Fatigue life estimation from bi-modal and tri-modal PSDs

1. Fatigue life estimation from bi-modal and tri-modal PSDs Frank Sherratt

2. Design methods using the power spectral density (PSD) of a stress history to estimate fatigue life are now accepted, with some reservations. Some of these reservations are analytical and some depend on the physics of the fatigue process

3. Analytical difficulties vary with the form of the PSD. One common solution, the narrow-band assumption, ignores these variations and provides a simple calculation, but is known to give an un-economic design in many cases.

4. Particularly high penalties occur when the PSD has components concentrated at only two or three frequencies (bi-modal and tri-modal histories).

6. False cycles generated by the narrow-band assumption when dealing with a bi-modal history.

7. Many reported tests using variable-amplitude loading have failures earlier than estimated if very simple analysis is used, such as applying Miner’s Hypothesis without modification. It is often found that low amplitude cycles are more damaging when they are part of a mixed range of amplitudes than they are when applied in isolation

8. Codes of Practice for fusion welds in metals, for instance, often use constant-amplitude stress-life (S/N) test data but assume a modified form beyond a certain life, attributing damage at amplitudes where the tests showed none.

9. Hypothetical S/N relationship allowed in some Codes of practice

10. (Predicted life)/(Test life) computed using the measured 1e7 CA stress, ref ( 1)

11. (Predicted life)/(Test life) using two allowed modifications to Miner, ref (1 )

12. The evidence shows that cycles of amplitude less than the measured constant amplitude value giving a life of 10 million cause damage. Either of the recognised empirical ways of correcting this is moderately successful. Note that the range of lives being considered in this particular report was > 1e6

13. Similar evidence from other sources establishes that: (1) When estimating the fatigue life of welds in structural metals Miners Hypothesis has to be modified if the loading is specified by PSD. (2) Modifications already accepted by Codes of Practice give major improvement.

14. Questions then are:- (1) Does Miners Hypothesis have similar weaknesses when used with other components. (2) Do similar modifications to the computation give similar improvement. Because of the major benefits of successful prediction when the loading is a bi-modal or tri-modal PSD, tests using these forms are likely to be the most interesting.

15. One programme reported by Booth (5) used four bi- modal and one tri-modal PSDs, and included tests on small, un-notched, steel specimens to verify the predictions. Although no measurements were made it is unlikely that crack propagation took up much of specimen life.

17. Component Frequency content f1 2.5 N0/sec centre 0.2 Hz bandwidth f2 10.8 N0/sec centre 0.2 Hz bandwidth f3 50 N0/sec centre 0.2 Hz bandwidth

18. Relative amplitude of frequencies f1 : f2 : f3 Total RMS Total Frequency N0/sec Irreg. Factor N0/NP Vanmarke Factor B 0 : 1.0 : 0.25 1.031*sMAX 13.5 0.527 0.645 C 0 : 1.0 : 0.5 1.118*sMAX 21.0 0.629 0.613 D 0 : 0.5 : 1.0 1.118*sMAX 42.5 0.835 0.448 E 0 : 1.0 : 1.0 1.414*sMAX 31.5 0.739 0.543 F 1.0: 0.5 : 0.25 1.146*sMAX 9.5 0.415 0.811

19. Signal B, Irregularity = 0.527 Signal F, Irregularity = 0.415 Short time histories of two of the signals.

21. A single loading station, ref ( )

22. The critical, un-notched, section of the test specimen. KT is about unity. Constant-amplitude fatigue tests had a negative slope of 9 on a log/log plot. Material EN 19 steel UTS 725 Mpa Yield 640 MPa

23. The tests allow an appraisal of the two simplest assumptions:- • that the measured CA fatigue limit applies • that the limit is zero • Taking Signal B and Signal F as examples gives:-

24. (Limit 463 Mpa) (Limit zero) Amplitudes f1 : f2 : f3 RMS Mpa Test peaks /1e6 Dirlik Ratio Dirlik Ratio Irreg factor 0 : 1.0 : 0.25 207 0.702 0.209 3.359 0.195 3.600 0.527 0 : 1.0 : 0.25 191 1.603 0.455 3.524 0.400 4.009 " 0 : 1.0 : 0.25 175 1.954 1.098 1.780 0.875 2.234 " 0 : 1.0 : 0.25 159 2.732 3.081 0.887 2.061 1.326 " 0 : 1.0 : 0.25 143 11.518 10.950 1.052 5.314 2.167 " Ratio of (Test life)/(Estimated life) for bi-modal Signal B using the Dirlik expression and Miner

25. (Limit 463 Mpa) (Limit Zero) Amplitudes f1 : f2 : f3 RMS Mpa Test Peaks /1e6 Dirlik Ratio Dirlik Ratio Irreg factor 1.0 : 0.5 : 0.25 230 0.424 0.180 2.356 0.174 2.437 0.415 1.0 : 0.5 : 0.25 212 0.643 0.381 1.689 0.358 1.797 " 1.0 : 0.5 : 0.25 195 0.848 0.873 0.972 0.781 1.086 " 1.0 : 0.5 : 0.25 177 1.733 2.28 0.760 1.847 0.938 " 1.0 : 0.5 : 0.25 159 3.398 7.15 0.475 4.784 0.710 " 1.0 : 0.5 : 0.25 141 15.759 30.120 0.523 13.87 1.136 " Ratio of (Test life)/(Estimated life) for tri-modal Signal F using the Dirlik expression and Miner.

27. Assuming zero limit reduces allowable design life compared to adopting the CA value. The magnitude of this may be calculated and compared with the reduction caused by using the narrow-band formula.

28. Assumption Signal B Signal C Signal D Signal E Signal F Narrow band 63 59 56 49 84 Zero limit 7-51 4-37 2-22 5-14 3-54 Percentage reduction in estimated life caused by two possible assumptions. High figures for "Zero limit" are at long lives.

29. Both assumptions allocate more damage to low- amplitude cycles than CA testing indicates. If crack propagation is a significant part of component life the effect of these assumptions is easily explained because low amplitude cycles may propagate cracks started by ones of high amplitude.

30. Although it is likely that the Booth tests had very little crack propagation, no measurements were taken.

31. Tests by Fisher (6) report crack initiation measurements on specimens fatigued by PSD histories. These included signals which were wide-band, but not bi-modal. Plots of the ratio (life to initiation)/(total life) were produced. Separate Miner fractions for initiation and propagation phases could then be estimated.

32. The applied histories were (I) 47 Hz narrow-band (ii) flat over 25-52 Hz (iii) flat over 5-52 Hz. Amplitude probability density distributions were Gaussian.

33. The specimen used in ref (6 ) (Cantilever in plane bending)

34. The notch in the specimen used in ref (6 ) Stress concentration factor, KT = 1.593 Slope of CA log/log tests = -6

35. Proportion of life spent initiating a crack; constant amplitude (CA) loading

36. Proportion of life spent initiating a crack; all random loading PSDs

37. Limit 217 Mpa Limit zero Signal RMS, Mpa Total Initiation Total Initiation 47 Hz 216 3.610 5.255 3.610 5.255 47 Hz 185 3.040 4.424 3.049 4.438 47 Hz 154 2.481 3.612 2.494 3.630 47 Hz 124 1.912 2.783 1.949 2.837 47 Hz 93 1.267 1.845 1.418 2.065 47 Hz 62 0.377 0.549 0.907 1.321 25/52 Hz 216 3.300 4.804 3.300 4.804 25/52 Hz 185 2.786 4.055 2.786 4.055 25/52 Hz 154 2.268 3.301 2.278 3.316 25/52 Hz 124 1.745 2.540 1.783 2.595 25/52 Hz 93 1.155 1.681 1.297 1.888 25/52 Hz 62 0.343 0.499 0.830 1.208 5/52 Hz 216 2.212 3.220 2.212 3.220 5/52 Hz 185 1.706 2.484 1.709 2.488 5/52 Hz 154 1.520 2.212 1.529 2.226 5/52 Hz 124 1.167 1.698 1.195 1.739 5/52 Hz 93 0.772 1.123 0.870 1.267 5/52 Hz 62 0.230 0.335 0.556 0.810 Ratios (Test life/predicted life) from Fisher (6); life estimates by the Dirlik formula.

38. Appraisal (i) Problems seem to occur when estimation of long lives is attempted (ii) They come from uncertainty about the role of low amplitude cycles. (iii) Most current methods seem to need a numerical value of some stress amplitude, conveniently called a "fatigue limit". (iv) Experimental determination of a "Fatigue limit" is difficult (v) The effect of cycles with amplitudes less than this "fatigue limit" is not well known.

39. (i) Problems seem to occur when estimation of long lives is attempted. (ii) They come from uncertainty about the role of low amplitude cycles. (iii) Most current methods seem to need a numerical value of some stress amplitude, conveniently called a "fatigue limit". (iv) Experimental determination of a "Fatigue limit" is difficult (v) The effect of cycles with amplitudes less than this "fatigue limit" is not well known.

40. (i) Problems seem to occur when estimation of long lives is attempted. (ii) They come from uncertainty about the role of low amplitude cycles. (iii) Most current methods seem to need a numerical value of some stress amplitude, conveniently called a "fatigue limit". (iv) Experimental determination of a "Fatigue limit" is difficult (v) The effect of cycles with amplitudes less than this "fatigue limit" is not well known.

41. (i) Problems seem to occur when estimation of long lives is attempted. (ii) They come from uncertainty about the role of low amplitude cycles. (iii) Most current methods seem to need a numerical value of some stress amplitude, conveniently called a "fatigue limit". (iv) Experimental determination of a "Fatigue limit" is difficult (v) The effect of cycles with amplitudes less than this "fatigue limit" is not well known.

42. (i) Problems seem to occur when estimation of long lives is attempted. (ii) They come from uncertainty about the role of low amplitude cycles. (iii) Most current methods seem to need a numerical value of some stress amplitude, conveniently called a "fatigue limit. (iv) Experimental determination of a "Fatigue limit" is difficult (v) The effect of cycles with amplitudes less than this "fatigue limit" is not well known.

43. Requirement A technique which gives safe but economical design but does not need a value for a "fatigue limit"

44. Possibility A If the band of RMS values which cause damage can be identified there is no need to define a fatigue limit. Tests from ref (5) allow this.

45. High-pass filtering As part of the ref. (5) programme tests were performed using narrow-band histories with two different levels of RMS removed. Bands 0-2.0 and 0-2.5 were chosen

46. RMS bands included Tests showing that band 0-2 x RMS, and possibly band 0-2.5 x RMS of a narrow-band history are non-damaging.

47. This figure shows that, surprisingly, the cut- off point below which cycles cause no damage does not have a fixed value, but depends on the RMS of the applied loading.

48. Test life/1e6 ------- ------------- ------------- ------------> RMS MPa 47 Hz 25/52 Hz 5/52 Hz 154 0.18 0.163 0.145 123 0.544 0.493 0.437 93 2.264 2.055 1.82 62 16.906 15.338 13.59 Estimated life using assumption ------------> RMS MPa 47 Hz 25/52 Hz 5/52 Hz 154 0.377 0.376 0.499 123 1.44 1.434 1.903 93 8.088 8.059 10.69 62 90.61 91.8 121.76 Ratio test/est. ----- ------------- ------------- ------------> 0.48 0.43 0.29 0.38 0.34 0.23 0.28 0.25 0.17 0.19 0.17 0.11 Trial of possibility A Data from Fisher (6) Assuming that only cycles of amplitude 2xRMS to 4xRMS are damaging gives unsafe predictions.

49. Possibility B Add data. Tests under narrow-band loading may give the information needed for:- (a) the location of the "limit" (b) the nature of the change in damage.

50. A possible assumption is that:- The form of the contribution made to damage by cycles of low amplitude is independent of the form of the PSD. Consequence If a hypothetical RMS has to be assumed in order to make test and prediction match for one PSD, using this RMS in life estimates for other PSDs will give correct results.