what’s the point of modelling uncertainty in engineering

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[ISUME, CTU Prague, May 2, 2011 ]

What’s the point of modellinguncertainty in engineering?Optimal decision making under uncertainty

Daniel StraubEngineering Risk Analysis GroupTU München

What is the value of a probabilistic analysis?

• Probabilistic analysis providesa more accurate description of the systema more accurate description of the system

• For engineers, this is not an inherent benefit

• The improved system description might supportthe identification of better engineering solutions

• This is the benefit

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A basic engineering problem

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• Basic failure definition:

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Design A is the optimal choice(as identified with a probabilistic analysis)

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The alternative is a deterministic code-based approach

• Select the cheapest design complying with the code:

Code criterion

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Code criterion

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Code criterion

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• Both comply Design A is selected

Code criterion

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The probabilistic analysisprovides no benefit in this case

• Both analyses lead to the same design

• Conditional value of information (CVI) of the probabilistic analysis is zero

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Value of informationconditional on different design situations

• Same design options A and B• Loading environment can change: Vary and (= 0 25 )• Loading environment can change: Vary S and S (= 0.25S )

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• Same design options A and B• Loading environment can change: Vary and (= 0 25 )• Loading environment can change: Vary S and S (= 0.25S )

• For given S=S, find the optimal design

– according to thedeterministic analysis:

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– according to theprobabilistic analysis:


The conditional value of information is:

Expected cost withdeterministic design

Expected cost withprobabilistic design

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The conditional value of information is:

Expected cost withdeterministic design

Expected cost withprobabilistic design

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• The CVI cannot be negative!

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Value of information of the probabilistic analysis

• The value of information is the expected value of the CVI with respect to all possible design situations:with respect to all possible design situations:

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Value of information of the probabilistic analysis

• The value of information is the expected value of the CVI with respect to the possible design situations:with respect to the possible design situations:

• Assuming a uniform distribution of S in the range[50kN 150kN] we obtain

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[50kN,150kN], we obtain

• (The cost of the design/construction is in the order of 10-3)

When is a probabilistic analysis useful in practice?(Some lessons to be learnt from the example)

• The analysis must be able to identify better solutions than a deterministic analysisdeterministic analysis

• The benefit of the better solution must be significantly higher than the cost of the analysis

• Useful for problems– In which the phenomena cannot be adequately modeled

deterministically

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deterministically• In the presence of large uncertainties and/or non-linear effects• When dealing with collecting information

– Where the potential benefit is huge (e.g. optimization of aircraft design)

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When is a probabilistic analysis useful in practice?(Some lessons to be learnt from the example)

• The analysis must be able to identify better solutions than a deterministic analysisdeterministic analysis

• The benefit of the better solution must be significantly higher than the cost of the analysis

• Useful for problems– In which the phenomena cannot be adequately modeled

deterministically

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deterministically• In the presence of large uncertainties and/or non-linear effects• When dealing with collecting information

– Where the potential benefit is huge (e.g. optimization of aircraft design)

Value of information theory:

• Raiffa H., and R. Schlaifer (1961), Applied Statistical Decision Theory, Cambridge University Press, Cambridge.Theory, Cambridge University Press, Cambridge.

• Benjamin, J. R., and C. A. Cornell (1970), Probability, statistics, and decision for civil engineers, McGraw-Hill, New York.

• Straub, D. (2004), Generic Approaches to Risk Based Inspection Planning for Steel Structures, PhD thesis, ETH Zürich.

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What are we doing?Decisions in complex systems under conditions of uncertainty

Aging of the infrastructuresystem:‐Monitoring & Inspection

Natural hazards in the system„built environment“‐ Prevention

Safety in the system „society“‐ Target reliability‐ Prescriptive limits

‐Maintenance‐ Replacement / redesign

‐ Emergency response‐ Rehabilitation

‐ Service life duration

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Three applications

a. Avalanche risk analysisb Dependence in earthquake fragility modellingb. Dependence in earthquake fragility modelling c. Planning of inspections in offshore structures

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Avalanche riskassessment

• Where is it safe to build?• Where should protection• Where should protection

measures beimplemented?

• When should roads beclosed / buildings beevacuated?

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Source: Kt. St. Gallen, Switzerland

Avalanche risk analysis

Avalanche model:

31Straub D., Grêt‐Regamey A. (2006). Cold Regions Science and Technology, 46(3) , pp. 192‐203.

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• Parameter uncertainty

• E.g. frictionparameter



• Parameter uncertainty

• E.g. frictionparameter


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• Observationsavailable(here 50 years)



• Observationsavailable(here 50 years)


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Avalanche risk analysis – Bayesian updating


Results in a probabilistic hazard map


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Bayesian networks for avalanche risk assessment

38Grêt‐Regamey A., Straub D. (2006). Natural Hazards and Earth System Sciences, 6(6), pp. 911‐926.

Implementation of the BN modelsin software is straightforward

• Implementation in a GIS environmentGIS environment

• Regional risk analysis

39Grêt‐Regamey A., Straub D. (2006). Natural Hazards and Earth System Sciences, 6(6), pp. 911‐926.

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Modelling dependence in Earthquake fragility(Statistical dependence is not captured by simple analyses)

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• Tsunami warning example:

Bayesian network is a powerful modeling tool

41Straub D., (2010). Lecture Notes in Engineering Risk Analysis. TU München

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Bayesian network in a nutshell

• Probabilistic models based on directed acyclic graphsdirected acyclic graphs

• Models the joint probability distribution of a set of variables

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• Efficient factoring of the joint probability distribution intoprobability distribution into conditional (local) distributions given the parents

)|()|()|()(

),,,(

3413121

4321

xxpxxpxxpxp

xxxxp

Here:

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3413121

])(|[)(1

n

iii xpaxpp x

General:


• Facilitates Bayesian updating when additional information (evidence)additional information (evidence) is available

)(

),()|(

2

3223 ep

xepexp

E.g.:

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2

1

)|()(

)|()|()(

121

13121

X

X

xepxp

xxpxepxp e

Straub D., (2010). Lecture Notes in Engineering Risk Analysis. TU München

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Modelling with BN: System dependence through common factors

• Performance of an electrical substation during an EQ

0.5

0.6

0.7

0.8

0.9

1

gilit

y

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

0.1

0.2

0.3

0.4

0.5

PGA [g]

Fra

gil

Can we observe the statistical dependence ?

1 20 Number of failures in 20 components

Failures are statistically independent

0.4

0.6

0.8

Frag

ilit

y

5

10

15 Failures are statistically dependent

Failures are statistically independent

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0 0.3 0.6 0.90

0.2

PGA [g]

0

5

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When accounting for dependence, the system fragility strongly increases

• Redundant system:(parallel system with 100 Parallel system TR 1

5 components)

10− 4

10− 3

10− 2

10− 1

Syst

em fr

agili

ty

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.910− 6

10− 5

10

PGA [g]

Including dependenceNeglecting dependence

Straub D., Der Kiureghian A. (2008). Structural Safety, 30(4), pp. 320‐366.

EQ: Modeling systems and portfolio of structures

M4

M5

Q1

R5

R1

UR

R3

R2

R4

V

R4a‘

R4b‘

R5a‘

R5b‘

Q

Q2

Q20

E(1) E(2) E(20)

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H1(1) H

1(2) H

1(20)

UH1

UH2

UH20

UH

H(1) H(2) H(20)

Straub D., Der Kiureghian A., (2010). Journal of Engineering Mechanics

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Risk-based inspection, maintenance, repair planning

• Structures deteriorate with time• Deterioration is associated with large uncertainty

f d d• Inspections are performed to reduce uncertainty The effect of inspections (and monitoring) can only be

appraised probabilistically

• Applications:– Offshore structures subject to fatigue, corrosion,

scour, ship impact, …Process systems subject to corrosion erosion

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– Process systems subject to corrosion, erosion, SCC, etc…

– Concrete structures (tunnels, bridges) subject to corrosion of the reinforcement

– Aircraft structures

Optimizing inspection strategies

• Deterioration of offshore steel structures and pipelines• Goal: Optimize sub-sea inspections• Goal: Optimize sub-sea inspections

Zona de plataformas

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Plan and optimize inspections

• We model the entire service life through event trees:

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• Fracture mechanics based probabilistic models of crack growth:

Probabilistic deterioration modelling

Fatigue loads Structural response Crack growth

d

b

17

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,

,

,

,

fm

fm

m

P a a

m

P c c

daC K a c

dNdc

C K a cdN

S

4 6 8 10 12 147

8

9

10

11

12

13

14

15

16

HS [m]

TP [

s]

1/pF = 25yr

1/pF = 100yr

1/pF = 250yr

1/pF = 1000yr

Inspection modeling

• Inspections are also modeled qualitatively

Probability of Detection on tubulars, underwater

0.8

1ACFM

MPI

0

0.2

0.4

0.6

0 2 4 6 8 10

Crack depth [mm]

PO

D

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Probability of failure as a function of time and the influence of inspection

58Straub D., Faber M.H. (2006). Computer‐Aided Civil and Infrastructure Engineering, 21(3), pp. 179‐192.

Plan and optimize inspections

• We model the entire service life through event trees:

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The maximum probability of failure determines the number of inspections

10-3

pF

pFT = 10-3 yr-1

10-4

An

nu

al p

rob

abili

ty o

f fa

ilure

pFT = 10-4 yr-1

60

0 10 20 30 40 50 60 70 80 90 10010-5

Year t

A

Inspectiontimes:

t

Optimization

61Straub D., Faber M.H. (2004). J. of Offshore Mechanics and Arctic Engineering, 126(3), pp. 265‐271.

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IT implementation (iPlan)

• Calculating inspection plans using the generic approach:

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Structural importance

Remove elements

• How to dermineo to de eredundancy?

• Deterministic approach is not sufficient (most components are not part of the dominant mechanism)

63Straub D., Der Kiureghian A. (2011) J. of Structural Engineering, in print.

)

• (Simplified) probabilistic approach is needed

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Optimize inspections in the structural systems

Single component: 1-5 Decision variables Structure: 100 -1000 Decision variables Heuristic method based on Value of Information Heuristic method based on Value-of-Information

64Straub D., Faber M.H. (2005). Structural Safety, 27(4), pp 335‐355.

DBN model for deterioration modeling

m m1 m2 m3 mTC

q1 q2

a0 a1 a2

q3

a3

qT

aT

qS

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Inspection

Failure/survival E1

Z1

E2

Z2

E3

Z3

ET

ZT

Straub D. (2009). Journal of Engineering Mechanics, 135(10), pp. 1089‐1099

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Bayesian updating is robust AND efficient

66Straub D. (2009). Journal of Engineering Mechanics, 135(10), pp. 1089‐1099

Monitoring, Inspection and Maintenance for Concrete Structures

Zone A

Zone B

67Straub D., et al. (2009). Structure and Infrastructure Engineering,

t,1 t,i t,n. . . . . .

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Therefore,

… simplified (engineering) models are often sufficient formaking optimal decisionsmaking optimal decisions

… but probabilistic analysis can provide useful insights and help making better decisions

… if we ensure that the benefit of the analysis outweighs thef h l

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cost of the analysis

questions or comments?

[email protected] / www.era.bv.tum.de

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(D. Straub, July 2006)

what’s the point of modelling uncertainty in engineering

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