hreaded fastener is used as the pin,

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The Effect of User Decisions on the Accuracy ofFatigue Analysis from FEA.

I Mercer, G Malton and J Draper

Safe Technology Limited

Abstract: Fatigue analysis from large FEA models is becoming increasingly common. This paperlooks at four main decision areas processing the loading histories, materials data, mesh densityand choice of analysis algorithm.

1. Introduction

Many fatigue analyses from FE models use an elastic FEA for a unit applied load. The fatiguesoftware uses a description of the service load history to scale the results.The finite element load case will consist of a linear elastic FEA solution for the stresses at eachnode, calculated for a single applied load most conveniently a unit load. These results will bewritten to the FEA results file as a step. At each node, the elastically-calculated stress tensor ismultiplied by the load history to give a time history of the stress tensor.On the surface of the model, the fatigue software will calculate the time histories of the in-planeprincipal stresses, and their directions. The time history of the principal stresses can be convertedinto elastic-plastic stress-strains using a multiaxial cyclic plasticity model. This strain-time historycan be used in a strain-life fatigue calculation and the associated stresses can be used to apply amean stress correction. This procedure is repeated for each node on the model.Components with multiple load directions can be analysed. Each load direction is modelledseparately in the FEA, and the fatigue software uses the principle of superimposition.For some components, the sequence of stresses may be calculated in the FE analysis. For example,an engine crankshaft FE analysis may model the stresses for each 5o of rotation through two orthree complete revolutions of the crankshaft. The fatigue software follows this sequence ofstresses.The accuracy of the fatigue analysis results will depend on (i) the way in which the loadinginformation has been processed, (ii) the materials fatigue data, (iii) the FEA mesh and (iv) thefatigue analysis algorithm. These four items are discussed in this paper.

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2. Load processing

2.1 Sample frequencyThe fatigue analysis requires an accurate description of the peaks and valleys in the load history.Analogue signals must be sampled at an appropriate sample frequency. As an example, a sinewave sampled at four times its frequency could produce many different sets of sampled values.Two possible sets of samples are shown in Figure 1. In the upper example, the amplitude of thesignal has been determined correctly. In the lower example, the amplitude has beenunderestimated, and a fatigue life calculated from these samples would very non-conservative.

Figure 2 shows the effect of sample frequency on the accuracy of the subsequent fatigue analysis.Narrow band and broad band Gaussian random signals were used for this study. A samplefrequency of 100 points per cycle was used as a datum, and the effect of reducing this samplefrequency is shown in Figure 2. It can be seen that sampling at 10 times the signal frequency gavecalculated lives of 1.1 times the true value for a broad band signal, and up to 1.5 times the truevalue for a narrow band signal. A sample frequency of 10 points/cycle is now widely used inindustry, as it offers a reasonable compromise between accuracy of analysis and quantity of data(and hence analysis time).

2.2 Peak-valley extractionMeasured load histories can be truncated by extracting the peaks and valleys from the sampledsignal. Because real signals contain a large number of very small fluctuations, it may beconvenient to omit them during the peak/valley extraction. This process is known as cycleomission, or gating (Figure 3).The cycle omission criterion, or gate level, must be chosen with care. Many materials exhibit anendurance limit stress amplitude under constant amplitude testing. Under variable amplitudeloading the endurance limit may disappear or its amplitude may be very much reduced [Conle,1980],[ DuQuesnay, 1993]. Figure 4 shows a measured strain history from a truck steering arm(upper signal), and the strain history that is produced if all the cycles smaller than the constantamplitude endurance limit are removed. Fatigue testing using the truncated signal producedfatigue lives which were 9 times longer than those produced using the full signal [Kerr, 1992].Peak-valley extraction can also be carried out on multiaxial loading signals. In this case it isnecessary to retain the phase relationship between the signals. To do this, each time a peak orvalley occurs on one signal, the corresponding data points on the other signal are also retained.The principle is illustrated in Figure 5. Gating to omit small cycles can be integrated into thisprocessing operation. The danger in this procedure is illustrated by considering the way thesesignals are used in the fatigue analysis of a node in a finite element model.(a) The unit load stress tensor for each node is multiplied by its corresponding load history, toproduce time histories of each stress tensor.(b) The time histories of the stress tensors are superimposed.(c) The time histories of the principal stresses are calculated.(d) The damage parameter (for example the time history of the shear strains on a critical plane) iscalculated.

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The peak/valley procedure in Figure 5 therefore assumes that a peak or valley in the principalstrains will always coincide with a peak or valley in one of the load histories. In general this is farfrom being true and serious errors in the calculated fatigue lives can be produced by peak-valleyextraction of multiaxial loading histories. The increase in processing speed can be dramatic, butthe potential errors are great. Safe Technologys fe-safe software does not peak-valley multiaxialloading histories unless the user specifically requests it. A sensitivity analysis should always becarried out to asses the effect on the calculated fatigue lives.

2.2 Length of load historiesFigure 6 shows fatigue damage histograms for a fatigue analysis of the first 3 000, 30 000 and 300000 cycles of a long signal. Although the calculated fatigue lives (adjusted for the differentlengths of signal) were very similar, the fatigue damage distribution for the shortest signal isdominated by the largest few cycles. This is a characteristic of short signals. It is possible toobtain quite adequate calculated fatigue lives from relatively short lengths of signals, but theselives are much more dependant of the accuracy of measurement of the few largest cycles, and onthe statistical validity of their frequency of occurrence.

3. Materials data

Fatigue analysis requires the parameters for the relationship between strain amplitude and fatiguelife

(2 ) (2 )2

ff f f

b cN NE

where is the applied strain range2 fN is the endurance in reversals

f is the fatigue strength coefficient

f is the fatigue ductility coefficient

b is the fatigue strength exponent c is the fatigue ductility exponent

and the parameters for the stable cyclic stress-strain curve

1n

E K

where E is the cyclic elastic modulusK is the cyclic strain hardening coefficientn is the cyclic strain hardening exponent

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This data is widely available for many commonly-used steels, aluminium alloys and cast irons(see for example [Boller, 1987]). Where data is not available, approximation algorithms may beused. The Seeger approximation can give acceptable estimates for materials properties. Twoexamples, comparing the estimated and measured properties of a steel and an aluminium alloy, areshown in Figures 7 and 8.

4. FE mesh effects

Fatigue cracks often initiate from the surface of a component. The accuracy of the surface stressestherefore has a significant effect on the accuracy of the subsequent fatigue analysis. [Colquhoun,2000] compared calculated fatigue lives for a forged aluminium suspension component, using apreliminary and a final mesh, and found significant differences (Figure 9). The final meshproduced fatigue lives which correlated very well with the results of a fatigue test of thecomponent with a calculated life to crack initiation of 27 000 miles, compared to a test life of 41000 miles at which quite long fatigue cracks were discovered. A difference of less than 15%between un-averaged and averaged nodal stresses is a reasonable criterion for defining anadequate mesh density for fatigue analysis.In selecting the parameter for analysis, possible options are integration point stresses (Gausspoints), elemental averaged stresses, nodal averaged stresses, or un-averaged nodal stresses.Integration point and elemental averaged stresses do not normally give adequate estimates of thesurface stresses, and are not recommended.With an adequate mesh, there should be little difference in the lives calculated from nodalaveraged stresses, or un-averaged nodal stresses. In practice, mesh density is rarely ideal, andexperience has shown that fatigue lives calculated from un-averaged nodal stresses correlate mostclosely with test results. A recommended method of assessing mesh density is to compare fatiguelife contour plots, calculated from un-averaged nodal stresses, with different amounts of averagingset in the contour plot software.

5. Choice of fatigue analysis method

5.1 Uniaxial fatigueThe use of uniaxial fatigue methods to analyse biaxially stressed components can give veryoptimistic life estimates. In [Devlukia, 1985] a welded steel bracket from a passenger carsubjected to multiaxial loading developed fatigue cracks at a life much shorter than that predictedby uniaxial local strain fatigue analysis. The component had also been tested under two differentservice duties and uniaxial analysis failed to reproduce the relative severity of the two duties.[Bannantine, 1985] reported the following results from a multiaxial fatigue test programme (Table1). The three specimens were (i) simple bending, (ii) in-phase bending and torsion and (iii) axialand torsion loading with random phase relationship. Fatigue life predictions using uniaxialmethods were always non-conservative, with a predictions up to 19 times the achieved test life.

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5.2 Principal stress criterionEarly attempts to analyse biaxial fatigue were based on principal stresses, using a conventional S-

N curve. For a fatigue cycle, the stress range of 1 , or the stress amplitude 1

2

, would be

used with a stress-life curve obtained by testing an axially loaded specimen. The (false)assumption in this procedure is that the fatigue life is always determined by the amplitude of thelargest principal stress 1 , and therefore that the second principal stress 2 has no effect onfatigue life.

Consider a simple circular shaft loaded in pure torsion. If xy is the torsion stress, then theprincipal stresses are :

21,2 xy

i.e. the maximum principal stress is equal to the torsion stress. A fatigue cycle of xy will

produce a principal stress cycle of 1 xy . The use of the principal stresses thereforepredicts that the fatigue strength in torsion is the same as the fatigue strength under axial loading.This is not supported by test data, as Figure 11 shows.Figure 11 shows the results of fatigue tests on a commonly-used steel. It is clear that the torsionfatigue strength is much lower than the axial fatigue strength - the allowable principal stress intorsion is approximately 60% of the allowable axial stress. Calculating fatigue lives usingprincipal stress will clearly be grossly optimistic for torsion loading, and allowable torsion fatiguestresses will be overestimated by a factor of 1/0.6 = 1.66. This could mean the difference betweenidentifying and missing a potential fatigue 'hot spot'. (In 1927, Moore reported that From thequite considerable amount of test data available for fatigue tests in torsion the general statementmay be made that under cycles of reversed torsion the endurance limit for metals ranges from 40per cent to 70 per cent of the endurance limit under cycles of reversed flexure [Moore, 1927]).It has been shown over the past 20 years that principal stresses should only be used for fatigueanalysis of 'brittle' metals, for example cast irons and some very high strength steels. A fatigueanalysis using principal stresses tends to give very unsafe fatigue life predictions for more ductilemetals including most commonly-used steels and aluminium alloys.

5.3 Principal strain criterionThis criterion proposes that fatigue cracks initiate on planes which experience the largestamplitude of principal strain. The standard strain-life equation for unixial stresses is

(2 ) (2 )2

ff f f

b cN NE

where is the applied strain range2 fN is the endurance in reversals

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f is the fatigue strength coefficient

f is the fatigue ductility coefficient

b is the fatigue strength exponent c is the fatigue ductility exponent

Replacing the axial strain with the maximum principal strain gives :

1 (2 ) (2 )2

ff f f

b cN NE

The SAE multiaxial test programme [Tipton, 1989] used a 40mm diameter notched shaft with5mm fillet radii, machined from SAE1045 steel. The specimens were tested under pure bendingloads, pure torsion loads, and combined bending-torsion with various proportions of bending andtorsion. The test results have been compared with life estimates made from measured strains at thenotch. The maximum principal strain criterion produced life estimates which were non-conservative, particularly at lower values of endurance, and the scatter was large (Figure 12).Experience has shown that this criterion should be used only for fatigue analysis of brittle metals,for example as cast irons and some very high strength steels.

5.4 von Mises Equivalent StrainBecause the von Mises criterion provides an estimate of the onset of yielding, it has beenproposed as a criterion for fatigue life estimation.The strain-life equation in terms of von Mises equivalent strain is

(2 ) (2 )2

fEFFf f f

b cN NE

The von Mises equivalent strain, calculated from principal strains, is

0.52 2 2

1 2 2 3 3 1EFF

The value of is chosen so that EFF has the same value as the principal strain 1 for theuniaxial stress condition.For design analysis based on stresses, at high endurance where the plastic component is small, thevon Mises equivalent stress is

0.52 2 2

1 2 2 3 3 112EFF

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and fatigue lives could be calculated using (2 )2EFF

f fbN

or using EFF with a conventional S-N curve.

A major problem with the practical application of von Mises criteria to measured signals is thatthe von Mises stress or strain is always positive, even for negative values of stress or strain, and soRainflow cycle counting cannot be applied directly. Some approximations have been proposed,such as to assign the sign of the largest stress or strain to the von Mises stress or strain, oralternatively to assign the sign of the hydrostatic stress or strain to the von Mises stress or strain.These are termed signed von Mises criteria. The different methods of determining the sign cangive significantly different life estimates.The von Mises criteria correlate poorly with test data, particularly for biaxial stresses when thetwo in-plane principal stresses change their orientation during the fatigue loading.

5.5 Brown-Miller criterion.The Brown-Miller equation proposes that the maximum fatigue damage occurs on the plane whichexperiences the maximum shear strain amplitude, and that the damage is a function of both thisshear strain max and the strain normal to this plane, N

max 1.65 (2 ) 1.75 (2 )2 2

fNf f f

b cN NE

This formulation of the Brown-Miller parameter was developed by Kandil, Brown and Miller[Kandil, 1982].The Brown-Miller criterion is attractive because it uses standard uniaxial materials properties.Figure 13 shows the results from the SAE test programme [Tipton, 1989]. In general, test resultsand predictions agreed to within a factor of 3. The Brown-Miller criterion is widely acceptedfor the analysis of most metals with the exception of very brittle metals such as cast irons.More recently, Chu, Conle and Bonnen [Chu, 1993] have shown improved correlation if the meanshear stress is included, and have proposed the following extension to the Brown-Miller equation,using a mean stress correction similar to a Smith-Watson-Topper correction

2

max ,max21.02 (2 ) 1.04 (2 )

2 2fN

N f f f fb b cN N

E

where max is the maximum shear stress

and , maxN is the maximum normal stress.

Again, this equation uses standard uniaxial materials properties.

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Varvani-Farahani has further extended the Brown-Miller equation, by weighting the contributionof the normal and shear stress/strains using the axial and torsion fatigue strength coefficients.[Varvani-Farahani, 2000], [Varvani-Farahani, 2003].

,

maxmax

11 (2 )

2

N m

fN N f

f f f f

f N

where ,N m is the mean value of the normal stress on the critical plane

max is the range of maximum shear stress on the critical plane

max is the range of maximum shear strain on the critical plane

N is the range of normal stress on the critical plane

N is the range of normal strain on the critical plane

This equation has shown excellent correlation for constant amplitude loading where the two in-plane principal stresses have the same amplitude but different frequencies, and it is being assessedfor random loading. However, it requires both axial and torsion fatigue test data.

6. Concluding remarks

This paper has given some guidelines to be followed when planning a fatigue analysis of a finiteelement model. Many of the guidelines are set as defaults in fe-safe, allowing engineers withrelatively little fatigue experience to carry out successful analyses.Processing speeds are also impressive. To give two examples: the fatigue analysis in fe-safe of a700 000 element model (4-noded solid elements) containing two load steps, in a 3 GByte file,took 35 minutes on a UNIX workstation. For an 8 GByte FEA results file containing 36 loadsteps, the total fe-safe time for read-in, fatigue analysis of the 36 load steps in sequence, andexport of results, took 1 hour 15 minutes on a PC running Windows.

7. References

Bannantine J A, Socie D F. A variable amplitude multiaxial fatigue life prediction method.Fatigue under biaxial and multiaxial loading, Proc. Third International Conference onBiaxial/Multiaxial Fatigue, Stuttgart, 1989. EISI Publication 10, MEP, London.

Chu C-C, Conle F A and Bonnen J F. Multiaxial stress-strain modelling and fatigue lifeprediction of SAE axle shafts. American Society for Testing and Materials, ASTM STP 1191,1993 pp 37-54

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Colquhoun C, Draper J. Fatigue analysis of an FEA model of a suspension component, andcomparison with experimental data. Proc. NAFEMS Conference 'Fatigue analysis from finiteelement models', Wiesbaden, November 2000.

Conle A and Topper T.H. Overstrain effects during variable amplitude service history testing.International Journal of Fatigue, Vol 2, No.3, pp130-136, 1980

Devlukia J, Davies J. Fatigue analysis of a vehicle structural component under biaxial loading.Biaxial Fatigue Conference, Sheffield University, Dec 1985

DuQuesnay D.L, Pompetzki M.A, Topper T.H. Fatigue life prediction for variable amplitudestrain histories. SAE Paper 930400, Society of Automotive Engineers

Kandil F A, Brown M W, Miller K J. Biaxial low cycle fatigue fracture of 316 stainless steel atelevated temperatures. Book 280, The Metals Society, London, 1982

Kerr W. 1992. Final year undergraduate project. Unpublished

Moore H F. Manual Of Endurance Of Metals Under Repeated Stress. Engineering FoundationPublication Number 13, 1927.

Morton K, Musiol C, Draper J. Local stress-strain analysis as a practical engineering tool.Proc. SEECO 83 Digital Techniques in Fatigue. City University, London 1983. Society ofEnvironmental Engineers

Boller CHR, Seeger T. Materials Data for Cyclic Loading. Elsevier Materials ScienceMonographs, 1987 (5 volumes).

Tipton S M, Fash J W. Multiaxial fatigue life predictions for the SAE specimen using strainbased approaches. Multiaxial Fatigue: Analysis and Experiments, SAE AE-14, 1989

Varvani-Farahani A and Topper TH. A new energy-based multiaxial fatigue parameter.Fatigue 2000: Fatigue and Durability Assessment of Materials, Components and Structures.4th International Conference of the Engineering Integrity Society, Cambridge UK. pp313-322.Bache MR, Blackmore PA, Draper J, Edwards JH, Roberts P, Yates JR (eds.). EMAS 2000.

Varvani-Farahani A. Critical plane-energy based approach for assessment of biaxial fatiguedamage where the stress-time axes are at different frequencies . 6th International Conference onBiaxial/Multiaxial Fatigue and Fracture, Lisbon, Portugal, 2001. Carpinteri A, de Freitas M,Spagnoli A (eds.). pp203-221. ESIS Publication 31, Elsevier 2003.

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Table 1. Uniaxial fatigue life predictions for various multiaxial conditions.(Lives are repeats of the test signal).

TEST LIFE PREDICTIONUNIAXIAL FATIGUE

600 5000

200 450

4000 530001700 300001000 19000

Figure 1. Possible samples from a signal sampled at four times the signalfrequency

(i)

(ii)

(iii)

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Points/cycle

Relativelife

Narrow band

Broad band

Figure 2. Effect of sampling frequency on fatigue life estimation (Narrow banddata from [Morton,1983])

Figure 3. Measured signal (top) and the same signal after peak-valley and cycleomission (bottom)

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Figure 4. Measured truck steering arm loading (top) and the same signal afteromitting cycles below the endurance limit (bottom)

Figure 5. Multi-channel peak-valley

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Figure 6 Fatigue damage histograms from the first 3000 (top), 30000 (centre) and300000 cycles of a long load history

Damage

0

0.0002

0.0004

0.0006

0.0008

0.001

0.0012

0.0014

Mean:uERange:uE

0721

14432164

2885

-1417-703

11726

1440 Mean ()Range ()

Damage

0

0.00002

0.00004

0.00006

0.00008

0.0001

0.00012

0.00014

0.00016

Mean:uERange:uE

0703

14062109

2812

-1344-648

48744

1440 Mean ()Range ()

Damage

0

0.000005

0.00001

0.000015

0.00002

0.000025

0.00003

Mean:uERange:uE

0605

12091814

2419

-1167-568

30629

1228 Mean ()Range ()

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Figure 7. Actual and approximated strain-life curves (left), and cyclic stress-strainand hysteresis curves (right) for SAE 1005 steel.

Figure 8. Actual and approximated strain-life curves (left), and cyclic stress-straincurves (right) for 2014-T6 aluminium alloy.

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Figure 9. Effect of mesh refinement on calculated fatigue lives

Figure 10. Principal stresses for a shaft under axial load and torsion load

0

10

20

30

40

50

60

1.0E+04 1.0E+06 1.0E+08 1.0E+10 1.0E+12 1.0E+14 1.0E+16

Predicted Fatigue Life (repeats).

Appl

ied

% o

f Ser

vice

Loa

d

Standard Mesh - Max Shear Strain

Refined Mesh - Max Shear Strain

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Figure 11. Stress-life curves for axial and torsion loading

Figure 12. SAE notched shaft test results, principal strain theory

100

1000

1.0E+04 1.0E+05 1.0E+06 1.0E+07

Cycles

Stre

ss A

mpl

itude

MPa

Axial stress

Torsional stress

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Figure 13. SAE notched shaft, Brown-Miller parameter