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Reliability Analysis for Dams and Levees
Reliability Analysis for Dams and Levees
Thomas F. Wolff, Ph.D., P.E.Michigan State University
Grand Rapids Branch ASCESeptember 2002
Hodges Village DamHodges Village Dam
Walter F. George DamWalter F. George Dam
Herbert Hoover DikeHerbert Hoover Dike
Some BackgroundSome Background
Corps of Engineers moving to probabilistic benefit-cost analysis for water resource investment decisions (pushed from above)
Geotechnical engineers must quantify relative reliability of embankments and other geotechnical features
Initial implementation must build on existing programs and methodology and be practical within resource constraints
Some Practical ProblemsSome Practical Problems
Given possibility of an earthquake and a high pool, what is the chance of a catastrophic breach ? (Wappapello Dam, St. Louis District, 1985)
Given navigation structures of differing condition, how can they be ranked for investment purposes ? (OCE, 1991+ )
What is the annualized probability of unsatisfactory performance for components of Corps’ structures ? (1992 - 1997)
Some More Practical ProblemsSome More Practical Problems
For a levee or dam, how does Pr(f) change with water height ? (Levee guidance and Hodges Village Dam)
How to characterize the annual probability of failure for segments of very long embankments ? (Herbert Hoover Dike)
How to characterize the annual risk of adverse seepage in jointed limestone ? (Walter F. George Dam)
General Approaches: Event TreeGeneral Approaches: Event Tree
Sand Boilp = 0.5
Carries materialp=0.3
Doesn’tp = 0.7
Close to leveep = 0.6
Notclosep = 0.4
0.09
0.06
0.35
Most problems of interest involve or could be represented by an event tree..
given some water level :
Probabilities for the Event TreeProbabilities for the Event Tree
f (Uncertainty in parameter values) Monte Carlo method FOSM methods
point estimate Taylor’s Series
– Mean Value– Hasofer-Lind
Frequency Basis Exponential, Weibull, or other lifetime distribution
Judgmental Values Expert elicitation
Pr(f) = Function of Parameter UncertaintyPr(f) = Function of Parameter Uncertainty
Identify performance function and limit state, typically ln(FS) = 0
Identify random variables, X i
Characterize random variables, E[X], x,
Determine E[FS], FS
Determine Reliability Index, Assume Distribution and calculate
Pr(f) = f()
The Probability of FailureThe Probability of Failure
f
FS
f(FS)
1
Pr(f)
parameter distribution
slope stability model
integration
Answers the question, how accurately can FS be calculated?, given measure of confidence in input values
The Reliability Index, The Reliability Index,
Normal Distribution on ln FS
0.0000
0.5000
1.0000
1.5000
2.0000
2.5000
-0.2500 0.0000 0.2500 0.5000 0.7500 1.0000
ln FS
f(ln
FS
}
ln FS
E FS
FS
[ln ]
ln
Pr (U)
Taylor’s series, mean-value FOSM approachTaylor’s series, mean-value FOSM approach
E FS FS E X E X E Xn[ ] ( [ ], [ ],... [ ]) 1 2
Var FSFS
X
FS
X
FS
XiX
i jX X X Xi i j i j
[ ] ,
2
2 2
FS
X
FS X FS X
X Xi
i i
i i
( ) ( )
Var FSFS X FS Xi i[ ]
( ) ( )
2
2
Slope Stability Results, Lock & Dam No. 2Slope Stability Results, Lock & Dam No. 2
Run Case FS Variance Percent of Total Variance
1 Expected values 2.410
2 Clay strength + 2.901 0.2460 95.0%
3 Clay strength - 1.909
4 Sand strength + 2.514 0.0100 3.9%
5 Sand strength - 2.314
6 Clay thickness + 2.255 0.0030 1.1%
7 Clay thickness - 2.146
Total 0.259 100.0%
Lognormal distribution on FS, L&D 2Lognormal distribution on FS, L&D 2
0.00
0.20
0.40
0.60
0.80
1.00
0 1 2 3 4 5
Factor of Safety
E[FS] = 2.41FS = 0.51
= 4.11
Change in FS and Pr(f)Change in FS and Pr(f)
0
0.5
1
1.5
2
2.5
3
3.5
0.75 1 1.25 1.5 1.75 2 2.25 2.5
FS
f(F
S)
Evaluate shape change of probability density function due to drainage.
Provide enough drainage to obtain > 4
FS = 1.3, VFS - 10%
FS = 1.5, VFS = 10%
( Duncan’s Mine Problem from Uncertainty ‘96 Conference)
Pros and Cons of , Pr(U)Pros and Cons of , Pr(U)
Advantages “Plug and Chug” fairly easy to
understand with some training
provides some insight about the problem
Disadvantages Still need better practical
tools for complex problems Non-unique, can be
seriously in error No inherent time
component only accounts for
uncertainties related to parameter values and models
Physical Meaning of , Pr(f)Physical Meaning of , Pr(f)
Reliability Index, By how many standard deviations of the
performance functions does the expected condition exceed the limit state?
Pr(f) or Pr(U) If a large number of statistically similar structures
(were designed) (were constructed) (existed) in these same conditions (in parallel universes?), what fraction would fail or perform unsatisfactorily?
Has No Time or Frequency Basis !
Frequency-based ProbabilitiesFrequency-based Probabilities
Represent probability of event per time period
Poisson / exponential model well-recognized in floods and earthquakes
Weibull model permits increasing or decreasing event rates as f(t), well developed in mechanical & electrical appliactions
Some application in material deterioration Requires historical data to fit
Pros and Cons of Frequency ModelsPros and Cons of Frequency Models
Advantages Can be checked
against reality and history
Can obtain confidence limits on the number of events
Is compatible with economic analysis
Disadvantages Need historical data Uncertainty in
extending into future Need
“homogeneous” or replicate data sets
Ignores site-specific variations in structural condition
Judgmental ProbabilitiesJudgmental Probabilities
Mathematically equivalent to previous two, can be handled in same way
Can be obtained by Expert Elicitation a systematic method of quantifying
individual judgments and developing some consensus, in the absence of means to quantify frequency data or parameter uncertainty
Pros and Cons of Judgmental ProbabilitiesPros and Cons of Judgmental Probabilities
Advantages Gives you a number
when nothing else will May be better reality
check than parameter uncertainty approach
permits consideration of site-specific information
Some experience in application to dams
Disadvantages Distrusted by some
(including some within Federal Agencies)
Some consider values “less accurate” than calculated ones
Non-unique values Who is an expert?
An Application:Levee Reliability = f (Water Level)An Application:Levee Reliability = f (Water Level)
Previous Corps’ policy treated substandard levees as not present for benefit calculations
New policy assumes levee present with some probability, a function of water level
First approach by Corps took relationship linear, R = 1 at base, R = 0 at crown
New research to develop functional shape
Levee Failure ModesLevee Failure Modes
Underseepage Slope Stability Internal erosion from through-seepage External erosion
through-seepage current velocity wave attack animal burrows, cracking, etc., may require
judgmental models Combine using system reliability methods
Pervious Sand Levee ExamplePervious Sand Levee Example
440
420
400
380
360
0-100 100
10' crown at el. 420
1V on 2.5 side slopes8 ft clay top blanket
80 ft thick pervious sand substratum
Extends to el. 312.0
Sand levee with clay face
FOSM Underseepage AnalysisFOSM Underseepage Analysis
Run kf
cm/s
kb
cm/s
z
ft
ic iexit FS Variance Percent of
Total
1 0.11 0.0001 10.0 0.843 0.245 3.441
2 0.131 0.0001 10.0 0.843 0.249 3.386
3 0.089 0.0001 10.0 0.843 0.239 3.527 0.0050 0.2%
4 0.11 0.00012 10.0 0.843 0.240 3.513
5 0.11 0.00008 10.0 0.843 0.250 3.372 0.0056 0.3%
6 0.11 0.0001 14.4 0.843 0.175 4.187
7 0.11 0.0001 5.6 0.843 0.411 2.051 1.9127 94.2%
8 0.11 0.0001 10.0 0.923 0.245 3.767
9 0.11 0.0001 10.0 0.763 0.245 3.114 0.1066 5.3%
Total 2.0299 100.0%
Pr (underseepage failure) vs HPr (underseepage failure) vs H
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 5 10 15 20
H, ft
Pr(
failu
re)
Probabilistic Case HistoryHodges Village DamProbabilistic Case HistoryHodges Village Dam
A dry reservoir
Notable seepage at high water events
Very pervious soils with no cutoff
Probabilistic Case HistoryHodges Village DamProbabilistic Case HistoryHodges Village Dam
Required probabilistic analysis to demonstrate economic justification
Random variablesRandom variables horizontal conductivity conductivity ratio critical gradient
FASTSEEPFASTSEEP analyses using Taylor’s series to obtain probabilistic moments of FS
Probabilistic Case HistoryHodges Village DamProbabilistic Case HistoryHodges Village Dam
Probabilistic Case HistoryHodges Village DamProbabilistic Case HistoryHodges Village Dam
Pr (failure) = Pr (FS < 1)Pr (FS < 1)
This is a conditional conditional probabilityprobability, given the modeled pool, which has an annual probability of occurrence
Probabilistic Case HistoryHodges Village DamProbabilistic Case HistoryHodges Village Dam
Annual Pr (failure)
= Pr [(FS < 1)|pool level] * Pr (pool level)
Integrated over all possible pool levels
Probabilistic Case HistoryHodges Village DamProbabilistic Case HistoryHodges Village Dam
Probabilistic Case HistoryWalter F. George Lock and DamProbabilistic Case HistoryWalter F. George Lock and Dam
Probabilistic Case HistoryWalter F. George Lock and DamProbabilistic Case HistoryWalter F. George Lock and Dam
Has had several known seepage eventsseepage events in 40 year history
From Weibull or Poisson frequency frequency analysisanalysis, can determine the probability distribution on the number of future events
Probabilistic Case HistoryWalter F. George Lock and DamProbabilistic Case HistoryWalter F. George Lock and Dam
Probabilistic Case HistoryWalter F. George Lock and DamProbabilistic Case HistoryWalter F. George Lock and Dam
Probabilistic Case HistoryHerbert Hoover DikeProbabilistic Case HistoryHerbert Hoover Dike
128 mile long128 mile long dike surrounds Lake Okeechobee, FL
Built without cutoffs or filtered seepage control system
Boils and sloughing occur at high pool levels
Failure expectedFailure expected in 100 yr event (El 21)
Probabilistic Case HistoryHerbert Hoover DikeProbabilistic Case HistoryHerbert Hoover Dike
Random variablesRandom variables hydraulic conductivities and ratio piping criteria
Seepage Seepage analysisanalysis FASTSEEP
Probabilistic modelProbabilistic model Taylor’s series
Probabilistic Case HistoryHerbert Hoover DikeProbabilistic Case HistoryHerbert Hoover Dike
Pr (failure) = Pr (FS < 1)Pr (FS < 1) Similar to Hodges Village, this is a
conditional probabilityconditional probability, given the occurrence of the modeled pool, which is has an annual probability
Consideration of length effectslength effects long levee is analogous to system of discrete links
in a chain; a link is hundreds of feet or meters
QuestionsQuestions
Yes Comparative reliability
problems Water vs. Sand vs. Clay
pressures on walls, different for same FS
Event tree for identifying relative risks
No Tools for complex
geometries Absolute reliability Spatial correlation where
data are sparse Time-dependent change
in geotechnical parameters
Accurate annual risk costs
Has the theory developed sufficiently for use in practical applications?
QuestionsQuestions
FOSM Reliability Index Reliability Comparisons
structure to structure component to component before and after a repair relative to desired target value
Insight to Uncertainty Contributions
When and where are the theories used most appropriately?
QuestionsQuestions
Frequency - Based Probability Earthquake and Flood recurrence, with
conditional geotechnical probability values attached thereto
Recurring random events where good models are not available: scour, through-seepage, impact loads, etc.
Wearing-in, wearing-out, corrosion, fatigue
When and where are the theories used most appropriately?
QuestionsQuestions
Expert Elicitation “Hard” problems without good frequency
data or analytical models seepage in rock likelihood of finding seepage entrance likelihood of effecting a repair before distress is
catastrophic
When and where are the theories used most appropriately?
QuestionsQuestions
Define purpose of analysis Select simplest reasonable approach consistent
with purpose Build an event tree Fill in probability values using whichever of three
approaches is appropriate to that node Understand and admit relative vs absolute
probability values
What Methods are Recommended for Reliability Assessments of Foundations and Structures ?
QuestionsQuestions
YES Conditional probability values tied to time-
dependent events such as earthquake acceleration or water level
NO variation of strength, permeability, geometry
(scour), etc; especially within resource constraints of planning studies
Are time-dependent reliability analysis possible for geotechnical problems? How?
NeedsNeeds
A Lot of Training Develop familarity and feeling for techniques by
practicing engineers Research
Computer tools for practical probabilistic seepage and slope stability analysis for complex problems
Characterizing and using real mixed data sets, of mixed type and quality, on practical problems, including spatial correlation issues
Approaches and tools for Monte Carlo analysis
How accurately can Pr(f) be calculated?How accurately can Pr(f) be calculated?
Not very accurately (my opinion) --Many ill-defined links in process: variations in deterministic and probabilistic models different methods of characterizing soil parameters - c strength envelopes are difficult slope is a system of slip surfaces- distributions of permeability and permeability ratio difficult to quantify spatial correlation in practice difficult to account for length of embankments difficult to account for independence vs correlation of multiple
monoliths, multiple footings, etc.