Safety Factors and Reliability: Friends or Foes?

303 Pages·2004·9.682 MB·English

by  Isaac Elishakoff (auth.)

Safety Factors and Reliability: Friends or Foes?
Safety Factors and Reliability:
Friends or Foes?
by
ISAAC ELISHAKOFF
J.M. Rubin Professor of Structural Reliability, Safety and Security,
Department of Mechanical Engineering,
Florida Atlantic University,
Boca Raton, Florida, U.S.A.
SPRINGER SCIENCE+BUSINESS MEDIA, B.V.
A c.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN 978-90-481-6500-1 ISBN 978-1-4020-2131-2 (eBook)
DOI 10.1007/978-1-4020-2131-2
Printed on acid-free paper
AII Rights Reserved
© 2004 Springer Science+Business Media Dordrecht
OriginalIy published by Kluwer Academic Publishers in 2004
Softcover reprint ofthe hardcover lst edition 2004
No part of this work may be reproduced, stored in a retrieval system, or transmitted
in any form or by any means, electronic, mechanical, photocopying, microfilming,
recording or otherwise, without written permis sion from the Publisher, with the
exception of any material supplied specificalIy for the purpose of being entered
and executed on a computer system, for exclusive use by the purchaser of the work.
Dedicated to the blessed memory
ofm y beloved brother Moshe
Contents
Chapter 1 Prologue
Chapter 2 Reliability of Structures
2.1 Introductory Comments ................................................................................................... 16
2.2 Basic Concepts ................................................................................................................ 19
2.3 How Accurate Is Minimum Distance Reliability Index? ................................................ 28
2.4 Safety Factors as Discussed in Literature ........................................................................ 33
2.5 About the Acceptable Probability of Failure ................................................................... 39
2.6 A Priority Question .......................................................................................................... 42
2.7 Concluding Comments on the Stress-Strength Interference Method .............................. 44
Chapter 3 Safety Factors and Reliability: Random Actual Stress & Deterministic
Yield Stress
3.1 Introductory Comments ................................................................................................... 47
3.2 Four Different Probabilistic Definitions of a Safety Factor ............................................ 48
3.3 Case 1: Stress Has an Uniform Probability Density, Strength Is Deterministic .............. 52
3.4 Case 2: Stress Has an Exponential Probability Density, Yield Stress Is Deterministic. 55
3.5 Case 3: Stress Has a Rayleigh Probability Density, Yield Stress is Deterministic ......... 56
3.6 Case 4: Stress Has a Normal Probability Density, Yield Stress Is Deterministic ........... 58
3.7 Case 5: Actual Stress Has a Log-Normal Probability Density, Yield Stress is
Deterministic ................................................................................................................... 59
3.8 Case 8: Actual Stress Has a Weibull Probability Density, Strength is Deterministic .... 61
3.9 Actual Stress Has a Frechet Probability Distribution, Yield Stress Is Deterministic .... 63
3.10 Actual Stress Has Two Parameter Weibull Probability density, Yield Stress Is
Deterministic ................................................................................................................... 67
3.11 Actual Stress Has a Three Parameter Weibull Probability Density, Yield Stress Is
Deterministic ................................................................................................................... 69
3.12 Discussion: Augmenting Classical Safety Factors, via Reliability ................................. 72
Chapter 4 Safety Factors and Reliability: Deterministic Actual Stress & Random
Yield Stress
4.1 Yields Stress Has an Uniform Probability Density, Actual Stress is Deterministic ........ 76
4.2 Yield Stress Has an Exponential Probability Density, Actual Stress Is Deterministic ... 79
4.3 Strength Has a Rayleigh Probability Density, Stress Is Deterministic ............................ 83
4.4 Various Factors of Safety in Buckling ............................................................................ 85
4.5 Yield Stress Has a Weibull Probability Density, Actual Stress Is Deterministic ............ 88
4.6 Yield Stress Has a Frechet Distribution, and Actual Stress Is Deterministic .................. 90
4.7 Yield Stress has a Three Parameter Weibull Distribution, and Actual Stress Is
Deterministic ................................................................................................................... 92
vii
Vlll
4.8 Yield Stress Has a Two Parameter Weibull Distribution, and Actual Stress is
Deterministic ................................................................................................................... 93
4.9 Concluding Comments on Proper Distribution Functions .............................................. 95
Chapter 5 Safety Factor and Reliability: Both Actual Stress and Yield Stress Are
Random
5.1 Introductory Comments ................................................................................................... 97
5.2 Both Actual Stress and Yield Stress Have Normal Probability Density ......................... 98
5.3 Actual Stress Has an Exponential Density, Yield Stress Has a Normal Probability
Density ........................................................................................................................... 101
5.4 Actual Stress Has a Normal Probability Density, Strength Has an Exponential
Probability Density ........................................................................................................ 104
5.5 Both Actual Stress and Yield Stress Have Log-Normal Probability Densities ............. 104
5.6 The Characteristic Safety Factor and the Design Safety Factor .................................... 106
5.7 Asymptotic Analysis ...................................................................................................... 107
5.8 Actual Stress and Yield Stress Are Correlated .............................................................. 109
5.9 Both Actual Stress and Yield Stress Follow the Pearson Probability Densities ........... 109
5.1 0 Conclusion: Reliability and Safety Factor Can Peacefully Coexist .............................. 111
Chapter 6 Non-Probabilistic Factor of Safety
6.1 Introductory Comments ................................................................................................. 118
6.2 Sensitivity of Failure Probability ................................................................................... 120
6.3 Remarks on Convex Modeling of Uncertainty .............................................................. 127
6.4 "Worst-Case" Probabilistic Safety Factor ..................................................................... 133
6.5 Which Concept Is More Feasible: Non-Probabilistic Reliability or Non-Probabilistic
Safety Factor? ................................................................................................................ 138
6.6 Concluding Comments on How to Treat Uncertainty in a Given Situation .................. 142
Chapter 7 Stochastic Safety Factor by Birger and Maymon
7.1 Introductory Comments ................................................................................................. 145
7.2 Definition of Stochastic Safety Factor ........................................................................... 146
7.3 Implication of the Stochastic Safety Factor ................................................................... 147
7.4 Cantilever Beam with Restricted Maximum Displacement .......................................... 150
7.5 Concluding Comments .................................................................................................. 152
Chapter 8 Safety Factor in Light of the Bienayme-Markov and Chebychev
Inequalities
8.1 Bienayme-MarkovInequality ........................................................................................ 154
8.2 Use of the Bienayme-Markov Inequality for Reliability Estimation ............................ 155
8.3 Derivation of the Chebychev's Inequality ..................................................................... 157
ix
8.4 Application of the Chebychev's Inequality: Mischke's Bound ..................................... 159
8.5 Application of the Chebychev's Inequality by My Dao-Thien and Massoud ............... 165
8.6 Examples ....................................................................................................................... 171
8.7 Conclusion: Other Bounds of Probability of Failure ..................................................... 173
Chapter 9 Japanese Contributions to the Interrelating Safety Factor and Reliability
9.1 Introduction ................................................................................................................... 176
9.2 Ichikawa's Formula ....................................................................................................... 176
9.3 Reiser's Correction ........................................................................................................ 178
9.4 Another Set of Formulas by Ichikawa and Reiser ......................................................... 179
9.5 Application of the Camp-Meidell Inequality ................................................................. 180
9.6 Series Representation of the Probability Density Functions ......................................... 185
9.7 Use of the Edgeworth Series by Murotsu et al ............................................................. 187
9.8 Hoshiya's Distinction of Seemingly Equivalent Designs ............................................. 192
9.9 Contribution by Konishi et al: Proof Loads ................................................................... 194
9.10 Concluding Remarks ..................................................................................................... 195
Chapter 10 Epilogue
Bibliography ............................................................................................................................ 204
Appendix A Accuracy of the Hasofer-Lind Method
A.l Introductory Comments ................................................................................................. 255
A.2 Beam Subjected to a Concentrated Force ...................................................................... 257
A.3 Approximate Solutions .................................................................................................. 258
A.4 Exact Solution ................................................................................................................ 259
A.5 Design of Structural Element ........................................................................................ 260
Appendix B Biographical Notes
I-J. Bienayme ........................................................................................................................... 264
P.L. Chebychev ........................................................................................................................ 264
Ch. A. de Coulomb .................................................................................................................. 265
A. M. Freudenthal .................................................................................................................... 267
A.M. Kakushadze .................................................................................................................... 268
G. Kazinczy ............................................................................................................................. 269
M. Mayer ................................................................................................................................. 270
G.M. Mukhadze ....................................................................................................................... 272
L.M.H. Navier ......................................................................................................................... 273
A.R. Rzhanitsyn ....................................................................................................................... 274
N.S. Streletskii ......................................................................................................................... 275
x
Author Index ................................................................................2 78
Subject Index ...............................................................................2 89
Chapter 1
Prologue
"The safety of the building constructions is a matter of calculating
probabilities. "
M. Mayer (1926)
"The values of safety factors, as well as closely associated values of
design loads and design resistances, were improved and modified
mainly empirically, by the way of generalization of multi-year
experience and exploitation of the structures. Yet, as is seen of the
essence of the problem in principle, there are also theoretical
approaches possible with wide application of the apparatus of theory of
probability and mathematical statistics."
V.Y. Bolotin (1965)
"Probability theory provides a more accurate engineering
representation of reality. Many leading civil engineers in many
countries have written of the statistical nature of loads and of material
properties."
C.A. Cornell (1969)
"The times of straightforward structural design, when the structural
engineer could afford to be fully ignorant of probabilistic approaches to
analysis, are definitely over."
A.M. Lovelace (1972)
"As a person who was brought up on factors of safety and used them all
his professional life, their simplicity appeals to me. However, if we are
to make any progress the bundle has to be unbundled, and each of the
constituents correctly modeled ... "
A.D.S. Carter (1997)
It is a conventional wisdom to maintain that the scientists and engineers, who earn a
living by being engaged in applied mechanics, are divided into two groups: those in the
first, traditional group deal with deterministic mechanics, whereas the representatives of
the second, more recent, group devote themselves to non-deterministic mechanics. The
traditionalists neglect uncertainties in the loading conditions, in the mechanical properties
of structures, in boundary conditions, in geometric characteristics and in other parameters
entering into the description of the problem at hand.
The second group embraces itself with various analyses of uncertainty. Within
this group, there are those who are active in probabilistic mechanics. They maintain that
uncertainty is identifiable with randomness, and hence methods of classical or modem
probability theories should be applied. Other investigators developed fuzzy sets based
theories, formulating their analysis on the principle developed by Lofti Zadeh:
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I. Elishakoff, Safety Factors and Reliability: Friends or Foes?
© Springer Science+Business Media Dordrecht 2004
2
"As the complexity of a system increases, our ability to make precise and yet
significant statements about its behavior diminishes until a threshold is reached
beyond which precision and significance (or relevance) become almost mutually
exclusive characteristics."
The theorists utilizing the fuzzy sets approach base their approach on the notion
of the membership function, in contrast to the concept of the probability density function
utilized by probabilists. Whereas the two approaches seem to be radically different, they
have one thing in common: They both use an uncertainty measure. Engineers developed
also an uncertainty analysis that does not demand a measure. The latter method is known
by its various appellations: method of accumulation of disturbances, unknown-but
bounded uncertainty, convex modeling of uncertainty, anti-optimization and more
recently information-gap methodology.
The set of scientists belonging to the groups dealing with the deterministic
mechanics or with the non-deterministic mechanics are not mutually exclusive. A very
limited number of scholars simultaneously deal with both approaches. Even smaller
numbers of researchers are engaged in seemingly competitive uncertainty analyses
techniques. Worldwide, these investigators apparently can be counted on the fingers of
one hand. The latter researchers appear to exercise tolerance by holding several
contradicting opinions since they find similarities or analogies between above
contradictory approaches, indeed; according to F.S. Fitzgerald,
"the test of a first-rate intelligence is the ability to hold to opposite ideas in the
mind at the same time, and still retain the ability to function."
The main goal of this prologue is not a classification of researches in applied
mechanics, but rather, to demonstrate that the above characterization, in essence, is
imprecise ifnot altogether wrong. The main thesis that we would like to propagate is this:
In actuality there is not such thing as deterministic mechanics.
Such a claim may appear, in the superficial reading, to be highly controversial.
But once we review the main premises of what is known as a deterministic mechanics,
we will acknowledge that the above statement is quite transparent. Indeed, although it is
assumed within the deterministic paradigm, that uncertainty is absent, at the latest stage
of the deterministic analysis, after stresses, deformations and displacements are found by
quite sophisticated analytical and/or numerical techniques, somehow, and nearly
miraculously, the neglected uncertainties are taken into account. These uncertainties are
enveloped by the concept of SAFETY FACTOR. Thus, the uncertainty is introduced via
the back door.
According to Vanmarke (1979),
"the format of many existing codes and design specifications impedes rational
risk assessment or communication among designers and owners about risk. For
example, the conventional safety factor of safety format considers a design
acceptable if the computed factor of safety exceeds prescribed allowable value.
Such a criterion characterizes structures as either safe or unsafe. It leads many
engineers to embracing the concept that all alternative designs which satisfy the
criteria are absolutely safe. Consequently, a little thought is given to the ever-