Utilizzo di un approccio Markoviano per la manutenzione di ponti in acciaio

Elsa Garavaglia Dipartimento di Ingegneria Civile e Ambientale

Utilizzo di un approccio Markoviano per la manutenzione

di ponti in acciaio

Franco Bontempi Sapienza Università di Roma

Elsa Garavaglia Politecnico di Milano

Luca Sgambi Politecnico di Milano

IABMAS-ITALY

2°Workshop

15 – 16 Dicembre 2014

Università di Padova, Aula Magna di Ingegneria


The problem

• Reliability Problem Solving

• loss of performance > prescribed threshold

value = failure

• failed system: state transition (lower level of

performance)

• structural performance improvement

(maintenance and/or rehabilitation): state

transition (higher level of performance)

• failure process defined as a transition

process

Structures deterioration due to environmental aggressions


To set up a strategic preventive maintenance

• need for basic knowledge about deterioration process (“historic” information)

and structural parameters’ evolution

• uncertainty affects deterioration process modeling

• probabilistic approach: Monte Carlo Simulation & semi-Markov modeling


Possible approaches

Monte Carlo Method

simulates the behavior of deteriorated

structures

samples structure-like systems

presenting the whole deterioration array

and following a probable deterioration

law

Probabilistic Model

considers Monte Carlo simulation’s

output data

estimates the occurrance of a certain

damage degree over time

+


Aggressive environment deteriorates structures decreasing

• cross-sectional area A

• material strength

“0” index denotes initial undamaged state

Time-variant damage index

measures the deterioration over time t

General damage model:

τ = t/Tc

Tc = normalized time instant when the failure

threshold δ=1 is reached

ρ , ω = damage parameters defining the shape of the

damage curve

a = damage associated with environmental aggression

b = damage associated with loading effects

ξ = ratio between level of acting stress and limit state value

The deterioration law


Structural parameter : stress level in each member

Basic assumptions: Structure and/or component member undamaged at the initial

time t0

Simulation process:

ρa, ρb, ωa, ωb (damage parameters) and Tc (time to reach failure) modeled as random

variables with prescribed probability distribution

For each simulation cycle

• time-variant structural analysis

• transition probability assessment for both structural members and the whole

structural system

• probability of failure PF estimation

MC Simulation

Distribution used in the Monte Carlo procedure.

__________________________________________________________________

Variable Distribution mean value standard deviation

__________________________________________________________________

a Normal 1.20 0.2

b Normal 1.20 0.2

a Normal 0.35 0.2

b Normal 0.35 0.2

TC Weibull 60 10

__________________________________________________________________


• Initial conditions: initial state (i.e. state occupied by

the system at starting time t=0) and time t0, (time

already spent in initial state at time t=0)

• Probability density function (p.d.f.) fik(x) of holding

time τik (i.e. time spent in state i if next state is k)

• Transition probability matrix, pik=Pr {next state k |

present state i}

* * ** *t

x = 0 x

t0 t = 0

i k

ik

initial state transitionto state i

transitionto state k

^

The MRP is defined when the following quantities are known:

Problem:

Definition of size

Problem:

Choice of density

memory

The deterioration model:

Markovian Renewal Process


Assumptions:

i = present state

t0 = time already spent by the system into i-state

Fij(t) = well-defined distribution functions

τij = t0

probability to move on to state j, in next interval Δt :

probability pij obtained by experimental evidence trough the ratio:

Probability of transition in the next Δt

1. whole system’s transition probability

evaluation

2. every structural members transition

probability evaluation


The bridge steel truss

Environmental condition

• Industrial area (Bellinzago

Lombardo, Milan, Italy)

• Aggressive environment

(pollution)

Structure

• 2 simply supported span

decks

• 2 main beams (Warren

truss structure)

• RC piles

• Welded joints


The structural model

Statically indeterminate truss system

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

27

45

26

46

25

47

24

48

23

49

22

50

21

51

20

52

19

28

53

29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44

Q3

Q1 Q2 Q2 Q1

2.6m

28.8m

1.6m


MRP probabilistically investigates crossing time for each member from i-state

at t0-time to j-state in the next interval Δt

MC simulation performs on population of 5000 samples belonging to the

structural system

Each structural member suffers by a random deterioration process

The practical approach and the state bounders

Weibull distribution models both members’ crossing times and whole system’s

failure time

State Bounders:


Transition probability

Probability of transition evaluated for the whole system

for each m members of the system (m=53)

risk of transition from i to j assumed upcoming

maintenance at instants close to the crossing time

0.0000

0.0002

0.0004

0.0006

0.0008

0.0010

0.0012

0.0014

0.0016

0.0018

0.0020

1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

P0

1D

t|t0

Dt [Years]

t0=3years

t0=2years

dmax

dmin

t0=1year

t*dmaxt*

dmin 0.0000

0.0002

0.0004

0.0006

0.0008

0.0010

0.0012

0.0014

0.0016

0.0018

0.0020

0.0 5.0 10.0 15.0 20.0 25.0

Pfa

il Dt|

t0

Dt [Years]

t0=20years

t0=10years

dmax

t0=1year

dmin

t*dmin t*

dmax

Probability of transition from state 0 to sate 1 Probability of collapse

Considering the whole system


• Replacing all damaged members

• Repairing all damaged members

• Replacing/repairing several damaged members

Selective maintenance scenarios

Scenario 1) t0=1 year

renew/replace the whole system (heavy maintenance)

Δt [t*min=19.0years; t*max=22.5years]

cyclical repetition every 19-22.5 years

prevents the system failure

Scenario 2) t0=1 year

repair the whole system (light maintenance)


cyclical repetition every 3-4 years

delays the transition from state 0 to state 1

Relevant maintenance scenario comparison:

Scenario 3) t0>3 year

repair/replace the whole system (moderate to heavy maintenance)


cyclical repetition every 6-7years

prevents the transition of the system from state 1 to state 2

0.0000

0.0002

0.0004

0.0006

0.0008

0.0010

0.0012

0.0014

0.0016

0.0018

0.0020

0.0 5.0 10.0 15.0 20.0 25.0

Pfa

il Dt|

t0

Dt [Years]

t0=20years

t0=10years

dmax

t0=1year

dmin

t*dmin t*

dmax

0.0000

0.0002

0.0004

0.0006

0.0008

0.0010

0.0012

0.0014

0.0016

0.0018

0.0020

1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

P0

1D

t|t0

Dt [Years]

t0=3years

t0=2years

dmax

dmin

t0=1year

t*dmaxt*

dmin



Probability of transition from

state 0 to state 1

t0 = 1 year

(“other members”=members 26-53)

Probability of transition from state

1 to state 2t0 = 1 year

(“other members”=members 26-53)

Transition 1-2 occurs always m years after transition 0-1

Transition 0-2 has never occurred

Considering each m member of the system (m=53)

0.0000

0.0002

0.0004

0.0006

0.0008

0.0010

0.0012

0.0014

0.0016

0.0018

0.0020

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

Dt [Years]

P0

1|t

0=1

year

m22

m23

m21

m24

m20

m25

other

members

dmax

dmin

t*dmin t*

dmax

m06

m13

0.0000

0.0002

0.0004

0.0006

0.0008

0.0010

0.0012

0.0014

0.0016

0.0018

0.0020

0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00

Dt [Years]

P1

2|t

0=1

year

other

members

dmax

t*dmin t*

dmax

m22

m23

m21

m24

m20

m25

dmin



Scenario A: Long-term preservation of state 0 Cyclical repetition of light maintenance

t0=1 year:

1. repairing of the deteriorated members 21, 22, 23, and 24 every 4 years.

2. repairing of the deteriorated members 21, 22 23, 24, 20 and 25 every 12-15 years.

3. repairing/ replacement of the whole system every 20-26 years.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

27

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26

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25

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24

48

23

49

22

50

21

51

20

52

19

18

53

28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43

Maintenance every 4 years.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

27

45

26

46

25

47

24

48

23

49

22

50

21

51

20

52

19

18

53

28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43

Maintenance every 12-15 years

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

27

45

26

46

25

47

24

48

23

49

22

50

21

51

20

52

19

18

53

28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43

Maintenance every 20-26 years



Considering a moderate risk

Scenario B: Long-term preservation of state 0

t0=1 year

1. repairing of the deteriorated members 21, 22 23, 24, 20 and 25 every 6 years

2. repairing/replacement of the whole system every 20 years



Scenario C: Preserve the system into state 1 (moderate damage)

t0>6 years: members 21, 22, 23, and 24 still could be in the state 1

1. repairing/replacement of the deteriorated members 21, 22 23, 24, every 11 years

2. repairing/replacement of the whole system every 22 years.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

27

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26

46

25

47

24

48

23

49

22

50

21

51

20

52

19

18

53

28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43

Maintenance every 11 years.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

27

45

26

46

25

47

24

48

23

49

22

50

21

51

20

52

19

18

53

28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43

Maintenance every 22 years

Conclusion

The MRP methodology supports:

•the definition of the best maintenance strategy for a long-lasting life of a system

•the definition of design changes for extending the maintenance interval

0.0000

0.0002

0.0004

0.0006

0.0008

0.0010

0.0012

0.0014

0.0016

0 5 10 15 20 25 30 35 40 45 50

instant of maintenance

P(m

,12) D

t|t0

=1

yr

d min

d max


1. Monte Carlo simulation & semi-Markov modeling investigate the probability

to exceed a certain damage threshold, member by member, at a specific t-time

2. Combined probabilistic approach supports maintenance decision-

making process (which member?) and financial resources

investments

3. Constraints: state definition & distribution choice

4. Application to a real case study

5. Improving monitoring data collection & phenomenon physical knowledge

potentially enhance the method

6. Economical evaluation & comparison of each scenario (still on going)

Conclusion

Engineering

Utilizzo di un approccio Markoviano per la manutenzione di ponti in acciaio