Engineering Risk Benefit Analysis

Transcription

1 Engineering Risk enefit nalysis.55, 2.943, 3.577, 6.938, 0.86, 3.62, 6.862, ESD.72J, ESD.72 RPR. The Logic of ertainty George E. postolakis Massachusetts Institute of Technology Spring 2007 RPR. The Logic of ertainty

2 Event Definition Event: statement that can be true or false. It may rain tonight is not an event. ccording to our current state of knowledge, we may say that an event E is TRUE, FLSE, or POSSILE (UNERTIN). Eventually, E will be either TRUE or FLSE. RPR. The Logic of ertainty 2

3 True Event False Possible RPR. The Logic of ertainty 3

4 Venn Diagrams Sample Space: The set of all possible outcomes of an experiment. Each elementary outcome is represented by a sample point. Examples: Die {,2,3,4,5,6} Failure Time {0, } collection of sample points is an event. S Venn Diagram E RPR. The Logic of ertainty 4

5 Indicator Variables,If E j is T j = Important Note: k =, k:, 2, 0, If E j is F S Venn Diagram E E RPR. The Logic of ertainty 5

6 Union (OR operation) = = ( )( j ) RPR. The Logic of ertainty 6

7 Intersection (ND operation) = = j Mutually Exclusive Events: = RPR. The Logic of ertainty 7

8 Simple Systems Reliability lock Diagram for the Series System.... N System Failure failure: N = ( j ) N j success : Y = N Yj... N RPR. The Logic of ertainty 8

9 Reliability lock Diagram for the Parallel System 2 i i+ Y N = = TOP N j Y j N 2 i i+ N RPR. The Logic of ertainty 9

10 Event-Tree nalysis IE SUESS RRIER RRIER 2 (OK) 2 (R) FILURE 3 (R2) RPR. The Logic of ertainty 0

11 Fault-Tree nalysis Reliability lock Diagram for the 2-out-of-3 System 2/3 RPR. The Logic of ertainty

12 T = ( Y = ( = ( )( Y 2 ) ){ [ ( ){ [ ( Z )( Z 2 )( Z )( 3 )]} )( )]} Expanding and using k = we get T = ( )( )( ) RPR. The Logic of ertainty 2

13 ut sets and minimal cut sets UT SET: ny set of events (failures of components and human actions) that cause system failure. MINIML UT SET: cut set that does not contain another cut set as a subset. RPR. The Logic of ertainty 3

14 RPR. The Logic of ertainty 4 New fault tree: M M M = = = 3 2,,, ( )( )( ) ) )( )( ( j T M M M M = = = Minimal cut sets: System Failure

15 T = φ(, 2, n ) φ() φ() is the structure or switching function. It maps an n-dimensional vector of 0s and s onto 0 or. Disjunctive Normal Form: T = N ( Sum-of-Products Form: T = N i = M i N N i= j= i+ M M i j M ) i ( ) N N + M i = RPR. The Logic of ertainty 5 N i M i

16 For the 2-out-of-3 System: T =-(- ) (- ) (- ) T = (M +M 2 +M 3 ) - (M M 2 +M 2 M 3 +M 3 M ) + M M 2 M 3 ut, M M 2 = 2 = Therefore, the sum-of-products expression is: T = ( + + ) - 2 RPR. The Logic of ertainty 6

17 The ridge Network { 2 }, { 3 4 }, { }, { 4 5 } Disjunctive Normal Form: T =-(- 2 )(- 3 4 )( )(- 4 5 ) Sum-of-Products Form: T = RPR. The Logic of ertainty 7

18 auses of Failure. Primary failure ("hardware" failure) 2. Secondary failure (external, environmental) 3. "ommand" failure (no input; no power) No Output from omponent Primary Failure Secondary Failure ommand Failure RPR. The Logic of ertainty 8

19 Reliability lock Diagram for the Fuel-Supply System T Fuel Source P ontrol Valve V Pump Train T2 Fuel Source P2 ontrol Valve V2 Pump Train 2 Emergency Diesel Engine Electric Power Source, E ontrol System, ooling System, O RPR. The Logic of ertainty 9

20 Fault tree elements TOP EVENT OR Gate INTERMEDITE EVENT, asic Event ND Gate 2 asic Event 2 INOMPLETELY DEVELOPED EVENT, 2 Transfer in from Sheet 2 Note: It s helpful to start the fault-tree development from the output of the system (the top event) and work backwards. RPR. The Logic of ertainty 20

21 LOSS OF FUEL FLOW, T LOSS OF TRIN E LOSS OF TRIN 2 E 2 MEHNIL LOSS OF TRIN 2 M 2 Loss of Electricity Loss of ontrol Loss of ooling E O T 2 P 2 V 2 Loss of Electricity Loss of ontrol Loss of ooling MEHNIL LOSS OF TRIN M E O T P V RPR. The Logic of ertainty 2

22 simpler fault tree No Fuel is Delivered When Needed E Fails Fails O Fails Pumping ranches Fail Train Fails Train 2 Fails T Fails to Supply Fuel P Fails to Pump Fuel V Fails losed T2 Fails to Supply Fuel P2 Fails to Pump Fuel V2 Fails losed RPR. The Logic of ertainty 22

23 Development of T Tank T Failure to Supply Fuel Tank is Intact ut Empty and Undetected Tank (and Supply Pipe) is Not Intact Supply Pipe is Plugged Tank is Emptied Inadvertantly (human error) Tank is Empty Tank is Emptied in Use and Not Refilled Tank Drain Valve is Left Open Fuel Level Detection Fails Human ction Sludge uildup Earthquake Induced Failure Missile Impact Induced Failure Internal Fire/Explosion Induced Failure orrosion Induced Failure orrosion Faulty Manufacture & ontrol Program Fatique Induced Failure RPR. The Logic of ertainty 23

24 System min cut sets ny combination of an element of T, Tank P, Pump V, Valve and of T2, Tank P2, Pump V2, Valve plus E O ontrol System or Electric Power Source or ooling System RPR. The Logic of ertainty 24

25 RPR. The Logic of ertainty 25

26 Examples of Initiating Events Loss of oolant Transients Human Error Loss of Power Fires irplane rashes Earthquakes RPR. The Logic of ertainty 26