what’s the point of modelling uncertainty in engineering
TRANSCRIPT
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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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A basic engineering problem
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A basic engineering problem
• Basic failure definition:
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A basic engineering problem
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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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The alternative is a deterministic code-based approach
• Select the cheapest design complying with the code:
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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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• 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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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 )
• For given S=S, find the optimal design
– according to thedeterministic analysis:
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– according to theprobabilistic analysis:
Value of informationconditional on different design situations
The conditional value of information is:
Expected cost withdeterministic design
Expected cost withprobabilistic design
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Value of informationconditional on different design situations
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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Avalanche risk analysis
• Parameter uncertainty
• E.g. frictionparameter
32Straub D., Grêt‐Regamey A. (2006). Cold Regions Science and Technology, 46(3) , pp. 192‐203.
Avalanche risk analysis
• Parameter uncertainty
• E.g. frictionparameter
33Straub D., Grêt‐Regamey A. (2006). Cold Regions Science and Technology, 46(3) , pp. 192‐203.
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Avalanche risk analysis
• Observationsavailable(here 50 years)
34Straub D., Grêt‐Regamey A. (2006). Cold Regions Science and Technology, 46(3) , pp. 192‐203.
Avalanche risk analysis
• Observationsavailable(here 50 years)
35Straub D., Grêt‐Regamey A. (2006). Cold Regions Science and Technology, 46(3) , pp. 192‐203.
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Avalanche risk analysis – Bayesian updating
36Straub D., Grêt‐Regamey A. (2006). Cold Regions Science and Technology, 46(3) , pp. 192‐203.
Results in a probabilistic hazard map
37Straub D., Grêt‐Regamey A. (2006). Cold Regions Science and Technology, 46(3) , pp. 192‐203.
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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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Bayesian network in a nutshell
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Bayesian network in a nutshell
• Efficient factoring of the joint probability distribution intoprobability distribution into conditional (local) distributions given the parents
)|()|()|()(
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xxpxxpxxpxp
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Here:
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General:
Bayesian network in a nutshell
• Facilitates Bayesian updating when additional information (evidence)additional information (evidence) is available
)(
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xepexp
E.g.:
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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
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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
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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
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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)