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

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.

Particularly high penalties occur when the PSD has

components concentrated at only two or three

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

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

-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.

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

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.

Hypothetical S/N relationship allowed in some Codes of practice

(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)

(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 )

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

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.

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.

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.

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

   

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

Signal B, Irregularity = 0.527

Signal F, Irregularity = 0.415

Short time histories of two of the signals.

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

A single loading station, ref ( )

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

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:-

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 "

       

(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.

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

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.

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

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.

Although it is likely that the Booth tests had very little

crack propagation, no measurements were taken.

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.

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.

The specimen used in ref (6 )

(Cantilever in plane bending)

The notch in the specimen used in ref (6 )

Stress concentration factor, KT = 1.593

Slope of CA log/log tests = -6

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

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

    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.

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.

(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.

(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.

(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.

(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.

Requirement

A technique which gives safe but economical design but

does not need a value for a "fatigue limit"

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.

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

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

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.

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)

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.

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.

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

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)

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.