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E-raamat: Reliability and Risk Models: Setting Reliability Requirements

(Cranfield University, UK)
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Reliability and Risk Models: Setting Reliability RequirementsSuurem pilt
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A comprehensively updated and reorganized new edition. The updates include comparative methods for improving reliability; methods for optimal allocation of limited resources to achieve a maximum risk reduction; methods for improving reliability at no extra cost and building reliability networks for engineering systems.

Includes:

• A unique set of 46 generic principles for reducing technical risk

• Monte Carlo simulation algorithms for improving reliability and reducing risk

• Methods for setting reliability requirements based on the cost of failure

• New reliability measures based on a minimal separation of random events on a time interval

• Overstress reliability integral for determining the time to failure caused by overstress failure modes

• A powerful equation for determining the probability of failure controlled by defects in loaded components with complex shape

• Comparative methods for improving reliability which do not require reliability data

• Optimal allocation of limited resources to achieve a maximum risk reduction

• Improving system reliability based solely on a permutation of interchangeable components

Series Preface xvii
Preface xix
1 Failure Modes: Building Reliability Networks
1(20)
1.1 Failure Modes
1(4)
1.2 Series and Parallel Arrangement of the Components in a Reliability Network
5(1)
1.3 Building Reliability Networks: Difference between a Physical and Logical Arrangement
6(4)
1.4 Complex Reliability Networks Which Cannot Be Presented as a Combination of Series and Parallel Arrangements
10(1)
1.5 Drawbacks of the Traditional Representation of the Reliability Block Diagrams
11(10)
1.5.1 Reliability Networks Which Require More Than a Single Terminal Node
11(2)
1.5.2 Reliability Networks Which Require the Use of Undirected Edges Only, Directed Edges Only or a Mixture of Undirected and Directed Edges
13(3)
1.5.3 Reliability Networks Which Require Different Edges Referring to the Same Component
16(1)
1.5.4 Reliability Networks Which Require Negative-State Components
17(4)
2 Basic Concepts
21(26)
2.1 Reliability (Survival) Function, Cumulative Distribution and Probability Density Function of the Times to Failure
21(2)
2.2 Random Events in Reliability and Risk Modelling
23(10)
2.2.1 Reliability and Risk Modelling Using Intersection of Statistically Independent Random Events
23(2)
2.2.2 Reliability and Risk Modelling Using a Union of Mutually Exclusive Random Events
25(2)
2.2.3 Reliability of a System with Components Logically Arranged in Series
27(2)
2.2.4 Reliability of a System with Components Logically Arranged in Parallel
29(2)
2.2.5 Reliability of a System with Components Logically Arranged in Series and Parallel
31(1)
2.2.6 Using Finite Sets to Infer Component Reliability
32(1)
2.3 Statistically Dependent Events and Conditional Probability in Reliability and Risk Modelling
33(3)
2.4 Total Probability Theorem in Reliability and Risk Modelling. Reliability of Systems with Complex Reliability Networks
36(7)
2.5 Reliability and Risk Modelling Using Bayesian Transform and Bayesian Updating
43(4)
2.5.1 Bayesian Transform
43(1)
2.5.2 Bayesian Updating
44(3)
3 Common Reliability and Risk Models and Their Applications
47(40)
3.1 General Framework for Reliability and Risk Analysis Based on Controlling Random Variables
47(1)
3.2 Binomial Model
48(5)
3.2.1 Application: A Voting System
52(1)
3.3 Homogeneous Poisson Process and Poisson Distribution
53(3)
3.4 Negative Exponential Distribution
56(2)
3.4.1 Memoryless Property of the Negative Exponential Distribution
57(1)
3.5 Hazard Rate
58(3)
3.5.1 Difference between Failure Density and Hazard Rate
60(1)
3.5.2 Reliability of a Series Arrangement Including Components with Constant Hazard Rates
61(1)
3.6 Mean Time to Failure
61(2)
3.7 Gamma Distribution
63(2)
3.8 Uncertainty Associated with the MTTF
65(2)
3.9 Mean Time between Failures
67(1)
3.10 Problems with the MTTF and MTBF Reliability Measures
67(1)
3.11 BX% Life
68(1)
3.12 Minimum Failure-Free Operation Period
69(1)
3.13 Availability
70(2)
3.13.1 Availability on Demand
70(1)
3.13.2 Production Availability
71(1)
3.14 Uniform Distribution Model
72(1)
3.15 Normal (Gaussian) Distribution Model
73(4)
3.16 Log-Normal Distribution Model
77(2)
3.17 Weibull I Distribution Model of the Time to Failure
79(2)
3.18 Extreme Value Distribution Model
81(1)
3.19 Reliability Bathtub Curve
82(5)
4 Reliability and Risk Models Based on Distribution Mixtures
87(16)
4.1 Distribution of a Property from Multiple Sources
87(2)
4.2 Variance of a Property from Multiple Sources
89(2)
4.3 Variance Upper Bound Theorem
91(2)
4.3.1 Determining the Source Whose Removal Results in the Largest Decrease of the Variance Upper Bound
92(1)
4.4 Applications of the Variance Upper Bound Theorem
93(10)
4.4.1 Using the Variance Upper Bound Theorem for Increasing the Robustness of Products and Processes
93(4)
4.4.2 Using the Variance Upper Bound Theorem for Developing Six-Sigma Products and Processes
97(2)
Appendix 4.1 Derivation of the Variance Upper Bound Theorem
99(2)
Appendix 4.2 An Algorithm for Determining the Upper Bound of the Variance of Properties from Sampling Multiple Sources
101(2)
5 Building Reliability and Risk Models
103(16)
5.1 General Rules for Reliability Data Analysis
103(4)
5.2 Probability Plotting
107(6)
5.2.1 Testing for Consistency with the Uniform Distribution Model
109(1)
5.2.2 Testing for Consistency with the Exponential Model
109(1)
5.2.3 Testing for Consistency with the Weibull Distribution
110(1)
5.2.4 Testing for Consistency with the Type I Extreme Value Distribution
111
5.2.5 Testing for Consistency will? the Normal Distribution
111(2)
5.3 Estimating Model Parameters Using the Method of Maximum Likelihood
113(1)
5.4 Estimating the Parameters of a Three-Parameter Power Law
114(5)
5.4.1 Some Applications of the Three-Parameter Power Law
116(3)
6 Load--Strength (Demand-Capacity) Models
119(20)
6.1 A General Reliability Model
119(1)
6.2 The Load--Strength Interference Model
120(2)
6.3 Load--Strength (Demand-Capacity) Integrals
122(2)
6.4 Evaluating the Load--Strength Integral Using Numerical Methods
124(1)
6.5 Normally Distributed and Statistically Independent Load and Strength
125(5)
6.6 Reliability and Risk Analysis Based on the Load-Strength Interference Approach
130(9)
6.6.1 Influence of Strength Variability on Reliability
130(4)
6.6.2 Critical Weaknesses of the Traditional Reliability Measures `Safety Margin' and `Loading Roughness'
134(2)
6.6.3 Interaction between the Upper Tail of the Load Distribution and the Lower Tail of the Strength Distribution
136(3)
7 Overstress Reliability Integral and Damage Factorisation Law
139(8)
7.1 Reliability Associated with Overstress Failure Mechanisms
139(4)
7.1.1 The Link between the Negative Exponential Distribution and the Overstress Reliability Integral
141(2)
7.2 Damage Factorisation Law
143(4)
8 Solving Reliability and Risk Models Using a Monte Carlo Simulation
147(22)
8.1 Monte Carlo Simulation Algorithms
147(4)
8.1.1 Monte Carlo Simulation and the Weak Law of Large Numbers
147(2)
8.1.2 Monte Carlo Simulation and the Central Limit Theorem
149(1)
8.1.3 Adopted Conventions in Describing the Monte Carlo Simulation Algorithms
149(2)
8.2 Simulation of Random Variables
151(18)
8.2.1 Simulation of a Uniformly Distributed Random Variable
151(1)
8.2.2 Generation of a Random Subset
152(1)
8.2.3 Inverse Transformation Method for Simulation of Continuous Random Variables
153(1)
8.2.4 Simulation of a Random Variable following the Negative Exponential Distribution
154(1)
8.2.5 Simulation of a Random Variable following the Gamma Distribution
154(1)
8.2.6 Simulation of a Random Variable following a Homogeneous Poisson Process in a Finite Interval
155(1)
8.2.7 Simulation of a Discrete Random Variable with a Specified Distribution
156(1)
8.2.8 Selection of a Point at Random in the N-Dimensional Space Region
157(1)
8.2.9 Simulation of Random Locations following a Homogeneous Poisson Process in a Finite Domain
158(1)
8.2.10 Simulation of a Random Direction in Space
158(2)
8.2.11 Generating Random Points on a Disc and in a Sphere
160(2)
8.2.12 Simulation of a Random Variable following the Three-Parameter Weibull Distribution
162(1)
8.2.13 Simulation of a Random Variable following the Maximum Extreme Value Distribution
162(1)
8.2.14 Simulation of a Gaussian Random Variable
162(1)
8.2.15 Simulation of a Log-Normal Random Variable
163(1)
8.2.16 Conditional Probability Technique for Bivariate Sampling
164(1)
8.2.17 Von Neumann's Method for Sampling Continuous Random Variables
165(1)
8.2.18 Sampling from a Mixture Distribution
166(1)
Appendix 8.1
166(3)
9 Evaluating Reliability and Probability of a Faulty Assembly Using Monte Carlo Simulation
169(12)
9.1 A General Algorithm for Determining Reliability Controlled by Statistically Independent Random Variables
169(1)
9.2 Evaluation of the Reliability Controlled by a Load--Strength Interference
170(3)
9.2.1 Evaluation of the Reliability on Demand, with No Time Included
170(1)
9.2.2 Evaluation of the Reliability Controlled by Random Shocks on a Time Interval
171(2)
9.3 A Virtual Testing Method for Determining the Probability of Faulty Assembly
173(4)
9.4 Optimal Replacement to Minimise the Probability of a System Failure
177(4)
10 Evaluating the Reliability of Complex Systems and Virtual Accelerated Life Testing Using Monte Carlo Simulation
181(8)
10.1 Evaluating the Reliability of Complex Systems
181(2)
10.2 Virtual Accelerated Life Testing of Complex Systems
183(6)
10.2.1 Acceleration Stresses and Their Impact on the Time to Failure of Components
183(2)
10.2.2 Arrhenius Stress--Life Relationship and Arrhenius-Type Acceleration Life Models
185(1)
10.2.3 Inverse Power Law Relationship and Inverse Power Law-Type Acceleration Life Models
185(1)
10.2.4 Eyring Stress--Life Relationship and Eyring-Type Acceleration Life Models
185(4)
11 Generic Principles for Reducing Technical Risk
189(46)
11.1 Preventive Principles: Reducing Mainly the Likelihood of Failure
191(26)
11.1.1 Building in High Reliability in Processes, Components and Systems with Large Failure Consequences
191(1)
11.1.2 Simplifying at a System and Component Level
192(1)
11.1.2.1 Reducing the Number of Moving Parts
193(1)
11.1.3 Root Cause Failure Analysis
193(1)
11.1.4 Identifying and Removing Potential Failure Modes
194(1)
11.1.5 Mitigating the Harmful Effect of the Environment
194(1)
11.1.6 Building in Redundancy
195(2)
11.1.7 Reliability and Risk Modelling and Optimisation
197(1)
11.1.7.1 Building and Analysing Comparative Reliability Models
197(1)
11.1.7.2 Building and Analysing Physics of Failure Models
198(1)
11.1.7.3 Minimising Technical Risk through Optimisation and Optimal Replacement
199(1)
11.1.7.4 Maximising System Reliability and Availability by Appropriate Permutations of Interchangeable Components
199(1)
11.1.7.5 Maximising the Availability and Throughput Flow Reliability by Altering the Network Topology
199(1)
11.1.8 Reducing Variability of Risk-Critical Parameters and Preventing them from Reaching Dangerous Values
199(1)
11.1.9 Altering the Component Geometry
200(1)
11.1.10 Strengthening or Eliminating Weak Links
201(1)
11.1.11 Eliminating Factors Promoting Human Errors
202(1)
11.1.12 Reducing Risk by Introducing Inverse States
203(1)
11.1.12.1 Inverse States Cancelling the Anticipated State with a Negative Impact
203(1)
11.1.12.2 Inverse States Buffering the Anticipated State with a Negative Impact
203(1)
11.1.12.3 Inverting the Relative Position of Objects and the Direction of Flows
204(1)
11.1.12.4 Inverse State as a Counterbalancing Force
205(1)
11.1.13 Failure Prevention Interlocks
206(1)
11.1.14 Reducing the Number of Latent Faults
206(2)
11.1.15 Increasing the Level of Balancing
208(1)
11.1.16 Reducing the Negative Impact of Temperature by Thermal Design
209(2)
11.1.17 Self-Stability
211(1)
11.1.18 Maintaining the Continuity of a Working State
212(1)
11.1.19 Substituting Mechanical Assemblies with Electrical, Optical or Acoustic Assemblies and Software
212(1)
11.1.20 Improving the Load Distribution
212(1)
11.1.21 Reducing the Sensitivity of Designs to the Variation of Design Parameters
212(4)
11.1.22 Vibration Control
216(1)
11.1.23 Built-in Prevention
216(1)
11.2 Dual Principles: Reduce Both the Likelihood of Failure and the Magnitude of Consequences
217(12)
11.2.1 Separating Critical Properties, Functions and Factors
217(1)
11.2.2 Reducing the Likelihood of Unfavourable Combinations of Risk-Critical Random Variables
218(1)
11.2.3 Condition Monitoring
219(1)
11.2.4 Reducing the Time of Exposure or the Space of Exposure
219(1)
11.2.4.1 Time of Exposure
219(1)
11.2.4.2 Length of Exposure and Space of Exposure
220(1)
11.2.5 Discovering and Eliminating a Common Cause: Diversity in Design
220(2)
11.2.6 Eliminating Vulnerabilities
222(1)
11.2.7 Self-Reinforcement
223(1)
11.2.8 Using Available Local Resources
223(1)
11.2.9 Derating
224(1)
11.2.10 Selecting Appropriate Materials and Microstructures
225(1)
11.2.11 Segmentation
225(1)
11.2.11.1 Segmentation Improves the Load Distribution
225(1)
11.2.11.2 Segmentation Reduces the Vulnerability to a Single Failure
225(1)
11.2.11.3 Segmentation Reduces the Damage Escalation
226(1)
11.2.11.14 Segmentation Limits the Hazard Potential
226(1)
11.2.12 Reducing the Vulnerability of Targets
226(1)
11.2.13 Making Zones Experiencing High Damage/Failure Rates Replaceable
227(1)
11.2.14 Reducing the Hazard Potential
227(1)
11.2.15 Integrated Risk Management
227(2)
11.3 Protective Principles: Minimise the Consequences of Failure
229(6)
11.3.1 Fault-Tolerant System Design
229(1)
11.3.2 Preventing Damage Escalation and Reducing the Rate of Deterioration
229(1)
11.3.3 Using Fail-Safe Designs
230(1)
11.3.4 Deliberately Designed Weak Links
231(1)
11.3.5 Built-in Protection
231(1)
11.3.6 Troubleshooting Procedures and Systems
232(1)
11.3.7 Simulation of the Consequences from Failure
232(1)
11.3.8 Risk Planning and Training
233(2)
12 Physics of Failure Models
235(34)
12.1 Fast Fracture
235(16)
12.1.1 Fast Fracture: Driving Forces behind Fast Fracture
235(6)
12.1.2 Reducing the Likelihood of Fast Fracture
241(1)
12.1.2.1 Basic Ways of Reducing the Likelihood of Fast Fracture
242(2)
12.1.2.2 Avoidance of Stress Raisers or Mitigating Their Harmful Effect
244(1)
12.1.2.3 Selecting Materials Which Fail in a Ductile Fashion
245(2)
12.1.3 Reducing the Consequences of Fast Fracture
247(1)
12.1.3.1 By Using Fail-Safe Designs
247(3)
12.1.3.2 By Using Crack Arrestors
250(1)
12.2 Fatigue Fracture
251(14)
12.2.1 Reducing the Risk of Fatigue Fracture
257(1)
12.2.1.1 Reducing the Size of the Flaws
257(1)
12.2.1.2 Increasing the Final Fatigue Crack Length by Selecting Material with a Higher Fracture Toughness
257(1)
12.2.1.3 Reducing the Stress Range by an Appropriate Design
257(1)
12.2.1.4 Reducing the Stress Range by Restricting the Springbuck of Elastic Components
258(1)
12.2.1.5 Reducing the Stress Range by Reducing the Magnitude of Thermal Stresses
259(2)
12.2.1.6 Reducing the Stress Range by Introducing Compressive Residual Stresses at the Surface
261(1)
12.2.1.7 Reducing the Stress Range by Avoiding Excessive Bending
262(1)
12.2.1.8 Reducing the Stress Range by Avoiding Stress Concentrators
263(1)
12.2.1.9 Improving the Condition of the Surface and Eliminating Low-Strength Surfaces
263(1)
12.2.1.10 Increasing the Fatigue Life of Automotive Suspension Springs
264(1)
12.3 Early-Life Failures
265(4)
12.3.1 Influence of the Design on Early-Life Failures
265(1)
12.3.2 Influence of the Variability of Critical Design Parameters on Early-Life Failures
266(3)
13 Probability of Failure Initiated by Flaws
269(14)
13.1 Distribution of the Minimum Fracture Stress and a Mathematical Formulation of the Weakest-Link Concept
269(5)
13.2 The Stress Hazard Density as an Alternative of the Weibull Distribution
274(2)
13.3 General Equation Related to the Probability of Failure of a Stressed Component with Complex Shape
276(2)
13.4 Link between the Stress Hazard Density and the Conditional Individual Probability of Initiating Failure
278(1)
13.5 Probability of Failure Initiated by Defects in Components with Complex Shape
279(1)
13.6 Limiting the Vulnerability of Designs to Failure Caused by Flaws
280(3)
14 A Comparative Method for Improving the Reliability and Availability of Components and Systems
283(10)
14.1 Advantages of the Comparative Method to Traditional Methods
283(2)
14.2 A Comparative Method for Improving the Reliability of Components Whose Failure is Initiated by Flaws
285(4)
14.3 A Comparative Method for Improving System Reliability
289(1)
14.4 A Comparative Method for Improving the Availability of Flow Networks
290(3)
15 Reliability Governed by the Relative Locations of Random Variables in a Finite Domain
293(14)
15.1 Reliability Dependent on the Relative Configurations of Random Variables
293(1)
15.2 A Generic Equation Related to Reliability Dependent on the Relative Locations of a Fixed Number of Random Variables
293(4)
15.3 A Given Number of Uniformly Distributed Random Variables in a Finite Interval (Conditional Case)
297(1)
15.4 Probability of Clustering of a Fixed Number Uniformly Distributed Random Events
298(4)
15.5 Probability of Unsatisfied Demand in the Case of One Available Source and Many Consumers
302(2)
15.6 Reliability Governed by the Relative Locations of Random Variables following a Homogeneous Poisson Process in a Finite Domain
304(3)
Appendix 15.1
305(2)
16 Reliability and Risk Dependent on the Existence of Minimum Separation Intervals between the Locations of Random Variables on a Finite Interval
307(20)
16.1 Applications Requiring Minimum Separation Intervals and Minimum Failure-Free Operating Periods
307(2)
16.2 Minimum Separation Intervals and Rolling MFFOP Reliability Measures
309(1)
16.3 General Equations Related to Random Variables following a Homogeneous Poisson Process in a Finite Interval
310(2)
16.4 Application Examples
312(5)
16.4.1 Setting Reliability Requirements to Guarantee a Specified MFFOP
312(1)
16.4.2 Reliability Assurance That a Specified MFFOP Has Been Met
312(2)
16.4.3 Specifying a Number Density Envelope to Guarantee Probability of Unsatisfied Random Demand below a Maximum Acceptable Level
314(1)
16.4.4 Insensitivity of the Probability of Unsatisfied Demand to the Variance of the Demand Time
315(2)
16.5 Setting Reliability Requirements to Guarantee a Rolling MFFOP Followed by a Downtime
317(3)
16.6 Setting Reliability Requirements to Guarantee an Availability Target
320(3)
16.7 Closed-Form Expression for the Expected Fraction of the Time of Unsatisfied Demand
323(4)
17 Reliability Analysis and Setting Reliability Requirements Based on the Cost of Failure
327(18)
17.1 The Need for a Cost-of-Failure-Based Approach
327(1)
17.2 Risk of Failure
328(2)
17.3 Setting Reliability Requirements Based on a Constant Cost of Failure
330(2)
17.4 Drawbacks of the Expected Loss as a Measure of the Potential Loss from Failure
332(1)
17.5 Potential Loss, Conditional Loss and Risk of Failure
333(3)
17.6 Risk Associated with Multiple Failure Modes
336(2)
17.6.1 An Important Special Case
337(1)
17.7 Expected Potential Loss Associated with Repairable Systems Whose Component Failures Follow a Homogeneous Poisson Process
338(3)
17.8 A Counterexample Related to Repairable Systems
341(1)
17.9 Guaranteeing Multiple Reliability Requirements for Systems with Components Logically Arranged in Series
342(3)
18 Potential Loss, Potential Profit and Risk
345(12)
18.1 Deficiencies of the Maximum Expected Profit Criterion in Selecting a Risky Prospect
345(1)
18.2 Risk of a Net Loss and Expected Potential Reward Associated with a Limited Number of Statistically Independent Risk-Reward Bets in a Risky Prospect
346(2)
18.3 Probability and Risk of a Net Loss Associated with a Small Number of Opportunity Bets
348(3)
18.4 Samuelson's Sequence of Good Bets Revisited
351(1)
18.5 Variation of the Risk of a Net Loss Associated with a Small Number of Opportunity Bets
352(1)
18.6 Distribution of the Potential Profit from a Limited Number of Risk--Reward Activities
353(4)
19 Optimal Allocation of Limited Resources among Discrete Risk Reduction Options
357(16)
19.1 Statement of the Problem
357(2)
19.2 Weaknesses of the Standard (0-1) Knapsack Dynamic Programming Approach
359(10)
19.2.1 A Counterexample
359(1)
19.2.2 The New Formulation of the Optimal Safety Budget Allocation Problem
360(1)
19.2.3 Dependence of the Removed System Risk on the Appropriate Selection of Combinations of Risk Reduction Options
361(4)
19.2.4 A Dynamic Algorithm for Solving the Optimal Safety Budget Allocation Problem
365(4)
19.3 Validation of the Model by a Recursive Backtracking
369(4)
Appendix A
373(18)
A.1 Random Events
373(2)
A.2 Union of Events
375(1)
A.3 Intersection of Events
376(2)
A.4 Probability
378(1)
A.5 Probability of a Union and Intersection of Mutually Exclusive Events
379(1)
A.6 Conditional Probability
380(3)
A.7 Probability of a Union of Non-disjoint Events
383(1)
A.8 Statistically Dependent Events
384(1)
A.9 Statistically Independent Events
384(1)
A.10 Probability of a Union of Independent Events
385(1)
A.11 Boolean Variables and Boolean Algebra
385(6)
Appendix B
391(8)
B.1 Random Variables: Basic Properties
391(1)
B.2 Boolean Random Variables
392(1)
B.3 Continuous Random Variables
392(1)
B.4 Probability Density Function
392(1)
B.5 Cumulative Distribution Function
393(1)
B.6 Joint Distribution of Continuous Random Variables
393(1)
B.7 Correlated Random Variables
394(1)
B.8 Statistically Independent Random Variables
395(1)
B.9 Properties of the Expectations and Variances of Random Variables
396(1)
B.10 Important Theoretical Results Regarding the Sample Mean
397(2)
Appendix C Cumulative Distribution Function of the Standard Normal Distribution
399(2)
Appendix D Χ2 Distribution
401(6)
References 407(6)
Index 413
Michael Todinov Oxford Brookes University, UK
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