# Reliability Engineering, 1st edition

• Singiresu S. Rao

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## Overview

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.

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

7.5.3 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

13.3.3 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