Reliability Engineering

  Reliability Engineering is intended for use as an introduction to reliability engineering, including the aspects analysis, design, testing, production and quality control of engineering components and systems.   Numerous analytical and numerical examples and problems are used to illustrate the principles and concepts. Expanded explanations of the fundamental concepts are given throughout the book, with emphasis on the physical significance of the ideas. The mathematical background necessary in the area of probability and statistics is covered briefly to make the presentation complete and self-contained. Solving probability and reliability problems using MATLAB and Excel is also presented.

Table of Contents:
Chapter 1 Introduction 1 What You Will Learn 1 1.1 Uncertainty in Engineering 1 1.2 Definition of Reliability 2 1.3 Importance of Reliability 3 1.4 Pattern of Failures 4 1.4.1 Component Failures 4 1.4.2 Mechanical and Structural Failures 11 1.5 Factor of Safety and Reliability 15 1.6 Reliability Analysis Procedure 18 1.7 Reliability Management 18 1.8 History of Reliability Engineering 19 1.9 Some Examples of System Failures 21 1.9.1 Collapse of Tacoma Narrows Bridge in 1940 21 1.9.2 Crash of El Al Boeing 747-200 in 1992 22 1.9.3 Disaster of Space Shuttle Challenger in 1986 22 1.9.4 Chernobyl Nuclear Power Plant Accident in 1986 24 1.9.5 Mississippi River Bridge 9340 Collapse in 2007 24 1.9.6 Fukushima Nuclear Accident in 2011 25 1.9.7 Explosion of the First Jet Airplane Comet 26 1.9.8 Breaking of the Tanker S. S. Schenectady 27 1.9.9 Crash of the Supersonic Aircraft Concorde 27 1.10 Numerical Solutions Using Matlab and Excel 29 1.11 Reliability Literature 32 References and Bibliography 32 Review Questions 44 Problems 45   Chapter 2 Basic Probability Theory 50 What You Will Learn 50 2.1 Introduction 50 2.2 Mutually Exclusive Events 51 2.3 Set Theory 51 2.4 Sample Points and Sample Space 52 2.5 Definition of Probability 55 2.5.1 Relative Frequency (Statistical) Definition 55 2.5.2 Axiomatic Definition 55 2.6 Laws of Probability 56 2.6.1 Union and Intersection of Two Events 56 2.6.2 Mutually Exclusive Events 56 2.6.3 Complementary Events 59 2.6.4 Conditional Probability 62 2.6.5 Statistically Independent Events 65 2.6.6 General Laws 66 2.7 Total Probability Theorem 68 2.8 Bayes’ Rule 73 References and Bibliography 75 Review Questions 76 Problems 79   Chapter 3 Random Variables and Probability Distributions 87 What You Will Learn 87 3.1 Introduction 87 3.2 Probability Mass Function for Discrete Random Variables 88 3.3 Cumulative Distribution Function for Discrete Random Variables 88 3.4 Probability Density Function for Continuous Random Variables 90 3.5 Mean, Mode, and Median 94 3.5.1 Mean 95 3.5.2 Mode 96 3.5.3 Median 96 3.6 Standard Deviation and Skewness Coefficient 98 3.6.1 Standard Deviation 99 3.6.2 Skewness Coefficient 103 3.7 Moments of Random Variables 105 3.8 Importance of Moment Functions– Chebyshev Inequality 106 3.9 Jointly Distributed Random Variables 108 3.9.1 Joint Density and Distribution Functions 108 3.9.2 Obtaining the Marginal or Individual Density Function from the Joint Density Function 109 3.10 Moments of Jointly Distributed Random Variables 111 3.11 Probability Distributions 112 3.11.1 Binomial Distribution 116 3.11.2 Poisson Distribution 119 3.11.3 Normal Distribution 122 3.11.3 Lognormal Distribution 130 3.12 Central Limit Theorem 134 3.13 Normal Approximation to Binomial Distribution 134 3.14 Numerical Solutions Using MATLAB and Excel 135 3.14.1 MATLAB Functions for Discrete and Continuous Probability Distributions 135 3.14.2 Random Numbers, Fitting Data to Distributions and Confidence Intervals 140 3.14.3 Solutions Using Excel 142 References and Bibliography 144 Review Questions 145 Problems 148   Chapter 4 Extremal Distributions 161 What You Will Learn 161 4.1 Introduction 161 4.2 Extreme Value Distributions in Terms of Parent Distribution 163 4.3 Asymptotic Distributions 166 4.4 Type-I Asymptotic Distributions 167 4.4.1 Maximum Value 167 4.4.2 Smallest Value 167 4.5 Type-II Asymptotic Distributions 168 4.5.1 Maximum Value 168 4.5.2 Smallest Value 169 4.6 Type-III Asymptotic Distributions 170 4.6.1 Maximum Value 170 4.6.2 Smallest Value 170 4.7 Return Period 171 4.8 Characteristic Value 172 4.9 Fitting Extremal Distributions to Experimental Data 173 4.9.1 Least-squares Fit 174 4.10 Generalized Extreme Value Distribution 176 4.11 Numerical Solutions Using MATLAB and Excel 178 References and Bibliography 183 Review Questions 184 Problems 186   Chapter 5 Functions of Random Variables 191 What You Will Learn 191 5.1 Introduction 191 5.2 Functions of a Single Random Variable 192 5.3 Functions of Two Random Variables 197 5.3.1 Sum of Two Random Variables 198 5.3.2 Product of Two Random Variables 202 5.3.3 Quotient of Two Random Variables 203 5.4 Function of Several Random Variables 205 5.5 Moments of a Function of Several Random Variables 205 5.5.1 Mean and Variance of a Linear Function 206 5.5.2 Mean and Variance of Sum of Two Random Variables 207 5.5.3 Mean and Variance of Product of Two Random Variables 207 5.5.4 Mean and Variance of Quotient of Two Random Variables 208 5.5.5 Mean and Variance of a General Nonlinear Function of Several Random Variables 208 5.6 Moment-Generating Function 212 5.6.1 Moments of Normally Distributed Variables 213 5.7 Functions of Several Random Variables 215 5.8 Numerical Solutions Using MATLAB 217 References and Bibliography 220 Review Questions 220 Problems 222   Chapter 6 Time-Dependent Reliability of Components and Systems 232 What You Will Learn 232 6.1 Introduction 232 6.2 Failure Rate versus Time Curve 233 6.3 Reliability and Hazard Functions 234 6.4 Modeling of Failure Rates 236 6.5 Estimation of Failure Rate from Empirical Data 237 6.6 Mean Time to Failure (MTTF) 239 6.7 Reliability and Hazard Functions for Different Distributions 241 6.7.1 Exponential Distribution 241 6.7.2 Normal Distribution 244 6.7.3 Lognormal Distribution 246 6.7.4 Weibull Distribution 251 6.7.5 Gamma Distribution 256 6.7.6 Rayleigh Distribution 258 6.7.7 Uniform Distribution 260 6.8 Expected Residual Life 262 6.9 Series Systems 265 6.9.1 Failure Rate of the System 267 6.9.2 MTBF of the System 267 6.10 Parallel Systems 268 6.10.1 Failure Rate of the System 270 6.10.2 MTBF of the System 270 6.11 (k, n) Systems 271 6.11.1 MTBF of the System 272 6.12 Mixed Series and Parallel Systems 272 6.13 Complex Systems 273 6.13.1 Enumeration Method 274 6.13.2 Conditional Probability Method 276 6.13.3 Cut-set Method 278 6.14 Reliability Enhancement 280 6.14.1 Series System 280 6.14.2 Parallel System 282 6.15 Reliability Allocation–AGREE Method 283 6.16 Numerical Solutions Using MATLAB and Excel 286 References and Bibliography 289 Review Questions 289 Problems 292   Chapter 7 Modeling of Geometry, Material Strength, and Loads 301 What You Will Learn 301 7.1 Introduction 301 7.2 Modeling of Geometry 302 7.2.1 Tolerances on Finished Metal Products 303 7.2.2 Assembly of Components 303 7.3 Modeling of Material Strength 308 7.3.1 Statistics of Elastic Properties 308 7.3.2 Statistical Models for Material Strength 309 7.3.3 Model for Brittle Materials 309 7.3.4 Model for Plastic Materials 311 7.3.5 Model for Fiber Bundles 312 7.4 Fatigue Strength 314 7.4.1 Constant-Amplitude Fatigue Strength 314 7.4.2 Variable-Amplitude Fatigue Strength 317 7.5 Modeling of Loads 319 7.5.1 Introduction 319 7.5.2 Dead Loads 320 7.5.3 Live Loads 320 7.5.4 Wind Loads 321 7.5.5 Earthquake Loads 326 7.6 Numerical Solutions Using MATLAB and Excel 331 References and Bibliography 333 Review Questions 337 Problems 339   Chapter 8 Strength-Based Reliability 343 What You Will Learn 343 8.1 Introduction 343 8.2 General Expression for Reliability 345 8.3 Expression for Probability of Failure 348 8.4 General Interpretation of Strength and Load 349 8.5 Reliability for Known Probability Distributions of S and L 349 8.5.1 Reliability When S and L Follow Normal Distribution 350 8.5.2 Approximate Expressions of Reliability for Normal Distribution 352 8.5.3 Reliability When S and L Follow Lognormal Distribution 356 8.5.4 Reliability When S and L Follow Exponential Distribution 361 8.5.5 Reliability When S and L Follow Extreme Value Distributions 363 8.5.6 When S and L Follow Type-III Extremal Distributions 364 8.5.7 Reliability in Terms of Experimentally Determined Distributions of S and L 365 8.6 Factor of Safety Corresponding to a Given Reliability 369 8.7 Reliability of Systems Involving More Than Two Random Parameters 373 8.8 First-Order Second-Moment (FOSM) Method 380 8.9 Hasofer-Lind Reliability Index with Two Normally Distributed Variables 383 8.10 Hasofer-Lind Reliability Index with Several Normally Distributed Variables 385 8.11 Reliability of Weakest-Link and Fail-Safe Systems 389 8.11.1 Introduction 389 8.11.2 Reliability of the Fundamental Problem 390 8.11.3 Reliability of Weakest-Link (or Series) Systems 392 8.11.4 Reliability Analysis of Fail-Safe (or Parallel) Systems 398 8.12 Numerical Solutions Using MATLAB and Excel 400 References and Bibliography 405 Review Questions 407 Problems 411   Chapter 9 Design of Mechanical Components and Systems 425 What You Will Learn 425 9.1 Introduction 425 9.2 Design of Mechanical Components 426 9.3 Fatigue Design 431 9.3.1 Deterministic Design Procedure 432 9.3.2 Probabilistic Design Procedure 435 9.4 Design of Mechanical Systems 439 9.4.1 Reliability-Based Design of Gear Trains 439 9.5 Reliability Analysis of Mechanical Systems 445 9.5.1 Cam-Follower Systems 445 9.5.2 Four-Bar Mechanisms 450 9.6 Numerical Solutions Using MATLAB and Excel 457 References and Bibliography 459 Review Questions 459 Problems 461   Chapter 10 Monte Carlo Simulation 465 What You Will Learn 465 10.1 Introduction 465 10.2 Generation of Random Numbers 466 10.2.1 Generation of Random Numbers Following Standard Uniform Distribution 468 10.2.2 Random Variables with Nonuniform Distribution 469 10.2.3 Generation of Discrete Random Variables 472 10.3 Generation of Jointly Distributed Random Numbers 475 10.3.1 Independent Random Variables 475 10.3.2 Dependent Random Variables 475 10.3.3 Generation of Correlated Normal Random Variables 478 10.4 Computation of Reliability 483 10.4.1 Sample Size and Error in Simulation 483 10.4.2 Example: Reliability Analysis of a Straight-Line Mechanism 485 10.5 Numerical Solutions Using MATLAB and Excel 489 References and Bibliography 491 Review Questions 492 Problems 494   Chapter 11 Reliability-Based Optimum Design 504 What You Will Learn 504 11.1 Introduction 504 11.2 Optimization Problem 505 11.3 Formulation of Optimization Problems 507 11.3.1 Reliability Allocation Problems 507 11.3.2 Structural and Mechanical Design Problems 509 11.4 Solution Techniques 516 11.4.1 Graphical-Optimization Method 516 11.4.2 Lagrange Multiplier Method 520 11.4.3 Penalty Function Method (SUMT) 523 11.4.4 Dynamic Programming 532 11.5 Numerical Solutions Using MATLAB 538 References and Bibliography 546 Review Questions 546 Problems 548   Chapter 12 Failure Modes, Event-Tree, and Fault-Tree Analyses 554 What You Will Learn 554 12.1 Introduction 555 12.2 System-Safety Analysis 555 12.3 Failure Modes and Effects Analysis (FMEA) 557 12.4 Event-Tree Analysis 558 12.5 Fault-Tree Analysis (FTA) 564 12.5.1 Concept 565 12.5.2 Procedure 565 12.6 Minimal Cut-Sets 572 12.6.1 Probability of the TOP Event 574 References and Bibliography 582 Review Questions 583 Problems 585   Chapter 13 Reliability Testing 594 What You Will Learn 594 13.1 Introduction 595 13.1.1 Objectives of Reliability Tests 595 13.1.2 Details of a Reliability Test 596 13.2 Analysis of Failure Time 596 13.2.1 Analysis of Individual Failure Data 596 13.2.2 Analysis of Grouped Failure Data 599 13.3 Accelerated Life Testing 601 13.3.1 Testing Until Partial Failure 601 13.3.2 Magnified Loading 602 13.3.3 Sudden-death Testing 605 13.4 Sequential Life Testing 608 13.5 Statistical Inference and Parameter Estimation 610 13.5.1 Maximum-likelihood Method 611 13.6 Confidence Intervals 613 13.6.1 Confidence Interval on the Mean of a Normal Random Variable of Known Standard Deviation 615 13.6.2 Confidence Interval on the Mean of a Normal Random Variable of Unknown Standard Deviation 616 13.6.3 Confidence Interval on the Standard Deviation of a Normal Random Variable with Unknown Mean 618 13.7 Plotting of Reliability Data 620 13.7.1 Least-Squares Technique 620 13.7.2 Linear Rectification 621 13.7.3 Plotting Positions 621 13.7.4 Exponential Distribution 621 13.7.5 Normal Distribution 623 13.7.6 Lognormal Distribution 626 13.7.7 Weibull Distribution 626 13.8 Numerical Solutions Using MATLAB 630 13.8.1 Parameter Estimation and Confidence Intervals 630 13.8.2 Plotting of Data 632 References and Bibliography 634 Review Questions 635 Problems 638   Chapter 14 Quality Control and Reliability 642 What You Will Learn 642 14.1 Introduction 642 14.2 Importance of Controlling Dimensions of Products 644 14.3 Important Discrete Probability Distributions 647 14.3.1 Binomial Distribution 647 14.3.2 Hypergeometric Distribution 648 14.3.3 Poisson Distribution 649 14.3.4 Relationship Between Poisson and Exponential Distributions 650 14.4 Six Sigma Approach and Reliability 650 14.4.1 Implementation of the Six Sigma Approach 657 14.5 Acceptance Sampling 658 14.5.1 Characteristics of Sampling Plans 659 14.6 Process Capability 659 14.7 Quality Control Charts 664 14.7.1 The p-Chart 665 14.7.2 The X-Chart 667 14.7.3 The R-Chart 670 14.7.4 The c-Chart 672 14.8 Risks 673 14.9 Operating Characteristic (OC) Curve 674 14.9.1 OC Curve 675 14.9.2 Construction of OC Curve 675 14.9.3 Designing a Single Sampling Plan with a Specified OC Curve 677 14.10 T aguchi Method 678 14.10.1 Basic Concept 678 14.10.2 Loss Function 679 14.10.3 Noise Factors 681 14.10.4 On-Line Versus Off-Line Quality Control 682 14.10.5 Three-Step Design Approach 683 14.10.6 Experimental Design 683 14.10.7 Signal-To-Noise Ratio 687 14.10.8 Experimental Design in the Presence of Noise Factors 689 14.11 Numerical Solutions Using MATLAB 697 References and Bibliography 698 Review Questions 699 Problems 701   Chapter 15 Maintainability and Availability 706 What You Will Learn 706 15.1 Introduction 706 15.2 Maintainability 707 15.2.1 Overview 707 15.2.2 Preventive Maintenance 708 15.2.3 Imperfect Maintenance 712 15.2.4 Repair-time Distributions 713 15.2.5 Unrepaired Failures 716 15.2.6 Optimal Replacement Strategy 717 15.2.7 Spare Parts Requirement 719 15.3 Availability 720 15.3.1 Definitions [15.1, 15.3] 720 15.3.2 Availability Analysis 721 15.3.3 Development of the Model 722 15.3.4 Systems with a Single Component 723 15.3.5 Series Systems 726 15.3.6 Parallel Systems 729 15.4 Optimization Approaches 730 15.5 Numerical Solutions Using MATLAB and Excel 731 References and Bibliography 733 Review Questions 734 Problems 736   Chapter 16 Warranties 739 What You Will Learn 739 16.1 Introduction 740 16.2 T ypes of Warranties 742 16.3 Warranty Cost Based on a Single Failure During the Warranty Period 742 16.3.1 Free Replacement Warranty 742 16.3.2 Pro-rata Warranty 744 16.3.3 Combined Free Replacement Warranty and Pro-rata Warranty (FRW/PRW) Policy 747 16.3.4 FRW Policy Equivalent to a FRW/PRW Policy 749 16.3.5 Lump-sum Payment Type of Warranty 750 16.4 Warranty Costs Considering the Time Value of Money 752 16.4.1 FRW Policy 752 16.4.2 PRW Policy 753 16.5 Warranty Reserve Fund Considering the Time Value of Money and Future Changes in the Price of the Product 754 16.6 Warranty Analysis Considering Multiple Failures During the Warranty Period 757 16.6.1 Renewal Process 758 16.6.2 Computation and Use of Renewal Functions 759 16.7 Optimum Warranty Period 764 16.8 Two-dimensional Warranties 768 16.9 Numerical Solutions Using MATLAB 770 References and Bibliography 772 Review Questions 773 Problems 775   Appendix A Standard Normal Distribution Function 779 Appendix B Values of ta, n for Specific Values of a and n of t Distribution 782 Appendix C Values of x2n, a Corresponding to Specific Values of a and n of x2-Distribution 784 Appendix D Product Liability 787 Answers to Selected Problems 791 Index 795

About the Author :
Dr. Singiresu S. Rao is a Professor in the Mechanical and Aerospace Engineering Department at the University of Miami College of Engineering.