Department of Mechanics and Machine Design, Opole University of Technology, ul. Miko!ajczyka 5, 45-271 Opole, Poland

a r t i c l e i n f o

Article history:

Received 15 April 2009

Received in revised form

18 May 2009

Accepted 20 May 2009Available online 30 May 2009

Keywords:

Service loading

Fatigue test

d bysts oftigue

effects on the material as service loading. The main problem during fatigue tests of machine parts is the selection

the ways to preserve real-loading conditions, it is to use registered service loading during the tests. However, it leads to

Contents lists available at ScienceDirect

Mechanical Systems and Signal Processing

ARTICLE IN PRESS

Mechanical Systems and Signal Processing 23 (2009) 271227210888-3270/$ - see front matter & 2009 Elsevier Ltd. All rights reserved.

doi:10.1016/j.ymssp.2009.05.010 Tel.: +487740 06 379; fax: +487740 06 343.E-mail address: a.nieslony@po.opole.plof suitable loading level and their character. Because of the kind of tested element, its geometry and environmentwhere the element is going to work, there are various factors that should be taken into account while experiments. One ofFatigue life analysis plays an important role in a modern designing process. Such kind of analysis is performesimulations at the early designing stage, during so-called virtual prototyping or in laboratories performing real temachine components. In both cases, appropriate loading history must be applied, and it should provide the same fa& 2009 Elsevier Ltd. All rights reserved.

1. IntroductionMultiaxial fatigue

Rainow counting

Standard loadinga b s t r a c t

The paper presents a method for determination of multiaxial load segments from original

service histories, where the loaded machine part is meaningfully subjected to fatigue

damage. These load segments are directly separated from a service-loading history, which

can be of a random character. Typical procedures used for fatigue life assessment under

multiaxial random loading are implemented to perform this task. While reduction from

the multiaxial stress state to the equivalent uniaxial one, application of the linear

multiaxial fatigue failure criteria was proposed. The equivalent stress history is subjected

to the rainow cycle-counting method, which allows to determine the amplitudes and

mean values of counted cycles, their occurrence moment and time of duration. Inuence of

the stress mean value was taken into consideration with the Morrows model. On the

assumption of the linear PalmgrenMiner hypothesis of damage accumulation and the

stresslife fatigue characteristics of the material, the damage-time function was

determined. The load segments, where the inuence on the material fatigue was

signicant, were determined on the basis of the xed damage-intensity level and the

proposed damage-intensity function. The presented method was studied on a hook loaded

with two independent forces. FEM program which has the possibility to perform fatigue

analysis was used during the computation for determining the expected place of crack

initiation. The service-loading course was compressed to shorter one, so that only a small

decrease of the fatigue damage in comparison with full length of the original service

loading was observed. The proposed method seems to be right for preparing multiaxial

loading histories in order to cut down the fatigue tests, especially in the case when the

correlation between particular loading channels is very important.Determination of fragments of multiaxial service loading stronglyinuencing the fatigue of machine components

Adam Nies"ony

journal homepage: www.elsevier.com/locate/jnlabr/ymssp

amplitude distribution is changed [35]. In consequence of the changes, the compressed loading history is required to

ARTICLE IN PRESS

Nomenclature

F1(t), F2(t) histories of forces loading the hook consid-ered in the example

rF1,F2 correlation coefcient between histories F1(t)and F2(t)

C cumulative damageI damage intensityIlim limit level of damage intensityD damageDi damage for ith cyclet timeT time to failure

Nf number of cycles to failureNi number of cycles computed from uniaxial

fatigue characteristic for known mean valuesmi and amplitude sai

sa cycle amplitudesm cycle mean values0f, b fatigue strength coefcient and exponent

respectively in the Morrow stress-life charac-teristic

seq(t) equivalent stress history obtained from multi-axial fatigue failure criterion

sij(t) stress history of particular stress tensor com-ponents, where ij (xx,yy,zz,xy,xz,yz)

A. Nies!ony / Mechanical Systems and Signal Processing 23 (2009) 27122721 2713generate much greater damages in the material at the same time but the general characteristic of the loading should bekept. However, in some cases it is necessary to apply non-standard loading, or to perform tests under near to real-loadingconditions. In such a case, the only way to shorten time of the tests is neglecting the loading intervals when no signicantdamage can be observed [3,6]. This can be acceptable for tests performed in so-called high-cycle regime, where thesequence of loading is unimportant and can be neglected.

While fatigue tests of machine elements or structures under multiaxial loading, the stress state at a given point dependson a considerable degree on geometry. In such a case, application of standardized loadings which do not include correlationbetween components of multiaxial loading can cause generation of the stress state strongly different than that observedwhile real-service loading. It can be hazardous to use loading prepared in such a way, especially in the case of new solutionsbased on foredesigns. Therefore, also the geometry of the tested element should be taken into account while preparation ofloading history for fatigue tests.

The main purpose of this paper is to present a method for preparation of multiaxial service loading for performingfatigue tests. By elaborating of the method, special emphasis was placed on preserving the original character of themultiaxial service loading.

1.1. Multiaxial fatigue life assessment

In the case of multiaxial random loadings, fatigue life assessment on the basis of uniaxial material characteristicstime-consuming fatigue tests, often equal to real life to the tested object. Of course, such a solution is unacceptable fromthe economical and practical points of view. Thus, many standardized loading histories are proposed for application at thetest stands [13]. They are usually obtained by numerical processing of the registered service histories, where the

To length of the loading historyni number of cycles with sai and smi counted

from random history

ln,mn,nn direction cosines used in multiaxial fatiguefailure criterionrequires specialised tasks, which have been discussed in many books and papers [713]. One of the well-known methods offatigue life calculations under multiaxial stress state includes reduction of the three-dimensional stress state to theequivalent uniaxial state with use of a suitable fatigue failure criterion [7,913]. The idea of this method is shown in Fig. 1.Application of the multiaxial fatigue failure criterion allow to obtain equivalent stress history. Next, the obtained histories

Equivalent uniaxial stress stateMultiaxial stress state

Multiaxialfatiguefailure

criterion

Time history ofthe equivalentstress

Damageaccumulation,e.g. Palmgren-Miner rule

Cyclecounting,e.g. rainflow

Stress tensor components

xx(t)yy(t)zz(t)xy(t)xz(t)yz(t)

(a-Nf)

eq(t)

Fig. 1. Transformation of the multiaxial stress state to the uniaxial one during fatigue life assessment.

of the equivalent stress are analysed in the sameway as those for the uniaxial case, i.e. the cycle counting and accumulationof the fatigue damage can be performed. The algorithm shown in Fig. 1 can use only those criteria, which realize reductionof the three-dimensional stress state at any time, continuously, and do not change frequency range of the equivalent stressin relation to the stress-state components. These requirements are satised by linear criteria using the concept of thecritical plane, proposed among others by Macha [911] or agoda and Ogonowski [11,12]. The three-dimensional stressstate should be correctly reduced to the uniaxial equivalent one. If the stress-state components are random, what can beobserved for a large majority of service loadings, also equivalent stress history of random character is expected. In order tocalculate damage on the basis of such equivalent stress history, the number of load cycles must be distinguished. There areseveral algorithms for cycle counting from a random history [14]. The rainow algorithm seems to be the most popular andwidely used in practice. It allows to determine half- and full-cycles with their amplitudes sai and mean values smi. It isgenerally accepted, that two equal half-cycles counted by rainow method form one full-cycle. Damage from the ith cycleof stress can be computed from

Di 1

Nf i sai ;smi , (1)

where Nf i sai ; smi is the function returning a number of cycles to failure according to the amplitude sai and mean valuesmi form the determined stress cycle and uniaxial fatigue characteristics of the material. After performing the cyclecounting at the xed time interval To of the random history a certain number of cycles with various amplitudes andmean values is distinguished. Computation of the fatigue damage caused by the considered loading interval To isusually realised with the assumption of hypothesis of the linear fatigue damage accumulation. The widely usedPalmgrenMiner hypothesis assumes that the total damage can be expressed as the sum of particular damages caused bydistinguished cycles

D Xki1

Di, (2)

ARTICLE IN PRESS

A. Nies!ony / Mechanical Systems and Signal Processing 23 (2009) 271227212714where k is a number of cycles determined from the history using cycle-counting algorithm. The expected life time of themachine component T can be calculated from the accumulated damage D caused by distinguished cycles in time interval Toof the stress history

T ToD. (3)

Selection of a suitable multiaxial fatigue failure criterion, fatigue characteristics of the material and hypothesis ofdamage accumulation allows to determine fatigue life under multiaxial random loading rightly.

stre

ss, M

Pa

stre

ss, M

Pa

Part of time history Part of time history

t, s t, s

Rainflow matrix Rainflow matrix

Fig. 2. A part of the time history and rainow matrices of the stresses registered under service conditions: stresses registered at the bus frame duringtypical road drive (a), and stress history of the arm of the excavator (b).

1.2. Rainow counting

The rainow algorithm is the most used algorithm of cycle counting [13,14]. It makes possible to assign amplitudes andmean values of distinguished cycles from random loading history. Under high-cycle fatigue, it is assumed that the effect ofthe material memory connected with plastic strains generated while successive loading cycles can be neglected. From theassumption, it appears that time of the cycle occurrence and its period is not important while determination of fatigue life.This assumption is widely applied in considerations on fatigue, for example, in the damage accumulation hypothesisproposed by PalmgrenMiner. Neglecting the inuence of the loading sequence while fatigue life assessment makes itpossible to present the distinguished cycles as the rainow matrix. The exemplary rainow matrices determined for stresshistories from measurements at real-machine components are presented in Fig. 2. A number of counted cycles with thegiven amplitude and mean value distinguished in the considered stress history can be readout from this matrix. Rainowmatrices are widely applied in fatigue calculations because their compact form allows to reduce time necessary for suchcalculations, and their visualization gives general information about the loading character. The rainow matrix does notinclude information when the distinguished loading cycle occurred in the considered history. It can be easily seen that it ispossible to write down the time of the cycle starting, and approximate its duration (period). Fig. 3 shows an exemplary timeinterval subjected to cycle counting. The obtained values have been presented in Table 1. The software used in this paperallows to count the cycles according to the rainow algorithm, and to state time of the cycle occurrence and the cycleperiods. Suitable software function was elaborated by the author of this paper in the MATLAB environment and it isavailable free of charge on the web [15].

ARTICLE IN PRESS

0 1 2 3 4 5 6 7 8 9 106

4

2

0

2

4

6

6

signalturning points

time, s

A. Nies!ony / Mechanical Systems and Signal Processing 23 (2009) 27122721 27150 1 2 3 4 5 6 7 8 9 1010

8

6

4

2

0

2

4turning points connected with line

C7

C6

C4

C3

C5

C1

C2

time, s

Fig. 3. Exemplary signal with turning points unequally distributed at time (a), and turning points with cycles and half-cycles counted using the rainowalgorithm (b) [11].

2. The proposed algorithm

The calculation algorithm is formulated for determination of fragments of loading of strong inuence on the structurefatigue, taking into account geometry of the element, fatigue properties of the material and time relations betweenmultiaxial loading histories. Thus, correlations between the stress tensor components formed during the service-loadingaction are preserved. The algorithm shown in Fig. 4 includes three basic blocks.

In the rst block, the input data for calculations are dened. The typical input data are material constants characterizingstatic and fatigue properties of the material, and the registered service-loading histories. The registered loading historiesshould be of good quality and subjected to low-pass ltration eliminating noises generated while measurements andregistration. Next, the loading levels, total mean values, length of the interval and physical quantities represented by theparticular loading histories should be precisely determined.

In the next block, a suitable criterion of multiaxial fatigue failure should be selected depending on a material of thetested machine element. Details concerning selection of a criterion can be found in the literature [913]. Next, the criticalpoint related to the material fatigue should be determined. It is usually the place where the fatigue crack initiation isexpected it can be dened from the service data for machine elements working under real conditions. At the designingstage, it is difcult to point out this place on the machine elements because of the lack of such data. In this case, specialsoftware for fatigue analyses based on nite element method can be used, for example, MSC Fatigue, Fe-Safe,LMS.Durability, FemFat, Ansys Workbench, etc. Such software allows to nd the place where the fatigue crack could initiatewith satisfactory accuracy for further calculations of signal edition.

At the third block, stress histories are determined in the element point dened while fatigue analysis. This should bedone for the successive loading values by FEM analysis. Next, the chosen criterion of multiaxial fatigue failure is applied forcalculations of the reduced stress history. It can be, for example, the criterion of maximum normal stress on the criticalplane [911], which is dedicated for a wide range of brittle materials. According to this criterion, the reduced stress historyis calculated from

seqt l2nsxxt m2nsyyt n2nszzt 2lnmnsxyt 2lnnnsxzt 2mnnnsyzt, (4)

where ln, mn and nn are suitable direction cosines of the unit vector ~n ln;mn;nn normal to the critical plane.

ARTICLE IN PRESS

Table 1Values readout and computed for the counted cycles presented in Fig. 3.

A. Nies!ony / Mechanical Systems and Signal Processing 23 (2009) 271227212716Number of the cycle or half-cycle C1 C2 C3 C4 C5 C6 C7

Amplitude 1 3 1.5 0.5 5 2.5 1

Mean value 1 2 0.5 2.5 0 2.5 1Cycle or half-cycle 1 0.5 1 1 0.5 0.5 0.5

Cycle beginning time (TCB, s) 1 0 5.5 7.5 4 8.5 9.5

Period (TCP, s) 2 8 2 1 9 2 1

Accepted occurrence time (TC, s) 2 4 6.5 8 8.5 9.5 10

Block1. Setting the input data- geometry of the structure or machine part- multiaxial service loading histories,- parameters and characteristics determining material behaviour,

Block2. Pre-calculation procedures- determining the area on the geometry for which the calculation will be done,

- selection of suitable multiaxial fatigue failure criterion,

Blok 3. Calculation- determination of the stress tensor component histories,- computation of thee quivalent uniaxial stress history,- advance drain flow cycle counting,- calculation of damage value for each extracted cycle / half cycle,- estimation of cumulative damage function,- estimation of damage intensity function,- finding the damaged fragments of histories on the basis of fixed

damage intensity level.

Fig. 4. Proposed algorithm for determination of multiaxial loading history fragments that strongly inuence on the material fatigue.

At the next step, cycles are distinguished from equivalent stress history seq(t). The general form of the proposedalgorithm, Fig. 4, allows to apply any cycle-counting algorithm, which enables to count cycles/half-cycles with theiroccurrence times. However, the author recommends the extended rainow algorithm presented in Section 1 where thecycle occurrence time is calculated as the sum of the cycle occurrence time and a half-time of the cycle period. Next, theparticular values of damages Di are determined for each cycle or half-cycle

Di niNf i

for i 1; . . . ; k, (5)

where ni is equal to 1 for a cycle and 0.5 for a half-cycle, k is a number of cycles and half-cycles distinguished from thehistory, Nf i is a number of cycles to the material failure computed from a uniaxial fatigue characteristic. If the equivalentstress history with mean value is considered, the well-known Morrow stress-life characteristic can be used [8]

sa s0f sm2Nf b, (6)from which

Nf i 1

2

sais0 f smi

1=b, (7)

where s0f is the fatigue strength coefcient (intercept at 1 reversal), and b is the fatigue strength exponent. In the next step,

ARTICLE IN PRESS

A. Nies!ony / Mechanical Systems and Signal Processing 23 (2009) 27122721 2717particular values of damage Di are classied according to time of cycle/half-cycle occurrence. The cumulative damagefunction is estimated on the assumption that damage accumulates in the material in a linear way

Ci Xin1

Dn for i 1; . . . ; k. (8)

Let us note that for time ti a value of the accumulated damage is equal to Ci, and the total damage caused by one loadingblock is equal to Ck. Assuming that the cumulative damage function C(t) is continuous, differentiable and dependent ontime t the damage-intensity function I(t) is estimated

It ddt

Ct. (9)

Let us note that the function C is obtained in a numerical way, and its value changes in jumps because of summation ofsuccessive damages. However, numerical determination of the damage-intensity function I(t) is unproblematic, sincediscrete values of the function Ci and their occurrence times ti are known. There are many algorithms that can be efcientlyand precisely used for such task solving [16]. The loading intervals not inuencing the material damage are dened at thenext step. In this order, the limit level of damage-intensity Ilim is established. The fragments of the service-loading historywith damage intensity lower than the limit one are understood as the unimportant intervals their inuence on thematerial fatigue is small.

3. Example: hook loaded with two forces

Fig. 5(a) shows geometry of the hook for which calculations were performed. It was assumed that the top surface of thehook was fully restrained, and the loading was realized by two independent forces F1(t) and F2(t) acting at a part of hole

F1(t)

F2(t)

constrainedplane

45

Fig. 5. Model of the hook with marked restrained plane and direction of loading forces action (a), and the FEM mesh applied while calculations (b).

surfaces. Direction of force action was xed so as the fatigue crack initiation occurred in another place, depending on thecorrelation between forces F1(t) and F2(t). In such a case, it is difcult to show the place of fatigue crack initiation on thebasis of the stresses obtained with use of static loading. Thus, this place can be found in a proper way only with use ofspecial software [17]. In this paper, the software Comsol and Matlab was applied. Results obtained from the fatigue analysisof the hook are presented in Fig. 6, where the fatigue damage maps are shown. Taking the advantage of damage maps it iseasy to determine the places of expected fatigue crack initiation. For further calculations, the case where the forcecorrelation coefcient rF1,F2E0 was chosen. In this case, it is expected that the fatigue failure should occur at the edge of thetop surface of the hook, Fig. 6(a). Thus, while determination of loading intervals strongly inuencing fatigue of the material,the stress state occurring only in that place should be considered. The forces loading the hook, Fig. 7, were generated so astheir mean value and variance varied at time and the correlation coefcient rF1,F2E0.

It was assumed that fatigue properties were described by the stress-life relationship (saNf), and the material constantsfor Morrow characteristic, Eq. (6) were: s0f 1200MPa, b 0.12. Numerical calculations were performed strictlyaccording to the algorithm shown in Fig. 4 with use of the Matlab program.

ARTICLE IN PRESS

F1(t) F2(t)rF1,F2 0

F1(t) = F2(t)rF1,F2 = 1

Fig. 6. Results of the fatigue calculations for the hook with the marked places of the maximum material damage; the calculations was done foruncorrelated (a) and correlated forces (b).

0

50

100

150signalmeanmean + standard deviationmean - standard deviation

F 1 (t

), kN

0 100 200 300 400 500 600 700 800 900 1000time, s

A. Nies!ony / Mechanical Systems and Signal Processing 23 (2009) 2712272127180 100 200 300 400 500 600 700 800 900 10000

50

100

150

F 2 (t

), kN

time, s

signalmeanmean + standard deviationmean - standard deviation

Fig. 7. Force histories applied for calculations with their variable mean value and standard deviation.

ARTICLE IN PRESS

-400

-200

0

200

400

10-30

10-20

10-10

100

0

0.5

1

1.5

2

2.5x10-3

0

0.5

1

1.5x10-4

0 100 200 300 400 500 600 700 800 900 1000-400

-200

0

200

400

Limit level of damage intensityIlim = 0.05 max[I(t)]

eq

(t), M

Pa

I (t)

C (t

)D

i e

q (t)

, MP

a

time, s

0 100 200 300 400 500 600 700 800 900 1000time, s

0 100 200 300 400 500 600 700 800 900 1000time, s

0 100 200 300 400 500 600 700 800 900 1000time, s

0 100 200 300 400 500 600 700 800 900 1000time, s

Fig. 8. Equivalent stress (a), damage values for particular cycles (b), cumulative function of damage (c), damage-intensity function (d), stress course withmarked fragments strongly inuencing the fatigue of hook material (e).

A. Nies!ony / Mechanical Systems and Signal Processing 23 (2009) 27122721 2719

ARTICLE IN PRESS

lim

A. Nies!ony / Mechanical Systems and Signal Processing 23 (2009) 271227212720At rst, the equivalent stress history was computed with use of the criterion of multiaxial fatigue failure. That historywas shown in Fig. 8(a). The criterion of maximum normal stress on the critical plane was applied, Eq. (4).While determination of the critical plane position, the mean value of the stress history was also taken into account.Next, the turning points of the equivalent stress history were determined, and cycles were counted with therainow algorithm [14,15]. The fatigue damage value for the counted cycles was calculated directly by rearrangement ofEqs. (5) and (7)

Di 2nisai

s0f smi

1=bfor i 1; . . . ; k. (10)

Time values of the beginning and period of the counted cycles were used for calculation of their time of occurrence ticorresponding to the damage values Di. It allowed to present particular damage values in Fig. 8(b). Estimation of thecumulative damage function C(t) was done with the numerical integration by the trapezoid method [16]. The obtainedfunction C(t) is presented in Fig. 8(c) after interpolation and smoothing.

The damage-intensity function I(t) was determined as a derivative of smooth and interpolated cumulative damagefunction C(t), Fig. 8(d). The intervals of important inuence on the material fatigue were distinguished for the assumednine values of the damage-intensity level Ilim. Summary of the calculation results was shown in Table 2. For the value Ilim/max[I(t)] 0.05 eight intervals were obtained and selected for presentation in Fig. 8(e). The marked fragments correspondto loading histories F1(t) and F2(t) determining the fragments strongly inuencing the fatigue of the material. As a result, inconsidered case, the loading history was shortened from 1000 to 175.6 s. Using this shortened multiaxial loading historyduring the calculation of the fatigue life, 26% of the initial fatigue life is expected.

0.9 3 10.9 0.79079 3.8

0.8 3 15.7 0.79080 5.5

0.7 3 19.8 0.79328 6.9

0.6 3 23.9 0.87601 7.6

0.5 4 28.8 1.1880 6.7

0.4 5 45.0 1.5000 8.3

0.3 5 57.9 1.5001 10.7

0.2 5 71.7 1.5021 13.3

0.1 7 105.9 1.5513 18.9

0.05 8 175.6 1.6243 30.1

0.01 9 293.7 1.7122 47.6

0.005 9 337.2 1.7913 52.2

0.0a 1 1000.0 2.3675 117.3

a Limit level value Ilim 0.0 lead to classify the whole loading history as a fragment strongly inuencing the fatigue of machine components.Table 2Results obtained for nine Ilim values.

Limit level

(I /max[I(t)])

Number of determined

fragments

Total length of determined

fragments (s)

Computed damage

value D

Calculated fatigue

life T (h)4. Conclusions

From the performed calculations it appears that the proposed algorithm for determination of fragments of serviceloading strongly inuencing the material fatigue is a good tool for preparation of fatigue tests at the early designing stage.The proposed method includes the element geometry and time relations between channels of multiaxial loading history.Namely, calculations are performed in the place where the fatigue crack initiation is expected. It can be easily adapted toactual needs and requirements, i.e. the selection of algorithm of cycle counting, suitable fatigue characteristics, inuence ofthe loading mean value. The nal result is also inuenced by a way of smoothing of the accumulated damage function,necessary for correct calculation of its derivative. Let us note that the algorithm programmed in Matlab acted veryefciently, and the computation was performed in a short time.

The proposed algorithm could be applied while preparation of the registered service histories for direct application atthe fatigue test stands or while preliminary preparation for further processing. A shortened service history has the samefrequency characteristic as the service history, and it is an important advantage of the method.

Acknowledgment

Article is co-nanced by the European Union within European Social Fund within the framework of the IV priorityTertiary Education and Science under the Human Capital Operational Programme.

References

[1] C.M. Sonsino, H. Klatschke, W. Schuetz, M. Hueck, Standardized load sequence for offshore structures, LBF-Report no. FB-181, Darmstadt, Germany,1988.

[2] D. Schutz, H. Klatschke, H. Steinhilber, P. Heuler, W. Schutz, Standardized load sequences for car wheel suspension components, LBF-Report no. FB-191, Darmstadt, Germany, 1990.

[3] J.H. Morrill, T. Achatz, A. Khosrovaneh, An application for fatigue damage analysis using power spectral density from road durability events, SAETechnical Paper Series, no. 980689, Reprinted from: Advancements in Fatigue Research and Applications, SP-1341 (1998) 5562.

[4] R.I. Stephens, P.M. Dindingert, J.E. Gunger, Fatigue damage editing for accelerated durability testing using strain range and SWT parameter criteria,International Journal of Fatigue 19 (89) (1997) 599606.

[5] T. Palin-Luc, A. Banvillet, J.-F. Vittori, How to reduce the duration of multiaxial fatigue tests under proportional service loadings, International Journalof Fatigue 28 (2006) 554563.

[6] A. Nies"ony, J. Fletcher, Determination of service loading parts of small inuence on material fatigue, in: Proceedings of the Conference on ResidualFatigue Life and Life Extension of In-Service Structures, JIP 2006, Societe Franc-aise de Metallurgie et de Materiaux, Paris, France, 2006, pp. 251258.

[7] A. Nies"ony, C.M. Sonsino, Comparison of some selected multiaxial fatigue assessment criteria, LBF Report no. FB-234, ISSN 0721-5320, FraunhoferGesellschaft, Germany, 2008.

[8] R.C. Rice, B.N. Leis, H.D. Berns, D.V. Nelson, D. Lingeneser, M.R. Mitchell, Fatigue Design Handbook, SAE, Warrendale, 1988.[9] E. Macha, Fatigue failure criteria for materials under random triaxial state of stress, advances in fracture research, in: S.R. Valluriin, et al. (Eds.),

Proceedings of the Sixth International Conference on Fracture (ICF6), New Delhi, Vol. 3, Pergamon Press, Oxford, 1984, pp. 18951902.[10] E. Macha, Generalization of fatigue fracture criteria for multiaxial sinusoidal loadings in the range of random loadings, in: M.W. Brown, K.J. Miller

(Eds.), Biaxial and Multiaxial Fatigue, EGF 3, Mechanical Engineering Publications, London, 1989, pp. 425436.[11] A. Karolczuk, E. Macha, A review of critical plane orientations in multiaxial fatigue failure criteria of metallic materials, International Journal of

Fracture 134 (2005) 267304.[12] T. agoda, P. Ogonowski, Criteria of multiaxial random fatigue based on stress, strain and energy parameters of damage in the critical plane,

Materialwissenschaft und Werkstofftechnik 36 (2005) 429437.[13] D. Socie, G.B. Marquis, Multiaxial Fatigue, Society of Automotive Engineers, USA, 1999.[14] ASTM E 1049-85 (Reapproved 1997), Standard practices for cycle counting in fatigue analysis, in: Annual Book of ASTM Standards, vol. 03.01, ASTM,

Philadelphia, 1999, pp. 710718.[15] A. Nies"ony, Rain ow counting method, set of functions with user guide for use with MATLAB, /http://www.mathworks.com/matlabcentral/

leexchange/3026S.[16] Signal Processing Toolbox: Users Guide, MathWorks Inc., Ver. 5, 2000.[17] A. Nies"ony, C.M. Sonsino, Calculating the fatigue crack initiation in machine parts under random multiaxial loading, In: F. Breitenecker (Ed.),

COMSOL Multiphysics Konferenz 2006-Neue Wege der Multiphysik Simulation, Gottingen, FEMLAB, 2006, pp. 106111.

ARTICLE IN PRESS

A. Nies!ony / Mechanical Systems and Signal Processing 23 (2009) 27122721 2721

Determination of fragments of multiaxial service loading strongly influencing the fatigue of machine componentsIntroductionMultiaxial fatigue life assessmentRainflow counting

The proposed algorithmExample: hook loaded with two forcesConclusionsAcknowledgmentReferences