Extrapolation%20of%20Fatigue%20Loads

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Extrapolation of Fatigue Loads
Pär Johannesson
Extrapolated load spectrum
Göteborg, Sweden
August 16, 2005
4th Conference on Extreme Value Analysis
Gothenburg, August 15-19, 2005
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What is Fatigue?

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

Clients tous différents
Routes de qualités variables
Contraintes
Fatigue Design in Automotive Industry PSA
(Peugeot Citroën)
Conception fiable
Résistances
Dispersion matériau
Dispersion de production
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Fatigue Life and Damage

• SN-curve (Wöhler, 1860s Basquin, 1910)
• Can resist N cycles of amplitude S
• a, ß 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 DTgt1.

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Rainflow Cycle Counting

• Definition of rainflow cycles by Rychlik (1987)

• From each local maximum one shall try to reach
  above the same level with as small a downward
  excursion as possible. The ith rainflow cycle is
  defined as (mirfc,Mi), where mirfcmax(mi,mi-).

• Equivalent to counting crossings of intervals.
• Equivalence upcrossings of u,v mirfcltu,
  Migtv
• Intensity of upcrossings µ(u,v) µrfc(u,v)

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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.

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Fatigue Tests Turning Points and Rainflow Filter
Load Measurement
Turning Points
Turning Points
TP-filter
RFC-filter
Remove small cycles
Extract peaks valleys

• Assumptions

• Frequency content not important.
• Small cycles give negligible damage.

Fatigue test

• Repeat block load until failure.

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Generation of Load Histories Extrapolation in
Time Domain

• Random Generation of block loads
• Statistical extreme value theory Peak Over
  Threshold (POT) model.
• Randomly change high peaks and low valleys.

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

block 1
block 2
block 3

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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.

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Peak Over Threshold Analysis General Model
Excesses over threshold level u Z Max - u

• Model for excesses
• Asymptotic extreme value theory.
• Possible distributions GPD Generalized Pareto
  Distribution.

• Special case of GPD (k0) ExpExponential
  distribution.

• Comments
• GPD corresponds to GEV for global maxima.
• Exp corresponds to Gumbel.

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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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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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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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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

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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.)

Approximate Rainflow matrix
Kernel Smoothing
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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

• Let u?-? and v?? when T??, such that

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

• 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

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Asymptotics for Large Rainflow Cycles

• Approximation of intensity of rainflow cycles
  with large amplitudes.

Intensity of rainflow cycles

• 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).

Iso-lines 10 30 50 70 90 99 99.9 99.99
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Example Limiting Shape for Markov Load

• Approximation of intensity of rainflow cycles
  with large amplitudes.

Intensity of rainflow cycles

• 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).

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

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Comparison of Extrapolation Methods
Extrapolated Load Spectra
100-fold extrapolation Measured
Extrapolated TP Extrapolated RFM
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Comparison of Extrapolation Methods
Extrapolated Load Spectra
100-fold extrapolation Measured
Extrapolated TP Extrapolated RFM
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Conclusions Comparison of Methods

• 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.

• 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.

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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.