1. ANCOLD/NZSOLD Conference 2004 Melbourne, Victoria, Australia
16 November 2004
RAC
Engineers & Economists
IDSRM
Practical Considerations in Developing F-N Curves for Dam Safety Risk Assessment
David S. Bowles and Sanjay S. Chauhan
Institute for Dam Safety Risk Management - Utah State University
and RAC Engineers & Economists

2. ANCOLD (2003) Societal Risk Guidelines
10-3
Unacceptable risks
10-4
Limit of Tolerability
EXISTING DAM
Probability of expected life loss > N per year
10-5
10-6
10-7 1
102
103
104 10
N, number of fatalities due to dam failure

3. ANCOLD (2003) Societal Risk Guidelines
10-3
Unacceptable risks
10-4
Limit of Tolerability
EXISTING DAM
Probability of expected life loss > N per year
10-5
Risks tolerable if
“as low as
reasonably
practicable”
- ALARP
10-6
10-7 1
102
103
104 10
N, number of fatalities due to dam failure

4. ANCOLD (2003) Societal Risk Guidelines
10-3
Unacceptable risks
10-4
Limit of Tolerability
NEW DAMS & MAJOR
AUGMENTATIONS
Probability of expected life loss > N per year
10-5
Risks tolerable if
“as low as
reasonably
practicable”
- ALARP
10-6
10-7 1
102
103
104 10
N, number of fatalities due to dam failure

5. ANCOLD (2003) Societal Risk Guidelines
10-3
Limit of Tolerability
EXISTING DAM
10-4
Probability of expected life loss > N per year
10-5
Limit of
Tolerability
NEW DAMS & MAJOR
AUGMENTATIONS
10-6
10-7 1
102
103
104 10
N, number of fatalities due to dam failure

6. Risk = (Scenario, Probability, Consequence)
f1,N1
f2,N2
f3,N3
Risk = (Scenario, Probability, Consequence)

7. Risk = (Scenario, Probability, Consequence)
flife loss,N
ffailure
Risk = (Scenario, Probability, Consequence)

8. ANCOLD (2003) Societal Risk Guidelines (F-N)

9. Outline
Theoretical Background
An inconsistency in ANCOLD Guidelines
Correctly Calculating F-N Curve
Risk Analysis Considerations
Numerical Considerations
Incorporating Uncertainty
Conclusions

10. 1) Theoretical Background

11. Discrete Probability Distribution
Probability Mass Function (PMF), f(n)

12. Discrete Probability Distribution
Probability Mass Function (PMF) f(n)
Cumulative Distribution Function (CDF), F'(N ≤ n)

13. Discrete Probability Distribution
Probability Mass Function (PMF), f(n)
Cumulative Distribution Function (CDF), F'(N≤n)
Complementary Cumulative Distribution Function (CDF), F(N > n)

14. Discrete Probability Distribution
Probability Mass Function (PMF), f(n)
F(N > 5) = 0
Cumulative Distribution Function (CDF), F'(N≤n)
Complementary Cumulative Distribution Function (CDF), F(N > n)

15. Discrete Probability Distribution
Probability Mass Function (PMF), f(n)
F(N > 4) = 0.1
Cumulative Distribution Function (CDF), F '(N≤n)
Complementary Cumulative Distribution Function (CDF), F(N > n)

16. Discrete Probability Distribution
Probability Mass Function (PMF), f(n)
F(N > 3) = = 0.2
Cumulative Distribution Function (CDF), F'(N≤n)
Complementary Cumulative Distribution Function (CDF), F(N > n)

17. Discrete Probability Distribution
CDF + CCDF = 1.0
F'(N≤n)+F(N>n)=1.0
Probability Mass Function (PMF), f(n)
Cumulative Distribution Function (CDF), F'(N≤n)
Complementary Cumulative Distribution Function (CDF), F(N > n)

18. Discrete Probability Distribution
Probability Mass Function (PMF), f(n)
Cumulative Distribution Function (CDF), F'(N≤n)
Incorrect F(N ≥ n)
Complementary Cumulative Distribution Function (CDF), F(N > n)
Correct F(N > n)

19. 2) ANCOLD (2003) Guidelines on Risk Assessment

20. 2) ANCOLD (2003) Guidelines on Risk Assessment
F, probability of failure per dam per year with expected loss of life > N

21. 2) ANCOLD (2003) Guidelines on Risk Assessment
F, probability of failure per dam per year with expected loss of life > N
By convention: F(N ≥ n)

22. 2) ANCOLD (2003) Guidelines on Risk Assessment
Definition of CCDF in Glossary: … probability that a random variable takes on values greater than or equal to a particular value …

23. 2) ANCOLD (2003) Guidelines on Risk Assessment
Correct F(N ≥ n)
Definition of CCDF in Glossary: … probability that a random variable takes on values greater than or equal to a particular value …

24. 2) ANCOLD (2003) Guidelines Appendix I F-N Example

25. 2) ANCOLD (2003) Guidelines Appendix I F-N Example
Correct F(N ≥ n)

26. 2) ANCOLD (2003) Guidelines Appendix I F-N Example
Correct F(N ≥ n)
Incorrect F(N > n)

27. 3) Correctly calculating F-N Curve
Correct F(N ≥ n)
F(N > 3) = = 0.2
Incorrect F(N > n)

28. Discrete Probability Distribution
CDF + CCDF = 1.0
F'(N≤n)+F(N≥n)≠1.0
Probability Mass Function (PMF), f(n)
Cumulative Distribution Function (CDF), F'(N≤n)
Correct F(N ≥ n)
Complementary Cumulative Distribution Function (CDF), F(N > n)
Incorrect F(N > n)

29. ANCOLD (2003) Guidelines Appendix I F-N Example
Correct F(N ≥ n)
Incorrect F(N > n) ≥

30. Correct F(N ≥ n) Incorrect F(N > n)
Unacceptable Risk
Incorrect F(N > n) ≥ ≥

31. Approximate F-N Plotting Approaches
Correct F(N ≥ n)
Incorrect F(N > n) ≥

32. Approximate F-N Plotting Approaches
Incorrect F(N ≥ n)
Correct F(N > n) ≥

33. Approximate F-N Plotting Approaches
Makes little difference if F-N Curve looks like this ≥

34. Approximate F-N Plotting Approaches
But the difference can be significant if F-N Curve looks like this – RISK ANALYSIS MAY BE TOO COARSE ≥

35. 4) Risk Analysis Considerations
Too coarse a level of detail can affect position of F-N curves
Ignoring seasonal changes in PAR
Making a conservative assumption about rate of seepage-piping failures
Consider a range of rates
Using main dam consequences estimates for a saddle dam failure mode

36. 5) Numerical Considerations
Adjust number of loading intervals to control numerical precision errors
Build event tree for representative interval
Protocols to assign probabilities and consequences
Risk reduction alternatives
Same partitioning as for existing dam
Loading intervals

37. 5) Numerical Considerations
Sensitivity of F-N Curve to Partitioning of Loading [Hill et al 2002]

38. 6) Incorporating Uncertainty Chauhan and Bowles (2001)
Best Estimate F-N
ANCOLD Societal Risk Limit (for New Dams)

39. 6) Incorporating Uncertainty
Best Estimate F-N
ANCOLD Societal Risk Limit (for New Dams)

40. 6) Incorporating Uncertainty
Best Estimate F-N
ANCOLD Societal Risk Limit (for New Dams)

41. 6) Incorporating Uncertainty
~80%
~47%
ANCOLD Societal Risk Limit (for New Dams)
~43%
~78%

42. 6) Incorporating Uncertainty
Risk Reduction Measure
Existing Dam
~80%
~47%
Confidence in Meeting ANCOLD Societal Risk Limit rather simply getting below the line
~43%
~78%

43. 6) Incorporating Uncertainty
Risk Reduction Measure
Existing Dam
~80%
~47%
Confidence in meeting ANCOLD Societal Risk Limit rather than simply getting below the line
~43%
~78%
Draft Corps of Engineers Tolerable Risk Guidelines

44. Confidence in satisfying ALARP Justification - Adjusted CSLS
DETERMINISTIC
Widen S/W (4th gate):
US$143M
Moderate/Poor
80th percentile
RCC S/W:
US$120M
Moderate
75th percentile
UNCERTAINTY
Widen & RCC S/W:
V. Strong: 30% confid
Strong: 20% confid
Moderate: 30% confid
Poor: 20% confid
Confidence in satisfying ALARP Justification - Adjusted CSLS

45. Calculate F-N as F(N ≥ n) NOT F(N > n)
7) Conclusions
Calculate F-N as F(N ≥ n) NOT F(N > n)
Select level of detail in Risk Analysis that is:
“Fit for purpose”
Initial PRA, progressive improvement, final sign off, etc
To obtain a representative estimated F-N relationship
Failure modes, exposure conditions, response cases etc.
“Art of risk analysis”
Control numerical precision errors through adequate partitioning
Consider & Communicate the uncertainties in the Societal Risk evaluation
Even if not performing an Uncertainty Analysis

46. E-mail: David.Bowles@usu.edu
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