Probabilistic Modelling of Multiphasic Degradations for...

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

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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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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

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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

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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

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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

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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


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)


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


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2 – Modelling analysis:State-dependant shape functions

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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





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Summary

Positioned the problem

Definition of the characteristics

Construction of the degradation

meta-model

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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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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



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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



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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


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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



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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



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Conclusions on the use of meta-models

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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!

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