04.05.2011
[ISUME, CTU Prague, May 2, 2011 ]
What’s the point of modelling
uncertainty in engineering?
Optimal decision making
under uncertainty
Daniel Straub
Engineering Risk Analysis Group
TU München
What is the value of a probabilistic analysis?
• Probabilistic analysis provides
a more accurate description of the system
• For engineers, this is not an inherent benefit
• The improved system description might support
the 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
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
Code criterion
• Both comply  Design A is selected
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The probabilistic analysis
provides 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 information
conditional on different design situations
• Same design options A and B
• Loading environment can change: Vary S and S (= 0.25
0 25S )
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Value of information
conditional on different design situations
• Same design options A and B
• Loading environment can change: Vary S and S (= 0.25
0 25S )
• For given S=S, find the optimal design
– according to the
deterministic analysis:
– according to the
probabilistic analysis:
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Value of information
conditional on different design situations
The conditional value of information is:
Expected cost with
deterministic design
Expected cost with
probabilistic design
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Value of information
conditional on different design situations
The conditional value of information is:
Expected cost with
deterministic design
Expected cost with
probabilistic design
• 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:
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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:
• Assuming a uniform distribution of S in the range
[50kN 150kN] we obtain
[50kN,150kN],
• (The cost of the design/construction is in the order of 10-3)
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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 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
• 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 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
• 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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Value of information theory:
• Raiffa H., and R. Schlaifer (1961), Applied Statistical Decision
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 infrastructure
system:
‐ Monitoring & Inspection
‐ Maintenance
‐ Replacement / redesign
Natural hazards in the system
„built environment“
‐ Prevention
‐ Emergency response
‐ Rehabilitation
Safety in the system „society“
‐ Target reliability
‐ Prescriptive limits
‐ Service life duration
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Three applications
a. Avalanche risk analysis
b Dependence in earthquake fragility modelling
b.
c. Planning of inspections in offshore structures
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Avalanche risk
assessment
• Where is it safe to build?
• Where should protection
measures be
implemented?
• When should roads be
closed / buildings be
evacuated?
Source: Kt. St. Gallen, Switzerland
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Avalanche risk analysis
Avalanche model:
Straub 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. friction
parameter 
Straub 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. friction
parameter 
Straub D., Grêt‐Regamey A. (2006). Cold Regions Science and Technology, 46(3) , pp. 192‐203.
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Avalanche risk analysis
• Observations
available
(here 50 years)
Straub D., Grêt‐Regamey A. (2006). Cold Regions Science and Technology, 46(3) , pp. 192‐203.
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Avalanche risk analysis
• Observations
available
(here 50 years)
Straub D., Grêt‐Regamey A. (2006). Cold Regions Science and Technology, 46(3) , pp. 192‐203.
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Avalanche risk analysis – Bayesian updating
Straub D., Grêt‐Regamey A. (2006). Cold Regions Science and Technology, 46(3) , pp. 192‐203.
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Results in a probabilistic hazard map
Straub 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
Grêt‐Regamey A., Straub D. (2006). Natural Hazards and Earth System Sciences, 6(6), pp. 911‐926.
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Implementation of the BN models
in software is straightforward
• Implementation in a
GIS environment
• Regional risk analysis
Grêt‐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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Bayesian network is a powerful modeling tool
• Tsunami warning example:
Straub 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 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 into
conditional (local) distributions
given the parents
Here:
p ( x1 , x2 , x3 , x4 ) 
 p( x1 ) p( x2 | x1 ) p( x3 | x1 ) p( x4 | x3 )
General:
n
p (x)   p[ xi | pa ( xi )]
i 1
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Bayesian network in a nutshell
• Facilitates Bayesian updating when
additional information (evidence)
is available
E.g.:
p ( x3 | e2 ) 
p (e2 , x3 )
p(e2 )
 p ( x ) p (e
1

X1
2
| x1 ) p ( x3 | x1 )
 p ( x ) p (e
1
2
e
| x1 )
X2
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
1
0.9
0.8
Fragil
gility
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
PGA [g]
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Can we observe the statistical dependence ?
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Number of failures in 20 components
Failures are statistically independent
Fragility
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Failures are statistically dependent
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0.6
10
0.4
5
0.2
0
0
0
0.3
0.6
0.9
PGA [g]
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When accounting for dependence,
the system fragility strongly increases
• Redundant system:
(parallel system with
5 components)
Parallel system TR 1
0
10
−1
System fragility
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−2
10
−3
10
−4
10
−5
Including dependence
Neglecting dependence
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−6
10
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
PGA [g]
Straub D., Der Kiureghian A. (2008). Structural Safety, 30(4), pp. 320‐366.
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EQ: Modeling systems and portfolio of structures
UR
R1
R2
R3
V
M4
M5
R4a‘
R5a‘
R4
R5
R4b‘
R5b‘
Q
Q1
Q2
Q20
E(1)
E(2)
E(20)
H1(1)
H1(2)
H1(20)
H(1)
H(2)
H(20)
UH1
UH2
UH20
UH
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
Inspections are performed
f
d to reduce
d
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
corrosion, erosion
erosion,
SCC, etc…
– Concrete structures (tunnels, bridges) subject to
corrosion of the reinforcement
– Aircraft structures
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Optimizing inspection strategies
• Deterioration of offshore steel structures and pipelines
• 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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Probabilistic deterioration modelling
• Fracture mechanics based probabilistic models of crack growth:
Fatigue loads
Structural response
Crack growth
b
d
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16
15
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S
12
m fm
da
 CP ,a  K a  a, c  
dN
m fm
dc
 CP ,c  K c  a, c  
dN
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10
1/pF = 25yr
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1/pF = 100yr
1/pF = 250yr
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1/pF = 1000yr
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6
8
10
12
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H [m]
S
Inspection modeling
Probability of Detection on tubulars, underwater
• Inspections are also
modeled qualitatively
1
ACFM
MPI
0.8
POD
TP [s]
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0.6
0.4
0.2
0
0
2
4
6
8
10
Crack depth [mm]
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Probability of failure as a function of time
and the influence of inspection
Straub D., Faber M.H. (2006). Computer‐Aided Civil and Infrastructure Engineering, 21(3), pp. 179‐192.
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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
pFT = 10-3 yr-1
Annual probability of failure pF
A
10-3
pFT = 10-4 yr-1
-4
10
10-5
0
10
Inspection
times:
20
30
40
50
60
70
80
90
100
Year t
t
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Optimization
Straub D., Faber M.H. (2004). J. of Offshore Mechanics and Arctic Engineering, 126(3), pp. 265‐271. 61
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IT implementation (iPlan)
•
Calculating inspection plans using the generic approach:
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Structural importance
Remove elements
• How
o to de
dermine
e
redundancy?
• Deterministic approach
is not sufficient (most
components are not
part of the dominant
mechanism))
• (Simplified) probabilistic
approach is needed
Straub D., Der Kiureghian A. (2011) J. of Structural Engineering, in print. 63
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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
Value of Information
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Straub D., Faber M.H. (2005). Structural Safety, 27(4), pp 335‐355.
DBN model for deterioration modeling
C
m
m1
m2
m3
mT
S
q
q1
q2
q3
qT
a0
a1
a2
a3
aT
Inspection
Z1
Z2
Z3
ZT
Failure/survival
E1
E2
E3
ET
Straub D. (2009). Journal of Engineering Mechanics, 135(10), pp. 1089‐1099
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Bayesian updating is robust AND efficient
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Straub D. (2009). Journal of Engineering Mechanics, 135(10), pp. 1089‐1099
Monitoring, Inspection and Maintenance for Concrete Structures
Zone A
Zone B

t,1
Straub D., et al. (2009). Structure and Infrastructure Engineering, ...
t,i
...
t,n
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Therefore,
… simplified (engineering) models are often sufficient for
making optimal decisions
… but probabilistic analysis can provide useful insights and
help making better decisions
… if we ensure that the benefit of the analysis outweighs the
cost off the
h analysis
l
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questions or comments?
[email protected] / www.era.bv.tum.de
(D. Straub, July 2006)
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