Reliability and risk models: setting reliability requirements

Contents PREFACE xv 1 SOME BASIC RELIABILITY CONCEPTS 1 1.1 Reliability (survival) Function, Cumulative Distribution and Probability Density Function of the Times to Failure 1 1.2 Random Events in Reliability and Risk Modelling 3 1.2.1 Reliability of a System with Components Logically Arranged in Series 5 1.2.2 Reliability of a System with Components Logically Arranged in Parallel 8 1.2.3 Reliability of a System with Components Logically Arranged in Series and Parallel 10 1.3 On Some Applications of the Total Probability Theorem and the Bayes Transform 12 1.3.1 Total Probability Theorem. Applications 12 1.3.2 Bayesian Transform. Applications 16 1.4 Physical and Logical Arrangement of Components 18 2 COMMON RELIABILITY AND RISK MODELS AND THEIR APPLICATIONS 19 2.1 General Framework for Reliability and Risk Analysis Based on Controlling Random Variables 19 2.2 Binomial Model 20 2.3 Homogeneous Poisson Process and Poisson Distribution 24 2.4 Negative Exponential Distribution 26 2.4.1 Memoryless Property of the Negative Exponential Distribution 28 2.4.2 Link Between the Poisson Distribution and the Negative Exponential Distribution 28 2.4.3 Reliability of a Series Arrangement, Including Components with Constant Hazard Rates 29 2.5 Hazard Rate 30 2.5.1 Difference between a Failure Density and Hazard Rate 32 2.6 Mean Time to Failure (MTTF) 33 2.7 Gamma Distribution 34 2.8 Uncertainty Associated with the Mean Time to Failure 37 2.9 Mean Time Between Failures (MTBF) 39 2.10 Uniform Distribution Model 40 2.11 Gaussian (Normal) Model 41 2.12 Log-Normal Model 45 2.13 The Weibull Model 46 2.14 Reliability Bathtub Curve for Non-Repairable Components/Systems 49 2.15 Extreme Value Models 50 3 RELIABILITY AND RISK MODELS BASED ON MIXTURE DISTRIBUTIONS 53 3.1 Distribution of a Property from Multiple Sources 53 3.2 Variance of a Property from Multiple Sources 55 3.3 Variance Upper Bound Theorem 59 3.3.1 Determining the Source Whose Removal Results in the Largest Decrease of the Variance Upper Bound 60 3.3.2 Modelling the Uncertainty of the Charpy Impact Energy at a Specified Test Temperature 61 3.4 Variation and Uncertainty Associated with the Charpy Impact Energy at a Specified Test Temperature 62 Appendix 3.1 64 Appendix 3.2 An Algorithm for Determining the Upper Bound of the Variance of Properties from Sampling from Multiple Sources 67 4 BUILDING RELIABILITY AND RISK MODELS 69 4.1 General Rules for Reliability Data Analysis 69 4.2 Probability Plotting 72 4.2.1 Testing for Consistency with the Uniform Distribution Model 75 4.2.2 Testing for Consistency with the Exponential Model 75 4.2.3 Testing for Consistency with the Weibull Distribution 76 viii Contents 4.2.4 Testing for Consistency with the Type I Extreme Value Model 77 4.2.5 Testing for Consistency with the Normal Distribution 77 4.3 Estimating Model Parameters Using the Method of Maximum Likelihood 78 4.4 Estimating the Parameters of a Three-Parameter Power Law 80 4.4.1 Some Applications of the Three-Parameter Power Law 82 5 LOAD–STRENGTH (DEMAND–CAPACITY) MODELS 85 5.1 A General Reliability Model 85 5.2 The Load–Strength Interference Model 86 5.3 Load–Strength (Demand Capacity) Integrals 88 5.4 Calculating the Load–Strength Integral Using Numerical Methods 90 5.5 Normally Distributed and Statistically Independent Load and Strength 91 5.6 Reliability and Risk Analysis Based on the Load–Strength Interference Approach 95 5.6.1 Influence of Strength Variability on Reliability 95 5.6.2 The Importance of the Shape of the Lower Tail of the Strength Distribution and the Upper Tail of the Load Distribution 99 6 SOLVING RELIABILITY AND RISK MODELS USING A MONTE CARLO SIMULATION 105 6.1 Monte Carlo Simulation Algorithms 105 6.2 Simulation of Random Variables 108 6.2.1 Simulation of a Uniformly Distributed Random Variable 108 6.2.2 Generation of a Random Subset 109 6.2.3 Inverse Transformation Method for Simulation of Continuous Random Variables 110 6.2.4 Simulation of a Random Variable Following the Negative Exponential Distribution 111 6.2.5 Simulation of a Random Variable Following the Gamma Distribution 112 6.2.6 Simulation of a Random Variable Following a Homogeneous Poisson Process in a Finite Interval 112 6.2.7 Simulation of a Discrete Random Variable with a Specified Distribution 113 6.2.8 Selection of a Point at Random in Three-dimensional Space Region 114 Contents ix 6.2.9 Simulation of Random Locations Following a Homogeneous Poisson Process in a Finite Domain 115 6.2.10 Simulation of a Random Direction in Space 116 6.2.11 Simulation of a Random Variable Following the Three-parameter Weibull Distribution 117 6.2.12 Simulation of a random variable following the maximum extreme value distribution 118 6.2.13 Simulation of a Gaussian Random Variable 118 6.2.14 Simulation of a Log-normal Random Variable 119 6.2.15 Conditional Probability Technique for Bivariate Sampling 120 6.2.16 Von Neumann’s Method for Modelling Continuous Random Variables 121 6.2.17 Random sampling from a mixture distribution 123 6.3 Monte Carlo Simulation Algorithms for Solving Reliability and Risk Models 123 6.3.1 An Algorithm for Solving the General Reliability Model 123 6.3.2 Monte Carlo Evaluation of the Reliability on Demand during a Load–Strength Interference 124 6.4 Monte Carlo Simulation of the Lower Tail of the Strength Distribution for Materials Containing Flaws 127 Appendix 6.1 130 7 ANALYSIS OF THE PROPERTIES OF INHOMOGENEOUS MEDIA USING MONTE CARLO SIMULATIONS 133 7.1 Analysis of Inhomogeneous Microstructures Using Random Transects 134 7.2 Empirical Cumulative Distribution of the Intercepts 135 7.3 Intercept Variance 139 7.4 Lower Tail of Material Properties, Monotonically Dependent on the Amount of the Intercepted Fraction 142 8 MECHANISMS OF FAILURE 145 8.1 Overstress Failures 145 8.1.1 Brittle Fracture 145 8.1.2 Ductile Fracture 147 8.1.3 Ductile-to-brittle Transition Region 147 8.1.4 Yielding 149 8.2 Wear-out Failures 149 8.2.1 Fatigue Failures 149 8.2.2 Failures due to Corrosion and Erosion 151 x Contents 8.3 Early-life Failures 153 8.3.1 Influence of the Design on Early-life Failures 153 8.3.2 Influence of the Variability of Critical Design Parameters on Early-life Failures 154 9 OVERSTRESS RELIABILITY INTEGRAL AND DAMAGE FACTORISATION LAW 159 9.1 Reliability Associated with Overstress Failure Mechanisms 159 9.2 Damage Factorisation Law 162 10 DETERMINING THE PROBABILITY OF FAILURE FOR COMPONENTS CONTAINING FLAWS 165 10.1 Background 165 10.2 General Equation Related to the Probability of Failure of a Stressed Component with Internal Flaws 168 10.3 Determining the Individual Probability of Triggering Failure, Characterising a Single Flaw 170 10.4 Parameters Related to the Statistics of Fracture Triggered by Flaws 173 10.5 Upper Bounds of the Flaw Number Density and the Stressed Volume Guaranteeing Probability of Failure below a Maximum Acceptable Level 174 10.6 A Stochastic Model Related to the Fatigue Life Distribution of a Component Containing Defects 176 11 UNCERTAINTY ASSOCIATED WITH THE LOCATION OF THE DUCTILE-TO-BRITTLE TRANSITION REGION OF MULTI-RUN WELDS 179 11.1 Modelling the Systematic and Random Component of the Charpy Impact Energy 179 11.2 Determining the Uncertainty Associated with the Location of the Ductile-to-brittle Transition Region 181 11.3 Risk Assessment Associated with the Location of the Transition Region 186 12 MODELLING THE KINETICS OF DETERIORATION OF PROTECTIVE COATINGS DUE TO CORROSION 191 12.1 Statement of the Problem 191 12.2 Modelling the Quantity of Corroded Protective Coating with Time 192 12.3 An Illustrative Example 194 Contents xi 13 MINIMISING THE PROBABILITY OF FAILURE OF AUTOMOTIVE SUSPENSION SPRINGS BY DELAYING THE FATIGUE FAILURE MODE 199 14 RELIABILITY GOVERNED BY THE RELATIVE LOCATIONS OF RANDOM VARIABLES IN A FINITE DOMAIN 205 14.1 Reliability Dependent on the Relative Configurations of Random Variables 205 14.2 A Generic Equation Related to Reliability Dependent on the Relative Locations of a Fixed Number of Random Variables 206 14.2.1 An Illustrative Example. Probability of Clustering of Random Demands within a Critical Interval 209 14.3 A Given Number of Uniformly Distributed Random Variables in a Finite Interval (Conditional Case) 210 14.4 Applications 212 14.5 Reliability Governed by the Relative Locations of Random Variables Following a Homogeneous Poisson Process in a Finite Domain 216 Appendix 14.1 217 15 RELIABILITY DEPENDENT ON THE EXISTENCE OF MINIMUM CRITICAL DISTANCES BETWEEN THE LOCATIONS OF RANDOM VARIABLES IN A FINITE INTERVAL 221 15.1 Problems Requiring Reliability Measures Based on Minimum Critical Intervals (MCI) and Minimum Failure-free Operating Periods (MFFOP) 221 15.2 The MCI and MFFOP Reliability Measures 224 15.3 General Equations Related to Random Variables Following a Homogeneous Poisson Process in a Finite Interval 225 15.4 Application Examples 227 15.4.1 Setting Reliability Requirements to Guarantee a Specified MFFOP 227 15.4.2 Reliability Assurance that a Specified MFFOP has been met 228 15.4.3 Specifying a Number Density Envelope to Guarantee Probability of Clustering below a Maximum Acceptable Level 230 xii Contents 15.5 Setting Reliability Requirements to Guarantee a Minimum Failure-free Operating Period before Failures Followed by a Downtime 231 15.5.1 A Monte Carlo Simulation Algorithm for Evaluating the Probability of Existence of an MFFOP of Specified Length 233 15.6 A Numerical Example 234 15.7 Setting Reliability Requirements to Guarantee an Availability Target 235 15.7.1 Monte Carlo Evaluation of the Average Availability Associated with a Finite Time Interval 236 15.7.2 A Numerical Example 237 16 RELIABILITY ANALYSIS AND SETTING RELIABILITY REQUIREMENTS BASED ON THE COST OF FAILURE 239 16.1 General Models for Setting Cost-of-failure-driven Reliability Requirements for Non-repairable Components/Systems 240 16.1.1 A Constant Cost of Failure 240 16.2 Cost of Failure from Mutually Exclusive Failure Modes 242 16.3 Some Important Counterexamples 246 16.4 Time-dependent Cost of Failure 249 16.4.1 Calculating the Risk of Failure in case of Cost of Failure Dependent on Time 249 16.5 Risk of Premature Failure if the Cost of Failure Depends on Time 251 16.6 Setting Reliability Requirements to Minimise the Total Cost. Value from the Reliability Investment 252 16.7 Guaranteeing Multiple Reliability Requirements for Systems with Components Logically Arranged in Series 259 16.8 Risk of Premature Failure for Systems whose Components are not Logically Arranged in Series 261 16.8.1 Expected Losses from Failures for Repairable Systems whose Components are not Arranged in Series 264 16.9 Reliability Allocation to Minimise the Total Losses 264 APPENDIX A 267 1. Random Events 267 2. Union of Events 270 3. Intersection of Events 270 Contents xiii 4. Probability 274 5. Probability of a Union and Intersection of Mutually Exclusive Events 275 6. Conditional Probability 276 7. Probability of a Union of Non-disjoint Events 279 8. Statistically Dependent Events 281 9. Statistically Independent Events 281 10. Probability of a Union of Independent Events 282 11. Boolean Variables and Boolean Algebra 282 APPENDIX B 289 1. Random Variables. Basic Properties 289 2. Boolean Random Variables 290 3. Continuous Random Variables 291 4. Probability Density Function 291 5. Cumulative Distribution Function 292 6. Joint Distribution of Continuous Random Variables 292 7. Correlated Random Variables 293 8. Statistically Independent Random Variables 294 9. Properties of the Expectations and Variances of Random Variables 295 10. Important Theoretical Results Regarding the Sample Mean 297 APPENDIX C CUMULATIVE DISTRIBUTION FUNCTION OF THE STANDARD NORMAL DISTRIBUTION 299 APPENDIX D c2-DISTRIBUTION 301 REFERENCES 307 INDEX 313