probabilistic safety in a multiscale and time dependent model

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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. - PowerPoint PPT Presentation

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  • Probabilistic safety in a multiscale and time dependent model

    for suspension cables S. M. Elachachi1,2, D. Breysse1 and S. Yotte1

    1CDGA, University of Bordeaux I (France)2LM2SC, University of Sciences and Technology of Oran (Algeria)

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

  • Outline of the presentation Introduction Experimental aspects Multiscale approach wire scaleStrand scaleCable scaleConclusions

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

  • Introduction

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

  • Mechanical loadsDead loads,Live loads,Accidental loads,Environmental loadsTemperature gradient,Humidity.Introduction

    CablesAquitaine Bridge (Bordeaux, France)Corrosion

    Workshop 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

    Workshop 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 ...).

    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. Introduction

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

  • Experimental aspects

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

  • Strand tension test

    before test

    after testF

    Displacement(LCPC Nantes)

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

  • 0200400600800100012001400160000,00020,00040,00060,00080,0010,00120,0014c_3-1c_3-2c_3-3c_4-1c_4-2Stress (MPa) Strain Wire constitutive lawvariability of MechanicalcharacteristicsWire tension test

    Workshop 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,

    Multiscale aspect (geometrical and mechanics rules of assemblage)

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

  • Multiscale Approach

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

  • Cable scaleThree scale system

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

  • CableStrandStrand's sectionWire layersWireUncoupled approachAquitaine Bridge : 37 strands, 1,750 strand's sections and 217 wires per strand's section 14,000,000 wires.Global descriptionLocal descriptionparallel-serie sub-System

    Parallel sub-system

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

  • CableStrandStrand's sectionWire layersWireUncoupled approachGlobal descriptionLocal descriptionparallel-serie sub-System

    Parallel sub-system

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

  • Wire Scale

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

  • Wire scale

    Constitutive wire law

    seeeRamdom local variables{X}= {ee,se , eu,su}

    Stress (MPa)su Straineu

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

    Graph3

    0000

    138.3329518952138.4615384615120.641025641120.641025641

    276.6659037903276.9230769231241.2820512821241.2820512821

    414.9988556855415.3846153846361.9230769231361.9230769231

    553.3318075807553.8461538462482.5641025641482.5641025641

    691.6647594759692.3076923077723.8461538462603.2051282051

    829.997711371830.7692307692844.4871794872723.8461538462

    900965.7101343988941844.4871794872

    954.65645097141078.9317507418968.4920517101964.6974739875

    1065.34125688111173.01539390281037.67005540371071.5335684684

    1182.94386316021252.43529084331106.84805909721162.1865094673

    1272.87526796181320.37180480251176.02606279081240.0744614763

    1335.13547128611379.14760414161245.20406648441307.7162714983

    1397.39567461031430.49853372431314.3820701781367.0089784173

    14401475.7479312541383.56007387161378

    contrainte (MPa)

    dformation

    exp C3N2

    sim C3N2

    exp C4N1

    sim C4N1

    Couche3_N1

    epsForce (N)Sig (Mpa)Donnes et +

    00sige9.00E+0200

    0.0001138.4615384615epse6.50E-040.0001138.4615384615

    0.0002276.9230769231epsu1.32E-030.0002276.9230769231

    0.0003415.3846153846sigu1.44E+030.0003415.3846153846

    0.0004553.8461538462E1384615.384615380.0004553.8461538462

    0.0005692.3076923077C1.07E+030.0005692.3076923077

    0.0006830.76923076920.0006830.7692307692

    0.000659000.0007965.7101343988

    0.000716562.76954.65645097140.00081078.9317507418

    0.000818483.081065.34125688110.00091173.0153939028

    0.000920523.421182.94386316020.0011252.4352908433

    0.00122083.681272.87526796180.00111320.3718048025

    0.001123163.861335.13547128610.00121379.1476041416

    0.001224244.041397.39567461030.00131430.4985337243

    0.001322500014400.001321440

    VALEURS EXPERIMENTALESVALEURS SIMULEES

    x (mm)y(mm)juste la partie non linairecoeff_chelle

    32.565

  • Data Base : (675 + 20) tension wire tests Wire scale

    [r]= ee se eu su

    identified Parameters descriptors ParametersaverageStandard deviationcoef. varTypeObs.se(MPa)872 1540.177lognormall: 13,663x: 0,175correl to ee, eu and suE (Mpa)2 1051040.05lognormall: 19,1x: 0,05ee4.36.10-37.73 10-40.177lognormaldefined by se and E Correl. to se, eu and sueu1.246.10-22.54 10-30.204Weibullm: 4,15emin: 6,95.10-3e0: 5,4. 10-3Correl. to ee, se and susu(MPa)151063.20.04Weibullm: 3,173smin: 1330s0: 204Healthy populat.12402080.168Weibullm: 3,526smin: 576,4s0: 735Corroded populat.

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

  • Wire scale

    c.d.f of suof the corroded population(3p Weibull model)c.d.f of suof the Healthy population(3p Weibull model)

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

  • 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)

    Wire scaleAssumption: Linear distribution

    layer i1234567P(i, T= 36 yrs)0.9990.80.60.40.200p(i, Dt = 1 yr)0.17460.04370.02510.01410.006200

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

  • Corrosion kinetics

    c.d.f of c(t=36ans)suc (mm)

    t0 initiation time (random),b corrosion tendancy, a corrosion rate (random).

    reduction of wire diameter

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

  • Corrosion kinetics

    suc (mm) Temporal evolution of c.d.f of su

    t0 initiation time (random),b corrosion tendancy, a corrosion rate (random).

    reduction of wire diameter

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

  • Strand Scale

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

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

    Where :

    Monte Carlo SimulationLocal descriptionw,i

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

  • Ftrc u (average) c.d.f of Ftrc max c.d.f of broken wiresFtrc (kN)u (m)Monte CarloNormal LognormalMonte CarloNormal LognormalLocal description

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

  • CableStrandStrand's sectionWire layersWireUncoupled approachGlobal descriptionLocal descriptionparallel-serie sub-System

    Parallel systemGlobal description

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

  • Ftrc(u,t) = (1-D(u)).(1- d(t)).u Damage Indicator Corrosion IndicatorF (eu) = Analytical model :Global description

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

  • Ftrc (kN)u (m)u (m)ModelMonte CarloModelMonte CarloAverage Standard deviationGlobal description

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

  • Cable Scale

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

  • P1 P1+P2Fcab (kN)Fcab (kN)u (m)u (m)Length cable: 8 m60 strands, 10 sections per strandyrsyrsTwo types of corroded populations P1 and P2

    PopulationType of corrosionBP1General1,9.10-39,5.10-50,050,95P2Pitting9,86.10-32,46.10-30,050,747

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

  • Rc (kN)t (yrs)Rc= max(Fcab(u) I tfixed)Case 2Case 1

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

  • Rc (kN)t (yrs)Rc vs time for different ratios of P3Introduction of a third population P3

    CaseP1(%)P

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