Reliability Engineering

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

• Chapter 1 Introduction 1What You Will Learn 11.1 Uncertainty in Engineering 11.2 Definition of Reliability 21.3 Importance of Reliability 31.4 Pattern of Failures 41.4.1 Component Failures 41.4.2 Mechanical and Structural Failures 111.5 Factor of Safety and Reliability 151.6 Reliability Analysis Procedure 181.7 Reliability Management 181.8 History of Reliability Engineering 191.9 Some Examples of System Failures 211.9.1 Collapse of Tacoma Narrows Bridge in 1940 211.9.2 Crash of El Al Boeing 747-200 in 1992 221.9.3 Disaster of Space Shuttle Challenger in 1986 221.9.4 Chernobyl Nuclear Power Plant Accident in 1986 241.9.5 Mississippi River Bridge 9340 Collapse in 2007 241.9.6 Fukushima Nuclear Accident in 2011 251.9.7 Explosion of the First Jet Airplane Comet 261.9.8 Breaking of the Tanker S. S. Schenectady 271.9.9 Crash of the Supersonic Aircraft Concorde 271.10 Numerical Solutions Using Matlab and Excel 291.11 Reliability Literature 32References and Bibliography 32Review Questions 44Problems 45  Chapter 2 Basic Probability Theory 50What You Will Learn 502.1 Introduction 502.2 Mutually Exclusive Events 512.3 Set Theory 512.4 Sample Points and Sample Space 522.5 Definition of Probability 552.5.1 Relative Frequency (Statistical) Definition 552.5.2 Axiomatic Definition 552.6 Laws of Probability 562.6.1 Union and Intersection of Two Events 562.6.2 Mutually Exclusive Events 562.6.3 Complementary Events 592.6.4 Conditional Probability 622.6.5 Statistically Independent Events 652.6.6 General Laws 662.7 Total Probability Theorem 682.8 Bayes’ Rule 73References and Bibliography 75Review Questions 76Problems 79  Chapter 3 Random Variables and Probability Distributions 87What You Will Learn 873.1 Introduction 873.2 Probability Mass Function for Discrete Random Variables 883.3 Cumulative Distribution Function for Discrete Random Variables 883.4 Probability Density Function for Continuous Random Variables 903.5 Mean, Mode, and Median 943.5.1 Mean 953.5.2 Mode 963.5.3 Median 963.6 Standard Deviation and Skewness Coefficient 983.6.1 Standard Deviation 993.6.2 Skewness Coefficient 1033.7 Moments of Random Variables 1053.8 Importance of Moment Functions– Chebyshev Inequality 1063.9 Jointly Distributed Random Variables 1083.9.1 Joint Density and Distribution Functions 1083.9.2 Obtaining the Marginal or Individual Density Function from the Joint Density Function 1093.10 Moments of Jointly Distributed Random Variables 1113.11 Probability Distributions 1123.11.1 Binomial Distribution 1163.11.2 Poisson Distribution 1193.11.3 Normal Distribution 1223.11.3 Lognormal Distribution 1303.12 Central Limit Theorem 1343.13 Normal Approximation to Binomial Distribution 1343.14 Numerical Solutions Using MATLAB and Excel 1353.14.1 MATLAB Functions for Discrete and Continuous Probability Distributions 1353.14.2 Random Numbers, Fitting Data to Distributions and Confidence Intervals 1403.14.3 Solutions Using Excel 142 References and Bibliography 144Review Questions 145Problems 148  Chapter 4 Extremal Distributions 161What You Will Learn 1614.1 Introduction 1614.2 Extreme Value Distributions in Terms of Parent Distribution 1634.3 Asymptotic Distributions 1664.4 Type-I Asymptotic Distributions 1674.4.1 Maximum Value 1674.4.2 Smallest Value 1674.5 Type-II Asymptotic Distributions 1684.5.1 Maximum Value 1684.5.2 Smallest Value 1694.6 Type-III Asymptotic Distributions 1704.6.1 Maximum Value 1704.6.2 Smallest Value 1704.7 Return Period 1714.8 Characteristic Value 1724.9 Fitting Extremal Distributions to Experimental Data 1734.9.1 Least-squares Fit 1744.10 Generalized Extreme Value Distribution 1764.11 Numerical Solutions Using MATLAB and Excel 178References and Bibliography 183Review Questions 184Problems 186  Chapter 5 Functions of Random Variables 191What You Will Learn 1915.1 Introduction 1915.2 Functions of a Single Random Variable 1925.3 Functions of Two Random Variables 1975.3.1 Sum of Two Random Variables 1985.3.2 Product of Two Random Variables 2025.3.3 Quotient of Two Random Variables 2035.4 Function of Several Random Variables 2055.5 Moments of a Function of Several Random Variables 2055.5.1 Mean and Variance of a Linear Function 2065.5.2 Mean and Variance of Sum of Two Random Variables 2075.5.3 Mean and Variance of Product of Two Random Variables 2075.5.4 Mean and Variance of Quotient of Two Random Variables 2085.5.5 Mean and Variance of a General Nonlinear Function of Several Random Variables 2085.6 Moment-Generating Function 2125.6.1 Moments of Normally Distributed Variables 2135.7 Functions of Several Random Variables 2155.8 Numerical Solutions Using MATLAB 217References and Bibliography 220Review Questions 220Problems 222  Chapter 6 Time-Dependent Reliability of Components and Systems 232What You Will Learn 2326.1 Introduction 2326.2 Failure Rate versus Time Curve 2336.3 Reliability and Hazard Functions 2346.4 Modeling of Failure Rates 2366.5 Estimation of Failure Rate from Empirical Data 2376.6 Mean Time to Failure (MTTF) 2396.7 Reliability and Hazard Functions for Different Distributions 2416.7.1 Exponential Distribution 2416.7.2 Normal Distribution 2446.7.3 Lognormal Distribution 2466.7.4 Weibull Distribution 2516.7.5 Gamma Distribution 2566.7.6 Rayleigh Distribution 2586.7.7 Uniform Distribution 2606.8 Expected Residual Life 2626.9 Series Systems 2656.9.1 Failure Rate of the System 2676.9.2 MTBF of the System 2676.10 Parallel Systems 2686.10.1 Failure Rate of the System 2706.10.2 MTBF of the System 2706.11 (k, n) Systems 2716.11.1 MTBF of the System 2726.12 Mixed Series and Parallel Systems 2726.13 Complex Systems 2736.13.1 Enumeration Method 2746.13.2 Conditional Probability Method 2766.13.3 Cut-set Method 2786.14 Reliability Enhancement 2806.14.1 Series System 2806.14.2 Parallel System 2826.15 Reliability Allocation–AGREE Method 2836.16 Numerical Solutions Using MATLAB and Excel 286References and Bibliography 289Review Questions 289Problems 292  Chapter 7 Modeling of Geometry, Material Strength, and Loads 301What You Will Learn 3017.1 Introduction 3017.2 Modeling of Geometry 3027.2.1 Tolerances on Finished Metal Products 3037.2.2 Assembly of Components 3037.3 Modeling of Material Strength 3087.3.1 Statistics of Elastic Properties 3087.3.2 Statistical Models for Material Strength 3097.3.3 Model for Brittle Materials 3097.3.4 Model for Plastic Materials 3117.3.5 Model for Fiber Bundles 3127.4 Fatigue Strength 3147.4.1 Constant-Amplitude Fatigue Strength 3147.4.2 Variable-Amplitude Fatigue Strength 3177.5 Modeling of Loads 3197.5.1 Introduction 3197.5.2 Dead Loads 3207.5.3 Live Loads 3207.5.4 Wind Loads 3217.5.5 Earthquake Loads 3267.6 Numerical Solutions Using MATLAB and Excel 331References and Bibliography 333Review Questions 337Problems 339  Chapter 8 Strength-Based Reliability 343What You Will Learn 3438.1 Introduction 3438.2 General Expression for Reliability 3458.3 Expression for Probability of Failure 3488.4 General Interpretation of Strength and Load 3498.5 Reliability for Known Probability Distributions of S and L 3498.5.1 Reliability When S and L Follow Normal Distribution 3508.5.2 Approximate Expressions of Reliability for Normal Distribution 3528.5.3 Reliability When S and L Follow Lognormal Distribution 3568.5.4 Reliability When S and L Follow Exponential Distribution 3618.5.5 Reliability When S and L Follow Extreme Value Distributions 3638.5.6 When S and L Follow Type-III Extremal Distributions 3648.5.7 Reliability in Terms of Experimentally Determined Distributions of S and L 3658.6 Factor of Safety Corresponding to a Given Reliability 3698.7 Reliability of Systems Involving More Than Two Random Parameters 3738.8 First-Order Second-Moment (FOSM) Method 3808.9 Hasofer-Lind Reliability Index with Two Normally Distributed Variables 3838.10 Hasofer-Lind Reliability Index with Several Normally Distributed Variables 3858.11 Reliability of Weakest-Link and Fail-Safe Systems 3898.11.1 Introduction 3898.11.2 Reliability of the Fundamental Problem 3908.11.3 Reliability of Weakest-Link (or Series) Systems 3928.11.4 Reliability Analysis of Fail-Safe (or Parallel) Systems 3988.12 Numerical Solutions Using MATLAB and Excel 400References and Bibliography 405Review Questions 407Problems 411  Chapter 9 Design of Mechanical Components and Systems 425What You Will Learn 4259.1 Introduction 4259.2 Design of Mechanical Components 4269.3 Fatigue Design 4319.3.1 Deterministic Design Procedure 4329.3.2 Probabilistic Design Procedure 4359.4 Design of Mechanical Systems 4399.4.1 Reliability-Based Design of Gear Trains 4399.5 Reliability Analysis of Mechanical Systems 4459.5.1 Cam-Follower Systems 4459.5.2 Four-Bar Mechanisms 4509.6 Numerical Solutions Using MATLAB and Excel 457References and Bibliography 459Review Questions 459Problems 461  Chapter 10 Monte Carlo Simulation 465What You Will Learn 46510.1 Introduction 46510.2 Generation of Random Numbers 46610.2.1 Generation of Random Numbers Following Standard Uniform Distribution 46810.2.2 Random Variables with Nonuniform Distribution 46910.2.3 Generation of Discrete Random Variables 47210.3 Generation of Jointly Distributed Random Numbers 47510.3.1 Independent Random Variables 47510.3.2 Dependent Random Variables 47510.3.3 Generation of Correlated Normal Random Variables 47810.4 Computation of Reliability 48310.4.1 Sample Size and Error in Simulation 48310.4.2 Example: Reliability Analysis of a Straight-Line Mechanism 48510.5 Numerical Solutions Using MATLAB and Excel 489References and Bibliography 491Review Questions 492Problems 494  Chapter 11 Reliability-Based Optimum Design 504What You Will Learn 50411.1 Introduction 50411.2 Optimization Problem 50511.3 Formulation of Optimization Problems 50711.3.1 Reliability Allocation Problems 50711.3.2 Structural and Mechanical Design Problems 50911.4 Solution Techniques 51611.4.1 Graphical-Optimization Method 51611.4.2 Lagrange Multiplier Method 52011.4.3 Penalty Function Method (SUMT) 52311.4.4 Dynamic Programming 53211.5 Numerical Solutions Using MATLAB 538References and Bibliography 546Review Questions 546Problems 548  Chapter 12 Failure Modes, Event-Tree, and Fault-Tree Analyses 554What You Will Learn 55412.1 Introduction 55512.2 System-Safety Analysis 55512.3 Failure Modes and Effects Analysis (FMEA) 55712.4 Event-Tree Analysis 55812.5 Fault-Tree Analysis (FTA) 56412.5.1 Concept 56512.5.2 Procedure 56512.6 Minimal Cut-Sets 57212.6.1 Probability of the TOP Event 574References and Bibliography 582Review Questions 583Problems 585  Chapter 13 Reliability Testing 594What You Will Learn 59413.1 Introduction 59513.1.1 Objectives of Reliability Tests 59513.1.2 Details of a Reliability Test 59613.2 Analysis of Failure Time 59613.2.1 Analysis of Individual Failure Data 59613.2.2 Analysis of Grouped Failure Data 59913.3 Accelerated Life Testing 60113.3.1 Testing Until Partial Failure 60113.3.2 Magnified Loading 60213.3.3 Sudden-death Testing 60513.4 Sequential Life Testing 60813.5 Statistical Inference and Parameter Estimation 61013.5.1 Maximum-likelihood Method 61113.6 Confidence Intervals 61313.6.1 Confidence Interval on the Mean of a Normal Random Variable of Known Standard Deviation 61513.6.2 Confidence Interval on the Mean of a Normal Random Variable of Unknown Standard Deviation 61613.6.3 Confidence Interval on the Standard Deviation of a Normal Random Variable with Unknown Mean 61813.7 Plotting of Reliability Data 62013.7.1 Least-Squares Technique 62013.7.2 Linear Rectification 62113.7.3 Plotting Positions 62113.7.4 Exponential Distribution 62113.7.5 Normal Distribution 62313.7.6 Lognormal Distribution 62613.7.7 Weibull Distribution 62613.8 Numerical Solutions Using MATLAB 63013.8.1 Parameter Estimation and Confidence Intervals 63013.8.2 Plotting of Data 632References and Bibliography 634Review Questions 635Problems 638  Chapter 14 Quality Control and Reliability 642What You Will Learn 64214.1 Introduction 64214.2 Importance of Controlling Dimensions of Products 64414.3 Important Discrete Probability Distributions 64714.3.1 Binomial Distribution 64714.3.2 Hypergeometric Distribution 64814.3.3 Poisson Distribution 64914.3.4 Relationship Between Poisson and Exponential Distributions 65014.4 Six Sigma Approach and Reliability 65014.4.1 Implementation of the Six Sigma Approach 65714.5 Acceptance Sampling 65814.5.1 Characteristics of Sampling Plans 65914.6 Process Capability 65914.7 Quality Control Charts 66414.7.1 The p-Chart 66514.7.2 The X-Chart 66714.7.3 The R-Chart 67014.7.4 The c-Chart 67214.8 Risks 67314.9 Operating Characteristic (OC) Curve 67414.9.1 OC Curve 67514.9.2 Construction of OC Curve 67514.9.3 Designing a Single Sampling Plan with a Specified OC Curve 67714.10 T aguchi Method 67814.10.1 Basic Concept 67814.10.2 Loss Function 67914.10.3 Noise Factors 68114.10.4 On-Line Versus Off-Line Quality Control 68214.10.5 Three-Step Design Approach 68314.10.6 Experimental Design 68314.10.7 Signal-To-Noise Ratio 68714.10.8 Experimental Design in the Presence of Noise Factors 68914.11 Numerical Solutions Using MATLAB 697References and Bibliography 698Review Questions 699Problems 701  Chapter 15 Maintainability and Availability 706What You Will Learn 70615.1 Introduction 70615.2 Maintainability 70715.2.1 Overview 70715.2.2 Preventive Maintenance 70815.2.3 Imperfect Maintenance 71215.2.4 Repair-time Distributions 71315.2.5 Unrepaired Failures 71615.2.6 Optimal Replacement Strategy 71715.2.7 Spare Parts Requirement 71915.3 Availability 72015.3.1 Definitions [15.1, 15.3] 72015.3.2 Availability Analysis 72115.3.3 Development of the Model 72215.3.4 Systems with a Single Component 72315.3.5 Series Systems 72615.3.6 Parallel Systems 72915.4 Optimization Approaches 73015.5 Numerical Solutions Using MATLAB and Excel 731References and Bibliography 733Review Questions 734Problems 736  Chapter 16 Warranties 739What You Will Learn 73916.1 Introduction 74016.2 T ypes of Warranties 74216.3 Warranty Cost Based on a Single Failure During the Warranty Period 74216.3.1 Free Replacement Warranty 74216.3.2 Pro-rata Warranty 74416.3.3 Combined Free Replacement Warranty and Pro-rata Warranty (FRW/PRW) Policy 74716.3.4 FRW Policy Equivalent to a FRW/PRW Policy 74916.3.5 Lump-sum Payment Type of Warranty 75016.4 Warranty Costs Considering the Time Value of Money 75216.4.1 FRW Policy 75216.4.2 PRW Policy 75316.5 Warranty Reserve Fund Considering the Time Value of Money and Future Changes in the Price of the Product 75416.6 Warranty Analysis Considering Multiple Failures During the Warranty Period 75716.6.1 Renewal Process 75816.6.2 Computation and Use of Renewal Functions 75916.7 Optimum Warranty Period 76416.8 Two-dimensional Warranties 76816.9 Numerical Solutions Using MATLAB 770References and Bibliography 772Review Questions 773Problems 775  Appendix A Standard Normal Distribution Function 779Appendix B Values of ta, n for Specific Values of a and n of t Distribution 782Appendix C Values of x2n, a Corresponding to Specific Values of a and n of x2-Distribution 784Appendix D Product Liability 787Answers to Selected Problems 791Index 795