1. 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

2. Pär Johannesson 16-Aug-2005 Extrapolation of Fatigue Loads 2 What is Fatigue? Fatigue is the phenomenon that a material gradually deteriorates when it is subjected to repeated loadings. Clients tous différentsRoutes de qualités variables Dispersion matériauDispersion de production Contraintes Résistances Conception fiable Fatigue Design in Automotive Industry PSA (Peugeot Citroën)

3. Pär Johannesson 16-Aug-2005 Extrapolation of Fatigue Loads 3 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 S 1, S 2, S 3, … Palmlgren-Miner damage accumulation rule (1924, 1945) –Each cycle of amplitude S i uses a fraction 1/N i of the total life. –Damage in time [0,T]: –Failure occurs at time T f when all life is used, i.e when D T >1. Fatigue Life and Damage time 2S

4. Pär Johannesson 16-Aug-2005 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 (m i rfc,M i ), where m i rfc =max(m i +,m i - ). Definition of rainflow cycles by Rychlik (1987): Equivalent to counting crossings of intervals. –Equivalence: #{upcrossings of [u,v]} = #{m i rfc v} –Intensity of upcrossings: μ(u,v) = μ rfc (u,v)

5. Pär Johannesson 16-Aug-2005 Extrapolation of Fatigue Loads 5 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. X Y Z

6. Pär Johannesson 16-Aug-2005 Extrapolation of Fatigue Loads 6 Fatigue Tests – Turning Points and Rainflow Filter Assumptions: Load MeasurementTurning Points TP-filterRFC-filter Remove small cycles Extract peaks & valleys Fatigue test: Frequency content not important. Small cycles give negligible damage. … Repeat block load until failure.

7. Pär Johannesson 16-Aug-2005 Extrapolation of Fatigue Loads 7 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 1block 2block 3

8. Pär Johannesson 16-Aug-2005 Extrapolation of Fatigue Loads 8 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.

9. Pär Johannesson 16-Aug-2005 Extrapolation of Fatigue Loads 9 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. Special case of GPD (k=0): Exp Exponential distribution.

10. Pär Johannesson 16-Aug-2005 Extrapolation of Fatigue Loads 10 Extrapolated Turning Points – 10 load blocks Example: Bombardier Train Load

11. Pär Johannesson 16-Aug-2005 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.

12. Pär Johannesson 16-Aug-2005 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

13. Pär Johannesson 16-Aug-2005 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

14. Pär Johannesson 16-Aug-2005 Extrapolation of Fatigue Loads 14 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, T life = N · T. Simple scaling method: F life = N · F,F = “rainflow matrix” Limiting shape of rainflow matrix –Definition: The shape of the rainflow matrix for a very long observation. Proposed method: G life = N · G,G = “limiting rainflow matrix” n = 100n = 1 000 n =  n = 10 000

15. Pär Johannesson 16-Aug-2005 Extrapolation of Fatigue Loads 15 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

16. Pär Johannesson 16-Aug-2005 Extrapolation of Fatigue Loads 16 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 (U T,V T ) converges in distribution to two independent Poisson processes (U,V) when (1) holds as T . Then Let u  -  and v  +  when T , such that

17. Pär Johannesson 16-Aug-2005 Extrapolation of Fatigue Loads 17 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). Iso-lines: 10% 30% 50% 70% 90% 99% 99.9% 99.99% Intensity of rainflow cycles

18. Pär Johannesson 16-Aug-2005 Extrapolation of Fatigue Loads 18 Example: Limiting Shape for Markov Load 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 Markov load. –Limiting rainflow matrix (blue lines). –Asymptotic approximation (red lines). Intensity of rainflow cycles Iso-lines: 10% 30% 50% 70% 90% 99% 99.9% 99.99% 99.999%

19. Pär Johannesson 16-Aug-2005 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%

20. Pär Johannesson 16-Aug-2005 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:

21. Pär Johannesson 16-Aug-2005 Extrapolation of Fatigue Loads 21 Comparison of Extrapolation Methods 100-fold extrapolation – Measured – Extrapolated TP – Extrapolated RFM Extrapolated Load Spectra

22. Pär Johannesson 16-Aug-2005 Extrapolation of Fatigue Loads 22 Comparison of Extrapolation Methods 100-fold extrapolation – Measured – Extrapolated TP – Extrapolated RFM Extrapolated Load Spectra

23. Pär Johannesson 16-Aug-2005 Extrapolation of Fatigue Loads 23 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.

24. Pär Johannesson 16-Aug-2005 Extrapolation of Fatigue Loads 24 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.