Chapter 18 Section 8.5 Fault Trees Analysis (FTA) Don t get caught out on a limb of your fault tree. C. Ebeling, Intro to Reliability & Maintainability Engineering, 2 nd ed. Waveland Press, Inc. Copyright 2010
Characteristics Graphical design technique Alternative to reliability block diagrams Broader in scope Perspective on faults rather than reliability Model events rather than components Faults include failures Focus on a catastrophic event (top event) Top-down deductive analysis FTA 2
The Four Steps to a FTA (1) Define the system, its boundaries, and the top event, (2) Construct the fault tree representing symbolically the system and its relevant events, (3) Perform a qualitative evaluation by identifying those combinations of events which will cause the top event, (4) Perform a quantitative evaluation by assigning failure probabilities or unavailabilities to the basic events and computing the probability of the top event. FTA 3
Fault Tree Symbols gate - a logic gate where an output event occurs only when all the input events have occurred. gate - a logic gate where an output event occurs if at least one of the input events have occurred. Resultant event - a fault event resulting from the logical combination of other fault events and usually an output to a logic gate. Basic event - an elementary event representing a basic fault or component failure. Incomplete event - an event that has not been fully developed because of lack of knowledge or its unimportance. FTA 4
General Structure of a Fault Tree Top Event System Failure Resultant Events / gates Basic Events FTA 5
Example of / Gates Tank Ruptures Overpressure Overpressure Wall Fatigue Failure Excessive Temperature Relief Valve Fails (a) (b) FTA 6
Example of / Gates Tank Ruptures Overpressure Wall Fatigue Failure Excessive Temperature Relief Valve Fails (c) FTA 7
Alarm System Example meter observer switch alarm backup power power source automatic sensor FTA 8
Alarm System Fault Tree T- Alarm Failure A-Power Failure B-Sensor Failure C-Alarm Failure D Secondary alarm failure E-Primary power fails F-Backup fails G-Manual Alarm Failure H-Auto Sensor Fails I-Human Error J-Switch fails K-Meter Fails FTA 9
Boolean representation of Top Event T- Alarm Failure A-Power Failure B-Sensor Failure C-Alarm Failure D-Secondary alarm failure E-Primary power fails F-Backup fails G-Manual Alarm Failure H-Auto Sensor Fails T = A B C D = ( E F ) ( G H ) C D I-Human Error J-Switch fails K-Meter Fails = ( E F ) [ (I J K ) H ] C D FTA 10
Example A Top Event Event B Event C Event D T = A B = A ( C D ) = A [ (E F ) ( E A )] = A [ E (F A)] = A E since A (F A) = A E F E A FTA 11
Example Equivalent Fault Tree T = A U E Top Event Event A Event E FTA 12
Minimal Cut Sets A cut set is a collection of basic events which will cause the top event. A minimal cut set is one with no unnecessary events. That is, all the events within the cut set must occur to cause the top event M1 Top Event M2 MK E1 E2 En T = M 1 M 2... M k where M i = E 1 E 2... E ni and E i are basic events. FTA 13
Example T- Alarm Failure 1 2 3 A-Power Failure E-Primary power fails F-Backup fails I-Human Error B-Sensor Failure G-Manual Alarm Failure J-Switch fails H-Auto Sensor Fails C-Alarm Failure K-Meter Fails D-Secondary alarm failure A B C E,F G,H C E,F I,H J,H K,H C D D D T = (E F) (I H) (J H) (K H) C D FTA 14
Example Cut Sets 1 2 3 4 5 A A A A A B C, D E, D E, E E F, D E, A F, E F, A A Top Event Event B Event C Event D E F E A since E E = E, E A A, F E E, and F A A. Therefore T = A E FTA 15
Quantitative Analysis If cut sets are mutually exclusive: If P(M) < 10-3 P(T) = P( M 1 U M 2 U U M k ) = P(M 1 ) + P(M 2 ) +... + P(M k ) If not: P(T) = P(A E) = P(A) + P(E) - P(A E). P(T) = P(M 1 ) +P(M 2 ) - P(M 1 ) P(M 2 ) + etc. P(M i ) = P(E 1 E 2... E ni ) = P(E 1 ) P(E 2 )... P(E n1 ) Then P(M 1 ) P(M 2 ) < 10-6 if independent FTA 16
Example P(T) = P{ ( E F ) [ (I J K ) H ] C D } P( E F) + P [ (I J K ) H ] + P(C) +P(D) P(E) P(F) + [ P(I) + P(J) +P(K) ] P(H) + P(C) + P(D) If each basic event has a probability of.01, then P(T) (.01) 2 + (.01+.01+.01) (.01) +.01 +.01 =.0204 FTA 17
One Last Example Top Event T = A B C = (D E) (H I) (L M N) D A B C E H I L M N = [D (F G) [H (J k)] [L (P Q) N] = [D (F G) [H (J k)] [L (R S Q) N] F G J K Q P R S P(T) P(D) [P(F) + P(G)] + P(H) + P(J)P(K) + P(L) [ P(R) + P(S) + P(Q)] P(N) FTA 19
The Cut Sets Top Event #1 #2 #3 #4 A B C A D,E D,F D,F D,G D F E G H I L M N J K Q P B C H I L,M,N H J,K L,P,N L,Q,N H J,K L,R,N L,S,N L,Q,N R S FTA 20