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International Journalof Fatigue

International Journal of Fatigue 29 (2007) 1080–1089


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 durability 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 damage content of a spectrum with the size Ls can be determined Xn ¼ Dspec ð1Þ N i and with this value the real damage sum is calculated from the experimental results: Dreal ¼

Dspec  N exp Ls


A broad investigation on cumulative fatigue [2] displays the scattering of the real damage sum over almost three decades, Fig. 2. About 90% of all results are below the convenE-mail address: [email protected] 0142-1123/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijfatigue.2006.10.011

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 distribution 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 destinations or a defined driving distance. The first variable amplitude loading spectrum was introduced 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.

C.M. Sonsino / International Journal of Fatigue 29 (2007) 1080–1089


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 amplitude 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, T N fatigue life scatter between Ps = 10% and 90%, for constant and variable amplitude loading  r, r stress, constant and variable amplitude loading


Damagesum of the spectrum:

Stressamplitude σa, σa

N calc. = Dreal =

Woehler curve

Cumulative frequency distribution(spectrum) a,max

l m n sN t D

knee point of the S–N curve strain amplitude equivalent frequency, failure risk factor safety factor slope of the S–N curve, slope of the prolongation longitudinal mean number of tests, number of cycles, nominal standard deviation (sN = 0.39 lg (1/TN)) time range

ni = D Spec. Ni

∑ i=1


rak e a eq f jR,C jN k, k 0

slope k

1 2

Ls ⋅ Dreal D spec. D spec. Ls

Gassner curve


σk (knee point)

N calc .



i = 1: steel, aluminium



⋅ N exper.


k' = 2k - i

N3 4 n4

Nk Ls


k' = k

i = 2: cast and sintered materials

Cycles N, N

Fig. 1. Modification of the S–N curve and calculation of fatigue life (schematic).

As load–time histories depend on the particular application (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 development. 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) Ls and – 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 spectrum, the irregularity factor I, and by the conventional cycle counting methods level crossings and range pairs. The maximum value of the level crossings counting indicates 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 information about present mean-load (stress) fluctuations, which have an additional damaging influence [3]; this is displayed if the spectra for both counting methods are not


C.M. Sonsino / International Journal of Fatigue 29 (2007) 1080–1089

Fig. 2. Real damage sum distributions for steel and aluminium.

without mean-load fluctuations has to be selected for a fatigue 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 damage mechanics founded approach is necessary. Figs. 7 and 8 present 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 electronical 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. 3. Different load–time histories.

identical. 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 Dreal than for a spectrum a. Load sequence

b. Cumulative frequency distribution Max. stress in individual steps















4 70

5. 5000



2. 3.





3. 2. 70

1. 4. 680



Mean stress (is constant for all steps)





Min. stress in individual steps

Sequence length Ls = 5 · 105 cycles

N (log)

Number of cycles


min /


Fig. 4. Ernst Gassner’s Eight-Block-Programme sequence.

Ls = 5 · 105

C.M. Sonsino / International Journal of Fatigue 29 (2007) 1080–1089


Fig. 5. Gaussian load spectrum.

Table 1 Overview of existing uniaxial variable amplitude loading standards Name


Structural detail


Eight-Block Programme Twist Gaussian Falstaff MiniTwist Helix, Felix Helix/32, Felix/28 Cold turbistan Wisper Wash I Wawesta Carlos

General purpose, block-wise variable amplitude loading Transport aircraft wing General purpose random sequence Fighter aircraft Shortened version Helicopters, hinged and fixed rotors Shortened versions Tactical aircraft engine discs Wind turbines Offshore structures Teel mill drive Car loading standard Sequence (uniaxial)

Components of transportation vehicles, heavy machinery components, etc. Wing root bending moment Narrow-band, medium-band, wide-band random Wing root As above Blade bending As above Bore Blade out-of-plane bending Structural members of oil platforms Drive train components Vertical, lateral, longitudinal forces on front suspension parts

1939 1973 1974 1975 1979 1984 1984 1985 1988 1989 1990 1990

Table 2 Overview of existing multi-channel variable amplitude loading standards Name


Structural detail


Eurocycle I Eurocycle II Enstaff Hot turbistan

passenger car wheels truck wheels Alstaff + temperature Tactical aircraft engine discs Cold turbistan + temperature Car loading standard (multiaxial) Car power train (manual shift) torques + speeds + gear pos. Car power train (autom. shift) torques + speeds + gear pos. Car trailer coupling (multiaxial)

Vertical and lateral loads wheels, wheel/hub/bearing units Vertical and lateral loads wheels, wheel/hub/bearing units Wing root Rim (hot section)

1981 1983 1987 1989

4-Channel load components for front suspension parts Power train components, e.g. clutch, gear-wheels, shafts, bearings, and universal joints Ower train components e.g. gear-wheels, shafts, bearings, and universal joints Trailer coupling device and vehicle supporting structure

1994 1997

Carlos multi Carlos PTM Carlos PTA CarloS TC

about the frequency spectrum, e.g. the power spectral density (PSD) [4], Fig. 10. The sequence length Ls of a test spectrum may be a value obtained after an omission of small, as non-damaging assumed amplitudes. However, in case of an omission it must be noted that the obtained test cycles to failure correspond to service cycles to failure given by N service ¼ N test 

Ls;before omission Ls


3. Performance of variable amplitude loading tests Variable amplitude loading (VAL) tests are principally carried out like constant amplitude loading tests (CAL)

2002 2003

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 Ls of 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

C.M. Sonsino / International Journal of Fatigue 29 (2007) 1080–1089

Load or (stress) amplitupe (linear)


Preload (prestress) Fp(σp) and load (stress) values Fmax (σmax ) and Fmin (σmin ) from which the maximum load (stress) amplitude Fa (σa ) and the load(stress) ratio R = Fmin / Fmax = σmin / σmax are derived

F (σ ) a a

Amplitude distribution (shape)

R, Po

Spectrum size Ls (total number of cycles N), probability of occurance Po

Cycles N (log)


Fig. 6. Main parameters of a spectrum.

Fig. 7. Gaussian spectrum with constant mean load.

Fig. 8. Truck spectrum with fluctuating mean load.

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 specimen, 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

C.M. Sonsino / International Journal of Fatigue 29 (2007) 1080–1089


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 differences [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 considered to be sufficient, to determine the slope k, the scatter T and the standard deviation s:

Fig. 9. Cut-out of a peak (turning point) sequence from a load time history.

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].

Fig. 10. Power spectral density and joint density distribution.

Repeated constant amplit udes


+1.0 +0.5


0 -0.5


Amplit ude σa/ σa,max(normalized) (log)

Constantamplitude loading-Woehler curve-

Variable amplitude loading -Gassner curve -

Rectangular spectrum

k 0






-0.5 -1.0

+0.5 0







Repeated sequence (linear amplitude distribution)

Repeated loadtime history






t 0




N 1 = x ⋅ Ls



N 2 = y ⋅ Ls

Ls : sequence lengt h


x, y : number of repetit ions

0.1 N1




Cycles to failure

Fig. 11. Principles of constant and variable amplitude loading tests.

Nf , N f (log)


C.M. Sonsino / International Journal of Fatigue 29 (2007) 1080–1089

For the durability proof of big structures the amount of test objects is very restricted; in the worst case an assessment may be required by only one test. In such cases the risk given by a few amount of tests must be covered by statistics; the mean value of fatigue lives obtained by few tests must be reduced by the risk factor jR, C to obtain the ‘‘real’’ mean value [1]:

f = const.

F 0



t2 Δt

F 0

t f2 > f Fig. 12. Acceleration of tests by frequency adaptation.

lgðN 2 =N 1 Þ lgðD r1 =D r2 Þ xðP s ¼ 10%Þ Tx ¼ 1 : xðP s ¼ 90%Þ

a ; D r or N x¼r 1 1 lg sx ¼ 2:56 T x

ð4Þ ð5Þ

N mean;tests jR;C ffi  ð1=pffiffiffi 4nÞ 1 ¼ TN

N P s ¼50% ¼




while n is the amount of tests performed and TN is the scatter which would be obtained for a high amount of tests (basic 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 confidence 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]: N P s >50% ¼

ð6Þ 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.

 jN ¼

N P s ¼50% jN " 

1 exp TN

ð10Þ 2:36

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jlgð1  P s Þj 1 2:56

# ð11Þ

In case of few testing objects, the durability proof can also be conducted on such a way that the spectrum for the required life cycle (the spectrum can be composed by a high amount of repeated sequences, e.g. for 25 years life of an

Fig. 13. Documentation example of a multichannel variable amplitude loading.

C.M. Sonsino / International Journal of Fatigue 29 (2007) 1080–1089


Fig. 14. Woehler- and Gassner-curves of a laserbeam welded hat profile.

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 linearly 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 arguments 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 critical 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. 15. Testing requirement for covering the risk of a low number of determining fatigue life with few tests.


C.M. Sonsino / International Journal of Fatigue 29 (2007) 1080–1089

Fig. 16. Influence of spectrum – shape on fatigue life and component dimensions.

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 diameter 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: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u h u X ni Drki k Dreq ¼ t ð12Þ Ls 1

 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.

This kind of presentation for comparing the VAL – results with the CAL – results assumes on one hand a damage 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 global 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.

In comparison to an already existing ISO-draft [23], this paper gives more information, especially on testing details and presentation of results.

6. Summary The lack in fatigue life assessment despite more then 70 cumulative damage hypothesis [21] necessitates experimentally based knowledge for the design practice [22]. However, as the performance of variable amplitude fatigue tests are not as simple as constant amplitude tests, a guidance 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:

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C.M. Sonsino / International Journal of Fatigue 29 (2007) 1080–1089 [7] Schijve J. Fatigue of structures and materials. AH Dordrecht: Kluwer Academic Publishers; 2001. [8] Sonsino CM. Versuchszeitverku¨rzung in der Betriebsfestigkeit (Reduction of testing time for structural durability proof). Materialpru¨fung 2003;45(4):133–44. [9] Kla¨tschke H, Schu¨tz D. Das Simultanverfahren zur Extrapolation und Raffung von mehraxialen Belastungs-Zeitfunktionen fu¨r Schwingfestigkeitsversuche (The simultaneous method for extrapolation and shortening of multiaxial load–time histories for fatigue testing). Mat-wiss U Werkstofftechnik 1995;8:404–15. [10] Schneider CRA, Maddox SJ. Best practice guide on statistical analysis of fatigue Data Report No. 13604.01/02/1157.02, 2002 The Welding Institute (TWI), Abington Hall, UK. [11] Bastenaire F. New method for the statistical evaluation of constant stress amplitude fatigue test results. In: Probabilistic aspects of fatigue. ASTM STP, vol. 511. Philadelphia: ASTM; 1972. p. 3–28. [12] Castillo E, Fernandez Canteli A. A general regression model for lifetime evaluation and prediction. Int J Fracture 2001;107:117–37. [13] Fuchs HO, Johns MV. The risks of extrapolations of metal fatigue data. J Test Eval, JTEVA 1988;16(3):276–9. [14] Filippini M, Dieterich K. An approximate formula for calculating the probability of failure. Fraunhofer-Institute for Structural Durability (LBF) Technical Information TM-No. 111; 1997. [15] Grubisic V. Determination of load spectra for design and testing. J Vehicle Des 1994;15(1/2):8–26.


[16] Grubisic V. Fatigue evaluation of vehicle components- State of the art, restrictions and requirements -Keynote address to section ‘‘Fatigue research and application’’ SAE International Congress, Detroit, February 24–27; 1997 [LBF-Publication No. 633]. [17] Sonsino CM. Limitations in the use of RMS-values and equivalent stresses in variable amplitude loading. Int J Fatigue 1989;11(3):142–52. [18] Lagoda T, Sonsino CM. Comparison of different methods for Presentino variable amplitude loading fatigue results. Mat-wiss U Werkstofftechnik 2003;34(11):13–20. [19] Eurocode No. 3. Part 1: General design rules for steel constructions. DIN-V ENV 1993-1-1/A2. [20] Hobbacher A. Fatigue design of welded joints and components. IIWDoc. XIII-1539-96/XV-845-96 1996 Abington Hall, UK. [21] Fatemi A, Yang L. Cumulative fatigue damage and life prediction theories: a survey of the state of the art for homogeneous materials. Int J Fatigue 1998;20(1):9–34. [22] Sonsino CM, Maddox SJ, Hobbacher A. Fatigue life assessment of welded joints under variable amplitude loading – State of the present knowledge and recommendations for fatigue design regulations. In: Proceedings of the IIW international conference on technical trends and future prospectives of welding, 2004. p. 84–99. [23] ISO/WD 12110-1. Metallic materials – fatigue testing – variable amplitude fatigue testing – Part 1: General principles test method and reporting requirements document state, September 2006.

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