An introduction to the basics of reliability and risk analysis

1 . Introduction .......................................................................................... 1 2 . Basic concepts of safety and risk analysis ........................................... 3 2.1 A qualitative definition of risk .................................................. 3 2.2 A quantitative definition of risk ................................................ 4 2.3 Risk analysis ............................................................................. 6 3 . Methods for hazard identification ...................................................... 11 3.1 Hierarchical ............................................................................ 12 3.2 Systematic identification of release points (SIRP) ................. 13 3.3 Failure mode and effect analysis (FMEA) .............................. 14 3.4 Hazard and operability analysis (HAZOP) ............................. 19 analysis ............................................................................................... 21 4.1 Definitions .............................................................................. 21 4.2 Boolean logic operations ........................................................ 22 4.3 Logic of uncertainty: definition of probability ....................... 27 4.3.1 Axiomatic Definition .............................................. 27 4.3.2 Empirical Frequentist Definition ............................. 28 4.3.3 Classical Definition ................................................. 29 4.3.4 Probability space ..................................................... 30 4.4.1 Union of non-mutually exclusive events ................. 31 4.4.2 Conditional Probability ........................................... 33 4.4.4 Bayes Theorem ....................................................... 37 4 . Basics of probability theory for applications to reliability and risk 4.4 Probability laws ...................................................................... 31 4.4.3 Theorem of Total Probability .................................. 35 ix X Contents 4.5 Random variables ................................................................... 39 4.5.1 Probability functions ............................................... 40 variance ................................................................... 41 4.5.3 The hazard fbnction ................................................ 49 4.6 Probability distributions ......................................................... 51 4.6.1 Univariate discrete distributions ............................. 51 4.6.2 Univariate continuous distributions ........................ 55 4.7 Regression and correlation analyses ....................................... 65 4.7.1 Regression with constant variance .......................... 65 4.5.2 Summary measures: percentiles, median. mean. 4.7.2 Regression with non-constant variance ................... 68 4.7.3 Multiple linear regression ....................................... 69 4.7.5 Correlation Analysis ............................................... 71 4.7.4 Non Linear Regression ............................................ 71 5 . Reliability of simple systems .............................................................. 77 5.1 Simple system configurations ................................................. 77 5.2 Series system .......................................................................... 78 5.3 Parallel system ........................................................................ 79 5.5 Standby systems 81 5.4 r-out-of-N systems ................................................................. 80 5.5.1 Cold Standby ........................................................... 82 5.5.2 Hot Standby ............................................................ 85 6 . Availability and maintainability .......................................................... 89 6.1 Introduction ............................................................................ 89 6.3 Contributions to unavailability ............................................... 91 6.4 The availability of an unattended component (no repairs) ..... 92 6.5 The availability of a continuously monitored component ...... 92 maintenance ............................................................................ 97 ..................................................................... 6.2 Availability definition. ............................................................ 89 6.6 The availability of a component under periodic test and Contents xi 6.6.1 Single component under periodic maintenance: a more realistic case .............................................. 100 6.8 A policy of preventive and corrective maintenance ............. 106 6.9 A policy of preventive replacement with economical optimization ......................................................................... 110 7 . Fault tree analysis ............................................................................. 115 7.1 Introduction .......................................................................... 115 7.2 Fault tree construction .......................................................... 116 7.3 Qualitative analysis: coherent structure functions and minimal cut sets ................................................................... 128 7.3.1 Structure functions ................................................ 128 7.3.2 Coherent structure functions and minimal cut sets .................................................................. 131 7.4 Quantitative analysis ............................................................. 132 7.5 Comments ............................................................................. 135 8 . Event tree analysis ........................................................................... 137 8.1 Introduction .......................................................................... 137 8.2 Event tree construction ......................................................... 138 8.3 Event tree evaluation ............................................................ 141 9 . Estimation of reliability parameters from experimental data ........... 147 9.1 Estimation of equipment reliability from tests ..................... 147 9.1.1 Complete data set .................................................. 147 9.1.2 Censored data sets ................................................. 148 9.1.3 Test plans .............................................................. 149 to test components lifetimes ................................ 151 without replacement .......................................... 152 9.1.6 Confidence limits for reliability parameters ......... 157 9.2 Accelerated Life Testing ...................................................... 167 6.7 Maintainability ..................................................................... 104 9.1.4 The method of maximum likelihood applied 9.1.5 Statistics of exponential components with or xii Contents 9.2.1 Introduction ........................................................... 167 9.2.2 Experimental designs for ALT .............................. 168 9.2.3 Parametric models used in step-stress accelerated tests .................................................... 171 9.2.4 Exponential distribution under design I ................ 172 9.2.5 Inverse gaussian fatigue failure time distribution under design I11 ..................................................... 176 9.3 Empirical determination of distribution models ................... 179 9.3.1 Probability paper ................................................... 179 9.3.2 The normal probability paper ................................ 180 9.3.4 Construction of a probability plotting paper ......... 182 9.3.5 Testing the validity of an assumed distribution .... 185 9.4 Kaplan-Meier estimator of the survivor function ................. 189 9.5 Reliability growth ................................................................. 197 9.5.2 Least square estimation ......................................... 200 9.3.3 The log-normal probability paper ......................... 181 9.5.1 Maximum likelihood estimation ........................... 199 9.6 Reliability prediction from stress-strength models ............... 208 9.6.1 Introduction ........................................................... 208 9.6.2 Internal and external causes of stress .................... 213 9.6.3 Physics of failures ................................................. 214 9.6.4 Reliability from stress-strength distributions ........ 215 Appendix A: Table of Standard Normal Cumulative Distribution ....... 217 Appendix B: Table of Chi-square Cumulative Distribution ................ 222