reliability analysis for dams and levees

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.