risk assessment of systems

Transcription

1 12735: Urban Systems Modeling Lec. 05 risk assessment of instructor: Matteo Pozzi C 1 C 1 C 2 C 1 C 2 C 3 C 2 C 3 C 3 1

2 outline definition of system; classification and representation; two state ; cut and link sets; system reliability; bounds for system probability of failure; multivariate cumulative distribution. 2

3 what is a system components in a single structure mechanisms components in a network M 3 M M F 4 press.com/2011/11/ carbine and hook with rope instone.jpg?w=400 Der Kiureghian, A. (2005) "First and Second Order Reliability Methods", in book: The Engineering design reliability handbook, CRC Press LLC. Google map 5 3

4 system classification and representation components: single, two, multi states. e.g. working/not working e.g. working at 0%, 25%, 50%... : single, two, multi states. representation: block diagram C 1 C 3 C 6 C 9 fault trees, event trees source C 5 C 2 C 4 C 7 C 10 sink C 8 vs event trees.html 4

5 two state system with two state components: example component state C i 1 component works 0 component fails convention is opposite respect to component analysis. C 1 C 3 C 6 C 9 source C 5 sink C 2 C 4 C 7 C 10 C 8 system state 1 system works 0 system fails 5

6 two state system with two state components vector of component state systemcomponents relation Boolean variable 2 2 number of components for components, there are 2 possible relations. coherent system: is non decreasing in. That is, if goes safe, it cannot degrade system operation. C 1 C 2 C 3 a states of components a a series a s parallel a s coherent a s non coherent a s C 1 C 2 C 3 C 1 C C 3 6

7 famous series system: C 1 C 2 C n min,,, or parallel system: max,,, C 1 or C k out of n system: true 1 indicator: false 0 C n 1: parallel system : series system 7

8 general system: cut and link sets example: C 2 C 1 C 3 C 4 cut set: a set of components whose joint failure implies failure of the system. E.g. C 1,2,3,4. minimum cut set: a cut set with no extra component : C 1,4 and C 2,3,4. link set: a set of components whose joint survival implies survival of the system. E.g. C 1,4. minimum link set: a link set with no extra component : C 1,2, C 1,3 and C 4. Algorithms are available for identifying cut sets and link sets. C 2,3, C 2,4 and C 3,4 are NOT cut sets C 1 and C 2 are NOT link sets 8

9 min cut set representation example: C 1 C 2 C 3 C 4 function relation for cut set 1 1 C 2 C 1 C 4 C 3 C 4 2 cut sets a system can be seen a series of its cut sets

10 min link set representation example: C 1 C 2 C C 4 function relation for link set C 1 C 1 C 2 C 3 3 link sets C 4 a system can be seen a parallel system of its link sets

11 component and system reliability component: probability of failure: P 0 1 reliability: 1 P 1 1 P 1 0 P 0 P 1 system: probability of failure: P 0 1 reliability: 1 P 1 11

12 reliability of series of indep. comp. C 1 C 2 C n independent failures P=20% P=10% P=5% P=2% P=1% equally reliable components: P s n 12

13 reliability of parallel of indep. comp. C 1 independent failures equally reliable components: 1 C 2 C n 1 1 P s P=20% P=10% P=5% P=2% P=1% n 13

14 general system: example of network of indep. comp. source 2 A B link set A 3 sink C 1 C 3 independence reliability probability of failure 1 1 C 2 link set B C 4, ,2 for equally reliable components 14

15 general system: example of network of indep. comp. source 2 A B C 3 D 2 reliability sink 3 probability of failure independent failures, : fail. prob. of C or D., for equally reliable components 15

16 general system: example of network of indep. comp. source sink source increment of reliability after adding segment 5: 3 sink Δ example: 1%, 99%, 99.96%, 99.98%, Δ 0.02% from: Kottegoda and Rosso (2008) "Applied statistics for civil and environmental engineers", Blackwell Pub. 16

17 two component series system equally reliable components: C 1 C 2 30% 1 70% independent failures identical failures positive correlation 1 0 ; opposite failures negative correlation % 17

18 limit cases for a two component series system equally reliable components: C 1 C 2 30% system failure 1 70% independent failures identical failures positive correlation 1 0 ; 1 51% 30% opposite failures negative correlation % 30% 70% % For series system, positive correlation decrease the probability of failure. 18

19 limit cases for a two component parallel system C 1 equally reliable components: 30% 170% system failure C independent failures identical failures positive correlation 1 0 ; 9% 30% opposite failures negative correlation % 0% For parallel system, positive correlation increase the probability of failure. 19

20 bounds for series two failure events (mechanisms):,0,,,, 0,min, max, min,1 many failure events (mechanisms): 1,2,, max min when the joint probability for each pair of events,, is known, you can use Ditlevsen s bounds (see Sørensen, J.D. (2004) "Notes in Structural Reliability Theory And Risk Analysis"). three failure events (mechanisms):,1,,,,, Ditlevsen s bounds require second order probability ( ) and define max and min values for higher order probability (e.g. third order: ). 20

21 bounds for parallel two failure events (mechanisms):,0,,,, 0,min, 0 min, many failure events (mechanisms): 1,2,, 0 min second order bounds are also available (see Sørensen, J.D. (2004) "Notes in Structural Reliability Theory And Risk Analysis"). 21

22 system reliability problem limit state functions,, 0 standard normal space component fails mechanism active the system components relation defines the failure of the system, depending on the failures of the components. 22

23 system reliability problem: series system Ω in a series system, the system fails if at least one component fails. failure domain: Ω standard normal space

24 system reliability problem: parallel system Ω in a series system, the system fails if at least one component fails. failure domain: Ω 0, 0, 0 0 standard normal space

25 system reliability problem: general system Ω e.g. failure domain: Ω 0, standard normal space C 1 C 3 C 2 25

26 approaches for system reliability problem Ω Monte Carlo Uniform Sampling: it can be easily implemented, it provides an unbiased estimate. Monte Carlo Important Sampling: it is not easy to find an appropriate proposal distribution. 0 standard normal space FORM: the design point (defined as the failure closest to the origin) may be not differentiable no linear approximation. Newton s method may not work. In any case, linear approximation may be not accurate. 26

27 FORM for series series system 0 standard normal space, : design points for and P 0 limit state functions approximated around the corresponding design point: 1 new coordinates: 27

28 FORM for series, : design points for and ;, ;, limit values for each coordinate: 0 standard normal space limit state function: matrix of directions : 1,1 1 probability of failure: 1P ;, 28

29 multivariate cumulative normal distribution Φ ;, P,, 1 = 4, 2 = 6, 1 = 2, 2 = 3, 12 = 0.4 PDF , p(x 1,x 2 ) x x x x 1 1 =4, 2 =6, 1 =2, 2 =3, 12 =0.4 Φ ;, ;, Φ ;, 0,1 F(x 1,x 2 ) CDF : Φ ;, 0 Matlab F=mvncdf(v_x',v_mu',m_Var) x x

30 mult. cum. norm. for series Φ ;,P,, 3 2 1, P safe Φ ;, Pfailure 1Φ ;, s s 1 recipe: solve component FORM for each limit state function, ; compute matrix and build ; compute. Formulation in normal space, with linear limit state functions: ;, ;, Φ ;, 1Φ ;, 30

31 FORM for parallel Design points may be too far from failure domain. To improve the quality of the approximation: linearization around this point 0 standard normal space

32 FORM for parallel 3 failure domain 3 s , equal probability s Φ ;, -1-1, s 1 s 1 Formulation in normal space, with linear limit state functions: ;, ;, Φ ;, 32

33 example 20 1 = 10, 2 = 15, 1 = 2, 2 = 1.5, 12 = 0.3 ;, x Φ ;, 1.4% x 1 ;, MC : 0.67%, 1.40% 33

34 references Ditlevsen, O. and H.O. Madsen. (1996). Structural reliability methods. J. Wiley & Sons, New York, NY. Downloadable from HOM StrucRelMeth Ed2.3.7 June September.pdf. Chapter 14. Faber, M. (2009) Risk and Safety in Engineering, lecture notes, Lecture 8, available at Sørensen, J.D. (2004) "Notes in Structural Reliability Theory And Risk Analysis", notes 6 and 7, avail. at ng_waterbouwkunde/sectie_waterbouwkunde/people/personal/gelder/publications/citations/ doc/citatie215.pdf 34