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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. What is Fatigue?.

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1. 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

2. 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 IndustryPSA (Peugeot Citroën) Conception fiable Résistances Dispersion matériau Dispersion de production Extrapolation of Fatigue Loads

3. 2S time Fatigue Life and Damage • 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. Extrapolation of Fatigue Loads

4. 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 (mirfc,Mi), where mirfc=max(mi+,mi-). • Definition of rainflow cycles by Rychlik (1987): • Equivalent to counting crossings of intervals. • Equivalence: #{upcrossings of [u,v]} = #{mirfc<u, Mi>v} • Intensity of upcrossings: μ(u,v) = μrfc(u,v) Extrapolation of Fatigue Loads

5. Z X Y 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. Extrapolation of Fatigue Loads

6. Fatigue Tests – Turning Points and Rainflow Filter Load Measurement Turning Points Turning Points Assumptions: TP-filter RFC-filter Extract peaks & valleys Remove small cycles • Frequency content not important. • Small cycles give negligible damage. Fatigue test: • Repeat block load until failure. … Extrapolation of Fatigue Loads

7. 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 … Extrapolation of Fatigue Loads

8. Peak Over Threshold Analysis Excesses over threshold level u: Z = Max - u 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. Comment: • The exponential excesses corresponds to the Gumbel distribution for global maxima. Extrapolation of Fatigue Loads

9. 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 (k=0): ExpExponential distribution. Comments: • GPD corresponds to GEV for global maxima. • Exp corresponds to Gumbel. Extrapolation of Fatigue Loads

11. Example: Train Load • Measured stress signal at a location just above the bogie. • The train is running from Oslo to Kristiansand in Norway. Extrapolation of Fatigue Loads

12. 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 Extrapolation of Fatigue Loads

13. 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 Extrapolation of Fatigue Loads

14. n = 100 n = 1 000 n = 10 000 n =  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” Extrapolation of Fatigue Loads

15. Extrapolate level crossings 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 Extrapolation of Fatigue Loads

16. 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 Extrapolation of Fatigue Loads

17. 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% Extrapolation of Fatigue Loads

18. 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% Extrapolation of Fatigue Loads

19. 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% Extrapolation of Fatigue Loads

20. 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: Extrapolation of Fatigue Loads

21. Comparison of Extrapolation Methods Extrapolated Load Spectra 100-fold extrapolation – Measured – Extrapolated TP – Extrapolated RFM Extrapolation of Fatigue Loads

22. Comparison of Extrapolation Methods Extrapolated Load Spectra 100-fold extrapolation – Measured – Extrapolated TP – Extrapolated RFM Extrapolation of Fatigue Loads

23. 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. Extrapolation of Fatigue Loads

24. References • 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. • Johannesson, P. and Thomas, J.-J. (2001) Extrapolation of Rainflow Matrices, Extremes Vol. 4, 241-262. Extrapolation of Fatigue Loads

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