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
