Reliability and Risk Models

Michael Todinov Oxford Brookes University, UK

• Series Preface xviiPreface xix1 Failure Modes: Building Reliability Networks 11.1 Failure Modes 11.2 Series and Parallel Arrangement of the Components in a Reliability Network 51.3 Building Reliability Networks: Difference between a Physical and Logical Arrangement 61.4 Complex Reliability Networks Which Cannot Be Presented as a Combination of Series and Parallel Arrangements 101.5 Drawbacks of the Traditional Representation of the Reliability Block Diagrams 111.5.1 Reliability Networks Which Require More Than a Single Terminal Node 111.5.2 Reliability Networks Which Require the Use of Undirected Edges Only,Directed Edges Only or a Mixture of Undirected and Directed Edges 131.5.3 Reliability Networks Which Require Different Edges Referring to the Same Component 161.5.4 Reliability Networks Which Require Negative‐State Components 172 Basic Concepts 212.1 Reliability (Survival) Function, Cumulative Distribution and Probability Density Function of the Times to Failure 212.2 Random Events in Reliability and Risk Modelling 232.2.1 Reliability and Risk Modelling Using Intersection of Statistically Independent Random Events 232.2.2 Reliability and Risk Modelling Using a Union of Mutually Exclusive Random Events 252.2.3 Reliability of a System with Components Logically Arranged in Series 272.2.4 Reliability of a System with Components Logically Arranged in Parallel 292.2.5 Reliability of a System with Components Logically Arranged in Series and Parallel 312.2.6 Using Finite Sets to Infer Component Reliability 322.3 Statistically Dependent Events and Conditional Probability in Reliability and Risk Modelling 332.4 Total Probability Theorem in Reliability and Risk Modelling. Reliability of Systems with Complex Reliability Networks 362.5 Reliability and Risk Modelling Using Bayesian Transform and Bayesian Updating 432.5.1 Bayesian Transform 432.5.2 Bayesian Updating 443 Common Reliability and Risk Models and Their Applications 473.1 General Framework for Reliability and Risk Analysis Based on Controlling Random Variables 473.2 Binomial Model 483.2.1 Application: A Voting System 523.3 Homogeneous Poisson Process and Poisson Distribution 533.4 Negative Exponential Distribution 563.4.1 Memoryless Property of the Negative Exponential Distribution 573.5 Hazard Rate 583.5.1 Difference between Failure Density and Hazard Rate 603.5.2 Reliability of a Series Arrangement Including Components with Constant Hazard Rates 613.6 Mean Time to Failure 613.7 Gamma Distribution 633.8 Uncertainty Associated with the MTTF 653.9 Mean Time between Failures 673.10 Problems with the MTTF and MTBF Reliability Measures 673.11 BX% Life 683.12 Minimum Failure‐Free Operation Period 693.13 Availability 703.13.1 Availability on Demand 703.13.2 Production Availability 713.14 Uniform Distribution Model 723.15 Normal (Gaussian) Distribution Model 733.16 Log‐Normal Distribution Model 773.17 Weibull Distribution Model of the Time to Failure 793.18 Extreme Value Distribution Model 813.19 Reliability Bathtub Curve 824 Reliability and Risk Models Based on Distribution Mixtures 874.1 Distribution of a Property from Multiple Sources 874.2 Variance of a Property from Multiple Sources 894.3 Variance Upper Bound Theorem 914.3.1 Determining the Source Whose Removal Results in the Largest Decrease of the Variance Upper Bound 924.4 Applications of the Variance Upper Bound Theorem 934.4.1 Using the Variance Upper Bound Theorem for Increasing the Robustness of Products and Processes 934.4.2 Using the Variance Upper Bound Theorem for Developing Six‐Sigma Products and Processes 97Appendix 4.1: Derivation of the Variance Upper Bound Theorem 99Appendix 4.2: An Algorithm for Determining the Upper Bound of the Variance of Properties from Sampling Multiple Sources 1015 Building Reliability and Risk Models 1035.1 General Rules for Reliability Data Analysis 1035.2 Probability Plotting 1075.2.1 Testing for Consistency with the Uniform Distribution Model 1095.2.2 Testing for Consistency with the Exponential Model 1095.2.3 Testing for Consistency with the Weibull Distribution 1105.2.4 Testing for Consistency with the Type I Extreme Value Distribution 1115.2.5 Testing for Consistency with the Normal Distribution 1115.3 Estimating Model Parameters Using the Method of Maximum Likelihood 1135.4 Estimating the Parameters of a Three‐Parameter Power Law 1145.4.1 Some Applications of the Three‐Parameter Power Law 1166 Load–Strength (Demand‐Capacity) Models 1196.1 A General Reliability Model 1196.2 The Load–Strength Interference Model 1206.3 Load–Strength (Demand‐Capacity) Integrals 1226.4 Evaluating the Load–Strength Integral Using Numerical Methods 1246.5 Normally Distributed and Statistically Independent Load and Strength 1256.6 Reliability and Risk Analysis Based on the Load–Strength Interference Approach 1306.6.1 Influence of Strength Variability on Reliability 1306.6.2 Critical Weaknesses of the Traditional Reliability Measures ‘Safety Margin’ and ‘Loading Roughness’ 1346.6.3 Interaction between the Upper Tail of the Load Distribution and the Lower Tail of the Strength Distribution 1367 Overstress Reliability Integral and Damage Factorisation Law 1397.1 Reliability Associated with Overstress Failure Mechanisms 1397.1.1 The Link between the Negative Exponential Distribution and the Overstress Reliability Integral 1417.2 Damage Factorisation Law 1438 Solving Reliability and Risk Models Using a Monte Carlo Simulation 1478.1 Monte Carlo Simulation Algorithms 1478.1.1 Monte Carlo Simulation and the Weak Law of Large Numbers 1478.1.2 Monte Carlo Simulation and the Central Limit Theorem 1498.1.3 Adopted Conventions in Describing the Monte Carlo Simulation Algorithms 1498.2 Simulation of Random Variables 1518.2.1 Simulation of a Uniformly Distributed Random Variable 1518.2.2 Generation of a Random Subset 1528.2.3 Inverse Transformation Method for Simulation of Continuous Random Variables 1538.2.4 Simulation of a Random Variable following the Negative Exponential Distribution 1548.2.5 Simulation of a Random Variable following the Gamma Distribution 1548.2.6 Simulation of a Random Variable following a Homogeneous Poisson Process in a Finite Interval 1558.2.7 Simulation of a Discrete Random Variable with a Specified Distribution 1568.2.8 Selection of a Point at Random in the N‐Dimensional Space Region 1578.2.9 Simulation of Random Locations following a Homogeneous Poisson Process in a Finite Domain 1588.2.10 Simulation of a Random Direction in Space 1588.2.11 Generating Random Points on a Disc and in a Sphere 1608.2.12 Simulation of a Random Variable following the Three‐Parameter Weibull Distribution 1628.2.13 Simulation of a Random Variable following the Maximum Extreme Value Distribution 1628.2.14 Simulation of a Gaussian Random Variable 1628.2.15 Simulation of a Log‐Normal Random Variable 1638.2.16 Conditional Probability Technique for Bivariate Sampling 1648.2.17 Von Neumann’s Method for Sampling Continuous Random Variables 1658.2.18 Sampling from a Mixture Distribution 166Appendix 8.1 1669 Evaluating Reliability and Probability of a Faulty Assembly Using Monte Carlo Simulation 1699.1 A General Algorithm for Determining Reliability Controlled by Statistically Independent Random Variables 1699.2 Evaluation of the Reliability Controlled by a Load–Strength Interference 1709.2.1 Evaluation of the Reliability on Demand, with No Time Included 1709.2.2 Evaluation of the Reliability Controlled by Random Shocks on a Time Interval 1719.3 A Virtual Testing Method for Determining the Probability of Faulty Assembly 1739.4 Optimal Replacement to Minimise the Probability of a System Failure 17710 Evaluating the Reliability of Complex Systems and Virtual Accelerated Life Testing Using Monte Carlo Simulation 18110.1 Evaluating the Reliability of Complex Systems 18110.2 Virtual Accelerated Life Testing of Complex Systems 18310.2.1 Acceleration Stresses and Their Impact on the Time to Failure of Components 18310.2.2 Arrhenius Stress–Life Relationship and Arrhenius‐Type Acceleration Life Models 18510.2.3 Inverse Power Law Relationship and Inverse Power Law‐Type Acceleration Life Models 18510.2.4 Eyring Stress–Life Relationship and Eyring‐Type Acceleration Life Models 18511 Generic Principles for Reducing Technical Risk 18911.1 Preventive Principles: Reducing Mainly the Likelihood of Failure 19111.1.1 Building in High Reliability in Processes, Components and Systems with Large Failure Consequences 19111.1.2 Simplifying at a System and Component Level 19211.1.2.1 Reducing the Number of Moving Parts 19311.1.3 Root Cause Failure Analysis 19311.1.4 Identifying and Removing Potential Failure Modes 19411.1.5 Mitigating the Harmful Effect of the Environment 19411.1.6 Building in Redundancy 19511.1.7 Reliability and Risk Modelling and Optimisation 19711.1.7.1 Building and Analysing Comparative Reliability Models 19711.1.7.2 Building and Analysing Physics of Failure Models 19811.1.7.3 Minimising Technical Risk through Optimisation and Optimal Replacement 19911.1.7.4 Maximising System Reliability and Availability by Appropriate Permutations of Interchangeable Components 19911.1.7.5 Maximising the Availability and Throughput Flow Reliability by Altering the Network Topology 19911.1.8 Reducing Variability of Risk-Critical Parameters and Preventing them from Reaching Dangerous Values 19911.1.9 Altering the Component Geometry 20011.1.10 Strengthening or Eliminating Weak Links 20111.1.11 Eliminating Factors Promoting Human Errors 20211.1.12 Reducing Risk by Introducing Inverse States 20311.1.12.1 Inverse States Cancelling the Anticipated State with a Negative Impact 20311.1.12.2 Inverse States Buffering the Anticipated State with a Negative Impact 20311.1.12.3 Inverting the Relative Position of Objects and the Direction of Flows 20411.1.12.4 Inverse State as a Counterbalancing Force 20511.1.13 Failure Prevention Interlocks 20611.1.14 Reducing the Number of Latent Faults 20611.1.15 Increasing the Level of Balancing 20811.1.16 Reducing the Negative Impact of Temperature by Thermal Design 20911.1.17 Self‐Stability 21111.1.18 Maintaining the Continuity of a Working State 21211.1.19 Substituting Mechanical Assemblies with Electrical, Optical or Acoustic Assemblies and Software 21211.1.20 Improving the Load Distribution 21211.1.21 Reducing the Sensitivity of Designs to the Variation of Design Parameters 21211.1.22 Vibration Control 21611.1.23 Built‐In Prevention 21611.2 Dual Principles: Reduce Both the Likelihood of Failure and the Magnitude of Consequences 21711.2.1 Separating Critical Properties, Functions and Factors 21711.2.2 Reducing the Likelihood of Unfavourable Combinations of Risk‐Critical Random Variables 21811.2.3 Condition Monitoring 21911.2.4 Reducing the Time of Exposure or the Space of Exposure 21911.2.4.1 Time of Exposure 21911.2.4.2 Length of Exposure and Space of Exposure 22011.2.5 Discovering and Eliminating a Common Cause: Diversity in Design 22011.2.6 Eliminating Vulnerabilities 22211.2.7 Self‐Reinforcement 22311.2.8 Using Available Local Resources 22311.2.9 Derating 22411.2.10 Selecting Appropriate Materials and Microstructures 22511.2.11 Segmentation 22511.2.11.1 Segmentation Improves the Load Distribution 22511.2.11.2 Segmentation Reduces the Vulnerability to a Single Failure 22511.2.11.3 Segmentation Reduces the Damage Escalation 22611.2.11.4 Segmentation Limits the Hazard Potential 22611.2.12 Reducing the Vulnerability of Targets 22611.2.13 Making Zones Experiencing High Damage/Failure Rates Replaceable 22711.2.14 Reducing the Hazard Potential 22711.2.15 Integrated Risk Management 22711.3 Protective Principles: Minimise the Consequences of Failure 22911.3.1 Fault‐Tolerant System Design 22911.3.2 Preventing Damage Escalation and Reducing the Rate of Deterioration 22911.3.3 Using Fail‐Safe Designs 23011.3.4 Deliberately Designed Weak Links 23111.3.5 Built‐In Protection 23111.3.6 Troubleshooting Procedures and Systems 23211.3.7 Simulation of the Consequences from Failure 23211.3.8 Risk Planning and Training 23312 Physics of Failure Models 23512.1 Fast Fracture 23512.1.1 Fast Fracture: Driving Forces behind Fast Fracture 23512.1.2 Reducing the Likelihood of Fast Fracture 24112.1.2.1 Basic Ways of Reducing the Likelihood of Fast Fracture 24212.1.2.2 Avoidance of Stress Raisers or Mitigating Their Harmful Effect 24412.1.2.3 Selecting Materials Which Fail in a Ductile Fashion 24512.1.3 Reducing the Consequences of Fast Fracture 24712.1.3.1 By Using Fail-Safe Designs 24712.1.3.2 By Using Crack Arrestors 25012.2 Fatigue Fracture 25112.2.1 Reducing the Risk of Fatigue Fracture 25712.2.1.1 Reducing the Size of the Flaws 25712.2.1.2 Increasing the Final Fatigue Crack Length by Selecting Material with a Higher Fracture Toughness 25712.2.1.3 Reducing the Stress Range by an Appropriate Design 25712.2.1.4 Reducing the Stress Range by Restricting the Springback of Elastic Components 25812.2.1.5 Reducing the Stress Range by Reducing the Magnitude of Thermal Stresses 25912.2.1.6 Reducing the Stress Range by Introducing Compressive Residual Stresses at the Surface 26112.2.1.7 Reducing the Stress Range by Avoiding Excessive Bending 26212.2.1.8 Reducing the Stress Range by Avoiding Stress Concentrators 26312.2.1.9 Improving the Condition of the Surface and Eliminating Low-Strength Surfaces 26312.2.1.10 Increasing the Fatigue Life of Automotive Suspension Springs 26412.3 Early‐Life Failures 26512.3.1 Influence of the Design on Early‐Life Failures 26512.3.2 Influence of the Variability of Critical Design Parameters on Early‐Life Failures 26613 Probability of Failure Initiated by Flaws 26913.1 Distribution of the Minimum Fracture Stress and a Mathematical Formulation of the Weakest‐Link Concept 26913.2 The Stress Hazard Density as an Alternative of the Weibull Distribution 27413.3 General Equation Related to the Probability of Failure of a Stressed Component with Complex Shape 27613.4 Link between the Stress Hazard Density and the Conditional Individual Probability of Initiating Failure 27813.5 Probability of Failure Initiated by Defects in Components with Complex Shape 27913.6 Limiting the Vulnerability of Designs to Failure Caused by Flaws 28014 A Comparative Method for Improving the Reliability and Availability of Components and Systems 28314.1 Advantages of the Comparative Method to Traditional Methods 28314.2 A Comparative Method for Improving the Reliability of Components Whose Failure is Initiated by Flaws 28514.3 A Comparative Method for Improving System Reliability 28914.4 A Comparative Method for Improving the Availability of Flow Networks 29015 Reliability Governed by the Relative Locations of Random Variables in a Finite Domain 29315.1 Reliability Dependent on the Relative Configurations of Random Variables 29315.2 A Generic Equation Related to Reliability Dependent on the Relative Locations of a Fixed Number of Random Variables 29315.3 A Given Number of Uniformly Distributed Random Variables in a Finite Interval (Conditional Case) 29715.4 Probability of Clustering of a Fixed Number Uniformly Distributed Random Events 29815.5 Probability of Unsatisfied Demand in the Case of One Available Source and Many Consumers 30215.6 Reliability Governed by the Relative Locations of Random Variables following a Homogeneous Poisson Process in a Finite Domain 304Appendix 15.1 30516 Reliability and Risk Dependent on the Existence of Minimum Separation Intervals between the Locations of Random Variables on a Finite Interval 30716.1 Applications Requiring Minimum Separation Intervals and Minimum Failure‐Free Operating Periods 30716.2 Minimum Separation Intervals and Rolling MFFOP Reliability Measures 30916.3 General Equations Related to Random Variables following a Homogeneous Poisson Process in a Finite Interval 31016.4 Application Examples 31216.4.1 Setting Reliability Requirements to Guarantee a Specified MFFOP 31216.4.2 Reliability Assurance That a Specified MFFOP Has Been Met 3120002547085.indd 13 8/18/2015 6:29:01 PMxiv Contents16.4.3 Specifying a Number Density Envelope to Guarantee Probabilityof Unsatisfied Random Demand below a Maximum Acceptable Level 31416.4.4 Insensitivity of the Probability of Unsatisfied Demand to the Variance of the Demand Time 31516.5 Setting Reliability Requirements to Guarantee a Rolling MFFOP Followed by a Downtime 31716.6 Setting Reliability Requirements to Guarantee an Availability Target 32016.7 Closed-Form Expression for the Expected Fraction of the Time of Unsatisfied Demand 32317 Reliability Analysis and Setting Reliability Requirements Based on the Cost of Failure 32717.1 The Need for a Cost‐of‐Failure‐Based Approach 32717.2 Risk of Failure 32817.3 Setting Reliability Requirements Based on a Constant Cost of Failure 33017.4 Drawbacks of the Expected Loss as a Measure of the Potential Loss from Failure 33217.5 Potential Loss, Conditional Loss and Risk of Failure 33317.6 Risk Associated with Multiple Failure Modes 33617.6.1 An Important Special Case 33717.7 Expected Potential Loss Associated with Repairable Systems Whose Component Failures Follow a Homogeneous Poisson Process 33817.8 A Counterexample Related to Repairable Systems 34117.9 Guaranteeing Multiple Reliability Requirements for Systems with Components Logically Arranged in Series 34218 Potential Loss, Potential Profit and Risk 34518.1 Deficiencies of the Maximum Expected Profit Criterion in Selecting a Risky Prospect 34518.2 Risk of a Net Loss and Expected Potential Reward Associated with a Limited Number of Statistically Independent Risk–Reward Bets in a Risky Prospect 34618.3 Probability and Risk of a Net Loss Associated with a Small Number of Opportunity Bets 34818.4 Samuelson’s Sequence of Good Bets Revisited 35118.5 Variation of the Risk of a Net Loss Associated with a Small Number of Opportunity Bets 35218.6 Distribution of the Potential Profit from a Limited Number of Risk–Reward Activities 35319 Optimal Allocation of Limited Resources among Discrete Risk Reduction Options 35719.1 Statement of the Problem 35719.2 Weaknesses of the Standard (0‐1) Knapsack Dynamic Programming Approach 35919.2.1 A Counterexample 35919.2.2 The New Formulation of the Optimal Safety Budget Allocation Problem 36019.2.3 Dependence of the Removed System Risk on the Appropriate Selection of Combinations of Risk Reduction Options 36119.2.4 A Dynamic Algorithm for Solving the Optimal Safety Budget Allocation Problem 36519.3 Validation of the Model by a Recursive Backtracking 369Appendix A 373A.1 Random Events 373A.2 Union of Events 375A.3 Intersection of Events 376A.4 Probability 378A.5 Probability of a Union and Intersection of Mutually Exclusive Events 379A.6 Conditional Probability 380A.7 Probability of a Union of Non‐disjoint Events 383A.8 Statistically Dependent Events 384A.9 Statistically Independent Events 384A.10 Probability of a Union of Independent Events 385A.11 Boolean Variables and Boolean Algebra 385Appendix B 391B.1 Random Variables: Basic Properties 391B.2 Boolean Random Variables 392B.3 Continuous Random Variables 392B.4 Probability Density Function 392B.5 Cumulative Distribution Function 393B.6 Joint Distribution of Continuous Random Variables 393B.7 Correlated Random Variables 394B.8 Statistically Independent Random Variables 395B.9 Properties of the Expectations and Variances of Random Variables 396B.10 Important Theoretical Results Regarding the Sample Mean 397Appendix C: Cumulative Distribution Function of the Standard Normal Distribution 399Appendix D: χ2‐Distribution 401References 407Index 413