Table Of ContentSTUDIES IN APPLIED MECHANICS
1. Mechanics and Strength of Materials (Skalmierski)
2. Nonlinear Differential Equations (FuSik and Kufner)
3. Mathematical Theory of Elastic and Elαstico-Plastic Bodies.
An Introduction (NeCas and HlavαCek)
4. Variational, Incremental and Energy Methods in Solid Mechanics and Shell
Theory (Mason)
5. Mechanics of Structured Media, Parts A and Β (Selvadurai, Editor)
6. Mechanics of Material Behavior (Dvorak and Shield, Editors)
7. Mechanics of Granular Materials: New Models and Constitutive Relations
(Jenkins and Satake, Editors)
8. Probabilistic Approach to Mechanisms (Sandler)
9. Methods of Functional Analysis for Application in Solid Mechanics (Mason)
10. Boundary Integral Equation Methods in Eigenvalue Problems of Elastodynamics
and Thin Plates (Kitahara)
11. Mechanics of Material Interfaces (Selvadurai and Voyiadjis, Editors)
12. Ljocal Effects in the Analysis of Structures (Ladeveze, Editor)
13. Ordinary Differential Equations (Kurzweil)
14. Random Vibration — Status and Recent Developments (Elishakoff and
Lyon, Editors)
15. Computational Methods for Predicting Material Processing Defects
(Predeleanu, Editor)
16. Developments in Engineering Mechanics (Selvadurai, Editor)
17. The Mechanics of Vibrations of Cylindrical Shells (Markup)
18. Theory of Plasticity and Limit Design of Plates (Markup)
19. Buckling of Structures—Theory and Experiment. The Josef Singer
Anniversary Volume (Elishakoff, Babcock, Arbocz and Libai, Editors)
20. Micromechanics of Granular Materials (Satake and Jenkins, Editors)
21. Plasticity. Theory and Engineering Applications (Kaliszky)
22. Stability in the Dynamics of Metal Cutting (Chiriacescu)
23. Stress Analysis by Boundary Element Methods (Bala§, SIαdek and SIαdek)
24. Advances in the Theory of Plates and Shells (Voyiadjis, Karamanlidis, Editors)
25. Convex Models of Uncertainty in Applied Mechanics (Ben-Haim, Elishakoff)
STUDIES IN APPLIED MECHANICS 25
C o n v ex M o d e ls of U n c e r t ; a i n t :y
in A p p l i ed M e c h a n i cs
Yakov Ben-Haim
Faculty of Mechanical Engineering, Technion - Israel Institute of Technology,
Haifa, Israel
and
Isaac Elishakoff
Center for Applied Stochastics Research and Department of Mechanical Engineering,
Florida Atlantic University, Boca Raton, Florida, U.S.A.
EL5EVIER
Amsterdam — Oxford — New York — Tokyo 1990
ELSEVIER SCIENCE PUBLISHERS B.V.
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Library of Congress Cataloging-1n-PublIcatlon Data
Ben-Haim, Yakov. 1952-
Convex models of uncertainty In applied mechanics / Yakov Ben-Halm
and Isaac Elishakoff.
p. cm. — (Studies in applied mechanics ; 25)
Includes bibliographical references.
ISBN 0-444-88406-8
1. Convex sets. 2. Probabilities. 3. Mechanics, Applied.
I. Elishakoff, Isaac. II. Title. III. Series.
QA640.B45 1990
620. 1—dc20 89-77748
CIP
ISBN 0-444-88406-8 (Vol. 25)
ISBN 0-444-41758-3 (Series)
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Y.B.-H.
To
Esther, Ben and Orly
I.E.
vil
Foreword
Professors Ben-Haim and Elishakoff honored me with their request
that I add some introductory remarks to those they make themselves
in this book, and make better. It is perhaps disquieting that any such
remarks should be desirable. Their book is timely, even overdue, and
should be widely welcomed. It treats a problem that recurs in most
applications of mathematics to the real world, namely that of how to
accommodate uncertainties in the empirical and theoretical information
which underlies such applications.
That an accommodation is absolutely necessary has been recognized
for a long time in the natural sciences as well as in engineering, and
the traditional way of achieving it has been to equate uncertainty with
probability. It is an idea that was, without question, spectacularly
successful. In physics, it produced statistical mechanics and quantum
theory. It entered into engineering perhaps mainly through the work
of S.O. Rice on noise in electronic devices and of N. Wiener on radar
fire control, and it led to C.E. Shannon's information theory, among
others.
Not all of these successes were unqualified, however, and engineers
in particular did not accept them without questioning. Chap. 1 of this
book quotes from a large literature that documents the soul-searching
that went on. (The development by L.A. Zadeh of the theory of fuzzy
sets and of its extensions is in a way another testimonial to this tech
nological queasiness.) An engineer, as opposed to a physicist, must
present a client with a design that embodies certain performance as
surances. To give these, at least ideally, he should utilize only facts and
information that are known to him on a comparable confidence level.
The trouble, of course, is that in practice much of his information is of
lower quality and hence of much greater uncertainty. Chap. 1 exhibits
a gallery of cases in point. These examples also show that probability
viii FOREWORD
theory cannot be relied upon for help. There are situations in which
it merely compounds the difficulty, even when its application makes
superficial sense. In others, it makes no sense in the first place.
The customary way out of the dilemma for the engineer has been
his trusted safety factor. He has done so on the theory that it is better
to over-design, and feel comfortable with one's performance assurances,
than to under-design and run the risk of a failure. In other words, it is
better to be wasteful than to court embarrassment or danger.
Professors Ben-Haim and Elishakoff have a better way. They point
out in this book that the designer should make clear to himself what he
knows on a sufficiently high confidence level, and then use this knowl
edge to place bounds on the design data and parameters that he can
properly utilize. In effect, this will isolate for him a set of possible
designs. The book deals with the case in which this set is convex and
demonstrates that a very broad range of problems in applied mechan
ics actually satisfies the convexity requirement. Their approach is novel
and highly welcome. In my opinion, it is inevitable that it, and its ex
tensions, will dominate the future practice of engineering.
Professor-Emeritus Rudolf F. Drenick
Polytechnic Institute of New York
Huntington, New York
August, 1989
XUl
Preface
If one thing today is certain, it is a feeling of uncertainty —
a premonition that the future cannot be a simple extension of
the present.^
Dear Reader:
The monograph before you deals with analysis of uncertainty in
applied mechanics. It is an ambitious book. It is not just one more
rendition of the traditional treatment of the subject, nor is it intended
to supplement existing structural engineering books. The ultimate goal
of this volume is to fill what the writers are confident is a significant
gap in the available professional literature. This book is about non-
probabilistic modelling of uncertainty. We have no ambition to supplant
probabilistic ideas. Rather, we wish to provide an alternative avenue
for analysis of uncertainty when only a limited amount of information is
available. Whether this far-reaching goal has been successfully achieved
is for you to judge.
Applied mechanics monographs and textbooks and associated uni
versity courses have historically focussed on the basic principles of me
chanics, analysis, numerical evaluation and design. These books and
courses almost exclusively assume that the material properties, geomet
ric dimensions, and loading of the structure are precisely known.
Nobody questions the ever-present need for a solid foundation in
applied mechanics. Neither does anyone question nowadays the fun
damental necessity to recognize that uncertainty exists, to learn to
evaluate it rationally, and to incorporate it into design.
But what do we mean by "uncertain"? Webster's Ninth New Colle-
^ Queen Beatrix of the Netherlands, speech of allegiance to the Dutch constitu
tion, April 30, 1980.
xiv PREFACE
giate Dictionary provides the following definitions: "Indefinite, indeter
minate; not certain to occur; problematical; not reliable; untrustwor
thy; not known beyond doubt; dubious; not having certain knowledge;
doubtful; not clearly identified or defined; not constant; variable, fitful".
Among the numerous synonyms to "uncertainty" one finds (Rodale,
1978): unsureness, unpredictability, randomness, haphazardness, arbi
trariness, indefiniteness, indeterminacy, ambiguity, variability, change
ability, irregularity, and so forth.
Recognition of the need to introduce the ideas of uncertainty in
a wide variety of scientific fields today reflects in part some of the
profound changes in science and engineering over the last decades.
The natural question arises: how to deal with uncertainty? Since
one of the meanings of uncertainty is randomness, a natural answer to
this question was and is to apply the theory of probability and random
functions. Indeed, since the pioneering work by Meier in Germany, and
by Freudenthal in Israel and later in the United States, probabilistic
structural mechanics has achieved a high degree of sophistication. The
power of probabilistic methods has been demonstrated beyond doubt
in numerous publications on engineering subjects of paramount im
portance. Probabilistic modelling requires extensive knowledge of the
random variables or functions involved. It leads to evaluation of the
probability of successful performance by the structure, called reliability,
or to its complement, the probability of failure. Is probabilistic mod
eUing the only way one could deal with uncertainty? It turns out that
the answer is negative. Indeed, the indeterminacy about the uncertain
variables involved could be stated in terms of these variables belonging
to certain sets, such as:
1. The uncertain parameter ÷ is bounded, |x| < a.
2. The uncertain function has envelope bounds,
íClower(í) < X{t) < »upper(0
where a;iower(0 and Xupper(i) are deterministic functions which de
limit the range of variation of x{t).
3. The uncertain function has an integral square bound:
PREFACE XV
OO
I x\t)dt<a
More examples will be found throughout the book. Instead of precise
information on the probability content of random events, we possess im
perfect, scanty knowledge on the uncertain quantities. This description
of uncertainty is a set-theoretic, non-probabilistic one. We will limit
ourselves, in this monograph, to representation of uncertain phenomena
by convex sets, and will refer to this approach as convex modelling.
This set-theoretic approach was independently and almost simul
taneously pioneered by Schweppe and by Drenick in 1968, the former
in control theory and the latter in applied mechanics, namely in mod
elling of the response of structures to earthquake excitation. Schweppe
summarized much of the early work in non-probabilistic modelling of
uncertainty in his monograph Uncertain Dynamic Systems (1973). In
dependently, and without knowledge of these works, convex models
of uncertainty were applied by Ben-Haim (1985) in his monograph on
assay of material with uncertain spatial variation.
There is a similarity between probabilistic and convex modelling.
Probabilistic modelling provides the probability of successful perfor
mance which is, roughly speaking, the probability of avoidance of the
distribution tails of the random variables controlling the system. In
similar fashion, convex modelling provides an evaluation based on as
sessments of extremal or delimiting properties of the system. In both
circumstances, uncertain parameters lead to certain quantitative de
scriptions, but of different kinds.
Is there a contradiction between probabilistic and convex modelling?
We want to preface the answer to this question by more general con
siderations. Since the emergence of modern scientific methodology, to
understand a natural phenomenon has meant to formulate a conceptual
model which describes or predicts selected aspects of the phenomenon.
To understand is to control, and the scientific model is the pre-eminent
tool of the engineer in his quest for controlling man's environment. The
scientific model is a distillation of reality. In attempting to explain this
reality — to describe it qualitatively and quantitatively — we find that
the same phenomenon can be represented by more than one model.
xvi PREFACE
The validity of each model can be established by checking its compati
bility with other models or with experimental observations. This done,
the question naturally arises: which of these models is preferable? The
answer to this question lies in adopting a criterion and setting an ob
jective in constructing the model. The criterion in question is clearly
one of usefulness.
Different models may refer to different aspects of a phenomenon
and yield answers to specific questions in each case. The utilitarian
approach to evaluating the multiplicity of models is both pragmatic
and tolerant. Different models can "peacefully coexist" as long as they
serve different purposes.
So the answer to our question as to possible contradiction between
probabilistic and convex modelling is negative. The engineer is a most
pragmatic creature and willingly allows coexistence of both models, as
long as each fills a useful role. The engineer with professional respon
sibility, whose aim is in-depth study of complex phenomena, cannot
allow himself to discard any useful model.
The organization of the monograph is as follows. The first chapter
briefly reviews probabilistic methods and discusses the sensitivity of
the probability of failure to uncertain knowledge of the system. Chap
ter two discusses the mathematical background of convex modelling.
In the remainder of the book, convex modelling is applied to various
linear and nonlinear problems. Except for the first two chapters, the
material is almost exclusively based on our joint research, which con
tinuously stimulates us to further the development of the subject, in
nearly perfect cooperation and harmony (not free from an occasional
"fight" on this or that idea, sentence or word). Accordingly, the sole
responsibility for mistakes, misprints and misconceptions, as well as for
successful achievement of our goals (which we hope for and count on)
rests on both of us. One of our goals will be met if a serious reader,
after initial perusal, is encouraged to read the book and feels that "it
makes sense" despite the lack of esoteric mathematics.
Many disciplines originate in the efforts of a small group of re
searchers. When the ideas generated are seen to be related to real
problems, they spread to a larger group. New questions then arise,
stimulating the growth and spread of the discipline. If this happens
as a result this book, then our goal will have been more than fully
