probabilistic safety in a multiscale and time dependent model

Probabilistic safety in Probabilistic safety in a multiscale and time a multiscale and time

dependent modeldependent model

for suspension cablesfor suspension cables S. M. ElachachiS. M. Elachachi1,21,2, , D. BreysseD. Breysse11 and S. Yotte and S. Yotte11

11CDGA, University of Bordeaux I (France)CDGA, University of Bordeaux I (France)22LM2SC, University of Sciences and Technology of Oran LM2SC, University of Sciences and Technology of Oran

(Algeria)(Algeria)

Université des Sciences et

U.S.T.O de la Technologie d’Oran

2Workshop Probability and Materials: from Nano- to Macro-Scale, JHU 5-7/01/2005

Outline of the presentationOutline of the presentation

IntroductionIntroduction Experimental Experimental

aspectsaspects Multiscale Multiscale

approachapproach wire scalewire scale Strand scaleStrand scale Cable scaleCable scale

ConclusionsConclusions


IntroductionIntroduction

Probabilistic safety in a multiscale and time dependent modelProbabilistic safety in a multiscale and time dependent model


Mechanical loadsMechanical loads Dead loads,Dead loads, Live loads,Live loads, Accidental loads,Accidental loads,

……

Environmental Environmental loadsloads

Temperature Temperature gradient,gradient,

Humidity.Humidity.

IntroductioIntroductionn

Cables

Aquitaine Bridge (Bordeaux, France)

CorrosionCorrosion



Types of Corrosion

general localized (pitting)

Old and New strands

Visual Inspection

cracks


ObjectivesObjectives

determinate the bearing capacity by determinate the bearing capacity by integrating the complexity of the integrating the complexity of the mechanical description (non linear mechanical description (non linear behavior, load redistributions ...).behavior, load redistributions ...).

evaluate the effect of the factors evaluate the effect of the factors affecting the long-term performance of affecting the long-term performance of the cablethe cable, ,

develop a model of the residual lifespan develop a model of the residual lifespan (for a requirement of given service), (for a requirement of given service),

Evaluate the risk of failure. Evaluate the risk of failure.



Experimental Experimental aspectsaspects



Strand tension testStrand tension test

before before testtest

after testafter test

experimental aspectsexperimental aspects

F

Displacement

(LCPC Nantes)(LCPC Nantes)


0

200

400

600

800

1000

1200

1400

1600

0 0,0002 0,0004 0,0006 0,0008 0,001 0,0012 0,0014

c_3-1

c_3-2

c_3-3

c_4-1

c_4-2

Stress (MPa)

Strain


W

ire c

on

stit

uti

ve l

aw

variability of Mechanicalcharacteristics

Wire tension test


The model must take into account :The model must take into account :

Random (probabilistic/stochastic) Random (probabilistic/stochastic) aspect of mechanical characteristics,aspect of mechanical characteristics,

Multiscale aspect (Multiscale aspect (geometrical and geometrical and mechanics rules of assemblagemechanics rules of assemblage))



Multiscale Multiscale ApproachApproach



Multiscale approachMultiscale approach

Cable scale

Strand scale

Wire scale

Three scale system



Cable

Strand

Strand's section

Wire layers Wire

Uncoupled approachAquitaine Bridge : 37 strands, 1,750 strand's sections and 217 wires per strand's section 14,000,000 wires.

Global description

Local description

para

llel-

seri

e s

ub

-Sys

tem

Para

llel

sub

-sys

tem



Cable

Strand

Strand's section

Wire layers

Wire

Uncoupled approach

Global description

Local description

para

llel-

seri

e s

ub

-Sys

tem

Para

llel

sub

-sys

tem


Wire ScaleWire Scale



Wire scaleWire scale

Constitutive wire law

e

si )eC(1

)eE(

e

e si E

σσεε

εεσσ

σσεσ

0

200

400

600

800

1000

1200

1400

1600

0 0,002 0,004 0,006 0,008 0,01 0,012 0,014 0,016

exp C3N2

sim C3N2

exp C4N1

sim C4N1

contrainte (MPa)

déformation

e

e

Ramdom local variables{X}=e, e , u, u}

Stress (MPa)

u

Strain

u


Data Base : (675 + 20) tension wire tests

identified Parameters descriptors Parameters

average Standard deviation

coef. var

Type Obs.

e

(MPa)

872 154 0.177 lognormal : 13,663: 0,175

correl to e, u and

u

E (Mpa)

2 105 104 0.05 lognormal : 19,1: 0,05

e4.36.10-3 7.73 10-4 0.177 lognormal defined by e

and E

Correl. to e, u and

u

u1.246.10-2 2.54 10-3 0.204 Weibull m : 4,15

min : 6,95.10-3

0 : 5,4. 10-3

Correl. to e, e and

u

u

(MPa)

1510 63.2 0.04 Weibull m : 3,173min : 1330

0 : 204

Healthy populat.

1240 208 0.168 Weibull m : 3,526min : 576,4

0 : 735

Corroded populat.


[]=

1

679,01

309,0276,01

309,0276,011

sym

eeuu



SAIN

0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1

1300 1400 1500 1600 1700

exp

identif 3p

Healthy

u

corrodé

0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1

600 800 1000 1200 1400 1600

exp

identif 3p

corroded

u

c.d.f of u

of the corroded population(3p Weibull model)

c.d.f of u

of the Healthy population(3p Weibull model)


Corrosion :Corrosion : Corrosion chart (initiation time)Corrosion chart (initiation time)

p(i, t) = p(i, n.p(i, t) = p(i, n.t), = 1 – (1 – pt), = 1 – (1 – ptt(i))(i))nn

Number of corroded wires:Number of corroded wires:

Iterative relation :Iterative relation :

p(i, t + p(i, t + t) = p(i, t) + [1 - p(i, t)] pt) = p(i, t) + [1 - p(i, t)] ptt(i)(i)


layer i 1 2 3 4 5 6 7

P(i, T= 36 yrs) 0.999 0.8 0.6 0.4 0.2 0 0

p(i, t = 1 yr) 0.1746 0.0437 0.0251 0.0141 0.0062 0 0

k-NikkNii ) t)p(i,-(1 t)p(i, C (t) Nc Assumption:

« Linear distribution »


0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1

0 0,2 0,4 0,6 0,8 1 1,2 1,4

identif

modèle loi N

modèle loi LN

IdentifiedTruncated NormalLognormal

β0t-t αc(t)

Corrosion kineticsCorrosion kinetics

c.d.f of c.d.f of

c(t=36ans)c(t=36ans)


u c (mm)

tt00 initiation time (random), initiation time (random), corrosion tendancy, corrosion tendancy, corrosion rate (random).corrosion rate (random).

reduction of reduction of wire diameterwire diameter


β0t-t αc(t)

Corrosion kineticsCorrosion kinetics


u c (mm)

Temporal evolution of c.d.f of u

tt00 initiation time (random), initiation time (random), corrosion tendancy, corrosion tendancy, corrosion rate (random).corrosion rate (random).

0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1

400 600 800 1000 1200 1400 1600

0 an

10 ans

36 ans

100 ans

sain

corrodé

0 yr

10 yrs

36 yrs

100 yrs

Healthy

Corroded

reduction of reduction of wire diameterwire diameter


Strand ScaleStrand Scale



Strand scaleStrand scale

t)(u,Ft)(u,FN

1iiw,trc

constitutive law of a strand's section (Ftrc vs displacement):

Where :

),,t,t(A).,,,,l

u()t(u,F 0uuee

rifil,

Monte Carlo Simulation

anchoring length

Local description

w,i


0

500

1000

1500

2000

2500

3000

3500

4000

0 0,002 0,004 0,006 0,008 0,01 0,012 0,014

0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1

3500 3550 3600 3650 3700 3750 3800 3850 3900 3950

C_Tr_Tor

loi Normale

loi LogNormale

0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1

0 2 4 6 8 10 12 14 16

C_Tr_Tor

loi Normale

loi LogNormale


Ftrc– u (average)

c.d.f of Ftrc max c.d.f of broken wires

Ftrc (kN)

u (m)

Monte CarloNormal Lognormal

Monte CarloNormal Lognormal

Local description


Cable

Strand

Strand's section

Wire layers

Wire

Uncoupled approach

Global description

Local description

para

llel-

seri

e s

ub

-Sys

tem

Para

llel

syst

em

Strand scaleStrand scaleGlobal description



Ftrc(u,t) = (1-D(u)).(1- (t)).u

tr0K

1D(u)0 1)t(0 Damage Indicator Corrosion Indicator

)H(u)1(u.K

)u(F-1D(u)

tr0

N

iifil,

B0t-t Aδ(t)

))l.(u(PH(u) ru F (u) =

m

0

minuexp1

Analytical model : Global description


0

0,05

0,1

0,15

0,2

0,25

0,3

0,35

0,4

0,45

0 20 40 60 80 100 120 140

C_Tr_Tor modèle

t (ans)

Monte Carlo

Model

t (yrs)0

500

1000

1500

2000

2500

3000

3500

4000

0 0,002 0,004 0,006 0,008 0,01 0,012 0,014

0 an

20 ans

40 ans

60 ans

80 ans

100 ans

120 ans

140 ans

160 ans

0 yr20 yrs40 yrs60 yrs80 yrs100 yrs120 yrs140 yrs160 yrs

Ftrc (kN)

0,00E+00

5,00E+02

1,00E+03

1,50E+03

2,00E+03

2,50E+03

3,00E+03

3,50E+03

4,00E+03

0 0,002 0,004 0,006 0,008 0,01 0,012 0,014

modèle

C_Tr_Tor (1000)

0,00E+00

2,00E+01

4,00E+01

6,00E+01

8,00E+01

1,00E+02

1,20E+02

1,40E+02

1,60E+02

1,80E+02

2,00E+02

0 0,002 0,004 0,006 0,008 0,01 0,012 0,014

modèle

C_Tr_Tor (1000) Ftrc (kN)

u (m)

u (m)


ModelMonte Carlo

ModelMonte Carlo

Average Standard deviationGlobal description


Cable ScaleCable Scale



Cable scaleCable scale

A A APopulatioPopulationn

Type of Type of corrosioncorrosion B

P1P1 GeneralGeneral 1,9.101,9.10-3-3 9,5.109,5.10-5-5 0,050,05 0,950,95

P2P2 PittingPitting 9,86.109,86.10-3-3 2,46.102,46.10-3-3 0,050,05 0,7470,747

P1 P1+P2

Fcab (kN) Fcab (kN)

u (m)

u (m)

Length cable: 8 m60 strands, 10 sections per strand

yrsyrs

Two types of corroded populations P1 and P2

0

50000

100000

150000

200000

250000

0 0,01 0,02 0,03 0,04 0,05 0,06

t=0

50 ans

150 ans

yrsyrs

0

50000

100000

150000

200000

250000

0 0,01 0,02 0,03 0,04 0,05 0,06

t=0

50 ans

150 ansyrsyrs


0

50000

100000

150000

200000

250000

0 20 40 60 80 100 120 140

cas 2

cas 1

Rc (kN)

t (yrs)


Rc= max(Fcab(u) I tfixed)

Case 2

Case 1


Case P1(%) P2(%) P3(%)

Out of collars Collars (20%)

1 80 20 0

2 80 16 4

3 80 12 8

4 80 8 12

5 80 4 16

10000

30000

50000

70000

90000

110000

130000

150000

170000

190000

210000

0 20 40 60 80 100 120 140

cas 1

cas 2

cas 3

cas 4

cas 5

²

Rc (kN)

t (yrs)

Rc vs time for different ratios of P3

Introduction of a third population P3 Cable scaleCable scale



0

0,005

0,01

0,015

0,02

0,025

0 20 40 60 80 100 120 140

a) strain =u/l

0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

0 20 40 60 80 100 120 140

b) corrosion indicator

0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1

0 20 40 60 80 100 120 140

c) mechanical damage indicator D

0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1

0 20 40 60 80 100 120 140

d) global degradation

elt_1 elt_2

elt_1

elt_3 elt_4

elt_3 elt_4 elt_2

(yrs) (yrs)

elt_1

elt_3 elt_4 elt_2 elt_3

elt_1

(yrs) (yrs)

elt_1 elt_2

elt_3 elt_4

Mechanics and corrosion coupling


50000

70000

90000

110000

130000

150000

170000

190000

210000

0 20 40 60 80 100 120 140

cas 1

cas 2

cas 3

cas 4

Rc (kN)

t (yrs)

Rc vs time for different rates of corrosion


Cas P1 P2 P3

Out of collars Collars

1 1,9.10-3 9,86.10-3 -

2 1,9.10-3 9,86.10-3 1,972.10-2

3 1,9.10-3 9,86.10-3 2,958.10-2

4 1,9.10-3 9,86.10-3 3,944.10-2

Values ofA

Corrosion kinetics effects


0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1

70000 90000 110000 130000 150000 170000 190000

t= 100 ans

t= 0 an

t=50 ans

0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1

70000 90000 110000 130000 150000 170000 190000

t= 100 ans

t= 50 ans

t= 0 an


c.d.f of Rc

(case 1)c.d.f of Rc

(case 2)Rc

Rc

CasCas P1P1 P2P2 P3P3

Out of collarsOut of collars CollarsCollars

11 1,9.101,9.10-3-3 9,86.109,86.10-3-3 1,972.101,972.10-2-2

22 1,9.101,9.10-3-3 9,86.109,86.10-3-3 3,944.103,944.10-2-2

p.d.f of Rp.d.f of Rcc: Gaussian !: Gaussian !

yrs

yrs

yr

yrs

yrs

yr

Values ofA


1,00E-14

1,00E-12

1,00E-10

1,00E-08

1,00E-06

1,00E-04

1,00E-02

1,00E+00

0 20 40 60 80 100 120 140

case 1

case 2

case 3

case 4

case 5


Case P1(%) P2(%) P3(%)

Out of collars

Collars (20%)

1 80 20 0

2 80 16 4

3 80 12 8

4 80 8 12

5 80 4 16

P

f

t (yrs)

Risk of failure


ConclusionsConclusions

The phenomena of corrosion induce strong The phenomena of corrosion induce strong modifications of the geometrical and modifications of the geometrical and mechanical characteristics of the mechanical characteristics of the components of suspension cables and thus components of suspension cables and thus causes a notable reduction of the bearing causes a notable reduction of the bearing capacity of the cable according to time, capacity of the cable according to time, whose consequences can sometimes lead to whose consequences can sometimes lead to its partial (or total) failure. its partial (or total) failure.

The main aspects of a mechanical modeling The main aspects of a mechanical modeling integrating the statistical distribution laws integrating the statistical distribution laws of the local variables relating at the wire of the local variables relating at the wire scale, in a parallel wire system to describe scale, in a parallel wire system to describe the behavior of the strand's section, were the behavior of the strand's section, were examined and numerically implemented. examined and numerically implemented.


The need for building a data base of the The need for building a data base of the state of corrosion (feeded with cable state of corrosion (feeded with cable inspections at more or less regular intervals) inspections at more or less regular intervals) of the cables seems to be priority if one wishes of the cables seems to be priority if one wishes to have really predictive forecasting.to have really predictive forecasting.

The anchoring length of wire is also an The anchoring length of wire is also an influential parameter.influential parameter.

The results obtained must be considered:The results obtained must be considered:• like qualitative indicators of the like qualitative indicators of the behavior, due to the incomplete behavior, due to the incomplete character of the data now available, character of the data now available, •like significant in terms of a like significant in terms of a hierarchical basishierarchical basis of the factors of of the factors of influence.influence.

Conclusions Conclusions (cnt’d)(cnt’d)


Thank youThank you

probabilistic safety in a multiscale and time dependent model

Documents

wire scaleworkshop probability

multiscale aspect geometrical

mechanical loadsdead

accidental loads

live loads

mechanical description

university of bordeaux

strands sections