Bayesian analysis applied to stochastic mechanics and reliability: Making the most of your data and observations

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1 [WOST 2012, November 30, 2012 ] Bayesian analysis applied to stochastic mechanics and reliability: Making the most of your data and observations Daniel Straub Engineering Risk Analysis Group TU München

2 Ever increasing amounts of information are available Sensor data Satelite data Spatial measurements on structures Advanced simulation BM 15 Strecke s BM 13 BM 14 BM 12 Sources: Frey et al. (in print); Gehlen et al. (2010); Michalski et al (2011); Schuhmacher et al. (2011) 2

3 Updating models and reliability computations with (indirect) information 3

4 How to compute the reliability of a geotechnical site conditional on deformation monitoring outcomes? -> Integrate Bayesian updating in structural reliability methods 4

5 Prior model in structural reliability Failure domain: Ω 0 Probability of failure: Pr Ω d 5

6 Information in structural reliability Inequality information: Ω 0 Conditional probability of failure: Pr Pr Pr Ω Ω Ω d d 6

7 Information in structural reliability Equality information: Ω 0 Conditional probability of failure: Pr Pr Pr 0 0? 7

8 In statistics, information is expressed as likelihood function Likelihood function for information event Z: Example: Measurement of system characteristic s(x) Additive measurement error Equality information:, Likelihood function:, 8

9 By expressing equality information as a likelihood function, it can be represented by an inequality domain F P (p) Pr 1 Pr Pr 9

10 it follows Pr Pr d Pr d the event is represented through the limit state function, and corresponding domain Ω, 0 thus Pr, Ω d d, Ω d d 10

11 accordingly Pr Pr Pr d, Ω Ω d d and finally Pr Pr Pr, Ω Ω, Ω d d 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

12 Application to spatially distributed systems an exploratory example 1, 2,, T 12

13 Demonstration example: Updating of a Gaussian process with 9 measurements Adaptive importance sampling 13

14 Application 2: How to compute the reliability of a geotechnical site conditional on monitoring? Papaioannou I., Straub D. (2012). Computers & Geotechnics, 42:

15 Reliability updating during construction Example: Geotechnical site Deformation should be limited 15

16 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) 16

17 Random field realizations (Homogenous, anisotropic random field) Realization ofthe Young s m odulus Realization ofthe friction angle 17

18 Solution strategy FEM model 18

19 Reliability index vs measured displacements m m [10-3 m] 19

20 Updated random soil parameters Mean of friction angle 32.7 Prior mean: 35 20

21 Updated random soil parameters Young s modulus E Prior mean: 125 MPa 105 MPa 21

22 Dynamical system identification (from Beck and Au 2002) 22

23 Bayesian parameter identification Simulation results X 2 k X 1 k 23

24 Simulation algorithm Subset simulation level 1 X 2 k X 1 k 24

25 Simulation algorithm Subset simulation level 2 X 2 k X 1 k 25

26 Simulation algorithm Subset simulation level 3 X 2 k X 1 k 26

27 Simulation algorithm Subset simulation level 4: final samples X 2 k X 1 k 27

28 Simulation algorithm Subset simulation level 4: final samples X 2 k X 1 k 28

29 Include all information in near-real-time over the lifetime Example: fatigue reliability Example deteriorating pipeline: Inspections Straub D. (2009). ASCE J Engineering Mechanics 29

30 Application: Corrosion of reinforcement in concrete Corrosion caused by ingress of chlorides Chloride profile measurements Straub D. & Fischer J. (2011) Proc. ICASP 30

31 Influence of chloride measurements on probability of corrosion 31

32 Potential in mechanical 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 2012) SHM of aircraft structures (EU project ROSA) Structural systems (Straub & Der Kiureghian 2010) Tunnel construction (Špačková & Straub in print) Aging ship structures (Luque & Straub, in preparation) 32

33 To conclude Bayesian updating enables to include any relevant observation into your prediction Presented methods based on structural reliability are efficient and simple Simple (robust) importance sampling and subset simulation schemes perform well for updating in large systems 33

34 Supported by Federal Roadway Agency 34

35 Contacts Daniel Straub Engineering Risk Analysis Group TU München 35

Managing risk. Špačková O., Straub D. (in print): Cost-benefit analysis for optimization of risk protection under budget constraints. Risk Analysis.

Managing risk. Špačková O., Straub D. (in print): Cost-benefit analysis for optimization of risk protection under budget constraints. Risk Analysis. 2 3 Managing risk Špačková O., Straub D. (in print): Cost-benefit analysis for optimization of risk protection under budget constraints. Risk Analysis. 4 Decision Straub D. (2014): Engineering Risk Assessment.