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, 2nd ed. Waveland Press, Inc. Copyright ©
2010
Characteristics
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•
•
•
•
•
•
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FTA
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
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
AND gate - a logic gate where an output event occurs only when all
the input events have occurred.
OR 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
AND/OR gates
Basic
Events
FTA
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Example of AND / OR Gates
Tank Ruptures
Overpressure
OR
Overpressure
(a)
FTA
AND
Wall
Fatigue
Failure
Relief
Valve
Fails
Excessive
Temperature
(b)
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Example of AND / OR Gates
Tank Ruptures
OR
Overpressure
AND
Excessive
Temperature
FTA
Wall
Fatigue
Failure
Relief
Valve
Fails
(c)
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Alarm System Example
meter
switch
alarm
observer
backup
power
FTA
power
source
automatic
sensor
8
Alarm System Fault Tree
T- Alarm Failure
OR
A-Power Failure
B-Sensor Failure
AND
AND
E-Primary
power fails
F-Backup
fails
G-Manual Alarm
Failure
C-Alarm
Failure
D Secondary
alarm failure
H-Auto
Sensor
Fails
OR
I-Human
Error
FTA
J-Switch
fails
K-Meter
Fails
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Boolean representation of Top Event
T- Alarm Failure
OR
A-Power Failure
B-Sensor Failure
AND
AND
E-Primary
power fails
F-Backup
fails
D-Secondary
alarm failure
H-Auto
Sensor
Fails
OR
T = A∪ B ∪ C ∪ D
= ( E ∩F ) ∪( G∩H) ∪C∪D
G-Manual Alarm
Failure
C-Alarm
Failure
I-Human
Error
J-Switch
fails
K-Meter
Fails
= ( E ∩ F ) ∪ [ (I ∪ J ∪ K ) ∩ H ] ∪ C ∪ D
FTA
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Example
T = A∪ B = A∪ ( C ∩ D )
Top Event
= A ∪ [ (E ∪ F ) ∩ ( E ∪ A )]
OR
= A ∪ [ E ∪ (F ∩ A)] = A ∪ E since A ∪ (F ∩ A) = A
Event B
A
AND
E
FTA
Event C
Event D
OR
OR
F
E
A
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Example Equivalent Fault Tree
Top Event
T=AUE
OR
Event A
FTA
Event E
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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
Top Event
OR
M1
M2
MK
AND
AND
AND
E1
E2
En
T = M1 ∪ M2 ∪ ... ∪ Mk where Mi = E1 ∩ E2 ∩ .... ∩ Eni
and Ei are basic events.
FTA
13
Example
T- Alarm Failure
1
2
3
A
E,F
E,F
B
G,H
C
C
I,H
J,H
K,H
C
D
D
D
OR
A-Power Failure
B-Sensor Failure
AND
AND
E-Primary
power fails
F-Backup
fails
G-Manual Alarm
Failure
C-Alarm
Failure
H-Auto
Sensor
Fails
D-Secondary
alarm failure
OR
I-Human
Error
J-Switch
fails
K-Meter
Fails
T = (E ∩ F) ∪ (I ∩ H) ∪ (J ∩ H) ∪ (K ∩ H) ∪ C ∪ D
FTA
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Example Cut Sets
1
A
B
2
A
C, D
3
A
E, D
F, D
4
A
E, E
E, A
F, E
F, A
Top Event
5
A
E
OR
Event B
A
AND
E
Event C
Event D
OR
OR
F
E
A
since E ∩ E = E, E ∩ A ⊆ A, F ∩ E ⊆ E, and F ∩ A ⊆ A. Therefore T = A ∪ E
FTA
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Quantitative Analysis
If P(M) < 10-3
If cut sets are mutually exclusive:
P(T) = P( M1 U M2 U …U Mk )
= P(M1 ) + P(M2 ) + ... + P(Mk )
If not:
P(T) = P(A ∪ E) = P(A) + P(E) - P(A ∩ E).
P(T) = P(M1) +P(M2) - P(M1) P(M2) + etc.
P(Mi ) = P(E1 ∩ E2 ∩ .... ∩ Eni ) = P(E1 ) P(E2 ) ... P(En1 )
if independent
Then P(M1) P(M2) < 10-6
FTA
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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
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One Last Example
T=A∪B∪C
= (D ∩ E) ∪ (H ∪ I) ∪ (L ∩ M ∩ N)
Top Event
OR
A
B
AND
D
E
L
I
G
J
N
M
K
Q
P
OR
R
FTA
= [D ∩ (F ∪ G) ∩ [H ∪ (J ∩ k)]
∪ [L ∩ (R ∪ S ∪ Q) ∩ N]
OR
AND
OR
F
AND
OR
H
= [D ∩ (F ∪ G) ∩ [H ∪ (J ∩ k)]
∪ [L ∩ (P ∪ Q) ∩ N]
C
P(T) ≈ P(D) [P(F) + P(G)] +
P(H) + P(J)P(K) + P(L) [ P(R) +
P(S) + P(Q)] P(N)
S
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The Cut Sets
Top Event
OR
A
B
AND
D
C
H
L
I
#3
#4
A
D,E
D,F
D,F
D,G
N
M
B
H
H
J,K
H
J,K
L,P,N
L,Q,N
L,R,N
L,S,N
L,Q,N
I
OR
AND
OR
#2
AND
OR
E
#1
C
F
G
J
K
Q
P
OR
R
FTA
L,M,N
S
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