Transcript 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 ASCE
September 2002
Hodges Village Dam
Walter F. George Dam
Herbert Hoover Dike
Some Background
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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 Problems
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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 Problems
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For a levee or dam, how does Pr(f) change
with water height ? (Levee guidance and
Hodges Village Dam)
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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
given some water level :
Sand Boil
p = 0.5
Close to levee
p = 0.6
0.09
Carries material
p=0.3
Doesn’t
p = 0.7
Not
close
p = 0.4
Most problems of interest involve or
could be represented by an event tree..
0.06
0.35
Probabilities for the Event Tree
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f (Uncertainty in parameter values)
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Monte Carlo method
FOSM methods
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point estimate
Taylor’s Series
–
–
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Frequency Basis
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Mean Value
Hasofer-Lind
Exponential, Weibull, or other lifetime distribution
Judgmental Values
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Expert elicitation
Pr(f) = Function of Parameter Uncertainty
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Identify performance function and
limit state, typically ln(FS) = 0
Identify random variables, X i
Characterize random variables,
 E[X], s x,
r
Determine E[FS], sFS
Determine Reliability Index, b
Assume Distribution and calculate
Pr(f) = f(b)
The Probability of Failure
Answers the question,
how accurately can FS
be calculated?, given
measure of confidence
in input values
parameter distribution
f 

f(FS)
integration
slope stability model
Pr(f)
1
FS
The Reliability Index, b
Normal Distribution on ln FS
b s ln FS
2.5000
E[lnFS]
b=
sln FS
f(ln FS}
2.0000
1.5000
1.0000
0.5000
Pr (U)
0.0000
-0.2500
0.0000
0.2500
0.5000
ln FS
0.7500
1.0000
Taylor’s series, mean-value FOSM approach
E[ FS] = FS ( E[ X1 ], E[ X 2 ],... E[ X n ])
 FS FS
 FS  2
 r X i ,X j s X i s X j
Var[ FS] =  
 s X i  2 
 X i 
 Xi X j 
2
FS FS( X i  )  FS( X i  )

X i
X i  X i
 FS( X i  )  FS( X i  ) 
Var[ FS] =  



2
2
Slope Stability Results, Lock & Dam No. 2
Run
Case
FS
1
Expected values
2.410
2
Clay strength +
2.901
3
Clay strength -
1.909
4
Sand strength +
2.514
5
Sand strength -
2.314
6
Clay thickness +
2.255
7
Clay thickness -
2.146
Total
Variance
Percent of Total Variance
0.2460
95.0%
0.0100
3.9%
0.0030
1.1%
0.259
100.0%
Lognormal distribution on FS, L&D 2
E[FS] = 2.41
s FS = 0.51
b = 4.11
Change in FS and Pr(f)
( Duncan’s Mine Problem from Uncertainty ‘96 Conference)
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3.5
FS = 1.3, VFS - 10%
3

f(FS)
2.5
2
Evaluate shape change of
probability density function due
to drainage.
Provide enough drainage to
obtain b > 4
1.5
FS = 1.5, VFS = 10%
1
0.5
0
0.75
1
1.25
1.5
1.75
FS
2
2.25
2.5
Pros and Cons of b, Pr(U)
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Advantages
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“Plug and Chug”
fairly easy to
understand with some
training
provides some insight
about the problem
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Disadvantages
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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 b, Pr(f)
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Reliability Index, b
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Pr(f) or Pr(U)
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By how many standard deviations of the
performance functions does the expected
condition exceed the limit state?
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 Probabilities
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Represent probability of event per time
period
Poisson / exponential model wellrecognized 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 Models
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Advantages
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Can be checked
against reality and
history
Can obtain
confidence limits on
the number of events
Is compatible with
economic analysis
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Disadvantages
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Need historical data
Uncertainty in
extending into future
Need
“homogeneous” or
replicate data sets
Ignores site-specific
variations in
structural condition
Judgmental Probabilities
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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 Probabilities
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Advantages
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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
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Disadvantages
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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)
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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 Modes
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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 Example
440
10' crown at el. 420
420
Sand levee with clay face
1V on 2.5 side slopes
8 ft clay top blanket
400
380
80 ft thick pervious sand substratum
Extends to el. 312.0
360
-100
0
100
FOSM Underseepage Analysis
ic
FS
iexit
kf
kb
z
cm/s
cm/s
ft
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
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
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
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
Run
Variance
Percent of
Total
Total
0.0050
0.2%
0.0056
0.3%
1.9127
94.2%
0.1066
5.3%
2.0299
100.0%
Pr (underseepage failure) vs H
1
0.9
0.8
Pr(failure)
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
5
10
H, ft
15
20
Probabilistic Case History
Hodges Village Dam
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A dry
reservoir
Notable
seepage at
high water
events
Very pervious
soils with no
cutoff
Probabilistic Case History
Hodges Village Dam
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Required probabilistic analysis to
demonstrate economic justification
Random variables
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horizontal conductivity
conductivity ratio
critical gradient
FASTSEEP analyses using Taylor’s
series to obtain probabilistic moments of
FS
Probabilistic Case History
Hodges Village Dam
Probabilistic Case History
Hodges Village Dam
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Pr (failure)
= Pr (FS < 1)
This is a
conditional
probability, given
the modeled pool,
which has an annual
probability of
occurrence
Probabilistic Case History
Hodges Village Dam
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Annual Pr (failure)
= Pr [(FS < 1)|pool level] * Pr (pool level)
Integrated over all possible pool levels
Probabilistic Case History
Hodges Village Dam
Probabilistic Case History
Walter F. George Lock and Dam
Probabilistic Case History
Walter F. George Lock and Dam
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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
Probabilistic Case History
Walter F. George Lock and Dam
Probabilistic Case History
Walter F. George Lock and Dam
Probabilistic Case History
Herbert Hoover Dike
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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)
Probabilistic Case History
Herbert Hoover Dike
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Random variables
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Seepage
analysis
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hydraulic conductivities and ratio
piping criteria
FASTSEEP
Probabilistic model
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Taylor’s series
Probabilistic Case History
Herbert Hoover Dike
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Pr (failure) = Pr (FS < 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
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long levee is analogous to system of discrete links
in a chain; a link is hundreds of feet or meters
Questions
Has the theory developed sufficiently
for use in practical applications?
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Yes
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Comparative reliability
problems
Water vs. Sand vs. Clay
pressures on walls,
different b for same FS
Event tree for identifying
relative risks
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No
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Tools for complex
geometries
Absolute reliability
Spatial correlation where
data are sparse
Time-dependent change
in geotechnical
parameters
Accurate annual risk
costs
Questions
When and where are the theories
used most appropriately?
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FOSM Reliability Index
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Reliability Comparisons
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structure to structure
component to component
before and after a repair
relative to desired target value
Insight to Uncertainty Contributions
Questions
When and where are the theories
used most appropriately?
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Frequency - Based Probability
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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
Questions
When and where are the theories
used most appropriately?
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Expert Elicitation
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“Hard” problems without good frequency
data or analytical models
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seepage in rock
likelihood of finding seepage entrance
likelihood of effecting a repair before distress is
catastrophic
Questions
What Methods are Recommended for Reliability
Assessments of Foundations and Structures ?
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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
Questions
Are time-dependent reliability analysis
possible for geotechnical problems? How?
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YES
 Conditional probability values tied to timedependent events such as earthquake
acceleration or water level
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NO
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variation of strength, permeability, geometry
(scour), etc; especially within resource
constraints of planning studies
Needs
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A Lot of Training
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Develop familarity and feeling for techniques by
practicing engineers
Research
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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?
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Not very accurately (my opinion) -Many ill-defined links in process:
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variations in deterministic and probabilistic models
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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.

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