Probability, Reliability, and Statistical

Learn the tools to assess product reliability! Haldar and Mahadevan crystallize the research and experience of the last few decades into the most up-to-date book on risk-based design concepts in engineering available. The fundamentals of reliability and statistics necessary for risk-based engineering analysis and design are clearly presented. And with the help of many practical examples integrated throughout the text, the material is made very relevant to today's practice. Key Features * Covers all the fundamental concepts and mathematical skills needed to conduct reliability assessments. * Presents the most widely-used reliability assessment methods. * Concepts that are required for the implementation of risk-based design in practical problems are developed gradually. * Both risk-based and deterministic design concepts are included to show the transition from traditional to modern design practice.

Achintya Haldar, PhD, is a professor in the Department of Civil Engineering and Engineering Mechanics at the University of Arizona, Tucson.

Sankaran Mahadevan, PhD, is Associate Professor of Civil and Environmental Engineering at Vanderbilt University in Nashville, Tennessee.

Basic Concept of Reliability

1

(9)

Introductory Comments

1

(1)

What Is Reliability?

1

(2)

Need for Reliability Evaluation

3

(1)

Measures of Reliability

4

(1)

Factors Affecting Reliability Evaluation

4

(2)

Sources of Uncertainty

5

(1)

Steps in the Modeling of Uncertainty

6

(1)

Concluding Remarks

7

(2)

Mathematics of Probability

9

(26)

Introductory Comments

9

(1)

Introduction to Set Theory

9

(7)

Elements of Set Theory

9

(3)

Venn Diagram

12

(1)

Combinations of Events

12

(2)

Operational Rules

14

(1)

De Morgan's Rule

15

(1)

Axioms of Probability

16

(4)

Multiplication Rule

20

(5)

Theorem of Total Probability

25

(1)

Bayes' Theorem

25

(3)

Review

28

(1)

Concluding Remarks

29

(1)

Problems

29

(6)

Modeling of Uncertainty

35

(28)

Introductory Comments

35

(1)

Steps in Quantifying Randomness

35

(5)

Data Collection

35

(1)

Descriptors of Randomness

36

(2)

Histogram and Frequency Diagram

38

(2)

Analytical Models to Quantify Randomness

40

(9)

Continuous Random Variables

40

(3)

Discrete Random Variables

43

(2)

General Definitions for Uncertainty Descriptors

45

(3)

Mode and Median

48

(1)

Percentile Value

48

(1)

Multiple Random Variables

49

(9)

Joint Distributions

49

(1)

Conditional PDF and PMF

50

(1)

Marginal PDF and PMF

51

(1)

Covariance and Correlation

51

(7)

Multivariate Distributions

58

(1)

Concluding Remarks

58

(1)

Problems

58

(5)

Commonly Used Probability Distributions

63

(43)

Introductory Comments

63

(1)

Continuous Random Variables

63

(11)

Normal or Gaussian Distribution

64

(4)

Lognormal Random Variables

68

(4)

Beta Distribution

72

(2)

Discrete Random Variables

74

(9)

Binomial Distribution

74

(2)

Geometric Distribution

76

(1)

Return Period

76

(1)

Poisson Distribution

77

(3)

Exponential Distribution

80

(3)

A Combination of Continuous and Discrete Random Variables: Hypergeometric and Hyperbinomial Distributions

83

(4)

Extreme Value Distributions

87

(11)

Introduction

87

(1)

Concept of Extreme Value Distributions

88

(1)

Asymptotic Distributions

89

(1)

The Type I Extreme Value Distribution

89

(2)

The Type II Extreme Value Distribution

91

(3)

The Type III Extreme Value Distribution

94

(3)

Special Cases of Two-Parameter Weibull Distribution

97

(1)

Other Useful Distributions

98

(1)

Concluding Remarks

98

(1)

Problems

99

(7)

Determination of Distributions and Parameters from Observed Data

106

(32)

Introductory Comments

106

(1)

Determination of Probability Distribution

106

(11)

Probability Papers

107

(1)

Construction of a Probability Paper

107

(5)

Statistical Tests

112

(5)

Estimation of Parameters of a Distribution

117

(3)

Method of Moments

117

(1)

Method of Maximum Likelihood

118

(2)

Interval Estimation of Mean and Variance

120

(14)

Interval Estimation for the Mean with Known Variance

121

(4)

Lower and Upper Confidence Limit for the Mean with Known Variance

125

(1)

Interval Estimation for the Mean with Unknown Variance

126

(3)

Lower and Upper Confidence Limit for the Mean with Unknown Variance

129

(1)

Sample Sizes in Estimating the Confidence Interval of Mean

130

(1)

Interval Estimation for the Variance

131

(3)

Concluding Remarks

134

(1)

Problems

134

(4)

Randomness in Response Variables

138

(43)

Introductory Comments

138

(1)

Known Functional Relationship Between the Response and a Single Basic Random Variable

139

(3)

Linear Relationship

139

(1)

Nonlinear Relationship

140

(2)

Response as a Known Function of Multiple Random Variables

142

(7)

Exact Solution

142

(7)

Central Limit Theorem

149

(1)

Partial and Approximate Solutions

149

(5)

Partial Uncertainty Analysis: Response as a Linear Function of Multiple Random Variables

149

(1)

Approximate Solution: Response as a General Function of Multiple Random Variables

150

(4)

Multiple Random Variables with Unknown Relationship

154

(2)

Repression Analysis

156

(17)

Simple Linear Regression Analysis

157

(2)

Coefficient of Determination

159

(2)

Residual Analysis

161

(3)

Multiple Linear Regression

164

(4)

Nonlinear Models

168

(5)

Concluding Remarks

173

(1)

Problems

174

(7)

Fundamentals of Reliability Analysis

181

(44)

Introductory Comments

181

(1)

Deterministic and Probabilistic Approaches

181

(1)

Risk and Safety Factors Concept

182

(1)

Risk-Based Design Concept and the Development of the Risk-Based Design Format

183

(10)

Load and Resistance Normal Variables: Single Load Case

184

(2)

Load and Resistance Normal Variables: Multiple Load Case

186

(4)

Load and Resistance Lognormal Variables: Single Load Case

190

(1)

Load and Resistance Lognormal Variables: Multiple Load Case

191

(2)

Fundamental Concept of Reliability Analysis

193

(2)

First-Order Reliability Methods (FORM)

195

(24)

First-Order Second-Moment Method (FOSM) or MVFOSM Method

195

(3)

AFOSM Method for Normal Variables (Hasofer-Lind Methods)

198

(6)

AFOSM Method for Nonnormal Variables

204

(15)

Risk-Based Design Format Using FORM

219

(3)

Concluding Remarks

222

(1)

Problems

223

(2)

Advanced Topics on Reliability Analysis

225

(25)

Introductory Comments

225

(1)

Second-Order Reliability Methods (SORM)

225

(6)

Reliability Analysis with Correlated Variables

231

(6)

Correlated Normal Variables

233

(1)

Correlated Nonnormal Variables

234

(3)

Probabilistic Sensitivity Indices

237

(1)

System Reliability Evaluation

238

(9)

Series Systems or Weakest Link Systems

240

(3)

Parallel Systems

243

(2)

Nonlinear System Reliability

245

(2)

Implicit Performance Functions

247

(1)

Concluding Remarks

248

(1)

Problems

248

(2)

Simulation Techniques

250

(25)

Introductory Comments

250

(1)

Monte Carlo Simulation Technique

251

(10)

Formulation of the Problem

251

(1)

Quantifying the Probabilistic Characteristics of Random Variables

251

(1)

Generation of Random Numbers

252

(4)

Numerical Experimentation

256

(2)

Extracting Probabilistic Information Using Simulation

258

(1)

Accuracy and Efficiency of Simulation

258

(3)

Variance Reduction Techniques

261

(5)

VRTs in Sampling Methods

262

(3)

Correlation-Based VRTs

265

(1)

Combined Conditional Expectation and Antithetic Variates Method

265

(1)

Simulation of Correlated Random Variables

266

(5)

Simulation of Correlated Normal Variables

268

(2)

Simulation of Correlated Nonnormal Variables

270

(1)

Concluding Remarks

271

(1)

Problems

272

(3)

Appendix 1 Table of the CDF of the Standard Normal Distribution

275

(3)

Appendix 2 Evaluation of Gamma Function

278

(2)

Appendix 3 Table of the CDF of the Chi-Square Distribution with f Degrees of Freedom

280

(2)

Appendix 4 Values of Dαn for the Kolkmogorov-Smirnov (K-S) Test

282

(1)

Appendix 5 Table of the CDF of Student's t-Distribution

283

(2)

Appendix 6 Gram-Schmidt Orthogonalization

285

(2)

Conversion Factors

287

(1)

References

288

(8)

Index

296

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Probability, Reliability, and Statistical Methods in Engineering Design

Summary

Author Biography

Table of Contents

Supplemental Materials