This document summarizes a comparative study on Bayesian updating of bridge safety models. It examines how the number of measurements and reliability of measurement data influence results for probability of failure. Bayesian updating was used to update distributions for bridge damage based on different measurement scenarios. Results showed that with more reliable measurement data (lower standard deviation), probability of failure estimates improved more with increasing numbers of measurements. However, even unreliable data (high standard deviation) allowed improvement when over 50 measurements were available. Bayesian updating effectively enhanced safety models, but quality and quantity of input data significantly impacted results.

1. … solutions for robust engineering
This project has received funding from the European Union’s Horizon 2020 research and innovation
programme under the Marie Skłodowska-Curie grant agreement No. 642453
Comparative study on Bayesian updating
of bridge safety model
Barbara Heitner, Eugene OBrien, Thierry Yalamas,
Franck Schoefs, Rodrigue Décatoire

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Scope
Introduction
Bridge safety model
Bayesian updating
Comparative study
Results
Conclusions
ESREL 2017 - Portoroz, 21/06/2017

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Introduction
Bridge safety assessment
• Probability of failure (level of safety)
• Advanced probabilistic theory
• Methodology that is flexible, yet robust
• Introduction of any deterioration process
• Information about the existence of damage
 increase the power of a bridge safety model
Bayesian updating
• Most important advantages:
 Works even with poor initial assumptions
 Works even with small number of or quite uncertain data available
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Introduction
Questions
• How effective an updating can be in case of bridge safety models?
• How the available number of measurements influences the results in
terms of probability of failure of the bridge?
• How the reliability of the measurement data (Damage Indicator data)
influences the results in terms of probability of failure of the bridge?
ESREL 2017 - Portoroz, 21/06/2017

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Scope
Introduction
Bridge safety model
Bayesian updating
Comparative study
Results
Conclusions
ESREL 2017 - Portoroz, 21/06/2017

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Bridge safety model
The scheme of the bridge safety model
ESREL 2017 - Portoroz, 21/06/2017

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Bridge safety model
Resistance and dead load models
• Reinforced concrete slab bridge
• Modelled as simply supported beam
• A localized damage is introduced
• Resistance = Bending moment capacity
• Deterministic and stochastic variables for geometrical and material
properties of the structure
• Moment capacity and dead load are slightly correlated through the
common geometrical stochastic variables
Localized damage
ESREL 2017 - Portoroz, 21/06/2017

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Bridge safety model
Traffic load
• Weigh-in-Motion database
• Extract daily maximum bending moment values segment-wise
• Generalized Extreme Value (GEV) distribution fitted to the tail
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ESREL 2017 - Portoroz, 21/06/2017

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Bridge safety model
Monte Carlo simulation
• Probability of failure:
where Nsim is the total number of simulations
and
where C is the capacity
Ed is the load effect from dead load
Et is the load effect from traffic load
i=1…k and k is the number of segment
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ESREL 2017 - Portoroz, 21/06/2017

10. ©PhimecaEngineering
Scope
Introduction
Bridge safety model
Bayesian updating
Comparative study
Results
Conclusions
ESREL 2017 - Portoroz, 21/06/2017

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Bayesian updating
Model deterioration process in probabilistic context
 Prior knowledge
Conduct real time measurement, which has to:
• Be easy and fast;
• Lead to no restriction on the traffic flow;
• Give relevant information about possible damage extent;
• And damage location.
Establish a damage indicator (DI) based on the measurement
Identify the connection between the damage indicator and the
possible deterioration
 ‘Measurement data’ for updating
ESREL 2017 - Portoroz, 21/06/2017

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Scope
Introduction
Bridge safety model
Bayesian updating
Comparative study
Results
Conclusions
ESREL 2017 - Portoroz, 21/06/2017

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Resistance and dead load
• Stochastic variables of the model:
Comparative study
Parameter (Distribution) Nominal value Unit Mean value STD* / CoV**
Slab depth (Normal) 1000 [mm] + 0.8 3.6
Concrete cover (N) 50 [mm] + 6 11.5
Concrete compressive strength (N) 45 [N/mm2] + 7.4 6
Steel yield strength (LogNormal) 400 [N/mm2] +36 21.3
Unit weight of concrete (N) 23.68 [kN/m3] x 1.05 x 0.11
Unit weight of asphalt (N) 22.73 [kN/m3] x 1.0 x 0.25
Thickness of asphalt (N) 80 [mm] + 0 40
Bar area loss (Gamma) [%] Shape: k=1.5 Scale: λ=8.9
*Standard deviation
**Coefficient of Variation
ESREL 2017 - Portoroz, 21/06/2017

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Comparative study
Traffic load
• One lane traffic
• 1 year Weigh-in-Motion (WIM) database from a site in Illinois (2011)
Daily maximum bending moments
Parameter Unit Mean CI95%- CI95%+
Speed [km/h] 105.95 94.94 117.47
GVW* [tons] 22.93 3.81 35.88
Length [m] 19.84 6.1 24.10
No. of axles [-] 4.57 2 5
Wheel base [m] 16.10 3.60 20.86
Gap behind [sec] 37.00 1.41 106.6
*Gross Vehicle Weight
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Comparative study
Bayesian updating
• Prior distribution
 Parameter of ‘bar area loss’ for the damaged segment (from literature)
 Gamma distribution (k=1.5; λ=8.9)
• Updating the two parameters (shape and scale) of ‘bar area loss’ initial
distribution using different levels of ‘measured’ damage
• Markov Chain Monte Carlo method is applied using the Metropolis-
Hastings algorithm
Measurement scenarios
• Related uncertainties of a DI may depend on:
 The accuracy and resolution of the applied sensor;
 The relevance of the human factor;
 The sensitivity to environmental conditions;
 The sensitivity to traffic conditions.
ESREL 2017 - Portoroz, 21/06/2017

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Comparative study
• These uncertainties influence the quality of DI, or with the other words
the ‘DI’s reliability’
The variance (or standard deviation) of the measured DI values at a
given bridge state
• The number of measurements possible (no. of data available for
updating) can also be an important burden
• Applied scenarios in the comparative study:
Number of
measurements
Standard deviation of DI
1.0 2.0 3.0 4.0
5 SC1 SC2 SC3 SC4
10 SC5 SC6 SC7 SC8
20 SC9 SC10 SC11 SC12
50 SC13 SC14 SC15 SC16
ESREL 2017 - Portoroz, 21/06/2017

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Comparative study
Prior and (example) posterior gamma distributions
Real value is 5%
Real value is 20%
ESREL 2017 - Portoroz, 21/06/2017

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Scope
Introduction
Bridge safety model
Bayesian updating
Comparative study
Results
Conclusions
ESREL 2017 - Portoroz, 21/06/2017

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Results
Results of the Bayesian updating regarding the bar area loss
distribution
• When the prior is poor (20% case) the updating shows clearer trends
with increasing no. of measurements regarding both hyperparameters
• When the prior is good (5% case) the shape (k) of the distribution
changes less, while the scale (𝜆) changes more
5 measurement data
10 measurement data
20 measurement data
50 measurement data
Triangle: STD=4
Star: STD=3
Diamond: STD=2
Circle: STD=1
Sample size for each
case = 10
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Results
Results regarding the probability of failure – Real value = 5%
• For all the case is possible
to improve the prior
assumption
• For STD=1 or 2 the trend of
getting better results with
increasing no. of
measurements is clear
• The uncertainty can be
visibly decreased by
increasing the no. of
measurements except for
STD=4
Pf using the prior distribution is 0.07
ESREL 2017 - Portoroz, 21/06/2017

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Results
Results regarding the probability of failure – Real value = 20%
• For any STD only for the 50
measurement case is
possible to improve the prior
assumption
• Only for STD=1 the trend of
getting better results with
increasing no. of
measurements is clear
• The uncertainty can be
visibly decreased by
increasing the no. of
measurements for STD=1 or
2
Pf using the prior distribution is 0.023
ESREL 2017 - Portoroz, 21/06/2017

22. ©PhimecaEngineering
Scope
Introduction
Bridge safety model
Bayesian updating
Comparative study
Results
Conclusions
ESREL 2017 - Portoroz, 21/06/2017

23. ©PhimecaEngineering
Conclusions
Bayesian updating can give significantly different results depending
on the quality and number of measurement (or DI) data as well as
on the prior assumption
Overall improvement of the bar area loss parameter does not
necessarily lead to improvement of the estimation on the probability
of failure
It is also important to acknowledge the effect of the order of
magnitude of the probability of failure and the type of distribution
defined
Wisely updating the bar area loss parameter according to the
different scenarios can lead to significant enhancement of the safety
model
ESREL 2017 - Portoroz, 21/06/2017

24. … solutions for robust engineering
This project has received funding from the European Union’s Horizon 2020 research and innovation
programme under the Marie Skłodowska-Curie grant agreement No. 642453
Thank you for your kind attention!
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