$128.00
Reliability & Statistical Analysis for Predicting Life, Safety, Risk, Support Costs, Failures, and Forecasting Warranty Claims, Substantiation and Accelerated Testing, Using Weibull, Log Normal, Crow-AMSAA, Probit and Kaplan-Meier Models
Copyright © 2022 by Nancy S. Embury
*Available in Hardcopy Only
Click here to view the preface and the table of contents.
Click here to view the first chapter.
This handbook will provide an understanding of life data analysis. Weibull and Log Normal analysis will be emphasized particularly for failure analysis. There are new applications of this technology in medical and dental implants, warranty analysis, life cycle cost, materials properties and production process control. Related quantitative models such as the binomial, Poisson, Kaplan-Meier, Gumbel extreme value and the Crow-AMSAA are included. The author intends that a novice engineer can perform Weibull analysis after studying this document. A secondary objective is to show the application of personal computers to replace the laborious hand calculations and manual plotting required in the past.
The New Weibull Handbook has become the national and international standard for Weibull analysis. It is in daily use throughout the world. Thousands of copies have been distributed. Users include military, automotive, medical, electronic, materials & composites research, aerospace, electrical power, nuclear power, dental research, advertising, bearings, compressors, v-belts, chain drives and on and on. Many organizations have adopted the Handbook and software implementing its methods as standard including the Federal Aviation Administration, Harley Davidson, Motorola, TVA, Honeywell, BICC Genereal, GKN Automotive, Dupont, Meritor Automotive, Teleflex, Guidant Cardiac Pacemaker (CPI), Daimler Chrysler, General Electric, Siemens, Behr, Borg-Warner, NASA, Rolls Royce, Gates Rubber, the US Naval Air Arm (NAVAIR), the US Air Force at SAALC and WPAFB, and Dow Chemical.
| Chapter 1 An Overview Of Weibull Analysis | 1-1 |
| 1.1 Objective | 1-1 |
| 1.2 Background | 1-1 |
| 1.3 Examples | 1-2 |
| 1.4 Scope | 1-2 |
| 1.5 Advantages of Weibull Analysis | 1-3 |
| 1.6 Data, Discrete Versus Life Data | 1-3 |
| 1.7 Failure Distribution | 1-4 |
| 1.8 Failure Forecasts And Predictions | 1-5 |
| 1.9 Engineering Change Test Substantiation | 1-6 |
| 1.10 Maintenance Planning | 1-6 |
| 1.11 System Analysis And Math Models | 1-7 |
| 1.12 Weibulls With Curved Data | 1-7 |
| 1.13 Weibulls With Corners And Doglegs | 1-9 |
| 1.14 Weibayes | 1-9 |
| 1.15 Small Sample Weibulls | 1-9 |
| 1.16 Updating Weibulls | 1-9 |
| 1.17 Deficient (Dirty) Data | 1-9 |
| 1.18 Establishing The Weibull Line, Choosing The Fit Method | 1-10 |
| 1.19 Related Methods And Problems | 1-10 |
| 1.20 Summary | 1-11 |
| Chapter 2 Plotting The Data And Interpreting The Plot | 2-1 |
| 2.1 Foreword | 2-1 |
| 2.2 Weibull Data | 2-1 |
| 2.3 The Weibull Plot Scales | 2-2 |
| 2.4 (Eta) and (Beta) | 2-2 |
| 2.5 Weibull Analysis – An Example | 2-4 |
| 2.6 Median Ranks | 2-5 |
| 2.7 The Weibull Plot | 2-6 |
| 2.8 “B” Life | 2-6 |
| 2.9 Suspended Test Items | 2-7 |
| 2.10 Bernard’s Approximation | 2-7 |
| 2.11 Suspensions Increase Eta | 2-8 |
| 2.12 Interpreting The Weibull Plot | 2-8 |
| 2.13 Beta < 1 Implies Infant Mortality | 2-9 |
| 2.14 Beta = 1-0 Implies Random Failures | 2-10 |
| 2.15 1-0 < Beta < 4-0 Implies Early Wear Out | 2-11 |
| 2.16 Beta > 4-0 Implies Old Age (Rapid) Wear Out | 2-11 |
| 2.17 Weibull Modes May Be “Covered” | 2-12 |
| 2.18 Weibull Paper And Its Construction | 2-12 |
| 2.19 Weibull Analysis – The Standard Method | 2-14 |
| 2.20 Problems | 2-13 |
| Chapter 3 Dirty Data, “Bad” Weibulls, And Uncertainties | 3-1 |
| 3.1 Foreword | 3-1 |
| 3.2 Small Sample Uncertainties | 3-1 |
| 3.2.1 Goodness Of Fit | 3-3 |
| 3.3 Suspensions | 3-6 |
| 3.4 Suspect Outliers | 3-6 |
| 3.5 Curved Weibulls And The t0 Correction | 3-7 |
| 3.6 Curved Weibulls And The Log Normal Distribution | 3-11 |
| 3.7 Data Inconsistencies And Multimode Failures | 3-14 |
| 3.7.1 Low-Time Failures | 3-14 |
| 3.7.2 Close Serial Numbers | 3-15 |
| 3.7.3 Mixtures Of Failure Modes | 3-16 |
| 3.8 Steep Slopes Hide Problems | 3-17 |
| 3.9 Bad Weibull Patterns | 3-18 |
| Conclusion | 3-18 |
| 3.10 Problems | 3-19 |
| Chapter 4 Failure Forecasting = Risk Analysis | 4-1 |
| 4.1 Situation | 4-1 |
| 4.2 Definition | 4-1 |
| 4.3 Forecasting Techniques | 4-1 |
| 4.4 Calculating Failure Forecasts | 4-1 |
| 4.4.1 Expected Failures Now | 4-1 |
| 4.1.2 Failure Forecast When Failed Units Are Not Replaced | 4-3 |
| 4.4.3 Failure Forecasts When Failed Units Are Replaced | 4-3 |
| 4.5 Failure Forecast Analysis-Summary | 4-4 |
| 4.5.1 Case Study 1: Bearing Cage Fracture | 4-5 |
| 4.5.2 Case Study 2: Bleed System Failures | 4-7 |
| 4.6 System Failure Forecast Without Simulation* | 4-12 |
| 4.6.1 Case Study 3: Aircraft In-Flight Engine Shutdowns* | 4-12 |
| 4.7 System Failure Forecasts With Simulation* | 4-15 |
| 4.7.1 Case Study 4: System Failure Forecast With Simulation* | 4-17 |
| 4.8 Optimal (Lowest Cost) And Block Replacement Intervals* | 4-19 |
| 4.9 Problems | 4-25 |
| Chapter 5 Maximum Likelihood Estimates & Other Alternatives | 5-1 |
| 5.1 Introduction | 5-1 |
| 5.2 Maximum Likelihood Estimation (MLE) | 5-1 |
| 5.3 MLE With Reduced Bias Adjustment (RBA) for Accurate Results | 5-3 |
| 5.3.1 The RBA Factor for Normal and Lognormal Distributions | 5-4 |
| 5.3.2 The RBA factor for the Weibull distribution | 5-5 |
| 5.3.3 Best Practice | 5-6 |
| 5.4 Median Rank Regression: X on Y Versus Y on X | 5-7 |
| 5.5 Plotting Positions | 5-9 |
| 5.5 Special Methods: MLE With Reduced Bias Adjustment (RBA) | 5-6 |
| 5.6 Special Methods: Gossett’s Student’s T | 5-10 |
| 5.7 The Dauser Shift – Unknown Suspension Times | 5-10 |
| 5.8 Special Methods For Inspection Interval Data And Coarse Data | 5-12 |
| 5.8.1 Inspection Option #1 | 5-12 |
| 5.8.2 & | |
| 5.8.3 Probit Analysis Inspection Options #2 & 3 | 5-13 |
| 5.8.4 Kaplan-Meier (KM) Inspection Option #4 | 5-14 |
| 5.8.5 Interval Maximum Likelihood Estimation (MLE) Inspection Option #5 | 5-14 |
| 5.9 Distribution Analysis | 5-16 |
| Chapter 6 Weibayes And Weibayes Substantiation Testing | 6-1 |
| 6.1 Foreword | 6-1 |
| 6.2 Weibayes Method | 6-2 |
| 6.3 Weibayes Without Failures | 6-2 |
| 6.4 Weibayes With Failures | 6-3 |
| 6.5 Unknown Failure Times | 6-4 |
| 6.6 Weibayes Worries And Concerns | 6-4 |
| 6.7 Weibayes Case Studies | 6-5 |
| 6.8 Substantiation And Reliability Testing | 6-9 |
| 6.9 Zero-Failure Test Plans For Substantiation Testing | 6-10 |
| 6.10 Zero-Failure Test Plans For Reliability Testing | 6-12 |
| 6.10.1 Re-Expression Of Reliability Goal To Determine ? | 6-12 |
| 6.10.2 Tailoring and Designing Test Plans | 6-14 |
| 6.11 Total Test Time | 6-15 |
| 6.12 Test-To-Failure Versus Weibayes Zero-Failure Tests | 6-16 |
| 6.13 One Or Zero Failure Test Plans | 6-19 |
| 6.14 Sudden Death Tests With Weibull And Weibayes | 6-20 |
| 6.15 Case Study: Cost Vs Uncertainty Trades | 6-23 |
| 6.16 Normal And Lognormal Tests | 6-22 |
| 6.17 Accelerated Testing | 6-24 |
| 6.17.1 Accelerated Step-Stress Test Data Analysis* | 6-25 |
| 6.17.2 Accelerated Testing: A Method For Estimating Test Acceleration Factor With No Existing In-Service Failures* |
6-26 |
| 6.18 System Deterioration | 6-28 |
| 6.19 Weibull Libraries And Lessons Learned | 6-29 |
| 6.19.1 A Weibull Library Case Study | 6-30 |
| 6.19.2 Weibull Libraries For End Users | 6-31 |
| 6.21 Problems | 6-32 |
| Chapter 7 Interval Estimates | 7-1 |
| 7.1 Interval Estimates | 7-1 |
| 7.2 Confidence Interval Concept | 7-1 |
| 7.3 Confidence Intervals For B Lives And Reliability | 7-2 |
| 7.3.1 Beta-Binomial Bounds | 7-3 |
| 7.3.2 Fisher’s Matrix Bounds | 7-4 |
| 7.3.3 Likelihood Ratio Bounds | 7-5 |
| 7.3.4 Pivotal Bounds Monte Carlo Bounds | 7-6 |
| 7.3.5 Reliability Assurance Interval and the “p” value | 7-7 |
| 7.3.6 Normal Distribution Confidence Bounds with Student’s t | 7-7 |
| 7.3.7 Summary Of Confidence Bounds For B Life And Reliability | 7-8 |
| 7.4 Confidence Intervals For Eta And Beta | 7-8 |
| 7.5 Are Two Weibull Data Sets Different Or From The Same Distribution | 7-9 |
| 7.5.1 Double Confidence Bounds Do Not Overlap | 7-10 |
| 7.5.2 Likelihood Ratio Test | 7-11 |
| 7.5.3 Likelihood Contour Plots | 7-11 |
| 7.6 Problems – True Or False? | 7-13 |
| Chapter 8 Related Math Models | 8-1 |
| 8.1 Introduction | 8-1 |
| 8.2 Binomial Distribution | 8-1 |
| 8.3 Poisson Distribution | 8-5 |
| 8.4 Binomial Becomes Poisson…Sometimes | 8-9 |
| 8.5 The Exponential Distribution | 8-11 |
| 8.6 Kaplan-Meier Survival Estimates | 8-12 |
| 8.7 Probabilistic Design | 8-17 |
| 8.7.1 Strength-Load And Life-Usage Interactions | 8-17 |
| 8.7.2 Total Life = Crack Life + Crack-To-Rupture Life | 8-18 |
| 8.7.3 Does Failure Mode A Cover Mode B? | 8-19 |
| 8.8 Production Process Reliability | 8-19 |
| 8.9 Extreme Value Statistics | 8-21 |
| 8.10 Batch Effects | 8-23 |
| 8.11 Problems | 8-24 |
| Chapter 9 Crow-AMSAA Modeling, Warranty Analysis & Life Cycle Costs | 9-1 |
| 9.0 The Crow-AMSAA-Duane Reliability Growth Model | 9-1 |
| 9.1 Background History | 9-2 |
| 9.2 CA Methods | 9-2 |
| 9.2.1 Simple Graphical and Regression Solution | 9-2 |
| 9.2.2 IEC Solutions for Time and Failure Terminated Data | 9-5 |
| 9.2.3 IEC MLE Solutions for Interval and Grouped Data | 9-7 |
| 9.3 Comparisons of the IEC and Regression CA Methods | 9-12 |
| 9.4 CA Input May Be Confusing | 9-14 |
| 9.5 Missing Early Data with CA | 9-14 |
| 9.6 First Time Failure Mode Analysis | 9-14 |
| 9.7 Warranty Claims Analysis | 9-15 |
| 9.8 Warranty Data Matrix | 9-16 |
| 9.9 Warranty Data Rules | 9-17 |
| 9.10 Warranty Data Matrix Conversion and Analysis | 9-18 |
| 9.11 Warranty Analysis Methods | 9-20 |
| 9.11.1 Inspection Option #1 | 9-20 |
| 9.11.2 Kaplan-Meier | 9-20 |
| 9.11.3 MLE Interval | 9-20 |
| 9.11.4 Crow AMSAA | 9-20 |
| 9.12 Case Studies | 9-21 |
| 9.13 Tracking Your Results | 9-21 |
| 9.14 Warranty Conclusions and Recommendations | 9-21 |
| 9.15 Life Cycle Cost | 9-21 |
| 9.16 Net Present Value (NPV) | 9-21 |
| 9.17 Discount Rate | 9-22 |
| 9.18 Life Cycle Cost and NPV | 9-22 |
| 9.19 LCC Calculations | 9-23 |
| 9.20 Case Studies | 9-24 |
| Chapter 10 Summary | 10-1 |
| 10.1 The Beginning Of The End | 10-1 |
| 10.2 Which Method? What Kind Of Data? | 10-1 |
| 10.3 Looking At The Plot, What Do You See? | 10-3 |
| 10.4 Which Distribution Is Best? | 10-4 |
| 10.5 Substantiation And Accelerated Testing | 10-6 |
| 10.6 Confidence Intervals | 10-6 |
| 10.7 Presentations And Reports | 10-6 |
| 10.8 Logic Diagram – Flowchart | 10-6 |
| 10.9 The End | 10-6 |
| 10.10 Best Practice Flow Chart | 10-7 |
| Chapter 11 Case Studies And New Applications | 11-1 |
| 11.1 Foreword | 11-1 |
| 11.2 Stress Corrosion Failure Forecasting | 11-2 |
| 11.3 Optimal Component Replacement – Voltage Regulators | 11-3 |
| 11.4 Locomotive Power Units Overhaul Life | 11-7 |
| 11.5 Cost Effective Calibration Intervals | 11-8 |
| 11.6 Florida Power & Light Turbogenerator Failure | 11-10 |
| 11.7 TVA Bull Run Fossil Plant – Controller Cards | 11-11 |
| 11.8 Repairable Systems Reliability Growth Assessment | 11-13 |
| 11.9 Front Jounce Bumpers | 11-14 |
| 11.10 Transfer Case Seal | 11-15 |
| 11.11 Dental Acrylic Adhesive Fatigue | 11-16 |
| 11.12 Duane-Crow-AMSAA Reliability Modeling | 11-17 |
| 11.13 Weibull Analysis Of Boiler Tube Failures | 11-20 |
| 11.14 Gas Turbine Seal Failures – A Batch Problem | 11-23 |
| 11.15 Challenger Space Shuttle Weibull | 11-25 |
| Appendix A: Glossary | A-1 |
| Appendix B: Rank Regression And Correlation Method Of Weibull Analysis | B-1 |
| B-1 Method | B-1 |
| B-2 Example And Step-By-Step Procedure | B-1 |
| Appendix C: Maximum Likelihood Estimation* | C-1 |
| C-1 Foreword | C-1 |
| C-2 Statistics, Probability And Likelihood | C-1 |
| C-3 The Likelihood Function | C-1 |
| C-4 Maximizing The Likelihood Function | C-2 |
| C.5 Maximum Likelihood Example | C-3 |
| C.6 Interval MLE | C-6 |
| C.7 Maximum Likelihood Versus Median Rank Regression Estimates | C-8 |
| Appendix D: Goodness of Fit | D-1 |
| Appendix E: Weibayes Analysis | E-1 |
| E.1 Foreword | E-1 |
| E.2 Weibayes Equation With No Failures | E-1 |
| E.3 Weibayes With Failures | E-2 |
| Appendix F: Batch Failures Using The Aggregated Cumulated Hazard Function | F-1 |
| F.1 Batch Failures On Weibull Plots | F-1 |
| F.2 Batch Problems With The “Present-Risk” Method | F-2 |
| F.3 The ACH Method | F-3 |
| F.4 A Case Study: Aero-Engines – (Lp Turbine Strap Failures) | F-4 |
| F.5 Concluding Remarks | F-5 |
| Appendix G: Weibull And Log Normal Mean And Variance | G-1 |
| G.1 Rth Moments | G-1 |
| G.2 Weibull Mean | G-2 |
| G.3 Weibull Variance | G-3 |
| G.4 Weibull Mode | G-3 |
| G.5 Weibull Median | G-3 |
| G.6 Log Normal Mean And Standard Deviation | G-3 |
| G.7 Log Normal Variance | G-4 |
| Appendix H: Weibull Graph Paper For Manual Plots | H-1 |
| Appendix I: Median Ranks | I-1 |
| Appendix J – Mixtures Of Populations And Failure Modes | J-1 |
| J.1 Competing Risk: | J-1 |
| J.3 Competing Risk Mixture: | J-2 |
| J.4 Compound Competing Risk Mixture: | J-2 |
| J.5 Weibath Model: | J-2 |
| J.7 Curve Shape. | J-3 |
| Appendix K: Answers To Problems | K-1 |
| Appendix L: The C4 Factor | L-1 |
| Appendix M: Graphical Repair Analysis | M-1 |
| Appendix N: Waloddi Weibull | N-1 |
| References | R-1 |
| Index | I-1 |
