Unformatted text preview: ETL 11102556
28 May 99 Appendix B
Evaluating the Reliability of Existing Levees Table of Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B9
Conversion Factors, NonSI to SI Units of Measurement . . . . . . . . . . . . . . . . . B10
1—Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Limitations of Engineering Reliability Methods . . . . . . . . . . . . . . . . . . . . .
Accuracy of probabilistic measures . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Calibration of procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Application to economic analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Past practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Current practice for navigation rehabilitation studies . . . . . . . . . . . . . .
The Conditional Probability of Failure Function . . . . . . . . . . . . . . . . . . . .
Study Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B11
B11
B11
B11
B11
B12
B12
B12
B13
B13
B15 2—Current Corps of Engineers’ Guidance: . . . . . . . . . . . . . . . . . . . . . . . . . . .
Policy Guidance Letter No. 26, Benefit Determination Involving
Existing Levees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
EM 111021913, “Design and Construction of Levees” . . . . . . . . . . . . .
Components of an Improved Probabilistic Assessment Procedure . . . . . . B16 3—Related Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Comprehensive Works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Slope Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Underseepage, ThroughSeepage, and Piping . . . . . . . . . . . . . . . . . . . . . . .
Multiple Modes of Failure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B23
B23
B25
B25
B26 B16
B18
B21 4—Two Example Problems Defined . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B27
Problem 1: Sand Levee on Thin Uniform Clay Top Stratum . . . . . . . . . . . B27
Problem 2: Clay Levee on Thick Nonuniform Clay Top Stratum . . . . . . . B27
5—Characterizing Uncertainty in Geotechnical Parameters . . . . . . . . . . . . . . . B29
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B29
Unit Weight of Soil Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B30
Drained Strength of Sands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B31
Drained Strength of Clays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B31
Undrained Strength of Clays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B31
Estimation from test results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B31 B1 ETL 11102556
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Estimation from test results and consolidation stress . . . . . . . . . . . . . .
Permeability for Seepage Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Permeability of foundation sands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Permeability of top blanket clays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Permeability ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B32
B32
B32
B34
B34 6—Underseepage Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B35
Example Problem 1: Sand Levee on Thin Uniform Clay Top Stratum . . . B35
Example Problem 2: Clay Levee on Thick Nonuniform Clay Top
Stratum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B42
7—Slope Stability Analysis for ShortTerm Conditions . . . . . . . . . . . . . . . . .
Example Problem 1: Sand Levee on Thin Uniform Clay Top Stratum . . .
Problem modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Example calculation of probability values . . . . . . . . . . . . . . . . . . . . . . .
Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Example Problem 2: Clay Levee on Thick Irregular Clay Top Stratum . .
Problem modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B49
B49
B49
B50
B64
B65
B66
B67
B67
B67
B75 8—Slope Stability Analysis for LongTerm Conditions . . . . . . . . . . . . . . . . . B76
9—ThroughSeepage Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Design practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Deterministic models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Erosion model of Khilar, Folger, and Gray . . . . . . . . . . . . . . . . . . . . . .
Rock Island District procedure for sand levees . . . . . . . . . . . . . . . . . . .
Extension of Khilar’s model to sandy materials . . . . . . . . . . . . . . . . . .
Example Problem 1: Sand Levee on Thin Uniform Clay Top Stratum . . .
Rock Island District method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Khilar equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Example Problem 2: Clay Levee on Thick Nonuniform Clay Top
Stratum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B77
B77
B77
B77
B78
B79
B80
B82
B83
B84
B91
B91 10—Surface Erosion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B94
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Erosion Due to Current Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Analytical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Critical velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Calculation of reliability index and probability of failure . . . . . . . . . . .
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Erosion Due to WindGenerated Waves . . . . . . . . . . . . . . . . . . . . . . . . . . B2 B94
B94
B94
B94
B95
B95
B96
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11—Combining Conditional Probability Functions and Other
Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B99
Combining Probability Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B99
Judgmental evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B100
Combinatorial probabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B101
Flood Duration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B104
Length of Levee and Spatial Correlation . . . . . . . . . . . . . . . . . . . . . . . . . B104
12—Summary, Conclusions, and Recommendations . . . . . . . . . . . . . . . . . . . . B106
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B106
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B106
Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B108
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B109
Annex A: Brief Review of Probability and Reliability Terms and
Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B113
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B113
Reliability Analysis Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B114
The probability of failure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B114
Contexts of reliability analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B114
Reliability Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B115
Accuracy of Reliability Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B115
The CapacityDemand Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B116
Steps in a Reliability Analysis Using the CapacityDemand Model . . . . . B117
Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B118
Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B118
Moments of random variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B118
Mean value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B118
Expected value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B118
Variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B119
Standard deviation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B119
Coefficient of variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B119
Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B120
Probability Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B120
Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B120
Estimating Probabilistic Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . B121
Normal distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B122
Lognormal distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B122
Calculation of the Reliability Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B123
Integration of the Performance Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B126
The Taylor’s series method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B126
Independent random variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B126
Correlated random variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B128
The Point Estimate Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B128
Independent random variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B129
Correlated random variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B130
Monte Carlo simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B130
Determining the Probability of Failure . . . . . . . . . . . . . . . . . . . . . . . . . . . . B131
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Overall System Reliability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B131
Series system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B131
Simple parallel system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B131
Parallel series systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B132
A practical approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B132
Target Reliability Indices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B132
Annex B: Example Calculations of Functions of Random Variables . . . . . . . B134
Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B134
Taylor’s Series with Exact Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . B135
Taylor’s Series with Numerically Approximated Derivatives . . . . . . . . . . B135
Point Estimate Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B137 List of Figures
Figure 1. Figure 2. Alternative conditional probabilityoffailure functions . . . . . . . . B19 Figure 3. Levee section for example problem 1 . . . . . . . . . . . . . . . . . . . . . . B28 Figure 4. Levee section for example problem 2 . . . . . . . . . . . . . . . . . . . . . . B28 Figure 5. Spreadsheet for underseepage analysis of example
problem 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B37 Figure 6. Calculation of probability of failure for underseepage . . . . . . . . . B40 Figure 7. Conditional probability of failure function: Underseepage
for example problem 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B41 Figure 8. Spreadsheet for underseepage analysis of example problem 2
(H = 20 ft) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B44 Figure 9. Spreadsheet for underseepage analysis of example
problem 2 (H = 17.5 ft) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B45 Figure 10. Spreadsheet for underseepage analysis of example problem 2
(H =15 ft) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B46 Figure 11. Spreadsheet for underseepage analysis of example problem 2
(H =15 ft) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B47 Figure 12. B4 Possible reliability versus floodwater elevation functions for
existing levees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B14 Conditional probability of failure function: Underseepage
for example problem 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B48 ETL 11102556
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Figure 13. Failure surfaces for example problem 1, water
elevation = 400 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B53 Figure 14. Failure surfaces for example problem 1, water
elevation = 410 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B54 Figure 15. Failure surfaces for example problem 1, water
elevation = 420 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B55 Figure 16. Reliability calculations for undrained slope stability,
example problem 1, water height = 0, water
elevation = 400 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B56 Figure 17. Reliability calculations for undrained slope stability,
example problem 1, water height = 5, water
elevation = 405 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B57 Figure 18. Reliability calculations for undrained slope stability,
example problem 1, water height = 10, water
elevation = 410 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B58 Figure 19. Reliability calculations for undrained slope stability,
example problem 1, water height = 15, water
elevation = 415 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B59 Figure 20. Reliability calculations for undrained slope stability,
example problem 1, water height = 17.5, water
elevation = 417.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B60 Figure 21. Reliability calculations for undrained slope stability,
example problem 1, water height = 20, water
elevation = 420 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B61 Figure 22. Conditional probability function for undrained slope failure,
example problem 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B62 Figure 23. Conditional probability function for undrained slope failure,
example problem 1, enlarged view . . . . . . . . . . . . . . . . . . . . . . . . B63 Figure 24. Calculation of probability of failure for slope stability . . . . . . . . . B65 Figure 25. Failure surfaces for example problem 2, water
elevation = 400 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B69 Figure 26. Failure surfaces for example problem 2, water
elevation = 420 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B70 B5 ETL 11102556
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Figure 27. Reliability calculations for undrained slope stability,
example problem 2, water height = 0, water
elevation = 400 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B71 Figure 28. Reliability calculations for undrained slope stability,
example problem 2, water height = 20, water
elevation = 420 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B72 Figure 29. Conditional probability function for undrained slope failure,
example problem 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B73 Figure 30. Conditional probability function for undrained slope failure,
example problem 2, enlarged view . . . . . . . . . . . . . . . . . . . . . . . . B74 Figure 31. Rock Island District berm criteria and linear approximation
of limit state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B82 Figure 32. Spreadsheet for throughseepage analysis . . . . . . . . . . . . . . . . . . . B85 Figure 33. Reliability calculations for throughseepage, example
problem 1, h = 5 ft . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B86 Figure 34. Reliability calculations for throughseepage, example
problem 1, h = 10 ft . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B87 Figure 35. Reliability calculations for throughseepage, example
problem 1, h = 15 ft . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B88 Figure 36. Reliability calculations for throughseepage, example
problem 1, h = 17.5 ft . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B89 Figure 37. Reliability calculations for throughseepage, example
problem 1, h = 20.0 ft . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B90 Figure 38. Conditional probability of failure function for throughseepage, example problem 1 (modified) . . . . . . . . . . . . . . . . . . . . B92 Figure 39. Reliability calculations for internal erosion analysis using
modified Khilar’s equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B93 Figure 40. Example spreadsheet for surface erosion analysis . . . . . . . . . . . . . B97 Figure 41. Combined conditional probability of failure function for
example problem 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B102 Figure 42. Combined conditional probability of failure function for
example problem 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B103 Figure A1. The capacitydemand model . . . . . . . . . . . . . . . . . . . . . . . . . . . . B116
B6 ETL 11102556
28 May 99
Figure A2. Alternative definitions of the reliability index . . . . . . . . . . . . . . . B124
Figure A3. Point estimate method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B129 List of Tables
Table 1. Coefficients of Variation for Geotechnical Parameters . . . . . . . . . B30 Table 2. Example Statistical Analysis of Undrained Tests on Clay,
Unconfined Compression Tests on Undisturbed Samples . . . . . . B33 Table 3. Random Variables for Example Problem 1 . . . . . . . . . . . . . . . . . . B35 Table 4. Problem 1, Underseepage Taylor’s Series Analysis Water at
Elevation 420 (H = 20 ft) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B38 Table 5. Random Variables for Example Problem 2 . . . . . . . . . . . . . . . . . . B42 Table 6. Problem 2, Underseepage Taylor’s Series Analysis, Water at
Elevation 420 (H = 20 ft) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B43 Table 7. Random Variables for Example Problem 1 . . . . . . . . . . . . . . . . . . B49 Table 8. Problem 1, Undrained Slope Stability, Taylor’s Series
Analysis, Water at Elevation 400 (H = 0 ft) . . . . . . . . . . . . . . . . . B50 Table 9. Problem 1, Undrained Slope Stability, Results for All Runs . . . . B51 Table 10. Problem 1, Slope Stability for ShortTerm Conditions,
Summary of Probabilistic Analyses . . . . . . . . . . . . . . . . . . . . . . . . B66 Table 11. Random Variables for Example Problem 2 . . . . . . . . . . . . . . . . . . B67 Table 12. Problem 2, Undrained Slope Stability, Results for All Runs . . . . B68 Table 13. Calculated Critical Gradients for Three Granular Soils
Using Khilar’s Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B83 Table 14. Random Variables for Internal Erosion Analysis, Example
Problem 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B84 Table 15. Results of Internal Erosion Analysis, Example Problem 1
(Modified to Flatter Slopes) H = 20 ft, Rock Island District
Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B84 B7 ETL 11102556
28 May 99
Table 16. Table A1. B8 Assigned Conditional Probability of Failure Functions
for Judgmental Evaluation of Observed Conditions . . . . . . . . . . B101
Target Reliability Indices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B133 ETL 11102556
28 May 99 Preface
This report is a product of the U. S. Army Corps of Engineers’ Risk Analysis
for Water Resources Investments Research Program managed by the Institute
for Water Resources (IWR), Water Resources Support Center (WRC). The work
was performed under Work Unit 32835, “Risk Analysis for Stability Evaluation
of Levees.” Dr. Edward B. Perry of the U.S. Army Engineer Waterways Experiment Station (WES) managed the work unit and Dr. David A. Moser of IWR
manages the Risk Analysis Program. Dr. Perry works under the direct supervision of Mr. W. Milton Myers, Chief, Soil Mechanics Branch, Soil and Rock
Mechanics Division (S&RMD), Geotechnical Laboratory (GL), and the general
supervision of Dr. Don C. Banks, Chief, S&RMD, and Dr. William F. Marcuson
III, Director, GL, WES. Dr. Moser works under the direct supervision of Mr.
Michael R. Krouse, Chief of the Technical Analysis and Research Division and
the general supervision of Mr. Kyle E. Shilling, Director of the IWR.
Mr. Robert Daniel, Chief, Plan Formulation and Evaluation Branch, Policy
and Planning Division; Mr. Earl Eiker, Chief, Hydrology and Hydraulics Branch,
Engineering Division; and Mr. James E. Crews, Deputy Chief, Operations, Construction and Readiness Division; all within the Civil Works Directorate, Headquarters, U.S. Army Corps of Engineers, serve as Technical Monitors for the Risk
Analysis Program.
This report was written by Dr. Thomas F. Wolff, Associate Professor, Department of Civil and Environmental Engineering, Michigan State University, under
Contract No. DACW3994M4226 to WES. Dr. Wolff was assisted in the work
by Dr. Mostafa Ashoor, Ms. Cynthia Ramon, and Mr. Todd Richter.
At the time of publication of this report, Director of WES was Dr. Robert W.
Whalin. Commander was COL Bruce K. Howard, EN. Mr. Kyle E. Schilling
was Acting Director of the WRC.
Note that, in Chapter 2, the comments on the rescinded Corps document that
appeared in the original report have been edited and deleted during preparation of
this ETL. The contents of this report are not to be used for advertising, publication,
or promotional purposes. Citation of trade names does not constitute an
official endorsement or approval of the use of such commercial products. B9 ETL 11102556
28 May 99 Conversion Factors,
NonSI to SI Units of
Measurement
NonSI units of measurement used in this report can be converted to SI units
as follows:
Multiply By To Obtain degrees 0.0174533 radians feet 0.3048 meters inches 2.54 centimeters feet per second 30.48 pounds (force) per square foot 0.04788 pounds (force) per square foot 478.802631 pounds (mass) per cubic foot B10 0.1570873 centimeters per second
kilopascals
dynes per square centimenter
kilonewtons per cubic meter ETL 11102556
28 May 99 1 Introduction Purpose
The purpose of the research effort leading to this report was to develop, test,
and illustrate procedures that can be used by geotechnical engineers to assign
conditional probabilities of failure for existing levees as functions of floodwater
elevation. Such functions are in turn to be used by economists when estimating
benefits to be derived from proposed levee improvements. Limitations of Engineering Reliability Analysis
Accuracy of probabilistic measures
Before proceeding, it is important to define a context in which to place engineering reliability analysis and its relationship to flood control levees. The application of probabilistic analysis in geotechnical engineering and other areas of
civil engineering is still an emerging technology. Much experience with such
procedures remains to be gained, and the appropriate form and shape of
probability distributions for the relevant parameters are not known with certainty.
The methods described herein should not be expected to provide “true,” or
"absolute" probabilityoffailure values but can provide consistent measures of
relative reliability when reasonable assumptions are employed. Such comparative
measures can be used to indicate, for example, which reach (or length) of levee,
which typical section, or which alternative design may be more reliable than
another. They also can be used to determine which of several performance modes
(seepage, slope stability, etc.) governs the reliability of a particular levee. All of
the levee reaches analyzed are considered independent and unrelated. Calibration of procedures
Any reliabilitybased evaluation must be calibrated; i.e., tested against a
sufficient number of wellunderstood engineering problems to ensure that it provides reasonable results. Performance modes known to be problematical (such as
seepage) should be found to have a lower reliability than those for which problems are seldom observed; larger and more stable sections should be found to be
B11 ETL 11102556
28 May 99
more reliable than smaller, less stable sections, etc. This study provides a beginning point on such calibration studies by performing example analyses on two
hypothetical levee sections. As additional analyses are performed, by both
researchers and practitioners, on a wide range of real levee cross sections using
real data, it is inevitable that adjustments and refinements in the procedures will
be required. Application to economic analysis
When the developed functions are used in an economic analysis, one may
perceive a greater degree of precision than really exists, not unlike longterm
projections of uncertain costs and benefits. Users are cautioned that functions
developed using the presented methods still retain some inherent uncertainty in
the absolute sense. Nevertheless, they also contain more information than deterministic approaches to the same problem. The use of a consistent probabilistic
framework, with personal judgment checks for reasonableness, should have the
advantage and appeal of consistency when compared to the alternative method of
trying to identify a single flood elevation at which a levee changes from being
reliable to unreliable. Background
When the Corps of Engineers proposes construction of new flood control
levees or improvement of existing levees (typically by raising the height), economic studies are required to assess the relative benefits and costs of the work.
Where an existing levee is already present, the project benefits accrue from a
difference in the degree of protection. Economic assessment of the levee
improvement in turn requires an engineering determination of the probable level
of protection afforded by the existing levee. Past practice
In the past, existing levees that had not been designed or constructed to Corps
of Engineers' standards were sometimes, if not often, taken to be nonexistent in
economic analysis or taken to afford protection to some low and rather arbitrary
elevation. This is no longer permitted; costbenefit studies for water resource
projects are increasingly being cast in a probabilistic framework wherein it is
recognized that neither costs nor benefits have precise, predictable values, but
rather can assume a range of values associated with a range of likelihoods. Hence,
an existing levee is considered to afford protection with some associated
probability. B12 ETL 11102556
28 May 99
Current practice for navigation rehabilitation studies
For similar economic studies involving the rehabilitation of Corps’ navigation locks and dams, possible adverse events that would demand expenditures
(e.g. sliding of a lock monolith that would impede navigation) are now analyzed
in a probabilistic framework. Investments in rehabilitation work to forestall
adverse structural performance are evaluated based on the reliability of components, the probability of adverse performance, and the probable cost of the consequences. Several studies have been conducted to develop procedures (Wolff and
Wang 1992a, 1992b; Shannon and Wilson, Inc., and Wolff 1994) and to promulgate guidance (ETL 11102532, U.S. Army Corps of Engineers 1992) for probabilistic analysis of hydraulic structures. The Conditional Probability of Failure Function
For an existing levee subjected to a flood, the probability of failure Pf can be
expressed as a function of the floodwater elevation and other factors including
flood duration, soil strength, permeability, embankment geometry, foundation
stratigraphy, etc. This study will focus on developing the conditional probability
of failure function for the floodwater elevation, which will be constructed using
engineering estimates of the probability functions or moments of the other
relevant variables.
The conditional probability of failure can be written as: Prf Pr (failure
FWE) f (FWE, X1, X2, .. Xn) (1) In the above expression, the first term (denoting probability of failure) will be
used as a shorthand version of the second term. In the second term, the symbol
“” is read given and the variable FWE is the floodwater elevation. In the third
term, the random variables X1 through Xn denote relevant parameters such as soil
strength, permeability, top stratum thickness, etc. Equation 1 can be restated as
follows: “The probability of failure, given the floodwater elevation, is a function
of the floodwater elevation and other random variables.”
Two extreme values of the function can be readily estimated by engineering
judgment:
a. For floodwater at the same level as the landside toe (base elevation) of
the levee, Pf = 0.
b. For floodwater at or near the levee crown (top elevation), Pf Ú1.00. It may be argued that the probability of failure value may be something less
than 1.0 with water at the crown, as additional protection can be provided by
emergency measures. The question of primary economic interest, however, is the
shape of the function between these extremes. Quantifying this shape is the focus B13 ETL 11102556
28 May 99
of this study; how reliable might the levee be for, say, a 10 or 20year flood event
that reaches half or threequarters the height of the levee?
Reliability (R) is defined as: R 1 Pf (2) hence, for any floodwater elevation, the probability of failure and reliability must
sum to unity.
For the case of floodwater partway up a levee, R could be very near zero or
very near unity, depending on engineering factors such as levee geometry, soil
strength and permeability, foundation stratigraphy, etc. In turn, these differences
in the conditional reliability function could result in very different economic
scenarios. Four possible shapes of the reliability versus floodwater elevation are
illustrated in Figure 1.
As illustrated by these example curves, the conditional probability of failure
function could have a wide range of shapes. For a “good” levee, the probability of
failure may remain low and the reliability remain high until the floodwater elevation is rather high. In contrast, a “poor” levee may experience greatly reduced
reliability when subjected to even a small flood head. It is hypothesized that
some real levees may follow the intermediate curve, which is similar in shape to
the “good” case for small floods, but reverses to approach the “poor” case for
floods of significant height. Finally, a straight line function is shown in Figure 1,
representing a linear relation between reliability and flood height. Although such
a linear approximation is shown in current Corps guidance (Policy Guidance
Letter No. 26, U.S. Army Corps of Engineers 1991), linearity would not be
expected to be the general case. Figure 1. Possible reliability versus floodwater elevation functions for existing
levees B14 ETL 11102556
28 May 99 Study Approach
To assess the differences in benefits between an existing levee and a proposed improved levee, an economist desires the engineering assessment of the
levee reliability quantified in a probabilistic form such as Figure 1. However,
geotechnical engineers are commonly much better versed in deterministic methods than in probabilistic methods, and are generally more experienced and comfortable designing a structure to be safe with some appropriate conservatism than
when making numerical assessments of the condition of existing and perhaps
marginal structures. To provide some initial methodology for the latter problem,
the approach of this study is to:
a. Review the performance modes of concern to existing levees loaded by
floods and the related deterministic models for assessing performance.
b. Review the use of probabilistic methods in geotechnical engineering,
hydraulic structures, and related areas.
c. Recommend procedures for developing reliability curves or conditional
probability of failure functions similar to Figure 1 that are sufficiently
simple for use in practice with limited data and a modest level of effort,
but reflect a geotechnical engineer’s understanding of the underlying
mechanics and uncertainty in the governing parameters.
d. Test and illustrate the procedures through two comprehensive example
problems. B15 ETL 11102556
28 May 99 2 Current Corps of Engineers’
Guidance In this chapter, current Corps of Engineers' guidance regarding levee planning and design is reviewed in order to begin to define the component parts of,
and the constraints on, a probabilistic procedure to evaluate existing levees. One
policy letter has been issued which defines a beginning point for these studies:
Policy Guidance Letter No. 26, Benefit Determination Involving Existing
Levees (23 Dec. 1991).
A second document, Engineer Manual (EM) 111021913, Design and Construction of Levees (U.S. Army Corps of Engineers 1978), is the primary source
of Corps policy on the engineering aspects of levee design. However, probabilistic methods are not considered in this engineering manual. In addition to the EM,
there exists a voluminous collection of research reports, flood performance
reports, and Division regulations, (all developed by the Corps), as well as journal
papers and reference books, that deal with the analysis and design of levees. Policy Guidance Letter No. 26, Benefit
Determination Involving Existing Levees
(23 Dec 1991)
This letter sets forth the need (of the planner to receive from the engineer) for
a function relating levee reliability to floodwater elevation, or at least two points
on this function. Several specific items in the letter are especially relevant to the
present study. These are quoted below and followed by a commentary.
Quote: Investigations ... involving the evaluation ...of existing levees
and the related effect on the economic analysis shall use a systematic
approach to resolving indeterminate, or arguable, degrees of reliability.
Comment: This language sets forth the requirement for applying the
principles of reliability analysis to the problem. B16 ETL 11102556
28 May 99
Quote: Studies ...will focus on the sources of uncertainty ... surface
erosion, internal erosion (piping), underseepage, and slides...
Comment: This wording summarizes the most commonly expected
modes of adverse performance prior to overtopping. These will be
considered in the developed methods.
Quote: The question to be answered is: what percent of the time will a
given levee withstand water at height x?
Comment: This wording provides the specific requirement for
developing the conditional probability of failure function defined in
Chapter 1.
Quote: ...commands...(i.e. Corps district and division offices) making
reliability determinations should gather information to enable them to
identify two points... The highest vertical elevation on the levee such that
it is highly likely that the levee would not fail if the water surface would
reach this level... shall be referred to as the Probable NonFailure Point
(PNP)... The lowest vertical elevation on the levee such that it is highly
likely that the levee would fail... shall be referred to as the Probable
Failure Point (PFP).. As used here, "highly likely" means 85+ percent
confidence...
Comment: The definition of two specific points, the PNP and the PFP,
implies the assumption of linearity noted later in the letter. The defined
levels of reliability (0.85 / 0.15 and 0.15 / 0.85) assigned to these points,
along with illustrated definitions (Figure 2a), permit an economist, in the
absence of any further engineering analysis, to quantify reliability as a
linear function based on two points derived from engineering analysis or
engineers' intuition and judgment. The engineer needs only to, by some
means, identify floodwater elevations for which he or she considers the
levee to be 15 and 85 percent reliable.
Quote: The requirement that as the water surface height increases the
probability of failure increases, incorporates the reasonable assumption
that as the levee is more and more stressed, it is more and more likely to
fail.
Comment: While this would often be the case, it should be noted that
there may be some cases, notably riverside slope stability, where a levee
may be more reliable or safe when loaded with floodwater than before or
after flooding.
Quote: If the form of the probability distribution is not known, a linear
relationship as shown in the enclosed example, is an acceptable
approach for calculating the benefits associated with the existing levees. B17 ETL 11102556
28 May 99
Comment: The assumption of linearity is certainly expedient, and is the
leastbiased assumption in a case where two and only two points on a
function are known and no other information is present. However, the
assumption of linearity may or may not be acceptable once some
additional information is known. One of the objectives of this research is
to determine what is in fact a reasonable function shape based on the
results of some engineering analyses for typical levee cross sections and
typical parameter values.
The attachment to the Policy Guidance Letter provides an illustration of the
assumed linear conditional probability of failure function. In Figures 2a, 2b, and
2c, respectively, of this report are sketched the linear version, a trilinear version
that could be extended from the linear version, and the general curves from Figure 1. The latter have been redrawn to show Prf , the dependent variable, on the
yaxis. In the Policy Guidance Letter, the shape of the curve below the 0.15 value
and above the 0.85 value is not defined; the trilinear version shown is merely a
representation of one possible interpretation. It will be seen from the results of
the example analyses that the conditionalprobabilityoffailure functions
generally take the shape of the middle curve in Figure 2c and can be approximated by a piecewise linear approach using three or more pieces similar to
Figure 2b. EM 111021913, Design and Construction of
Levees
The current primary source of levee design guidance in the Corps of Engineers is EM 111021913, Design and Construction of Levees (U.S. Army Corps
of Engineers 1978). Guidance in EM 111021913 relevant to the reliability
assessment of existing levees includes the following:
a. Q tests (UU tests) are recommended for determining the strength of
foundation clays.
b. Q, R, and S tests (UU, CU, and CD tests) are recommended for determining strength of borrow materials compacted to water contents and
densities consistent with expected field compaction.
c. For familiar foundation conditions, undrained strength of finegrained
soil may be estimated from consolidation stresses and Atterberg limits
(c/p = f(PI)) and drained strength may be estimated from Atterberg limits
data (' = f(PI)) .
d. Strength of pervious soils is estimated from S (CD) tests on similar soils
or correlations such as those given by NAVFAC DM7.
e. Permeability of pervious soils is estimated from grain size information,
specifically D10 size.
B18 ETL 11102556
28 May 99 Figure 2. Alternative conditional probabilityoffailure functions
B19 ETL 11102556
28 May 99
f. Berms of 40ft width (riverside) to 100ft width 1 (landside) are recommended to be left at natural ground elevation between the levee and
borrow areas. g. At least 2 ft of impervious cover should be left over pervious materials
in borrow areas.
h. Although underseepage control is discussed, no criteria are given. The
reader is referred to TM 3424 (U.S. Army Corps of Engineers 1956).
i. Throughseepage and defensive works such as toe drains and internal
drains are described; however, no design criteria are presented and it is
noted that provision of such defenses is usually uneconomical. Underseepage and throughseepage for dams are discussed in EM 111021901, “Seepage Analysis and Control for Dams” (U.S. Army Corps of
Engineers 1986). A design procedure for toe berms to provide stability
against throughseepage for sand levees has been developed by Schwartz
(1976) and the Rock Island District.1 j. A 1V:2.5H slope is considered the steepest that can be maintained with
mowing equipment. k. Freeboard (crest height above design flood) is recommended to be at
least 2 ft in agricultural areas and 3 ft in urban areas, with additional
height in critical areas.
l. Crown width is recommended to be a minimum width of 10 to 12 ft for
floodfighting operations. m. Slope stability analyses may be in accordance with the Modified
Swedish Method or the wedge method from EM 111021902, or the
simpler Swedish Slide Method (ordinary method of slices). It would be
expected that current practice may also be to use Spencer's method from
computer programs UTEXAS2 or UTEXAS3 and not to use the simpler
Swedish Slide Method. In the EM, five stability cases are identified; of
these, Case I (endofconstruction) and Case V (earthquake) are not considered of interest for economic assessment of existing levees; the
remaining cases (sudden drawdown, intermediate river stage, and steady
seepage) are to be considered.
n. Embankment construction deficiencies leading to poor performance are
summarized in Table 72 of the EM. Relevant items include organic
material not stripped from the foundation, highly organic fill, excessively 1 A table of factors for converting nonSI units of measurement to SI units is presented on
page B11.
1 Personal Communication, 1993, S. Zaidi, U.S. Army Engineer District, Rock Island; Rock Island,
IL. B20 ETL 11102556
28 May 99
wet or dry fill, pervious layers through the embankment, and inadequate
compaction.
o. Erosion protection for riverside slopes is discussed in general terms,
but no quantitative criteria are given except where riprap is to be used,
where another EM is referenced. Components of an Improved Probabilistic
Assessment Procedure
The current guidance for assessing the reliability of existing levees essentially
consists of the following:
a. Using the template method and/or slope stability analysis to determine
stable slopes that meet accepted criteria.
b. Defining the PNP and PFP from these slope stability considerations
c. Adjusting the PNP and PFP, if necessary, by some judgmental means,
based on the sum total of information gleaned from the field inspection.
It is proposed that a more rational and consistent assessment procedure
should include the following components.
a. Develop a set of conditional probabilityoffailure versus floodwater
elevation functions, one for each of the following performance
considerations:
(1) Underseepage using established Corps methods (closedform equations or numerical methods such as program LEVEEMSU) and
engineering reliability analysis. Geometry may be based on field
surveys, minimal borings, and geologic experience; permeability
values may be based on correlations with grain size and experience.
(2) Slope stability for shortterm conditions, where undrained
strengths related to consolidation stresses are used for impervious
materials and drained strengths for pervious materials, using a slope
stability program and engineering reliability analysis. Strengths may
be based on field data where available or on correlations and experience for preliminary studies.
(3) Slope stability for longterm conditions, where flood duration is
expected to be sufficiently long that pore pressures adjust to flood
conditions, using drained strengths, infinite slope analysis or slope
stability programs, and engineering reliability analysis. Strengths
may be based on correlations and experience. B21 ETL 11102556
28 May 99
(4) Throughseepage leading to internal erosion (piping) or surface
erosion of the landside slope. For sand levees, several methods are
considered in Chapter 9; these can likely be further refined based on
additional studies. Results may be modified based on engineering
judgment and observations from the field inspection regarding materials, geometry, vegetation in the levee, crown width, likelihood of
animal burrows, cracks, roots, defects, etc.
(5) Surface erosion due to current and wave attack on the riverside
slope, using engineering judgment and observations from the field
inspection regarding soil cover, vegetative cover, river characteristics, wave exposure, etc. As techniques are further developed, these
analyses can be based on probabilistic definitions of current velocities, wave properties, and the properties of levee cover materials.
b. Systematically combine these functions into one composite
conditionalprobabilityof failure function for a given floodwater
elevation, using accepted methods from probability theory.
c. Using the results of steps a and b for a few selected levee reaches,
incorporate length effects to estimate the conditionalprobabilityoffailure function for the entire levee system.
Such a scheme will be developed and illustrated in Chapters 4 through 11.
Before doing so, related research work by others will be briefly reviewed in
Chapter 3. B22 ETL 11102556
28 May 99 3 Related Research Before developing the procedures and examples herein, a brief review of the
engineering literature on levees, their primary modes of performance (i.e., slope
stability, seepage, etc.), and the application of probabilistic methods thereto was
made to provide a basis for model development and to take advantage, if possible,
of previous work in the field. This section summarizes recent (within the past
20 years) work relevant to the topic. It is not intended to be a comprehensive
review of levee engineering. In making the review, it became clear that nearly all
work on levees and flood control embankments published in English derives from
the experiences of three sources: the Corps of Engineers in the United States,
Dutch engineers involved in sea dike construction, and Czech engineers involved
in protection from flooding along the Danube. Comprehensive Works
Peter (1982), in Canal and River Levees, provides the most complete and
recent reference book treatise on levee design, based on work in the former
Czechoslovakia. Notable among Peter’s work is a more uptodate and extended
treatment of mathematical and numerical modeling than in most other references.
(His numerical treatment of the underseepage problem was part of the inspiration
for the numerical approach used in LEVEEMSU.) Peter also considers underseepage safety as a function of particle size and size distribution, and not just
gradient alone. Although Peter’s work was not directly used in this study, it bears
consideration and rereview as the probabilistic approach to levee assessment is
further extended and developed by the Corps of Engineers.
Vrouwenvelder (1987), in Probabilistic Design of Flood Defenses, provides
a very thorough treatise on a probabilistic approach to the design of dikes and
levees in the Netherlands. At this time, the report does not have the status of a
code, but reviews the status of research activities and provides worked examples
illustrating how dike design can be cast as a risk management problem. Highlights of Vrouwenvelder's work potentially relevant to this effort include the
following: B23 ETL 11102556
28 May 99
a. It is recognized that exceedance frequency of the crest elevation is not
taken as the frequency of failure; there is some probability of failure for
lower elevations, and there is some probability of no failure or inundation
above this level if an effort is made to raise the protection.
b. A problemspecific review of probabilistic concepts such as event trees,
fault trees, reliability analysis (limit state, performance function, etc.),
and series and parallel systems is provided.
c. In his example, eleven parameters are taken as random variables
which are used in conjunction with relatively simple mathematical
physical models.
d. Performance modes considered are overflowing and overtopping,
macroinstability (deep sliding), microinstability (shallow sliding or
erosion of the landside slope due to seepage), and piping (as used,
equivalent to underseepage as termed by the Corps).
e. Aside from overtopping, piping (underseepage) is found to be the
governing mode for the section studied; slope stability is of little
significance to probability of failure.
f. Surface erosion due to wave attack or parallel currents is not
considered. g. For analysis of macroinstability (deep sliding), the Bishop method is
used, and previous data from Alonso (1976) is cited that indicates pore
pressure and cohesion dominate the uncertainty. This is consistent with
findings of this writer in the study of Corps' dams (Wolff 1985, 1989).
h. For analysis of microinstability (shallow landside sloughing), a limit
equilibrium derivation, essentially equivalent to the "infinite slope"
method of EM 111021902 (U.S. Army Corps of Engineers 1970) is
used.
i. For analysis of piping (underseepage), the Lane and Bligh creep ratio
approaches were originally used and then supplanted by an empirical
model test procedure that incorporates the D50 size and coefficient of
uniformity of the foundation sands. Research is under way toward the
development of a graintransport model and the consideration of timedependent effects. j. The "length problem" (longer dikes are less reliable than equivalent
short ones) is discussed. k. An example probabilistic design is provided for a 20kmlong river dike
constructed of sand with a cover of clay. Random variables include: B24 ETL 11102556
28 May 99
Water height and duration
Soil permeability k
Soil friction angle
Soil cohesion c'
Equivalent permeability of the (top blanket) clay k k.eq
Equivalent thickness of the (top blanket) clay dk.eq
Equivalent leakage factor of the clay facing eq = kk.eq / dk.eq
Model uncertainty factor for piping, based on Lane's creep ratio
l. The probabilistic procedure is aimed at optimizing the height and
slope angle of new dikes with respect to total costs for construction and
expected losses, including property and life. Macroinstability (slope
failure) of the inner slope was found to have a low risk, much less than
8 x 104 per year. Piping was found to be sensitive to seepage path
length; probabilities of failure varied but were several orders of magnitude higher (102 to 103 per year). Microinstability (landside sloughing
due to seepage) was found to have very low probabilities of failure.
Based on these results, it was determined that only overtopping and
piping need be considered in the combined reliability evaluation. Slope Stability
Termaat and Calle (1994) describe studies made to evaluate the shortterm
acceptable risk of slope failure of levees being reconstructed along rivers in
Holland. Using a slope stability analysis procedure (Calle 1985) that considers a
random field model of spatial fluctuation of shear strength combined with a
Bishop type slope stability model cast in a secondmoment probabilistic analysis,
the factor of safety is determined as a Gaussian random function in the direction
of the length of the levee. The expected value, standard deviation, and auto
correlation function for the factor of safety are determined by the random field
statistics of the shear strength functions. From these, estimates of the probability
of occurrence of a zone where the factor of safety is below 1.0 somewhere along
the slope axis can be obtained along with an indication of the width of such a
zone. The authors conclude that probabilities of failure for the endofconstruction condition are on the order of 1 in 200, which is consistent with the
findings of a number of other researchers. Although the spatial correlation
considerations used by Termaat, Calle and others are beyond the scope of this
preliminary study of levee reliability, these are important factors that should be
considered as the methodology is further developed. Underseepage, ThroughSeepage, and Piping
Calle et al. (1989), all with Delft Geotechnics in The Netherlands, developed
a probabilistic procedure for analyzing the likelihood of piping beneath sea dikes
and river levees. Whereas Corps models for underseepage (U.S. Army Corps of
Engineers 1956) are based on considerations of equilibrium necessary to initiate a
B25 ETL 11102556
28 May 99
sand boil, Calle's model considers the dynamic equilibrium necessary to
accelerate or terminate erosion and material movement once piping has initiated.
The latter phenomenon is related to the creep ratio, originally defined by Bligh
(1910) and Lane (1935). The critical creep ratio defines a limit state which
explicitly depends on geometrical and physical parameters of the aquifer and its
sand material. These parameters, which are modeled as seven random variables
and one deterministic variable, include the D10 and D70 grain sizes, the permeability, the length of the structure, and the soil friction angle. Using the HasoferLind (1974) reliability formulation, the reliability index can be calculated for a
levee and foundation system under consideration. This in turn is used to calculate the partial factors of safety on the creep ratio necessary to make the
probability of piping small relative to the annual risk of overtopping (1 in 12,500
for the Dutch structures considered). In doing so, it was found that creep ratios
on the order of twothirds those recommended by Bligh would provide adequate
reliability against uncontrolled movement of material. Multiple Modes of Failure
Duckstein and Bogardi (1981) applied reliability theory to levee design,
considering the combined effects of overtopping, boiling, slope sliding, and wind
wave erosion. However, specific models for geotechnical aspects such as boiling
or slope sliding are not developed in detail. Instead, each performance mode i is
characterized by a critical height Hi for which failure would occur, and the Hi
values are taken as a set of random variables. The combined probability is
obtained as a union of the conditional probabilities, similar in concept to the
scheme used in Chapter 11 of this report.
Duncan and Houston (1981) summarize studies performed for the
Sacramento District to estimate failure probabilities for California levees
constructed of a heterogeneous mixture of sand, silt, and peat, and founded on
peat of uncertain strength. Stability failure was analyzed using a horizontal
sliding block model driven by the riverside water load. The factor of safety is
expressed as a function of the shear strength, which is a random variable due to
its uncertainty, and the water level, for which there is a defined annual exceedance probability. Using elementary probability theory, values for the annual
probability of failure for 18 islands in the levee system were calculated by
numerically integrating over the joint events of high water levels and insufficient
shear strength. At this point, the obtained probability of failure values were
adjusted based on several practical considerations; first, they were normalized
with respect to length of levee reach modeled (longer reaches should be more
likely to have a failure) and secondly, they were adjusted from relative probability
values to more absolute values by adjusting them with respect to the observed
number of failures. These practical concepts are of significance to many or most
ongoing developments in applying probabilistic procedures to practical problems
by the Corps of Engineers. B26 ETL 11102556
28 May 99 4 Two Example Problems
Defined In this chapter, hypothetical levee cross sections for two example problems
are defined. These are considered to represent two points along a broad range of
levee problems that may be encountered by an engineer in practice. In subsequent chapters, these two sections will be used to illustrate analyses for slope
stability, seepage, and erosion. The examples involve:
a. A sand levee with a thin topsoil facing on a thin uniform clay top
stratum.
b. A clay levee on a thick, nonuniform clay top stratum.
For each example section, the semipervious clay top stratum is assumed to be
underlain by a thick pervious substratum. Problem 1: Sand Levee on Thin Uniform Clay Top
Stratum
Example problem 1 consists of a 20fthigh sand levee with 1V:2.5H side
slopes and a 20ftwide crown. It is founded on an 8ftthick clay top blanket
which is in turn underlain by an 80ftthick pervious sand substratum. The crown
width of 10 ft is between the 8ft and 12ft values corresponding to the PFP and
PNP templates. The 1V:2.5 slopes are steeper than recommended for either
template and represent a slope at the margin of maintainability. A levee section
for example problem 1 is shown in Figure 3. Problem 2: Clay Levee on Thick NonUniform Clay
Top Stratum
Example problem 2 consists of a 20fthigh clay levee with 1V:2H side slopes
and a 10ftwide crown. It is founded on a semipervious clay top blanket which
is 20 ft thick on the riverside of the levee. On the landside, the clay thickness
increases to 30 ft at the levee toe where a plugged channel parallels the levee.
B27 ETL 11102556
28 May 99 Figure 3. Levee section for example problem 1 Landside of the levee toe, the ground elevation drops 5 ft in 40 ft and the clay
blanket thins to 15 ft, creating a location for a potential seepage concentration
80 ft landside of the levee center line. The top stratum is underlain by a pervious
sand substratum extending to elevation 312.0. Figure 4 is a levee section for
example problem 2. Figure 4. Levee section for example problem 2 The crown width of 10 ft corresponds to the PNP template (PFP is 6 ft) and
the 1V:2H side slopes correspond to the PFP template and the margin of
maintainability. B28 ETL 11102556
28 May 99 5 Characterizing Uncertainty
in Geotechnical Parameters Introduction
The capacitydemand model, described in Annex A and used herein to
calculate probabilities of failure, requires that the engineer assign values for the
probabilistic moments of the random variables considered in analysis. This
chapter reviews information regarding the observed variability of geotechnical
parameters and can be used as a guide when characterizing random variables for
the analysis of levees.
Any parameter used in a geotechnical analysis can be modeled as a random
variable, and any variables that are expected to contribute uncertainty regarding
the expected performance of the structure or system should be so modeled.
Typically these include soil strength and soil permeability. In the Taylor’s Series
firstorder second moment (FOSM) approach used herein, random variables are
quantified by their expected values, standard deviations, and correlation
coefficients, commonly referred to as probabilistic moments. These moments are
defined in Annex A. Depending on the quantity and quality of available
information, values for probabilistic moments may be estimated in one of several
ways:
a. From statistical analysis of test data measuring the desired parameter.
b. From index test data which may be correlated to the desired parameter.
c. Simply based on judgment and experience where test data are not
available.
Each step from the top to the bottom in the above list implies increasing uncertainty. When designing a new structure, the move from using test data to using
index data or from using index data to using experience only would likely be
accompanied by an increase in the factor of safety or an adjustment in the value
of a design parameter (e.g. reducing the design strength). The corresponding
action in reliability analysis would be to assume a larger coefficient of variation. B29 ETL 11102556
28 May 99
Table 1 provides a summary of typical reported values for the coefficients of
variation of commonly encountered geotechnical parameters. More detailed comment regarding the observed variability of relevant parameters is provided in the
subsequent sections. Table 1
Coefficients of Variation for Geotechnical Parameters
Parameter Coefficient of
Variation, percent Unit weight 3
4 to 8 Drained strength of sand ' 3.7 to 9.3 12 Reference
Hammitt (1966),
cited by Harr (1987)
assumed by Shannon and Wilson, Inc., and
Wolff (1994)
Direct shear tests, Mississippi River Lock
and Dam No. 2, Shannon and Wilson, Inc.,
and Wolff (1994)
Schultze (1972), cited by Harr (1987) Drained strength of clay ' 7.5 to 10.1 S tests on compacted clay at Cannon Dam,
Wolff (1985) Undrained strength of clay su 40 Fredlund and Dahlman (1972) cited by Harr
(1987)
Assumed by Shannon and Wilson, Inc., and
Wolff (1994)
Q tests on compacted clay at Cannon Dam,
Wolff (1985) 30 to 40
11 to 45
Strengthtoeffective stress ratio
su / 'v 31 Clay at Mississippi River Lock and Dam
No. 2, Shannon and Wilson, Inc., and Wolff
(1994) Coefficient of permeability k 90 For saturated soils, Nielson, Biggar, and
Erh (1973) cited by Harr (1987) Permeability of top blanket clay
kb 20 to 30 Derived from assumed distribution,
Shannon and Wilson, Inc., and Wolff (1994) Permeability of foundation sands
kf 20 to 30 For average permeability over thickness of
aquifer, Shannon and Wilson, Inc., and
Wolff (1994) Permeability ratio kf / kb 40 Derived using 30% for kf and kb ; see
Annex B Permeability of embankment
sand 30 Assumed by Shannon and Wilson, Inc., and
Wolff (1994) Unit Weight of Soil Materials
The coefficient of variation of the unit weight of soil material is usually on the
order of 3 to 8 percent. In slope stability problems, uncertainty in unit weight
usually contributes little to the overall uncertainty, which is dominated by soil
strength. For stability problems, it can usually be taken as a deterministic variable
in order to reduce the number of random variables and simplify calculations. It B30 ETL 11102556
28 May 99
may, however, require consideration for underseepage problems, where the
critical exit gradient is directly proportional to the unit weight. Drained Strength of Sands
Reported coefficients of variation for the friction angle () of sands are in the
range of 3 to 12 percent. Lower values can be used where there is some
confidence that the materials considered are of consistent quality and relative
density, and the higher values should be used where there is considerable
uncertainty regarding material type or density. For the direct shear tests on sands
from Lock and Dam No. 2 cited in Table 1 (Shannon and Wilson, Inc., and Wolff
1994), the lower coefficients of variation correspond to higher confining stresses
and viceversa. Drained Strength of Clays
As the drained strength (') of clays is essentially a physical phenomenon
similar to the drained strength for sands, similar coefficients of variation (3 to
12 percent) would be expected. Evaluation of S test data on compacted clays at
Cannon Dam (Wolff 1985) showed coefficients of variation in the range of 7.5 to
10 percent.
A common method in practice to estimate drained strength is by correlation to
the plasticity index. Correlations developed by the Corps of Engineers are shown
in the engineering manual on design and construction of levees (U.S. Army Corps
of Engineers 1978). Holtz and Kovacs (1981) summarize correlations developed
by Kenney (1959), Bjerrum and Simons (1960) and Ladd et al. (1977). Using
such correlations, the observed variation in plasticity index for a clay deposit can
be combined with the observed data scatter of the correlations in order to estimate
coefficients of variation for drained strength parameters. Undrained Strength of Clays
Estimation from test results
Where undrained tests are available on soils considered to be “representative”
of a considered project area, the expected value and standard deviation of the
undrained strength, su or c, may be estimated directly from statistical analysis of
test data. An example is given in Table 2, which illustrates a statistical analysis of
unconfined compression test data furnished by the St. Louis District. The
resulting mean value and standard deviation of c, 1,234 and 798 lb/ft2,
respectively, might be rounded to the following estimated moments: B31 ETL 11102556
28 May 99
Expected value:
Standard deviation:
Coefficient of variation: E[c] = 1,200 lb/ft2
c
= 800 lb/ft2
Vc
= 66.7 percent Note, however, that the calculated coefficient of variation is very large, even
larger than typical values cited in Table 1. In the case considered, samples were
taken from a range of depths from about 2 to 20 ft, and hence had been consolidated under different effective overburden stresses. Where reasonable estimates
of consolidation stress can be made, the uncertainty can be reduced if the
undrained strength is normalized with respect to effective overburden stress as
described in the next section. However, for the St. Louis data, even a regression
analysis of strength versus sample depth did not reveal any trend. This suggests a
“mixed population” of samples from different soil formations. Smaller
coefficients of variation might be obtained if the soil samples can be separated
into different strata based on visual examination, index property tests, and an
understanding of the surficial geology. Estimation from test results and consolidation stress
Ladd et al. (1977) and others have shown that the undrained strength su (or c)
of clays with a given geologic origin can be “normalized” with respect to overburden stress (v) and overconsolidation ratio (OCR) and defined in terms of the
ratio su/v. Analysis of test data on clay under the overflow dike for Mississippi
River Lock and Dam No. 2 (Shannon and Wilson, Inc., and Wolff 1994) showed
that it was reasonable to characterize uncertainty in clay strength in terms of the
probabilistic moments of the su/v parameter. The ratio of su/v for 24 tested
samples was found to have a mean value of 0.35, a standard deviation of 0.11,
and a coefficient of variation of 31 percent. Permeability for Seepage Analysis
Permeability of foundation sands
Permeability of sand samples can vary quite considerably; coefficients of
variation of more than 100 percent have been reported. These large values are
apparently the result of analyzing the variability of sand permeability from sample
to sample. However, in an underseepage analysis, the variable of interest is not
the permeability at the location of a specific sample, but the average permeability
over the vertical extent of an aquifer at a selected cross section. For levee
underseepage investigations, it is common to perform grain size analyses and
obtain values for the D10 sizes at a number of points in a single boring. If these
are used to estimate a set of permeability values using standard correlations (e.g.,
U.S. Army Corps of Engineers 1956), the expected value of the average permeability over the depth of the aquifer at the boring site can be taken as the mean
value of the permeability estimates. The uncertainty in the average permeability
over the section is smaller than the uncertainty in the permeability at a random
point, and can be expressed as the standard error of the mean, which is the
B32 ETL 11102556
28 May 99
Table 2
Example Statistical Analysis of Undrained Tests on Clay, Unconfined Compression
Tests On Undisturbed Samples
(Wtest  Wavg)2 Wtest Wtest  Wavg VPC20191U T1 24.8 2.07 4.277376 750 484.0909 234,344 VPCS0191U T6 23.3 0.57 0.322831 1,600 365.9091 133,889.5 VPCS0191U T7B 25.5 2.77 7.662831 1,350 115.9091 VPCS0291U T2 20.5 2.23 4.981012 750 484.0909 234,344 VPCS0291U T3 21.2 1.53 2.346467 1,800 565.9091 320,253.1 VPCS0291U T4 20.5 2.23 4.981012 650 584.0909 341,162.2 VPCS0291U T6 20.4 2.33 5.437376 650 584.0909 341,162.2 VPCS0391U T3B 24.1 1.37 1.871921 2,500 1,265.909 1,602,526 VPCS0391U T4B 20.5 2.23 4.981012 2,250 1,015.909 1,032,071 VPCS0391U T5 21.9 0.83 0.691921 2,850 1,615.909 2,611,162 VPPS0291U T1 19.7 3.03 9.191921 2,750 1,515.909 2,297,980 VPPS0291U T3 18.5 4.23 17.90829 800 434.0909 VPGD0191U ST2 19.5 3.23 10.44465 1,350 115.9091 VPGD05091U ST1 23.9 1.17 900 334.0909 VPL1091U S1 19.5 3.23 1,350 115.9091 VPL1991U ST2 21.5 1.23 1.517376 500 734.0909 538,889.5 VPL1991U ST3 23.3 0.57 0.322831 400 834.0909 695,707.6 VPL1991U ST5 31.1 8.37 70.02647 250 984.0909 968,434.9 VPL2291U S1 17.6 5.13 26.33556 2,100 865.9091 749,798.6 VPL2291U S3 23.8 1.07 350 884.0909 781,616.7 VPL2291U S5 27.4 4.67 21.79192 450 784.0909 614,798.6 VPL2291U S7 31.6 8.87 78.64465 800 434.0909 188,434.9 Sum = 500.1 N= 22 W avg = 22.73 1.364649
10.44465 1.141012 286.6877
22
Var =
Std. Dev. =
N1=
Var =
Std. Dev. = 13.03126
3.610
21
13.6518
3.695 c (ccbar)2 Boring/Sample ccbar 27,150 188,434.9
13,434.92
111,616.7
13,434.92 14,026,932 22
Cbar = 1,234.0 13,434.92 22
Var =
Std. Dev. =
N1=
Var =
Std. Dev. = 637587.8
798.491
21
667949.1
817.282 B33 ETL 11102556
28 May 99
standard deviation of the sample values divided by the square root of the number
of samples: k k
n (3) From detailed analysis of a number of borings near Lock and Dam No. 25 on the
Mississippi River, the author (Shannon and Wilson, Inc., and Wolff 1994)
measured coefficients of variation for the average sand permeability on the order
of 20 to 30 percent. Permeability of top blanket clays
Although intact clays may have coefficients of permeability in the range 106
to 109 cm/sec, values used to model the global permeability of a semipervious top
stratum (kb) are typically much larger, commonly on the order of 104 cm/sec, to
reflect the effects of seepage through surface cracks, animal holes, and other
defects. As the appropriate values have traditionally been estimated semiempirically, using numbers backcalculated from observations during floods,
typical values of the coefficient of variation are not accurately known. For studies
of dikes along the Mississippi River, a coefficient of variation of 20 percent was
assumed (Shannon and Wilson, Inc., and Wolff 1994), based on judgmental
evaluation of the shape of trial probability distributions. For the underseepage
studies in Chapter 6, a coefficient of variation of 30 percent was assumed for the
top blanket. Permeability ratio
The residual head landside of a levee and hence the potential for piping or
boiling is in fact related to the ratio of the permeability of the pervious substratum
to the permeability of the top blanket, kf/kb , and not to the absolute value of either
permeability. If the expected values and standard deviations of the two
parameters are known, the expected value and permeability of the ratio can be
found as shown by example in Annex B. B34 ETL 11102556
28 May 99 6 Underseepage Analysis In this chapter, levee underseepage analyses are illustrated for the two
example problems defined in Chapter 4. The maximum exit gradient landside of
the levee is taken as the performance function, and the value of the critical
gradient, assumed to be 0.85, is taken as the limit state. As example problem 1
involves uniform foundation geometry, the classical methods of underseepage
analysis given in TM3424 (U.S. Army Corps of Engineers 1956a) are used to
calculate the exit gradient at the levee toe. For example problem 2, which has an
irregular foundation, the program LEVEEMSU (Wolff 1989) is used to calculate
the maximum value of the exit gradient along a cross section perpendicular to the
levee. Piezometric head profiles from these analyses are in turn used in the slope
stability analyses of the next chapter. Example Problem 1: Sand Levee on Thin Uniform
Clay Top Stratum
The levee cross section for example problem 1 was illustrated in Figure 3.
Four random variables are considered, the horizontal permeability of the pervious
substratum kf, the vertical permeability of the semipervious top blanket kb, the
thickness of the top blanket z, and the thickness of the pervious substratum d. The
assigned probabilistic moments for these variables are given in Table 3. Table 3
Random Variables for Example Problem 1
Coefficient of
Variation, Percent Parameter Expected Value Standard Deviation Substratum permeability, kf 1000 x 104 cm/sec 300 x 104 cm/sec 30 4 4 Top blanket permeability, kb 1 x 10 cm/sec 0.3 x 10 cm /sec 30 Blanket thickness, z 8.0 ft 2.0 ft 25 Substratum thickness, d 80 ft 5 ft 6.25 B35 ETL 11102556
28 May 99
The coefficients of variation of the top blanket and foundation permeability
values (each 30 percent) were assigned based on the typical values summarized
in Chapter 5.
As borings are not available at every possible cross section, there is some
uncertainty regarding the thicknesses of the soil strata at the critical location.
Hence, d and z are modeled as random variables. Their deviations are set to
match engineering judgment regarding the probable range of actual values. For
the blanket thickness z, assigning the standard deviation at 2.0 ft models a high
probability that the actual blanket thickness will be between 4.0 and 12.0 ft ( + 2
standard deviations) and a very high probability that the blanket thickness will be
between 2.0 and 14.0 ft ( + 3 standard deviations). For the aquifer thickness d,
the twostandarddeviation range is 70 to 90 ft and the threestandarddeviation
range is 65 to 95 ft. For analysis of real levee systems, it is suggested that the
engineer review the geologic history and stratigraphy of the area and assign a
range of likely strata thicknesses that are considered the thickest and thinnest
probable values. These can then be taken to correspond to ± 2.5 to 3.0 standard
deviations from the expected value.
As it is known that the exit gradient and stability against underseepage
problems are functions of the permeability ratio kf /kb and not the absolute
magnitude of the values, the number of calculations required for analyses can be
reduced by treating the permeability ratio as a single random variable. To do so,
it is necessary to determine the coefficient of variation of the permeability ratio
given the coefficient of variation of the two permeability values. In Annex B of
this report, example calculations are provided for three methods of calculating the
moments of functions of random variables: the Taylor’s series method with both
exact and approximate derivatives, and the point estimate method. Based on
these three examples, it appears reasonable to take the expected value of the
permeability ratio as 1,000 and its coefficient of variation as 40 percent. This
corresponds to a standard deviation of 400 for kf /kb.
To facilitate calculations, a spreadsheet (shown in Figure 5) was developed
that accomplishes the following:
a. Solves for the exit gradient using the methods in TM3424 (U.S. Army
Corps of Engineers 1956a).
b. Repeats the solution for seven combinations of the input parameters
required in the Taylor’s series method.
c. Determines the expected value and standard deviation of the exit gradient.
d. Calculates the expected value and standard deviation of the natural
logarithm of the exit gradient.
e. Calculates the probability that the exit gradient is above a critical value. B36 Figure 5. Spreadsheet for underseepage analysis of example problem 2 ETL 11102556
28 May 99 B37 ETL 11102556
28 May 99
Table 4
Problem 1, Underseepage Taylor's Series Analysis Water at
Elevation 420 (H = 20 ft)
Run kf /kb 1 1,000 2
3 600
1,400 4
5 1,000
1,000 6
7 z 1,000
1,000 Variance Percent of
Total Variance d ho i 8.0 80.0 9.357 1.170 8.0
8.0 80.0
80.0 9.185
9.451 1.148
1.181 0.000276 0.30 6.0
10.0 80.0
80.0 9.265
9.421 1.544
0.942 0.090606 99.69 8.0
8.0 75.0
85.0 9.337
9.375 1.167
1.172 0.000006 0.01 Total 0.090888 100.0 Results from the spreadsheet for a 20ft total head on the levee are summarized in
Table 4. The details of the calculations follow.
For the first analysis (Run 1), the three random variables are all taken at their
expected values. From TM3424, first the effective exit distance x3 is calculated
as: x3 kf
kb zd 1000 8 80 800 ft (4) As the problem is symmetrical, the distance from the riverside toes to the effective source of seepage entrance x1 is also 800 ft.
From the geometry of the given problem, the base width of the levee x2
is 110 ft.
The distance from the landside toe to the effective source of seepage entrance
is: s x1 x2 800 110 910 ft (5) The net residual head at the levee toe is: h0 Hx3 sx3 20 800 9.357 ft
910 800 And the landside toe exit gradient is:
B38 (6) ETL 11102556
28 May 99 i h0
z 9.357 1.170 (7) 8.0 For the second and third analyses, the permeability ratio is adjusted to the
expected value plus and minus one standard deviation while the other two variables are held at their expected values. These are used to determine the component of the total variance related to the permeability ratio: i (kf /kb ) 2kf /kb) ( 2 i i 2k /k
f 2f kb =
k b (8) i  i 2 2 = 1.181  1.148
2 2 = 0.000277 A similar calculation is performed to determine the variance components
contributed by the other random variables.
When the variance components are summed, the total variance of the exit
gradient is obtained as 0.090888. Taking the square root of the variance gives the
standard deviation of 0.301.
The exit gradient is assumed to be a lognormally distributed random variable
with probabilistic moments E[i] = 1.170 and i = 0.301. Using the properties of
the lognormal distribution described in Annex A, the equivalent normally
distributed random variable has moments E[ln i] = 0.124 and ln i = 0.254.
The critical exit gradient is assumed to be 0.85. The probability of failure is
then:
(9) Prf = Pr (ln i > ln 0.85) This probability was evaluated using a normal distribution function built into
the spreadsheet. It can be solved using standard tables by first calculating the
standard normalized variate z: z 1n icrit E[1n i] 1ni 0.16252 0.12449 1.132 (10) 0.253629 For this value, the cumulative distribution function F(z) is 0.129, and
represents the probability that the gradient is below critical. The probability that
the gradient is above critical is B39 ETL 11102556
28 May 99
(11) Prf = 1  F(z) = 1  0.129 = 0.871 Note that the z value is analogous to the reliability index , and it could be stated
that = 1.13.
The probability calculation is illustrated in Figure 6. The exit gradient is
taken to be lognormally distributed, making the natural log of the exit gradient
normally distributed. The expected value of ln i (0.124) exceeds the limit state
value (ln i = 0.163) by 0.287, or 1.132 standard deviations. The probability of
having an exit gradient above critical is the area shaded. For a normal distribution, the probability of a value less than 1.132 standard deviations below the
expected value or mean is 0.129; hence the probability of being above this point
is 0.871.
Once the spreadsheet was complete, the analysis could be readily repeated for
a range of heads on the levee from 0 to 20 ft. This was accomplished and the
resulting conditional probability of failure function was plotted as shown in
Figure 7. The shape of the function is similar to that suggested in Chapter 1. The
probability of failure is very low until the head on the levee exceeds about 8 ft,
after which it curves up sharply. It reverses curvature when heads are in the
range 14 to 16 ft and the probability of failure is near 50 percent. When the
floodwater elevation is near the top of the levee, the conditional probability of
failure approaches 87 percent. E[ln i] = 0.124 ln i crit = 0.163 ln i
1.132 sigma ln i
Figure 6. Calculation of probability of failure for underseepage The results of one intermediate calculation in the analysis are worthy of note.
As indicated by the relative size of the variance components shown in Table 4,
virtually all of the uncertainty is in the top blanket thickness. A similar effect was
found in other underseepage analyses by the writer reported in the Upper
Mississippi River report (Shannon and Wilson, Inc., and Wolff 1994); where the
B40 B41 ETL 11102556
28 May 99 Figure 7. Conditional probability of failure function: Underseepage for example problem 1 ETL 11102556
28 May 99
top blanket thickness was treated as a random variable, its uncertainty dominated
the problem. This has two implications:
a. Probability of failure functions for preliminary economic analysis might be
developed using a single random variable, the top blanket thickness z.
b. In expending resources to design levees against underseepage failure,
adding more data to the blanket thickness profile may be more justified
than obtaining more data on material properties. Example Problem 2: Clay Levee on Thick NonUniform Clay Top Stratum
Underseepage for example problem 2 was analyzed using the computer
program LEVEEMSU (Wolff 1989), which is capable of analyzing irregular
foundation geometry. Random variables were assigned the probabilistic
moments shown in Table 5.
The permeability ratio kf /kb was modeled in LEVEEMSU by setting the top
stratum permeability to 1 x 104 cm/sec and analyzing the foundation permeability
at values of 1,000 x 104, 600 x 104, and 1400 x 104 cm/sec for the expected
value, plus one standard deviation, and minus one standard deviation analyses,
respectively.
Table 5
Random Variables for Example Problem 2
Parameter Expected
Value Standard
Deviation Coefficient of
Variation Permeability ratio, kf/kb 1,000 40 40% Blanket thickness, z As shown in
Figure 6 2.0 ft NA Base of substratum elevation 312.0 5 ft NA Uncertainty in the blanket thickness was modeled by specifying the base of
the blanket profile as shown in Figure 4 for the expected value and then moving it
up and down 2 ft. This implies that the top blanket is assumed to be of the
general shape shown and that there is a high probability that the blanket thickness
is within + 4 ft of the thickness shown and a very high probability that it is within
+ 6 ft of the thickness shown. B42 ETL 11102556
28 May 99
Uncertainty in the base of the pervious substratum was likewise modeled by
specifying it as shown and then moving it up and down 5 ft. This implies that
there is a high probability that the base of the substratum is between elevation
302 and 322 (two standard deviations), and a very high probability that it is
between elevations 297 and 327 (three standard deviations).
Results of the analyses for the maximum 20ft head on the levee are as shown
in Table 6.
A spreadsheet similar to that for problem 1 was developed to perform
probability of failure calculations (Figure 8). For the maximum head of 20 ft on
the levee, the expected value of the maximum exit gradient is 0.718 and its
standard deviation is 0.0898. This corresponds to a probability of failure of
0.078, or almost 8 percent.
For lesser heads on the levee, it was assumed that the exit gradient is linear
with respect to levee head, and the same spreadsheet was used with scaled exit
gradient values (Figures 9 through 11) to calculate the probability of failure for
lesser heads. At a 17.5ft head, the probability of failure drops to 0.006, and at a
15ft head, to 0.000097.
Table 6
Problem 2, Underseepage Taylor's Series Analysis Water at
Elevation 420 (H = 20 ft)
Run kf /kb z Base of
Substratum ho
at toe I max 1 1000 E[z] 312.0 2 600 E[z] 312.0 .729 3 1400 E[z] 312.0 .699 4 1000 +2.0 312.0 .640 5 1000 2.0 312.0 .817 6 1000 E[z] 317.0 .715 7 1000 E[z] 307.0 .721 Variance Percent of
Total Variance .718
.000225 2.8 .007832 97.1 .000009 0.1 .008066 100.0 Total The conditional probability of failure versus floodwater elevation is shown in
Figure 12.
As was previously observed for example problem 1, examination of the variance terms indicates that virtually all of the uncertainty in the levee performance
with respect to underseepage traces to uncertainty in the thickness of the top
blanket: the thicker the top blanket or the more certain one is regarding the
thickness of the blanket, the more reliable the levee can be considered. B43 ETL 11102556
28 May 99 B44
Figure 8. Spreadsheet for underseepage analysis of example problem 2 (H = 20 ft) Figure 9. Spreadsheet for underseepage analysis of example problem 2 (H = 17.5 ft) ETL 11102556
28 May 99 B45 ETL 11102556
28 May 99 B46
Figure 10. Spreadsheet for underseepage analysis of example problem 2 (H = 15 ft) Figure 11. Spreadsheet for underseepage analysis of example problem 2 (H = 12.5 ft) ETL 11102556
28 May 99 B47 ETL 11102556
28 May 99 B48
Figure 12. Conditional probability of failure function: Underseepage for example problem 2 ETL 11102556
28 May 99 7 Slope Stability Analysis for
ShortTerm Conditions In this chapter, slope stability analyses are illustrated for the two example
problems defined in Chapter 4 assuming undrained conditions prevail in the clay
soils present in the profiles. This in turn implies that pore pressure conditions in
the clay are dependent only on initial conditions prior to a flood and pore pressure
changes due to shear, and that pore pressures have not equilibrated with flood
water to develop steadystate seepage conditions in clay soils. These assumptions
are consistent with shortterm flood loadings. Slope stability analyses were
performed using the computer program UTEXAS2 (Edris and Wright 1987). For
the cases analyzed, similar results would be expected with the more recent
program UTEXAS3. Example Problem 1: Sand Levee on Thin Uniform
Clay Top Stratum
Problem modeling
The levee cross section for example problem 1 was illustrated in Figure 3.
For slope stability analysis, three random variables were defined; these variables,
along with their assigned probabilistic moments, are summarized in Table 7. Table 7
Random Variables for Example Problem 1
Parameter Expected
Value Standard
Deviation Coefficient
of Variation Friction angle of sand levee embankment, emb 30 deg 2 deg 6.7% Undrained strength of clay foundation, c or su 800 lb/ft2 320 lb/ft2 40% Friction angle of sand foundation, found 34 deg 2 deg 5.9% B49 ETL 11102556
28 May 99
For slope stability analysis, the piezometric surface in the embankment sand was
approximated as a straight line from the point where the floodwater intersects the
riverside slope to the landside levee toe. For the internal erosion and throughseepage analyses in Chapter 9, this assumption is refined using Casagrande
s
basic parabola solution. The piezometric surface in the foundation sands was
taken as that obtained for the expected value condition in the underseepage
analysis reported in Chapter 6. If desired, the piezometric surface could be
modeled as an additional random variable using the probabilistic moments of the
residual head developed from the underseepage analysis. Results
Using the Taylor
s Series  Finite Difference method described in Annexes A
and B, seven runs of the slope stability program are required for each floodwater
level considered; one for the expected value case, and two runs to determine the
variance component of each random variable. For the first water elevation
considered (el. 400, or water at the natural ground surface), eleven runs were in
fact made as several starting centers for the circular search option were checked
to ensure that the critical failure surface was found. The results of the required
seven runs are summarized in Table 8. Table 8
Problem 1, Undrained Slope Stability, Taylor's Series
Analysis Water at Elevation 400 (H = 0 ft)
levee c
clay Run found FS 12 32 800 34 1.568 4 30 800 34 1.448 5 34 800 34 1.693 6 32 480 34 1.365 7 32 1120 34 1.568 8 32 800 32 1.568 9 32 800 36 1.567 Total Variance Percent of
Total Variance 0.015 006 59.29 0.010 302 40.71 2.5 x 107 0.00 0.025 309 100.0 The results for all runs for all water elevations are summarized in Table 9.
Critical failure surfaces for the cases of floodwater at elevation 400, 410, and 420
are illustrated in Figures 13 through 15. The reliability index and probability of
failure for each water elevation were calculated using the spreadsheet templates
illustrated in Figures 16 through 21. The resulting conditional probability of
failure function is illustrated in Figure 22 and enlarged in Figure 23. B50 Table 9
Problem 1, Undrained Slope Stability, Results for All Runs
Material Properties Initial Values Final Critical Surface Initial Values Final Critical Surface Run # (Emb) Water
c (clay) (Fnd) Elev X 1A 32 800 34 400 50 450 400 1.568 63 444 400 50 450 390 2.009 31.6 432.6 392 4A 30 800 34 400 50 450 400 1.449 63 444 400 50 450 390 1.448 41.6 461.4 409.2 5A 34 800 34 400 50 450 400 1.693 63 444 400 50 450 390 2.042 31.6 432.4 392 6A 34 480 34 400 50 450 400 1.568 63 444 400 50 450 390 1.365 31.4 431.4 392 7A 32 1120 34 400 50 450 400 1.568 63 444 400 50 450 390 2.548 41.2 439.6 388 8A 32 800 32 400 50 450 400 1.568 63 444 400 50 450 390 2.150 41.2 441 389.6 11A 32 800 36 400 50 450 400 1.568 63 444 400 50 450 390 1.567 41.6 461.4 409.2 12A 32 800 34 410 50 450 400 1.568 63 444 400 50 450 390 1.961 43.4 442.2 388.4 13A 30 800 34 410 50 450 400 1.449 63 444 400 50 450 390 1.936 43.6 443 388.6 14A 34 800 34 410 50 450 400 1.449 63 444 400 50 450 390 1.987 43.2 441.8 388.4 15A 32 480 34 410 50 450 400 1.693 63 444 400 50 450 390 1.332 31.2 431.6 392 16A 32 1120 34 410 50 450 400 1.568 63 444 400 50 450 390 2.248 43 440.2 386.4 17A 32 800 32 410 50 450 400 1.568 63 444 400 50 450 390 1.897 43.2 441.4 388 18A 32 800 36 410 50 450 400 1.568
63 444 400 50 450 390 2.023 43.6 443.4 389 19A 32 800 34 405 50 450 400 1.568 63 444 400 50 450 390 2.105 42.4 442 389.2 20A 30 800 34 405 50 450 400 1.449 63 444 400 50 450 390 2.077 42.4 442.4 389.4 21A 34 800 34 405 50 450 400 1.693 63 444 400 50 450 390 2.134 42.2 441.4 389.2 22A 32 480 34 405 50 450 400 1.568 63 444 400 50 450 390 1.359 31.4 431.4 392 23A 32 1120 34 405 50 450 400 1.568 63 444 400 50 450 390 2.410 42 439.8 387.2 24A 32 800 32 405 50 450 400 1.568 63 444 400 50 450 390 2.038 42.2 440.8 388.6 25A 32 800 36 405 50 450 400 1.568 63 432.6 400 50 450 390 2.169 42.4 442.6 389.8 Y Tang FS X Y Tang X Y Tang FS X Y Tang B51 ETL 11102556
28 May 99 (Continued) Material Properties Initial Values Final Critical Surface Initial Values Final Critical Surface Water
Run # (Emb) c (clay) (Fnd) Elev X Y Tang FS X Y Tang X Y Tang FS X Y Tang 26A 32 800 34 415 50 450 400 1.502 50 452 400 50 450 390 1.774 44.6 443.6 387.8 27A 30 800 34 415 50 450 400 1.387 50 452.8 400 50 450 390 1.753 44.6 444.4 388 28A 34 800 34 415 50 450 400 1.584 48.6 454 400 50 450 390 1.794 44.6 443.2 387.6 29A 32 480 34 415 50 450 400 1.502 50 452.8 400 50 450 390 1.420 45.8 445.8 390 30A 32 1120 34 415 50 450 400 1.502 50 452.8 400 50 450 390 2.049 44 441.6 386 31A 32 800 32 415 50 450 400 1.502 50 452.8 400 50 450 390 1.717 44.4 442.4 387.2 32A 32 800 36 415 50 450 400 1.502 50 452.8 400 50 450 390 1.829 45 444.8 388.2 33A 32 800 34 420 50 450 400 1.044 50 454 400 50 450 390 1.504 46.8 449.2 387.4 34A 30 800 34 420 50 450 400 0.995 50.8 452.8 400 50 450 390 1.490 47.0 449.4 387.4 35A 34 800 34 420 50 450 400 1.162 50.8 452.8 400 50 450 390 1.515 46.4 448.6 387.4 36A 32 480 34 420 50 450 400 1.044 50 454 400 50 450 390 1.158 48.8 454.8 389.4 37A 32 1120 34 420 50 450 400 1.044 50 454 400 50 450 390 1.775 45.4 445.6 385.8 38A 32 800 32 420 50 450 400 1.044 50 454 400 50 450 390 1.457 46.2 448 387 39A 32 800 36 420 50 450 400 1.044 50 454 400 50 450 390 1.546 47.2 450.4 387.8 50A 32 800 34 417.5 50 450 400 1.339 50.0 444.4 400.0 50 450 390 1.601 45.0 444.4 387.0 51A 30 800 34 417.5 50 450 400 1.237 50.0 444.4 400.0 50 450 390 1.636 45.6 445.8 387.6 52A 34 800 34 417.5 50 450 400 1.446 50.0 444.8 400.0 50 450 390 1.670 45.2 444.8 387.4 53A 32 480 34 417.5 50 450 400 1.339 50.0 444.4 400.0 50 450 390 1.307 47.0 449.4 389.6 54A 32 1120 34 417.5 50 450 400 1.339 50.0 444.4 400.0 50 450 390 1.926 44.6 442.8 385.8 55A 32 800 32 417.5 50 450 400 1.339 50.0 444.4 400.0 50 450 390 1.601 45.0 444.4 387.0 56A 32 800 36 417.5 50 450 400 1.339 50.0 444.4 400.0 50 450 390 1.703 45.8 446.6 388.0 ETL 11102556
28 May 99 B52 Table 9 (Concluded) Figure 13. Failure surfaces for example problem 1, water elevation = 400 ETL 11102556
28 May 99 B53 ETL 11102556
28 May 99 B54
Figure 14. Failure surfaces for example problem 1, water elevation = 410 Figure 15. Failure surfaces for example problem 1, water elevation = 420 ETL 11102556
28 May 99 B55 ETL 11102556
28 May 99 B56
Figure 16. Reliability calculations for undrained slope stability, example problem 1, water height = 0, water elevation = 400 Figure 17. Reliability calculations for undrained slope stability, example problem 1, water height = 5, water elevation = 405 ETL 11102556
28 May 99 B57 ETL 11102556
28 May 99 B58
Figure 18. Reliability calculations for undrained slope stability, example problem 1, water height = 10, water elevation = 410 Figure 19. Reliability calculations for undrained slope stability, example problem 1, water height = 15, water elevation = 415 ETL 11102556
28 May 99 B59 ETL 11102556
28 May 99 B60
Figure 20. Reliability calculations for undrained slope stability, example problem 1, water height = 17.5, water elevation = 417.5 Figure 21. Reliability calculations for undrained slope stability, example problem 1, water height = 20, water elevation = 420 ETL 11102556
28 May 99 B61 ETL 11102556
28 May 99 B62
Figure 22. Conditional probability function for undrained slope failure, example problem 1 Figure 23. Conditional probability function for undrained slope failure, example problem 1, enlarged view ETL 11102556
28 May 99 B63 ETL 11102556
28 May 99
A discontinuity in Prf is observed as the flood height is increased from 10 ft to
15 ft; Prf abruptly decreases, then begins to rise again. This illustrates an
interesting facet of probability analysis; Prf is a function not only of the expected
values of the factor of safety and the underlying parameters, but also of their
coefficients of variation. In the present case, at a flood height between 10 ft and
15 ft, some of the critical surfaces move from the foundation clay, with a high
coefficient of variation for its strength, to the embankment sands, for which the
coefficient of variation is smaller. This decreases and Prf . Even though the
safety factor may decrease as the flood height increases, if the value of the smaller
safety factor is more certain, due to the lesser strength uncertainty, Prf may
decrease. Example calculation of probability values
The calculation of the probability values for the case of water at elevation 400
is summarized as follows.
The expected value of the factor of safety is the factor of safety calculated using
the expected values of all variables:
E[FS] = 1.568 (12) The variance of the factor of safety, calculated in the same manner as previously
illustrated for the exit gradient in underseepage in the previous chapter, is:
Var[FS] = 0.025309 (13) and the standard deviation of the factor of safety is: FS = 0.159 (14) While the factor of safety is expected to be adequate (1.568), its exact value is
uncertain. The factor of safety is assumed to be a lognormally distributed random
variable with E[FS] = 1.568 and FS = 0.159. From the properties of the
lognormal distribution given in Annex A, VFS FS
E[FS] 0.159 21.568 0.1015 2
InFS 1n(1 VFS) 1n(1 0.10152) 0.1012 B64 (15) (16) ETL 11102556
28 May 99 E[1nFS] 1n E[FS] 2nFS
1
2 1n1.568 0.0102 0.4447
2 (17) The reliability index is then: E [1n FS] 1nFS 0.447 0.1012 4.394 (18) From the cumulative distribution function of the standard normal distribution
evaluated at , the conditional probability of failure for water at elevation 400 is:
Prf = 6 x 106 (19) The calculation of the reliability index is illustrated in Figure 24. E[ln FS] = 0.445
limit state
ln 1 = 0.0
Pr(f) = 0.000006
ln FS
ln FS
Figure 24. Calculation of probability of failure for slope stability Interpretation
Note that the calculated probability of failure infers that the existing levee is
taken to have approximately a six in one million probability of not being stable
under the condition of floodwater to its base elevation of 400, even though it may
in fact be existing and observed stable under such conditions. The capacitydemand / reliability index model was developed for the analysis of yetunconstructed structures. When applied to existing structures, it will provide
probabilities of failure greater than zero. This can be interpreted as follows:
given a large number of different levees, each with the same geometry and with
the variability in the strength of their soils distributed according to the same
B65 ETL 11102556
28 May 99
density functions as those assigned by the engineer to characterize uncertainty in
the soil strength, about six in one million of those levees might be expected to
have slope stability problems. The expression of reliability of existing structures
in this manner provides a consistent probabilistic framework for use in economic
evaluation of improvements to those structures. Discussion
The results of the probabilistic analyses are summarized in Table 10.
Table 10
Problem 1, Slope Stability for ShortTerm Conditions, Summary of
Probabilistic Analyses
Water Elevation E[FS] FS Prf 400.0 1.568 0.159 4.394 6 x 106 405.0 1.568 0.161 4.351 7 x 106 410.0 1.568 0.170 4.114 1.9 x 105 415.0 1.502 0.107 5.699 6 x 109 420.0 1.044 0.084 0.499 0.3087 As would be expected, the anticipated value of the factor of safety decreases
with increasing floodwater elevation. Contrary to what might be expected, the
reliability index increases and the probability of failure decreases with increasing
floodwater elevation until the floodwater exceeds elevation 415.0, or three
quarters the levee height. This occurs because the uncertainty in the factor of
safety decreases along with the expected value, and the probability of failure
reflects both measures. Although the factor of safety becomes smaller as the
floodwater rises, its value becomes more dependent on the shear strength of the
embankment sands and less dependent on the shear strength of the foundation
clays. This is evident in Figure 14 where the failure surface moves down into the
foundation clay for the case of weak clay, and in Figure 15 where the failure
surfaces move up into the embankment sand for all cases. As there is more
certainty regarding the strength of the sand (the coefficient of variations are about
6 percent versus 40 percent for the clay), this amounts to saying that a sand
embankment with a low factor of safety can be more reliable than a clay
embankment with a higher factor of safety. Similar findings were observed by
Wolff (1985) and others.
Review of the relative magnitudes of the variance components indicates that
40 to 48 percent of the problem uncertainty is related to the shear strength of the
foundation clay, until the floodwater elevation exceeds 415, at which the contribution of the foundation clay abruptly drops to about 15 percent and then continues to drop as the embankment sand becomes the dominant random variable. B66 ETL 11102556
28 May 99 Example Problem 2: Clay Levee on Thick Irregular
Clay Top Stratum
Problem modeling
The levee cross section for example problem 2 was illustrated in Figure 4.
For slope stability analysis, four random variables were defined; these variables
along with their assigned probabilistic moments are shown in Table 11.
Table 11
Random Variables for Example Problem 2
Parameter Expected
Value Standard
Deviation Coefficient
of Variation Undrained strength of clay levee, c or su 800 lb/ft2 240 lb/ft2 30% 2 50 lb/ft 2 Undrained strength at top of clay foundation, c or
su (CPROFL) 500 lb/ft 10% Rate of increase of undrained strength of clay
foundation, (RATEIN) 18 lb/ft2/ft 2 lb/ft2/ft 11% Friction angle of sand foundation, +found 34 deg 2 deg 5.9% The linearly varying strength option of UTEXAS2 was used to model strength
of the clay foundation. The variable CPROFL models the undrained strength at
the top of the clay foundation and the variable RATEIN models the rate of
increase of the undrained strength with respect to depth. Combination of these
two parameters permits the uncertainty in strength to increase with depth.
Coefficients of variation were chosen to give a reasonable value for the total
uncertainty. Waterfilled cracks were specified to a depth of 2c/, where the
value of c was runspecific.
The piezometric surface in the foundation sands was taken as that obtained
for the expected value condition in the underseepage analysis reported in
Chapter 6. Results
The results for all runs for all water elevations are summarized in Table 12.
Critical failure surfaces for the cases of floodwater at elevations 400 and 420 are
illustrated in Figures 25 and 26. Calculation of the reliability index and
probability of failure for each water elevation were accomplished using the
spreadsheet templates illustrated in Figures 27 and 28. The resulting conditional
probability of failure function is illustrated in Figure 29 and enlarged in
Figure 30. B67 ETL 11102556
28 May 99 B68 Table 12
Problem 2, Undrained Slope Stability, Results for All Runs
Run #
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118 c(levee) Material Properties
c(profl)
rate of c 800
560
1040
800
800
800
800
800
800
800
560
1040
800
800
800
800
800
800 500
500
500
450
550
500
500
500
500
500
500
500
450
550
500
500
500
500 18
18
18
18
18
16
20
18
18
18
18
18
18
18
16
20
18
18 +(Found) 34
34
34
34
34
34
34
32
36
34
34
34
34
34
34
34
32
36 Water
Elevation
400
400
400
400
400
400
400
400
400
420
420
420
420
420
420
420
420
420 Xstart
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30 RunTime Seed Values
Ystart
Ytanin
430
430
430
430
430
430
430
430
430
430
430
430
430
430
430
430
430
430 392
380
360
392
392
392
392
392
380
360
392
392
392
392
392
392
392
392 Ylimit
300
300
300
300
300
300
300
300
300
300
300
300
300
300
300
300
300
300 FS
1.525
1.405
1.615
1.425
1.625
1.490
1.558
1.525
1.525
1.517
1.401
1.603
1.419
1.616
1.481
1.552
1.517
1.517 Final Critical Surface
XCenter
Ycenter
31.0
28.6
33.4
30.8
31.2
32.0
30.4
31.0
31.0
31.2
28.6
33.8
31.0
31.6
32.4
30.2
31.2
31.2 430.8
432.6
430.4
430.6
430.8
430.6
430.6
430.8
430.8
433.2
434.0
433.8
433.2
433.4
433.8
433.4
433.2
433.2 Radius
43.6
40.6
47.2
43.6
43.8
45.0
42.6
43.6
43.6
47.4
42.4
52.6
47.0
48.0
49.4
46.0
47.4
47.4 Figure 25. Failure surfaces for example problem 2, water elevation = 400 ETL 11102556
28 May 99 B69 ETL 11102556
28 May 99 B70
Figure 26. Failure surfaces for example problem 2, water elevation = 420 Figure 27. Reliability calculations for undrained slope stability, example problem 2, water height = 0, water elevation = 400 ETL 11102556
28 May 99 B71 ETL 11102556
28 May 99 B72
Figure 28. Reliability calculations for undrained slope stability, example problem 2, water height = 20, water elevation = 420 Figure 29. Conditional probability function for undrained slope failure, example problem 2 ETL 11102556
28 May 99 B73 ETL 11102556
28 May 99 B74
Figure 30. Conditional probability function for undrained slope failure, example problem 2, enlarged view ETL 11102556
28 May 99
Discussion
As none of the critical failure surfaces for problem 2 for any of the analysis
cases cut into the underlying foundation sands, all of the probability of failure
values are low, on the order of 106, and are essentially insensitive to floodwater
elevation. This is in general agreement with engineering experience; failures of
clay slopes are not, in general, related to pool level during the time of inundation.
They may, however, be related to pore pressures remaining in an embankment
after a flood has receded. B75 ETL 11102556
28 May 99 8 Slope Stability Analysis for
LongTerm Conditions “Longterm conditions” are defined as the conditions prevailing at the time
when any excess pore pressures due to shear have had sufficient time to dissipate,
and stability analyses may be modeled using drained strength parameters in both
clay and sand. No examples for slope stability analysis using drained strength
parameters for clays are presented in this report. In general, levees subjected to
flood loadings would be expected to be loaded for a sufficiently short time that
undrained conditions would prevail in clayey materials. Where it is considered
that flood durations could be of long enough duration that drained (steady
seepage) conditions could develop in clayey embankments or foundations,
analyses similar to those in Chapter 7 could be performed. Alternatively, the
Taylor’s series method could be applied to the infinite slope method of analysis.
As the coefficients of variation for drained strength parameters are typically
considerably smaller than those for undrained strength parameters, the probability
of failure would be expected to be less than for the undrained case. Wolff (1985)
(also cited in Harr (1987)) showed that for welldesigned dam embankments, the
probability of failure for longterm, steady seepage conditions analyzed using
drained strengths can be several orders of magnitude lower than for shortterm
(after construction) conditions analyzed using undrained strengths. B76 ETL 11102556
28 May 99 9 ThroughSeepage Analysis Introduction
Definition
Three types of internal erosion or piping can occur as a result of seepage
through a levee:
a. If there are cracks in the levee due to hydraulic fracturing, tensile stresses,
decay of vegetation, or animal activity along the contours of hydraulic
structures, etc., where the water will have a preferential path of seepage,
piping may occur. For piping to occur, the tractive shear stress exerted by
the flowing water must exceed the critical tractive shear stress of the soil.
b. High exit gradients on the downstream face of the levee may cause piping
and possible progressive backward erosion. This is the same phenomenon
which was addressed in Chapter 6 and piping occurs when the exit
gradient exceeds the critical exit gradient.
c. Internal erosion (suffusion) or removal of fine grains by excessive seepage
forces may occur. This type of piping occurs when the seepage gradient
exceeds a critical value. Design practice
Quantitative erosion analyses are not routinely performed for levee design in
the Corps of Engineers, although erodibility is implicity considered in the
specification of erosionresistant embankment materials. For design of sand
levees, the procedures used by the Rock Island District based on research by
Schwartz (1976) do include some elements of erosion analysis. However, the
result of the method is to determine the need for providing toe berms according to
a semiempirical criterion rather than to directly determine the threshold of
erosion conditions or predict whether erosion will occur. Presumably, some
conservatism is present in the berm criteria and thus the criteria do not represent a
true limit state. Wellconstructed clay levees are generally considered resistant to
internal erosion, but such erosion can occur where there is a preexisting crack,
B77 ETL 11102556
28 May 99
defect, or discontinuity and the clay is erodible or dispersive under the effect of a
locally high internal gradient. Observed erosion problems in clay embankments
have occurred in cases such as poor compaction around drainage culverts and
where dispersive clays are present. Deterministic models
There is no single widely accepted analytical technique or performance
function in common use for predicting internal erosion. As probabilistic analysis
requires the selection of such a function upon which to calculate probability
values, it will be necessary to choose one or two for purposes of illustration
herein. Review of various erosion models indicates that erodibility is taken to be
a function of some set of the following parameters:
a. Permeability or hydraulic conductivity k.
b. Hydraulic gradient i.
c. Porosity n.
d. Critical stress c (the shear stress required for flowing water to dislodge a
soil particle).
e. Particle size, expressed as some representative size such as D50 or D85.
f. Friction angle 1 or angle of repose.
Essentially, the analyses use the gradient, critical tractive stress, and particle
size to determine whether the shear stresses induced by seepage head loss are
sufficient to dislodge soil particles, and use the gradient, permeability, and
porosity to determine whether the seepage flow rate is sufficient to carry away or
transport the particles once they have been dislodged. Grain size and pore size
information may also be used to determine whether soils, once dislodged, will
continue to move (piping) or be caught in the adjacent soil pores (plugging).
It is commonly known that very fine sands and siltsized materials are among
the most erosionsusceptible soils. This arises from their having a critical balance
of relatively high permeability, low particle weight, and low critical tractive
stress. Particles larger than fine sand sizes are generally too heavy to be moved
easily, as particle weight increases with the cube of size. Particles smaller than
silts (i.e., clay sizes), although of light weight, may have relatively large electrochemical forces acting on them, which can substantially increase the critical
tractive stress c, and also have sufficiently small permeability as to inhibit
particle transport in significant quantity. B78 ETL 11102556
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The models considered herein to illustrate probabilistic erosion analysis are:
a. Work by Khilar, Folger, and Gray (1985) for clay embankments.
b. The Rock Island District procedure.1
c. Extension of the work by Khilar, Folger, and Gray (1985).
In the event that other erosion models are adopted as Corps policy at some
later time, or in cases where geotechnical engineers have experience with other
erosion models, such models can be substituted for the illustrated methods, using
the same approach of defining the probability of failure as the probability that the
performance function crosses the limit state. Erosion model of Khilar, Folger, and Gray
Khilar, Folger, and Gray (1985) investigated the potential for clay soils to
pipe or plug under induced flow gradients using a mathematical analysis of a
cylindrical opening in the soil. In each element of the cylinder, the tendency for
soil dispersion depends on the dissolved solids content of the water (function of
the upgradient erosion) and the exchangeable sodium percentage (ESP), where
the latter parameter is defined as: ESP Na × 100% (20) CEC In the above equation, Na* is the exchangeable sodium and CEC is the cation
exchange capacity.
The tendency for plugging or piping depends on the capability for particle
capture at the pore throats. Soil and water samples from Corps of Engineers’
Districts throughout the United States were used in laboratory verification
studies. Khilar, Folger, and Gray defined two lumped parameters, NF , and NG .
For erosion to initiate, NF should initially be greater than NG , which means that
“the initial flow rate should be sufficient to produce a shear stress which is
greater than the critical shear stress c for the particular soilwater system.” When
these parameters are set equal to each other, the following expression for the
pressure gradient required to sustain erosion results: P L c n0 2.828 K0 1/2 (21) 1 Personal Communication, 1993, S. Zaidi, U.S. Army Engineer District, Rock Island; Rock Island,
IL. B79 ETL 11102556
28 May 99
where P/L = pressure gradient in units of pressure per length
c = critical tractive shear stress
nc = initial porosity
Ko = initial intrinsic permeability in units of length2 (for water at 20 C,
when k = 1 × 105 cm/sec, K = 1010 cm2)
as P/L = iw, the above expression can be rewritten as: ic c no 1/2 (22) 2.878w Ko which provides a measure of the critical gradient required to cause piping.
The critical shear stress c can vary widely, with values for clay ranging from
less than 0.2 to more than 20 dynes/cm2, depending on the soil pore fluid
concentration, dielectric dispersion, and sodium absorption ratio. These are
parameters not generally available to geotechnical engineers doing preliminary
economic analyses of existing levees. However, it can be shown that, in most
cases, the gradients required for clay soils are so high as to not be expected in
levee embankments and hence the probability of failure due to internal erosion
may be small in comparison to other more dominant modes. For example, Khilar,
Folger, and Gray (1985) use the following to check the criterion by Arulanandan
and Perry (1983) that soil can be considered nonerodible if c > 10 dynes/cm2.
Assume n = 0.4 and ko = 1010 cm2 (k = 105 cm/sec). Then, according to the
above equation, ic 10 dynes/cm2
2.828 980.7 dynes/cm3 0.4 10cm2
10 2 228 (23) As hydraulic gradients on the order of 200 seldom occur in earth
embankments, or in laboratory experiments such as the pinhole test, piping
erosion is generally not observed at such for materials with critical tractive
stresses as large as 10 dynes/cm2. Rock Island District procedure for sand levees
The Rock Island District procedure to ensure the erosion stability of the
landside slope of sand levees involves the calculation of two parameters, the
maximum erosion susceptibility M and the relative erosion susceptibility R. The
B80 ETL 11102556
28 May 99
calculated values are compared to critical combinations for which toe berms are
considered necessary. The parameters are functions of the embankment geometry
and soil properties. To analyze stability, first the vertical distance of the seepage
exit point on the downstream slope yc, is determined using the wellknown solution for “the basic parabola” by L. Casagrande. Two parameters l and l are then
calculated as: 1 cos w sin
sin tan( ) sat b b tan1 2 w sin0.7 n
1.49 0.6 [k tan( )]0.6 (24) (25) where = downstream slope angle = zero for a horizontal exit gradient
n = Manning’s coefficient for sand, typically 0.02 sat = saturated density of the sand in lb/ft3 b = submerged effective density of the sand in lb/ft3
k = permeability in ft/s 1 = friction angle
It is important to note that the parameter 2 is not dimensionless, and the units
stated above must be used.
The erosion susceptibility parameters are then calculated as: M R 2ye0.6
1 ye (26) 1 
2 1.67 (27) H B81 ETL 11102556
28 May 99
In the above equations,  is the critical tractive stress, which the Rock Island
District takes as typically about 0.03 lb/ft2 (14.36 dynes/cm2) for medium sand,
and H is the full embankment height, measured in feet. Again, it should be noted
that the parameters M and R values are not dimensionless, and must be calculated
using the units shown. According to the Rock Island design criteria, toe berms
are recommended when M and R values fall above the shaded region shown in
Figure 31. To simplify probabilistic analysis, Shannon and Wilson, Inc., and
Wolff (1994) suggested replacing this region with a linear approximation (also
shown in Figure 31), and taken to be the limit state. The linear approximation is
represented by the following equation:
M + 14.4R  13.0 = 0 (28) Figure 31. Rock Island District berm criteria and linear approximation of limit
state Positive values of the expression to the left of the equals sign indicate the need
for toe berms. Extension of Khilar’s model to sandy materials
Khilar’s model was developed for soils with a sufficient cohesive component
to sustain an open crack. For these soils, it has been shown that very high
gradients, much higher than would typically be found in flood control levees, are
necessary to initiate piping.
However, if the same equation given above is considered for silty and sandy
materials, reasonable results are obtained that are consistent with engineering
expectations of what gradients might initiate piping in such materials. Knowing
the D50 and D10 grain sizes, reasonable estimates of the permeability k and the
critical tractive stress c can be made and substituted in Khilar’s equation. The
B82 ETL 11102556
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critical tractive stress for granular materials can be estimated from the D50 size
(Lane 1935) as:
c (dynes/cm2) = 10 × D50 (in mm) (29) The permeability k can be estimated from the D10 grain size using the wellknown correlation developed for Mississippi River levees published in TM3424
(U.S. Army Corps of Engineers 1956a).
Table 13 summarizes the critical gradients calculated using the above procedure for three granular materials from which a levee might be constructed. It is
noted that the relative magnitudes of the calculated critical gradients appear reasonable and this procedure might be considered as a possible approach for initial
evaluation of the erodibility of existing granular levees. However, it should also
be noted that internal gradients in a pervious levee will generally be below these
values, and will seldom exceed 0.20, unless local discontinuities are present.
Table 13
Calculated Critical Gradients for Three Granular Soils Using
Khilar’s Equation
Soil
D50 , mm c , dynes/cm2 D10 , mm k, cm/sec Critical
gradient Uniform fine sand 0.1 0.09 150 × 104 0.59 1.0 Silty gravelly sand 0.4 4.0 Coarse to medium sand 1.8 18.0 4 0.005 10 × 10 9.1 0.3 2,000 × 104 2.9 Example Problem 1: Sand Levee on Thin Uniform
Clay Top Stratum
The erosion resistance of example problem 1 will be evaluated using two
techniques, as follows:
a. The Rock Island criteria.
b. The extended Khilar model.
The embankment soil will be taken to be a coarsetomedium sand similar to
that in the third row of Table 13. Random variables are characterized as shown
in Table 14.
The analysis for the Rock Island method and the Khilar equation method was
performed using a spreadsheet extended from one previously developed by
Shannon and Wilson, Inc., and Wolff (1994). An example of the spreadsheet is
shown in Figure 32.
B83 ETL 11102556
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Table 14
Random Variables for Internal Erosion Analysis, Example
Problem 1
Variable Expected Value Coefficient Rock Island
of Variation Model Mannings coefficient, n 0.02 10% * Unit weight, sat 125 lb/ft3 8% * Friction angle, 1 30 deg 6.7% * Coefficient of permeability, k 2,000 × 104 cm/s 30% * * Critical tractive stress,  18 dynes/cm2 10% * * Khilar’s Model Rock Island District method
For the Rock Island District method, which assesses erosion at the landside
seepage face, the method was numerically unstable (1 becomes negative) for the
slopes assumed in example problem 1. To make the problem stable, the slopes
had to be flattened to 1V:3H riverside and 1V:5H landside.
The results for the Taylor series analysis for a 20ft water height are
summarized in Table 15. Results for other heights are shown in the spreadsheets
in Figures 33 through 37.
Table 15
Results of Internal Erosion Analysis, Example Problem 1 (Modified
to Flatter Slopes) H = 20 ft, Rock Island District Method n k × 10 cm/sec Performance Variance
c
dynes/cm2 Function
Component 0.02 125 30 2000 18 125 30 2000 18 18.491 0.018 125 30 2000 18 16.515 0.02 135 30 2000 18 14.798 0.02 115 30 2000 18 23.667 0.02 125 32 2000 18 14.817 0.02 125 28 2000 18 22.179 0.02 125 30 2600 18 20.321 0.02 125 30 1400 18 14.339 0.02 125 30 2000 19.8 16.046 0.02 125 30 2000 16.2 19.369 Percent
of Total
Variance 17.524 0.022 B84 set
lb/ft3 4 0.9761 2.1 19.6648 42.9 13.5498 29.5 8.961 19.5 2.7606 6.0 ETL 11102556
28 May 99 Figure 32. Spreadsheet for throughseepage analysis It is noted that the most significant random variables, based on descending
order of their variance components, are the unit weight, the friction angle, and the
permeability. The effects of Manning’s coefficient and the critical tractive stress,
at least for the coefficients of variation assumed, are relatively insignificant. B85 ETL 11102556
28 May 99 B86
Figure 33. Reliability calculations for throughseepage, example problem 1, h = 5 ft Figure 34. Reliability calculations for throughseepage, example problem 1, h = 10 ft ETL 11102556
28 May 99 B87 ETL 11102556
28 May 99 B88
Figure 35. Reliability calculations for throughseepage, example problem 1, h = 15 ft Figure 36. Reliability calculations for throughseepage, example problem 1, h = 17.5 ft ETL 11102556
28 May 99 B89 ETL 11102556
28 May 99 B90
Figure 37. Reliability calculations for throughseepage, example problem 1, h = 20.0 ft ETL 11102556
28 May 99
When the probabilities of failure from the individual spreadsheet solutions are
plotted, the result is the conditional probability of failure function shown in
Figure 38. Again, it takes the expected reversecurve shape. Below heads of
10 ft, or about half the levee height, the probability of failure against throughseepage failure is virtually nil. The probability of failure becomes greater than
0.5 for a head of about 16.5 ft, and approaches unity at the full head of 20 ft. Khilar equation
The analysis was repeated using the original geometry for example problem 1
and using Equation 21 to predict the critical gradient for piping. The actual
gradient was estimated as the head loss from the riverside water elevation to the
landside slope exit point (based on the basic parabola) divided by the horizontal
distance between these two points. The factor of safety was taken as the critical
gradient divided by the actual gradient. As shown in the spreadsheets in
Figure 39, the reliability index values were greater than 12, even for a full head
on the levee, corresponding to a nil (<106) probability of failure. Example Problem 2: Clay Levee on Thick Nonuniform Clay Top Stratum
For any reasonable values of the critical tractive stress and permeability for
clays, the calculated factors of safety were extremely large, indicating that the
probability of failure against piping would be nil in wellconstructed clay
embankments. It is understood that piping may still occur at undetected areas of
poor construction or defects, but analytical models for such conditions are not
available, requiring that probability values be estimated judgmentally or based on
historical data. B91 ETL 11102556
28 May 99 B92
Figure 38. Conditional probability of failure function for throughseepage, example problem 1 (modified) Figure 39. Reliability calculations for internal erosion analysis using modified Khilar’s equation ETL 11102556
28 May 99 B93 ETL 11102556
28 May 99 10 Surface Erosion Introduction
As flood stages increase, the potential increases for surface erosion from the
following two sources:
a. Erosion due to excessive current velocities parallel to the levee slope.
b. Erosion due to wave attack directly against the levee slope.
The Corps of Engineers provides protection against these events for new
construction by providing adequate slope protection, typically a thick grass cover
for most levees, and stone revetment at locations expected to be susceptible to
wave attack. During flood emergencies, additional protection may be provided
where necessary using dumped rock, snow fence, or plastic sheeting. Erosion Due to Current Velocity
Analytical model
Although there are criteria for decisionmaking relative to the need for slope
protection and the design of slope protection, they are not in the form of a limit
state or performance function (i.e., one does not typically calculate a factor of
safety against scour). To perform a reliability analysis, one needs to define the
problem as a comparison between the probable velocity and the velocity that will
result in damaging scour. Considerable research could be undertaken to derive an
appropriate model. As a first approximation for the purpose of illustration, this
chapter will use a simple adaptation of Manning’s formula for average flow
velocity and assume that the critical velocity for a grassed slope can be expressed
by its expected value and coefficient of variation. Velocity. For channels that are very wide relative to their depth (width >
10×depth), the velocity can be expressed as: B94 ETL 11102556
28 May 99 V 1.486y 2/3 S 1/2
n (30) where
y = depth of flow
S = slope of the energy line
n = Manning’s roughness coefficient
For the purpose of illustration, it will be assumed that the velocity of flow
parallel to a levee slope for water heights from 0 to 20 ft can be approximated
using the above formula with y taken from 0 to 20 ft. For real levees in the field,
it is likely that better estimates of flow velocities at the location of the riverside
slope can be obtained by more detailed hydraulic models (see EM111021418
(U.S. Army Corps of Engineers 1994)).
For purposes of illustration, the following probabilistic moments are assumed.
More detailed and sitespecific studies would be necessary to determine
appropriate values. E[S] 0.0001 Vs 10% (31) E[n] 0.03 Vn 10% (32) Critical velocity. For purposes of illustration, it is assumed that the critical
velocity that will result in damaging scour can be expressed as: E[Vcrit] 5.0 ft / sec Vvcrit 20% (33) Further research is necessary to develop guidance on appropriate values for
prototype structures. Calculation of reliability index and probability of failure
The Manning equation is of the form G(x1 , x2 , x3 , . . .) a x1 x2 x3
g1 g2 g3 (34)
B95 ETL 11102556
28 May 99
For equations of this form, Harr (1987) shows that the probabilistic moments
can be easily determined using a special form of the Taylor’s series approximation he refers to as the vector equation. In such cases, the expected value of the
function is evaluated as the function of the expected values. The coefficient of
variation of the function can be calculated as: VG g1 V 2 (x1) g2 V 2 (x2) g3 V 2 x3) . . . .
2 2 2 2 (35) For the case considered, the coefficient of variation of the flow velocity is then: Vv 1
2
Vs
4 2
Vn (36) Note that, although the velocity increases with floodwater height y, the coefficient
of variation of the velocity is constant for all heights.
Knowing the expected value and standard deviation of the velocity and the
critical velocity, a performance function can be defined as the ratio of critical
velocity to the actual velocity, (i.e., the factor of safety) and the limit state can be
taken as this ratio equaling the value 1.0. If the ratio is assumed to be
lognormally distributed as described in Annex A, then the reliability index is: 1n E[C]
E[D] VC VD
2 2 1n E[Vcrit]
E[V] (37) Vvcrit Vv
2 2 and the probability of failure can be determined from the cumulative distribution
function for the normal distribution. Results
The assumed model and probabilistic moments were used to construct the
example spreadsheet in Figure 40, which calculates expected values and standard
deviations of the flow velocity, the reliability index, and the probability of failure,
all as functions of the flood water height y. It is again observed that a typical
levee may be highly reliable for water levels up to about onehalf the height, and
then the probability of failure may increase rapidly. B96 ETL 11102556
28 May 99 Figure 40. Example spreadsheet for surface erosion analysis Erosion Due to WindGenerated Waves
The height and frequency of windgenerated waves are dependent on wind
speed, duration of the wind, fetch (overwater distance wind travels while
generating waves), and depth of water. As flood stages increase, the potential for
wave attack increases due to the increase in fetch and depth of water. The
relative effect of wavecaused erosion is highly sitespecific, and will vary
significantly depending on such factors as direction of exposure to wind waves,
whether timber stands exist to shield the levee from wave attack, steepness of the
levee slope, and nature of the embankment material. B97 ETL 11102556
28 May 99
Wavecaused erosion during prolonged flooding has occurred on the upper
Mississippi River where appreciable fetch exists. This is especially a problem in
the Rock Island District where levees are constructed of dredged sand and to a
lesser degree in the St. Louis District at locations where specific site conditions
favorable to wavecaused erosion are present.
Wavecaused erosion is a complicated problem and has not at this time been
reduced to an appropriate model which could be used to perform a reliability
analysis. B98 ETL 11102556
28 May 99 11 Combining Conditional
Probability Functions and
Other Considerations Combining Probability Functions
Once a conditional probability of failure function has been obtained for each
considered failure mode, it is desired to combine them to determine the total
conditional probability of failure of all modes combined as a function of the
floodwater elevation (FWE).
As a first approximation, it may be assumed that each of the following four
failure modes are independent and hence uncorrelated:
a. Underseepage.
b. Slope stability.
c. Throughseepage and internal erosion.
d. Surface erosion.
This assumption is not necessarily true, as some of the conditions increasing
the probability of failure for one mode may likely increase the probability of
failure by another. However, there is insufficient research to better quantify such
possible correlation, and it is beyond the scope of the present project. Assuming
independence considerably simplifies the mathematics involved, which is also a
desired condition for studies at the level of economic analysis.
For underseepage, the probability of failure at each water elevation is taken
as that determined in Chapter 6; i.e., the probability of developing an upward
gradient sufficient to cause boiling throughout the top stratum.
For slope stability, the probability of failure is taken as the probability that the
factor of safety is less than unity, and it is assumed that the factor of safety is
lognormally distributed. It is necessary to determine whether modeling
B99 ETL 11102556
28 May 99
shortterm conditions only is sufficient, or whether it is necessary to also model
longterm conditions and postflood conditions in the analysis. For the two
examples given, only shortterm analyses are considered; however, the probability
of failure could also be evaluated for these other cases using the same techniques.
In such cases, they would not be combined with other failure modes as illustrated
in this section, as they are not concurrent events.
For throughseepage and internal erosion, the results of the Rock Island
District method will be used herein for example 1. The probability of failure is
taken as the probability that a function for which a zero value approximates the
Rock Island berm criteria in fact assumes a negative value. The performance
function is assumed to be normally distributed. It should be recalled that the
assumed slopes had to be flattened to make the method numerically stable and the
resulting conditional probability of failure function is thus not for the same levee
section as those for other modes. It is retained for illustrative purposes to show
how probability functions can be combined. For the assessment of internal erosion based on the Khilar, Folger, and Gray (1985) piping model, the probabilities
of failure appear to be so low as to be negligible.
For surface erosion, a conceptual example based on the Manning equation
for flow velocity was illustrated for this report. Additional research needs to be
performed to determine the most appropriate way to model the probability of
surface erosion, for both current and wave attack, considering the current stateofthepractice in the Corps of Engineers. Judgmental evaluation
It is required that a levee under consideration be field inspected. During such
an inspection, it is likely that the inspection team may encounter any number of
items and features, in addition to the three to four quantified failure modes, that
may compromise the confidence of the levee section during a flood event. These
might include animal burrows, cracks, roots, and poor maintenance that might
impede detection of defects or execution of floodfighting activities. To provide a
mathematical means to factor in such information, one may develop a judgmentbased conditional probability function by answering the following question:
Discounting the likelihood of failure accounted for in the quantitative
analyses, but considering observed conditions, what would an experienced
levee engineer consider the probability of failure of this levee for a range of
water elevations?
For the two example problems considered herein, the functions listed in
Table 16 were assumed. While this may appear to be “outright guessing,”
leaving out such information has the greater danger of not considering the
obvious. Formalized techniques for quantifying expert opinion (such as the
Delphi method) exist and merit further research for application to the economic
analysis of existing levees and existing structures. B100 ETL 11102556
28 May 99
+Table 16
Assigned Conditional Probability of Failure Functions for
Judgmental Evaluation of Observed Conditions
Floodwater Elevation Probability of Failure
Example 1 Probability of Failure
Example 2 400.0 0 0 405.0 0.01 0.005 410.0 0.02 0.01 415.0 0.20 0.02 417.5 0.40 0.05 420.0 0.80 0.10 Combinatorial probabilities
For N independent failure modes, the reliability, or probability of no failure
involving any mode, is the probability of no failure due to mode 1 and no failure
due to mode 2, and no failure due to mode 3, etc. As and implies multiplication,
the overall reliability at a given floodwater elevation is the product of the modal
reliability values for that flood elevation, or: R RUS RSS RTS RSE RJ (38) where the subscripts refer to the identified failure modes. Hence the probability
of failure at any floodwater elevation is: Pr(f) 1 R 1 (1 pUS ) (1 pSS ) (1 pTS) (1 pSE ) (1 pJ ) (39) The total conditional probability of failure functions calculated for the two
example problems are shown in Figures 41 and 42. It is observed that probabilities of failure are generally quite low for water elevations less than onehalf the
levee height, then rise sharply as water levels approach the levee crest. While
there are insufficient data to judge whether this is a general trend for all levees, it
has some basis in experience and intuition. B101 ETL 11102556
28 May 99 B102
Figure 41. Combined conditional probability of failure function for example problem 1 Figure 42. Combined conditional probability of failure function for example problem 2 ETL 11102556
28 May 99 B103 ETL 11102556
28 May 99 Flood Duration
As the duration of a flood extends, the probability of failure inevitably
increases, as extended flooding increases pore pressures, and increases the
likelihood and intensity of damaging erosion. The analyses herein essentially
assume that the flood has been of sufficient duration that steadystate seepage
conditions have developed in pervious substratum materials and pervious
embankment materials, but no pore pressure adjustment has occurred in
impervious clayey foundation and embankment materials. These are reasonable
assumptions for economic analysis of most levees. Further research will be
required to provide a rational basis for modifying these functions for flood
duration. Length of Levee and Spatial Correlation
The analyses illustrated herein are for a twodimensional levee cross section,
assumed representative of conditions of a reach of levee extending some unspecified length. Real levees may be a number of miles in length, and reaches are not
in fact discrete entities, but rather a continuum. The details of determining the
probability of failure for the entire length of levee are beyond the scope of this
preliminary report, but several firstcut approximations are noteworthy.
If the levee system were modeled as a series system of discrete independent
reaches, such as links in a chain, the reliability is the product of the reliabilities
for each link, and the same mathematics holds for combining probabilities as
noted above for modes; hence: R R1 R2 R3 . . . RN (40) where the subscripts refer to the separate reaches. Hence the probability of
failure for the system is: Pr(f) 1 R 1 (1 p1 ) (1 p2 ) (1 p3 ) . . . (1 pN ) (41) The problem thus degenerates to that of determining an “equivalent length” of
levee for which the soil properties can be taken as statistically independent of
adjacent reaches. Much research has been done in the areas of spatial correlation,
autocorrelation functions, variance reduction functions, etc., which have a direct
bearing on this problem. However, there are seldom sufficient data to quantify
such functions. B104 ETL 11102556
28 May 99
For practical purposes, pending further research, it seems reasonable to preidentify levee reaches that are likely to be low in reliability, analyze one or more
of these, and base the economic evaluation on the most critical reaches, as a levee
system is generally no more reliable than its weakest reach. B105 ETL 11102556
28 May 99 12 Summary, Conclusions,
and Recommendations Summary
This research effort and report provided a set of “firstcut” examples of the
application of reliability theory to the analysis of several modes of levee performance. Using the capacitydemand model, a conditional probability of failure function can be developed for each performance mode as a function of floodwater
elevation. Using elementary reliability theory and assuming an independent
series system, a composite conditionalprobabilityoffailure function can then be
calculated that reflects all considered failure modes. The developed methodology
is intended to be used as a component in the economic analysis of existing levees. Conclusions
This effort was the first by the Corps of Engineers to cast the problem of
predicted geotechnical performance of existing levees in a probabilistic framework. Full implementation of a probabilistic approach to levee performance prediction will undoubtedly require additional research, additional developmental
efforts, and experiencebuilding by practicing engineers in the Corps, and decisions by Corps’ policy makers. Nevertheless, a number of conclusions can be
drawn from the analyses of the two example problems presented herein:
a. b. B106 The template method presented in current guidance for estimating existing levee reliability does not explicitly account for the several modes of
levee performance (e.g., underseepage) as it does not incorporate
information regarding foundation conditions.
The probabilistic capacitydemand model can be used to develop
conditionalprobabilityoffailure functions for levees as functions of
floodwater elevation. In this approach, the probability of failure is taken
to be a function of the quantified uncertainty in the engineering parameters used in performance analysis of the levee. ETL 11102556
28 May 99
c. For underseepage analysis, relatively high probabilities of failure can be
present for some commonly encountered foundation conditions. In the
probabilistic analysis of the example problems, it was found that the top
blanket thickness (z) is the major contributor to the uncertainty in
performance. This is consistent with previous analyses of similar
problems (Shannon and Wilson, Inc., and Wolff (1994)). d. For slope stability analysis, probabilities of failure calculated for the
two example problems were considerably lower than those for seepage
analysis. This is also consistent with previous studies on similar
problems (Shannon and Wilson, Inc., and Wolff (1994)). In general,
floodwater elevation does not significantly affect the probability of slope
failure except for pervious levees where throughseepage may induce
slope instability. e. For throughseepage analysis, further review of available deterministic
analysis models is required. For the example analyses herein, an adaptation of Rock Island procedures was used to illustrate a probabilistic
approach. However, the procedure used is based on criteria for conditions at which the construction of berms is recommended, and does not
represent a true limit state or condition where erosion may result in levee
failure. f. For surface erosion, a conceptual example was presented based on
average current velocities determined from a simplified Manning equation and an assumed scour velocity. For actual levees under study,
better characterization of the actual current velocity can likely be
obtained from existing hydraulic models used by the Corps, and a better
characterization of the critical velocity that will induce damaging scour
can also likely be developed. Furthermore, the occurrence of damaging
scour does not necessarily imply that levee failure will occur, and some
adjustment of results may be necessary to account for this. g. Surface erosion can also be induced by wave attack. A similar analytical model should be developed for this condition, which was beyond the
scope of the present effort. h. Engineering judgment regarding the probability of failure for modes
other than those analyzed can be incorporated into the analysis so long
as it can be quantified. For example, deficiencies such as cracks or
animal burrows observed in a field inspection can be included by having
the engineer assign judgmental probabilityoffailure functions
reflecting observed conditions. i. As a first approximation, the several conditional probabilityoffailure
functions for the considered performance modes were combined assuming independence of performance modes and functioning as a series
system. However, there is undoubtedly some correlation between some B107 ETL 11102556
28 May 99
modes; for example, throughseepage and slope stability, which should
be considered in further development of the methodology.
Each analysis presented herein was based on a single formulation of the
problem (e.g., a defined set of random variables and the performance function
used with the Taylor’s series method). In order to be in a position to recommend
the best specific approaches for application in practice, further research and
refinement of these analyses are required to evaluate and compare a number of
alternative formulations in the probabilistic methods, the effect of various
assumptions, etc.
Incorporation of length effects requires further research. The example
analyses herein provide the combined probability of failure function for a twodimensional levee cross section representative of an unspecified length. Sections
very close to the analyzed section will be highly correlated with that section, and
hence the analyzed section can be considered to model some equivalent “statistically homogeneous” length of levee. Sections at some distance can be considered to represent another equivalent length of levee. The entire levee length
can then be considered as a chain, with each equivalent section an independent
link. Probabilistic techniques are readily available to analyze such a system once
the number and size of links and the distribution of their probabilities of failure
are determined; however, much work remains to be done in developing
methodology for that specific step. Recommendations
To continue development and implementation of a probabilistic approach to
assessment of existing levees, the following activities are recommended:
a. b. Additional research, with examples similar to those herein, on a wider
range of levee conditions and considering and testing possible
alternative approaches in characterizing variables, defining
performance functions, calculating probabilistic moments, etc. c. Additional research on length effects and spatial correlation effects,
as previously described. d. Initial research on the probabilistic frequency and categorization of
levee performance problems, to begin calibration of developed
procedures against observed performance. e. B108 Development and revision of software, to enhance practitioners’
capability to fit probability distribution or moments to random variables,
and to perform probabilistic seepage and stability analysis. Training of geotechnical engineers expected to use the developed
methodology. ETL 11102556
28 May 99 References Arulanandan, K., and Perry, E. B. (1983). “Erosion in relation to fitted design
criteria in earth dams,” Journal of Geotechnical Engineering Division, ASCE
109(5), 682697.
Bjerrum, L., and Simons, N. E. (1960). “Comparison and shear strength
characteristics of normally consolidated clays.” Proceedings of the ASCE
Research Conference on the Shear Strength of Cohesive Soils. Boulder, CO,
711726.
Calle, E. O. F. (1985). “Probabilistic analysis of stability of earth slopes,”
Proceedings of the 11th International Conference on Soil Mechanics and
Foundation Engineering. San Francisco, Vol 2, 809812.
Calle, E. O. F., Best, H., Sellmeijer, J. B., and Weijers, J. (1989). “Probabilistic
analysis of piping underneath water retaining structures. ” Proceedings of the
12th International Conference on Soil Mechanics and Foundation
Engineering. Rio de Janeiro, Vol 2, 819822.
Duckstein, L., and Bogardi, I. (1981). “Application of reliability theory to
hydraulic engineering design,” Journal of the Hydraulics Division, ASCE,
107(HY6), 799813.
Duncan, J. M., and Houston, W. N. (1981). “Estimating failure probability for
california levees,” Journal of Geotechnical Engineering, ASCE, 109(2), 260269.
Edris, E. V., Jr., and Wright, S. G. (1987). “User’s guide: UTEXAS2 slope
stability package, Volume I: User’s Manual,” Instruction Report GL871,
U.S. Army Engineer Waterways Experiment Station, Vicksburg, MS.
Fredlund, D. G., and Dahlman, A. E. (1972). “Statistical geotechnical properties
of glacial Lake Edmonton sediments.” Statistics and probability in civil
engineering. Hong Kong University Press (Hong Kong Int. Conf.), distributed by Oxford University Press, London. B109 ETL 11102556
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Hammitt, G. M. (1966). “Statistical analysis of data from a comparative
laboratory test program sponsored by ACIL,” Miscellaneous Paper 4785,
U.S. Army Engineer Waterways Experiment Station, Vicksburg, MS.
Harr, M. E. (1987). Reliability based design in civil engineering. McGrawHill,
New York.
Hasofer, A. A., and Lind, A. M. (1974). “An exact and invariant secondmoment
code format,” Journal of the Engineering Mechanics Division, ASCE, 100,
111121.
Holtz, R. D., and Kovacs, W. D. (1981). An introduction to geotechnical
engineering. PrenticeHall, Englewood Cliffs, NJ.
Kenney, T. C. (1959). Discussion of “Geotechnical Properties of Glacial Lake
Clays, by T. H. Wu,” Journal of the Soil Mechanics and Foundations
Division, ASCE, 85(SM3), 6779.
Khilar, K. C., Folger, H. S., and Gray, D.H. (1985). “Model for pipingplugging
in earthen structures,” Journal of Geotechnical Engineering, ASCE, 111(7).
Ladd, C. C., Foote, R., Ishihara, K., Schlosser, F., and Poulos, H.G. (1977).
“Stressdeformation and strength characteristics.” StateoftheArt Report,
Proceedings of the Ninth International Conference on Soil Mechanics and
Foundation Engineering. Tokyo, 2, 421494.
Lane, E. W. (1935). “Security from underseepage: Masonry dams on Earth
foundations.” Transactions of the American Society of Civil Engineers,
Vol 100.
Nielson, D. R., Biggar, J. W., and Erh, K. T. (1973). “Spatial variability of fieldmeasured soilwater properties,” Hilgardia, J. Agr. Sci. (Calif. Agr. Experiment Stu.) 42, 215260.
Peter, P. (1982). Canal and river levees. Elsevier Scientific Publishing
Company.
Rosenblueth, E. (1975). “Point estimates for probability moments.” Proceedings of the National Academy of Science. Vol 72, No. 10.
__________. (1981). “Twopoint estimates in probabilities,” Applied Mathematical Modeling 5.
Schultze, E. (1972). “Frequency distributions and correlations of soil properties.” Statistics and probability in civil engineering. Hong Kong University
Press (Hong Kong Int. Conf.), distributed by Oxford University Press,
London. B110 ETL 11102556
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Schwartz, P. (1976). “Analysis and performance of hydraulic sandfill levees,”
Ph.D. diss., Iowa State University.
Shannon and Wilson, Inc., and Wolff, T. F. (1994). “Probability models for
geotechnical aspects of navigation structures,” report to the St. Louis District,
U.S. Army Corps of Engineers.
Termaat, R. J., and Calle, E. O. F. (1994). “Short term acceptable risk of slope
failure of levees.” Proceedings of the 13th International Conference on Soil
Mechanics and Foundation Engineering, New Delhi, India.
U.S. Army Corps of Engineers. (1956a). “Investigation of underseepage, lower
Mississippi River Levees,” Technical Memorandum 3424, U.S. Army
Engineer Waterways Experiment Station, Vicksburg, MS.
__________. (1956b). “Investigation of underseepage, Mississippi River
Levees, Alton to Gale Illinois,” Technical Memorandum 3430, U.S. Army
Engineer Waterways Experiment Station, Vicksburg, MS.
__________. (1970). “Engineering designstability of earth and rockfill dams,”
Engineer Manual 111021902, Headquarters, U.S. Army Corps of Engineers,
Washington, DC.
__________. (1978). “Design and construction of levees,” Engineer Manual
111021913, Headquarters, U.S. Army Corps of Engineers, Washington,
D.C.
__________. (1980). “Development of a quantitative method to predict critical
shear stress and rate of erosion of natural undisturbed cohesive soils,”
Technical Report GL805, U.S. Army Engineer Waterways Experiment
Station, Vicksburg, MS.
__________. (1986). “Seepage analysis and control for dams,” Engineer
Manual 111021901, Department of the Army, Office of the Chief of
Engineers, Washington, DC.
__________. (1991). “Benefit determination involving existing levees,” Policy
Guidance Letter No. 26, Headquarters, U.S. Army Corps of Engineers,
Washington, DC.
__________. (1992a). “Reliability assessment of navigation structures,
Engineer Technical Letter 11102532, Washington, DC.
__________. (1992b). “Reliability assessment of navigation structures:
Stability assessment of existing gravity structures,” Engineer Technical Letter
11102310, Washington, DC.
__________. (1994). “Channel stability assessment for flood control projects,”
Engineer Manual 111021418, Washington, DC.
B111 ETL 11102556
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Vrouwenvelder. (1987). Probabilistic design of flood defenses,” Report No. B87404, IBBCTNO (Institute for Building Materials and Structures of the
Netherlands Organization for Applied Scientific Research), The Netherlands.
Wolff, T. F. (1985). “Analysis and design of embankment dam slopes: A
probabilistic approach,” Ph.D. diss., Purdue University, Lafayette, IN.
__________. (1989). “LEVEEMSU: A software package designed for levee
underseepage analysis,” Technical Report GL8913, U.S. Army Engineer
Waterways Experiment Station, Vicksburg, MS.
Wolff, T. F., and Wang, W. (1992a). “Engineering reliability of navigation
structures,” Research Report, Michigan State University, for U.S. Army Corps
of Engineers.
__________. (1992b). “Engineering reliability of navigation structures Supplement No. 1,” Research Report, Michigan State University, for
U.S. Army Corps of Engineers. B112 ETL 11102556
28 May 99 Annex A
Brief Review of Probability and
Reliability Terms and Concepts
Introduction
The objective of this annex is to introduce some basis elements of engineering
reliability analysis applicable to geotechnical structures for various modes of
performance. These reliability measures are intended to be sufficiently consistent
and suitable for application to economic analysis of geotechnical structures of
water resource projects. References are provided which should be consulted for
detailed discussion of the principles of reliability analyses.
Traditionally, evaluations of geotechnical adequacy are expressed by safety
factors. A safety factor can be expressed as the ratio of capacity to demand. The
safety concept, however, has shortcomings as a measure of the relative reliability
of geotechnical structures for different performance modes. A primary deficiency
is that parameters (material properties, strengths, loads, etc.) must be assigned
single, precise values when the appropriate values may in fact be uncertain. The
use of precisely defined single values in an analysis is known as the deterministic
approach. The safety factor using this approach reflects the condition of the
feature, the engineer's judgement, and the degree of conservatism incorporated
into the parameter values.
Another approach, the probabilistic approach, extends the safety factor
concept to explicitly incorporate uncertainty in the parameters. This uncertainty
can be quantified through statistical analysis of existing data or judgementally
assigned. Even if judgementally assigned, the probabilistic results will be more
meaningful than a deterministic analysis because the engineer provides a measure
of the uncertainty of his or her judgement in each parameter. B113 ETL 11102556
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The probability of failure
Engineering reliability analysis is concerned with finding the reliability R or
the probability of failure Pr(f) of a feature, structure, or system. As a system is
considered reliable unless it fails, the reliability and probability of failure sum to
unity:
R + Pr(f) = 1 (A1) R = 1  Pr(f) (A2) Pr(f) = 1  R (A3) In the engineering reliability literature, the term failure is used to refer to any
occurrence of an adverse event under consideration, including simple events such
as maintenance items. To distinguish adverse but noncatastrophic events (which
may require repairs and associated expenditures) from events of catastrophic
failure (as used in the dam safety context), the term probability of unsatisfactory
performance Pr(U) is sometimes used. An example would be slope stability
where the safety factor is below the required minimum safety factor but above
1.0. Thus for this case, reliability is defined as:
R = 1  Pr(U) (A4) Contexts of reliability analysis
Engineering reliability analysis can be used in several general contexts:
a. Estimation of the reliability of a new structure or system upon its
construction and first loading. b. Estimation of the reliability of an existing structure or system upon a
new loading. c. Estimation of the probability of a part or system surviving for a given
lifetime. Note that the third context has an associated time interval, whereas the first two
involve measures of the overall adequacy of the system in response to a load
event.
Reliability for the first two contexts can be calculated using the capacitydemand model and quantified by the reliability index . In the capacitydemand
model, uncertainty in the performance of the structure or system is taken to be a
function of the uncertainty in the values of various parameters used in calculating
some measure of performance, such as the factor of safety.
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In the third context, reliability over a future time interval is calculated using
parameters developed from actual data on the lifetimes or frequencies of failure
of similar parts or systems. These are usually taken to follow the exponential or
Weibull probability distributions. This methodology is wellestablished in
electrical, mechanical, and aerospace engineering, where parts and components
routinely require periodic replacement. This approach produces a hazard function
which defines the probability of failure in any time period. These functions are
used in economic analysis of proposed geotechnical improvements.
For reliability evaluation of most geotechnical structures, in particular
existing levees, the capacitydemand model will be utilized, as the question of
interest is the probability of failure related to a load event rather than the
probability of failure within a time interval. Reliability Index
The reliability index is a measure of the reliability of an engineering system
that reflects both the mechanics of the problem and the uncertainty in the input
variables. This index was developed by the structural engineering profession to
provide a measure of comparative reliability without having to assume or
determine the shape of the probability distribution necessary to calculate an exact
value of the probability of failure. The reliability index is defined in terms of the
expected value and standard deviation of the performance function, and permits
comparison of reliability among different structures or modes of performance
without having to calculate absolute probability values. Calculating the reliability
index requires:
a. A deterministic model (e.g., a slope stability analysis procedure). b. A performance function (e.g., the factor of safety from UTEXAS2). c. The expected values and standard deviations of the parameters taken as
random variables (e.g., E[ and ). d. A definition of the limit state (e.g., ln(FS) = 0). e. A method to estimate the expected value and standard deviation of the
limit state given the expected values and standard deviations of the
parameters (e.g., the Taylor's series or point estimate methods). Accuracy of Reliability Index
For rehabilitation studies of geotechnical structures, the reliability index is
used as a “relative measure of reliability or confidence in the ability of a structure
to perform its function in a satisfactory manner.” B115 ETL 11102556
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The analysis methods used to calculate the reliability index should be
sufficiently accurate to rank the relative reliability of various structures and
components. However, reliability index values are not absolute measures of
probability. Structures, components, and performance modes with higher indices
are considered more reliable than those with lower indices. Experience analyzing
geotechnical structures will refine these techniques. The CapacityDemand Model
In the capacitydemand model, the probability of failure or unsatisfactory
performance is defined as the probability that the demand on a system or
component exceeds the capacity of the system or component. The capacity and
demand can be combined into a single function (the performance function), and
the event that the capacity equals the demand taken as the limit state. Reliability
R is the probability that the limit state will not be achieved or crossed.
The concept of the capacitydemand model is illustrated for slope stability
analysis in Figure A1. Using the expected value and standard deviation of the
random variables c and in conjunction with the Taylor
s series method or the
point estimate method, the expected value and standard deviation of the factor of
safety can be calculated. If it is assumed that the factor of safety is lognormally
distributed, then the natural log of the factor of safety is normally distributed. f(c) f( φ ) φ c βσ lnFS E[lnFS] f(lnFS) ln(FS)
Figure A1. The capacitydemand model B116 ETL 11102556
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The performance function is taken as the log of the factor of safety, and the limit
state is taken as the condition ln(FS) = 0. The probability of failure is then the
shaded area corresponding to the condition ln(FS) < 0. If it is assumed that the
distribution on ln(FS) is normal, then the probability of failure can be obtained
using standard statistical tables.
Equivalent performance functions and limit states can be defined using other
measures, such as the exit gradient for seepage.
The probability of failure associated with the reliability index is a probability
per structure; it has no timefrequency basis. Once a structure is constructed or
loaded as modeled, it either performs satisfactorily or not. Nevertheless, the
value calculated for an existing structure provides a rational comparative
measure. Steps in a Reliability Analysis Using the CapacityDemand Model
As suggested by Figure A1 for slope stability, a reliability analysis includes
the following steps:
a. Important variables considered to have sufficient inherent uncertainty
are taken as random variables and characterized by their expected
values, standard deviations, and correlation coefficients. In concept,
every variable in an analysis can be modeled as a random variable as
most properties and parameters have some inherent variability and
uncertainty. However, a few specific random variables will usually
dominate the analysis. Including additional random variables may
unnecessarily increase computational effort without significantly
improving results. When in doubt, a few analyses with and without
certain random variables will quickly illustrate which are significant, as
will the examination of variance terms in a Taylor's series analysis. For
levee analysis, significant random variables typically include material
strengths, soil permeability or permeability ratio, and thickness of top
stratum. Material properties such as soil density may be significant, but
where strength and density both appear in an analysis, strength may
dominate. An example of a variable that can be represented
deterministically (nonrandom) is the density of water. b. A performance function and limit state are identified. c. The expected value and standard deviation of the performance function
are calculated. In concept, this involves integrating the performance
function over the probability density functions of the random variables.
In practice, approximate values are obtained using the expected value,
standard deviation, and correlation coefficients of the random variables
in the Taylor's series method or the point estimate method.
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d. The reliability index is calculated from the expected and standard
deviation of the performance function. The reliability index is a measure
of the distance between the expected value of ln (C/D) or ln (FS) and the
limit state. e. If a probability of failure value is desired, a distribution is assumed and
Pr(f) is calculated. Random Variables
Description
Parameters having significance in the analysis and some significant uncertainty are taken as random variables. Instead of having precise single values,
random variables assume a range of values in accordance with a probability
density function or probability distribution. The probability distribution quantifies the likelihood that its value lies in any given interval. Two commonly used
distributions, the normal and the lognormal, are described later in this appendix. Moments of random variables
To model random variables in the Taylor's series or point estimate methods,
one must provide their expected values and standard deviations, which are two of
several probabilistic moments of a random variable. These can be calculated
from data or estimated from experience. For random variables which are not
independent of each other, but tend to vary together, correlation coefficients must
also be assigned.
Mean value. The mean value µ x of a set of N measured values for the
random variable X is obtained by summing the values and dividing by N: MX
N µ x i
1 i (A5) N Expected value. The expected value E[X] of a random variable is the mean
value one would obtain if all possible values of the random variable were multiplied by their likelihood of occurrence and summed. Where a mean value can be
calculated from representative data, it provides an unbiased estimate of the
expected value of a parameter; hence, the mean and expected value are numerically the same. The expected value is defined as: P M Xp (X ) E X µ X Xf (X)dx B118 i (A6) ETL 11102556
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where f(X) is the probability density function of X (for continuous random
variables) and p(Xi) is the probability of the value Xi (for discrete random
variables).
Variance. The variance Var[X] of a random variable X is the expected value
of the squared difference between the random variable and its mean value.
Where actual data are available, the variance of the data can be calculated by
subtracting each value from the mean, squaring the result, and determining the
average of these values: P Var [X] E[(X µ X)2] (X µ X)2 f(X)dX M [(X µ ) ]
2 i N X (A7) The summation form above involving the Xi term provides the variance of a
population containing exactly N elements. Usually, a sample of size N is used to
obtain an estimate of the variance of the associated random variable which
represents an entire population of items or continuum of material. To obtain an
unbiased estimate of the population working from a finite sample, the N is
replaced by N1:
Var[X] M [(X µ ) ]
2 i N 1 X (A8) Standard deviation. To express the scatter or dispersion of a random
variable about its expected value in the same units as the random variable itself,
the standard deviation is taken as the square root of the variance; thus: X Var X (A9) Coefficient of variation. To provide a convenient dimensionless expression
of the uncertainty inherent in a random variable, the standard deviation is divided
by the expected value to obtain the coefficient of variation, which is usually
expressed as a percent:
VX X
EX ×100% (A10) The expected value, standard deviation, and coefficient of variation are interdependent: knowing any two, the third is known. In practice, a convenient way to
estimate moments for parameters where little data are available is to assume that
the coefficient of variation is similar to previously measured values from other
data sets for the same parameter. B119 ETL 11102556
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Correlation
Pairs of random variables may be correlated or independent; if correlated, the
likelihood of a certain value of the random variable Y depends on the value of the
random variable X. For example, the strength of sand may be correlated with
density or the top blanket permeability may be correlated with grain size of the
sand. The covariance is analogous to the variance but measures the combined
effect of how two variables vary together. The definition of the covariance is:
Cov[X,Y] E[(X µ X)(Y µ Y)] (A11) which is equivalent to:
Cov[X,Y] PP(X µ )(Y µ )f(X,Y)dYdX
X Y (A12) In the above equation, f (X, Y) is the joint probability density function of the
random variables X and Y. To calculate the covariance from data, the following
equation can be used:
Cov[X,Y] 1
N M (X µ )(Y µ )
i X i Y (A13) To provide a nondimensional measure of the degree of correlation between X and
Y, the correlation coefficient X, Y, is obtained by dividing the covariance by the
product of the standard deviations: X,Y Cov X,Y
XY (A14) The correlation coefficient may assume values from 1.0 to +1.0. A value of
1.0 or 1.0 indicates there is perfect linear correlation; given a value of X, the
value of Y is known and hence is not random. A value of zero indicates no linear
correlation between variables. A positive value indicates the variables increase
and decrease together; a negative value indicates that one variable decreases as
the other increases. Pairs of independent random variables have zero correlation
coefficients. Probability Distributions
Definition
The terms probability distribution, probability density function, pdf, or the
notation fX(X) refer to a function that defines a continuous random variable. The
Taylor's series and point estimate methods described herein to determine
moments of performance functions require only the mean and standard deviation
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of random variables and their correlation coefficients; knowledge of the form of
the probability density function is not necessary. However, in order to ensure that
estimates made for these moments are reasonable, it is recommended that the
engineer plot the shape of the normal or lognormal distribution which has the
expected value and standard deviation assumed. This can easily be done with
spreadsheet software.
Figure A1 illustrated probability density functions for the random variables c
and . A probability density function has the property that for any X, the value of
f (x) is proportional to the likelihood of X. The area under a probability density
function is unity. The probability that the random variable X lies between two
values X1 and X2 is the integral of the probability density function taken between
the two values. Hence:
X2 Pr (X1 < X < X2) fX(X)dx 2 (A15) X1 The cumulative distribution function CDF or FX (X) measures the integral of
the probability density function from minus infinity to X:
X FX(X) f (X)dx 2X (A16) Thus, for any value X, FX (X) is the probability that the random variable X is less
than the given x. Estimating Probabilistic Distributions
A suggested method to assign or check assumed moments for random
variables is to:
a. Assume trial values for the expected value and standard deviation and
take the random variable to be normal or lognormal. b. Plot the resulting density function and tabulate and plot the resulting
cumulative distribution function (spreadsheet software is a convenient
way to do this). c. Assess the reasonableness of the shape of the pdf and the values of the
CDF. d. Repeat the above steps with successively improved estimates of the
expected value and standard deviation until an appropriate pdf and CDF
are obtained. B121 ETL 11102556
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Normal distribution
The normal or Gaussian distribution is the most wellknown and widely
assumed probability density function. It is defined in terms of the mean X and
standard deviation X as:
fX(X) 1 x 2 exp (x µ x)2
2x 2 (A17) When fitting the normal distribution, the mean of the distribution is taken as the
expected value of the random variable. The cumulative distribution function for
the normal distribution is not conveniently expressed in closed form but is widely
tabulated and can be readily computed by numerical approximation. It is a builtin function in most spreadsheet programs. Although the normal distribution has
limits of plus and minus infinity, values more than 3 or 4 standard deviations
from the mean have very low probability. Hence, one empirical fitting method is
to take minimum and maximum reasonable values to be at approximately ±3
standard deviations. The normal distribution is commonly assumed to characterize many random variables where the coefficient of variation is less than about
30 percent. For levees, these include soil density and drained friction angle.
Where the mean and standard deviation are the only information known, it can be
shown that the normal distribution is the most unbiased choice. Lognormal distribution
When a random variable X is lognormally distributed, its natural logarithm,
ln X, is normally distributed. The lognormal distribution has several properties
which often favor its selection to model certain random variables in engineering
analysis:
a. As X is positive for any value of ln X, lognormally distributed random
variables cannot assume values below zero. b. It often provides a reasonable shape in cases where the coefficient of
variation is large ( >30 percent ) or the random variable may assume
values over one or more orders of magnitude. c. The central limit theorem implies that the distribution of products or
ratios of random variables approaches the lognormal distribution as the
number of random variables increases. If the random variable X is lognormally distributed, then the random variable
Y = ln X is normally distributed with parameters E[Y] = E[ln X] and Y = lnX . To
obtain the parameters of the normal random variable Y, first the coefficient of
variation of X is calculated: B122 ETL 11102556
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VX X (A18) E[X] The standard deviation of Y is then calculated as:
2
Y lnX ln(1VX ) (A19) The standard deviation Y is in turn used to calculate the expected value of Y:
E Y E lnX lnE X 2
Y (A20) 2 The density function of the lognormal variate X is:
f(X) 1 XY 2 exp 1 lnX E[Y]
Y
2 2 (A21) The shape of the distribution can be plotted from the above equation. Values on
the cumulative distribution function for X can be determined from the cumulative
distribution function of Y (E[Y], Y) by substituting the X in the expression Y =
ln X. Calculation of the Reliability Index
As illustrated in Figure A2, a simple definition of the reliability index is based
on the assumption that capacity and demand are normally distributed and the
limit state is the event that their difference, the safety margin S, is zero. The random variable S is then also normally distributed and the reliability index is the
distance by which E[S] exceeds zero in units of S :
E S E C D
S
2 2
C
D (A22) An alternative formulation (also shown in Figure A2) implies that capacity C and
demand D are lognormally distributed random variables. In this case, ln C and
ln D are normally distributed. Defining the factor of safety FS as the ratio C/D,
then ln FS = (ln C)  (ln D) and ln FS is normally distributed. Defining the
reliability index as the distance by which ln FS exceeds zero in terms of the
standard deviation of ln FS, it is:
E lnC lnD E ln C/D E lnFS
lnC lnD
ln C/D
lnFS (A23) B123 ETL 11102556
28 May 99 Figure A2. Alternative definitions of the reliability index From the properties of the lognormal distribution, the expected value of ln C
is:
12
E lnC lnE C lnC
2 (A24) where:
2
2 C ln 1VC
ln Similar expressions apply to E[lnD] and lnD .
The expected value of the log of the factor of safety is then:
B124 (A25) ETL 11102556
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2
2
1
1
E lnFS lnE C lnE D ln 1VC ln 1VD
2
2 (A26) As the secondorder terms are small when the coefficients of variation are not
exceedingly large (below approximately 30 percent), the equation above is
sometimes approximated as:
E lnFS lnE C lnE D ln E C
ED (A27) The standard deviation of the log of the factor of safety is obtained as: lnFS 2 2
lnC
lnD (A28) 2
2
lnFS ln 1VC ln 1VD (A29) Introducing an approximation,
ln 1VC VC
2 2 (A30) the reliability index for lognormally distributed C, D, and FS and normally
distributed ln C, ln D, and ln FS can be expressed approximately as: ln EC
ED (A31) VCVD
2 2 The exact expression is:
E[C] 1VD 2 ln E[D] 1VC 2 (A32) ln[1vC ] ln[1VD]
2 2 For many geotechnical problems and related deterministic computer programs,
the output is in the form of the factor of safety, and the capacity and demand are
not explicitly separated. The reliability index must be calculated from values of
E[FS] and FS obtained from multiple runs as later described in the next section.
In this case, the reliability index is obtained using the following steps:
VFS FS
E FS (A33) B125 ETL 11102556
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2
lnFS ln(1VFS) (A34) 2
1
E[lnFS] ln E[FS] ln(1VFS)
2 (A35) 2
ln E FS / 1VFS
E[lnFS] )lnFS
2
ln 1VFS (A36) Integration of the Performance Function
Methods such as direct integration, Taylor's series, point estimate methods,
and Monte Carlo simulation are available for calculating the mean and standard
deviation of the performance function. For direct integration, the mean value of
the function is obtained by integrating over the probability density function of the
random variables. A brief description of the other methods follows. The References section that follows the main text of this report should be consulted for
additional information.
The Taylor
s series method The Taylor's series method is one of several methods to estimate the moments
of a performance function based on moments of the input random variables [see
Harr (1987)]. It is based on a Taylor's series expansion of the performance
function about some point. For the Corps navigation rehabilitation studies, the
expansion is performed about the expected values of the random variables. The
Taylor's series method is termed a firstorder, secondmoment (FOSM) method,
as only firstorder (linear) terms of the series are retained and only the first two
moments (mean and the standard deviation) are considered. The method is
summarized below and illustrated by an example in Annex B. Independent random variables. Given a function Y = g(X1, X2, ... Xn), where
all Xi are independent, the expected value of the function is obtained by
evaluating the function at the expected values of the random variables:
E Y g E X1 ,E X2 ,...E Xn (A37) For a function such as the factor of safety, this implies that the expected value
of the factor of safety is calculated using the expected values of the random
variables:
E FS FS E 11 ,E c1 ,E 1 ,... B126 (A38) ETL 11102556
28 May 99 The variance of the performance function is taken as:
Var Y M Y
Xi 2 VarXi (A39) with the partial derivatives taken at the expansion point (in this case the mean or
expected value). Using the factor of safety as an example performance function,
the variance is obtained by finding the partial derivative of the factor of safety
with respect to each random variable evaluated at the expected value of that
variable, squaring it, multiplying it by the variance of that random variable, and
summing these terms over all of the random variables:
Var FS M FS
Xi 2 VarXi (A40) The standard deviation of the factor of safety is then simply the square root of the
variance.
Having the expected value and variance of the factor of safety, the reliability
index can be calculated as described earlier in this annex. Advantages of the
Taylor's Series method include the following:
a. The relative magnitudes of the terms in the above summation provide an
explicit indication of the relative contribution of uncertainty of each
variable. b. The method is exact for linear performance functions. Disadvantages of the Taylor's Series method include the following:
a. It is necessary to determine the value of derivatives. b. The neglect of higherorder terms introduces errors for nonlinear
functions. The required derivatives can be estimated numerically by evaluating the
performance function at two points. The function is evaluated at one increment
above and below the expected value of the random variable Xi and the difference
of the results is divided by the difference between the two values of Xi. Although
the derivative at a point is most precisely evaluated using a very small increment,
evaluating the derivative over a range of ±1 standard deviation may better capture
some of the nonlinear behavior of the function over a range of likely values. Thus,
the derivative is evaluated using the following approximation: B127 ETL 11102556
28 May 99
Y
Xi g E Xi X g E Xi X
i 2X i (A41) i When the above expression is squared and multiplied by the variance, the
standard deviation term in the denominator cancels the variance, leading to
Y
Xi 2 g X g X VarX 2 2 (A42) where X+ and X are values of the random variable at plus and minus one standard
deviation from the expected value. Correlated random variables. Where random variables are correlated,
solution is more complex. The expression for the expected value, retaining
secondorder terms is:
E Y g E X1 ,E X2 ,...E Xn 1
2
M X YX Cov X X
2 ij i (A43) j However, in keeping with the firstorder approach, the secondorder terms are
generally neglected, and the expected value is calculated the same as for
independent random variables.
The variance, however, is taken as:
Var[Y] M Y
Xi 2 YY
M X X Cov X X VarXi 2 ij i (A44) j where the covariance part contains terms for each possible combination of
random variables. The Point Estimate Method
An alternative method to estimate moments of a performance function based
on moments of the random variables is the point estimate method. Point estimate
methods are procedures where probability distributions for continuous random
variables are modeled by discrete “equivalent” distributions having two or more
values. The elements of these discrete distributions (or point estimates) have
specific values with defined probabilities such that the first few moments of the
discrete distribution match that of the continuous random variable. Having only a
few values over which to integrate, the moments of the performance function are
B128 ETL 11102556
28 May 99 easily obtained. A simple and straightforward point estimate method has been
proposed by Rosenblueth (1975, 1981) and is summarized by Harr (1987). That
method is briefly summarized below and illustrated by example in Annex B. Independent random variables As shown in Figure A3, a continuous random variable X is represented by two
point estimates, X+ and X, with probability concentrations P+ and P, respectively.
As the two point estimates and their probability concentrations form an
equivalent probability distribution for the random variable, the two P values must
sum to unity. The two point estimates and probability concentrations are chosen
to match three moments of the random variable. When these conditions are
satisfied for symmetrically distributed random variables, the point estimates are
taken at the mean ±1 standard deviation: f(x) P P+ x x+ x Figure A3. Point estimate method Xi E Xi Xi (A45) Xi E Xi Xi (A46) For independent random variables, the associated probability concentrations are
each onehalf:
Pi
Pi 0.50 (A47) Knowing the point estimates and their probability concentrations for each
variable, the expected value of a function of the random variables raised to any
power M can be approximated by evaluating the function for each possible
B129 ETL 11102556
28 May 99 combination of the point estimates (e.g., X1+ , X2 , X3+ , Xn ), multiplying each
result by the product of the associated probability concentrations
(e.g., P+ = P1+P2 P3) and summing the terms. For example, two random
variables result in four combinations of point estimates and four terms:
E[Y M] P g(X1, X2)M P g(X1, X2 )M P g(X1 , X2 )M P g(X1 , X2 )M (A48) For N random variables, there are 2N combinations of the point estimates and 2N
terms in the summation. To obtain the expected value of the performance
function, the function g(X1, X2) is calculated 2N times using all the combinations
and the exponent M in Equation A48 is 1. To obtain the standard deviation of the
performance function, the exponent M is taken as 2 and the squares of the
obtained results are weighted and summed to obtain E[Y2]. The variance can then
be obtained from the identity
Var Y E Y 2 E Y 2 (A49) and the standard deviation is the square root of the variance. Correlated random variables Correlation between symmetrically distributed random variables is treated by
adjusting the probability concentrations (P ±± .... ±). A detailed discussion is
provided by Rosenblueth (1975) and summarized by Harr (1987). For certain
geotechnical analyses involving lateral earth pressure, bearing capacity of shallow
foundations, and slope stability, often only two random variables (c and or tan
) need to considered as correlated. For two correlated random variables within a
group of two or more, the product of their concentrations is modified by adding a
correlation term:
Pij Pi j (Pi )(Pj) Pij Pi j (Pi)(Pj) 4 4 (A50) (A51) Monte Carlo simulation The performance function is evaluated for many possible values of the
random variables. A plot of the results will produce an approximation of the
probability distribution. Once the probability distribution is determined in this
manner, the mean and standard deviation of the distribution can be calculated. B130 ETL 11102556
28 May 99 Determining the Probability of Failure
Once the expected value and standard deviation of the performance function
have been determined using the Taylor
s Series or point estimate methods, the
reliability index can be calculated as previously described. If the reliability index
is assumed to be the number of standard deviations by which the expected value
of a normally distributed performance function (e.g., ln(FS)) exceeds zero, then
the probability of failure can be calculated as:
Pr(f) "( ) "( z) (A52) where #(z) is the cumulative distribution function of the standard normal
distribution evaluated at z, which is widely tabulated and available as a builtin
function on modern microcomputer spreadsheet programs. Overall System Reliability
Reliability indices for a number of components or a number of modes of
performance may be used to estimate the overall reliability of an embankment.
There are two types of systems that bound the possible cases, the series system
and the parallel system.
Series system In a series system, the system will perform unsatisfactorily if any one
component performs unsatisfactorily. If a system has n components in series, the
probability of unsatisfactory performance of the ith component is pi and its
reliability, Ri = 1  pi, then the reliability of the system, or probability that all
components will perform satisfactorily, is the product of the component
reliabilities.
R R1R2R3... Ri.... Rn (1 p1)(1 p2)(1 p3) ...(1 pi) ...(1 pn) (A53) Simple parallel system In a parallel system, the system will only perform unsatisfactorily if all
components perform unsatisfactorily. Thus, the reliability is unity minus the
probability that all components perform unsatisfactorily, or
R 1 p1p2p3 ...pi ...pn (A54) B131 ETL 11102556
28 May 99
Parallel series systems Solutions are available for systems requiring routofn operable components,
which may be applicable to problems such as dewatering with multiple pumps,
where r is defined as the number of reliable units. Subsystems involving
independent parallel and series systems can be mathematically combined by
standard techniques.
Upper and lower bounds on system reliability can be determined by
considering all components to be from subgroups of parallel and series systems,
respectively; however, the resulting bounds may be so broad as to be impractical.
A number of procedures are found in the references to narrow the bounds.
Engineering systems such as embankments are complex and have many
performance modes. Some of these modes may not be independent; for instance
several performance modes may be correlated to the occurrence of a high or low
pool level. Rational estimation of the overall reliability of an embankment is a
topic that is beyond the scope of this report. A practical approach The reliability of a few subsystems or components may govern the reliability
of the entire system. Thus, developing a means to characterize and compare the
reliability of these components as a function of time is sufficient to make
engineering judgements to prioritize operations and maintenance expenditures.
For initial use in reliability assessment of geotechnical systems, the target
reliability values presented in the following section should be used. The
objective of a rehabilitation program would be to keep the reliability index for
each significant mode above the target value for the foreseeable future. Target Reliability Indices
Reliability indices are a relative measure of the current condition and provide
a qualitative estimate of the expected performance. Embankments with relatively
high reliability indices will be expected to perform their function well.
Embankments with low reliability indices will be expected to perform poorly and
present major rehabilitation problems. If the reliability indices are very low, the
embankment may be classified as a hazard. The target reliability values shown in
Table A1 should be used in general. B132 ETL 11102556
28 May 99
Table A1
Target Reliability Indices
Expected Performance Level Beta Probability of Unsatisfactory Performance High 5 0.0000003 Good 4 0.00003 Above average 3 0.001 Below average 2.5 0.006 Poor 2.0 0.023 Unsatisfactory 1.5 0.07 Hazardous 1.0 0.16 Note: Probability of unsatisfactory performance is the probability that the value of performance
function will approach the limit state, or that an unsatisfactory event will occur. For example, if the
performance function is defined in terms of slope instability, and the probability of unsatisfactory
performance is 0.023, then 23 of every 1,000 instabilities will result in damage which causes a
safety hazard. B133 ETL 11102556
28 May 99 Annex B
Example Calculations of
Functions of Random Variables
In this annex, example calculations are provided for three approaches for
defining the expected value and standard deviation of a function given the
expected values and standard deviations of the input variables. Problem Statement
The example function considered is the permeability ratio kf / kb used in levee
underseepage analysis. Note that it could just as well be a performance function,
such as the factor of safety in a slope stability analysis. For simplicity of notation,
let the permeability ratio be denoted as PR; thus:
PR kf (B1) kb where kf is the horizontal permeability of the pervious substratum, and kb is the
vertical permeability of the semipervious top stratum. Given the following:
E[kf] 1000 x 10 4 cm/sec kf 300 x 10 4 cm/sec E[kb] 1 x 10 4 cm/sec kb 0.3 x 10 4 cm/sec Vkf Vkb 30% It is desired to estimate E[PR], PR, and VPR. B134 (B2a)
(B2b)
(B2c) ETL 11102556
28 May 99 Taylor’s Series with Exact Derivatives
The expected value of the function, retaining only firstorder terms, is the
function of the expected values:
1000 x 10 4 1000
1 x 10 4 E[PR] PR(E[kF], E[kb]) (B3) As the derivatives of the function are easily obtained, the exact derivatives can be
used to calculate the variance. The variance of the permeability ratio is:
Var[PR] PR 2 2
kf
kf Var[PR] 1 22
kf
kb PR 2 2
kb
kb kf 2
kb (B4a) 2
kb (B4b) The derivatives are evaluated at the expected values of the random variables,
giving: Var[PR] 1
10 4 2 (300 × 10 4)2 10 1 2
(0.3 × 10 4)2 8 10 (B5a) Var[PR] 90,000 90,000 180,000 (B5b) PR Var[PR] 180,000 424 (B5c) The coefficient of variation of the permeability ratio is then:
VPR PR
E[PR] 424 42.4%
1000 (B6) Taylor’s Series with Numerically Approximated
Derivatives
Where derivatives are difficult to precisely calculate, a finite difference
approximation can be used, approximating the derivatives using two points, one
standard deviation above and below the expected value of each random variable.
B135 ETL 11102556
28 May 99 The expected value of the function, retaining only firstorder terms, is the
function of the expected values:
E[PR] PR(E[kf ],E[kb]) 1000 × 10 4 1000
1 × 10 4 (B7) The variance term
PR
kf Var[PR] 2 2 PR
kb 2
kf 2
kb (B8) can be expressed using finite difference approximations of the derivatives as: Var[PR] PR(kf) PR(kf )
2kf PR(kb) PR(kb )
2kb 2 2
kf 2 2
kb (B9) where PR(kf+) refers to the permeability ratio evaluated with kf taken one standard
deviation above the expected value, i.e., kf+ = E[kf] + kf , and the expected value
of the other random variables are used. The other terms are developed similarly.
Substituting, one obtains
Var[PR] Var[PR] PR(kf) PR(kf ) 2 2kf 2
kf PR(kb) PR(kb ) 4
1300 × 10 4 700 × 104 4 1 × 10
1 × 10 4
600 × 10 2kb 2 2
kb 2 4
1000 × 10 4 1000 × 104
1.3 × 10 4
0.7 × 10 0.60 × 10 4 (300 × 10 4)2 (B10) 2 (0.30 × 10 4)2 Var[PR] 90,000 108,684 198,684
PR 445.7 The coefficient of variation is then: VPR B136 PR
E[PR] 445.7 44.6%
1000 (B11) ETL 11102556
28 May 99 Point Estimate Method
Using the point estimate method, the permeability of the foundation is
represented by two point estimates and two probability concentrations:
kf E[kf] kf 1300 × 10 4cm/sec
kf E[kf] kf 700 × 10 4cm/sec (B12) Pkf 0.50
Pkf 0.50 Likewise, the top blanket permeability is modeled by
kb E[kb] kb 1.30 × 10 4cm/sec
kb E[kb] kb 0.70 × 10 4cm/sec (B13) Pkb 0.50
Pkb 0.50 The expected value of the permeability ratio is then
E[PR] M Pkf±Pkb±PR±±
all combinations E[PR] 0.25(PR²²) 0.25(PR±) 0.25(PR ²) 0.25(PR ) (B14) 1 1300
1300 700 700
E[PR] 4 1.3
0.7
1.3
0.7 1 (1000 1857.1 701.3 1000)
4 1139 Note that the expected value is higher than that found using the Taylor’s series
method as it picks up some of the nonlinearity of the function which was
neglected when the terms above the first order were neglected.
To find the variance, first E[PR2] is calculated:
E[PR 2] 0.25(PR²²) 0.25(PR² ) 0.25(PR ²) 0.25(PR )
2 E[PR 2] 2 2 1
(10002 1857.12 701.32 10002)
4 2 (B15) 1,485,200
B137 ETL 11102556
28 May 99 The variance is then calculated by the identity:
Var[PR] E[PR 2] (E[PR])2 1,485,200 11392 (B16) 187,879
and the standard deviation and coefficient of variation are: PR 187,879 433
VPR PR
E[PR] 433 38%
1139 Note that the estimate of the standard deviation is similar to that for the two
Taylor’s series methods, but the coefficient of variation drops because the
expected value increased. B138 (B17) ...
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 Spring '11
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 Probability, The Land, Probability theory, probability density function, Corps of Engineers

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