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Probabilistic safety in a multiscale and time dependent model

Université des Sciences et U .S.T. O de la Technologie d’Oran. Probabilistic safety in a multiscale and time dependent model. for suspension cables.

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Probabilistic safety in a multiscale and time dependent model

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  1. Université des Sciences et U.S.T.O de la Technologie d’Oran Probabilistic safety in a multiscale and time dependent model for suspension cables S. M. Elachachi1,2, D. Breysse1 and S. Yotte11CDGA, University of Bordeaux I (France)2LM2SC, University of Sciences and Technology of Oran (Algeria)

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

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

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

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

  6. Introduction Objectives • 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, • develop a model of the residual lifespan (for a requirement of given service), • Evaluate the risk of failure. Workshop Probability and Materials: from Nano- to Macro-Scale, JHU 5-7/01/2005

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

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

  9. experimental aspects Wire tension test 1600 Stress (MPa) variability of Mechanical characteristics 1400 1200 1000 800 c_3-1 Wire constitutive law c_3-2 600 c_3-3 400 c_4-1 c_4-2 200 Strain 0 0 0,0002 0,0004 0,0006 0,0008 0,001 0,0012 0,0014 Workshop Probability and Materials: from Nano- to Macro-Scale, JHU 5-7/01/2005

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

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

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

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

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

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

  16. Wire scale Constitutive wire law • su Stress (MPa) • se • eu • ee Strain • Ramdom local variables • {X}= {ee, se , eu, su} Workshop Probability and Materials: from Nano- to Macro-Scale, JHU 5-7/01/2005

  17. Wire scale Data Base : (675 + 20) tension wire tests ee se eu su [r]= Workshop Probability and Materials: from Nano- to Macro-Scale, JHU 5-7/01/2005

  18. Healthy • su corroded • su Wire scale c.d.f of su of the Healthy population (3p Weibull model) c.d.f of su of the corroded population (3p Weibull model) Workshop Probability and Materials: from Nano- to Macro-Scale, JHU 5-7/01/2005

  19. Wire scale Corrosion : • Corrosion chart (initiation time) p(i, t) = p(i, n.Dt), = 1 – (1 – pDt(i))n Number of corroded wires: Iterative relation : p(i, t + Dt) = p(i, t) + [1 - p(i, t)] pDt(i) Assumption: « Linear distribution » Workshop Probability and Materials: from Nano- to Macro-Scale, JHU 5-7/01/2005

  20. Wire scale Identified Truncated Normal Lognormal • Corrosion kinetics c.d.f of c(t=36ans) • t0 initiation time (random), • b corrosion tendancy, • a corrosion rate (random). reduction of wire diameter • c (mm) • su Workshop Probability and Materials: from Nano- to Macro-Scale, JHU 5-7/01/2005

  21. Wire scale 0 yr 10 yrs 36 yrs 100 yrs Healthy Corroded • Corrosion kinetics • t0 initiation time (random), • b corrosion tendancy, • a corrosion rate (random). reduction of wire diameter Temporal evolution of c.d.f of su • c (mm) • su Workshop Probability and Materials: from Nano- to Macro-Scale, JHU 5-7/01/2005

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

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

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

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

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

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

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

  29. Cable scale yrs yrs yrs yrs Two types of corroded populations P1 and P2 Length cable: 8 m 60 strands, 10 sections per strand •  Fcab (kN) •  Fcab (kN) yrs yrs •  u (m) •  u (m) P1 P1+P2 Workshop Probability and Materials: from Nano- to Macro-Scale, JHU 5-7/01/2005

  30. Cable scale Rc= max(Fcab(u) I tfixed) •  Rc (kN) Case 2 Case 1 • t (yrs) Workshop Probability and Materials: from Nano- to Macro-Scale, JHU 5-7/01/2005

  31. Cable scale Introduction of a third population P3 •  Rc (kN) • t (yrs) Rc vs time for different ratios of P3 Workshop Probability and Materials: from Nano- to Macro-Scale, JHU 5-7/01/2005

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

  33. Cable scale Corrosion kinetics effects Values of •  Rc (kN) • t (yrs) Rc vs time for different rates of corrosion Workshop Probability and Materials: from Nano- to Macro-Scale, JHU 5-7/01/2005

  34. Cable scale p.d.f of Rc: Gaussian ! yrs c.d.f of Rc (case 1) yrs yr c.d.f of Rc (case 2) Rc yrs yrs yr Rc Values of Workshop Probability and Materials: from Nano- to Macro-Scale, JHU 5-7/01/2005

  35. Cable scale • t (yrs) Risk of failure • Pf Workshop Probability and Materials: from Nano- to Macro-Scale, JHU 5-7/01/2005

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

  37. Conclusions (cnt’d) • 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 significant in terms of a hierarchical basis of the factors of influence. Workshop Probability and Materials: from Nano- to Macro-Scale, JHU 5-7/01/2005

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

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