This document provides an overview of engineering risk analysis and event logic. It discusses how to define events, represent them using Venn diagrams and indicator variables, and perform logic operations like unions and intersections on events. Simple reliability models are presented, including series, parallel and 2-out-of-3 systems. Methods for analyzing events are covered, such as event trees, fault trees, cut sets, and minimal cut sets. Examples of initiating events in risk analysis are also listed.

1. Engineering Risk Benefit Analysis
1.155, 2.943, 3.577, 6.938, 10.816, 13.621, 16.862, 22.82
ESD.72J, ESD.721
RPRA 1. The Logic of Certainty
George E. Apostolakis
Massachusetts Institute of Technology
Spring 2007
RPRA 1. The Logic of Certainty 1

2. Event Definition
•Event: A statement that can be true or false.
•“It may rain tonight” is not an event.
•According to our current state of knowledge, we
may say that an event E is TRUE, FALSE, or
POSSIBLE (UNCERTAIN).
•Eventually, E will be either TRUE or FALSE.
RPRA 1. The Logic of Certainty 2

3. True
Event False
Possible
RPRA 1. The Logic of Certainty 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 {1,2,3,4,5,6} Failure Time {0, ∞}
• A collection of sample points is an event.
S
Venn Diagram E
RPRA 1. The Logic of Certainty 4

5. Indicator Variables
1,If E j is T
Xj =
0, If E j is F
Important Note: Xk = X, k: 1, 2, …
S
___
Venn Diagram E E
RPRA 1. The Logic of Certainty 5

6. Union (OR operation)
A∪ B = C
X C = 1 − (1 − X A )(1 − X B )
X C ≡ C X j
C
A B
A B
RPRA 1. The Logic of Certainty 6

7. Intersection (AND operation)
A∩ B = C XC = X AX B
XC ≡ ∏ X j
C
A B
A B
Mutually Exclusive Events: A∩ B = ∅
RPRA 1. The Logic of Certainty 7

8. Simple Systems
Reliability Block Diagram for the Series System
1 .... N
N N
System
Failure
failure: X = 1 − ∏ (1 − X j ) ≡ C X j
1 1
N
success : Y = ∏ Yj
1
1 ... N
RPRA 1. The Logic of Certainty 8

9. Reliability Block Diagram
for the Parallel System N
X =∏ X j
1
1
2
N
Y = CYj
i
1
i+1
TOP
N
1 2 i i+1 N
RPRA 1. The Logic of Certainty 9

10. Event-Tree Analysis
IE BARRIER 1 BARRIER 2
1 (OK)
SUCCESS
2 (R1)
3 (R2)
FAILURE
RPRA 1. The Logic of Certainty 10

11. Fault-Tree Analysis
Reliability Block Diagram for the 2-out-of-3 System
A
B 2/3
C
RPRA 1. The Logic of Certainty 11

12. X T = 1 − (1 − Y1 )(1 − Y2 )
= 1 − (1 − X A X B X C ){1 − [1 − (1 − Z1 )(1 − Z 2 )(1 − Z 3 )]}
= 1 − (1 − X A X B X C ){1 − [1 − (1 − X A X B )(1 − X B X C )(1 − X C X A )]}
Expanding and using Xk = X we get
X T = 1 − (1 − X A X B )(1 − X B X C )(1 − X C X A )
RPRA 1. The Logic of Certainty 12

13. Cut sets and minimal cut sets
• CUT SET: Any set of events (failures of components and
human actions) that cause system failure.
• MINIMAL CUT SET: A cut set that does not contain
another cut set as a subset.
RPRA 1. The Logic of Certainty 13

14. New fault tree:
S y s te m F a ilu r e
A B B C C A
Minimal cut sets:
M 1 = X A X B, M2 = X B XC ,, M3 = XC X A
3
X T = C M j ≡ 1 − (1 − M 1 ) (1 − M 2 ) (1 − M 3 ) =
1
= 1 − (1 − X A X B )(1 − X B X C )(1 − X C X A)
RPRA 1. The Logic of Certainty 14

15. XT = φ(X1, X2,…Xn) ≡ φ(X)
φ(X) is the structure or switching function.
It maps an n-dimensional vector of 0s and 1s onto 0 or 1.
Disjunctive Normal Form:
N N
XT = 1 − ∏ (1 − M i ) ≡ C M i
1 1
Sum-of-Products Form:
N N −1 N N +1 N
XT = ∑ Mi − ∑ ∑ M i M j + ... + (−1) ∏ M i
i =1 i =1 j =i +1 i =1
RPRA 1. The Logic of Certainty 15

16. For the 2-out-of-3 System:
XT=1-(1-XAXB) (1-XBXC) (1-XCXA)
XT = (M1+M2+M3) - (M1M2+M2M3+M3M1) + M1M2M3
But,
M1M2 = XAXB2XC = XAXBXC
Therefore, the sum-of-products expression is:
XT = (XAXB+XBXC+XCXA) - 2XAXBXC
RPRA 1. The Logic of Certainty 16

17. The Bridge Network
A 1 3 B
5
2 4
{X1X2}, {X3X4}, {X2X3X5}, {X1X4X5}
Disjunctive Normal Form:
XT=1-(1-X1X2)(1-X3X4)(1-X2X3X5)(1-X1X4X5)
Sum-of-Products Form:
XT = X1X2+ X3X4+ X2X3X5+ X1X4X5-
- X1X2 X3X4- X1X2X3X5- X1X2X4X5 -
-X2X3X4X5 - X1X3X4X5 + 2X1X2X3X4X5
RPRA 1. The Logic of Certainty 17

18. Causes of Failure
1. Primary failure ("hardware" failure)
2. Secondary failure (external, environmental)
3. "Command" failure (no input; no power)
N o O u tp u t fro m
C om ponent
P r im a r y
F a ilu r e
S e c o n d a ry C om m and
F a ilu r e F a ilu r e
RPRA 1. The Logic of Certainty 18

19. Reliability Block Diagram for the Fuel-Supply
System
T1 Control Valve
Fuel V1
Source P1 Pump Train 1
Emergency
Diesel
T2 Control Valve Engine
Fuel V2
Source P2
Pump Train 2
Electric
Power
Source, E
Control
System, C
Cooling
System,
CO
RPRA 1. The Logic of Certainty 19

20. Fault tree elements
TOP EVENT
“OR” Gate
INTERMEDIATE
EVENT, A
INCOMPLETELY
“AND” Gate DEVELOPED
EVENT, B 2
Transfer in
from Sheet 2
A1 A2
Basic Basic
Event Event
A1 A2
Note: It’s helpful to start the fault-tree development from the
output of the system (the top event) and work backwards.
RPRA 1. The Logic of Certainty 20

21. LOSS OF FUEL
FLOW , T
LOSS OF TRAIN E1 LOSS OF TRAIN E2
1 2
MECHANICAL M
LOSS OF TRAIN 2
2 Loss of Loss of Loss of
Electricity Control Cooling
E C CO
T2 P2 V2
MECHANICAL
LOSS OF TRAIN M 1
Loss of Loss of Loss of 1
Electricity Control Cooling
E C CO
T1 P1 V1
RPRA 1. The Logic of Certainty 21

22. A simpler fault tree
No Fuel is
Delivered
When Needed
E C CO Pumping
Fails Fails Fails Branches Fail
Train 1 Fails Train 2 Fails
T1 P1 V1 T2 P2 V2
Fails to Fails to Fails Fails to Fails to Fails
Supply Pump Closed Supply Pump Closed
Fuel Fuel Fuel Fuel
RPRA 1. The Logic of Certainty 22

23. Development of T1
Tank T1
Failure to
Supply Fuel
Tank is Intact Tank (and Supply Pipe
But Em pty Supply Pipe) is Plugged
and Undetected is Not Intact
Tank is Em pty
Fuel
Level Hum an Sludge
Detection Action Buildup
Tank is Tank is Tank Fails
Em ptied Em ptied Drain
Inadvertantly in Use and Valve is
(hum an Not Refilled Left O pen
error)
Corrosion Induced Failure Fatique
M issile Induced
Earthquake Internal Failure
Induced Im pact Fire/Explosion
Failure Induced Induced
Failure Failure
Faulty
Corrosion M anufacture
& Control
Program
RPRA 1. The Logic of Certainty 23

24. System min cut sets
T1, Tank T2, Tank
Any combination of P1, Pump and of P2, Pump
an element of V1, Valve V2, Valve
C Control System
or
plus
E Electric Power
Source
or
CO Cooling System
RPRA 1. The Logic of Certainty 24

25. RPRA 1. The Logic of Certainty 25

26. Examples of Initiating Events
• Loss of Coolant
• Transients
• Human Error
• Loss of Power
• Fires
• Airplane Crashes
• Earthquakes
RPRA 1. The Logic of Certainty 26