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Simple Probabilistic Simple Probabilistic Analysis of a Gravity Dam Analysis of a Gravity Dam
FEDERAL ENERGY REGULATORY COMMISSIONFEDERAL ENERGY REGULATORY COMMISSIONOFFICE OF ENERGY PROJECTSOFFICE OF ENERGY PROJECTS
DIVISION OF DAM SAFETY AND INSPECTIONSDIVISION OF DAM SAFETY AND INSPECTIONS
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Traditional Deterministic ApproachTraditional Deterministic Approach
For Sliding of the structure of the FoundationFor Sliding of the structure of the Foundation
Fixed parametersFixed parameters
Weight of concrete: Weight of concrete: 150 lb/ft150 lb/ft33
Weight of water:Weight of water: 62.4 lb/ft62.4 lb/ft33
Variable parametersVariable parameters
UpliftUplift
Foundation strengthFoundation strength
CohesionCohesion
AsperitiesAsperities
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Traditional AnalysisTraditional Analysis
Little or no cohesion usedLittle or no cohesion used
Conservative shear strength estimateConservative shear strength estimate
Full headwater to Full headwater to tailwatertailwater
uplift used, or based uplift used, or based
on on piezometerpiezometer
readings.readings.
Drain effectiveness must be estimatedDrain effectiveness must be estimated
Results are expressed as a factor of safety with a Results are expressed as a factor of safety with a pass/fail criterion.pass/fail criterion.
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Sources of UncertaintySources of Uncertainty
““One of the main sources of uncertainty in the One of the main sources of uncertainty in the analysis of gravity dam stability is the amount of analysis of gravity dam stability is the amount of cohesive bondcohesive bond
present at the dam foundation present at the dam foundation
interface. The FERC recognizes that cohesive bond interface. The FERC recognizes that cohesive bond is present, but it is very difficult to quantify through is present, but it is very difficult to quantify through borings and testing.borings and testing.””
FERC Engineering Guidelines FERC Engineering Guidelines
Chapter 3Chapter 3
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Sources of UncertaintySources of Uncertainty
USACE “Gravity Dam Design”
1974
43.5
43
52.5
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Sources of UncertaintySources of Uncertainty
Foundation properties Foundation properties may not always be clear may not always be clear cut.cut.
This SPT test log shows This SPT test log shows blow counts from 20blow counts from 20--
100.100.
What would be chosen What would be chosen based on engineering based on engineering judgementjudgement? ? μ
= 69.8 σ
= 29.0 or 40%
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Sources of UncertaintySources of Uncertainty
SPT SPT Penetration, Penetration,
NN--ValueValue
(blows/ foot)(blows/ foot)
Density of Density of SandSand
(degrees)(degrees)
<4<4 Very looseVery loose <29<29
4 4 --
1010 LooseLoose 29 29 --
3030
10 10 --
3030 MediumMedium 30 30 --
3636
30 30 --
5050 DenseDense 36 36 --
4141
>50>50 Very denseVery dense >41>41
SPT SPT Penetration, Penetration,
NN--ValueValue
(blows/ foot)(blows/ foot)
(degrees)(degrees)
<4<4 2525--3030
4 4 --
1010 2727--3232
10 10 --
3030 30 30 --
3535
30 30 --
5050 35 35 --
4040
>50>50 38 38 --
4343
For Sand For Granular Soils
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Traditional Analysis (cont.)Traditional Analysis (cont.)
AssumptionsAssumptionsC = 0C = 0Full UpliftFull Uplift ΦΦ= 32= 32°°
Shear
Normal
FFCA
SFFtan
SFF = 1.5SFF = 1.5
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Deterministic Approach Meets Deterministic Approach Meets Engineering GuidelinesEngineering Guidelines
Driving Force Resisting Force
Mean SFF =1.5
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Calculated SFF = 1.5Calculated SFF = 1.5
Factor of Safety >= 1.5
Guidelines Met
Actual Values
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Calculated SFF = 1.5Calculated SFF = 1.5
Factor of Safety > 1.0
Guidelines May be Met
Actual Values
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Calculated SFF = 1.5Calculated SFF = 1.5
Factor of Safety > 1.0
Guidelines May be Met
Actual Values
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Calculated SFF = 1.5Calculated SFF = 1.5
Factor of Safety < 1.5
Guidelines Not Met
Actual Values
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Calculated SFF = 1.5Calculated SFF = 1.5
Factor of Safety < 1.0
Dam Failure
Actual Values
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Which Dam is Safer?Which Dam is Safer?
ΦΦ= 32= 32°° SFF = 1.5SFF = 1.5σσSFFSFF
= 0.15= 0.15C = 0C = 0
ΦΦ= 40= 40°° SFF = 2.0SFF = 2.0σσSFFSFF
= 0.40= 0.40C = 0C = 0
Fenton and Griffiths, GeoRisk
2011
Shear
Normal
FFCA
SFFtan
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Chance of failure 0.65% or 1 in 153
Chance of failure 0.05% or 1 in 2000
7:30 AM
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Methods to address uncertainty in Methods to address uncertainty in Deterministic AnalysesDeterministic Analyses
Sensitivity Analysis on parametersSensitivity Analysis on parameters
Use of Conservative estimatesUse of Conservative estimates
Better sampling programBetter sampling program
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Sensitivity AnalysisSensitivity Analysis
Sensitivity tells you if analysis is sensitive to the Sensitivity tells you if analysis is sensitive to the variable.variable.
Does not provide information about uncertainty Does not provide information about uncertainty of variable.of variable.
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Conservatism vs. Conservatism vs. ProbabilismProbabilism
The Phi angle for a certain foundation is listed The Phi angle for a certain foundation is listed as between 30 and 50 degrees. Is choosing 32 as between 30 and 50 degrees. Is choosing 32 conservative?conservative?
It dependsIt depends……
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Conservatism does not change Conservatism does not change DistributionDistribution
40 degrees“Conservative estimate”
is 32 degrees
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Key PointsKey Points
Just because you have a high SFF doesnJust because you have a high SFF doesn’’t t necessarily mean you have a safe dam.necessarily mean you have a safe dam.
Reducing uncertainty can be more important Reducing uncertainty can be more important than the shear strength. than the shear strength.
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Key PointsKey Points
Probabilistic analysis results expressed in Probabilistic analysis results expressed in Probability of Failure, not SFF.Probability of Failure, not SFF.
Best Estimates, not conservative values are used.Best Estimates, not conservative values are used.
Estimates of range of values needed (i.e. Estimates of range of values needed (i.e. Standard Deviation or comparable).Standard Deviation or comparable).
Please note: Gravity dams in particular are more Please note: Gravity dams in particular are more susceptible to unknown weak seams than susceptible to unknown weak seams than foundation material variability.foundation material variability.