Extrapolation of Fatigue Loads 4th Conference on Extreme Value Analysis Gothenburg, August 15-19,...

Extrapolation of Fatigue Loads

4th Conference on

Extreme Value Analysis Gothenburg, August 15-19, 2005

Pär Johannesson

Göteborg, Sweden

August 16, 2005

Extrapolated load spectrum

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Pär Johannesson

What is Fatigue?

Fatigue is the phenomenon that a material gradually deteriorates when it is subjected to repeated loadings.

3s

3s

Clients tous différents Routes de qualités variables

Dispersion matériau Dispersion de production

Contraintes

Résistances

Conception fiableFatigue Design in

Automotive Industry

PSA (Peugeot Citroën)

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Pär Johannesson

• SN-curve (Wöhler, 1860s; Basquin, 1910)– Can resist N cycles of amplitude S

α, β material parameters.

• Rainflow cycle counting (Endo, 1967)– Convert a complicated load function to

equivalent load cycles.– Load X(t) gives amplitudes S1, S2, S3, …

• Palmlgren-Miner damage accumulation rule (1924, 1945)– Each cycle of amplitude Si uses a fraction 1/Ni of the total life.– Damage in time [0,T]:

– Failure occurs at time Tf when all life is used, i.e when DT>1.

Fatigue Life and Damage

i

ii i

T SN

D 1

SN

1

time

2S

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Pär Johannesson

Rainflow Cycle Counting

From each local maximum one shall try to reach above the same level with as small a downward excursion as possible. The i:th rainflow cycle is defined as (mi

rfc,Mi), where mirfc=max(mi

+,mi-).

• Definition of rainflow cycles by Rychlik (1987):

• Equivalent to counting crossings of intervals.– Equivalence: #{upcrossings of [u,v]} = #{mi

rfc<u, Mi>v}

– Intensity of upcrossings: μ(u,v) = μrfc(u,v)

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Pär Johannesson

Why Extrapolation?

• We measure fatigue loads for a limited period of time.– E.g. 100 km on a vehicle, or– 1 lap on the test track.

• We want to make a fatigue life assessment.– Predict the fatigue life of component.– FEM & damage calculations.– Fatigue tests of components.– Estimate the reliability of the construction

for a full design life.

• Hence there is a need to extrapolate the load history:– E.g. to a full design life representing 250 000 km, or– 1000 laps on the test track.

XY

Z

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Pär Johannesson

Fatigue Tests – Turning Points and Rainflow Filter

Assumptions:

Load Measurement Turning Points Turning Points

TP-filter RFC-filter

Remove small cycles

Extract peaks & valleys

Fatigue test:

• Frequency content not important.

• Small cycles give negligible damage.

…

• Repeat block load until failure.

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Pär Johannesson

Generation of Load Histories – Extrapolation in Time Domain

Method• Block load from measurement.• Turning points & rainflow filter.• Generate new block loads.• Repeat the new block loads.

Random Generation of block loads• Statistical extreme value theory:

Peak Over Threshold (POT) model.• Randomly change high peaks and low

valleys.

…

block 1 block 2 block 3

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Pär Johannesson

Peak Over Threshold Analysis

Model for excesses • Statistical extreme value theory. • Peak Over Threshold model. • Study the excesses over a

threshold level u.• Excesses are modelled by the

exponential distribution.

Excesses over threshold level u:

Z = Max - u

Comment:• The exponential excesses

corresponds to the Gumbel distribution for global maxima.

)/exp(1)( mzzFZ

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Pär Johannesson

Peak Over Threshold Analysis – General Model

Model for excesses • Asymptotic extreme value theory. • Possible distributions: GPD

Generalized Pareto Distribution.

Excesses over threshold level u:

Z = Max - u

Comments:• GPD corresponds to GEV for

global maxima.• Exp corresponds to Gumbel.

)/exp(1)( mzzFZ

• Special case of GPD (k=0): ExpExponential distribution.

kZ akzzF 1/11)(

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Extrapolated Turning Points – 10 load blocks

Example: Bombardier Train Load

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Example: Train Load

• Measured stress signal at a location just above the bogie.

• The train is running from Oslo to Kristiansand in Norway.

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Pär Johannesson

Extrapolated Load Spectrum – Time Domain Method

Extrapolation of Turning Points

• Generation of 10 different load blocks.

• 10-fold extrapolation.

Compared to ...• 10 repetitions of the

measured load.

Extrapolates ...• load spectrum in the

large amplitude area.• maximum load value.

– Measured

– Extrapolated

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Pär Johannesson

Extrapolated Load Spectrum – Time Domain Method

Extrapolation of Turning Points

• Generation of 10 different load blocks.

• 10-fold extrapolation.

Compared to ...• 10 repetitions of the

measured load.

Extrapolates ...• load spectrum in the

large amplitude area.• maximum load value.

– Measured

– Extrapolated

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Pär Johannesson

Extrapolation of Rainflow Matrices

• Why Extrapolation?– We measure fatigue loads on a vehicle for a limited period of time, T.

– We want to analyse the reliability for a full design life, Tlife = N · T.

• Simple scaling method: Flife = N · F, F = “rainflow matrix”

• Limiting shape of rainflow matrix– Definition: The shape of the rainflow matrix for a very long observation.

• Proposed method: Glife = N · G, G = “limiting rainflow matrix”

n = 100 n = 1 000 n = n = 10 000

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Pär Johannesson

Extreme Value Extrapolation of Rainflow Matrices

• Strategy: Use the limiting rainflow matrix when extrapolating.

• Main Method: Statistical extreme value theory.

• Result: Method for estimating the limiting rainflow matrix.– For large cycles:

• Approximate rainflow matrix from extreme value theory.• Valid for the extreme part of the rainflow matrix. • Need to extrapolate the level crossings.

– For other cycles:• Kernel smoothing. (Need to choose a smoothing parameter.)

Extrapolate level crossings

Approximate Rainflow matrix

Kernel Smoothing

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Pär Johannesson

where (u) is the intensity of u-upcrossings.

Asymptotics for Crossings of Large Intervals

• Aim: Find the asymptotic behaviour of μ(u,v) as u- and v+.• Define the time-normalized point processes of upcrossings of u and v:

• Theorem: Let X(t) be stationary, ergodic, and smooth sample paths. If (UT,VT) converges in distribution to two independent Poisson processes (U,V) when (1) holds as T. Then

1)()(

)()(),(

vu

vuvu

Bt/TX(t)vBV

Bt/TX(t)uBU

T

T

;by upcrossing-#)(

;by upcrossing-#)(Bset Borelany for

• Let u- and v+ when T, such that

vTuT vTuT )(and)(

T

T

vv

uuT

)1(

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Pär Johannesson

Asymptotics for Large Rainflow Cycles

• Approximation of intensity of rainflow cycles with large amplitudes.

• Simple formula since it only depends on the intensity of level upcrossings, (u).

• Example of approximation for Gaussian process.

– Accurate approximation (blue lines).

– Asymptotic approximation (red lines).

,)()(

)()(),(rfc

vu

vuvu

(maximum)

(minimum)

v

u

Iso-lines:10%30%50%70%90%99%99.9%99.99%

Intensity of rainflow cycles

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Example: Limiting Shape for Markov Load

• Approximation of intensity of rainflow cycles with large amplitudes.

,)()(

)()(),(rfc

vu

vuvu

• Simple formula since it only depends on the intensity of level upcrossings, (u).

• Example of approximation for Markov load.

– Limiting rainflow matrix(blue lines).

– Asymptotic approximation (red lines).

(maximum)

(minimum)

v

u

Intensity of rainflow cycles

Iso-lines:10%30%50%70%90%99%99.9%99.99% 99.999%

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Pär Johannesson

Example: rainflow matrix, PSA test track measurements

• The load is vertical forces on the front wheel of a prototype vehicle from PSA Peugeot Citroën.

– Measured rainflow matrix, 1 lap on the test track. (blue lines)– Estimated limiting rainflow matrix (red lines), combination of

• Large cycles: Approximate RFM, from estimated level crossing intensity.

• Elsewhere: Kernel smoothing of RFM.

Iso-lines:10%30%50%90%99%99.9%99.99%99.999%

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Pär Johannesson

Validation of Model Assumptions

Choice of thresholds• High enough to get

good extreme value approximation.

• Low enough to get sufficient number of exceedances.

Automatic choice• Difficult problem.• Suggested rule of

thumb:

0NCN

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Comparison of Extrapolation Methods

100-fold extrapolation

– Measured

– Extrapolated TP

– Extrapolated RFM

Extrapolated Load Spectra

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Comparison of Extrapolation Methods

100-fold extrapolation

– Measured

– Extrapolated TP

– Extrapolated RFM

Extrapolated Load Spectra

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Conclusions – Comparison of Methods

Rainflow domain:

• Result is a limiting rainflow matrix.• Use more extreme value theory.

(POT + asymptotic distribution)• Need to simulate time signal.• Efficient for generation of a

design load spectrum.

Time domain:

• Result is a time signal.• POT method.

(more robust ?!?)• Need to calculate rainflow matrix.• Efficient for generation of a

time signal for fatigue testing.

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Pär Johannesson

References

1. Johannesson, P. (2004) Extrapolation of Load Histories and Spectra, Proceedings of 15th European Conference on Fracture. Accepted for publication in Fatigue & Fracture of Engineering Materials & Structures.

2. Johannesson, P. and Thomas, J.-J. (2001) Extrapolation of Rainflow Matrices, Extremes Vol. 4, 241-262.

Extrapolation of Fatigue Loads 4th Conference on Extreme Value Analysis Gothenburg, August 15-19,...

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