Fatigue life estimation from bi-modal and tri-modal PSDs
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Transcript of Fatigue life estimation from bi-modal and tri-modal PSDs
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Fatigue life estimation from bi-modal and tri-modal PSDs
Frank Sherratt
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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
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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.
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Particularly high penalties occur when the PSD has
components concentrated at only two or three
frequencies (bi-modal and tri-modal histories).
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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
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-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.
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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
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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.
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Hypothetical S/N relationship allowed in some Codes of practice
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(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)
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(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 )
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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
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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.
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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.
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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.
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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
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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
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Signal B, Irregularity = 0.527
Signal F, Irregularity = 0.415
Short time histories of two of the signals.
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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
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A single loading station, ref ( )
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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
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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:-
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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 "
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(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.
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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
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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.
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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
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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.
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Although it is likely that the Booth tests had very little
crack propagation, no measurements were taken.
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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.
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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.
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The specimen used in ref (6 )
(Cantilever in plane bending)
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The notch in the specimen used in ref (6 )
Stress concentration factor, KT = 1.593
Slope of CA log/log tests = -6
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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
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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
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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.
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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.
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(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.
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(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.
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(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.
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(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.
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Requirement
A technique which gives safe but economical design but
does not need a value for a "fatigue limit"
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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.
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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
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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
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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.
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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)
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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.
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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.
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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
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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)
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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.