IVW Cursus 7 Oktober 2003 Betrouwbaarheid van systemen en elementen

1. IVW Cursus7 Oktober 2003Betrouwbaarheid van systemen en elementen Pieter van Gelder TU Delft

2. Inhoud • Tijdsafhankelijke faalkansen • Faalkansberekening op systeem niveau • Parallel en serieel • Afhankelijkheden • Faalkansberekening op element niveau • Niveau III, II, en I

3. 1 0.8 P(Xx) 0.6 (x) X F 0.4 0.2 0 x 0.5 0.4 P(x < X x+d x) 0.3 fX(x) 0.2 0.1 0 x x+d x

4. Random Variable T:Time to Failure

5. J.K. Vrijling and P.H.A.J.M. van Gelder, The effect of inherent uncertainty in time and space on the reliability of flood protection, ESREL'98: European Safety and Reliability Conference 1998, pp.451-456, 16 - 19 June 1998, Trondheim, Norway.

8. T = time to failure • The Hazard Rate, or instantaneous failure rate is defined as: • h(t) = f(t) / [1 - F(t) ] = f(t) / R(t) • f(t) probability density function of time to failure, • F(t) is the Cumulative Distribution Function (CDF) of time to failure, • R(t) is the Reliability function (CCDF of time to failure). • From:     f(t) = d F(t)/dt , it follows that: • h(t) dt = d F(t) / [1 - F(t) ] = - d R(t) / R(t) = - d ln R(t)

9. Integrating this expression between 0 and T yields an expression relating the Reliability function R(t) and the Hazard Rate h(t):

10. Bathtub Curve

11. Constant Hazard Rate • The most simple Hazard Rate model is to assume that: h(t) = λ , a constant. This implies that the Hazard or failure rate is not significantly increasing with component age. Such a model is perfectly suitable for modeling component hazard during its useful lifetime. • Substituting the assumption of constant failure rate into the expression for the Reliability yields: • R(t) = 1 - F(t) = exp (- λt) • This results in the simple exponential probability law for the Reliability function.

12. Non-Constant Hazard Rate • One of the more common non-constant Hazard Rate models used for evaluation of component aging phenomenon, is to assume a Weibull distribution for the time to failure: • Using the definition of the Hazard function and substituting in appropriate Weibull distribution terms yields: • h(t) = f(t) / [1 - F(t) ] = β t β -1 / t β

13. For the specific case of:  β = 1.0 , the Hazard Rate h(t)reverts back to the constant failure rate model described above, with:  t = 1/ λ . The specific value of the β parameter determines whether the hazard is increasing or decreasing.

14. β values, 0.5, 1.0, and 1.5.

15. β values, 0.5, 1.0, and 1.5.

16. Effect of inherent uncertainty on decrease in hazard rate • Example: PAC dike and sea dike

20. ReliabilityMaintainabilityAvailability

21. Reliability Reliability is the probability that a process or a system will operate without failure for a given period and under given operating conditions. R(t) = e-lt This equation is the exponential reliability function, it applies only for cases of “constant failure rate”, where l is failure rate.

22. Maintainability Maintainability is the probability that a process or a system that has failed will be restored to operation effectiveness within a given time. M(t) = 1 - e-mt where m is repair (restoration) rate

23. The Concept of Availability Reliability Maintainability Availability

24. Availability Availability is the proportion of the process or system “Up-Time” to the total time (Up + Down) over a long period. Up-Time Up-Time + Down-Time Availability =

25. Process Requirements Process Effectiveness Capability Performance Availability Reliability Maintainability Dependability

26. Capability: A measure of the ability of a process to satisfy given requirements (A measure of Quality - no time dependency) Availability: A measure of the ability of a process to complete a mission without excessive down time (Depends on Reliability and Maintainability) Dependability: A measure of the ability of a process to commence and complete a mission without failure (Depends on Reliability and Maintainability)

27. System Operational States B1 B2 B3 Up t Down A1 A2 A3 Up: System up and running Down: System under repair

28. Mean Time To Fail (MTTF) MTTF is defined as the mean time of the occurrence of the first failure after entering service. B1 + B2 + B3 3 MTTF = B1 B2 B3 Up t Down A1 A2 A3

29. Mean Time Between Failure (MTBF) MTBF is defined as the mean time between successive failures. (A1 + B1) + (A2 + B2) + (A3 + B3) 3 MTBF = B1 B2 B3 Up t Down A1 A2 A3

30. Mean Time To Repair (MTTR) MTTR is defined as the mean time of restoring a process or system to operation condition. A1 + A2 + A3 3 MTTR = B1 B2 B3 Up t Down A1 A2 A3

31. Availability Availability is defined as: Up-Time Up-Time + Down-Time A = Availability is normally expressed in terms of MTBF and MTTR as: MTBF MTBF + MTTR A =

32. Reliability/Maintainability Measures Reliability R(t) (Failure Rate) l = 1 / MTBF R(t) = e-lt Maintainability M(t) (Maintenance Rate) m = 1 / MTTR M(t) = 1 - e-mt

33. Types of Redundancy • Active Redundancy • Standby Redundancy

34. Active Redundancy A Input Div Output B Divider Both A and B subsystems are operative at all times Note: the dividing device is a Series Element

35. Standby Redundancy A SW Input Output B Switch Standby The standby unit is not operative until a failure-sensing device senses a failure in subsystem A and switches operation to subsystem B, either automatically or through manual selection.

36. Series System Input A1 A2 An Output ps = p1 + p2 +……. + pn - (-1)n joint probabilities For identical and independent elements: ps ~ 1 - (1-p)n < np (>p) ps : Probability of system failure pi : Probability of component failure

37. Parallel System A Input Output B Multiplicative Rule ps = p1.p2 …pn ps : Probability of system failure

38. Haringvliet outlet sluices t t t t Time start Modellering Lifetime distribution for one component Replacement strategies of large numbers of similar components in hydraulic structures

39. Series / Parallel System A1 A2 Input Output C B1 B2

40. System with Repairs Let MTBF = q and system MTBF = qs A Input Output B For Active Redundancy (Parallel or duplicated system) qs = ( 3l + m )/ ( 2l2 ) l << m qs = m / 2l2 = MTBF2 / 2 MTTR

41. A SW Input Output B Switch Standby Note: The switch is a series element, neglect for now. Note: The standby system is normally inactive. For Standby Redundancy qs = ( 2l + m )/ (l2 ) qs = m / l2 = MTBF2 / MTTR

42. System without Repairs For systems without repairs, m = 0 For Active Redundancy qs = ( 3l + m )/ ( 2l2 ) qs = 3l / ( 2l2 ) = 3/ ( 2l) qs = (3/2) q where q = 1/l qs = 1.5 MTBF For Standby Redundancy qs = ( 2l + m )/ (l2 ) qs = 2l/ l2 = 2/ l qs = 2q where q = 1/l qs = 2 MTBF

43. Summary Type With Repairs Without Repairs Active MTBF2 / 2 MTTR 1.5 MTBF Standby MTBF2 / MTTR 2 MTBF Redundancy techniques are used to increase the system MTBF

45. R S Example: wire • Limit state function: • Z = R - S • with: variable distribution mean standarddeviation R normal 6 0 kN 5 kN S normal 4 0 kN 10 kN

46. Probability densities 0.08 0.07 R 0.06 0.05 S Probability density (1/N) 0.04 0.03 0.02 0.01 0 0 20 40 60 80 R,S (N)

47. Joint probability density 80 70 60 50 Z>0 R (kN) 40 failure: Z<0 30 20 10 0 0 20 40 60 80 S (kN)

48. Analytical 80 • Failure probability • Independent • R and S: 70 r dr 50 Z>0 R (kN) 40 30 falen: Z<0 20 10 0 0 20 40 r 80 S (kN)

49. Analytical • Elaborate: • R and S normally distributed: • Fill in and calculate

50. Analytical • In this case simple approach possible: • variables normally distributed • Z is linear in variables • Then Z also normally distributed. • Mean • Standard deviation