Fatigue testing under variable amplitude loading
C.M. Sonsino
Fraunhofer-Institute for Structural Durability and System Reliability LBF, Darmstadt, Germany
Received 22 May 2006; received in revised form 7 September 2006; accepted 4 October 2006 Available online 28 November 2006
Abstract
There are many publications about variable amplitude test results. However, very often information on load–time histories, spectra and testing details are missing. This fact does not allow the interpretation of test results with regard to fatigue lifing and structural dura-bility design. Therefore, this paper aims at presenting how spectra and test conditions should be clearly described and how statistics can be applied when variable amplitude test results are presented.
Ó 2006 Elsevier Ltd. All rights reserved.
Keywords: Variable amplitude loading; Constant amplitude loading; Cumulative damage; Load–time histories; Multichannel loading; Presentation of spectrum; Level crossings; Range pairs; Rainflow matrix; Safety; Risk
1. Introduction
The major reason for carrying out variable amplitude loading (VAL) tests is the fact that a prediction of fatigue life under this complex loading is not possible by any cumulative damage hypothesis. Therefore, for the purpose of fatigue lifing, experiences must be gained by such tests which allow to derive real damage sums by comparing
Woehler- and Gassner-lines,Fig. 1.
Applying the because of its simplicity still mostly used
Palmgren–Miner-Rule modified by Haibach [1], the
dam-age content of a spectrum with the size Ls can be
determined X n
N
i¼ Dspec ð1Þ
and with this value the real damage sum is calculated from the experimental results:
Dreal¼
Dspec
Ls
Nexp ð2Þ
A broad investigation on cumulative fatigue[2]displays the
scattering of the real damage sum over almost three
dec-ades,Fig. 2. About 90% of all results are below the
conven-tionally used value D = 1.0, i.e. a fatigue life estimation with D = 1.0 is in these cases at the unsafe side.
This knowledge justifies the need of variable amplitude testing, necessary on one hand for the investigation of cumulative damage behaviour of components or structures and on the other hand for the structural durability proof
[3]. For this, the most important prerequisite, the load–time
history, must be given[1,4]. The cumulative frequency
dis-tribution of load amplitudes or ranges (spectrum) is derived afterwards from the load–time history.
Generally, load–time histories applied in testing are
derived from service load–time histories,Fig. 3, compiled
to load sequences corresponding to a defined mission, e.g. wave spectrum for one year, a flight between two des-tinations or a defined driving distance.
The first variable amplitude loading spectrum was intro-duced by Gassner for aeronautical structures, the historical
Eight-Block-Programme Test,Fig. 4 [5]. The reason of the
blocking was that random loading processes could not be yet simulated by existing simple testing machines at that time.
In the 1960s due to the access of servo hydraulic testing machines random processes could be simulated fairly well and the historical Eight-Block-Programme Test could be substituted by a more realistic load–time process, e.g. the
Gaussian random load distribution,Fig. 5.
0142-1123/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijfatigue.2006.10.011
E-mail address:[email protected]
www.elsevier.com/locate/ijfatigue International Journal of Fatigue 29 (2007) 1080–1089
Journalof
Fatigue
As load–time histories depend on the particular
appli-cation (offshore, aeronautics, railways, automotive,
bridges etc.) and function of the components, in the past 65 years different application related standard spectra were
developed,Tables 1 and 2 [6], and are still under
develop-ment. Thus, this paper will not address the methodologies for deriving testing spectra, but the principles to be respected, when tests have to be performed with a given spectrum.
2. Documentation and presentation of the loading
A testing spectrum is characterized mainly by following
parameters,Fig. 6:
– Maximum and minimum values,
– load (stress) ratio R of the maximum values,
– spectrum (sequence) length (size) Lsand
– shape.
These parameters must be documented and presented by a cut-out of the load–time history, by the rainflow matrix, by the load ratio R of the maximum load values of the spec-trum, the irregularity factor I, and by the conventional cycle counting methods level crossings and range pairs. The maximum value of the level crossings counting indi-cates the level of the maximum spectrum stress with regard to the yield strength of the material as well as how far the high-cycle fatigue strength is exceeded. The comparison of the spectra accounting to both counting methods, level crossings and range pairs, in term of ranges gives the infor-mation about present mean-load (stress) fluctuations,
which have an additional damaging influence[3]; this is
dis-played if the spectra for both counting methods are not Nomenclature
C probability of confidence
D damage sum
F load
Ls sequence length
LT, LTR spectrum size, test, test prolonged by risk factor
N, N number of cycles, constant and variable
ampli-tude loading
Nk fatigue life at knee point
Ps, Pf, Po probability of survival, failure, occurrence
Rx, Rx load, stress or strain ratio Rx= Xmin/Xmax for
constant and variable amplitude loading
TN, TN fatigue life scatter between Ps= 10% and 90%,
for constant and variable amplitude loading
r, r stress, constant and variable amplitude
loading
rak knee point of the S–N curve
e strain a amplitude eq equivalent f frequency, failure jR,C risk factor jN safety factor
k, k0 slope of the S–N curve, slope of the
prolonga-tion
l longitudinal
m mean
n number of tests, number of cycles, nominal
sN standard deviation (sN= 0.39 lg (1/TN)) t time D range i = 1: steel, aluminium i = 2: cast and sintered materials exper. s . spec real real . spec s calc. N L D D D D L N ⋅ = ⋅ = . calc N k' = 2k - i Cumulative frequency distribution(spectrum) Woehler curve Gassner curve slope k Damagesum of the spectrum:
n1 n2 n3 n4 Ls Nk . Spec n 1 i i i D N n =
∑
= 1 2 3 4 N1 N2 N3 N4 max , a σ Stressamplitude σa ,σ a σk(knee point) Cycles N, N k' = kidentical. If in term of cycles or cumulative frequency the ratio of 1:3 between the two counting methods is exceeded,
a much lower real damage sum Drealthan for a spectrum
without mean-load fluctuations has to be selected for a
fati-gue life assessment [3,4], e.g. Dreal= 0.2 versus 0.5 for
welded joints, 0.1 versus 0.3 for not welded components basing on experiences. However, more research for a dam-age mechanics founded approach is necessary.
Figs. 7 and 8present the documentation of two different
spectra, one without a mean-load fluctuation (narrow band) and the other one with a large mean-load fluctuation (wide band).
The load history for performing a variable amplitude test is stored usually as a peak (turning point) sequence,
Fig. 9, and by the appertaining rainflow matrix. (In the
past also the Markovian matrix was used.)
Generally, a spectrum does not contain the information about the loading frequency. Often, the testing frequency depends on the interaction between the testing machine and the stiffness of the test object, as well as on the elec-tronical control possibilities of the frequency. However, for variable amplitude tests of dynamic (swinging) as well as non-linear systems, e.g. mass-damper-systems, where the frequency content is required, the storage of the load–time history has to contain also the information Fig. 2. Real damage sum distributions for steel and aluminium.
Fig. 3. Different load–time histories.
a. Load sequence b. Cumulative frequency distribution
-1.0 1.0 0 Ls= 5 · 105 N (log) S tr e ss (li n ) 1. 2. 3. 4. 5. 6. 7. 8. 2. 3. 4. 7. 6. 5. 4. Step Number of cycles 1 4 70 680 4 70 68 0 500 0 230 00 700 0 0 3 0250 0 7 0000 23000 500 0 min m max
Max. stress in individual steps Mean stress (is constant for all steps)
Min. stress in individual steps
Sequence length Ls= 5 · 105cycles R = min / max
S tr e ss (li n ) 1. 2. 3. 4. 5. 6. 7. 8. 2. 3. 4. 7. 6. 5. 4. Step Number of cycles 1 4 70 680 4 70 68 0 500 0 230 00 700 0 0 3 0250 0 7 0000 23000 500 0 min min m m max max
Max. stress in individual steps Mean stress (is constant for all steps)
Min. stress in individual steps
Sequence length Ls= 5 · 105cycles R = R = min min // maxmax
about the frequency spectrum, e.g. the power spectral
den-sity (PSD)[4],Fig. 10.
The sequence length Ls of a test spectrum may be a
value obtained after an omission of small, as non-damag-ing assumed amplitudes. However, in case of an omission it must be noted that the obtained test cycles to failure cor-respond to service cycles to failure given by
Nservice¼ Ntest
Ls;before omission
Ls
ð3Þ
3. Performance of variable amplitude loading tests
Variable amplitude loading (VAL) tests are principally carried out like constant amplitude loading tests (CAL)
on different load levels,Fig. 11. The only difference is that
in case of VAL a given sequence must be continuously repeated until a failure is obtained, while under CAL the amplitude (or range) remains unchanged. For a valid VAL test, the sequence must be repeated at least 5–10 times
in order to achieve a service-like load mixing[7]. There are
different failure criteria which must be defined according to the particular application: a crack with a defined depth, a defined decrease of stiffness, a total rupture etc.
The difference between the load levels is only a linear amplification of the amplitudes (or ranges) of the spectrum;
shape and length Lsof the spectrum remain independent of
the load level.
As long as the frequency does not affect the fatigue life, or particular attention of the frequency content is not Fig. 5. Gaussian load spectrum.
Table 1
Overview of existing uniaxial variable amplitude loading standards
Name Purpose Structural detail Year
Eight-Block Programme
General purpose, block-wise variable amplitude loading
Components of transportation vehicles, heavy machinery components, etc.
1939
Twist Transport aircraft wing Wing root bending moment 1973
Gaussian General purpose random sequence Narrow-band, medium-band, wide-band random 1974
Falstaff Fighter aircraft Wing root 1975
MiniTwist Shortened version As above 1979
Helix, Felix Helicopters, hinged and fixed rotors Blade bending 1984
Helix/32, Felix/28 Shortened versions As above 1984
Cold turbistan Tactical aircraft engine discs Bore 1985
Wisper Wind turbines Blade out-of-plane bending 1988
Wash I Offshore structures Structural members of oil platforms 1989
Wawesta Teel mill drive Drive train components 1990
Carlos Car loading standard Sequence (uniaxial) Vertical, lateral, longitudinal forces on front suspension parts 1990
Table 2
Overview of existing multi-channel variable amplitude loading standards
Name Purpose Structural detail Year
Eurocycle I passenger car wheels Vertical and lateral loads wheels, wheel/hub/bearing units 1981
Eurocycle II truck wheels Vertical and lateral loads wheels, wheel/hub/bearing units 1983
Enstaff Alstaff + temperature Wing root 1987
Hot turbistan Tactical aircraft engine discs Cold turbistan + temperature
Rim (hot section) 1989
Carlos multi Car loading standard (multiaxial) 4-Channel load components for front suspension parts 1994
Carlos PTM Car power train (manual shift)
torques + speeds + gear pos.
Power train components, e.g. clutch, gear-wheels, shafts, bearings, and universal joints
1997
Carlos PTA Car power train (autom. shift)
torques + speeds + gear pos.
Ower train components e.g. gear-wheels, shafts, bearings, and universal joints
2002
required, the frequency can be increased for shortening the testing time. However, depending on the interaction between the testing machine and the stiffness of the speci-men, the overall testing frequency can be limited. In such cases, especially low load amplitudes can be accelerated
by an amplitude and frequency adaptive control,Fig. 12
[8].
During the testing, control and real signals must be compared and registered with regard to turning points, amplitude distribution, rainflow matrix and if required Fig. 7. Gaussian spectrum with constant mean load.
Fig. 8. Truck spectrum with fluctuating mean load.
Load or (stress) amplitupe (linear)
) a ( a F σ o P , R Cycles N (log) L (σ values from which the maximum load (stress) amplitude
and the load(stress) ratio are derived ) ( F and ) (
Fmax σmax minσmin
Amplitude distribution (shape)
Spectrum size Ls
(total number of cycles N), probability of occurance Po max min max min/F / F R= =σ σ ) ( Fa σa s Fig. 6. Main parameters of a spectrum.
the power spectral density. In case of multichannel VAL, especially of dynamic or/and of non-linear systems, e.g. automotive suspensions or car body structures, also the time order and the phase differences between the particular
channels must be controlled,Fig. 13 [9].
For the durability proof of components or structures tests are carried out usually only on the service–load level. 4. Statistics and required amount of tests
The statistics applied to VAL tests is principally the
same as the statistics applied to CAL tests [1,4,10,11].
The mostly assumed distribution type is the Gaussian Log-Normal-Distribution, but other distributions, e.g.
according to Student[10]or Weibull[12,13]can be applied,
too, for defining the course of the Gassner curve with the
probability of survival Ps= 50% and the scatter band with
Ps= 10% and 90% or 2.5% and 97.5%. (Within Ps= 10%
and 90% the type of distribution does not influence the position of the curves significantly, but an extrapolation to much lower or higher probabilities results significant
dif-ferences[13].)
As VAL tests are more complicated and often more time consuming than CAL tests, at least two levels with each five tests or three levels with each three tests can be consid-ered to be sufficient, to determine the slope k, the scatter T and the standard deviation s:
Fig. 10. Power spectral density and joint density distribution.
s
1 xL
N= ⋅ Constantamplitude
loading--Woehler
curve-N1 N2 Cycles to failure Nf,Nf(log)
Amplit u d e σa /σ a,m ax (norma li z e d) ( log) t 0 +0.5 -0.5 0 +0.5 -0.5 +1.0 -1.0 N1 N2 0 +0.5 -0.5 +1.0 -1.0 0 +0.5 -0.5 t t t s 2 yL N= ⋅ x, y : number of repet it ions
Variable amplitude loading Gassner curve
-Repeated load-time history
Repeated sequence (linear amplit ude distribution) 1 N 0.1 0.5 1.0 2.0 0.2 k k Rectangular spectrum Ls y Ls x 2 N 0 0 0 0 Ls: sequence lengt h Repeated constant amplit udes a m plit ud e
Fig. 11. Principles of constant and variable amplitude loading tests. Fig. 9. Cut-out of a peak (turning point) sequence from a load time
k¼ lgðN2=N1Þ lgðDr1=Dr2Þ ð4Þ Tx¼ 1 : xðPs¼ 10%Þ xðPs¼ 90%Þ ð5Þ x¼ ra; Dr or N ð6Þ sx¼ 1 2:56lg 1 Tx ð7Þ
In Fig. 14 Woehler- and Gassner-curves are displayed
with their mean values (Ps= 50%), the appertaining scatter
bands between Ps= 10% and 90% and the particular slopes
k.
For the durability proof of big structures the amount of test objects is very restricted; in the worst case an assess-ment may be required by only one test. In such cases the risk given by a few amount of tests must be covered by sta-tistics; the mean value of fatigue lives obtained by few tests
must be reduced by the risk factor jR, Cto obtain the ‘‘real’’
mean value[1]: NPs¼50%¼ Nmean;tests jR;C ð8Þ jR;C¼ 1 TN ð1=pffiffiffiffi4nÞ ð9Þ
while n is the amount of tests performed and TNis the
scat-ter which would be obtained for a high amount of tests (ba-sic population); it is not the scatter resulting from few tests. It can be estimated by testing experiences with a larger number of specimens manufactured in a comparable way.
The risk factor in Eq.(8)is valid for a probability of
con-fidence Pc= 90%.
To calculate a fatigue life for an allowable probability of
survival Ps> 50%, the ‘‘real’’ mean value must be reduced
by the safety factor jN [1,14]:
NPs>50%¼ NPs¼50% jN ð10Þ jN¼ 1 TN exp 2:36 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi lgð1 PsÞ j j p 2:56 1 " # ð11Þ In case of few testing objects, the durability proof can also be conducted on such a way that the spectrum for the re-quired life cycle (the spectrum can be composed by a high amount of repeated sequences, e.g. for 25 years life of an
F 0 t F 0 t t1 t2 Δt f = const. f2> f
Fig. 12. Acceleration of tests by frequency adaptation.
offshore rig 25 repetitions of the 1 years sequence) has to be
repeated according to the risk factor[15,16], Fig. 15. If a
failure is not caused, the durability is proved. 5. Documentation and presentation of test results
As mentioned before, test results must be documented in following way:
Description of the spectrum by its rainflow matrix, sequence length, visualization by level crossings and range pairs counting; for dynamic or/and non-linear behaving test objects additionally the power spectral density.
Storage of the peak (turning point) sequence.
Tabulation of applied maximum load levels of the sequence (all other amplitudes or ranges are related lin-early to the maximum value) and the number of cycles to failure or the number of repetitions of the spectrum. Definition of the failure criterion, e.g. crack, break
through, total failure, and stiffness loss.
Testing frequency.
Environmental conditions, e.g. temperature, corrosion. For the graphical presentation of the test results in the double-logarithmic plot the maximum load (stress) of the spectrum versus the number of cycles to failure should be
preferred [3–5,17,18]. This is justified by following
argu-ments which are important for the design of structures: Distance between the maximum spectrum stress and the
structural yield strength can be evaluated. However, this requires the determination of the local stress in the crit-ical area of the component.
Exceedance of the Woehler-curve can be evaluated with regard to exploitable light-weight design potential in
dependency of the spectrum applied [3],Fig. 16.
In case of a spectrum with a Gaussian distribution of the
amplitudes for achieving a fatigue life of e.g. N ¼ 1 108
cycles the constant amplitude high-cycle fatigue strength can be exceeded by a factor of 1.50, in case of a straight line Fig. 14. Woehler- and Gassner-curves of a laserbeam welded hat profile.
distribution even more, 1.90. Light-weight design can be performed by allowing higher-stresses in the structure and thus reaching the required fatigue life: compared to constant amplitude design of a steering rod with a diameter of d = 22 mm, by considering the spectrum shape a diame-ter of d = 18 mm for a Gaussian distribution, and d = 16 mm for a straight-line distribution is obtained. This diameter reduction renders a weight decrease of 50%.
In some design codes or recommendations [19,20] the
calculation of an equivalent stress or load of the spectrum is suggested: Dreq¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Xh 1 niDrki Ls k v u u t ð12Þ
This kind of presentation for comparing the VAL – results with the CAL – results assumes on one hand a dam-age sum of D = 1.0 which is mostly on the unsafe side,
Fig. 2, [17,18]and on the other hand it does not allow to
recognize at one glance the light-weight design potential (exceedance of the Woehler-curve) as well as the risk of glo-bal plastification (distance of the maximum value of the spectrum from the yield strength). The equation assumes also the same slope for the Woehler- and Gassner-curves, which is seldom the case.
6. Summary
The lack in fatigue life assessment despite more then 70
cumulative damage hypothesis[21]necessitates
experimen-tally based knowledge for the design practice [22].
How-ever, as the performance of variable amplitude fatigue tests are not as simple as constant amplitude tests, a guid-ance on the particular testing principles, the documentation of testing details and results and finally the presentation of the results is needed.
The major points to be respected in variable amplitude loading (VAL) tests are:
Description of the load spectrum (maximum values, shape, sequence length) and documentation by storage of the peak (turning point) sequences as well as by the rainflow matrix; in case of systems with dynamic response or/and non-linear testing objects additionally the power spectral density (frequency content).
Definition of the failure criterion (crack length, total failure, stiffness loss, etc.).
Description of the experimental devices and conditions (frequency, environment).
Presentation of the maximum spectrum loads versus cycles to failure or/and number of repetitions of the sequence length in a double-logarithmic plot as well as in a table.
In comparison to an already existing ISO-draft[23], this
paper gives more information, especially on testing details and presentation of results.
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