Reliability Analysis for Dams and Levees

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Reliability Analysis for Dams and Levees

• Thomas F. Wolff, Ph.D., P.E.
• Michigan State University
• Grand Rapids Branch ASCE
• September 2002

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Hodges Village Dam
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Walter F. George Dam
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Herbert Hoover Dike
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Some 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

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Some 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)

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Some 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)

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General Approaches Event Tree
Close to levee p 0.6
given some water level
0.09
Carries material p0.3
Sand Boil p 0.5
Not close p 0.4
0.06
Doesnt p 0.7
0.35
Most problems of interest involve or could be
represented by an event tree..
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Probabilities for the Event Tree

• f (Uncertainty in parameter values)
• Monte Carlo method
• FOSM methods
• point estimate
• Taylors Series
• Mean Value
• Hasofer-Lind
• Frequency Basis
• Exponential, Weibull, or other lifetime
  distribution
• Judgmental Values
• Expert elicitation

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Pr(f) Function of Parameter Uncertainty

• Identify performance function and limit state,
  typically ln(FS) 0
• Identify random variables, X i
• Characterize random variables,
• EX, s x, r
• Determine EFS, sFS
• Determine Reliability Index, b
• Assume Distribution and calculate
• Pr(f) f(b)

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The Probability of Failure
Answers the question, how accurately can FS be
calculated?, given measure of confidence in input
values
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The Reliability Index, b
E
FS
ln

b s ln FS
b

s
FS
ln
Pr (U)
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Taylors series, mean-value FOSM approach
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Slope Stability Results, Lock Dam No. 2
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Lognormal distribution on FS, LD 2
EFS 2.41 s FS 0.51 b 4.11
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Change in FS and Pr(f)
( Duncans Mine Problem from Uncertainty 96
Conference)

• Evaluate shape change of probability density
  function due to drainage.
• Provide enough drainage to obtain b gt 4

FS 1.3, VFS - 10
FS 1.5, VFS 10
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Pros and Cons of b, 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

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Physical Meaning of b, Pr(f)

• Reliability Index, b
• 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 !

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Frequency-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

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Pros 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

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Judgmental 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

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Pros 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?

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An ApplicationLevee 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

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Levee 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

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Pervious Sand Levee Example
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FOSM Underseepage Analysis
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Pr (underseepage failure) vs H
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Probabilistic Case HistoryHodges Village Dam

• A dry reservoir
• Notable seepage at high water events
• Very pervious soils with no cutoff

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Probabilistic Case HistoryHodges Village Dam

• Required probabilistic analysis to demonstrate
  economic justification
• Random variables
• horizontal conductivity
• conductivity ratio
• critical gradient
• FASTSEEP analyses using Taylors series to obtain
  probabilistic moments of FS

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Probabilistic Case HistoryHodges Village Dam
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Probabilistic Case HistoryHodges Village Dam

• Pr (failure) Pr (FS lt 1)
• This is a conditional probability, given the
  modeled pool, which has an annual probability of
  occurrence

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Probabilistic Case HistoryHodges Village Dam

• Annual Pr (failure) Pr (FS lt 1)pool level
  Pr (pool level)Integrated over all possible
  pool levels

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Probabilistic Case HistoryHodges Village Dam
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Probabilistic Case HistoryWalter F. George Lock
and Dam
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Probabilistic Case HistoryWalter F. George Lock
and Dam

• Has had several known seepage events in 40 year
  history
• From Weibull or Poisson frequency analysis, can
  determine the probability distribution on the
  number of future events

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Probabilistic Case HistoryWalter F. George Lock
and Dam
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Probabilistic Case HistoryWalter F. George Lock
and Dam
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Probabilistic Case HistoryHerbert Hoover Dike

• 128 mile long dike surrounds Lake Okeechobee, FL
• Built without cutoffs or filtered seepage control
  system
• Boils and sloughing occur at high pool levels
• Failure expected in 100 yr event (El 21)

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Probabilistic Case HistoryHerbert Hoover Dike

• Random variables
• hydraulic conductivities and ratio
• piping criteria
• Seepage analysis
• FASTSEEP
• Probabilistic model
• Taylors series

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Probabilistic Case HistoryHerbert Hoover Dike

• Pr (failure) Pr (FS lt 1)
• Similar to Hodges Village, this is a conditional
  probability, given the occurrence of the modeled
  pool, which is has an annual probability
• Consideration of length effects
• long levee is analogous to system of discrete
  links in a chain a link is hundreds of feet or
  meters

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Questions
Has the theory developed sufficiently for use in
practical applications?

• Yes
• Comparative reliability problems
• Water vs. Sand vs. Clay pressures on walls,
  different b 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

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Questions
When and where are the theories used most
appropriately?

• 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

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Questions
When and where are the theories used most
appropriately?

• 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

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Questions
When and where are the theories used most
appropriately?

• 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

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Questions
What Methods are Recommended for Reliability
Assessments of Foundations and Structures ?

• 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

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Questions
Are time-dependent reliability analysis possible
for geotechnical problems? How?

• 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

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Needs

• 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

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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
• f - 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.