01.06.2012. 1. Bayesian Updating in. Structural Reliability. Daniel Straub.
Engineering Risk Analysis Group. TU München. [APSSRA 2012, May 25, 2012 ].
2.
01.06.2012
[APSSRA 2012, May 25, 2012 ]
Bayesian Updating in Structural Reliability Daniel Straub Engineering Risk Analysis Group TU München
Ever increasing amounts of information are available
Sensor data
Satelite data
Spatial measurements on structures
Advanced simulation
Sources: Frey et al. (in print); Gehlen et al. (2010); Michalski et al (2011); Schuhmacher et al. (2011)
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1973:
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Probabilistic Updating of Flaw Information Tang (1973) • Imperfect information through inspection modeled by probability-ofdetection:
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Probabilistic Updating of Flaw Information Tang (1973)
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Updating models and reliability computations with (indirect) information • Bayes‘ rule:
∝
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How to compute the reliability of a geotechnical site conditional on deformation monitoring outcomes? -> Integrate Bayesian updating in structural reliability methods
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Prior model in structural reliability
• Failure domain: Ω
0
• Probability of failure: Pr
d ∈Ω
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Information in structural reliability
• Inequality information: Ω
0
• Conditional probability of failure: Pr
|
Pr ∩ Pr
d
∈ Ω ∩Ω ∈Ω
d 9
Information in structural reliability
• Equality information: Ω
0
• Conditional probability of failure:
Pr
|
Pr ∩ Pr
0 0
? 10
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In statistics, information is expressed as likelihood function • Likelihood function for information event Z:
∝ Pr |
• Example: – Measurement of system characteristic s(X) – Additive measurement error • Equality information:
,
• Likelihood function:
, 11
By expressing equality information as a likelihood function, it can be represented by an inequality domain • Let – P be a standard uniform random variable – c be a constant, such that 0 1 for any x
• then
FP(p) 1
Pr 1
• and Pr
| Pr 12
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Pr
it follows
Pr
|
d d
Pr
the event function
is represented through the limit state
,
and corresponding domain Ω
,
0
thus Pr
d d , ∈Ω
d d , ∈Ω 13
accordingly
Pr
∩
Pr
|
|
Pr
d
d d , ∈ Ω ∩Ω
and finally Pr
|
Pr ∩ Pr
d d
, ∈ Ω ∩Ω , ∈Ω
d d
Both terms can be solved by any Structural Reliability Method, since all domains are described by inequalities Straub D. (2011). Reliability updating with equality information. Probabilistic Engineering Mechanics, 26(2), pp. 254–258.
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Application to spatially distributed systems – an exploratory example • Observations of a Gaussian process • Failure at location t:
1,
2, … ,
T
2
• Measurements:
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Demonstration example: Updating of a Gaussian process with 9 measurements Adaptive importance sampling
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Application 2: How to compute the reliability of a geotechnical site conditional on monitoring?
•
Papaioannou I., Straub D. (2012). Reliability updating in geotechnical engineering including spatial variability of soil. Computers & Geotechnics, 42: 44–51. 17
Reliability updating during construction Example: Geotechnical site Deformation should be limited
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Reliability updating during construction Example: Geotechnical site Deformation at intermediate stage can be measured Probability (of the final stage) is updated
Probabilistic FEM model (random field, non-linear)
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Random field realizations (Homogenous, anisotropic random field) Realization of the Young’s modulus
Realization of the friction angle
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Solution strategy FEM model
• Limit state function for failure: 0.1 • Limit state function for observation:
• Evaluated with Subset simulation (2 3 ∙ 10 LSF calls)
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Reliability index vs measured displacements
2 10-3 m 1 10-3 m
[10-3 m]
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Application: Corrosion of reinforcement in concrete
• Corrosion caused by ingress of chlorides • Chloride profile measurements
Straub D. & Fischer J. (2011) Proc. ICASP
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Corrosion model
• Surface of 5 x 10m, discretized in 200 elements of 0.5 x 0.5m • Ingress of corosion at one location described by
0
erf
√4
• Failure criterion (onset of corrosion): ,
,
,
⋅ erf
4
• Parameters are modelled as random fields with correlation lengths of 1 – 2m 24
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Measurements
• Cores are taken, chloride contents at depths zm are measured ,
,
1
1 √2
erf
exp
4
1 2
,
,
2
• Here: cores are taken at two locations and cloride content is evaluated at two depths (20mm, 40mm) for each core 25
Case a
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Case b
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Alternative: Bayesian networks (Straub & Der Kiureghian, J Eng Mech 2010) Structural model:
Failure mechanisms:
V
H
• Model problem as a Bayesian network • Precompile the BN using structural reliability methods • Perform updating with the BN algorithms
R2
R3
R4
2) beam mechanism
5m R1
R5
5m
5m
1) sway mechanism
3) combined mechanism
• Advantage: fast and robust • Disadvantage: large computational efforts in precompiling 28
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Potential in structural and civil engineering is huge Additional past/current projects on Bayesian updating Updating probabilistic models with observations for: • • • • • • •
Acoustic emission (Schumacher & Straub 2011) Avalanche risk (Straub & Grêt-Regamey 2006) Flood damage assessment (Frey, Butenuth & Straub in print) Rock-fall loads on protection systems (Straub & Schubert 2008) SHM of aircraft structures (EU project ROSA) Structural systems (Straub & Der Kiureghian 2010) Tunnel construction (Špačková & Straub in print)
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To conclude…
• Reliability updating with any observation can be performed with any structural reliability method • Simple (robust) importance sampling schemes perform well for updating in spatially distributed systems
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Contacts
Daniel Straub Engineering Risk Analysis Group TU München www.era.bv.tum.de
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