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Probabilistic safety in a multiscale and time dependent model Probabilistic safety in a multiscale and time dependent model for suspension cables S. M.

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Presentation on theme: "Probabilistic safety in a multiscale and time dependent model Probabilistic safety in a multiscale and time dependent model for suspension cables S. M."— Presentation transcript:

1 Probabilistic safety in a multiscale and time dependent model Probabilistic safety in a multiscale and time dependent model for suspension cables S. M. Elachachi 1,2, D. Breysse 1 and S. Yotte 1 1 CDGA, University of Bordeaux I (France) 2 LM2SC, University of Sciences and Technology of Oran (Algeria) S. M. Elachachi 1,2, D. Breysse 1 and S. Yotte 1 1 CDGA, University of Bordeaux I (France) 2 LM2SC, University of Sciences and Technology of Oran (Algeria) Université des Sciences et U.S.T. O de la Technologie d’Oran

2 2Workshop Probability and Materials: from Nano- to Macro-Scale, JHU 5-7/01/2005 Outline of the presentation Introduction Introduction Experimental aspects Experimental aspects Multiscale approach Multiscale approach wire scale wire scale Strand scale Strand scale Cable scale Cable scale Conclusions Conclusions

3 3Workshop Probability and Materials: from Nano- to Macro-Scale, JHU 5-7/01/2005 Introduction Probabilistic safety in a multiscale and time dependent model

4 4Workshop Probability and Materials: from Nano- to Macro-Scale, JHU 5-7/01/2005 Mechanical loads Dead loads, Live loads, Accidental loads,… Environmental loads Temperature gradient, Temperature gradient, Humidity. Humidity. Introduction Cables Aquitaine Bridge (Bordeaux, France) Corrosion

5 5Workshop Probability and Materials: from Nano- to Macro-Scale, JHU 5-7/01/2005 Introduction Types of Corrosion  general  localized (pitting) Old and New strands Visual Inspection cracks

6 6Workshop Probability and Materials: from Nano- to Macro-Scale, JHU 5-7/01/2005 Objectives determinate the bearing capacity by integrating the complexity of the mechanical description (non linear behavior, load redistributions...). determinate the bearing capacity by integrating the complexity of the mechanical description (non linear behavior, load redistributions...). evaluate the effect of the factors affecting the long- term performance of the cable, evaluate the effect of the factors affecting the long- term performance of the cable, develop a model of the residual lifespan (for a requirement of given service), develop a model of the residual lifespan (for a requirement of given service), Evaluate the risk of failure. Evaluate the risk of failure. Introduction

7 7Workshop Probability and Materials: from Nano- to Macro-Scale, JHU 5-7/01/2005 Experimental aspects Probabilistic safety in a multiscale and time dependent model

8 8Workshop Probability and Materials: from Nano- to Macro-Scale, JHU 5-7/01/2005 Strand tension test before test after test experimental aspects F Displacement (LCPC Nantes)

9 9Workshop Probability and Materials: from Nano- to Macro-Scale, JHU 5-7/01/2005 0 200 400 600 800 1000 1200 1400 1600 00,00020,00040,00060,00080,0010,00120,0014 c_3-1 c_3-2 c_3-3 c_4-1 c_4-2 Stress (MPa) Strain experimental aspects Wire constitutive law variability of Mechanical characteristics Wire tension test

10 10Workshop Probability and Materials: from Nano- to Macro-Scale, JHU 5-7/01/2005 The model must take into account : Random (probabilistic/stochastic) aspect of mechanical characteristics, Random (probabilistic/stochastic) aspect of mechanical characteristics, Multiscale aspect (geometrical and mechanics rules of assemblage) Multiscale aspect (geometrical and mechanics rules of assemblage) experimental aspects

11 11Workshop Probability and Materials: from Nano- to Macro-Scale, JHU 5-7/01/2005 Multiscale Approach Probabilistic safety in a multiscale and time dependent model

12 12Workshop Probability and Materials: from Nano- to Macro-Scale, JHU 5-7/01/2005 Multiscale approach Multiscale approach Cable scale Strand scaleWire scale Three scale system

13 13Workshop Probability and Materials: from Nano- to Macro-Scale, JHU 5-7/01/2005 Multiscale approach Multiscale approach Cable Strand Strand's section Wire layersWire Uncoupled approach Aquitaine Bridge : 37 strands, 1,750 strand's sections and 217 wires per strand's section 14,000,000 wires. Global description Local description parallel-serie sub-System Parallel sub-system

14 14Workshop Probability and Materials: from Nano- to Macro-Scale, JHU 5-7/01/2005 Multiscale approach Multiscale approach Cable Strand Strand's section Wire layers Wire Uncoupled approach Global description Local description parallel-serie sub-System Parallel sub-system

15 15Workshop Probability and Materials: from Nano- to Macro-Scale, JHU 5-7/01/2005 Wire Scale Probabilistic safety in a multiscale and time dependent model

16 16Workshop Probability and Materials: from Nano- to Macro-Scale, JHU 5-7/01/2005 Wire scale Wire scale Constitutive wire law  e e ee Ramdom local variables {X}=  e,  e,  u,  u } Stress (MPa)  u u Strain uu

17 17Workshop Probability and Materials: from Nano- to Macro-Scale, JHU 5-7/01/2005 Data Base : (675 + 20) tension wire tests identified Parameters descriptors Parameters averageStandard deviation coef. var TypeObs.  e (MPa) 8721540.177lognormal : 13,663  : 0,175 correl to  e,  u and  u E (Mpa) 2 10 5 10 4 0.05lognormal : 19,1  : 0,05 ee 4.36.10 -3 7.73 10 -4 0.177lognormal defined by  e and E Correl. to  e,  u and  u uu 1.246.10 -2 2.54 10 -3 0.204Weibullm : 4,15  min : 6,95.10 -3  0 : 5,4. 10 -3 Correl. to  e,  e and  u  u (MPa) 151063.20.04Weibullm : 3,173  min : 1330  0 : 204 Healthy populat. 12402080.168Weibullm : 3,526  min : 576,4  0 : 735 Corroded populat. Wire scale Wire scale [  ]=  e  e  u  u

18 18Workshop Probability and Materials: from Nano- to Macro-Scale, JHU 5-7/01/2005 Wire scale Wire scale Healthy  u u corroded  u 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)

19 19Workshop Probability and Materials: from Nano- to Macro-Scale, JHU 5-7/01/2005 Corrosion : Corrosion chart (initiation time) p(i, t) = p(i, n.  t), = 1 – (1 – p  t (i)) n Number of corroded wires: Iterative relation : p(i, t +  t) = p(i, t) + [1 - p(i, t)] p  t (i) Wire scale Wire scale layer i1234567 P(i, T= 36 yrs)0.9990.80.60.40.200 p(i,  t = 1 yr) 0.17460.04370.02510.01410.006200 Assumption: « Linear distribution »

20 20Workshop Probability and Materials: from Nano- to Macro-Scale, JHU 5-7/01/2005 Identified Truncated Normal Lognormal Corrosion kinetics Corrosion kinetics c.d.f of c.d.f of c(t=36ans) c(t=36ans) Wire scale Wire scale  u u  c (mm) t 0 initiation time (random), t 0 initiation time (random),  corrosion tendancy,  corrosion tendancy,  corrosion rate (random).  corrosion rate (random). reduction of wire diameter

21 21Workshop Probability and Materials: from Nano- to Macro-Scale, JHU 5-7/01/2005 Corrosion kinetics Corrosion kinetics Wire scale Wire scale  u u  c (mm) Temporal evolution of c.d.f of  u t 0 initiation time (random), t 0 initiation time (random),  corrosion tendancy,  corrosion tendancy,  corrosion rate (random).  corrosion rate (random). 0 yr 10 yrs 36 yrs 100 yrs Healthy Corroded reduction of wire diameter

22 22Workshop Probability and Materials: from Nano- to Macro-Scale, JHU 5-7/01/2005 Strand Scale Probabilistic safety in a multiscale and time dependent model

23 23Workshop Probability and Materials: from Nano- to Macro-Scale, JHU 5-7/01/2005 Strand scale constitutive law of a strand's section ( F trc vs displacement): Where : Monte Carlo Simulation anchoring length Local description w,i

24 24Workshop Probability and Materials: from Nano- to Macro-Scale, JHU 5-7/01/2005 Strand scale F trc – u (average) c.d.f of F trc max c.d.f of broken wires  F tr c (kN)  u (m) Monte Carlo Normal Lognormal Monte Carlo Normal Lognormal Local description

25 25Workshop Probability and Materials: from Nano- to Macro-Scale, JHU 5-7/01/2005 Cable Strand Strand's section Wire layers Wire Uncoupled approach Global description Local description parallel-serie sub-System Parallel system Strand scale Global description

26 26Workshop Probability and Materials: from Nano- to Macro-Scale, JHU 5-7/01/2005 Strand scale F trc (u,t) = (1-D(u)).(1-  (t)).u Damage Indicator Corrosion Indicator F (  u ) = Analytical model : Global description

27 27Workshop Probability and Materials: from Nano- to Macro-Scale, JHU 5-7/01/2005 Monte CarloModel  t (yrs) 0 yr 20 yrs 40 yrs 60 yrs 80 yrs 100 yrs 120 yrs 140 yrs 160 yrs  F tr c (kN)  u (m) Strand scale Model Monte Carlo Model Monte Carlo Average Standard deviation Global description

28 28Workshop Probability and Materials: from Nano- to Macro-Scale, JHU 5-7/01/2005 Cable Scale Probabilistic safety in a multiscale and time dependent model

29 29Workshop Probability and Materials: from Nano- to Macro-Scale, JHU 5-7/01/2005 Cable scale Population Type of corrosion BP1General 1,9.10 -3 9,5.10 -5 0,050,95 P2Pitting 9,86.10 -3 2,46.10 -3 0,050,747 P1 P1+P2  F cab (kN)  u (m) Length cable: 8 m 60 strands, 10 sections per strand yrs Two types of corroded populations P1 and P2 yrs

30 30Workshop Probability and Materials: from Nano- to Macro-Scale, JHU 5-7/01/2005  R c (kN)  t (yrs) Cable scale R c = max(F cab (u) I t fixed ) Case 2 Case 1

31 31Workshop Probability and Materials: from Nano- to Macro-Scale, JHU 5-7/01/2005 CaseP1(%)P2(%)P3(%) Out of collarsCollars (20%) 180200 280164 380128 480812 580416  R c (kN)  t (yrs) Rc vs time for different ratios of P3 Introduction of a third population P3 Cable scale

32 32Workshop Probability and Materials: from Nano- to Macro-Scale, JHU 5-7/01/2005 Cable scale elt_1 elt_2 elt_3 elt_4 Mechanics and corrosion coupling

33 33Workshop Probability and Materials: from Nano- to Macro-Scale, JHU 5-7/01/2005  R c (kN)  t (yrs) Rc vs time for different rates of corrosion Cable scale CasP1P2P3 Out of collarsCollars 11,9.10 -3 9,86.10 -3 - 21,9.10 -3 9,86.10 -3 1,972.10 -2 31,9.10 -3 9,86.10 -3 2,958.10 -2 41,9.10 -3 9,86.10 -3 3,944.10 -2 Values of Corrosion kinetics effects

34 34Workshop Probability and Materials: from Nano- to Macro-Scale, JHU 5-7/01/2005 Cable scale c.d.f of R c (case 1) c.d.f of R c (case 2) RcRc RcRcCasP1P2P3 Out of collars Collars 1 1,9.10 -3 9,86.10 -3 1,972.10 -2 2 1,9.10 -3 9,86.10 -3 3,944.10 -2 p.d.f of R c : Gaussian ! yrs yr yrs yr Values of

35 35Workshop Probability and Materials: from Nano- to Macro-Scale, JHU 5-7/01/2005 Cable scale CaseP1(%)P2(%)P3(%) Out of collars Collars (20%) 180200 280164 380128 480812 580416 PfPf  t (yrs) Risk of failure

36 36Workshop Probability and Materials: from Nano- to Macro-Scale, JHU 5-7/01/2005 Conclusions  The phenomena of corrosion induce strong modifications of the geometrical and mechanical characteristics of the components of suspension cables and thus causes a notable reduction of the bearing capacity of the cable according to time, whose consequences can sometimes lead to its partial (or total) failure.  The main aspects of a mechanical modeling integrating the statistical distribution laws of the local variables relating at the wire scale, in a parallel wire system to describe the behavior of the strand's section, were examined and numerically implemented.

37 37Workshop Probability and Materials: from Nano- to Macro-Scale, JHU 5-7/01/2005  The need for building a data base of the state of corrosion (feeded with cable inspections at more or less regular intervals) of the cables seems to be priority if one wishes to have really predictive forecasting.  The anchoring length of wire is also an influential parameter.  The results obtained must be considered: like qualitative indicators of the behavior, due to the incomplete character of the data now available, like qualitative indicators of the behavior, due to the incomplete character of the data now available, like significant in terms of a hierarchical basis of the factors of influence.like significant in terms of a hierarchical basis of the factors of influence. Conclusions (cnt’d)

38 38Workshop Probability and Materials: from Nano- to Macro-Scale, JHU 5-7/01/2005 Thank you

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