Risk & Safety in Engineering. Dr. Jochen Köhler

Transcription

1 Risk & afety in Engineering Dr. Jochen Köhler

2 Contents of Today's Lecture Introduction to Classical Reliability Theory tructural Reliability The fundaental case afety argin

3 Introduction to Classical Reliability Theory Initially the reliability theory was developed for systes with a large nuber of (sei-) identical coponents subject to the sae exposure conditions - electrical systes (bulbs, switches,..) and later on to - nuclear power installations (valves, pipes, pups,..) - cheical plants (pipes, pressure vessels, valves, pups,..) - anufacturing plants (pups, copressors, conveyers,..)

4 Introduction to Classical Reliability Theory For such systes the probability of a coponential failure ay be assessed in a frequentistic anner fro observed failure rates - nuber of failures per coponent operation hours Due to the characteristics of the failure echanis a steady deterioration as function of tie/use for the considered coponents the ain concern was centred around the statistical odelling of the tie till failure

5 Introduction to Classical Reliability Theory The failure rate function is typically use to represent the hazard of failure over tie. z(t) First there is a rate of failure due to birth defects. Then there is a steady state. Then deterioration starts and the failure rate increases. t

6 Introduction to Classical Reliability Theory For technical systes such as structures and structural coponents the classical reliability analysis is of liited use because - all structural coponents are unique - the failure echanis tends to be related to extree load events exceeding the residual capacity of the coponent not the direct effect of deterioration alone For such systes a different approach is thus required, naely - an individual odelling of both the resistance as a function of tie and the loading as a function of tie

7 Introduction to Classical Reliability Theory Classical Reliability Analysis tructural Reliability Analysis Data on tie to failures Distribution odel for tie to failure T Estiation of distribution paraeters Data on load characteristics Data on aterial characteristics Distribution odels for load Distribution odels for resistance R tatistics on tie to failures T Estiation of distribution paraeters R and Probability of R-

8 tructural Reliability Analysis Can be generally used for probles in Civil- and Environental Engineering if the event of intrest can be represented by Deand exceeding Capacity e.g. required water volue exceeding existing water volue traffic volue exceeding road capacity stress exceeding resistance The Failure event is thereby defined by F = R

9 tructural Reliability Analysis The fundaental case: The Failure event is defined by F = R The failure probability is defined as Pr F = Pr(R ) With known stress s Pr F is Pr F = s = Pr R s = F R s According to the total probability theore the failure probability for unknown s: Pr F = Pr F = s f s ds = F R (s)f s ds or, siilarly Pr F = 1 F (r)f R r dr R

10 tructural Reliability Analysis afety argin: = R Pr F = Pr 0 = F (0) If R and are Noral distributed: E = E R E μ = μ R μ Var = Var R + Var σ = σ R 2 σ 2 The failure probability is: Pr F = F 0 = Φ 0 μ σ = Φ μ R μ σ R 2 σ 2

11 tructural Reliability Analysis The failure probability is: Pr F = F 0 = Φ 0 μ σ = Φ μ R μ σ R 2 σ 2 = Φ β Failure f () afe

12 Failure: In general defined as: capacity < deand resistance < load effect e.g. a bea in bending: f W r s Gl 4 BUT - resistance and load effect are uncertain!!! Assuing r and s are Noral distributed rando variables with known ean values and standard deviations.

13 Probability Density Probability of failure defined as: R f W; F The probability, that resistance < load effect Gl 4 0 P P R P R ~ N, R~ N, R R

14 Probability Density Probability of failure defined as: R f W; F The probability, that resistance < load effect Gl 4 0 P P R P R ~ N, R~ N, R R ~ N, R R The probability, that

15 Probability Density Probability of failure defined as: The probability, that resistance < load effect f R f W; F Gl P P R P R P F 0 0 P f d The probability, that

16 Probability Density Probability of failure defined as: The probability, that resistance < load effect f R f W; F Gl P P R P R P 0 0 PF f d The probability, that Reliability Index

17 Probability Density Reliability Index defined as: P F Geoetrical interpretation:

18 Probability Density Reliability Index defined as: P F Geoetrical interpretation: P F E E E E E

19 Exaple Reliability assessent: Tiber bea: strength class C24, span l = 8, 16 x 16 c. Load: 1 Person iple probabilistic odelling: Bending strength noral distributed The 5% fractile value is 24 pa, the assued CoV (stdev/ean) is f,05 f f 24 Pa; CoV f, CoV 40.8 Pa 10.2 Pa Person load noral distributed G G 0.8kN 0.15kN f W r s Gl 4

20 Exaple Reliability assessent: Tiber bea: strength class C24, span l = 8, 10 x 10 c. Load: 1 Person iple probabilistic odelling: oent Resistance: f f 40.8Pa 10.2 Pa W R 6.8kN 1.7 kn R Person load effect: G 0.8kN G 0.15kN 1.6 kn 0.3kN f W r s Gl 4

21 Exaple Reliability assessent: oent Resistance: R 6.8kN 1.7 kn R Person load effect: 1.6kN 0.3kN afety argin : R kN kn f W r s Gl 4

22 Exaple Reliability assessent: oent Resistance: R 6.8kN 1.7 kn R afety argin : R kN kn Person load effect: 1.6kN 0.3kN Reliability Index β: 3.01 Probability of failure: PF f W r s Gl 4

23 Exaple Reliability assessent: Failure probability and reliability index are estiates conditional on the inforation we have! a. Let s assue that we know that the person weights 1 kn. oent Resistance: R 6.8kN 1.7 kn R afety argin : R s kN 1.7 kn Person load effect: g s 1kN 2kN Reliability Index β: 2.82 Probability of failure: PF

24 tructural Reliability Analysis Linear Liit tate Function with uncorrelated noral distributed rando variables: g X = c + a 1 X 1 + a 2 X a n X n = c + a T X E g X = c + a 1 E X 1 + a 2 E X a n E[X n ] Var g X = a 1 2 Var X 1 + a 2 2 Var X a n 2 Var[X n ] β = E g X = c+a 1E X 1 +a 2 E X 2 + +a n E[X n ] Var g X a 2 1 Var X 1 +a 2 2 Var X 2 + +a 2 n Var[X n ] Pr F = Φ( β)