Probabilistic assessment of ageing structures has become an important research area as it attracts the interest from not only researchers but also investors, municipalities, and governments. The most commonly used material for many important structures and infrastructure is reinforced concrete. Various degradations of such structures are manifest in the form of direct loss of reinforcement area. In this study, a time-dependent stochastic model of the reinforcement loss (in [%]) due to corrosion is presented, which has a crucial role in the estimation of the lifetime and the time-dependent health state of the structure. Bayesian updating is applied in multiple steps during the lifetime of the structure in order to improve the estimate of the reinforcement loss. An example application is shown where updating is applied in two steps.

1. Reducing Uncertainty in Structural Safety
Special Session SS6
Ghent, Belgium
28-31 October 2018

2. Franck Schoefs, Thierry Yalamas, Barbara Heitner,
Eugene OBrien, Guillaume Causse

3. Scope
Introduction
Multi-step Bayesian updating
Example application
Results and conclusions

4. Introduction
Problems of ageing bridges
How to estimate the level of safety of these bridges?
How to plan inspection/maintenance?

5. Introduction
Problems of ageing bridges
How to estimate the level of safety of these bridges?
How to plan inspection/maintenance?
Different approaches/models exist
Involving stochastic parameters (e.g. Monte Carlo, FORM, SORM)
Incorporating measurement data (e.g. FE model calibration, Bayesian
updating)
Bayesian Networks
Stochastic processes etc.

6. Introduction
Problems of ageing bridges
How to estimate the level of safety of these bridges?
How to plan inspection/maintenance?
Different approaches/models exist
Involving stochastic parameters (e.g. Monte Carlo, FORM, SORM)
Incorporating measurement data (e.g. FE model calibration, Bayesian
updating)
Bayesian Networks
Stochastic processes etc.
Crucial to estimate/model as good as possible
The original geometrical/material/structural properties
The loading (history)
The level of deterioration/ health state

7. Introduction
Problems of ageing bridges
How to estimate the level of safety of these bridges?
How to plan inspection/maintenance?
Different approaches/models exist
Involving stochastic parameters (e.g. Monte Carlo, FORM, SORM)
Incorporating measurement data (e.g. FE model calibration, Bayesian
updating)
Bayesian Networks
Stochastic processes etc.
Crucial to estimate/model as good as possible
The original geometrical/material/structural properties
The loading (history)
The level of deterioration/ health state

8. Introduction
Corrosion is one of the most common and dangerous
phenomena related to the deterioration of RC bridges
Loss of capacity
Loss of elasticity
Increased deformations
Cracking
Spalling
Aesthetical aspects

9. Introduction
Corrosion is one of the most common and dangerous
phenomena related to the deterioration of RC bridges
Loss of capacity
Loss of elasticity
Increased deformations
Cracking
Spalling
Aesthetical aspects
In this work
Corrosion is modelled in a simplified context
Bayesian updating is applied on the distribution of reinforcement area
loss due to corrosion at different points in time
Parametric study is conducted based on:
The quality of measurement data and
The number of data available

10. Multi-step Bayesian updating
Estimating the value
of interest*
Developing the
physical model of
deterioration at t = 0
Estimating the
deterioration level
until t = service life
* Can be inspection/maintenance planning,
probability of failure, remaining service life
etc.

11. Multi-step Bayesian updating
Estimating the value
of interest*
Collecting relevant
data at time t = xi
Developing the
physical model of
deterioration at t = 0
Estimating the
deterioration level
until t = service life
* Can be inspection/maintenance planning,
probability of failure, remaining service life
etc.

12. Multi-step Bayesian updating
Estimating the value
of interest*
Collecting relevant
data at time t = xi
Estimating the
deterioration level at
t = xi
Developing the
physical model of
deterioration at t = 0
Estimating the
deterioration level
until t = service life
i = 1
* Can be inspection/maintenance planning,
probability of failure, remaining service life
etc.

13. Multi-step Bayesian updating
Estimating the value
of interest*
Collecting relevant
data at time t = xi
Estimating the
deterioration level at
t = xi
Developing the
physical model of
deterioration at t = 0
Estimating the
deterioration level
until t = service life
Updating the
deterioration level at
t = xi
i = 1
* Can be inspection/maintenance planning,
probability of failure, remaining service life
etc.

14. Multi-step Bayesian updating
Estimating the value
of interest*
Collecting relevant
data at time t = xi
Estimating the
deterioration level at
t = xi
Developing the
physical model of
deterioration at t = 0
Estimating the
deterioration level
until t = service life
Updating the
deterioration level at
t = xi
Updated estimation of
the deterioration level
until t = service life
i = i+1
i = 1
* Can be inspection/maintenance planning,
probability of failure, remaining service life
etc.

15. Multi-step Bayesian updating
Some ideas about what is
going on
Physical model can be built
High level of uncertainty
Environmental conditions
Material and geometrical
imperfections etc.
Why Bayesian updating?

16. Multi-step Bayesian updating
Some ideas about what is
going on
Physical model can be built
High level of uncertainty
Environmental conditions
Material and geometrical
imperfections etc.
Monitoring bridges has become
commonplace
periodical inspections (visual
inspections, NDT)
SHM systems
Valuable new information BUT
Limit on the amount of data
High level of uncertainty
Indirectly related data
Why Bayesian updating?

17. Multi-step Bayesian updating
Some ideas about what is
going on
Physical model can be built
High level of uncertainty
Environmental conditions
Material and geometrical
imperfections etc.
Bayesian inference: prior hypothesis + new evidence/information
Why Bayesian updating?
Monitoring bridges has become
commonplace
periodical inspections (visual
inspections, NDT)
SHM systems
Valuable new information BUT
Limit on the amount of data
High level of uncertainty
Indirectly related data

18. Example application
Corrosion model prior knowledge
To calculate the residual reinforcement area
5.13
27
c
fc
w
th
corr
c
c
w
Tti
64.1
1
)1(378
)(
29.0
11
85.0)()( pcorrpcorr
tTtitTti
ptT
T
corr
dttiDtD
1
1
)(232.0)( 0
For one reinforcing bar
Where:
w/c : water cement ration of concrete
fc : compressive strength of concrete in
[N/mm2]
icorr(t) : corrosion rate in [µA/cm2] at time t
t : time in [year]
Cth : cover thickness in [cm]
T1 : corrosion time initiation (=0)
tp : time since T1 in [year]
D0 : initial reinforcing bar diameter in [mm]
D(t) : reinforcing bar diameter in [mm] at time
t

19. Example application
Corrosion model prior knowledge
2D histogram of reinforcement area loss
(RAL) based on 100 000 sample size
(Monte Carlo simulation)

20. Example application
Corrosion model prior knowledge
2D histogram of reinforcement area loss
(RAL) based on 100 000 sample size
(Monte Carlo simulation) Mean, SD and 95% CI

21. Example application
Bayesian updating of LN distribution
At a given time RAL can be modelled using Log-Normal distribution
3 hyper parameters to update:
µlog
log

22. Example application
Bayesian updating of LN distribution
At a given time RAL can be modelled using Log-Normal distribution
3 hyper parameters to update:
µlog
log

23. Example application
Bayesian updating of LN distribution
At a given time RAL can be modelled using Log-Normal distribution
3 hyper parameters to update:
µlog
log
Markov Chain Monte Carlo (MCMC) method
Numerical method
Sampling directly from the posterior distribution using Metropolis-
Hastings algorithm
Convergence has to be ensured
The mean values of the posterior distribution of hyper parameters
are chosen to define the posterior LN distribution

24. Example application
Measurement scenarios
Obtaining new data
at t = 20 years
at t = 40 years
9 different cases:
no. of measurements: 10,20 or 50
(Coefficient of variation of the
measurement data / damage
indicator): 0.05, 0.1 or 0.2

25. Example application
Measurement scenarios
Obtaining new data
at t = 20 years
at t = 40 years
9 different cases:
no. of measurements: 10,20 or 50
(Coefficient of variation of the
measurement data / damage
indicator): 0.05, 0.1 or 0.2
A realization is randomly
chosen in order to simulate
the measurements

26. Example application
Measurement scenarios
Obtaining new data
at t = 20 years
at t = 40 years
9 different cases:
no. of measurements: 10,20 or 50
(Coefficient of variation of the
measurement data / damage
indicator): 0.05, 0.1 or 0.2
t = 20 years
A realization is randomly
chosen in order to simulate
the measurements

27. 10 20 50
0.2
0.1
0.05
Results and conclusions

28. 10 20 50
0.2
0.1
0.05
Results and conclusions

29. Results and conclusions
Number of data: 50
CoV of data: 0.05

30. Results and conclusions
Comparing the results based on
One updating at t = 20 years (blue)
Involving a second updating at t = 40 years (red)

31. Results and conclusions
A methodology is presented for better estimating reinforcement
loss due to corrosion in RC structures based on periodically
collected data
relatively simple, low computational cost
requires limited workforce and traffic disruption
can be used in case of any other deterioration process, where relevant
data can be collected from time to time

32. Results and conclusions
A methodology is presented for better estimating reinforcement
loss due to corrosion in RC structures based on periodically
collected data
relatively simple, low computational cost
requires limited workforce and traffic disruption
can be used in case of any other deterioration process, where relevant
data can be collected from time to time
In all the studied cases it is possible to improve the estimation
of the reinforcement loss

33. Results and conclusions
A methodology is presented for better estimating reinforcement
loss due to corrosion in RC structures based on periodically
collected data
relatively simple, low computational cost
requires limited workforce and traffic disruption
can be used in case of any other deterioration process, where relevant
data can be collected from time to time
In all the studied cases it is possible to improve the estimation
of the reinforcement loss
More measurements or better quality of measurements both
significant

34. Results and conclusions
A methodology is presented for better estimating reinforcement
loss due to corrosion in RC structures based on periodically
collected data
relatively simple, low computational cost
requires limited workforce and traffic disruption
can be used in case of any other deterioration process, where relevant
data can be collected from time to time
In all the studied cases it is possible to improve the estimation
of the reinforcement loss
More measurements or better quality of measurements both
significant
Collecting less data multiple times leads to higher rate of
improvement than collecting more data at a single point in time

35. The TRUSS ITN project (http://trussitn.eu) has
received funding from the European
Horizon 2020 research and innovation
programme under the Marie -Curie
grant agreement No. 642453

36. MCMC
Sample size of each posterior: 2000
Hyper-priors are normally distributed (truncated at 0) with
CoV = 0.5
Burn-in period: 100
Thinning: 20
Acceptance rate: 0.15 - 0.25
For each case measurement data sampling and updating
have been done 100 times to have some statistical
understanding of the results

37. MCMC
Example for the posterior distribution random walk samples for the
three hyper-prior
µlog log

38. MCMC
Results of MCMC for all 9 scenarios
Mean and SD for the 3 hyper prior parameters
µlog log