijenko, 1997). Laminated plates are also optimized considering uncertainty in material properties (Adali, Elishakoff, Richter and Verijenko, 1994a) and external loads (Adali, Lene, Duvaut and Chiaruttini, 2003). The convexity of the stability region described in Sec. 5.6 can be effectively used for optimization under the worst linear buckling load (de Faria and Hansen, 2001a,b). De Faria and de Almeida (2003b) presented the optimization method of a plate with variable thickness. They extended their methodology to optimization for fundamental frequency considering geometrical stiffness under the worst loading scenario (de Faria and de Almeida, 2006). For flexible structures such as spherical shells and latticed domes in civil and architectural engineering, the effect of prebuckling deformation should be incorporated for accurate estimation of the buckling load. Ohsaki (2000, 2002c, 2005)

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Λ

Ui Fig. 9.2

Equilibrium path of a geometrically nonlinear structure.

optimized shallow space trusses to find that imperfection sensitivity does not always increase as a result of optimization, but is often reduced. Furthermore, optimization of a shallow shell-type structure often leads to a coincident critical point, called the hilltop branching point, as a coincidence of bifurcation point(s) and a limit point, which is not always imperfection-sensitive (Ohsaki and Ikeda, 2007a). Therefore, the critical load considering its reduction due to initial imperfection can be effectively increased by optimizing the perfect system. However, to be more comfortable with optimization against buckling, we can directly incorporate the reduction of the critical load due to imperfection into the process of optimization (Ramm, Bletzinger and Reitinger, 1993; Reitinger, Bletzinger and Ramm, 1994; Reitinger and Ramm, 1995). In this process, the worstcase scenario should be taken into account to evaluate the lower bound of the critical load for the given magnitude of imperfection. Since optimization may lead to imperfection-sensitive structures with multiple (coincident) buckling loads, if some bifurcation points coincide, the imperfection sensitivity of such structures has been extensively investigated (Ohsaki, 2002a,b; Ikeda, Ohsaki and Kanno, 2005; Ohsaki and Ikeda, 2006). Ohsaki, Uetani and Takeuchi (1998b) presented an optimization method for symmetric systems considering reduction of the maximum load due to the worst imperfection. El Damatty and Nassef (2001) optimized a ribbed shell by using a genetic algorithm. Ohsaki (2003a) defined the worst load case of flexible structures by the strain energy stored at the deformation with the specified norm, and optimized trusses under constraint such that the initial undeformed state is recovered after releasing the loads corresponding to the worst-case scenario. 9.5.2

Problem formulation

Consider an elastic structure subjected to a set of proportional loads P = Λp parameterized by the load factor Λ and the constant load pattern vector p. The nodal displacement vector U nonlinearly depends on Λ as shown in Fig. 9.2, where Ui is the representative displacement component.

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Λ

Λ

Perfect Perfect

Imperfect

Imperfect

Imperfect

Imperfect

q

q (a) Unstable-symmetric bifurcation point

(b) Stable-symmetric bifurcation point

Fig. 9.3 Classification of bifurcation points and associated equilibrium paths of perfect and imperfect systems; ◦: bifurcation point.

Suppose the structure reaches a critical point at Λ = Λc as Λ is increased from 0 (see Sec. 5.3 for the definition of the critical point). Note that the critical load factor Λc is also called the buckling load factor or simply the buckling load. It is known that optimization against nonlinear buckling is very difficult if the buckling loads coincide (Ohsaki, 2000, 2005). However, only the simple critical point is considered in this section; i.e., 0 = λc1 < λc2 ≤ · · · ≤ λcn

(9.21)

where λci is the ith eigenvalue of the tangent stiffness matrix at the critical point, and n is the number of degrees of freedom. Let x = (x1 , . . . , xm )> denote the vector of design variables. The first critical load Λc (x) on the fundamental path is the function of x. The optimization problem of minimizing the total structural volume V (x) under nonlinear buckling constraint is simply formulated as minimize

V (x)

subject to Λc (x) ≥ Λ xLi

c

≤ xi ≤

(9.22a) c

xU i ,

(9.22b) (i = 1, . . . , m)

(9.22c)

where Λ is the specified lower bound for the critical load, and xLi and xU i are the lower and upper bounds for xi , respectively. The major disadvantage of this formulation is that the possible reduction of the maximum (critical) load due to initial imperfection is not considered. Figures 9.3(a) and (b) illustrate the equilibrium paths of the structures exhibiting an unstable-symmetric bifurcation and stable-symmetric bifurcation, respectively, where q represents the generalized displacement in the direction of the buckling mode, or the generalized antisymmetric component of the displacements. As is

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Λ

Λ c

Λ

b

Optimization

Λ (q, x)

b

c

Λ (q, x)

Λ

q

Fig. 9.4

q

Stabilization of the bifurcation path by optimization.

seen, for the unstable-symmetric bifurcation, the critical point of an imperfect system turns out to be a limit point, which may be far below the bifurcation load of the perfect system. However, for the stable-symmetric bifurcation point, the critical point disappears due to the presence of imperfection. For structures exhibiting unstable-symmetric bifurcation point, the critical load reduced due to the worst initial imperfection with specified bound may be used instead of Λc in Eq. (9.22b). However, the maximum load cannot be defined by a critical load for the case of stable-symmetric bifurcation. In this case, the maximum load may be defined by the upper-bound constraints on the responses such as stresses and displacements. For example, consider a truss and let σ = (σ1 , . . . , σN m )> denote the vector of member stresses, where N m is the number of members. The maximum load can be defined with the upper and lower bounds denoted by σiU and σiL , respectively, of σi as Λm = max{Λ|σiL ≤ σi (Λ) ≤ σiU , i = 1, . . . , N m } (9.23) Λ

Hence, the optimization problem may be defined as follows under constraint on the maximum load Λm (x): minimize V (x) (9.24a) subject to Λm (x) ≥ Λ m

xLi

≤ xi ≤

m

xU i ,

(9.24b) (i = 1, . . . , m)

(9.24c)

where Λ is the specified lower bound for the maximum load (see Sec. 5.5 for antioptimization of the linear buckling load of a braced frame using this definition of the maximum load). The maximum load of a structure exhibiting stable bifurcation can be alternatively defined by the displacement constraint (Ohsaki, 2002b). Constraints on the antisymmetric components of displacements can also be given for the case of unstable bifurcation point, because the antisymmetric deformation rapidly increases with decreasing load after reaching the bifurcation load (Pietrzak, 1996). The optimization problem can be alternatively formulated under the condition such that the bifurcation point is stable; i.e., the structure is stabilized by optimization so that the perfect system has the bifurcation path (initial postbuckling

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path) with increasing load factor as illustrated in Fig. 9.4. Design sensitivity analysis of the responses on the postbuckling path is presented by Godoy and Taroco (2000). Schranz, Krenn and Mang (2006a) investigated the characteristics of the bifurcation path based on Koiter’s perturbation approach (Koiter, 1945), and presented a numerical approach in Schranz, Krenn and Mang (2006b) to convert an imperfection-sensitive structure into an imperfection-insensitive one by placing additional supports. Bochenek (2003) investigated the effect of the stiffness of spring support on the shape of the postbuckling path in the load–displacement space. Let Λb (q, x) denote the load factor along the bifurcation path of the perfect system, which is a function of the generalized antisymmetric component q of the displacement. Note that q = 0 corresponds to the bifurcation point. The curvature κb (x) of the bifurcation path at the bifurcation point is defined by ∂ 2 Λb (q, x) b (9.25) κ (x) = ∂q 2 q=0

Then the optimization problem under curvature constraint is formulated as minimize

V (x) c

subject to Λ (x) ≥ Λ

(9.26a) c

b

κ (x) ≥ 0

(9.26b) (9.26c)

By solving Problem (9.24), the structure is stabilized as illustrated in Fig. 9.4. Bochenek (1997, 2003) also presented a problem of converting an asymmetric bifurcation point to a symmetric one, which seems to be difficult because the symmetry of the bifurcation point generally depends on the geometrical property of the structure. Bochenek (2003) formulated the following optimization problem, considering extended local stability, as minimize

V (x) c

subject to Λ (x) ≥ Λ b

(9.27a) c

(9.27b) b

Λ (qi , x) − Λ (qi+1 , x) ≤ 0, (i = 1, . . . , r)

(9.27c)

where q1 , . . . , qr+1 satisfying 0 = q1 < q2 < · · · < qr+1

(9.28)

are the specified values of generalized displacement in the direction of the bifurcation mode. By enforcing the constraint (9.27c), the structure with a non-decreasing bifurcation path is obtained, and the limit point (snapthrough) in the bifurcation path is also avoided. Bochenek (2003) formulated similar problems considering multiple buckling loads.

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ΛP 7 (5)

(6)

(12)

5

6

(4)

(3) (11)

3

(7)

4

(8) y

H

(2) 2

1

x

W

Fig. 9.5

9.5.3

H

(10)

(9)

(1)

H

W

A 12-bar tower truss.

Numerical examples

The effect of geometrical imperfection on the maximum load is investigated, and stabilization by design modification is demonstrated for a simple truss. Consider a 12-bar tower truss as shown in Fig. 9.5 subjected to a vertical load ΛP at the top node 7, where P = 1 kN for simplicity. The member number and node number are indicated by numbers with and without parentheses, respectively. The dimension of the structure is given as W = 250 mm and H = 1000 mm. The intersecting pair of diagonal members are not connected at their centers. Young’s 2 modulus is 200 kN/mm , and yielding or buckling of members is not considered. In the following, the units of force and length are kN and mm, respectively, which are omitted for brevity. Geometrical nonlinearity is incorporated, and the strain of a member is measured from the member lengths before and after deformation. The displacement increment method is used for tracing the equilibrium path. No iteration is carried out at each step, but the unbalanced load is canceled at the subsequent step. The properties of the equilibrium paths are first investigated for different designs with the same total structural volume V = 1.0 × 106 . The design parameter is the ratio α of the cross-sectional area of the diagonal members to that of the remaining members, i.e., the cross-sectional areas are scaled for each value of α so that V = 1.0 × 106 is satisfied. The nonlinear buckling load Λc of the perfect system is 3678.0

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4000

Load factor

3000

2000

1000

0 0

100

200

300

400

Horizontal displacement of node 7 Fig. 9.6

Relation between Λ and U for ∆ = 0.01; solid line: α = 1.0; dashed line: α = 0.01.

for the truss with uniform cross-sectional area; i.e., α = 1.0, and is 3842.1 for α = 0.01. Therefore, the truss with a given total structural volume has a larger buckling load if the diagonal members have smaller cross-sectional areas than the vertical and horizontal members. An imperfection ∆ is given in the x-direction of the top node 7. Consider first the case where ∆ = 0.01. The relation between P and the x-directional displacement U of node 7 for α = 1.0 is shown in the solid line in Fig. 9.6. As is seen, the horizontal displacement suddenly increases as P approaches Λc (= 3678.0). Note that P is a monotonically increasing function of U . For α = 0.01, the equilibrium path is plotted in a dashed line in Fig. 9.6. The load factor for α = 0.01 is larger than that for α = 1.0 in the range of small U . However, Λ gradually decreases as U is increased after approaching the bifurcation load of the perfect system. In fact, the value of Λ for α = 0.01 is smaller than that for α = 1.0 for the range U ≥ 160. Therefore, a stable bifurcation with smaller bifurcation load is preferred to an unstable bifurcation with larger bifurcation load if the allowable displacement is moderately large. Figure 9.7 shows the equilibrium paths of imperfect systems for ∆ = 1.0, where the solid and dashed lines correspond to the designs with α = 1.0 and 0.01, respectively. As is seen, the equilibrium path has a limit point for α = 0.01, and the load factor monotonically increases for α = 1.0. The load factor for α = 0.01 is smaller than that for α = 1.0 in the range U ≥ 185. Optimal design is next found for ∆ = 1.0, where α is chosen as the design parameter. The allowable displacement is given as U U = 200; i.e., the maximum load Λm is defined as the limit point load in the range U < U U or the load factor m at which U = U U is satisfied, and the specified maximum load factor Λ is 4000.0. The relation between the design parameter α and the total structural volume for m satisfying Λ = 4000.0 is plotted in Fig. 9.8. The optimal design that minimizes the total structural volume exists at α ' 0.027, and the corresponding optimal

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4000

Load factor

3000

2000

1000

0 0

100

200

300

400

Horizontal displacement of node 7 Fig. 9.7

Relation between Λ and U for ∆ = 1.0; solid line: α = 1.0; dashed line: α = 0.01.

Total structural volume

1400000

1200000

1000000

800000 0

0.02

0.04

0.06

0.08

0.1

Design parameter Fig. 9.8

Relation between design parameter α and total structural volume.

objective value is about 9.0467 × 105 . This optimal solution with a small value of α agrees with the fact that a tower-type structure can be effectively stabilized by adding very slender braces. Figure 9.9 shows the relation between Λ and U for designs satisfying Λm = 4000.0. As is seen, the maximum load is achieved at a limit point for α = 0.01 and at U = U U = 200 for α = 1.0. Consequently, for the optimal design α = 0.027, Λ has the maximum value at a limit point that satisfies U = U U = 200.

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4500

Load factor

4000

3500

3000 0

100

200

300

400

Horizontal displacement of node 7 Fig. 9.9 Relation between Λ and U for designs satisfying Λm = 4000.0 (∆ = 1.0); solid line: α = 0.027; dashed line: α = 0.1; dotted line: α = 1.0.

(1)

2

(7)

4

5

x

P1 2.0 m Fig. 9.10

(6) (10) (4)

(3)

9.6

3

(5)

(8) y

(2)

2.0 m

1

(9) 6 P2 P3

2.0 m A 10-bar truss.

Stress and Displacement Constraints

An example of optimization of trusses under stress and displacement constraints is presented in this section considering the worst load scenario. Consider a 10-bar truss as shown in Fig. 9.10. The examples below are basically the same as those in Elishakoff, Haftka and Fang (1994c); however, the geometry and mechanical properties have been modified to use SI units. The total structural volume is minimized under constraints on stresses and displacements. Let σi denote the stress of the ith member. The upper and lower bounds denoted by σiU and σiL , respectively, 2 2 are ±0.2 kN/mm for all members except ±0.6 kN/mm for member 9 defined in Fig. 9.10. The external loads are P1 = 1.0, P2 = 4.0, P3 = 1.0 (kN). Young’s mod2 ulus is 200.0 kN/mm . The units for force and length are kN and mm, respectively,

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Table 9.1 Cross-sectional areas and the structural volume of the optimal solution (deterministic loads). Member number

1 2 3 4 5 6 7 8 9 10 Total volume (mm3 )

Ai (mm2 ) Case 1

Case 2

5.3536 1.0000 5.3536 16.0000 3.6464 1.0000 13.6421 1.0000 2.8284 1.4142

6.0910 1.0000 5.3502 15.3359 4.9652 1.0000 13.6469 1.0000 7.5264 1.0000

1.1812 × 105

1.3303 × 105

in the following examples. The objective here is to minimize the total structural volume under stress and displacement constraints. Optimization is carried out by SNOPT Ver. 7 (Gill, Murray and Saunders, 2002) based on the sequential quadratic programming algorithm. The cross-sectional areas of all members are independent variables; i.e., we have ten design variables. The lower bound ALi for the cross-sectional area Ai is 1.0 for all members, and the upper bounds are not given. Consider first the optimization under deterministic loads, where the loads P1 , P2 , and P3 are applied simultaneously as shown in Fig. 9.10. Optimal solutions are found for two cases: without displacement constraints (Case 1), and with upper bound U U = 5.0 for negative y-directional displacement U6 at node 6 (Case 2). The optimal cross-sectional areas and the total structural volume are listed in Table 9.1. As is seen, the members 2, 6, and 10 connected to node 3, and member 8 have small cross-sectional areas. Since the displacement constraint is active at the optimal solution for Case 2, it has a larger objective value than Case 1. Consider next the case with uncertain loads. Let the nominal loads be defined as P10 = 1.0, P20 = 4.0, P30 = 1.0. The lower and upper bounds PiL and PiU , respectively, for Pi are given as PiL = 0.9Pi0 , PiU = 1.1Pi0 , (i = 1, 2, 3)

(9.29)

Since the stresses and displacements are linear functions of the loads, the maximum and minimum values can be found by enumerating the eight sets of loads P V k (k = 1, . . . , 8) at the vertices of the feasible region defined by P1L ≤ P1 ≤ P1U , P2L ≤ P2 ≤ P2U , P3L ≤ P3 ≤ P3U

(9.30)

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Therefore, the optimization problem with the worst load scenario is formulated as m X minimize V (A) = A i Li (9.31a) i=1

U subject to σiL ≤ σi (A, P V k ) ≤ σi , (i = 1, . . . , m; k = 1, . . . , 8) U U6 (A, P V k ) ≤ U , (k = 1, . . . , 8) ALi ≤ Ai , (i = 1, . . . , m)

(9.31b) (9.31c)

(9.31d) where Li is the length of the ith member, and m is the number of members that is equal to 10. In this formulation, we have 168 inequality constraints in addition to the side constraints (9.31d) for Ai . Alternatively, the maximum and minimum stresses and displacements for the feasible set of loads defined by Eq. (9.30) can be found by combining the responses against each load P1 , P2 , and P3 applied independently. Let σij denote the stress of member i against the load Pj0 . From the definition (9.29) of load uncertainty, the maximum σimax and minimum σimin of σi are found as 3 X σimax = (1 + 0.1 sign σij )σij , j=1

σimin

3 X = (1 − 0.1 sign σij )σij

(9.32)

j=1

For example, if σij is negative, the maximum and minimum values of the stress of the ith member to the jth load are (1 − 0.1)σij = 0.9σij and (1 + 0.1)σij = 1.1σij , respectively. Then the constraints (9.31b) can be written as σiL ≤ σimin , σimax ≤ σiU , (i = 1, . . . , m) (9.33) This way, the number of constraints is reduced to 21, including the displacement constraint that can be defined in a similar manner as Eqs. (9.32) and (9.33). Table 9.2 shows the optimal solutions without displacement constraint (Case 1) and with displacement constraint (Case 2). It is seen by comparing the results in Tables 9.1 and 9.2 that the optimal objective value significantly increases, especially for Case 2, as a result of load uncertainty. Table 9.3 shows the values of stresses at the eight vertices of the feasible region of the optimal solution for Case 1, where ‘U’ and ‘L’ denote that the load takes upper- and lower-bound values, respectively. For example, (U,L,U) corresponds to the vertex defined by (P1 , P2 , P3 ) = (P1U , P2L , P3U ). The underline indicates that the stress is equal to the lower or upper bound; i.e., the member is fully stressed. As is seen, each member is fully stressed for at least one loading condition, except members 8 and 9. Note that the cross-sectional area of member 8 is equal to the lower bound and the stress constraint is satisfied with inequality. Since member 9 has a larger range of allowable stress, the stress is not equal to either the upper or lower bound; i.e., the optimal solution under multiple loading conditions is not fully stressed when the load uncertainty is considered and the bounds of the stresses are not given uniformly.

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Table 9.2 Optimal cross-sectional areas and the structural volume of the optimal solution (uncertain loads). Member number 1 2 3 4 5 6 7 8 9 10 Total volume (mm3 )

Table 9.3

Ai (mm2 ) Case 1

Case 2

6.0932 1.0000 8.7550 18.2481 4.1565 1.0000 14.8966 1.0000 3.3680 1.4142

14.3939 1.0000 10.4616 17.6119 4.2834 1.0000 11.9314 7.0584 12.7440 1.0000

1.3699 × 105

1.9009 × 105

Stresses at vertices of the optimal solution (Case 1).

Member number (U,U,U) (U,U,L) (U,L,U) (U,L,L) (L,U,U) (L,U,L) (L,L,U) 1 2 3 4 5 6 7 8 9 10

9.7 9.7.1

(L,L,L)

0.1958 0.1589 0.2000 0.1631 0.1930 0.1561 0.1972 0.1602 0.1570 0.1919 0.1488 0.1992 0.1562 0.1927 0.1496 0.2000 0.1363 0.1791 0.0478 0.0906 0.1572 0.2000 0.0687 0.1115 0.1479 0.1565 0.1914 0.2000 0.1480 0.1566 0.1914 0.2000 −0.1960 −0.1643 −0.1882 −0.1564 −0.2000 −0.1683 −0.1921 −0.1604 0.1919 0.1488 0.1992 0.1562 0.1927 0.1496 0.2000 0.1570 0.1834 0.1976 0.1810 0.1827 0.1660 0.1802 0.1636 0.2000 −0.1319 −0.0965 −0.1679 −0.1324 −0.1076 −0.0721 −0.1434 −0.1080 0.3813 0.3154 0.3783 0.3123 0.3810 0.3151 0.3779 0.3120 −0.1918 −0.1488 −0.1992 −0.1562 −0.1926 −0.1496 −0.2000 −0.1569

Compliance Constraints Introduction

It is well known that the interval analysis for anti-optimization of structures under bounded uncertainty leads to a too conservative result if the dependency among the uncertain variables are not appropriately incorporated; see Sec. 3.3 for details. One strategy to avoid this unrealistic situation is to reduce the number of independent uncertain parameters using Fourier transformation or modal decomposition where the bounds are to be given for the coefficients of the modes, or simply by linking the variables (Elishakoff, 1999). However, in these approaches, explicit bounds cannot be assigned for each variable. Therefore, bounds should be given as additional linear constraints, because the coefficient itself has no physical meaning related to uncertainty in external loads, material parameters, geometrical properties, and so on.

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Consider, for example, a finite-dimensional elastic structure subjected to earthquake loads. The standard approach to reduce the computational cost of the evaluation of the maximum elastic response may be the response spectrum approach (Clough and Penzien, 1975), which utilizes modal decomposition of the responses. After computing the maximum response of the each dominant mode, the maximum responses of structure can be found by the square-root-of-sum-of-squares (SRSS) method or complete quadratic combination (CQC) method (Wilson, Der Kiureghian and Bayo, 1982). However, in the practical situation of civil engineering, dynamic analysis is not carried out for the design of low-rise building frames, and the equivalent static loads are usually used for evaluating the maximum elastic and elastoplastic responses by the so-called static pushover analysis; see, e.g., FEMA (2000); Han and Chopra (2006). Therefore, in this section, an example is shown for static anti-optimization with additional linear constraints on uncertain parameters. Let U (t) denote the time-dependent vector of nodal displacements of a finitedimensional structure, which is decomposed to modal responses as U (t) =

n X

ci (t)Φi

(9.34)

i=1

where Φi is the ith eigenmode of undamped free vibration, ci (t) is the modal response that is a function of time, and n is the number of degrees of freedom. The equivalent static load P i corresponding to Φi is defined as P i = Ω i M Φi

(9.35)

where M is the mass matrix, and Ωi is the ith eigenvalue of undamped free vibration. Suppose a single mode, say mode i, dominates in the response. For example, for a building frame, the response can be evaluated with good accuracy by using the first mode only. In this case, the maximum response U max corresponding to mode i i is computed from KU max = cmax Pi i i

(9.36)

where K is the stiffness matrix, and cmax is the maximum modal response obtained i from the displacement response spectrum, which is multiplied by the participation factor of mode i (see Sec. 6.6 for examples of response spectra). Let R denote the representative response, such as stress or strain of the specified member or element, which is a linear function of U , as R = b> U

(9.37)

where b is a constant vector. The maximum absolute value of the response corresponding to mode i against the equivalent static load cmax P i for seismic design, i called design load for simplicity, is obtained as Rimax = |b> U max | i

(9.38)

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Even when a few modes dominate in the response of a structure, only those dominant modes can be used for constructing the design loads. Suppose we use the three modes 1, 2 and 3, for simplicity, and assume the response R is positive for the maximum responses of all the modes. The worst possible static load pattern is then given as P worst = cmax P 1 + cmax P 2 + cmax P3 1 2 3

(9.39)

which corresponds to the absolute sum approach of the modal combination. However, in Eq. (9.39), all the three modal responses are assumed to take their maximum simultaneously at a certain time instance during vibration. Therefore, Eq. (9.39) overestimates the design load and the corresponding responses. The standard approach to avoid this overestimation is to use the SRSS combination. For example, the maximum displacements are estimated as q max )2 + (U max )2 + (U max )2 , (j = 1, . . . , n) (9.40) UjSRSS = (U1,j 2,j 3,j max where Ui,j is the jth component of U max . However, we can assign more explicit i constraints on cmax in Eq. (9.39) to give a set of design loads that consider unceri tainty in the combination of the coefficients; e.g.,

cL ≤ cmax + cmax + cmax ≤ cU 1 2 3

(9.41)

where cL and cU are the specified lower and upper bounds, respectively. More rigorous constraints can be incorporated based on the statistical data of maximum responses of structures to a set of recorded seismic motions. Zhang, Ohsaki and Uchida (2008) presented an approach to generating several design loads as combinations of multiple modes. The design load vector is alternatively defined by using only the most dominant mode, say Φ1 , and adding the uncertain load deviation vector P dev for representing the contribution of the higher modes: P = cmax P 1 + P dev , 1 dev dev Pj,L ≤ Pjdev ≤ Pj,U , (j = 1, . . . , n)

(9.42)

dev dev where Pj,L and Pj,U are the specified lower and upper bounds for Pjdev , respectively. However, also for this case, it is unrealistic that all Pjdev will take their maximum or minimum at the same time instance during vibration. Therefore, we need to incorporate additional constraints on Pjdev . In the following, we present a general procedure for anti-optimization of static responses with additional linear constraints. As an example, anti-optimization problem of a truss is formulated for maximizing the compliance (external work) under uncertain static nodal loads. The optimum design problem under compliance constraint is solved by a simple heuristic approach called the greedy method.

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Optimization problem and optimization algorithm

Let A = (A1 , . . . , Am )> and L = (L1 , . . . , Lm )> denote the vectors of crosssectional areas and the lengths of members of a truss. The objective function to be minimized is the total structural volume defined as V (A) = A> L

(9.43)

The truss is subjected to static nodal loads P , for which the nodal displacement vector U is found from U = CP

(9.44)

where C is the flexibility matrix; i.e., C = K −1 with K being the stiffness matrix. A constraint is given for the compliance W , defined by W (P , A) = U > P = P > CP

(9.45)

which is a quadratic convex function of P , because C is positive definite for a stable structure. The compliance is equivalent to the external work (twice of the strain energy), and smaller compliance under specified loads leads to a stiffer structure. For simplicity, the nodal loads are conceived as uncertain variables; i.e., let a = P and N = n in the general formulation in Sec. 9.3. The anti-optimization problem (lower-level problem) for finding the worst load scenario for maximizing the compliance is formulated as find

c (A) = max W (P , A) = P > C(A)P W

(9.46b)

PiL

(9.46c)

P

s subject to b> i P ≤ di , (i = 1, . . . , N )

≤ Pi ≤

PiU ,

(i = 1, . . . , n)

(9.46a)

where bi is a constant vector, di is a constant, and N s is the number of additional linear constraints. Note that C(A) depends on A which is the design variable of the upper-level problem. Since C(A) is positive definite, Problem (9.46) is a maximization problem of a convex function over a convex feasible region. Therefore, the anti-optimal solution exists at a vertex of the feasible region. The optimum design problem (upper-level problem) under constraint on compliance corresponding to the worst load scenario is formulated as minimize

V (A) = A> L

c (A) ≤ W U subject to W ALi

≤ Ai ≤

AU i ,

(9.47a) (9.47b) (i = 1, . . . , m)

(9.47c)

where W U is the specified upper bound for W , and ALi and AU i are the lower and upper bounds for Ai . Since the constraint function of (9.47b) is not generally continuously differentiable with respect to A, a simple heuristic approach called the greedy method is

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used for design optimization (see Sec. 2.6 for details of heuristic approaches). The design variable Ai is discretized as Ai ∈ {j∆A | j = 1, . . . , N D }

(9.48) D

where ∆A is the specified unit value of the cross-sectional area, and N is the number of possible values of each cross-sectional area. The algorithm is summarized as follows: (0)

Step 1 Assign initial value Aj for Aj (j = 1, . . . , m), and set the iteration counter k = 0. Step 2 Find the anti-optimal solution P (k) and the corresponding worst value c (A(k) ) of W , for A fixed at A(k) , by enumerating the values of W at all the W vertices of the feasible region defined by Eqs. (9.46b) and (9.46c). c > W U , increase Aj (j ∈ {1, . . . , m}) by ∆A for the member which Step 3 If W c to the satisfies Aj < N D ∆A and maximizes the ratio of the decrease of W U c increase ∆ALj of the structural volume. If W ≤ W , decrease Aj by ∆A for a member which satisfies Aj > ∆A and minimizes the ratio of the increase of c to the decrease of the total structural volume. W Step 4 Set k ← k + 1 and go to Step 3, if the termination condition is not satisfied. Step 5 Output the best solution satisfying the constraints, and terminate the process. 9.7.3

Numerical examples

Consider a 15-bar arch-type truss as shown in Fig. 9.11 subjected to static nodal loads. The nodal coordinates are listed in Table 9.4. Young’s modulus is 2 200.0 kN/mm . The units of length and force are mm and kN, respectively, which are omitted for simplicity in the following. The vertical loads at nodes 1, 2 and 3 are denoted by P1 , P2 and P3 , respectively, which are uncertain variables. Uncertainty is directly introduced to the loads, instead of using the nominal values and uncertain parameters also in this section, for simple presentation of problem formulation and numerical results. Linear constraints are given for P1 and P3 as −D ≤ P1 − P3 ≤ D

(9.49)

The upper bound D characterizes the asymmetry of the loads; i.e., the loads are symmetric if D = 0. Let P denote the vector of nonzero components of the loads as P = (P1 , P2 , P3 )> . The vertical displacements U1 , U2 and U3 at nodes 1, 2 and 3 are the linear functions of P . The anti-optimization problem is formulated as c (A) = max W (P , A) find W (9.50a) P

subject to − D ≤ P1 − P3 ≤ D PiL

≤ Pi ≤

PiU ,

(i = 1, 2, 3)

(9.50b) (9.50c)

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5

(6)

6

(7)

(5)

y (9) (8) 8

(11)

(10)

4

(1)

(12) 2

(2)

1

(3)

P2

7

(13) (14)

3

(15) P3 (4)

P1

9

x Fig. 9.11

A 15-bar arch-type truss.

Table 9.4 Nodal coordinates (mm) of the arch-type truss. Node number

x

y

1 2 3 4 5 6 7 8 9

2000.0 4000.0 6000.0 1000.0 3000.0 5000.0 7000.0 0.0 8000.0

1500.0 2000.0 1500.0 1750.0 2750.0 2750.0 1750.0 0.0 0.0

which is a maximization problem of a quadratic convex function over a convex region. Therefore, the anti-optimal solution exists at a vertex of the feasible region. First, we investigate the properties of the anti-optimization problem by fixing the cross-sectional areas of all members at 100.0 mm2 . The values of PiL and PiU , respectively, are −10.0 and 10.0 (kN) for i = 1, 2, 3, which lead to D ≤ 20. Therefore, for D < 20, we have 12 vertices (P1 , P2 , P3 ) = (10, 10, 10), (10−D, 10, 10), (10, 10, 10−D), (−10, 10, −10), (−10+D, 10, −10), (−10, 10, −10+ D), (−10, −10, −10), (−10 + D, −10, −10), (−10, −10, −10 + D), (10, −10, 10), (10 − D, −10, 10), (10, −10, 10 − D). The following properties hold between the compliance W (P , A) and nodal loads P for fixed cross-sectional areas: P1: W (P , A) takes the maximum value at a vertex of the feasible region of P defined by the constraints (9.50b) and (9.50c). P2: W (P , A) does not change by the simultaneous reversal of the signs of the loads as P → −P . P3: W (P , A) monotonically increases if the absolute values of all the components of P are monotonically and simultaneously increased.

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Table 9.5 Values of compliance W at the vertices of the feasible region, as well as the worst values and the corresponding worst load patterns, for various values of D. (P1 , P2 , P3 )

D=6

D = 12

D = 18

(10, 10, 10) (10 − D, 10, 10) (−10, 10, −10 + D) (−10, 10, −10)

123.67 96.270 56.704 67.150

123.67 102.31 79.699 67.150

123.67 141.79 136.14 67.150

Worst value

123.67 (10, 10, 10)

123.67 (10, 10, 10)

141.79 (2, 10, 10)

140

Compliance

130 120 110 100 P1* 90 -6

-4

-2

0

2

4

6

8

10

P1 Fig. 9.12

Variation of compliance against P1 for P2 = P3 = 10.0.

P4: From the symmetry of the truss, W (P , A) does not change by exchanging the values of P1 and P3 . From the properties P1 and P2, we can fix P2 at 10.0, because Eq. (9.50b) does not include P2 . From P4, we can assume that P1 ≤ P3 . Then utilizing the convexity of W with respect to P , the anti-optimal solutions are found by enumerating the vertices (P1 , P2 , P3 ) = (10, 10, 10), (10−D, 10, 10), (−10, 10, −10+ D) and (−10, 10, −10). Thus the number of vertices for which the responses are to be evaluated is reduced from 12 to 4. The values of W at the vertices corresponding to various values of D are listed in Table 9.5, along with the worst value among the four vertices. Variation of the compliance W is plotted in Fig. 9.12, for example, with respect to P1 for P2 = P3 = 10.0. The value of W for P1 = 10 is 123.67, and W = 123.67 is satisfied at P1 = P1∗ ' −5.833 as indicated in Fig. 9.12. Since P3 = 10.0, the lower bound of P1 is 10.0 − D from Eq. (9.50b). Hence, the anti-optimal value (worst load scenario) of P1 corresponding to the maximum value of W is 10.0 for D < 15.833,

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Fig. 9.13

Optimal cross-sectional areas of the arch-type truss.

and is 10.0 − D for D ≥ 15.833. We next find the optimal solution under constraints on the compliance corresponding to the worst load scenario for D = 10. Let A1 , A2 and A3 denote the cross-sectional areas of the lower chords, upper chords, and diagonals, respectively. The design variable Ai is discretized as Ai = j∆A, (i = 1, 2, 3; j = 1, . . . , 20)

(9.51)

where ∆A = 10. The worst load scenario is searched for at the vertices (P1 , P2 , P3 ) = (10, 10, 10), (0, 10, 10), (−10, 10, 0), and (−10, 10, −10) for 20 3 = 8000 designs to find the exact optimal solution (A1 , A2 , A3 ) = (170, 90, 60), where c = 98.391. The optimal solution is illustrated in Fig. 9.13, V = 2.8411 × 106 and W where the width of each member is proportional to its cross-sectional area. Note that the lower chords have large cross-sectional areas in the optimal solution. The worst load for this optimal design is P1 = P2 = P3 = 10. Note that the number of solutions to be enumerated can be reduced by using the current minimum value V ∗ of the solution satisfying the compliance constraint. For example, in the process of increasing A1 while fixing A2 and A3 , the process is terminated if V exceeds V ∗ , and A2 is next to be increased. The optimal solution has also been found by the greedy method presented in Sec. 9.7.2. The process is terminated in 30 iterations. Among the ten solutions found from ten randomly generated initial solutions, seven solutions coincided with the optimal solution (A1 , A2 , A3 ) = (170, 90, 60) found by enumeration. Therefore, for this example, optimal solution can be easily found by the greedy method, for which the total number of design evaluation is 30 × 10 = 300, that is far smaller than the 8000 for the enumeration approach. 9.8 9.8.1

Homology Design Introduction

A great deal of attention is paid to structure/control-integrated optimum design for advanced structures, such as space structures, adaptive structures and smart structures, which are required to fulfil complicated missions at high quality with

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Advanced structures Structure

High quality of performance Mechanism

Fig. 9.14

Control

Concept of advanced structure.

the aid of active control systems. The concept of advanced structures is illustrated in Fig. 9.14 (Kida and Komatsu, 1993; Thomas, Sepulveda and Schmit, 1992). The precise control on geometrical properties of these structures is stringently demanded for high quality performance, and a sophisticated structural design methodology considering the interaction with an active control system and mechanism should be devised. Homology design can be a candidate for such structural optimization of advanced structures, utilizing the concept of homologous deformation, in which a prescribed geometrical property at a part of the structure is maintained before, during, and after the deformation. The resolution of a huge and precise radio telescope was greatly enhanced by realizing homologous deformation, which kept, the shape of the reflector structure parabolic in spite of the deformation caused by its own weight (Morimoto, Kaifu, Takizawa, Aoki and Sakakibara, 1982; von Hoerner, 1967). The control cost of the active system to adjust the shape of the parabolic reflector is greatly reduced and the resolution quality is ensured easily. Two formulations based on the finite element method have been derived for homology design in static problems (Hangai, 1990; Yoshikawa and Nakagiri, 1990). However, the formulations were given in a purely deterministic manner, where the external loadings must be specified. The advanced structures are to be made very flexible owing to the lightweight requirement. Thus, the homologous deformation under specified loading appears to be easily disturbed by the uncertain fluctuation of loading. The sensitivity of homologous deformation with respect to such uncertainties should be estimated prior to the application of homology design. In this section, we closely follow the results of Yoshikawa, Elishakoff and Nakagiri (1998a). The formulation of homology design with an illustrative numerical example of an 11-bar truss is given, using finite element sensitivity analysis, prior to the discussion on uncertain loading. The uppermost deck subjected to uniformly distributed vertical load as nominal loading is kept both straight and horizontal, before and after the deformation by homology design. The methodology for the estimation of the disturbed homologous deformation is developed based on the convex model of uncertain loadings (Ben-Haim and Elishakoff, 1990; Elishakoff, Lin and Zhu, 1994e), in which uncertain variables are assigned to discretized nodal forces and the existence domain of the uncertain variables is confined within the convex

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hull of hyperellipse. The worst case of homology design is estimated on the boundary of the convex hull as the point that maximizes the error index. The latter is defined as the square of the Euclidean norm of displacements error from objective homologous deformation. The validity of the proposed method for the worst-case estimation of the homology design is demonstrated in a numerical example of the 11-bar truss after homology design. 9.8.2

Deterministic loading

Consider the static deformation of the linear and elastic structure discretized by the finite elements. The formulation for homology design under deterministic loading is summarized first. The structural response is governed by the following stiffness equation in matrix form after incorporation of the geometrical boundary condition: KU = P

(9.52)

where K is the n × n stiffness matrix, U is the unknown displacement vector, P is the external nodal load vector, and the number of degrees of freedom of the discretized structure is denoted by n. In the following, a simple iterative approach is presented without resort to a sophisticated nonlinear programming algorithm. For this, we need a trial design to satisfy the constraint of homologous deformation H(U ) = 0

(9.53)

which consists of J equations with respect to the nodal displacements. The way of representation is not unique even for the same homologous deformation. The success of the objective homology design greatly depends on the manner of determining an adequate trial design and neat representation of the homologous constraint. For the sake of simplicity of discussion, we treat only linear homologous deformation, the constraint for which is given by linear equations as follows with respect to the nodal displacements: CU = d

(9.54)

where C is the constant J ×n matrix, and d is the constant vector of J components. If we find an adequate trial design a priori, the nodal displacements obtained by solving the stiffness equation (9.52) satisfy the homologous constraint in (9.54) and there is no need to change the trial design. In general, however, it is difficult to find the transparent homology design. Therefore, we are interested in changing the trial design such that the objective homologous deformation is achieved under specific loading. For the design change, we judiciously select structural parameters p1 , . . . , pm and assign the design variables α1 , . . . , αm in the form of pi = pi (1 + αi ), (i = 1, . . . , m)

(9.55)

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The effect of the design change is linearly approximated and the change of the nodal displacement vector is expressed in the following linear form: U =U+

m X

Uα i αi

(9.56)

i=1

where U is the displacement vector of the current design, and U α i is the first-order displacement sensitivity vector with respect to αi , obtained by finite element sensitivity analysis (see Sec. 3.6.2 for details of static sensitivity analysis). Substituting the linearly approximated nodal displacement vector of Eq. (9.56) into the homologous constraint in Eq. (9.54), we obtain the governing equation of design variables in the form m X (9.57) CU α i αi = d − CU i=1

which can be rewritten in a matrix form as

Gα = b

(9.58) >

where G is a J × m rectangular matrix, α = (α1 , . . . , αm ) is the design variable vector to be determined, and b = d − CU is a constant vector. If m is larger than J, the governing equation (9.58) has solutions, which in general, are not unique. On the other hand, if J is larger than m, Eq. (9.58) scarcely has any solutions. The prediction about the existence of solutions based on the size of the coefficient matrix G mentioned above suggests that the increase of the number of design variables enlarges the possibility of success in homology design. The solution to Eq. (9.58) is efficiently handled with the Moore–Penrose generalized inverse (Rao and Mitra, 1971). The general solution of Eq. (9.58) is given as α = G− b + (I − G− G)h

(9.59)

when the condition of solution existence given by (I − G− G)b = 0

(9.60)

is satisfied, where G− is the Moore–Penrose generalized inverse of G, I is the J × J identity matrix, and h is an arbitrary vector. The first and second terms of the right-hand-side of Eq. (9.59) are called particular and complementary solutions, respectively. Equation (9.59) indicates that the way of changing the current trial design to realize objective homologous deformation in general is not unique, where arbitrariness creeps into the general solution of the design variable vector through the arbitrary vector h. We employ only the particular solution to determine the design variable vector for design change. The solution thus determined is the solution of least design change, because the Euclidean norm of design variable vector α becomes a minimum owing to the property of the Moore–Penrose generalized inverse.

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1

2

3

4 1500 mm

6 5

3000 mm

Fig. 9.15

7

An 11-bar truss structure.

0.1 mm

Fig. 9.16

Deformed shape of the initial trial design.

The objective homologous deformation is not realized strictly by the above mentioned design modification, since the governing equation (9.58) is derived based on the linear approximation of nodal displacements change. The deficient approximation can be compensated by the iterative update of the design variables, in which the updated structure according to the solution of Eq. (9.58) is used as the trial design for the next step. The iterative process is stopped when the homologous constraint (9.54) is regarded to be satisfied, that is, the Euclidean norm of CU − d becomes sufficiently small. 9.8.3

A numerical example

An instructive example of homology design is demonstrated by using the 11-bar truss illustrated in Fig. 9.15. The cross-sectional area of all the members is 100 mm 2 in the initial trial design, and Young’s modulus is 70.0 GPa. The structure is subjected to vertical nodal loads that are consistent with a uniformly distributed vertical load of 1.0 N/mm as nominal loading on the uppermost deck. The deformed structure of the initial trial design is depicted in Fig. 9.16. The measure for displacement is put at the upper right of Fig. 9.16. The maximum vertical displacement turns out to be 0.133 mm in the initial trial design. The objective homologous deformation is set so as to keep the uppermost deck straight and horizontal before and after deformation. Hence, the homologous constraint equation is given as     v1 − v m 0 v2 − vm  0     (9.61) v3 − vm  = 0 v4 − v m

0

optimization

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100 48.5

327

156

104 143

optimization

303

100 104 326 142

48.5

unit: mm2

Fig. 9.17

Realized homologous deformation and the final cross-sectional areas.

where vi is the vertical nodal displacement of node i, and vm is the average of vertical nodal displacements on the uppermost deck, that is, vm = (v1 + v2 + v3 + v4 )/4. The realized homologous deformation by changing the cross-sectional areas of all the members is illustrated in Fig. 9.17. The vertical displacement of the uppermost deck turns out to be 0.0516 mm. The manner of illustration of the deformation is the same as that of Fig. 9.16. In Fig. 9.17, the cross-sectional area of the member after homology design is indicated by the number attached to each member. The increase in total weight is 58.7% in this example. Three update steps are needed for the design variables to ensure a sufficiently small Euclidean norm of the vector on the left-hand-side of Eq. (9.61); i.e., less than 0.001 mm in this case. One of the transparent homology designs for this example is constructed so as to make the cross-sectional area of vertical and outer horizontal members extremely large. This method inevitably results in an extreme increase of the total weight. 9.8.4

Convex model of uncertain loading

The legitimate question arises of how sensitive the homology design is to uncertain fluctuation of the loadings. The behavior of the structure is described by the following stiffness equation in the state of the homology design: KHU = P 0

(9.62)

where K H corresponds to the stiffness matrix of the structure experiencing homologous deformation at the specified nominal loading P 0 . The fluctuation of loading is represented by introducing n uncertain variables ai assigned to n nodal force components as Pi = Pi0 (1 + ai ), (i = 1, . . . , n)

(9.63)

The amplitude of variation (P10 a1 , . . . , Pn0 an )> from the nominal loading is assumed to be unknown-but-bounded by a convex hull of a hyperellipse with respect to uncertain variables given as q 2 a> W a ≤ 1

(9.64)

where a = (a1 , . . . , an )> , and W is a symmetric and positive definite matrix that defines the shape of the convex hull. The magnification coefficient q governs the size of the convex hull. W and q are determined from the available information.

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9.8.5

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Worst-case estimation

In the context of convex analysis, the problem is posed as follows: • Find the most harmful (worst-case) amplitude of variation to maximize the distortion from objective homologous deformation. The fluctuation of nodal displacement vector is expressed in the following linear approximation form as the result of sensitivity analysis with respect to uncertain variables for the structure governed by the stiffness equation (9.62): n X U = U0 + U ai ai (9.65) i=1

0

where U is the displacement under nominal loading after homology design, and U ai is the sensitivity vector of U with respect to ai . Obviously the homologous constraint in Eq. (9.54) cannot be satisfied under the uncertain fluctuation of loading. The error in Eq. (9.54) after substitution of the fluctuating nodal displacement vector expressed by Eq. (9.65) is defined by an error vector e, defined as follows, from the exact homologous deformation: e = CU − d =C

0

U +

n X

U ai ai

i=1

0

= (CU − d) + =

n X

n X

!

−d

CU ai ai

(9.66)

i=1

CU ai ai

i=1

To evaluate the magnitude of the distortion from the objective homologous deformation, we introduce the concept of error index, which can be defined by using the components of the error vector. Here, we define the error index ∆2 as the square of the Euclidean norm of the error vector: ∆2 = e> e = a> Da

(9.67)

where D is an n×n symmetric matrix defined by Eq. (9.66) using the rate of change of nodal displacements with respect to the uncertain variables. The worst case of uncertain variables, which gives the maximum value of the error index, is searched for within the convex hull of Eq. (9.64). As the error index is expressed in quadratic form with respect to the uncertain variables and becomes a convex function, the anti-optimization problem turns out to be a maximization problem of a convex function over a convex domain, and the point of the worst case is located on the boundary of the convex hull. The search is carried out by the Lagrange multiplier method employing the following Lagrangian Π: Π = ∆2 − λ(q 2 a> W a − 1)

(9.68)

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where λ ≥ 0 is the Lagrange multiplier (see Sec. 4.1 for details of the Lagrange multiplier approach). The uncertain variables in the worst case are so determined as to satisfy the stationary condition of Π derived as ∂Π = 2(D − λq 2 W )a = 0 ∂a ∂Π = 1 − q 2 a> W a = 0 ∂λ

(9.69a) (9.69b)

The stationary condition (9.69a) results in the eigenvalue problem formed by the eigenvalue λq 2 , which is rewritten as µ = λq 2 and the eigenvector a. On the other hand, the stationary condition (9.69b) gives the normalization condition for the eigenvector. Generally, n eigenpairs are obtained by solving the eigenvalue problem (9.69a) under the normalization condition (9.69b). In view of Eqs. (9.69a) and (9.69b), the error index of Eq. (9.67) becomes ∆2 = a> Da = λq 2 a> W a = λ =

µ q2

(9.70)

as the magnification coefficient q is constant and the maximum value of the error index, i.e., the worst case, is estimated by the maximum eigenvalue µ obtained by solving Eq. (9.69a). The eigenvector corresponding to the maximum eigenvalue gives the uncertain variable vector for the worst case. 9.8.6

A numerical example of worst-case estimation of homology design

The 11-bar truss obtained from the homology design in Sec. 9.8.3 is employed in this numerical example to evaluate the worst case of the distortion from the objective homologous deformation caused by uncertain fluctuation of the uniformly distributed vertical load. Four uncertain variables are assigned to vertical load components for nodes 1, 2, 3, and 4 in Fig. 9.15. The fluctuation of the uncertain variables is bounded by hypersphere with radius 1.0, which is expressed by setting q equal to 1.0 and W as an identity matrix in Eq. (9.64). The nodal load components fluctuate from 0 to twice the nominal values in this case. The result of the worst-case estimation obtained by solving the eigenvalue problem of Eq. (9.69a) with Eq. (9.69b) is shown in Table 9.6. The error indices estimated by the eigenvalue, i.e., ∆2 = µ/q 2 , are listed in the leftmost column and corresponding uncertain variables, i.e., the eigenvector components, are listed in the right columns. As is seen, the maximum eigenvalue is duplicate, and the error index is maximized by two cases of uncertain variables, namely, (a1 , a2 , a3 , a4 ) = (0.707, 0.0, 0.0, −0.707), called worst case 1, and (a1 , a2 , a3 , a4 ) = (0.5, −0.5, −0.5, 0.5), called worst case 2. The actual error indices evaluated by the analyses employing the nodal force components corresponding to the worst cases 1 and 2 turn out to be equal to 2.82 × 10−3 mm2 in both cases, because the nodal

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Estimated error index by convex analysis.

Error index ∆2 (×10−3 mm2 )

2.82 2.82 1.94 0.00

Uncertain variables a1

a2

a3

a4

0.707 0.500 0.000 0.500

0.000 −0.500 −0.707 0.500

0.000 −0.500 −0.707 0.500

−0.707 −0.500 0.000 0.500

0.1 mm

Fig. 9.18

Deformed structure in worst case 1.

0.1 mm

Fig. 9.19

Deformed structure in worst case 2.

displacements are linear functions of the nodal loads. The deformations of the 11bar truss in the two worst cases are illustrated in Figs. 9.18 and 9.19, respectively, in a similar manner as Fig. 9.16. 9.8.7

Concluding remarks

A formulation of homology design based on finite element analysis has been reviewed. The change of displacements by design variables is approximated by means of first-order finite element sensitivity analysis and the governing equation for design variables is derived in the matrix form with a rectangular coefficient matrix. The equation is solved with the Moore–Penrose generalized inverse to obtain the solution with least design change. The illustrative numerical example of the 11-bar truss demonstrates that the uppermost deck remains straight and horizontal even after the deformation caused by a uniformly distributed vertical load. The worst case of homologous deformation caused by uncertain fluctuation of loadings is estimated by means of convex analysis. The uncertain variables are assigned to the discretized nodal force components, and the fluctuation of the uncertain variables is bounded within a convex hull. The worst case maximizing the error index of homologous deformation is searched for on the boundary of the convex

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hull by the Lagrange multiplier method. The numerical example using the 11-bar truss structure shows the validity of the method by demonstrating the identification of the worst case caused by uncertain variables confined within a hypersphere. 9.9

9.9.1

Design of Flexible Structures under Constraints on Asymptotic Stability Introduction

The approaches to designing structures against external loads are classified into two categories: stiff structure and flexible structure. In the stiff structure approach, the structure is designed so as to have enough stiffness and strength against the static and dynamic loads, and the deformation under such loads should be small enough. In some situations, however, the structure may be flexible and large deformation may be allowed so far as no failure or yielding occurs and the undeformed initial state can be recovered after releasing the external loads. Contrary to mechanical parts and structures, the loading condition for structures in the field of civil engineering is highly unpredictable. Therefore, it is not practically acceptable to design such structures by considering only static loads and proportional loading conditions with fixed load patterns. The magnitude of the input energy by dynamic motions such as earthquake excitations and wind loads can be characterized and bounded by the total energy applied to the structure. Hence, the strain energy that can be stored may be used as the performance measure of elastic structures in the civil engineering field. This way, the stability of the structure under dynamic seismic or wind loads with specified energy is assured (see Sec. 6.6 for anti-optimization with energy bounds of seismic excitation). Note that the energy dissipation capacity may be an important performance measure also for the case where inelastic responses are allowed (Ohsaki, Kinoshita and Pan, 2007; Pan, Ohsaki and Tagawa, 2007). This section summarizes the optimization–anti-optimization approach developed by Ohsaki (2003a) to designing elastic flexible structures. The structure is designed allowing large deformation under the external loads with specified energy. A constraint is given for the storable strain energy under the specified norm of deformation. Furthermore, the structure should recover the undeformed state after releasing the external loads. The criterion for asymptotic stability is used for formulating the optimization problem. An illustrative example is shown using a 2-bar truss, and the performances of the optimum design is verified by carrying out dynamic analysis of a 24-bar dome-type truss.

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H(U)

Ui Fig. 9.20 A quasi-convex function H(U ) that has a stationary point at the origin U = 0; U i : a representative displacement component.

9.9.2

Definition of asymptotic stability

Consider a finite-dimensional elastic structure. The vector of nodal displacements is denoted by U = (U1 , . . . , Un )> , where n is the number of degrees of freedom. In the following, a component of a vector is indicated by a subscript. The performance of the structure is defined by the strain energy H(U ) that can be stored in the structure within the specified norm of the nodal displacements. Lagrangian formulation is used for the strain-displacement relation allowing large deformation. The nodal force vector F (U ) that is equivalent to the deformation U is defined by Fj (U ) =

∂H , (j = 1, . . . , n) ∂Uj

(9.71)

In order to ensure that the undeformed state U = 0 is recovered after releasing the external loads, the condition of asymptotic stability is to be satisfied at U = 0. There are several definitions for asymptotic stability of an elastic system. From the definition of Liapunov’s direct method (Salle and Lefschetz, 1961), if the undeformed state is the only stationary point of H(U ) in a bounded domain Ω containing the origin U = 0, then it is asymptotically stable. Hence, H(U ) should be a strictly quasi-convex function (Boyd and Vandenberghe, 2004) as shown in Fig. 9.20 that has only one stationary point U = 0 (see Ohsaki (2003a) for details). Let G(U ) be defined as G(U ) = U > ∇H(U )

(9.72)

where ∇H is the gradient of H(U ). By using Eq. (9.71), Eq. (9.72) is rewritten as G(U ) = U > F (U )

(9.73)

A strictly quasi-convex function H(U ) in a bounded domain Ω of U , which has only one stationary point at U = 0, is characterized by G(U ) > 0 for U ∈ Ω and U 6= 0

(9.74)

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309

Optimization problem

The strain energy that can be stored in the structure is used as a performance measure. Moreover, the structure should have enough flexibility. Therefore, the lower bound is given for the worst (minimum) value of the strain energy at the deformed state defined by the specified norm of displacements, and the undeformed state should be recovered as the external loads are removed from the deformed state. Let A denote the vector of design variables such as the cross-sectional areas of the members of a truss. The objective function to be minimized is denoted by V (A). The norm of the displacement vector is defined by p D(U ) = U > U (9.75)

The specified value for D(U ) is denoted by D; i.e., the feasible domain Ω for e , A) to indicate that U is the U is given by D(U ) ≤ D. A tilde is used as H(U variable vector, and A is regarded as a parameter vector, whereas a hat is used b e , A) to indicate a function of A only. The worst (minimum) value H(A) of H(U among the displacements satisfying D(U ) = D should not be less than the specified lower bound H. b Hence, H(A) is defined as the anti-optimal objective values of the following problem: find

b e , A) H(A) = min H(U U

subject to D(U ) = D

(9.76a)

(9.76b)

The undeformed initial state should be recovered by releasing the external loads at any state with U in Ω. Therefore, the condition (9.74) should be satisfied. Since a strict inequality condition cannot be given for an optimization problem, Eq. (9.73) is replaced by e , A) ≥ 0 G(U

(9.77)

Moreover, it is difficult to satisfy the equality constraint (9.76b) during the optimization process. Therefore, the equality constraint is relaxed to inequality D(U ) ≤ D. e , A) is simply searched for in a region Ω However, if the minimum value of G(U bounded by D(U ) ≤ D, an obvious and meaningless solution U = 0 will be found. e , A) of the anti-optimization problem is reTherefore, the objective function G(U e placed by G(U , A)/D(U ) to prevent convergence to U = 0. e , A)/D(U ) at the anti-optimal solution is denoted The minimum value of G(U b by G(A), which is a function of the design variable vector A. The anti-optimization problem is formulated as find

e , A) G(U b G(A) = min U D(U )

subject to D(U ) ≤ D

(9.78a) (9.78b)

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U1 h

U2 w Fig. 9.21

w A 2-bar truss.

b e , A) ≥ 0 throughout the region If G(A) ≥ 0, then D(U ) > 0 leads to G(U except U = 0, and H(U ) is a quasi-convex function that has the maximum value at D(U ) = D. Since the minimum value of H(U ) along the boundary D(U ) = D is b H(A), which is obtained by solving Problem (9.76), H(U ) is a quasi-convex function b in the region including the origin U = 0 bounded by H(U ) ≤ H(A). Therefore, b the undeformed state is recovered from any state U satisfying H(U ) ≤ H(A). Finally, the upper-level optimum design problem is formulated as minimize

V (A)

b subject to H(A) ≥H b G(A) ≥0

(9.79a) (9.79b) (9.79c)

where H is the specified lower bound of the strain energy. Suppose that the objective function V (A) is an increasing function of A, which is a natural assumption for the b case, e.g., V (A) is the total structural volume. If the strain energy H(A) is also an increasing function of A, e.g., for the case where A is the vector of cross-sectional area of a truss, the constraint (9.79b) is satisfied in equality at the optimal solution. In the following numerical examples, optimization is carried out by using a standard gradient-based nonlinear programming with line search, where the sensitivity coefficients are found by using the finite difference approach. The lower-level anti-optimization problems (9.76) and (9.78) are solved at each step of solving the upper-level optimum design problem (9.79). 9.9.4

Numerical examples

An example of a 2-bar truss is first shown to illustrate the validity of the problem formulation, and for verification of the optimization algorithm. Optimum designs are next found for a 24-bar shallow truss, and the results are verified by carrying out dynamic analysis. The Green strain is used for computing the strain energy. The objective function to be minimized in the upper-level problem is the total structural volume. Optimization is carried out by the library DOT Ver. 5.0 (VR&D, 1999), where sequential linear programming is used.

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311

Optimization results of the 2-bar truss. 100.0

200.0 105

300.0

3.8133 × 182.63

105

1.4896 × 71.338

1.1770 × 105 56.366

H(U I ) D(U I ) D(U II ) b G(A)

2.0060 × 107 100.00 99.700 3.2163 × 105

2.0060 × 107 200.00 200.00 5.0150 × 104

2.0060 × 107 300.00 300.00 0.0

U1I U2I

0.0 100.00

0.0 200.00

0.0 300.00

U1II U2II

0.0 99.700

0.0 200.00

0.0 300.00

V (A) A

optimization

9.9.4.1 A 2-bar truss Consider a 2-bar truss as shown in Fig. 9.21, where w = 1000 mm and h = 300 mm. The specified strain energy is H = 2.0 × 107 N · mm. In the following, the units of force and length are N and mm, respectively, which are omitted for brevity. Assuming the symmetry of the truss, the cross-sectional areas of the two members have the same value; i.e., A1 = A2 = A, and the number of design variables is 1. The optimization results for D = 100, 200 and 300 are as shown in Table 9.7. Note b that the constraint H(A) ≥ H is satisfied with equality for all the cases within the specified tolerance. Let U I and U II denote the anti-optimal solutions of lower-level problems (9.76) and (9.78), respectively. Displacement components U1 and U2 are defined as shown in Fig. 9.21; i.e., U I = (U1I , U2I )> , U II = (U1II , U2II )> . Since h < w, U1I = 0 should be obviously satisfied to minimize the strain energy for the specified displacement norm. Consequently, |U2I | = D is obtained from the constraint D(U ) = D, which agrees with the results in Table 9.7. The inequality constraint D(U ) ≤ D of Problem (9.78) is satisfied with equality for all the cases. b The minimum value G(A) of Problem (9.78) is positive for D = 100 and 200. Therefore, the undeformed state is asymptotically stable in the region defined by b = 0, i.e., the upper bound D of the displacement norm. For D = 300 (= h), G(A) e G(U , A) = 0, is satisfied at (U1 , U2 ) = (0, 300), because the two bars are colinear at (U1 , U2 ) = (0, 300), and a statical equilibrium state is attained without external load. If U2 > 300, the equilibrium point (U1 , U2 ) = (0, 600) without external loads will be reached as the external load is released. Therefore, D = 300 is the upper bound of D for the undeformed state to be asymptotically stable. 9.9.4.2 A 24-bar truss Consider next a 24-bar truss as shown in Fig. 9.22, where the nodal coordinates are listed in Table 9.8. The numbers with and without the parentheses in Fig. 9.22 are

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y 13 (23) 9

(20)

(13)

(12) 7

(6)

(7) (3)

(18)

(17) (16)

6

11 (22)

(21)

(15)

(14)

2

(24)

10

3

(19)

(10) 4 (5)

(4) (2)

(1)

x

8

(9)

(8)

12

(11) 5

1

z x Fig. 9.22

Table 9.8 truss.

A 24-bar truss.

Nodal coordinates (mm) of the 24-bar

Node Number

x

y

z

1 2 3 4 5 6 7 8 9 10 11 12 13

0.0 −4330.0 −1250.0 1250.0 4330.0 −2500.0 0.0 2500.0 −4330.0 −1250.0 1250.0 4330.0 0.0

−5000.0 −2500.0 −2165.0 −2165.0 −2500.0 0.0 0.0 0.0 2500.0 2165.0 2165.0 2500.0 5000.0

0.0 0.0 621.6 621.6 0.0 621.6 821.6 621.6 0.0 621.6 621.6 0.0 0.0

the member numbers and node numbers, respectively. The specified strain energy is H = 1.0 × 106 . The members are divided into three groups 1, 2 and 3, each of which has the members {8, 9, 12, 13, 16, 17}, {4, 7, 10, 15, 18, 21}, {1, 2, 3, 5, 6, 11, 14, 19, 20, 22, 23, 24}, respectively, where the cross-sectional areas of the members in group i have the same value Agi . Therefore, the number of design variables is three. Optimal solutions are found for various values of D. The optimization results

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Optimization results of the 24-bar truss. 50.0 3.5598 ×

100.0 107

1.0905 ×

180.0 107

5.2154 × 106

Ag1 Ag2 Ag3

795.19 370.69 476.60

161.62 168.13 156.93

110.25 165.88 281.74

H(U I ) D(U I ) D(U II ) b G(A)

1.0000 × 106 49.978 50.040 5.6848 × 104

1.0000 × 106 99.956 45.069 2.4705 × 104

1.0000 × 106 179.92 179.84 13.121

Fig. 9.23 Displacement U I that has the minimum value of strain energy for specified displacement norm D = 180.0; solid line: deformed shape; dotted line: undeformed shape.

b are listed in Table 9.9. For D = 200.0, a solution satisfying G(A) ≥ 0 could not be found; i.e., the undeformed state cannot be recovered from the worst state corresponding to D = 200.0. The constraints on strain energy and the displacement b norm are satisfied in equality for three cases with D = 50.0, 100.0 and 180.0. G(A) has the minimum value on the boundary D(U ) = D for D = 50 and 180. For b D = 100, however, G(A) has the minimum at the point inside the region. The total structural volume decreases as D is increased, which agrees with the intuitive observation that the amount of material may be reduced by allowing large deformation. The cross-sectional areas of the members in group 1 that connect to the center node have relatively large values if D is small. On the contrary, the cross-sectional areas of the members in group 3 that connect to the supports have large values if D is large. The displacement U I at the anti-optimal solution of Problem (9.76) for D = 180.0 is shown in solid lines in Fig. 9.23 in real scale, where the dotted lines indicate the undeformed shape. Note that U II at the anti-optimal solution of Problem (9.78) has a similar shape as U I . It is observed from Fig. 9.23 that the vertical displacement of the center node is very large with deformation of the members in group 1 only in this worst-case deformation. Asymptotic stability of the undeformed state may be verified by one of the following two approaches:

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12000000

Strain energy

1000000 800000 600000 400000 200000 0 0

1

2

3

4

Time Fig. 9.24

Time histories of strain energy from various initial conditions.

Norm of displacement

120 100 80 60 40 20 0 0

1

2

3

4

Time Fig. 9.25

Time histories of norm of displacement vector from various initial conditions.

(i) Randomly assign an initial static equilibrium state with specified strain energy, and carry out transient response analysis. (ii) Assign initial velocities at the nodes so that the initial kinetic energy is equivalent to the specified strain energy, and carry out transient response analysis. Assuming that the target application of the proposed method is seismic design of a latticed dome as shown in Fig. 9.25, the second approach is used for verification purposes; i.e., the specified strain energy represents the maximum energy supplied by the seismic motion. Dynamic analysis for specified initial velocity has been carried out by using MSC.Nastran Ver. 70.5 (Nastran, 2001). The solution type of Nastran is SOL129, which is a transient response analysis with geometrical nonlinearity. The time

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increment for integration with quasi-Newton iteration is 0.02 sec. Since the dome structures in civil engineering usually have large masses attached to the nodes, the member mass is neglected in the following analysis. Initial velocities are given randomly only in z-direction of the internal nodes. A random number ri is generated for each node, and let ri∗ =

ri

r

, rsum = sum

N X

rj

(9.80)

j=1

where N is the number of internal nodes, which is equal to 7 for this example. Let mi denote the mass at the ith internal node. Then the initial velocity vi is calculated from s N X 2H ∗ , M= (mj rj∗ ) (9.81) vi = r i M j=1 In the following, mi = 100.0 kg for all the internal nodes. Damping has not been considered in the problem formulation, in order to investigate the maximum values of strain energy and displacement norm that can be reached. Obviously, dynamic response does not converge to the undeformed state if no damping is included. It is very difficult, however, to estimate the correct damping matrix of the latticed dome, and also it is important to investigate the maximum response that can be reached without damping, because damping of structures in civil engineering is usually very small, and the real response with damping is bounded by the response without damping. Suppose mi for all the nodes are simultaneously scaled by α (> 1). Then M √ becomes αM , vi is replaced by vi / α, and the rth natural circular frequency ωr of √ the truss is also scaled to ωr / α, without changing the maximum strain energy, if the damping is negligibly small. Hence, the values of mi are not important provided that the ratios among mi are fixed. Figures 9.24 and 9.25 show the time histories of strain energy and the norm of displacement vector, respectively, through the vibration process of the optimal solution for D = 100.0. The specified strain energy 1.0 × 106 can be reached if the mode of vibration defined by the distribution of the initial nodal velocities is close to one of the eigenmodes of free vibration, and velocities of all the nodes vanish simultaneously at the state corresponding to the maximum strain. However, the situation stated above cannot be realized as a result of randomly generated initial excitation; hence, it is verified from Fig. 9.24 that the maximum value of the strain energy is a little below the specified value. The maximum value of the displacement norm for each case is far less than the specified value 100.0 as observed in Fig. 9.25, because the displacements increase rapidly with a small increment of strain energy near the final deformed shape of a flexible structure, and the displacements are more sensitive than the strain energy against the variation of initial velocity.

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Force identification of prestressed structures Introduction

Assessment and maintenance of member forces are of great importance for prestressed structures such as cable nets and tensegrity structures, which take advantage of prestress to enhance their stiffness. Therefore, the distribution of forces should be precisely identified and carefully adjusted in the construction process. The member forces should also be monitored and maintained after construction for the process of life-cycle management (see Sec. 4.6 for anti-optimization in force design of tensegrity structures). In this section, we present the methodology for the force identification of prestressed pin-jointed structures based on measurements of the member forces and nodal locations (Zhang, Ohsaki and Araki, 2004). An optimization problem is formulated for determination of the optimal placement of the measurement devices for the member forces, and a heuristic approach is presented for this combinatorial optimization problem. Because of the high ratio of stiffness to structural weight, prestressed structures are considered to be the ideal structural form for lightweight structures, and they attract much attention in theoretical studies as well as practical applications. We mainly consider the prestressed structures, of which members are pin-jointed. Many novel structural forms, such as cable nets, tensegrity structures, and tensile membrane structures modeled by cable nets, fall into this category (Kanno and Ohsaki, 2005; Ohsaki and Zhang, 2006). Since the prestressed pin-jointed structures are usually stabilized by the prestresses in the state of self-equilibrium, their configurations and member forces are highly interdependent (Kanno, Ohsaki and Ito, 2002; Vassart and Motro, 1999; Zhang and Ohsaki, 2006). Furthermore, the distribution of member forces is of great importance to the stiffness of the structures, because it has significant influence on the geometrical stiffness. In the design problem of this kind of structure, member forces are usually properly assigned to satisfy the self-equilibrium equations, and more importantly, to have positive effect on its stiffness so as to have higher capability of resisting external loads. However, it is very difficult to achieve the member forces as expected in practical applications, mainly due to (a) fabrication errors, (b) installation errors, (c) short and long term creep of materials, and (d) some other unexpected environmental influences such as temperature change, deformation of base and supporting structures, etc. Their capability to resist external loads may be significantly reduced or even lost due to unexpected change of member forces. Moreover, some structures make use of the interdependency between configurations and member forces to actively modify their configurations in order to satisfy certain requirements; see for example the work by Shea, Fest and Smith (2002). Hence, it is extremely important to accurately identify the current distribution of member forces.

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Member forces can be obtained with high accuracy by measurement devices. There have been many studies on the optimal placement of measurement devices for conventional structures; see for example Abdullah, Richardson and Hanif (2001), Pothisiri and Hjelmstad (2002), Sanayei, Onipede and Babu (1992), and Zavoni, Iturrizaga and Esteva (1999). However, it is not practically acceptable to measure the forces of all members due to the limitation in cost and availability of measurement devices. Therefore, in this section, we present a method for optimal placement of the limited number of measurement devices to minimize the errors in force identification. 9.10.2

Equations for self-equilibrium state

The equations for the self-equilibrium state are briefly summarized below for a prestressed pin-jointed structure (see Sec. 4.6.2 for the details of derivations of the equations). It is assumed that the self-weight of the members are neglected, and the external loads are applied only at the nodes. Therefore, the members of a prestressed pinjointed structure transmit only axial forces. Moreover, the cable members transmit only tensile forces, whereas the struts usually transmit only compressive forces. Pin-jointed structures consist of two types of nodes: fixed nodes (supports) and free nodes (internal unconstrained nodes). Suppose that a structure has m members, n free nodes and nf fixed nodes. Therefore, a structure in d-dimensional space (d = 2 or 3) has dn degrees of freedom. In the following, we assume a structure in three-dimensional space. The m × (n + nf ) matrix C s that has 0, 1 or −1 components represents the connectivity of the nodes and members defined by Eq. (4.113) in Sec. 4.6.2 (Kaveh, 2004). For convenience, the fixed nodes are preceded by the free nodes in the numbering sequence, so that the m × (n + nf ) connectivity matrix C s can be partitioned into two parts as C s = (C, C f )

(9.82) f

f

where the m × n matrix C and the m × n matrix C are the connectivity matrices corresponding to the free and fixed nodes, respectively. Let x, y, z and xf , y f , z f denote the nodal coordinate vectors of free and fixed nodes in x-, y- and z-directions, respectively. The coordinate difference vectors of members hx , hy and hz in x-, y- and z-directions, respectively, are calculated by hx = Cx + C f xf hy = Cy + C f y f z

(9.83)

f f

h = Cz + C z

Then the diagonal coordinate difference matrices H x , H y and H z in x-, y- and z-directions, respectively, are defined as H x = diag(hx ), H y = diag(hy ), H z = diag(hz )

(9.84)

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In the following, the components of vectors and matrices are indicated by subscripts, x e.g., as hx = (hxi ), H x = (Hij ). The length matrix L = (Lij ) is diagonal and given as q (9.85) Lii = (Hiix )2 + (Hiiy )2 + (Hiiz )2 , (i = 1, . . . , m) Let s denote the vector of member forces. In the state of self-equilibrium, the equilibrium equation of a pin-jointed structure is written as (Schek, 1974) Ds = 0

(9.86)

where the 3n × m matrix D is called equilibrium matrix, which is defined by  > x C H (9.87) D = C > H y  L−1 C>H z

The self-equilibrium equation (9.86) will be extensively used to identify the force distribution of the structure in the state of self-equilibrium, based on measurement data of the nodal coordinates and the forces of some members. It is easily observed from Eq. (9.86) that the measurement errors in nodal coordinates in D and some components of member forces in s have significant influence on the identification accuracy of the member forces that are not measured, because the self-equilibrium equation has to be exactly satisfied. Suppose that the structure consists of h independent modes of member forces that satisfy the self-equilibrium equation (9.86), where h = m − rank(D)

(9.88)

For prestressed structures, h ≥ 1 always holds so that there exist forces in the self-equilibrium state. As will be discussed in the following sections, at least hindependent components of member forces need to be measured so as to determine the forces of the remaining members based on the self-equilibrium equation. 9.10.3

Formulation of identification error

9.10.3.1

Force errors

Let s∗ = (s∗1 , . . . , s∗p )> denote the vector of the forces of p members, which are to be measured, while the vector of the forces of the remaining members, which are to be estimated, is denoted by e s = (e s1 , . . . , sem−p )> . Equation (9.86) can be rewritten as follows by classifying the member forces into the measured set s∗ and the estimated set e s: es = 0 D ∗ s∗ + De

(9.89)

e are constructed where the 3n × p matrix D ∗ and the 3n × (m − p) matrix D by assembling the columns in D that correspond to the members in s ∗ and e s, respectively.

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e is full-rank, i.e., rank(D) e = m − p, then the solution of e If D s with least square norm can be determined as e − D ∗ s∗ e s = −D

(9.90)

e has where ( · )− denotes the Moore–Penrose generalized inverse matrix. When D e e rank deficiency, i.e., rank(D) < m − p, there exist m − p − rank(D) independent modes of e s that satisfy Eq. (9.89), and Eq. (9.90) gives the solution that has the e should be full-rank to have minimum norm among the possible solutions. Hence, D accurate estimation of the member forces. Furthermore, h or more members should e so that D e can be be measured to exclude the h dependent columns in D from D full-rank. The nodal coordinates are combined as X = (x> , y > , z > )> . Suppose that there exist measurement errors in X and the member forces s∗ . Total differentiation of Eq. (9.90) with respect to X and s∗ leads to 3n X ∂D∗ i=1

∂Xi

∆Xi s∗ + D ∗ ∆s∗ +

3n X e ∂D e s=0 ∆Xi e s + D∆e ∂Xi i=1

(9.91)

where ∆Xi and ∆s∗ , respectively, are the measurement errors of nodal coordinates and member forces to be measured, and ∆e s is the resulting estimation error of e the forces of the remaining members. ∂D ∗ /∂Xi and ∂ D/∂X i can be derived by assembling the corresponding components in ∂D/∂Xi ; see details in Sec. 9.10.4. From Eqs. (9.90) and (9.91), ∆e s is written as " 3n ! # ∗ X ∂D e ∂ D ∗ ∗ − − ∗ ∗ e e D s ∆Xi + D ∆s ∆e s = −(D) − (D) (9.92) ∂Xi ∂Xi i=1

which is simplified to

∆e s = (H, G)



∆x ∆s∗



(9.93)

where H = (H 1 , . . . ,H i , . . . ,H 3n ) e H i = −D

−

! e − ∗ ∂D ∂D∗ e D e − D s ∂Xi ∂Xi

(9.94)

e − D∗ G = −(D)

By combining the measurement errors ∆s∗ and the estimation errors ∆e s of member forces as ∆s = (∆s∗> , ∆e s> )> , we have the following equation from Eq. (9.93):  ∗    ∆s O Ip ∆X ∆s = = (9.95) ∆e s H G ∆s∗ where I p is the p × p identity matrix. This way, the force error vector ∆s of all members is formulated as a function of measurement errors of nodal coordinates ∆X and member forces ∆s∗ .

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Identification error

Let ec and ef denote the parameters for defining the bounds of the measurement errors ∆X and ∆s∗ , respectively. The measurement errors of member forces and nodal coordinates with different dimensions are transformed into dimensionless variables by dividing them by ec and ef , respectively. Hence, Eq. (9.95) is rewritten as    O ef I p ∆X/ec ∆s = B∆r = (9.96) ec H ef G ∆s∗ /ef where B=



   O ef I p ∆X/ec , ∆r = ec H e f G ∆s∗ /ef

(9.97)

When the sets of members, for which the forces are to be measured and to be estimated, are specified, B is a constant matrix from the definition of H and G. Hence, ∆s is conceived as a function of ∆r. To incorporate the worst-case scenario, the worst p value of ∆r with unit norm is found to lead to the maximum Euclidean norm (∆s)> ∆s. Hence, the performance measure of the identification is defined as the identification error E, which can be obtained by solving the following antioptimization problem: q find E = max (B∆r)> B∆r (9.98a) ∆r q (∆r)> ∆r = 1 (9.98b) subject to

Note that E is equivalent to the 2-norm kBk2 of the matrix B (Horn and Johnson, 1990), which is equal to the square root of the largest eigenvalue λ max of the symmetric matrix B > B; i.e., p E = λmax (9.99)

Note that λmax is also defined as the maximum singular value of B. When the forces of all members are to be measured, i.e., p = m, and the errors in nodal coordinates are not considered, we have E = ef , because   O O > (9.100) B = (O, ef I m ), B B = O e2f I m where I m is the m × m identity matrix. 9.10.4

Sensitivity analysis with respect to nodal coordinates

Let xi denote the x-coordinate of free node i. The sensitivity coefficient of the coordinate difference vector hx in x-direction with respect to xi is given as ∂(Cx) ∂(C f xf ) ∂hx = + ∂xi ∂xi ∂xi

(9.101)

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Since C and C f are constant, and the measurement error of coordinates of fixed nodes is not considered, Eq. (9.101) becomes ∂hx = Ci (9.102) ∂xi where C i is the ith column of C. From H x = diag(hx ), we have ∂H x = diag(C i ) (9.103) ∂xi Since H y and H z do not depend on the x-coordinates, their sensitivity coefficients with respect to xi vanish as ∂H y ∂H z = 0, =0 (9.104) ∂xi ∂xi x y z The sensitivity coefficients of H , H and H with respect to yi and zi can be formulated similarly to Eq. (9.103) and Eq. (9.104) as   ∂H x ∂H x     =0 =0     ∂yi    ∂ziy    y ∂H ∂H (9.105) = diag(C j ) , =0   ∂y i  ∂zi      z z    ∂H  ∂H   =0 = diag(C k ) ∂yi ∂zi From Eq. (9.85), we have L2 = (H x )2 + (H y )2 + (H z )2 (9.106) Let t represent a variable xi , yi , or zi . The sensitivity coefficient of L with respect to t is derived as  y z x ∂L −1 y ∂H z ∂H x ∂H =L +H +H (9.107) H ∂t ∂t ∂t ∂t Since the sensitivity coefficient of L−1 with respect to t is ∂L−1 ∂L = −L−2 (9.108) ∂t ∂t the sensitivity coefficient of L−1 with respect to t is written as follows by substituting Eq. (9.107) into Eq. (9.108):   ∂H y ∂H z ∂H x ∂L−1 = −L−3 H x + Hy + Hz (9.109) ∂t ∂t ∂t ∂t For the structure with given topology, the connectivity matrix C is a constant matrix. From the definition of the equilibrium matrix D in Eq. (9.87), its sensitivity coefficient with respect to a variable t is derived as   −1 x −1 > x ∂L > ∂H C ∂t L + C H ∂t    −1  ∂D    > ∂H y −1 ∂L (9.110) = C  L + C>H y  ∂t ∂t ∂t     ∂L−1  ∂H z −1 L + C>H z C> ∂t ∂t

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Substituting ∂H x /∂t, ∂H y /∂t and ∂H z /∂t in Eqs. (9.104) and (9.105) into Eq. (9.109) and then to Eq. (9.110), we have the sensitivity coefficient ∂D/∂t of the equilibrium matrix D with respect to a nodal coordinate t. 9.10.5

Optimal placement of measurement devices

9.10.5.1

Problem formulation

In the process of force identification of a prestressed pin-jointed structure, the key factors to be considered are: (a) accuracy of measurement, (b) number of measurement devices, and (c) locations of measurement devices. We consider only the last two factors, since the accuracy of the measurement depends on the property of the measurement device. In practical applications of force identification, the nodal coordinates can be easily measured with relatively small cost, while the measurement devices for member forces are much more expensive. Hence, we assume that all nodal coordinates are measured, whereas only some of the member forces are measured. From the definition of identification error in Eq. (9.99), it is seen that the matrix B depends on the set of measurement members to be selected even for the case where the number of devices is kept constant. Therefore, our problem is to find the optimal locations of measurement devices for member forces, that lead to minimum identification error E with the fixed number p of devices. Let J = {J1 , . . . , Jp } denote the set of p members, for which the forces are to be measured. The variable J in the optimization problem is a parameter for the anti-optimization problem (9.98) that is reformulated as q b ) = max (B(J )∆r)> B(J )∆r (9.111a) find E(J ∆r q subject to (∆r)> ∆r = 1 (9.111b)

b ) is the worst identification error corresponding to the specified set J . where E(J Then the optimum location problem for the measurement devices is formulated as b ) minimize E(J (9.112a) subject to |J | = p

(9.112b)

where |J | is the number of elements in J . Note that there is actually another e has to be full-rank so as to have the least squares inherent constraint: the matrix D solution for the member forces to be estimated as in Eq. (9.90). 9.10.5.2

Solution process

Since the proposed problem (9.112) is a typical combinatorial optimization problem, a heuristic approach called Simulated Annealing (SA) and the stingy method are adopted (see Sec. 2.6.3 for the details of heuristic approaches).

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In SA, the initial solution and the cooling schedule play critical roles in finding the optimal solution with good accuracy. Hence, it is more reliable to start the SA from a better initial solution. Zhang, Ohsaki and Araki (2004) developed several simple heuristic approaches, including the stingy method, to investigate the identification accuracy of force distribution of prestressed pin-jointed structures, where only the measurement errors of member forces are considered. It has been demonstrated that the stingy method has relatively good accuracy with small computational cost. Therefore, the stingy method can be effectively used for producing a good initial solution, rather than random solutions, to reduce the computational cost as well as to improve the accuracy in SA. The stingy method is a basic heuristic approach, based on the local search, to combinatorial optimization problems. For the problem considered here, it starts from a complete set of the possible measurement members, and successively removes the member with least contribution to the objective function from the current set of solutions, under the condition that the removal does not lead to an infeasible e solution violating the requirement on rank deficiency of D. The stingy method used in this example is summarized as follows:

Step 0: Set J = {1, . . . , m}, i.e., the forces of all members are to be measured, and p = m. Step 1 Let Ep0 and Epj denote the identification errors by the sets I and I − {j}, respectively. Find the member k in I that minimizes Epj − Ep0 . Step 2 If |J | = p, then terminate the process; otherwise, set I ← I − {k} and |J | ← |J | − 1 accordingly, because it has the minimum contribution in I to reduction of the identification errors and go to Step 1. This way, we can find the p measurement members with the approximately minimal identification error for the optimization problem (9.112), which is used as the initial solution for SA. 9.10.6

Numerical examples

Figure 9.26 shows a three-dimensional cable dome, consisting of 12 struts and 48 cables, which are shown in thick and thin lines, respectively (Ohsaki and Kanno, 2003). The nodes and supports of the structure are located on three circles with different radii, 5.0 m, 10.0 m, and 15.0 m, and the four different elevations of the circles are −1.0 m, 0.0 m, 1.0 m, and 2.0 m. Computation is carried out using MATLAB (Borse, 1997). The variance of measurement error of the nodal coordinate is set as ec = 1.0 mm, and that of the member force is ef = 0.5 kN, which is about 1.7% of the largest member force of the cables. In practical applications, it might be more convenient to measure the forces in the cables rather than those in the struts, because the measurement devices can be embedded into the cables more easily. Suppose that

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10 5 0 −5

−10 −15

−10

−5

0

5

10

15

(a) top view

(b) diagonal view

2 1 0 −1 −15

−10

−5

0

5

10

15

(c) side view Fig. 9.26

(a) stingy method only

A three-dimensional cable dome.

(b) SA with stingy method

Fig. 9.27 Optimal locations of measurement devices for cables (dotted line) of the threedimensional cable dome found by the stingy method only, and the SA with stingy method for generating the initial solution.

we have only 12 measurement devices, and the objective of this example is to find 12 cables, i.e., p = 12, to be measured so as to minimize the identification error. The dotted lines in Fig. 9.27(a) show the cables to be measured, which are determined by the stingy method only. The identification error in this case is 1.1988. As is seen, the members are not symmetrically located. We next use this result as the initial solution for SA, and obtain the optimal locations of cables to be measured as shown in the dotted lines in Fig. 9.27(b). The identification error in this case is reduced to 1.0418, which is smaller than that of using the stingy method only.

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Some Further References

For additional references on hybrid optimization and anti-optimization, the reader may consult, for example, the works by Adali, Richter and Verijenko (1995a), Adali, Verijenko and Richter (1994b), Banichuk and Ivanova (2009), Chen and Wu (2004b), Chen, Wu and Chen (2004), Cheng, Au, Tham and Zeng (2004), Cherkaev and Cherkaev (1999, 2003), Cherkaev and Kucuk (1999), Dube and Hausen (2004), de Faria (2000, 2002, 2004, 2007), de Faria and de Almeida (2003a, 2003b, 2004, 2006), de Faria and Hansen (2001a, 2001b), Fatemi, Mooij, Gurav, Andreykiv and van Keulen (2005), Ganzerli and de Palma (2007), Ganzerli and Pantelides (2000), Ganzerli, Pantelides and Reaveley (2000), Gu, Renaud, Batill, Brach and Budhiraja (2000), Gurav (2005), Gurav, Goosen and van Keulen (2005a), Gurav, Kasyap, Sheplak, Cattafesta and Haftka (2004), Gurav, Langhaar, Goosen and van Keulen (2005b), Gurav and van Keulen (2005), Koˇcvara and Stingl (2006), Lombardi (1995, 1998), Lombardi and Haftka (1998), Lombardi, Mariani and Venini (1998), Natke and Soong (1993), Pantelides and Booth (2000), Pantelides and Ganzerli (1997), Pownuk (2000, 2004), van Keulen, Damveld, Elishakoff and Toporov (2001), Venini (1998), Venter and Haftka (1996, 1999), and Zhang and Zhang (2000).

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Chapter 10

Concluding Remarks

‘ “Would you tell me, please, which way I ought to go from here?” “That depends a good deal on where you want to get to,” said the cat.’ (Lewis Carroll, Alice in Wonderland, Chapter 8) ‘You’ve got to be very careful if you don’t know where you are going, because you might not get there.’ (Yogi Berra) ‘Optimal design became a rich and rewarding field of research.’ (Niordson, 2001)

In this chapter we try to summarize this book and some thoughts on the optimization and probabilistic methods. It appears to us that total acceptance of the ever-present uncertainty may infuse the old subject of optimization with the needed ‘new blood’. The hybrid approach may help release both concepts from their confinement in the esoteric ivory tower and use them in the everyday quest for the best and safest designs. To encourage further thought, the sections hereinafter will be presented as questions in an attempt to provide replies in the form of first approximations. 10.1

Why Were Practical Engineers Reluctant to Adopt Structural Optimization?

Ever since structural optimization came into being, researchers have been expressing doubts as to its justification and/or practicality. The background can be described as follows: (1) Why should we look for the best design if an acceptable, hence, a good one, can be found? In short, adherence to the American maxim ‘If it ain’t broke, don’t fix it’ was recommended. (2) In his review article Ashley (1982) describes a poll conducted among 46 people about the merits and demerits of structural optimization. The responses were 327

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as follows: • an engineer experienced in civil and aeronautical structures: ‘One of the reasons that I stopped work in optimization was my dismay... that there were so very few applications.’ • a Dean of Engineering who has known the field for over a quarter of a century: ‘I do not recollect any applications.’ • a foremost specialist on synthesis with aeroelastic constraints: ‘I am sorry, but I don’t really have any...’ • a recently retired senior design engineer, describing events at his aerospace company: ‘For 15 years I beat my head against a stone wall... The end was: formal optimization techniques were never used in aircraft design (even to this day!). The company was forced to use [them] in its subsequent ICBM and space programs.’ • an interview with a distinguished European aerodynamicist regarding wing design: ‘Numerical optimization techniques were being used to some extent in prefeasibility studies, but [he was] unable to produce any specific or referenceable material.’ Did the situation change in the past quarter century? Venkataraman and Haftka (2004) address this topic. They write: ‘Yet, in 2004 structural optimization practitioners hear the same refrain from structural analysts that was heard in 1971; a single structural analysis takes several hours or even several days to run on my computer; how can I even afford to think about optimization?’ However, ‘... computing speed and storage capacity increased one million fold at a rate of about 100-fold increase every decade. In 1971, when structural optimization was still struggling to deal with any “reallife” problems, a one-million factor increase in computational speed would have seemed to solve all problems. After all, engineers were already analyzing complex structures with detailed finite element (FE) models, and the ability to do one million analyses is sufficient for most optimization problems.’ Others concluded that ‘the paradox of the continuing difficulty of applying optimization to real-life problems is, of course, due to increasing complexity of the structural models that analysts consider adequate’ for predicting structural response.’

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Venkataraman and Haftka ascribe this situation to the ‘computerized Parkinson’s law’. The classical law (Parkinson, 1959) states that the work done expands to fill up all the available time. Its computerized counterpart (Thimbleby, 1993) manifests itself, in the words of Venkataraman and Haftka, ‘by software applications growing to fill up increased computer memory, processing capacities and storage space.’ They further state: ‘In terms of structural analysis, it appears that the computer time required for an “adequate” structural analysis has been fixed at several hours. Essentially, if a single analysis cannot be computed overnight on the available computer, progress in terms of debugging models and improving structures can become intolerably slow. So a requirement that a working model can be analyzed in a few hours appears to be the main restraint to the desire of the structural analyst to have a high-fidelity structural analysis. In their paper on optimal design of a sandwich composite fuselage section, Ley and Peltier (2001) discuss how they had to limit the analysis model so as to be able to finish a local optimization overnight.’ Venkataraman and Haftka also stress the progress made by structural optimization: ‘In spite of increased appetite of structural analysis for more complex models and analyses, there is no question that structural optimization did benefit from faster computers and more efficient analysis and optimization algorithms. The turning point was in the mid-1980s. In 1981, Holt Ashley still lamented that optimization was used by researchers but not by industry... Since then [1985], the increasing number of optimization papers reflecting industrial applications attests to the acceptance of structural optimization as a design tool... one benefit of having faster computers is that engineering computations and challenging optimization are now routinely performed on much cheaper desktop computers.’ As Vanderplaats (1993) notes, ‘Structural optimization technology has now passed its thirtieth birthday. This technology has matured to the point that these methods have been added to many commercial finite element codes... it is clear that structural optimization technology may now be efficiently applied to a wide range of design tasks... As advanced software becomes more widely available, it will be used to dramatically improve design efficiency, as well as to aid in the allocation of scarce resources.’ In an earlier paper, Vanderplaats (1991) writes:

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‘Structural optimization technology has now matured to the point that these methods have been added to most commercial finite element codes. Various methods are employed, ranging from simple coupling between the analysis and optimization to reasonably sophisticated use of approximation techniques.’ The above paper concludes in an optimistic vein: ‘... it is clear that research and development will continue to expand and to continue the improve[ment of] the efficiency and reliability of the overall structural synthesis process.’ (3) There has been a lot of controversy since the 1960s about the effectiveness of optimizing structures against buckling. Thompson and Hunt (1974) come with apparently the most powerful argument ‘the thesis that a process of optimization leads almost inevitably to designs which exhibit the notorious failure characteristics often associated with the buckling of thin elastic shells. Here the idealized perfect structure exhibits unstable and often compound branching point and would fail by an explosive instability, while nominally perfect real structures containing inevitable initial imperfections fail at scattered loads which can be quite considerably lower than that of the idealization.’ They also emphasize that ‘an optimum design is by its very nature imperfection-sensitive, since any deviation from the idealized optimum design must yield a lower failure load.’ and conclude as follows: ‘Perhaps we can close with the question expressed by Pugsley (1966). Should the structures be designed to be just stable, or in some way more stable than stable?’ We arrive at the following conclusion: optimization may be a generator of severe imperfection sensitivity for some structures like thin-walled shells, but need not necessarily lead to an imperfection-sensitive structure as we have seen in Secs. 5.5 and 5.7. Optimization against buckling is very effective if instability phenomena are fully understood, as was clearly demonstrated by Ohsaki and Ikeda (2007a). (4) It appears to some investigators, and we tend to agree with them, that optimization theory, in order to become practical, ought to embrace the notion of uncertainty. Frangopol and Maute (2005) stress: ‘[A] major limitation of the deterministic optimization approach is that uncertainties in the loads, resistances, and structural responses are not included in the computation process. Therefore, irrespective

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of the level of sophistification in the approach used in the computational process, the optimum solutions are predicated on idealized deterministic assumptions. Consequently, the solutions are not computed under realistic conditions.’ Indeed, Thoft-Christensen (2006) writes: ‘Classical deterministic structural optimization has only in few cases been seriously used in designing real engineering structures.’ Pu, Das and Faulkner (1997) write: ‘Conventional optimization... is a deterministic process. It has been observed that the optimum structures obtained through deterministic optimization do not necessarily have high reliability. Hence it is likely that the failure probability of the optimum structure does not satisfy the reliability criteria. The optimization based on reliability concepts will lead to more consistent safety in the structure system.’ Frangopol and Maute (2005) stress: ‘... despite the demonstrated exponential growth in structural optimization research, most of it has been cast in deterministic format. In fact, a quick survey of about 2400 references (Burns, 2002) in structural optimization shows that more than 95% are deterministicoriented.’ Frangopol (1995) is a strong proponent of stochastic optimization, stating that ‘stochastical design based on both reliability and optimization represents the ultimate goal in design under uncertainty.’ This gives rise to the following question: If probabilistic (or stochastic) optimization is the remedy for the ills of structural optimization, why don’t theoreticians and practitioners alike jump on the wagon? An even more cardinal question is discussed in the next section. 10.2

Why Didn’t Practical Engineers Totally Embrace Probabilistic Methods?

Reservations regarding the various uncertainty analyses are not unlike their counterparts regarding the optimization approaches. Currently, the most utilized methods for incorporating uncertainty are the probabilistic and fuzzy-sets based methodologies. The book by Ben-Haim and Elishakoff (1990) devotes its first chapter to the pros and cons of probabilistic modeling, including numerous pertinent quotations in Sec.1.3; these will not be repeated here. Elishakoff (2000b) published a result of the poll that he conducted for 12 years among more than 40 engineering researchers

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extremely active in probabilistic analysis of structures. The possible limitations of the probabilistic methods are listed and discussed in some detail. It appears that probabilistic design of structures, based upon its fundamental notion of reliability, finds itself in a situation reminiscent of the state of structural optimization. Complaints in this context are nearly the same. In his Freudenthal Lecture at the ICOSSAR’93 conference, Yang (1994) stated in his concluding remarks: ‘To date, the reliability method has not been widely used as an analysis or design tool in industry.’ Schu¨eller (1996) emphasizes: ‘There appears to be no much disagreement, that, despite the significant progress made so far, particularly over the last 30 years, a large number of procedures are still in the stage of development, i.e., have not yet passed academic stage.’ Soong (1988) notes: ‘The gap between research and practice, it appears, is widening...’ Sexmith and Nelson (1969) state: ‘In spite of many notable contributions... probabilistic concepts have not found their way into practice or design codes in anything but a superficial way.’ Cohn (1994) emphasizes: ‘Along with the research optimism expressed by these publications, some reservations are raised on the philosophical, theoretical and practical implications of reliability-based design and optimization.’ Templeman (1988) notes: ‘... truly safe structures can only be assured by design and fabrication procedures which thoroughly examine and accommodate the real causes of structural failures. The probability of failure approach forms only a small part of these...’ Cohn’s (1994) ‘verdict’ on stochastic optimization is quite peremptory: ‘... Large-scale, practical use of reliability-based optimization is unlikely, and quite unjustified, at least in the foreseeable future.’ By contrast, it should be noted that reliability-based optimization is being extensively studied in the 21st century, and some industrial applications can be found (Kim and Choi, 2008; Cheng, Xu and Jiang, 2006; Kharmanda, Olhoff and El-Hami, 2004). It also should be mentioned that Working Groups on Reliability and Optimization of Structural Systems belonging to the International Federation for Information Processing (IFIP) conducts regular meetings since 1987. There are at least 12 volumes edited by Thoft-Christensen (1987, 1988), Der Kiureghian and Thoft-Christensen (1990), Rackwitz and ThoftChristensen (1991), Thoft-Christensen and Ishikawa (1993), Rackwitz, Augusti and Bori (1995), Frangopol, Corotis and Rackwitz (1996), Nowak and Szerszen (1998, 2000), Furuta, Dogaki and Sakano (2002), Maes and Huyse (2003), and Sørensen

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and Frangopol (2006). these volumes, except the latter one, are conveniently titles Reliability and Optimization of Structural Systems.

10.3

Why Don’t the Probabilistic Methods Find Appreciation among Theoreticians and Practitioners Alike?

Probabilistic modeling involves design with allowance for the aspect of reliability, defined as the probability of successful performance of the structure. This means that the engineer envisages some measure of the probability of failure, which is unity minus the reliability. The main proponent and developer of probabilistic methods, Alfred Freudenthal (1961) states: ‘... the notion that a finite (no matter how small) probability of failure or at least of unserviceability is repulsive to a majority of engineers.’ The persistent question arises: Has the situation changed since this observation was made over half a century ago? Vladimir Vasilievich Bolotin (see Elishakoff (2000b)) responds in the following manner in the poll: ‘Engineers, especially those at the top, do not trust statements and predictions of probabilistic character. They must make decisions on important projects, large structures and big investments, and they prefer to be completely sure that their decisions are true. Of course, a proper education in the theory of probability and mathematical statistics helps here, but there is a more profound, maybe subconscious, case of such an attitude.’ Grandori (1991) writes: ‘... the concept of structural safety will not leave the “realm of metaphysics” unless we devise a method for justifying the choice of risk acceptability levels.’ Kalman (1994) is even harsher: ‘I see enormous activity, seemingly aimless, I see fanatical devotion to ideas and principles which have grown into a quasi-religion... Probability is an intellectual construct. It does not exist in the real world... It is not of scientific concern today because it exists only in the self-interest of gamblers or the imagination of statisticians or the mind-reading of philosophers. There was and is no such thing as a probabilistic revolution in science...’ Schu¨eller (1996) responds with the following statement, expressing guarded optimism: ‘... it should be stated, that the consideration of uncertainties, inherent in the problems of the field of engineering mechanics, by

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stochastic methods certainly has not revolutionized our profession, but there are very good reasons to expect that the change of thinking will come about in a long-time evolution.’ Likewise, Rackwitz (2001) asks: ‘... Cornell (1981) called structural reliability a healthy adolescent. In 1990 structural reliability then certainly was an efficient executive in all his virulence. But has this man in the meantime just got more years, but no more wisdom?’ It appears that this controversy, sometimes very hot, will continue in the future. It is more pragmatic to pose a more practical question on how to model the unavoidable uncertainty in engineering practice. Bolotin (see Elishakoff (2000b)) provides a useful insight: ‘In my own experience, it is expedient to show to an engineer a set of samples of the system behavior, in particular the worst sample, the best sample and an average or a typical one... In general, in my opinion, a representative set of time histories obtained with numerical simulation of the probabilistic model presents to engineers more information than the final results of statistical treatment.’ It is this very methodology of finding the most favorable (best) responses along with the least favorable (worst) ones that is adopted in this monograph. Bolotin’s idea is overlaid with optimization study. Indeed, it may be overly conservative to use the worst possible response without optimization: in fact, Kharitonov (1997) tellingly titles his paper ‘Interval uncertainty structure: conservative but simple’. Optimization minimizes overdesign, thereby becoming a natural ally in looking for the worst-case scenario, which was dubbed here and elsewhere as anti-optimization. Therefore, this book deals with optimization (Chap. 2), anti-optimization (Chaps. 3–8) and their hybrid (Chap. 9).

10.4

Is the Suggested Methodology a New One?

Usually, researchers tend to characterize their approaches as ‘new’, or ‘novel’, or even ‘radically new’. However, in full agreement with the wisest of men, King Solomon, we feel that ‘There is nothing new under the sun’, and the antioptimization approach is not new. Ovid (43B.C.E.–18C.E.) writes in his Metamorphoses: ‘I see and approve better things, but follow worse.’ (VII, line 20) Rustem and Howe (2002) lead into their book with a quote from William Shakespeare (1564–1616): ‘ ... since the affairs of men rest still uncertain, Let’s reason with the worst that may befall.’ (Julius Caesar, Act 5, Scene 1)

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Sniedovich (2008a) and Burgman (2008) include Shakespeare’s name in the titles of their respective papers. Burgman (2008) notes: ‘We can again claim Shakespeare’s credentials as a decision theorist. He tells us that it is better to be safe than sorry, that “The better part of valour is discretion”.’ (Henry IV, Part 1, Act 5, Scene 4) Leonhard Euler (1707–1783) stressed that ‘For since the fabric of the universe is most perfect and the work of a most wise Creator, nothing at all takes place in the universe in which some rule of maximum or minimum does not appear.’ (Tikhomirov, 1990). In theoretical and applied mechanics, the idea of minimization of the upper bounds of mechanical characteristics apparently occurs first in the 1877 book by J. W. Strutt (Lord Rayleigh), Theory of Sound, in which a method of determining the natural frequencies of discrete and continuous structures was presented. The method now bears his name. Specifically, he addressed the well-known problem of the vibrating string. For the fundamental mode he assumed   n  2x cos ωt (10.1) w(x, t) = 1 − L where ω is the sought circular frequency, 2L is the length of the string, x is the axial coordinate (he chose x = 0 at the midpoint of the string). In the end, Rayleigh arrived at the following formula: ω2 =

2(n + 1)(2n + 1) T 2n − 1 ρL2

(10.2)

where T is the tension in the string, and ρ is the mass density. For n = 1, the formula yields λ = ω 2 ρL2 /T = 12, instead of the exact value λ = π 2 ' 9.8696; n = 2 yields a better result, λ = 10, whereas a minimum λ, or in √ Leissa’s (2005) words ‘the best possible result from Eq. (10.2) is 9.8990 when n = ( 6 + 1)/2 = 1.72474.’ Clearly, Rayleigh carried out minimization of the upper bound of the fundamental natural frequency. Apparently he was also the first to use (see p. 287 in his book) the combined term ‘maximum–minimum condition’. (A detailed description of Rayleigh’s method can be found in almost any text on vibration, and a careful paper on its history was published by Leissa (2005)). Use of a function containing a power-law term with an undetermined exponent was resurrected by Schmidt (1985), and reviewed extensively by Bert (1987). Note that minimization of the maximum response under a constraint on structural weight is equivalent to that of structural weight under a constraint on the response. Sniedovich (2006, 2007, 2008a, 2008b, 2008c) attributes the modern terminology of the ‘maximin principle’ to Abraham Wald (1902–1950) as follows: ‘Rank alternatives by their worst possible outcomes: adopt the alternative the worst outcome of which is at least as good as the worst outcome of the others.’ Moreover,

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‘Wald’s Maximin Principle remains one of the modeling pillars of decision-making under severe uncertainty,’ although he is not ‘advocating Maximin as a panacea for decision under uncertainty.’ And finally, ‘On the face of it, the Maximin paradigm seems to be something of a second nature of many of us: its basic attitude to uncertainty is summed up in the familiar, indeed widely held, maxim: When in doubt, assume the worst!’ Thus, the worst-case scenario design in mechanics appears to be contemporary with Wald (1945, 1950). Bulgakov’s and Boley’s problem (see Secs. 4.2 and 6.3, respectively) is a testimony to this. The idea of minimizing the maximum response apparently first appeared in a deterministic setting. As Manasyan (1962) notes ‘Usually, the dynamic analysis of an elastic body is reduced to static analysis via introducing the so called dynamic coefficient, the latter being the absolute value of the ratio of the maximum dynamic displacement and the maximum static one.’ Manasyan (1962) proposes to minimize the dynamic coefficient; in other words, the maximum dynamic displacement. Later on, Brach (1968) treats a class of simply supported beams with constrained total mass and minimizes the maximum dynamic displacement at midpoint. Plaut (1970) and Yau (1975) study optimal design of a simply supported beam with given total mass, minimizing the upper bound of its dynamic response. Fox and Kapoor (1970) develop a bounding technique for deriving the approximate peak response of planar truss frames under shock loads; this response is minimized by a feasible-direction technique of optimization. Adali (1982) deals with minimum-maximum deflection and stress design of beams subjected to harmonic excitation. A general minimax methodology is expounded in the monographs by Boca (1982), Demyanov and Malozemov (1990), and Du and Pardalos (1995). Minimization of maximum of some quantity, associated with an adverse effect, is suggested also by Bellman (1962), Besharati and Azarm (2006), Chang, Paez and Ju (1983), Chern, Dafalias and Martin (1973), Cherkaev and Cherkaev (1999, 2008), Cherkaev and Kucuk (1999), Gonzales and Elliot (1995), Guretskii (1969), Hakim and Fuchs (1996), Hilding, Klarbring and Pang (1999), Kalinowski and Pilkey (1975), Komkov and Dannon (1987), Nishihara, Asami and Watanabe (1999a, 1999b), Paez (1981), Papoulis (1970), Popplewell and Youssef (1979), Puig, Saludes and Quevedo (2003), Sevin (1957), Sevin and Pilkey (1967), Smallwood (1972b, 1972a, 1973, 1989), Tseng (2006), Westermo (1985), Witte (1974), Witte and Smallwood (1974), and Youssef and Popplewell (1978, 1979). In his correspondence with I.E., Sniedovich (2009) stresses: ‘I do not have any problem with the use of term “antioptimization”, provided that it is clear what it means. However, it is important to relate it to the state of the art. Personally, I

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am happy with the old terms “worst case” and “best-case” and “Maximin”. I have been using them for more that 40 years.’ Elishakoff (1999) anticipating, as it were, the above message, writes: ‘It will be no surprise to find that the idea of anti-optimization in its various forms stems from the times immemorial. This anticipation is strengthened by the fact that its main idea resonates well with a universal wisdom “Make the best out of the worst”. ’ Here the following personal anecdote seems to be instructive. Some years ago one of us (I.E.) lectured at a certain university; then, the idea of combined antioptimization and optimization was propageted. In the questions-and-answers period, one faculty member said: ‘I don’t like the term anti-optimization, in fact, had I gotten your manuscript for review, I would have rejected it, just for this term!’ The answer was that one of the leading optimization experts, Professor R. T. Haftka, had endorsed it. The faculty member responded, ‘If Haftka has embraced this notion, I feel and am defeated!’ It is pleasing to record that this term has been adopted by many researchers around the world. The above correspondence with Sniedovich helped us to spell out the need to discuss its various ‘brothers’ and ‘sisters’. We hope that in this section we have correlated the various origins, counterparts and equivalents of the term ‘optimization/anti-optimization hybrid’. As can be seen, applied mechanics provided development of ideas from various analyses, leading to terms ‘worst-case’. ‘least favorable’, ‘best-case’, ‘most favorable’, ‘minmax’ or ‘anti-optimization’ with a view to determining the worst-case scenarios. Then, combined optimization seeks to find the minimum of the worst response, or the least worst response. Hence, it is worthwhile to make comprehensive survey of the problem formulations and methodologies for optimization considering maximum (worst-case) responses. Although the idea is known independently in several fields of engineering, this book presents all the necessary knowledge and approaches, as well as the philosophy underlining the hybrid concept, and establishes a new direction based on computer technology. In Chap. 4 we analyze the worst, or anti-optimized, responses in a static setting, whereas Chap. 6 deals with analogous vibration problem. In the buckling context, ‘worst-case’ or anti-optimization entails finding the worst (or least favorable) response, i.e., the minimum buckling load. Combination with anti-optimization leads then to maximization of the minimum buckling load. Since the FEM is the central numerical method in applied mechanics, it made sense to discuss, in Chap. 7, how the anti-optimization is performed in a discretized environment. The topic is under active development and the final word has not been said about it, luckily for us researchers! Chapter 8 correlates the probabilistic and anti-optimization approaches, showing perhaps to the disappointment of ‘separationists’ (those who ‘divide and conquer’ the specific fields of research) and hopefully, to the satisfaction of ‘unionists’ (those

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who try to find similarities in seemingly different fields) – that the two approaches lead to close or even the same results. Our general conclusion is in line with the title of the paper by Bai and Andersland (1994) ‘Stochastic and Worst Case System Identification are not Necessarily Incompatible’; moreover, Bai and Andersland claim: ‘Clearly the two approaches [stochastic and worst case – I.E. and M.O.] are based on different philosophies. In the literature they are usually treated differently and separately, we believe that they are not necessarily incompatible, and, in fact, may be complementary.’ Likewise, Elishakoff (1999) in his paper ‘Are Probabilistic and Anti-Optimization Methods Interrelated?’ claims: ‘This study does not aim to make a bombastic statement like that of de Finetti (1974), “Probability does not exist.” We can take the liberty of suggesting to consider two possible scenarios: either probability does not exist or it does. Consider first the former case: if probability does not exist, then obviously, anti-optimization method is of utmost importance, and it must be utilized in design. Consider now the latter case: if probability exists, then for the structure to be acceptable it must possess high, near-unity reliability; for the case of bounded random variations, the probabilistic method tends to one furnished by the anti-optimization. Thus, irrespective of the existence or nonexistence of probability, the anti-optimization method must be incorporated.’ In this vein, in Sec. 1.3 we demonstrated the convergence of probabilistic and anti-optimization. Systematic comparison of these seemingly contradictory approaches was conducted in Chap. 8, establishing the identity of designs developed by the probabilistic and non-probabilistic analyses, constituting pacifying news: no need to fight and argue. We may conclude this section by quoting from Minsky (1987) ‘If you understand something in only one way, then you do not really understand it at all... The secret of what anything means to us depends on how we have connected it to all the other things we know.’ Likewise, Takao Fujisawa (see Sato (2006)) mentions: ‘It’s only the man who can look at the same problem from many different aspects that will make a true leader.’ It is noteworthy that stochasticians are tending more and more to invoke the fact of physically realizable random variables and processes. To review the current thinking of the stochasticians and the interested reader may like to consult

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the papers by Cai (2003), Cai and Lin (2006), Grigoriu (2006), Ma, Leng, Meng and Fang (2004), and Simiu and Heckert (1996) already quoted and discussed in Sec. 1.3. It appears therefore that the prediction of increased appreciation, of the anti-optimization method in the future, is not premature. The combination of the probabilistic and ellipsoidal analyses was facilitated in works by Elishakoff and Colombi (1993), Elishakoff, Lin and Zhu (1994e), Elishakoff and Li (1999), and Qiu, Yang and Elishakoff (2008b). In this context Drenick (1995) wrote in the correspondence with one of us (I.E.) ‘You mentioned that the problems you liked the best had both probabilistic and non-probabilistic aspects. I agreed with you because I felt that the treatment of such “mixed problems” might not run into the same unyielding resistance that is encountered by the pure non-probabilistic type.’ 10.5

Finally, Why Did We Write This Book?

We would like to end with the provocative question in the above subtitle. Indeed, with the abundance of texts on optimization and/or uncertainty – why bother with another? Our answer is: because it combines these two cardinal ideas, thereby ejecting them from their respective ivory-tower spheres and providing a friendly environment for the much needed communication and interaction. In such circumstances, the existential question posed by Laville (2000), ‘Shall we abandon optimization theory?’ can be answered in the negative, and the equally important one by Mongin (2000) ‘Does optimization imply rationality?’ in the affirmative. The thesis that the optimization/anti-optimization hybrid saves both optimization and uncertainty analysis from pure academism is an exaggeration. In fact, the reliability-based design optimization (RBDO) is an established field answering the above need of marrying uncertainty and optimization. As Chateauneuf and Aoues (2007) note: ‘A very intensive research activity is performed in the field of RBDO solution methods.’ ‘Although significant progress is performed in developing efficient numerical methods, the approach to practical engineering structures is still a challenge, knowing the complexity of realistic industrial systems.’ These authors also stress: ‘While the optimization problem is carried out in the space of design variables, the reliability analysis is performed in the space

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of random variables, where a lot of numerical calculations are required to evaluate the failure probability. Consequently, in order to search for the optimal structural configuration, the design variables are repeatedly changed, and each set of design variables corresponds to a new random variable space which they need to be manipulated to evaluate the structural reliability at the point (Murotsu, Shao and Watanabe, 1994). Because of the too many repeated searches needed in the above two spaces, the computational time for such an optimization becomes the main problem.’ Moreover, ‘... the progress in System Reliability-Based Design Optimization (SRBDO) is relatively slow because it depends on the system reliability analysis where the computational time, and the numerical instability lead to many difficulties in the SRBDO formulation which is mainly due to design variable changes at each iteration of the optimization procedure.’ Note that the overwhelming majority of works on RBDO and SRBDO deal in independent random variables. The need to allow for interdependence in stochastic analysis is discussed in numerous papers; the interested reader is referred to the papers of Ferson, Ginzburg and Akcakaya (1995) and Ferson, Nelsen, Hajagos, Berleant, Zhang, Tucker, Ginzburg and Oberkampf (2004). Ganzerli and de Palma (2007) write: ‘The book, Convex Models of Uncertainty in Applied Mechanics by Ben-Haim and Elishakoff, established the foundation for the convex model theory and application in 1990. Since then, much work has been done on convex models. Together with probabilistic and fuzzy sets, convex models can be considered part of the uncertainty triangle (Elishakoff, 1990). Convex models are especially useful for problems where data on the uncertain parameters is scarce, as in the case of severe uncertainties. One of the most widely used of the convex models is the uniform bounded model. Here the convex set that encloses the uncertain parameter, has the shape of a rectangle in two dimensions. The uniform bound convex model has been implemented in structural design using a technique called “anti-optimization” (Elishakoff, Haftka and Fang, 1994c).’ We here already showed the contiguity of probability and anti-optimization. Likewise, RBDO and the optimization/anti-optimization hybrid are expected to yield similar designs. However, we do not advocate replacement of RBDO by the

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hybrid. We think that they are complementary approaches capable of ‘peaceful coexistence’, with a preference for the hybrid due to its conceptual and numerical simplicity. Throughout this book, by adopting a hybrid of optimization and antioptimization instead of a more complex RBDO, we here tried to follow the sage advice of Ren´e Descartes (1596–1650): ‘Remove all redundancies from the problem and reduce it to the simplest elements.’ The simple hybrid method may attract more ‘deterministic’ engineers and researchers to the topic of uncertainty – especially those who, for some reason or other, doubt or deny the existence of probability. Please communicate to us your own ideas by e-mail: [email protected] and [email protected], or by airmail. We will try to respond to input! Sorry if we here omitted to quote some pertinent work: it was unintentional! And sorry for the excess of references, if any. Enjoy the book! This is the reason why we undertook it: we felt that both industry and academia needed it; no less a reason was our own pleasure!

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Tzan, S.-R. and Pantelides, C. P. (1996b). Convex models for seismic design of structures – 2: Design of conventional and active structures, Earthquake Engineering and Structural Dynamics 25, 9, pp. 945–963. Utkin, L. V. and Gurov, S. V. (1996). A general formal approach for fuzzy reliability analysis in the possibility context, Fuzzy Sets and Systems 83, pp. 203–213. van Dyke, M. (1975). Perturbation Methods in Fluid Mechanics (Parabolic Press, Stanford, CA), pp. 13–14. van Keulen, F., Damveld, H., Elishakoff, I. and Toporov, V. (2001). Bounded-but-unknown uncertainties in design optimization, in Proc. AIAA Structural Dynamics Conference (CD-ROM), AIAA Paper 2001-1242. Vanderplaats, G. N. (1991). Efficient structural optimization techniques, in Computer Application in Civil and Building Engineering, Proc. 4th International Conference (Kozo System Inc, Tokyo), pp. 151–158. Vanderplaats, G. N. (1993). Thirty years of modern structural optimization, Advances in Engineering Software 16, pp. 81–88. Vanderplaats, G. N. and Yoshida, N. (1985). Efficient calculation of optimum design sensitivity, AIAA Journal 23, pp. 1798–1803. Vanderplaats, G. N. (1999). Numerical Optimization Techniques for Engineering Design (Vanderplaats Research & Development, Inc., Colorado Springs, CO). Vanmarcke, E. H., Diaz-Padilla, J. and Roth, D. A. (1972). Reliability-based optimum design of simple plastic frames, Tech. Rep. R72-46, Department of Mechanical Engineering, Massachusetts Institute of Technology. Vassart, N. and Motro, R. (1999). Multiparametered formfinding method: application to tensegrity systems, International Journal of Space Structures 14, 2, pp. 147–154. Venini, P. (1998). Robust control of uncertain structures, Computers & Structures 67, pp. 165–174. Venkataraman, S. and Haftka, R. T. (2004). Structural optimization complexity: What has Moore’s law done for us? Structural and Multidisciplinary Optimization 28, pp. 375–387. Venkayya, V. B. (1993). Introduction: Historical perspective and future direction, in M. P. Kamat (ed.), Structural Optimization: Status and Promise (AIAA Press, Washington D.C.), pp. 1–10. Venter, G. and Haftka, R. T. (1996). A two species genetic algorithm for designing composite laminates subjected to uncertainty, in Proc. 37th AIAA/ASME/ASCE/ AHS/ASC Structures, Structural Dynamics, and Materials Conference and Exhibit: Part 3 (Salt Lake City, UT). Venter, G. and Haftka, R. T. (1999). Two species genetic algorithm for design under worst case condition, in Proc. 3rd World Congress of Structural and Multidisciplinary Optimization (Buffalo, N. Y.), (Also Evolutionary Optimization (Internet Journal), 2(1), 1–19). Venter, G., Haftka, R. T. and Starnes, Jr., J. H. (1998). Construction of response surface approximation for design optimization, AIAA Journal 36, 12, pp. 2242–2249. Vick, S. G. (2002). Degrees of Belief: Subjective Probability and Engineering Judgement (ASCE Press, Reston, VA). Vinot, P., Cogan, S. and Lallement, G. (2003). Approche non-probabiliste de fiabilit´e base sur les mode les convexes, Me´canique & Industries 4, 1, pp. 45–50, (in French). Vitali, F., Blanc, G., Larque, P., Duplay, J., Morvan, G., Nakagiri, S. and Suzuki, K. (1999). Finite element interval analysis of external loads identified by displacement input with uncertainty, Computer Methods in Applied Mechanics and Engineering 168, 1, pp. 63–72.

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Index

convex hull, 303 critical excitation, 176 critical point, 281 coincident, 281 criticality condition, 115, 116, 125 cycle-based method, 277 cylinder, 59

accelerogram admissible, 177 credible, 179 autocorrelation function, 195 ball, 59 Bernoulli–Euler beam, 191, 218, 221 bifurcation point, 135 stable, 283 stable-symmetric, 282 unstable-symmetric, 282 bound ellipsoidal, 115, 147, 275 energy, 149 interval, 49 Bubnov–Galerkin approach, 192 buckling Euler load, 230 linear, 117 linear analysis, 117, 280 mode, 282 linear, 117 nonlinear, 280–282

dependency problem, 55 design by experiment, 31 design load, 273 Dirac delta function, 153 disk, 59 distance Euclid, 33 humming, 33 Manhattan, 33 earthquake epicentral intensity, 184 spectrum-compatible, 181 eigenvalue problem, 67 eigenmode, 67, 292 eigenvalue, 67, 292 element-by-element approach, 215 ellipsoid principal axis, 59 semi-axis, 57 ellipsoidal model, 57, 61, 78, 85, 117, 166 enumeration, 122 vertex, 63, 109, 131, 278, 297 equilibrium path bifurcation, 115, 284 fundamental, 115, 282 postbuckling, 284 error

calculus of variation, 18 Cauchy–Schwartz inequality, 183 compliance, 294 constraint active, 24 bound, 24, 62, 80 box, 24 equality, 22 inactive, 24 inequality, 22 side, 24 387

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identification, 320 index, 304 round-off, 48 vector, 304 exact reanalysis, 69 feasible design, 20 feasible region, 23 finite element method, 211 stochastic, 190 fixed point, 213 flexible structure, 307 flutter velocity, 194 force identification, 316 forced vibration, 148 Fourier expansion, 184 series, 178 transformation, 291 frame braced, 118 shear, 217 function characteristic, 42 concave, 62, 129, 278 convex, 24, 107, 278, 294 impulse response, 164, 176 membership, 42 monotonic, 278 quasi-convex, 308 singularity, 229, 232 step, 229 fuzzy theory, 42 Gamma function, 4 gas-flow interaction, 192 generalized displacement, 135 Gram–Schmidt orthogonalization, 169 greedy method, 34, 294, 298 ground structure approach, 36 guaranteed approach, 57, 228 heuristics, 31 hilltop branching point, 134, 281 homologous deformation, 299 homology design, 298 hyperellipse, 303 hypersphere, 305 imperfection

initial, 229 parameter, 135 pattern, 135 random, 140 worst, 136, 281 interval analysis, 49, 50, 212, 245 arithmetic, 54 coefficient, 221 Gaussian elimination, 213 matrix radius, 217 number absolute value, 55 addition, 53 diameter, 56, 214 distance, 55 distributivity property, 55 division, 53 multiplication, 53 overestimation, 214 radius, 56, 219 subcancellation property, 55 subdistributity property, 55 subtraction, 53 width, 56 Karush–Kuhn–Tucker condition, 25 kinetic energy, 314 Kronecker delta, 67, 152 lack of knowledge, 49 Lagrange multiplier, 25, 62, 115, 147, 158, 166, 179, 215, 275, 304 Lagrangian, 25, 62, 78, 115, 269 Laplace transform, 153 Liapunov’s direct method, 308 limit point, 135, 283 load factor, 116, 281 bifurcation, 115 buckling, 115, 116, 282 critical, 115, 282 linear buckling, 117 multiple buckling, 118 local search, 32 matrix connectivity, 103, 317 flexibility, 294 interval, 87

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Index

mass, 67 negative definite, 183 perturbation, 86 positive definite, 24, 57, 59, 183, 196, 294, 303 positive semidefinite, 24, 26, 59 square root, 59 stability, 116, 134 tangent stiffness, 105 transformation, 57, 59 weight, 57 Minkowski sum, 60 modal decomposition, 292 mode interaction, 123, 124 Moore–Penrose generalized inverse, 301, 319 norm Chebychev, 33 Euclidean, 301, 320 Oettli–Prager lemma, 222 optimal solution global, 24 local, 24 Pareto, 29 optimality condition, 25 optimization fuzzy, 41 geometry, 36 layout, 36 probabilistic, 39 shape, 36 sizing, 36 topology, 18, 36 two level, 276 optimum design sensitivity, 27 orthogonality condition, 164 out-of-straightness, 120 outward rounding, 54 parallel computation, 277 parallelepiped, 167 peak ground acceleration, 177 penalty function, 34 exterior, 34 interior, 34 polytropic thermodynamic transformation, 192 possibilistic approach, 228

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prebuckling deformation, 116 prestress, 316 probabilistic method, 227 probability exponential, 3 joint, 250 log-normal, 6 marginal, 252 normal, 140 Rayleigh, 3 truncated exponential, 240 uniform, 10, 232 Weibull, 3 programming convex, 24 integer, 24 linear, 25, 79, 116, 148 multiobjective, 28 nonlinear, 22, 25 parametric, 26, 279 quadratic, 25, 78, 148 semidefinite, 26 sequential linear, 310 proportional load, 281 pseudo force method, 70 pseudodistortion method, 70 random search, 35 rank, 318 deficiency, 319 Rayleigh principle, 129 reliability, 3, 231, 251 reliability-based design, 274 response least favorable, 158 most favorable, 158 response spectrum approach, 292 response surface method, 31, 279 robust design, 274 Routh–Hurwitz criterion, 156 safety factor, 273 actual, 3 central, 4 required, 3 self-equilibrium force, 104 state, 317 sensitivity imperfection, 135

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coefficient, 22, 65, 117, 146 imperfection, 280 linear buckling load, 117 sensitivity analysis, 63, 117, 301, 304, 320 compliance, 107 design, 63, 65 direct differentiation method, 64 eigenvalue, 67 finite-difference approach, 66 shape, 63, 68 tangent stiffness matrix, 106 set convex, 24 crisp, 42 fuzzy, 42 Pareto optimal, 29 Sherman–Morrison–Woodbury formula, 70 simulated annealing, 35, 122 slenderness ratio, 126 solution complementary, 301 dominance, 29 neighborhood, 33 particular, 301 spectrum central, 180 Kanai–Tajimi, 180 power, 180 response, 180 response displacement, 186

stability asymptotic, 307 boundary, 129 static pushover analysis, 292 stiffness matrix geometrical, 116, 129 tangent, 116, 129, 282 strain energy, 308 stress concentration factor, 90 supersonic stream flow, 191 tensegrity structure, 102, 316 total potential energy, 124, 134, 215 uncertainty data, 49 model, 49 unknown-but-bounded, 57, 303 Vieta’s theorem, 193 virtual distortion method, 70 viscoelastic kernel, 151 worst load scenario, 275, 298 worst load-case, 79 worst-case design, 273 worst-case scenario, 320

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Author Index

Aarts, E. 32, 35 Abbas, A. M. 112, 189 Abdullah, M. M. 317 Adali, S. 45, 63, 144, 280, 325, 336 Adduri, P. R. 272 Adeli, H. 40 Ahrens, R. 153 Akcakaya, R. 340 Akg¨ un, M. A. 70 Alefeld, G. 52, 54, 56, 87, 220 Allen, M. 190 Alotto, P. 85 Amazigo, J. C. 227 Andersland, M. S. 338 Andreykiv, A. 325 Ang, A. H-S. 14 Ant´ onio, C. A. C. 26 Antonnson, E. K. xiii Aoki, K. 299 Aoues, Y. 339 Araiza, R. 225, 272 Arakawa, M. 43, 45 Araki, Y. 316, 323 Arbocz, J. 144 Argyris, J. H. 69 Ariaratnam, S. T. 227 Arora, J. S. 18, 27, 30, 127 Asami, T. 336 Ashida, F. 30, 279 Ashley, H. 190, 191, 327 Attoh-Okine, N. O. 61, 85

Au, F. T. K 325 Averin, A. M. 90 Avis, D. 109, 110 Ayyub, B. 46 Azarm, S. 336 Babu, S. R. 317 Babuˇska, I. vii, xiv, 49, 85 Badcock, K. J. 208 Bae, H-R. xiii Bagchi, G. 163, 164, 208 Bagley, R. L. 158 Bai, E-W. 338 Bainer, L. 190 Baitsch, M. 144, 279 Balakrishna, C. 43, 45 Balandin, D. V. 150 Balasubramanyam, S. 208 Banichuk, N. V. 85, 325 Baratta, A. 61, 145, 175, 177 Barbieri, E. 85 Barthelemy, J. 27 Batill, S. M. 325 Baykasoglu, A. 30 Bayo, E. P. 292 Beck, J. L. xiii, 272 Beer, M. xiii, 43, 45, 46 Bellman, R. 336 Beltzer, A. I. 92 Belytschko, T. vii Ben-Haim, Y. ix, xi–xiii, 2, 57, 61, 85, 391

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144, 151, 189, 208, 212, 227, 228, 247, 274, 299, 331 Ben-Tal, A. xi, 26, 85, 227 Bendsøe, M. P. 20, 36 Berke, L. 89 Berleant, D. J. 15, 340 Bernardini, A. xiii, 43, 46 Bert, C. W. 335 Besharati, B. 336 Bezdek, J. 45 Bickford, W. B. 70 Bishop, J. A. 190 Black, M. 41 Blanc, G. 225 Bland, D. R. 150 Bletzinger, K.-U. 144, 281 Blockley, D. I. 45, 46 Bloebaum, C. L. 27 Boca, T. J. 336 Bochenek, B. 284 Bogdanovich, A. E. 227 Boley, B. A. ix, 77, 80–84 Bolotin, V. V. viii, ix, xiv, 191, 194 Booth, B. C. 61, 211, 325 Borkowski, A. 61 Borse, G. J. 323 Boyd, S. 107, 308 Brach, R. M. 325, 336 Bricmont, J. x Brown, C. B. 45 Budhiraja, A. S. 325 Budiansky, B. 227 Bulgakov, B. V. 149, 150 Burgman, M. 335 Cai, G. Q. xii, 14, 15, 144, 228, 339 Cai, K.-Y. 45 Calafiore, G. 61 Capiez-Lernout, E. 208 Cattafesta, R. T. 325 C ¸ akmak, H. S. 85, 89, 217, 221 Ceberio, M. 272

Chakraborty, S. 45 Chang, F. C. 336 Chateauneuf, A. 339 Chen, G. 189 Chen, J. 86, 89 Chen, L. 225 Chen, M. D. 325 Chen, R. 85 Chen, S.-H. 45, 56, 61, 70, 85–89, 112, 145, 146, 208, 209, 225, 325 Chen, S. Q. 45 Chen, W-F. 121 Chen, W. Y. 208 Cheng, G. 274, 332 Cheng, Y. S. 61, 274, 325 Cherkaev, A. V. 85, 325, 336 Cherkaev, E. 85, 325, 336 Chern, J. M. 336 Chernousko, F. L. 47, 57–61, 228, 247 Chessa, J. 225 Chi, C. C. 43 Chi-Ti, H. 144 Chiang, W. L. xiii, 45 Chiaruttini, V. 63, 144, 280 Chleboun, J. 85 Cho, M. 85, 144 Choi, K. K. 23, 45, 63, 274, 332 Chopra, A. K. 292 Christensen, R. M. 150 Chryssanthopoulos, M. K. 144 Chu, F. L. 208 Cinquini, C. 85 Claudio, D. 54, 56 Clough, R. W. 292 Cogan, S. 61 Cohn, M. Z. 39, 332 Cohon, J. L. 28 Colombi, P. 61, 212, 272, 339 Conrado, A. C. 144 Corliss, G. 85 Cornell, C. A. 1, 334 Corotis, R. 190, 209

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Author Index

Corotis, R. B. 39, 70 Cozzarelli, F. A. 151 Craig, K. J. 208 Cremona, C. 45 Cudney, H. 45 Cudney, H. H. 45 Cuntze, R. 40 Dafalias, J. F. 336 Damveld, H. 325 Dannon, V. 336 Das, P. K. 331 Dash, P. K. 112 Datcheva, M. 85 Davies, P. 208 de Almeida, S. F. M. 114, 129, 144, 280, 325 de Faria, A. R. 114, 129, 144, 280, 325 de Finetti, B. 338 de Palma, P. 325, 340 Deif, A. 87 Deml, M. 144 Dempster, A. P. 45 Demyanov, V. M. 336 Deng, L. 70 Deodatis, G. 112, 190 Der Kiureghian, A. 113, 292 Dessombz, O. 56, 225 Diaz-Padilla, J. 39 Dimarogonas, A. D. 56, 89 Ditlevsen, O. 44 Dong, W. M. xiii, 45 Dorn, W. 18 Dovi, A. 27 Drenick, R. F. ix, x, 163, 164, 176, 177, 189, 208, 339 Du, D. Z. 336 Du, L. 45 Duan, D. H. 208 Dube, R. 325 Dubois, D. xiii, 43, 45, 46

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Duplay, J. 225 Duvaut, G. 63, 144, 280 Dwyer, P. S. 49 Dyne, S. J. C. 208 Dzhur, Y. 85 Eastep, F. E. 190 El Damatty, A. A. 114, 274, 281 El Ghaoui, L. 61, 227 El-Hami, A. 274, 332 Eliseeff, P. 151 Elishakoff, I. ix–xiii, 2, 9, 10, 12, 13, 15, 45, 46, 49, 56, 57, 61, 85, 90, 92, 94, 103, 144–146, 148, 151, 163, 164, 173, 175, 189, 190, 197, 198, 208, 209, 211, 212, 217, 218, 220, 225, 227, 228, 230, 247, 248, 272, 273, 280, 288, 291, 299, 325, 331, 333, 334, 337–340 Ellingwood, B. 190, 209 Elliot, S.J. 336 Elseifi, M. 144 Elseifi, M. A. 61, 144 Eschenauer, H. A. vii, 36 Esteva, L. 317 Fang, J. 273, 288, 340 Fang, J. J. xiii, 45, 49 Fang, T. 14, 189, 339 Fatemi, J. 325 Faulkner, D. 331 Feigen, M. 39 FEMA 292 Feng, Y. 49 Ferracuti, B. x, 45 Ferson, S. 15, 52, 340 Feshbach, H. 92 Fest, E. 316 Feynman, R. F. vii Floudas, C. A. 32 Fl¨ ugge, W. 150 Foley, C. 85 Fox, R. L. 336

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Frangopol, D. M. 39, 40, 330, 331 Frank, D. 227 Freiberger, P. 46 Freudenthal, A. M. ix, 41, 333 Frommer, A. 49, 61, 89, 209 Fr´ yba, L. 208, 209, 274 Fuchs, M. B. 217, 336 Fujisawa, K. 26 Fukuda, K. 109, 110 Fuller, R. B. 102 Furuta, H. 45 Gabriel, D. 208 Gal, T. 26 Gana-Shvili, J. 61, 85, 101 Gangadharan, S. N. 86 Ganzerli, S. 45, 58, 61, 85, 189, 274, 279, 325, 340 Gao, W. 208 Gao, Y. 45 Garcelon, J. H. 70 Gasser, M. 40 Gay, D. M. 213 Gelatt, C. D. 35 Genalo, L. J. 190 Ghosn, M. 70 Giavotto, V. 211, 274, 279 Gill, P. E. 289 Gindy, N. 30 Ginzburg, L. R. 15, 340 Gioffre, M. 14 Givoli, D. 61, 85, 90, 92, 94 Glegg, S. 151 Glover, F. 32 Gnoenskii, L. S. 150 Godoy, L. A. 284 Goldberg, D. E. 30, 32 Golla, D. F. 153 Gomory, R. 18 Goncharenko, V. M. 227 Gonzales, A. 336 Good, I. J. 45

Goodman-Strauss, C. 15 Goosen, J. F. L. 211, 277, 279, 325 Gordeyev, V. 85 Grab, W. 43 Graham, H. 70 Graham, L. 112, 190 Graham, W. B. 153 Grandhi, R. V. xiii Grandori, G. 333 Greenberg, H. 18 Griffin, O. H. 86 Grigoriu, M. 14, 339 Gu, D. W. 30 Gu, X. 325 Gu, Y. 89 Gubanova, I. I. 193 Guo, S-X. 225 Gurav, S. P. 211, 277, 279, 325 G¨ urdal, Z. 18, 144 Guretskii, V. V. 336 Gurov, S. V. 45 Guz, A. N. 97 Haftka, R. T. 18, 30, 31, 45, 46, 70, 85, 86, 273, 276, 277, 288, 325, 328, 340 Hagaki, H. 211 Haider, G. S. 39 Hajagos, J. G. 272, 340 Hajela, P. 27, 274 Hakim, S. 336 Haldar, A. xiii Hammond, J. K. 208 Han, S. W. 292 Han, W. Z. 85 Hangai, Y. 299 Hanif, J. 317 Hansen, E. 52 Hansen, J. S. 129, 144, 280, 325 Hanss, M. 46 Hartmann, D. 144, 279 Hasofer, A. M. 2 Hattori, A. 279

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Author Index

Haug, E. J. 63, 273, 279 Hausen, Y. S. 325 Hayes, B. 47, 49 He, C. Y. 189 He, I. Q. 208 Heckert, A. 15, 339 Hemp, W. S. 18 Henderson, H. V. 70–72 Hertz, T. H. 190 Herzberger, J. 52, 87, 220 Hilding, D. 336 Hilton, H. H. 39, 151 Hirota, K. 42 Hjelmstad, K. D. 317 Hlav´ aˇcek, I. 85, 208 Ho, D. 114 Hoff, N. J. 231 Hoh, S. 211 Holmes, J. D. 15 Horn, R. A. 320 Horst, R. 23, 32 Hoshi, Y. 225 Howe, M. 334 Hsiao, J. H. 43 Hsu, D. S. 43 Hsu, J. 151 Hu, B. 150 Hu, J. X. 209 Huang, W. N. 151 Hughes, P. C. 153 Hughes, T. Y. R. vii Hunt, G. W. 113, 118, 123–125, 133, 136, 330 Huseyin, K. 124, 129 Hutchinson, J. W. 227 Iankov, R. 85 Ikeda, K. 113, 114, 124, 125, 134, 136, 281, 330 Il’yushin, A. A. 191 Inman, D. 151, 153, 154 Ishida, M. 90

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Isshiki, Y. 36 Ito, J. 316 Iturrizaga, R. M. 317 Ivanova, S. Y. 325 Iwaya, E. 225 Iyengar, R. N. 112, 189 Jabbari, F. 190 James, B. 27 Jamison, D. 45 Jensen, H. 45, 56 J´ez´equel, L. 189, 208, 225 Jia, H. G. 56, 209 Jiang, L. 274, 332 Jih-Hua, C. 144 John, F. 60 Johnson, C. R. 320 Johnson, J. L. 45 Ju, F. 336 Jung, J. J. 30 Kaifu, N. 299 Kalinowski, A. 336 Kalman, R. E. 2, 333 Kam, T.-Y. 45 Kamat, M. P. 18 Kanda, J. 15 Kandathi, R. 272 Kanno, Y. 26, 85, 89, 134, 281, 316, 323 Kapoor, M. P. 336 Kapur, K. C. 39 Kasyap, A. 325 Katoh, N. 26, 36 Kaveh, A. 317 Kavlie, D. 70 Keafott, R. B. 52, 85 Khachaturian, N. 39 Khalessi, M. R. 61 Khariton, L. E. 40 Kharitonov, V. 334 Kharmanda, G. 274, 332 Khodaparasi, H. H. 208

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Kida, T. 299 ´ Kildibekov, I. G. 227 Kim, C. 274, 332 Kim, N-H. 23, 63 Kim, S-E. 121 Kim, T-U. 144 Kim, Y. V. 15, 228 Kinoshita, T. 30, 307 Kinser, D. E. 39 Kirby, J. S. 151 Kirkpatrick, S. 35 Kirsch, U. 36, 69, 70 Klarbring, A. 336 Klir, G. J. 15, 42 Kogiso, N. 144, 209, 274 Koiter, W. T. 284 Kolakowski, P. 70 Koltunov, M. A. 150 Komatsu, K. 299 Komkov, V. 63, 336 Korneev, A. B. 150 Korst, J. 32, 35 Kosko, B. xiii, 16, 42, 44–46 Koˇcvara, M. 325 K¨ oyl¨ uoˇ glu, H. U. 225 K¨ oyl¨ uoˇ glu, U. 217, 220 Kozovkov, N. T. 150 Kreinovich, V. 61, 225, 272 Krenn, B. 284 Krichevskii, I. Ya. 150 Krishna, L. G. 150 Krishnamurthy, T. 45 Kucuk, I. 325, 336 Kulpa, Z. 85, 89 Kumamoto, H. 336 Kurzhanski, A. B. 58, 61, 63, 228, 247 Kuttekeuler, J. 190 Kuwazuru, O. 61 Kwak, B. M. 273, 279 Kwakernaak, H. xiii Ladev`eze, P. 49

Lain´e, J.-P. 56, 225 Lallement, G. 61 Lalvani, H. 102 Langhaar, M. 325 Langley, R. S. 228 Lanzi, L. 211, 274, 279 Larque, P. 225 Laville, F. 339 Lee, J. 86 Lee, K. 86 Lee, T. H. 30 Lefschetz, S. 308 Leichtweiss, K. 60 Leissa, A. W. 335 Lene, F. 63, 144, 280 Leng, H. N. 208 Leng, X. L. 14, 189, 339 Lewis, R. W. 43 Ley, R. 329 Li, J. 89 Li, Q. 45, 228, 272, 339 Li-Wei, L. 144 Li, Y. W. ix, 61, 144, 189, 208, 248 Lian, H. D. 70, 85, 89, 112, 208 Liaw, D. G. 190 Libai, A. 227 Librescu, L. 190 Lignon, S. 189, 208 Lin, Y. K. ix, xiii, 14, 61, 272, 299, 339 Lindberg, H. E. 61, 114, 144, 227 Lindley, D. V. 15 Liu, S. 69 Liu, Z. S. 56, 85, 208 Livne, E. 190 Lockhart, D. 227 Lockwood, E. H. 94 Lodwick, W. 45 Loftus, J. J. 45 Lomakin, V. A. 90, 92, 96 Lombardi, M. 85, 276, 277, 325 Longpr´e, L. 272 Lu, Q. S. 209

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Author Index

L¨ u, Z-Z. 225 Luenberger, D. G. 18, 23, 248 Luik, R. 69 Ma, J. 208 Ma, L. H. xii, 61, 89, 144 Ma, X. P. 14, 189, 339 Ma, Y. 86, 89, 209 Maglaras, G. 45 Majumder, L. 89 Makode, P. V. 70 Malozemov, V. N. 336 Manasyan, A. A. 336 Mang, H. A. 284 Manohar, C. S. 112, 189, 190 Maplesoft 125, 139 Mariani, M. 85, 325 Marler, T. 30 Marques, S. 208 Marti, K. 40 Martin, J. B. 336 Masur, E. F. 114 Matheron, G. xiii, 46 Matsumoto, M. 225 Maturana, S. 45 Maute, K. 40, 190, 330, 331 Mayer, G. S. 54 Mayer, M. 1 Maymon, G. 227 McNeill, D. 46 McTavish, D. J. 153, 158 McWilliam, S. 85, 89 Melosh, R. J. 69 Meng, G. A. 14, 189, 339 Micaletti, R. 112, 190 Michalopoulos, C. D. 149, 150 Michell, A. G. M. 18 Miki, M. 274 Minciarelli, F. 14 Mineau, D. 190 Minsky, M. 338 Mironovkii, L. S. 150

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397

Mitra, S. K. 301 Modares, M. 145, 208 Moens, D. 45, 212 Molfino, P. 85 Molinari, G. 85 M¨ oller, B. xiii, 43, 45, 46 Mongin, P. 339 Montgomery, D. C. 30, 31, 279 Mooij, E. 325 Moore, R. E. 47–49, 52, 53, 56, 87 Moreau, J. J. 61 Morimoto, M. 299 Morrison, W. J. 70, 72 Morse, P. M. 92 Morvan, G. 225 Moses, F. 39 Motro, R. 102, 109, 316 Mottershead, J. E. 208 Mourelatos, Z. P. 45 Moustafa, A. 189 Muhanna, R. L. 52–54, 85, 89, 145, 208, 211–213, 215, 216, 225 Mulkay, E. 225 Mullen, R. L. 52–54, 85, 89, 145, 208, 211–213, 215, 216, 225 M¨ uller, P. C. 49, 61, 89, 209 Mungan, I. vii Murota, K. 113, 134 Murotsu, T. 144, 274 Murotsu, Y. 340 Murray, W. 289 Myers, R. H. 30, 31, 279 Naishul, A. B. 150 Nakagiri, S. 61, 208, 225, 299 Nakamura, T. 27, 36 Nakayama, H. 279 Nassef, A. O. 114, 274, 281 Nastran 314 Natke, H. G. 208, 325 Nayak, A. 272 Neittaanm¨ aki, P. 85

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Nelsen, R. B. 340 Nelson, M. I. 332 Nemirovski, A. xi, 26, 85, 227 Nemish, Y. N. 97 Neuber, M. 90 Neumaier, A. 45, 52, 85, 89, 213, 220, 222 Nguyen, T. B. 150 Nickel, K. 213 Nielsen, S. R. K. 85, 89, 217, 221 Nikolaidis, E. 45, 46, 86, 144 Niordson, F. I. 327 Nishihara, O. 336 Nishiwaki, S. 20 Nobile, F. xiv, 49, 85 Norton, R. L. viii, ix Novomestky, F. 163, 164, 208 Oberguggenberger, M. 45, 46 Oberkampf, W. L. 340 Oden, J. T. vii, 49 Ohishi, Y. 103, 108 Ohsaki, M. xii, 20, 26, 27, 30, 35–37, 70, 76, 102, 103, 105, 108, 113, 114, 118, 124, 125, 134, 136, 280–283, 293, 307, 308, 316, 323, 330 Ohta, Y. 45 Oide, K. 114, 134 Olhoff, N. vii, 114, 274, 332 Onipede, O. 317 Otsubo, H. 225 Ottl, D. 153 Ovseyevich, A. I. 15, 228 Owen, S. 30 Paez, T. L. 336 Pal, N. R. 46 Palassopoulos, G. V. 274 Palmov, ´ V. A. 90 Pan, P. 30, 307 Panagiotopoulos, P. D. 61 Pang, J.-S. 336

Panovko, Y. G. 193 Pantelides, C. P. 45, 58, 61, 85, 114, 144, 149, 189, 190, 211, 274, 279, 325 Papadopoulos, V. 112 Papadrakakis, M. 112 Papoulis, A. 336 Pardalos, P. M. 336 Parimi, S. R. 39 Park, C. B. 208 Park, J. 30 Park, Y. H. 274 Parkinson, C. N. 329 Parnes, R. 92 Peek, R. 114 Peltier, A. 329 Penmetsa, R. C. 272 Penzien, J. 177, 292 Peterson, R. E. 90 Pettit, C. L. xiii Philappacopoulos, A. J. 189, 208 Pierson, B. L. 190 Pietrzak, J. 283 Pilkey, W. D. 336 Plaut, R. H. 190, 336 Pletner, B. 148, 163, 164, 189 Pleˇsek, J. 208 Pochtman, Y. M. 40 Poggi, C. 144 Polak, E. 113 Popova, E. D. 85 Popplewell, N. 336 Postlethwaite, I. 30 Pothisiri, T. 317 Powell, G. H. 70 Pownuk, A. 85, 89, 325 Prade, H. xiii, 45, 46 Prager, W. 18, 107 Prakash, B. G. 43, 45 Prokhorov, V. N. 150 Pu, Y. 331 Puel, G. 49 Pugsley, A. G. 330

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Author Index

Puig, V. 336 Putresza, J. T. 70 Qiu, Z. P. xii, 45, 49, 56, 61, 85–89, 144, 146, 173, 208, 209, 272, 339 Quevedo, J. 336 Rackwitz, R. 40, 334 Ramirez, M. R. 70 Ramm, E. 144, 281 Rao, C. R. 301 Rao, S. S. 43, 45, 89, 225 Rasmussen, S. H. 114 Reaveley, L. D. 189, 325 Reddy, R. K. xiii Reeves, C. 23, 32 Reitinger, R. 144, 281 Ren, Y. J. 218 Renaud, J. E. 325 Reshetnyak, Y. N. 15, 228 Rhee, S. Y. 85, 144 Richardson, A. 317 Richter, A. 144, 280, 325 Riks, E. 116 Riley, T. A. 149, 150 Ringertz, U. T. 190 Rohn, J. 52 Roitenberg, Ya. N. 150 Romeuf, T. 49 Rosenblatt, F. 30 Roth, D. A. 39 Roux, W. J. 31 Roy, R. 69 Royset, J. O. 113 Rozvany, G. I. N. 18, 36 Rubinstein, M. F. 69 Ruiz, P. 177 Rump, S. M. 213 Russell, B. 43 Russell, S. 45 Russo, F. 45 Rustem, B. 334

Rzhanitsyn, A. R. ix Sage, A. P. 150 Sakakibara, O. 299 Sakata, S. 30, 279 Salle, J. L. 308 Saludes, Y. 336 Sam, P. C. 45 Sanayei, M. 317 Sandberg, I. W. 30 Sangalli, A. 46 Sarkar, A. 112, 189, 190 Sarma, K. 40 Sato, M. 338 Saunders, M. A. 289 Savin, G. N. 90 Savoia, M. xiii Saxena, V. 225 Sayles, R. S. 90 Schanz, T. 85 Schek, H. J. 104, 318 Schiehlen, W. 150 Schmidt, R. 335 Schmit, L. A. 299 Schmitendorf, W. E. 190 Schorrock, P. A. 134 Schranz, C. 284 Schu¨eller, G. I. 41, 332, 333 Schweppe, F. C. 57, 61, 247 Searle, R. 70–72 Sensmeier, M. D. 86 Sepulveda, A. E. 299 Sevin, E. 336 Sexmith, R. G. 332 Shao, S. 144, 274, 340 Sharuz, S. M. 150 Shea, K. 316 Sheinin, V. I. 90 Sheplak, M. 325 Sherman, J. 70, 72 Shih, C. J. 43 Shimanovsky, A. 85

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Shinozuka, M. ix, 112, 145, 151, 163, 164, 175, 177, 187–190, 208, 218, 248 Shiraishi, N. 45 Shirk, M. H. 190 Sigmund, O. 20, 36 Simiu, E. 14, 15, 339 Sin, H-C. 144 Skalna, I. 85, 86, 89 Slaev, V. A. 150 Smallwood, D. O. 336 Smith, I. F. C. 316 Smith, S. xiii, 45 Sniedovich, M. 334–336 Sobieszczanski-Sobieski, J. 27 Soize, C. 49, 208 Sokal, A. x Song, D. 56, 85–89, 208 Soong, T. T. 189, 190, 325, 332 Sophie, Q. C. 45 Soucy, Y. 153 Spunt, L. 273 Srinivasan, S. 190, 209 St. Clair, U. H. 42 Stander, N. 31, 208 Starks, S. A. 272 Starnes, Jr., J. H. ix, xii, 15, 30, 61, 144, 189, 208, 209, 228, 248 Stasenko, I. V. 150 Stingl, M. 325 Stix, R. 9 Stolpe, M. 107 Stoyanov, J. 7 Striz, A. G. 190 Stroud, W. J. 45 Sunaga, T. 49 Sundararaju, K. 43, 45 Supple, W. J. 123, 280 Suzuki, K. 208, 225 Svanberg, K. 107 Svetlitskii, V. A. 150 Sˇ olin, P. 225

Switzky, H. 39 Tagami, S. 90 Tagawa, H. 307 Takagi, J. 114 Takeuchi, M. 281 Takewaki, I. 85, 89, 190, 211 Takizawa, Y. 299 Taleb, N. 15 Tang, W. H. 14 Taroco, E. O. 284 Taylor, J. E. 18 Templeman, A. B. 332 Tempone, R. xiv, 49, 85 Terada, K. 114, 134 Tham, L. G. 61, 274, 325 Thimbleby, H. 329 Thoft-Christensen, P. 331 Thomas, H. L. 299 Thompson, J. M. T. 113, 118, 123–125, 133, 134, 136, 280, 330 Thouverez, F. 56, 225 Tikhomirov, V. M. 335 Tonon, F. xiii, 46 Toporov, V. 325 Torres, R. 272 Torvik, P. J. 158 Triantafyllidis, N. 114 Tseng, C. H. 127 Tseng, W. K. 336 Tucker, W. T. 340 Tunguskova, Z. G. 91, 99 Turner, M. J. 190 Tuy, H. 23, 32 Tzan, S.-R. 61, 149, 189, 190 Uchida, A. 293 Uetani, K. 281 Ulanov, G. M. 150 Utkin, L. V. 45 Valyi, I. 58, 61, 63, 247

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Author Index

van Dyke, M. 92 van Keulen, F. 211, 277, 279, 325 van Manen, S. 144 Vandenberghe, L. 107, 308 Vandepitte, D. 45, 212 Vanderplaats, G. N. 18, 27, 329 Vanmarcke, E. H. 39 Varaiya, P. 63 Vassart, N. 316 Vecchi, M. P. 35 Venini, P. 85, 209, 325 Venkataraman, S. 328 Venkayya, V. B. 18, 190 Venter, G. 30, 325 Verijenko, V. E. 144, 280, 325 Vermeulen, P. G. 144 Vick, S. G. 46 Vinot, P. 61 Vitali, F. 225 Vittal, S. 274 von Hoerner, S. 299 VR&D 120, 310 Wald, A. 336 Wall, F. J. 112 Walshan, D. 15 Walster, G. W. 52 Wang, D. 89 Wang, G. Y. 43 Wang, K. Y. 208 Wang, P. C. 208 Wang, W. Q. 43 Wang, X. B. 208 Wang, X. J. xii, 45, 61, 85, 86, 89, 144, 173, 208, 209 Ward, A. C. 85 Warmus, M. 49 Watanabe, A. 340 Watanabe, S. 336 Watson, L. T. 86 Weibull, W. 3 Weiner, J. H. 80

Weisshaar, T. A. 190 Wentzel, E. C. 1 Westermo, B. D. 336 Whidborne, J. F. 30 Wilson, E. L. 292 Witte, A. F. 336 Wolfram, S. 9 Wong, F.-S. xiii, 45 Wood, K. L. xiii Woodbury, M. A. 70, 72 Wu, C. F. 14 Wu, J. 145, 208, 325 Wu, T. S. 84 Wu, X. M. 70, 208 Wu, Z. 190 Wunderlich, W. 144 Xiang, G. 61, 225, 272 Xie, L-L. 190 Xu, C. 43 Xu, L. 274, 332 Yamada, T. 225 Yamaguchi, T. 209 Yamakawa, H. 43, 45, 209 Yan, S. O. 208 Yang, D. 45, 272, 339 Yang, J. L. 209 Yang, J. N. 190, 332 Yang, T. Y. 190 Yang, X. W. 70, 85, 89, 112, 208, 225 Yang, Z. J. 70, 208 Yao, J. T. P. 45, 46 Yau, W. 336 Yeh, Y. C. 43 Yoshida, N. 27 Yoshikawa, N. 61, 190, 208, 209, 225, 274, 299 Youn, B. D. 45, 274 Young, R. C. 49 Youssef, N. A. N. 336 Yuan, B. 42

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Yun, C. B. 208 Zadeh, L. A. 42, 44 Zaguskin, V. L. 60 Zako, M. 30, 279 Zartarian, C. 191 Zavoni, E. H. 317 Zeng, G. W. 325 Zhai, C-H. 190 Zhang, H. 53, 211–213, 215, 225 Zhang, J. 340 Zhang, J. H. 189, 325

Zhang, J. Y. 102, 103, 105, 108, 293, 316, 323 Zhang, N. 208 Zhang, T. Y. 316 Zhang, Z. Y. 325 Zharii, O. Y. 144 Zheng, G. W. 61, 274 Zheng, Y. 43 Zhou, J. 45 Zhu, L. P. ix, xiii, 61, 272, 299, 339 Zingales, M. 49, 190, 197, 198, 228, 248 Zuccaro, G. 61, 145, 175, 177

optimization