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

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

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

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

Frank Sherratt

Page 2: Fatigue life estimation from bi-modal and tri-modal PSDs Frank Sherratt

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

Page 3: Fatigue life estimation from bi-modal and tri-modal PSDs Frank Sherratt

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.

Page 4: Fatigue life estimation from bi-modal and tri-modal PSDs Frank Sherratt

Particularly high penalties occur when the PSD has

components concentrated at only two or three

frequencies (bi-modal and tri-modal histories).

Page 5: Fatigue life estimation from bi-modal and tri-modal PSDs Frank Sherratt

Ratio (Dirlik life)/(NB life) against percentile, 252 cases.

0.500

1.000

1.500

2.000

2.500

3.000

3.500

4.000

4.500

5.000

0 10 20 30 40 50 60 70 80 90 100

Percentile

Rat

io

Page 6: Fatigue life estimation from bi-modal and tri-modal PSDs Frank Sherratt

-6

-4

-2

0

2

4

6

6 8 10 12 14 16 18 20 22

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

Page 7: Fatigue life estimation from bi-modal and tri-modal PSDs Frank Sherratt

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

Page 8: Fatigue life estimation from bi-modal and tri-modal PSDs Frank Sherratt

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.

Page 9: Fatigue life estimation from bi-modal and tri-modal PSDs Frank Sherratt

Hypothetical S/N relationship allowed in some Codes of practice

Page 10: Fatigue life estimation from bi-modal and tri-modal PSDs Frank Sherratt

(Predicted life)/(Test life), WB Signal 2: effect of using constant amplitude limit.

0

10

20

30

40

50

60

70

6.6 6.7 6.8 6.9 7 7.1 7.2 7.3 7.4 7.5 7.6

Log test life

Rati

o P

red

icti

on

/Test

CA limit

Agreement

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

Page 11: Fatigue life estimation from bi-modal and tri-modal PSDs Frank Sherratt

(Predicted life)/(Test life), WB Signal 2: effect of S/N assumptions.

0

0.5

1

1.5

2

2.5

6.6 6.7 6.8 6.9 7 7.1 7.2 7.3 7.4 7.5 7.6

Log test life

Rati

o P

red

icti

on

/Test

Bi-slope

Zero limit

Agreement

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

Page 12: Fatigue life estimation from bi-modal and tri-modal PSDs Frank Sherratt

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

Page 13: Fatigue life estimation from bi-modal and tri-modal PSDs Frank Sherratt

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.

Page 14: Fatigue life estimation from bi-modal and tri-modal PSDs Frank Sherratt

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.

Page 15: Fatigue life estimation from bi-modal and tri-modal PSDs Frank Sherratt

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.

Page 16: Fatigue life estimation from bi-modal and tri-modal PSDs Frank Sherratt
Page 17: Fatigue life estimation from bi-modal and tri-modal PSDs Frank Sherratt

Component Frequencycontent

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

Page 18: Fatigue life estimation from bi-modal and tri-modal PSDs Frank Sherratt

   

Relative amplitude of frequenciesf1 : f2 : f3

 

TotalRMS

 

TotalFrequencyN0/sec

 

Irreg.FactorN0/NP

 

VanmarkeFactor

B 0 : 1.0 : 0.25 1.031*MAX 13.5 0.527 0.645

C 0 : 1.0 : 0.5 1.118*MAX 21.0 0.629 0.613

D 0 : 0.5 : 1.0

1.118*MAX 42.5 0.835 0.448

E 0 : 1.0 : 1.0 1.414*MAX 31.5 0.739 0.543

F 1.0: 0.5 : 0.25 1.146*MAX 9.5 0.415 0.811

Page 19: Fatigue life estimation from bi-modal and tri-modal PSDs Frank Sherratt

Signal B, Irregularity = 0.527

Signal F, Irregularity = 0.415

Short time histories of two of the signals.

Page 20: Fatigue life estimation from bi-modal and tri-modal PSDs Frank Sherratt

PRR distributions for Booth signals

0

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Relative range

PR

R n

orm

alis

ed

to

un

ity

0:1:0

0:1:0.25

0:1:0.5

0:1:1

0:0.5:1

1:0.5:0.25

Page 21: Fatigue life estimation from bi-modal and tri-modal PSDs Frank Sherratt

A single loading station, ref ( )

Page 22: Fatigue life estimation from bi-modal and tri-modal PSDs Frank Sherratt

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

Page 23: Fatigue life estimation from bi-modal and tri-modal PSDs Frank Sherratt

The tests allow an appraisal of the two simplest assumptions:-

(a) that the measured CA fatigue limit applies

(b) that the limit is zero

Taking Signal B and Signal F as examples gives:-

Page 24: Fatigue life estimation from bi-modal and tri-modal PSDs Frank Sherratt

Ratio of (Test life)/(Estimated life) for bi-modal

Signal B using the Dirlik expression and Miner

   

 

(Limit 

463 Mpa)

(Limit 

zero)

 

Amplitudes f1 : f2 : f3

 

RMS Mpa 

 

Testpeaks/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 "

Page 25: Fatigue life estimation from bi-modal and tri-modal PSDs Frank Sherratt

       

(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.

Page 26: Fatigue life estimation from bi-modal and tri-modal PSDs Frank Sherratt

Rainflow distributions for Narrow band and Signal F

0

0.002

0.004

0.006

0.008

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Relative range

PR

R

Narrow bandSignal F

Page 27: Fatigue life estimation from bi-modal and tri-modal PSDs Frank Sherratt

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.

Page 28: Fatigue life estimation from bi-modal and tri-modal PSDs Frank Sherratt

Percentage reduction in estimated life caused by two possible assumptions.

High figures for "Zero limit" are at long lives.

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

Page 29: Fatigue life estimation from bi-modal and tri-modal PSDs Frank Sherratt

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.

Page 30: Fatigue life estimation from bi-modal and tri-modal PSDs Frank Sherratt

Although it is likely that the Booth tests had very little

crack propagation, no measurements were taken.

Page 31: Fatigue life estimation from bi-modal and tri-modal PSDs Frank Sherratt

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.

Page 32: Fatigue life estimation from bi-modal and tri-modal PSDs Frank Sherratt

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.

Page 33: Fatigue life estimation from bi-modal and tri-modal PSDs Frank Sherratt

The specimen used in ref (6 )

(Cantilever in plane bending)

Page 34: Fatigue life estimation from bi-modal and tri-modal PSDs Frank Sherratt

The notch in the specimen used in ref (6 )

Stress concentration factor, KT = 1.593

Slope of CA log/log tests = -6

Page 35: Fatigue life estimation from bi-modal and tri-modal PSDs Frank Sherratt

Initiation life vs total; small specimens, CA

0

4000

8000

12000

16000

20000

0 10000 20000 30000 40000

Total life

Init

iati

on

lif

e

Exptl.0.5 line

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

Page 36: Fatigue life estimation from bi-modal and tri-modal PSDs Frank Sherratt

Initiation life vs Total, small specimens, random loading

0

4000

8000

12000

16000

20000

0 10000 20000 30000 40000

Total life

Init

iati

on

life

Exptl 0.3 line

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

Page 37: Fatigue life estimation from bi-modal and tri-modal PSDs Frank Sherratt

    Limit 217 Mpa Limit zero   

Signal RMS, Mpa Total Initiation Total Initiation 

47 Hz216 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.

Page 38: Fatigue life estimation from bi-modal and tri-modal PSDs Frank Sherratt

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.

Page 39: Fatigue life estimation from bi-modal and tri-modal PSDs Frank Sherratt

(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.

Page 40: Fatigue life estimation from bi-modal and tri-modal PSDs Frank Sherratt

(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.

Page 41: Fatigue life estimation from bi-modal and tri-modal PSDs Frank Sherratt

(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.

Page 42: Fatigue life estimation from bi-modal and tri-modal PSDs Frank Sherratt

(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.

Page 43: Fatigue life estimation from bi-modal and tri-modal PSDs Frank Sherratt

Requirement

A technique which gives safe but economical design but

does not need a value for a "fatigue limit"

Page 44: Fatigue life estimation from bi-modal and tri-modal PSDs Frank Sherratt

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.

Page 45: Fatigue life estimation from bi-modal and tri-modal PSDs Frank Sherratt

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

Page 46: Fatigue life estimation from bi-modal and tri-modal PSDs Frank Sherratt

High-pass filtering

4.5

5

5.5

6

6.5

7

2.15 2.2 2.25 2.3 2.35 2.4

log S (Mpa)

log

N 0-4 2-4 2.5-4 Line 0-4 Line 2-4 Line 2.5-4

Tests showing that band 0-2 x RMS, and possibly band

0-2.5 x RMS of a narrow-band history are non-damaging.

RMS bands

included

Page 47: Fatigue life estimation from bi-modal and tri-modal PSDs Frank Sherratt

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.

Page 48: Fatigue life estimation from bi-modal and tri-modal PSDs Frank Sherratt

Test life/1e6 -------

------------- ------------- ------------>

RMS MPa 47 Hz 25/52 Hz 5/52 Hz

154 0.18 0.163 0.145123 0.544 0.493 0.437

93 2.264 2.055 1.8262 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 0.499123 1.44 1.434 1.903

93 8.088 8.059 10.6962 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

Assuming that only cycles of amplitude 2xRMS to

4xRMS are damaging gives unsafe predictions.

Trial of possibility A Data from Fisher (6)

Page 49: Fatigue life estimation from bi-modal and tri-modal PSDs Frank Sherratt

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.

Page 50: Fatigue life estimation from bi-modal and tri-modal PSDs Frank Sherratt

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.

Page 51: Fatigue life estimation from bi-modal and tri-modal PSDs Frank Sherratt

Proposed method

1. Carry out tests on the component under narrow-band

loading, at low RMS values, say RA

2 Determine slope and intercept needed for a life estimate

(possibly by CA testing)

3 Use a life estimation algorithm (e.g. Dirlik) to determine the

hypothetical RMS level, RH which would have estimated life

correctly for RA under this form of PSD

4 In subsequent estimations using different forms of PSD, use

RH in place of RA

Page 52: Fatigue life estimation from bi-modal and tri-modal PSDs Frank Sherratt

Signal

RMS Mpa

 

Test n/1e6

 Assume Limit=2xRMS

  Ratio

 Assume Limit =zero

 

Ratio

 AssumeLimit zeroand RMS increased

Ratio

25/52 Hz 62 15.3 91.8 6.00 74.6 4.88 16.4 1.07

25/52 Hz 93 2.05 8.06 3.93 6.55 3.20 2.19 1.07

5/52 Hz 62 13.6 121.8 8.96 98.6 7.25 21.6 1.59

5/52 Hz 93 1.82 10.69 5.87 8.66 4.76 2.9 1.59

Comparison of assumptions for RMS values giving long lives.

(a) Limit is 2xRMS, (b) Limit is zero, (c) Limit is zero and RMS is increased.

Using a modified RMS determined previously is successful.

Trial of possibility B Data again from (6)

Page 53: Fatigue life estimation from bi-modal and tri-modal PSDs Frank Sherratt

Conclusions

(a) Life estimates for components loaded by histories specified by

PSD may be optimistic in some circumstances.

(b) The effect is more likely at low stresses and long lives.

(c) The effect is not confined to components whose life mainly

consists of crack propagation.

(d) Empirical methods of correction are successful in many

circumstances.

(e) In circumstances where these methods are unproven further

tests may be helpful.