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Probabilistic Modelling of Multiphasic Degradations for Maintenance Optimization
of Infrastructures in Civil Engineering:application for a submerged reinforced concrete
structure.
A dissertation submitted by Boutros EL HAJJ to the University of Nantes for the degree of Doctor of Philosophy in Civil Engineering
23rd of November 2015, Nantes, France
PhD committee: Franck Schoefs (Director)
Bruno Castanier
Thomas Yeung
PhD defence
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Probabilistic Modelling of Multiphasic Degradations for Maintenance Optimization
of Infrastructures in Civil Engineering:application for a submerged reinforced concrete
structure.
ProbabilisticMaintenanceDegradations
Application:
submerged RC
structure Chloride
rich environment
To take into account
uncertainties
In the aim for optimized
decision-making aid
tools
Multiphasic
Non-destructive testing (NDT)
Objective
Build degradation modelling tools that allow a better integration
in realistic dynamic maintenance optimization contexts
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Early examples of maintenance and rehabilitation management
Boutros EL HAJJ 3
Rome’s aqueducts were
rehabilitated and reused
Rialto Bridge, VeniceBuilt in 1181
1444 collapsed due to overload by spectators
during a wedding
Maintenance was vital for the timber bridge
1503: Wood to stone
Pont du Gard, NîmesBuilt around 40-60 BC
1703 repair works
Management of infrastructures is an old occupation
Now a touristic attractions
1743 open to cars
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Consequences of bad inspection/maintenance
Boutros EL HAJJ 4
February 1878: Inspections
Satisfactory results
Collapsed December 1878
Collapsed 1980: lack of inspection and maintenance
previous decade
Hayakawa wire bridge, Japan
Somerton Bridge, Australia Collapsed 2008: poor maintenance
Tay Rail Bridge, Scotland
Reichsbrücke, Austria Collapsed 1976: lack of inspection techniques
Collapsed 1983: inadequate inspection resourcesMianus River Bridge, USA
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Consequences of a failure
Boutros EL HAJJ 5
2007: collapsed killing 13 people and
injuring 145
2005: rated again "structurally
deficient” and in need for replacement
1990: rated "structurally deficient”
due to corrosion
Approximately 75000 bridges in the
US share the same rating(Anderson et al 2007)
I-35W Mississippi River Bridge, USA
Built 1967
$3.6 trillion/5 years to improve the US’
infrastructure to an acceptable level(ASCE report card, 2013)
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Maintenance impact
Boutros EL HAJJ 6
Europe: a large number of infrastructures
were built after WW2
Rotterdam, 1940
Many structures require maintenance
Many repaired structures display non-
satisfactory performances and need
rehabilitation
1/3 steel structures in the Atlantic area
built more than 100 years ago
Europe: 50 % of annual construction
budget is currently spent on
refurbishment and remediation
(Duratinet project)
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Introduction
I. How to improve the evaluation, modelling and prediction of degradation for maintenance purposes?
II. How to be realistic? What to do with missing inspections and lost information?
III. How to update and model the effect of a maintenance action after a decision?
IV. How to take the best decision throughout the operation time of the system?
Illustrations
Conclusions and perspectives
Boutros EL HAJJ 7
Degradation
model
Decision
model
Decide the
optimal times of
inspection and
maintenance
Predict the
process of ageing
in condition or in
reliability
Maintenance management
system
Summary
The importance of
maintenance
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I. How to improve the evaluation, modelling and prediction of degradation for maintenance purposes?
Boutros EL HAJJ 8
What qualities are required from a degradation model to be able to
respond to these advancements?
Inspections and
monitoring techniques
Maintenance management
systemsDecisions-making
processes
CorrectivePreventive
Time-based
Maintenance
Condition-based
Maintenance
Maintenance
action
Is it time for
maintenance?
Maintenance
decision
Condition
assessment
yes
NDT, SDT
Inc. Visual
SensorsInspections
Decision
making
Risk
assessment
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Physics-based
models
or
White box models
Statistics-based
models
or
Black box models
(lifetime models)
(Nicolai 2008)
(Frangopol et al. 2004)
Classical degradation modelling approaches
Meta-models
or
Grey box modelsDescribes the relation
between time and failure
Simulation of the
physics of measurable
deterioration
and failureBased on measurable
quantities indicating time-
dependent deterioration and
failure (e.g., stochastic
processes)
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Characteristics of a “good” degradation model
Boutros EL HAJJ 10
Modelling the pathology Fit to data
Implementation maintenance systemsPrognostic characteristic
Predict the degradation evolution spatially
Physical meanings into maintenance models Uncertainties
Missing data
or errors of
acquisitions
Predict the degradation evolution temporally
Decision-making
Choice of
indicators
Use all available information
Integrate new data issued from NDT
Un-observable
indicators
Non-stationarity
Maintenance
effects
(imperfect)
Different insp.
techniques
Integrate in dynamic maintenance platforms
physics meta-model statistics
physicsmeta-
model statistics
physicsmeta-
model statistics
physicsmeta-
model statistics
challenge benefit
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Degradation meta-model approaches
Boutros EL HAJJ 11
Markov chains
Lévy processes
Brownian motion
definition of the discrete states
identification of the one-step transition matrix
non-monotonous evolution
Gamma process natural candidate (monotonous)
self-explanatory parameters
extensions
(van Noortwijk 2009)
(Si et al. 2013) measurement error, fillers, etc.
problems related to non-stationarity
Maintain the most critical
aspects of the degradation
Ease of integration in complex
maintenance decision schemes
Model the evolution of the
degradation using observations
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Extensions to solve non-stationarity
Boutros EL HAJJ 12
State-dependant
degradation models
Age-dependant
degradation models(Nicolai, Dekker, and van
Noortwijk 2007)
Non-Stationary
evolution
Discrete-state(Markov Chains)
Continuous-state(Lévy processes)
State-based
Monovariate(Vatn 2012)
Multivariate(Zouch et al. 2012; Mercier
and Pham 2012)
Covariates(Paroissin and Salami
2009)
Un-observable degradations
Imperfect maintenance actions
Individualisation
a robust procedure for the
identification of input parameters
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Monovariate state-dependant gamma process
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Definition 1 – A stochastic process 𝑋 = 𝑋𝑡 ∶ 𝑡 > 0
is said to be a stationary gamma process with
parameters 𝛼 ⋅ 𝜏, 𝛽 , where 𝛼 > 0 and 𝛽 > 0, if it
satisfies the following properties:
a) 𝑋0 = 0
b) 𝑋𝑡 has independent positive increments
c) 𝑋𝑡 has stationary increment ∀ 𝑡 > 0
𝑋𝑡+𝜏 − 𝑋𝑡 ~ 𝐺𝑎 𝛼 ⋅ 𝜏, 𝛽 =𝛽𝛼𝜏
𝛤(𝛼𝜏𝑥𝛼𝜏−1 𝑒−𝛽.𝑥
Definition 2 – A stochastic process 𝐺 =
𝐺𝑡 ∶ 𝑡 > 0 is said to be SDGP with
parameters 𝛼 𝐺𝑡 ⋅ 𝜏, 𝛽 , where 𝛼 𝐺𝑡 > 0 and 𝛽 >
0, if it satisfies the following properties:
a) 𝐺0 = 0
b) 𝐺𝑡 has independent positive increments
c) For a time interval 𝜏 > 0, we have:
𝐺𝑡+𝜏 − 𝐺𝑡 ~ 𝐺𝑎 𝛼 𝐺𝑡 ⋅ 𝜏, 𝛽
The SDGP is not a Lévy process anymore
loses the infinite-divisibility property
Transform the non-stationary process into
pieces of stationary SDGP
𝑡
𝐺𝑎 𝛼 ⋅ 𝜏, 𝛽
𝑋
𝑡
𝐺𝑡
𝐺𝑎 𝛼 𝐺𝑡 ⋅ 𝜏, 𝛽
𝐺
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NDT
Maintenance
modelling analysis:
Decisions, inspections,
maintenance actions and
policies
Items of the meta-models
A small number of parameters
A probabilistic pertinence and physical expertise
Indicators of degradation and durability directly accessible through NDT
Boutros EL HAJJ 14
Modelling analysis:
Statistical degradation
modelling, stochastic
processes, etc.
Degradation analysis:
Physical mechanism,
degradation indicators,
accessibility through NDT
Physical meaning of
the main probabilistic
trends
State-dependent stochastic
processes using information given
by NDT
META MODEL
(Condition–based)
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Objectives
Degradation analysis Study the pathology:
here, chloride-induces corrosion Look into physical indicators
Modelling analysis Construction of the degradation model Propose estimation and calibration algorithm
Maintenance analysis Catalogue potential maintenance actions Modelling the effect of an action in the model Discuss decision scenarios
Boutros EL HAJJ 15
Maintenance
analysis:
Modelling
analysis
Degradation
analysisMM
Maintenance
analysis:
Modelling
analysis
Degradation
analysisMM
Maintenance
analysis:
Modelling
analysis
Degradation
analysisMM
Argue and promote the use of condition based meta-models
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1 – Degradation analysis:Chloride-induced corrosion of RC structures
Boutros EL HAJJ 16
Chloride-induced corrosion of RC Structures
[Cl-]
RC cross-section
As
𝐶𝑙−
[Cl-]
Phase 1: Diffusion
Diffusion of chlorides
Phase2: Corrosion
Initiation of corrosion[Cl-]
> 𝐶𝑙− 𝑠𝑒𝑢𝑖𝑙
Phase3: Propagation
Cracking propagation𝜎
l
> 𝜎𝑡
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Pe
rce
nta
ge
of cu
mu
lative
da
ma
ge
(%
)
10
20
30
40
50
60
70
80
0
90
100
Diffusion Corrosion Deterioration
0 5 10 15 20 25
End of functional service life, rehabilitation
I: corrosion initiation
C: cracking
Initial cracking
I C
1 – Degradation analysis:Choice of indicators
Boutros EL HAJJ 17
Diffusion of
chlorideCorrosion of
reinforcementCrack propagation
𝜌𝑡, 𝜃𝑡 ∀𝑡≥0
For each phase:
A bivariate process written
𝜌𝑡: condition indicator
𝜃𝑡: potential of evolution
Choice of indicators:
• Accessibility via. NDT
• Representation of the
degradation process
𝜌3,𝑡 ∀𝑡≥0
𝑎: Crack width
𝜃3,𝑡 ∀𝑡≥0
𝑖𝑐𝑜𝑟𝑟: corrosion current density
𝜌2,𝑡 ∀𝑡≥0
𝜎: Internal stress
𝜃2,𝑡 ∀𝑡≥0
𝑖𝑐𝑜𝑟𝑟: corrosion current density
𝜌1,𝑡 ∀𝑡≥0
𝐶𝑙− : Chloride concentration
𝜃1,𝑡 ∀𝑡≥0
𝑃𝐻
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1 – Degradation analysis:Indicators’ tendencies
Boutros EL HAJJ 18
Phase 1 :
Chloride diffusion
Phase 2:
Corrosion of the reinforcement
Phase 3:
Crack propagation
𝜎 𝑖𝑐𝑜𝑟𝑟
a (𝑚𝑚)
𝜌3,𝑡 ∀𝑡≥0
𝑖𝑐𝑜𝑟𝑟
𝜃3,𝑡 ∀𝑡≥0
𝐶𝑙−
𝜎 (𝑀𝑝𝑎)
𝜃2,𝑡 ∀𝑡≥0
𝜌2,𝑡 ∀𝑡≥0
𝑃𝐻
𝑃𝐻
𝜌1,𝑡 ∀𝑡≥0
𝜃1,𝑡 ∀𝑡≥0
𝐶𝑙− (%)
𝑎 𝑖𝑐𝑜𝑟𝑟
S-shaped
S-shaped S-shapedL-shaped
L-shaped L-shaped
(Angst et al. 2009)
𝑢𝑛𝑖𝑟𝑛𝑑 2.7 − 3.1𝑢𝑛𝑖𝑟𝑛𝑑 0.4 − 0.53
𝑖𝑐𝑜𝑟𝑟
𝜌𝑡: Condition indicator 𝜃𝑡: potential of evolution
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2 – Modelling analysis:Construction of Bivariate State-Dependant Gamma Process
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A uniform approach using SDGP applied to the third phase
cause-effect relation
(mechanical)
𝜃3,𝑡 ∀𝑡≥0models 𝑖𝑐𝑜𝑟𝑟
𝜌3,𝑡 ∀𝑡≥0represents a
(Δθ3 ,Δρ3) ~ Gamma distribution law 𝛼 ⋅ 𝜏 is the shape function (state-dependent)
𝛽 is the scale parameter (constant)
𝑖𝑐𝑜𝑟𝑟
𝑎
𝑡
𝜌0
𝜌3,𝑡 ∀𝑡≥0
𝜃3,𝑡 ∀𝑡≥0
𝑡 𝜃3
𝜌3
E(Δθ3)
E(Δρ3)
𝛼𝜃3𝜌3, 𝜃3 = 𝑔1(𝜌3 . 𝑐3. 𝑒
− 𝜃3−𝑐12
𝑐2
𝛼𝜌3𝜌3, 𝜃3, ∆𝜃3 = 𝑔2 𝜃3, ∆𝜃3 . 𝑐4. 𝑒
−𝑐5.𝜌3
c2
c1(𝑐3. 𝜌3 + 𝑐4
𝑐6. 𝜃3 +∆𝜃3
2+ 𝑐7
𝜃0
𝜌𝑡: Condition indicator
𝜃𝑡: potential of evolution
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2 – Modelling analysis:State-dependant shape functions
Boutros EL HAJJ 20
𝛼𝜃3𝜌3, 𝜃3 = (𝑐3. 𝜌3 + 𝑐4 . 𝑒
− 𝜃3−𝑐12
𝑐2 𝛼𝜌3𝜌3, 𝜃3, ∆𝜃3 = 𝑐6. 𝜃3 +
∆𝜃3
2+ 𝑐7 . 𝑒−𝑐5.𝜌3
𝑐1 = 1, 𝑐2 = 1, 𝑐3 = 1, 𝑐4 = 1.2, 𝑐5 = 0.8, 𝑐6 = 1.8, 𝑐7 = 2, 𝛽𝜌 = 0.3, 𝛽𝜃 = 0.3𝜌𝑡: Condition indicator
𝜃𝑡: potential of
evolution
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Question I
I. How to improve the evaluation, modelling and prediction of degradation for maintenance purposes?
II. How to be realistic? What to do with missing inspections and lost information?
III. How to update and model the effect of a maintenance action after a decision?
IV. How to take the best decision throughout the operation time of the system?
Boutros EL HAJJ 21
Summary
Positioned the problem
Definition of the characteristics
Construction of the degradation
meta-model
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II. How to be realistic? What to do with missing inspections and lost information?
Boutros EL HAJJ 22
𝑛 number of structures
𝑇 number of inspections
𝑁 = 𝑛 × 𝑇
Maximum likelihood estimation (+fixed-point)
𝑛 = 3
𝑇 = 6
𝑁 = 3 × 6 = 18
Size of the database
Truncated
Censored
Missing
Stochastic Estimation Maximization (SEM)
1 2 3 4 5 6
𝜌
𝑡
Benefit of non-homogenous databases
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III. How to update and model the effect of a maintenance action after a decision?
Boutros EL HAJJ 23
E(Δθ)
𝛼𝐿 𝜌, 𝜃 = k1 × 𝑔2 𝜃 . 𝑒−𝑎1.𝜌
S-shaped
1st [Cl−] - 𝜌2nd & 3rd icorr - 𝜃
L shaped
1st pH - 𝜃2nd stress - 𝜌
3rd crack width - 𝜌
𝛼𝑠 𝜌, 𝜃 = m1 × 𝑔1 (𝜌 . 𝑒− (𝜃−𝑚2 −𝑎1
2
𝑎2
𝜌
𝜃
E(Δ𝜌)
𝑚2
m1
k1
after maintenancebefore maintenance
Maintenance action
effects
Speed
Level
Surface
protection
𝐶𝑙−
Chlorides
extraction𝜌1,𝑡 ∀𝑡≥0
𝑡
𝜌𝑡: Condition indicator
𝜃𝑡: potential of evolution
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IV. How to take the best decision throughout the operation time of the system?
Boutros EL HAJJ 24
Assessment of the degradation
Decision model
Inspection Estimation
Define condition
indexes (𝐶𝐼)
Define an estimation
algorithm
𝑡
𝑡
𝜏𝑖𝑛𝑠 = 2 . 𝜏𝐷Decisions plan
Inspections plan
𝜏𝐷
𝜏𝑖𝑛𝑠
inter-inspection
time intervalinter-decision
time interval
Decisions
scenario
Decision based on the observed 𝐶𝐼
Decision based on the estimated 𝐶𝐼 Bi variate processDifferent inspection
plans for 𝜌 & 𝜃
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Illustration: decision scenarioSame inspection plans for 𝜌 and 𝜃
Boutros EL HAJJ 25
0 2 4 6 8 10 12 14 16 18 200
1
2
3
4
5Crack width
:
L (
mm
)
Time
0 2 4 6 8 10 12 14 16 180
1
2
3
4
5
6Corrosion current density
:
Ic (
A/c
m2)
Time
inspection/possible history
crack width limit
Time 0 1 2 3 4 5 6 7 8 9 10 11 12
CI Ins Ins Ins Ins Ins
0 0 0 0 0 0 0 0 0.01 0.04 0 0.12 0.34 1
1 0 0 0 0 0 0.02 0 0.05 0.18 0 0.7 0.64 0
2 0 0 0.02 0 0.05 0.1 0 0.79 0.78 1 0 0.02 0
3 1 1 0.98 1 0.95 0.81 1 0.16 0.01 0 0 0 0
𝐶𝐼 = 3
𝐶𝐼 = 2𝐶𝐼 = 1
𝐶𝐼 = 0
𝑠𝑞𝑟𝑡 (𝑥
𝜌𝑡: Condition indicator
𝜃𝑡: potential of evolution
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Phase 1
Chloride
extraction [CE]
Cathodic prevention
[CP1]
Cathodic protection
[CP2]
Concrete
replacement [CR1]
Concrete replacement
+ Steel cleaning
[CR2]
Concrete replacement
+ Steel replacement
[CR3]
263 €/𝑚²
323 €/𝑚² 353 €/𝑚²
Indirect cost: 2000 €/𝑚²
Cathodic protection
[CP3]
Inspection = 25 €/𝑚² Inspection = 25 €/𝑚² Inspection = 10 €/𝑚²
Phase 2 Phase 3
Illustration: Maintenance managementPossible maintenance actions
(Srifi 2012)
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Stress
t
[Cl−]
Crack
width
𝐶𝐼 = 9
𝐶𝐼 = 8
𝐶𝐼 = 7
𝐶𝐼 = 6
𝐶𝐼 = 5
𝐶𝐼 = 4
𝐶𝐼 = 3
𝐶𝐼 = 2
CI=0
𝐶𝐼 = 1
t t
Preventive
Maintenance
Corrective
Maintenance
5 years
[CR2]
Do nothing
Do nothing
[CR1]
[CR3]
Illustration: Maintenance managementperformance indexes and management policies
Expected Life-Costs and Condition indexes for PM and CM
Policy PM CM PM CM PM CM
Lifetime (years) 50 75 100
Annual cost
(€/m²/year)23.9 50 24 58 24.3 55
Condition Index 8.21 6.3 8.18 5.86 8.14 5.89
𝜌𝑡: Condition indicator
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Life-Costs (€/m²)
and Condition indexes
Policy PM CM Benefit
Lifetime (years) 75
Inspections 466 200 + 133%
Maintenance 1330 4142 - 68%
Total cost 1765 4342 - 59%
Annual cost (€/m²/year) 24 58 - 59%
Condition Index 8.18 5.86 2.32 points
PM: Preventive Maintenance
CM: Corrective Maintenance
Illustration: Maintenance managementBenefit
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DiscussionState-based Condition Indexing
Boutros EL HAJJ 29
ρ
t𝐶𝐼 = 9
𝐶𝐼 = 8
𝐶𝐼 = 7
ρ
θ
ρ
θ
State-based 𝐶𝐼s 𝐶𝐼 based on a 𝑃𝑓
𝑃𝑓 𝜌𝑖 , 𝜃𝑖 = 𝑃 ∆𝜌 + 𝜌𝑖 > 𝐿 𝜌 = 𝜌𝑖 , 𝜃 = 𝜃𝑖
=
𝐿−𝜌𝑖
+∞
𝜃𝑖
+∞
𝑔 𝑥, 𝑦; 𝜌𝑖 , 𝜃𝑖 𝑑𝑦𝑑𝑥
𝑝𝑓: probability of failure
before the next inspection
0 1 2 3 4 5 6 7 80
0.5
1
1.5
2
2.5
3
3.5
4
Decision graph for the 4th epoch
One simulation
𝜌3, 𝜃3
𝜌3
𝜃3
𝐶𝐼
= 2
𝐶𝐼 = 0
𝐶𝐼
= 1
𝐶𝐼
= 3
𝜌𝑡: Condition indicator
𝜃𝑡: potential of evolution
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Discussion A 𝑃𝑓 approach to state-based 𝐶𝐼s
Boutros EL HAJJ 30
Every iso-plan
Iso-curve of equal 𝑝𝑓
Ex: 𝑝𝑓 = 0.05
Different 𝐶𝐼s
𝜌𝑡: Condition indicator
𝜃𝑡: potential of evolution
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Conclusions on the use of meta-models
Boutros EL HAJJ 31
I – The description of the ageing model:
Physical meaning To probabilistic trends
Input
(NDT assessment)
Output
(decision parameters)
1.
2.
complex physical models increasing complexity of NDT
II – In a CBM context:
simple description, flexibility,
calibration and statistical calculation
implement and beneficial in a
risk management framework
III – Evaluation of the Meta-model is done through state-dependant stochastic
processes using NDT
Available information Model
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Perspectives
Different models of degradation (carbonation, etc.)
Future tests under real databases (Surffeol, COST action)
Boutros EL HAJJ 32
Maintenance decision
Pre-specifications of databases
Confront the problem
of lack of data
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Mathematical challenges
Consider spatial variability of inspections
Integration of measurement error
Non-homogeneous database
Integration of variability to 𝛽 (e.g., state-dependant)
Effect of the estimation process on low probabilities
Loss of infinite-divisibility
…
Boutros EL HAJJ 33
𝑎1 𝑎2
𝐿2𝐿1𝑑
1st pit 2nd pit
Perspectives
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Thank you!