Risk and Reliability Analysis - A Handbook for Civil and Environmental Engineers

812 Pages·2007·8.68 MB·English

by  Singh

Risk and Reliability Analysis
A Handbook for Civil and
Environmental Engineers
Vijay P. Singh
Sharad K. Jain
Aditya Tyagi
Library of Congress Cataloging-in-Publication Data
Singh, V. P. (Vijay P.)
Risk and reliability analysis : a handbook for civil and environmental engineers / Vijay P.
Singh, Sharad K. Jain, Aditya K. Tyagi.
p. cm.
Includes bibliographical references and index.
ISBN-13: 978-0-7844-0891-9
ISBN-10: 0-7844-0891-2
1. Engineering—Management—Handbooks, manuals, etc. 2. Reliability (Engineering)—
Handbooks, manuals, etc. 3. Risk assessment—Handbooks, manuals, etc. I. Jain, S. K. (Sharad
Kumar), 1960- II. Tyagi, Aditya K. III. Title.
TA190.S594 2007
620'.00452—dc22
2006038853
Published by American Society of Civil Engineers
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www.pubs.asce.org
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ISBN 13: 978-0-7844-0891-9
ISBN 10: 0-7844-0891-2
Manufactured in the United States of America.
Preface
Risk is an inherent part of any decision-making process. Risk and uncertainty
are unavoidable in the planning, design, construction, and management of engi-
neering systems. Until recently, a common way to account for risk and uncer-
tainty in engineering design was through a factor of safety. However, this does
not provide any quantitative idea of the risk in a particular situation and most
decision makers these days want to explicitly know the risk involved while
making a decision as well as the risk of operating an existing structure or project
and the consequences of a management action. Answers to these questions
require an understanding of the behavior of inputs to the system under study as
well as the consequences of a management action. Despite the importance and
relevance of the subject, there are very few universities and institutions where
risk and reliability analysis is taught or where it is part of either civil and envi-
ronmental engineering curricula or environmental and watershed sciences cur-
ricula. Commonly, these topics form the subject matter of a course in statistics or
systems analysis. A review of the literature in civil and environmental engineer-
ing shows that there are few books providing a comprehensive discussion of rel-
evant issues related to uncertainty, risk, and reliability. This constituted the
motivation for the proposed book.
The subject matter of the book is divided into four parts. The first part,
termed Preliminaries, contains three chapters. Introducing the basic theme of the
book, Chapter 1 provides a broad overview of the art of decision making under
uncertainty. Chapters 2 and 3 provide a preliminary background of probability
and random variables and moments and expectations of data that are needed to
grasp the topics that follow.
The second part of the book deals with Probability Distributions and Param-
eter Estimation, which is a bouquet of techniques organized into six chapters.
Discrete and continuous probability distributions form the subject matter of
Chapter 4. The distributions discussed include Bernoulli, binomial, geometric,
and negative binomial distributions. Another set of distributions is used when
interest lies in the number of times a specified event occurs over a certain dura-
tion. Poisson, exponential, and gamma distributions are used in such cases and
these are also described in Chapter 4. Chapter 5 deals with limit distributions
and other continuous distributions. Two frequently used distributions that form
the backbone of statistical analysis—the normal and log-normal distributions—
are discussed. These distributions are based on an important theorem in
xi
xii Preface
statistical analysis, the central limit theorem, which is discussed next. Distribu-
tions of extremes and other distributions found useful in environmental and
water engineering, such as uniform, triangular, beta, Pareto, logistic, Pearson
type III, and log-Pearson type III distributions, are also presented. Many envi-
ronmental processes can be described by using the concepts of probability and
physical laws. Chapter 6 focuses on impulse response functions as probability
distributions. These distributions have a quasi-physical basis. The concepts dis-
cussed include impulse response of a linear reservoir, cascade of linear reser-
voirs, the Muskingum model, the diffusion model, and the linear channel
downstream model. These concepts are widely used in hydrologic systems anal-
ysis. Many real-world decisions frequently involve more than one variable and
there may not be a one-to-one relationship among them. In such a situation, an
analysis of the joint probabilistic behavior of the variables involved would be
desirable. Chapter 7 presents multivariate distributions, with particular atten-
tion to bivariate distributions. A newly emerging copula methodology is pre-
sented and several bivariate distributions are discussed using this methodology.
The concept of return period is extended to more than one variable.
In statistical analysis, a considerable effort is devoted to deriving parameters
of a distribution, which constitutes the subject matter of Chapter 8. Many tech-
niques are available for this purpose; these include the method of moments, the
method of maximum likelihood, the method of probability weighted moments,
L-moments, and the method of least squares. Discussion of these methods is fol-
lowed by a treatment of the problems of parameter estimation. Besides point
estimates, interval estimation of parameters is carried out to determine the confi-
dence that can be placed in the point estimates and this chapter also includes a
description of interval estimates. The last chapter in the second part, Chapter 9,
deals with entropy. Originating in thermodynamics, the principle of information
theoretic entropy has found applications in many branches of engineering,
including civil and environmental engineering. The fundamental concepts of the
Shannon entropy theory are discussed. The methodology to derive parameters
of normal and gamma distributions by following the Lagrange multiplier
method and the parameter-space expansion method is described. This chapter
also provides a discussion of the fields where the entropy concept has proved to
be useful.
Part 3 of the book, comprising four chapters, deals with Uncertainty Analysis.
Chapter 10 discusses the concepts of error and uncertainty analysis. The focus of
this chapter is on a treatment of the types of uncertainties and analysis of errors.
The Monte Carlo method, a powerful tool to solve a range of problems, is dis-
cussed in Chapter 11. Generation of random numbers comprises an important
part of the Monte Carlo method. Therefore, this chapter provides a discussion of
many techniques that can be used to generate random numbers that follow a
given distribution. Several examples help illustrate the application of Monte
Carlo methods. Because many environmental processes are stochastic and can be
treated as stochastic processes, Chapter 12 gives a preliminary treatment of this
Preface xiii
topic. It includes a discussion of mean, variance, covariance, correlation, station-
arity, correlogram, and spectral density of stochastic processes. The chapter is
concluded with a discussion of time series analysis. Description of many environ-
mental processes requires the use of stochastic differential equations, which are
presented in Chapter 13. Several examples are presented to demonstrate the
application of these equations and the techniques to solve them.
The fourth and last part of the book, encompassing three chapters, focuses on
Risk and Reliability Analysis. The first chapter of this section, Chapter 14,
describes the various reliability measures, such as time to failure and the hazard
function. This is followed by a discussion of two concepts that are widely used in
reliability analysis: margin of safety and factor of safety. The highlight of this
chapter is the discussion on methods of reliability analysis, such as the first-order
approximation, the first-order second moment (FOSM) method, the mean-value
FOSM method, and point estimation methods. A number of real-world examples
are discussed to illustrate the concepts of reliability analysis and estimation.
Chapter 15 is related to risk analysis and management. In this chapter, a
broad view of risk is explained wherein risk is considered as a triplet involving
answering three questions: What can go wrong? What is the probability of
things going wrong? What are the consequences if something goes wrong? Con-
cepts of reliability and failure analysis and event and fault-tree analysis are
explained using suitable examples.
An important area that has drawn much attention and application of the
principles of reliability analysis is the design of water distribution networks
(WDNs). The final chapter of the book, Chapter 16, begins with a basic descrip-
tion of analysis of WDNs, followed by hydraulic reliability analysis of a WDN
for a range of conditions. The entropy method can be useful in hydraulic reliabil-
ity analysis of a WDN and the entropy-based methodology is described in this
chapter.
The chapters of this book are arranged by keeping in view the requirements
of a typical engineering student, who hopefully has a fundamental knowledge
of mathematics and statistics. This book is intended for senior undergraduate
and beginning graduate students as well as water resources practitioners. Those
who have an adequate background in probability and statistical analysis can
skip the first section of the book and do a quick reading of the first two chapters
of the second section. Numerous examples have been solved step by step and
this should help with understanding of computational procedures. Besides the
book being of value to students, it should also be useful to faculty members and
practitioners working in the fields of civil and environmental engineering,
watershed sciences, and biological and agricultural engineering. Much of the
material has been used for teaching a course on risk and reliability analysis in
civil and environmental engineering.
—Vijay P. Singh, College Station, Texas
—Sharad K. Jain, Roorkee, India
—Aditya Tyagi, Austin, Texas
Table of Contents
Preface xi
Acknowledgments xiv
Part I Preliminaries
Chapter 1 Rational Decision Making Under Uncertainty.........................................3
1.1 Problems Requiring Decision Making 5
1.2 Concept of Rationality 8
1.3 Evaluation of Alternatives 10
1.4 Dealing with Uncertainty 13
1.5 Decision-Making Policy 18
1.6 Sources of Uncertainties 30
1.7 Rational Decision Making 31
1.8 Questions 32
Chapter 2 Elements of Probability..............................................................................36
2.1 Random Variables 37
2.2 Elements of Probability 48
2.3 Conditional Probability and Independence 62
2.4 Total Probability and Bayes’s Formula 65
2.5 Probability Distribution 69
2.6 Function of a Random Variable 84
2.7 Questions 89
Chapter 3 Moments and Expectation.........................................................................95
3.1 Expectation 96
3.2 Moments 100
3.3 Moment-Generating Functions 103
3.4 Characteristic Functions 105
3.5 Characterization of a Single Random 1Variable 106
3.6 Two Random Variables 125
3.7 Functions of Random Variables 137
3.8 Questions 139
vii
viii Table of Contents
Part II Probability Distributions and Parameter Estimation
Chapter 4 Discrete and Continuous Probability Distributions.............................149
4.1 Simple Discrete Random Trials 150
4.2 Models for Random Occurrences 166
4.3 Summary of Distributions 180
4.4 Questions 180
Chapter 5 Limit and Other Distributions.................................................................187
5.1 Normal Distribution: Model of Sums 188
5.2 Central Limit Theorem 198
5.3 Log-Normal Distribution: Product of Variables 204
5.4 Distribution of Extremes 211
5.5 Other Distributions 230
5.6 Questions 247
Chapter 6 Impulse Response Functions as Probability Distributions.................252
6.1 Hypothesis 254
6.2 Impulse Responses of Linear Systems 255
6.3 Application 266
6.4 Summary 269
6.5 Questions 269
Chapter 7 Multivariate Probability Distributions...................................................272
7.1 Existing Multivariate Distributions 273
7.2 Return Period 294
7.3 Derivation of Multivariate Distributions 295
7.4 Questions 311
Chapter 8 Parameter Estimation...............................................................................313
8.1 Methods of Parameter Estimation 314
8.2 Assessment of Parameter Accuracy 336
8.3 Interval Estimation of Parameters 344
8.4 Questions 353
Chapter 9 Entropy Theory and Its Applications in Risk Analysis.......................356
9.1 History and Meaning of Entropy 357
9.2 Principle of Maximum Entropy 360
9.3 Derivation of Parameters of the Normal Distribution Using
Entropy 366
9.4 Determination of Parameters of the Gamma Distribution 374
9.5 Application of Entropy Theory in Environmental and Water
Resources 381
9.6 Closure 389
9.7 Questions 390
Table of Contents ix
Part III Uncertainty Analysis
Chapter 10 Error and Uncertainty Analysis..............................................................395
10.1 Types of Uncertainty 396
10.2 Sources of Uncertainty: Types of Errors 400
10.3 Analysis of Errors 401
10.4 Questions 429
Chapter 11 Monte Carlo Simulation...........................................................................437
11.1 Basics of Monte Carlo Simulation 438
11.2 Generation of Random Numbers 439
11.3 Continuous Random Variates 442
11.4 Discrete Random Variates 447
11.5 Jointly Distributed Random Variates 448
11.6 Simulation of Systems with Random Inputs 450
11.7 Monte Carlo Integration 474
11.8 Variance Reduction Techniques 475
11.9 Reducing Parameter Uncertainty 480
11.10 Uncertainty and Sensitivity Analysis Using Monte Carlo
Simulation 481
11.11 Questions 484
Chapter 12 Stochastic Processes..................................................................................486
12.1 Mean and Variance 487
12.2 Covariance and Correlation 488
12.3 Stationarity 494
12.4 Correlogram 496
12.5 Spectral Density 498
12.6 Stochastic Processes 503
12.7 Time Series Analysis 508
12.8 Questions 513
Chapter 13 Stochastic Differential Equations............................................................515
13.1 Ordinary Differential Equations 516
13.2 Partial Differential Equations 517
13.3 Stochastic Differential Equations 517
13.4 Fundamental Concepts in Solving SDEs 518
13.5 Questions 558
Part IV Risk and Reliability Analysis
Chapter 14 Reliability Analysis and Estimation.......................................................563
14.1 Approaches to Reliability Evaluation 564
14.2 Reliability Analysis Methods 576
14.3 Reliability Analysis of Dynamic Systems 634
x Table of Contents
14.4 Reliability Analysis of a Multicomponent System 643
14.5 Reliability Programming 647
14.6 Reliability-Based Design 657
14.7 Questions 664
Chapter 15 Risk Analysis and Management.............................................................669
15.1 Basic Definitions 669
15.2 Risk and Its Classification 675
15.3 Risk Criteria 680
15.4 Risk Assessment and Management 681
15.5 Risk Modeling 691
15.6 Decision Making Under Uncertainty 697
15.7 Questions 710
Chapter 16 Reliability Analysis of Water Distribution Networks..........................712
16.1 Relevant Principles of Hydraulics 713
16.2 Analysis of a WDN 720
16.3 Reliability of a WDN 726
16.4 Reliability Analysis of a WDN Using the Entropy Concept 744
16.5 Questions 748
References 751
Index 769
About the Authors 785
Part I
Preliminaries
1