Present standards for bearing life cal-culation are based on work carriedout at SKF in 1947 by Gustaf Lund-

berg and Arvid Palmgren. The life equationwas formulated using the Weibull probabil-ity theory of fatigue developed in 1936.This allowed the calculation to includebearing reliability. It marked a substantialstep forward in the achievement of predic-tive methods for the applications of thisessential machine element.

The ability to estimate the life of a bear-ing and to select, on a rational basis, a spe-cific bearing suitable for a particular appli-cation was a breakthrough for engineeringdesign. Since then, science and technology,in particular tribology, have made tremen-dous strides. The latest theoretical knowl-edge and calculation techniques supportedby powerful modern computers are nowapplied in bearing technology and designwork. Also, steel manufacturers can makecleaner steels to more accurate and consis-tent formulations. In addition, greatlyimproved lubricants allow better separa-tion than before of moving contacting sur-faces in rolling bearings, while modernmass production manufacturing methods

produce high precision bearings to ever-higher levels of quality.

Such technological progress requiresthat bearing life calculations take intoaccount the improved performance oftoday’s bearings. Over the years, perfor-mance improvements were accounted forlargely through increased dynamic loadratings, which yielded longer calculatedlives. Application or adjustment factorswere sometimes used to account for opera-tional conditions, such as special bearingmaterials and lubricants. Until recently,these life calculations served industry suffi-ciently well.

However, observations about the pre-dominant failure mechanisms in modernrolling bearings have added new knowl-edge in this field. It has been found thatfatigue failures are more frequently initi-ated at the surface rather than from cracksformed beneath the surface. Consequently,attention has been paid to the influence ofsurface finish and contamination on bear-ing performance. The existence of a fatiguelimit for modern bearing steels has alsobeen observed. Such considerations haveresulted in the publication of improved

bearing life calculations and papers study-ing the effects of contamination and sur-face finish on bearing life.

In particular, the rolling contact fatigue lifemodel published by Eustathios Ioannides andTedric Harris (1985) introduced two innova-tions to the Lundgren and Palmgren theory:

i) a threshold value of the local stress, afatigue limit, below which no fatigue isexpected to occur in the bearing,

ii) consideration of the stressed volumebelow a contact as an array of relativelysmall volume elements, each experienc-ing individual local stresses.

In this way, real local stresses and manyeffects stemming from surface stress con-centrations, such as edge stresses and con-tamination dents, can be introduced consis-tently into fatigue life predictions. Such acomprehensive approach requires access tocomplex computer programs for bearing

1 /0 1 evolution.skf .com E V O L U T I O N I 2 5

technology

The SKF formula forrolling bearing lifeThe demands for safe design and lighter, competitively

priced products have put a new emphasis on predictability

of rolling bearing performance. This has prompted

researchers to update standards for bearing life to take

into account progress in engineering technology.

by EUSTATHIOS IOANNIDES,SKF Engineering and Research Centre (ERC), Nieuwegein,

the Netherlands & Imperial College of Science,Technology and Medicine, London, UK;

GUNNAR BERGLING, AB SKF, Gothenburg, Sweden, and

ANTONIO GABELLI, SKF Engineering and Research Centre (ERC), Nieuwegein, the Netherlands.

Fig.1 Stress Life Factor for radial

ball bearings.

κ=4

1

0.6

0.45

0.30

0.15

0.1

0.005 0.01 0.02 0.05 0.1 0.2 0.5 1 2 5

50

20

10

5

2

1

0.5

0.2

0.1

0.05

ηc Pu/P

aS

KF

life calculations, as integration of the fail-ure risk over complicated stress fields isrequired. Many bearing users do not haveaccess to such resources. Instead they usu-ally apply simple formulae readily avail-able in bearing manufacturers’cataloguesor standards.

To support these needs, and reap thebenefits provided by advances in technol-ogy and knowledge, the SKF Life Equa-tion* was made available. This equationwas developed by introducing an effectivereal weighted stress that is averagedthroughout the risk volume. The fatiguelimit and the local stress were implementedinto equations similar to those used byLundberg-Palmgren. In practice, this isdone by introducing a stress factor that actson the fatigue limit of the bearing. This fac-tor accounts for the actual stress conditionof real bearings but is not included with theideal Hertzian stress field, as used in theLundberg-Palmgren classical analysis.This approach led to a simple expressionthat can be readily calculated in a similarway to the ISO 281 (1990) formulation.This equation retains much of the original

simplicity of the load-life equation but alsodescribes the performance of modern bear-ings more accurately than before.

Standardised life equations

Fatigue life modelling is central to all bear-ing life prediction. The traditional crackinitiation or cumulative damage modelinvokes a stress power law to account forthe portion of the life spent in the initiationof a crack, which dominates the completelifetime of rolling contacts. Probabilisticmethods were applied early in bearingendurance prediction because of the nat-ural dispersion of bearing life. Lundbergand Palmgren (1947) used the Weibull(1939) probability distribution of metalfatigue to establish the basic theory of thestochastic dispersion of bearing lives, asshown by the equation below:

c1 ~ τ0ln ----~ Ne ------az0l (1)S z h

0

By substituting the Hertzian contact para-meters (in terms of the applied load andcontact geometry), they obtained the load-life relationship for rolling bearings. This

evolution.skf .com 1 /0 1

List of symbols:

a Contact semi-axis in trans-verse direction, [m]

a1, a2, a3 Life adjustment factors:1 reliability, 2 material,3 operating conditions

a23, aSKF Life adjustment factor, SKFStress Life Factor

A Constant of proportionality,scaling factor

c Exponent in the stress-lifeequation

C Basic dynamic load rating, [N] dm Bearing pitch diameter, [m]e Weibull exponenth Exponent in the stress-life

equationl Length of raceway contact, [m]L Life, [Mrevs]L10 Basic rating life,

(10 % failure life), [Mrevs]L10aa Adjusted rating life according

to SKF Life Theory, [Mrevs]Ln Bearing life at (100-n) %

reliability, [Mrevs]Lna Adjusted rating life, [Mrevs]N Number of load cycles,

number of tests setsp Exponent in life equation P Equivalent dynamic bearing

load, [N]Pu Fatigue load limit, [N]S Survival probability [%] w Exponent in the load-stress

relationshipzo Depth of max. orthogonal shear

stress of Hertzian contact, [m]βcc Lubricant cleanliness degree

according to ISO 4406η Life factor for added stressηb Lubrication factorηc Contamination factorκ Viscosity ratio v,v1 Actual and required kinematic

viscosity at the operating tem-perature, [m2/s]

τ0 Maximum orthogonal shearstress amplitude in Hertziancontact, [Pa]

τu Fatigue limit shear stress, [Pa]

2 6 I E V O L U T I O N

Fig.2 Diagrams for calculating the contamination factor for oil circulation systems.

0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

1.0

0.8

0.6

0.4

0.2

0

1.0

0.8

0.6

0.4

0.2

00 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

1.0

0.8

0.6

0.4

0.2

0

κ

ηc

ηc

ηcκ

κ

20001000

500200100

5025

dmmm

20001000

500200100

5025

dmmm

20001000

500200100

5025

dmmm

Filter

β6 = 200

ISO 13/10

Filter

β12 = 200

ISO 15/12

Filter

β25 >–75

ISO 17/14

can be expressed in its final form in a verysimple way:

Cp

L10 = ------- (2)P

where C is the bearing basic dynamic loadrating, a factor that depends on the bearinggeometry, and P is the equivalent bearingload. The exponent p is 3 for ball bearingsand 10/3 for roller bearings. Equation 2was adopted by ISO in the recommenda-tion R281-(1962). In 1977, the multiplica-tive constants a1, a2 and a3 were intro-duced to account for different levels of reli-ability, material fatigue properties andlubrication, resulting in the standard usedtoday, ISO 281 (1990):

Cp

Lna = a1a2a3 ------- a1 = 1 if n =10 (3)P

Many bearing manufacturers, who haverecognised the interrelationship of materialand lubrication effects, use it as

Cp

Lna = a1a23 ------- a1 = 1 if n =10 (4)P

SKF life formula

A simple analytical formulation for bear-ing life that includes consideration of thefatigue strength of the material evolvedfrom the rolling contact fatigue model setforth by Ioannides and Harris (1985). Thiswas initially applied in the numerical solu-tions of the fatigue risk of 3-D stress fieldsof rolling contacts. Due to the self-similar-ity of Hertzian stress fields, it was foundthat in the case of strictly Hertzian con-tacts, simplification was possible. Thiscould be introduced by applying stress cri-teria based on the maximum alternatingshear stress amplitude τ0 above a thresholdvalue and of a fatigue limit stress τu of therolling contact*. Thus the introduction ofthe fatigue limit stress τu can be effectedthrough the replacement of the shear stressamplitude τ0 of the Hertzian stress field ofequation 1, by the difference τ0 – τu . Thenequation 1 becomes:

c1 ~τ0 –τu τu τ0ln ----~ Ne ------------- az0l = 1– ---- Ne ----- az0lS z h τ0 z h

0 0

(5)

In the above formulation, Macauleybracket notation is used; in other words, theterm is set to zero if the quantity enclosedis negative. To obtain the Life Equation forrolling bearings in the presence of a mater-

ial fatigue limit, one may use equation 5.This equation differs from the originalLundberg and Palmgren form (equation 1)only by the Macauley bracket on the rightside of equation 5. It is, therefore, possibleto work through the bearing capacity for-mulation very much as in Lundberg andPalmgren’s original work. Applying thewell-known relationship between inducedstresses and applied load for Hertzian con-tacts, either point or line, as done by Lund-berg and Palmgren (1947), an equationcorresponding to equation 5, but written interms of the equivalent load and the basicdynamic load rating, is obtained. For 10percent probability of failure, the corre-sponding bearing life is expressed as*:

c– ––Pu

w eC

p

L10aa = A 1 – η ------- -------- (6)P P

where the η parameter is a factor (corre-sponding to a stress factor) that is intro-duced to account for the actual stresses pre-sent in real bearing contacts. This parame-ter is not included with the ideal Hertzianstress field used in the derivation of theoriginal Lundberg and Palmgren (1947)life formula. It can be shown that, besidesthe bulk stresses due to the heat treatmentand mounting, the stress factor η dependsto a large extent on the lubrication condi-tion of the contact and on micro-scalestress concentrations due to denting orimperfections. Consequently, η is given asa function of the environmental conditions,(lubrication and contamination), and inrelation to the bearing size.

η (κ,dm,βcc) = ηb (κ)brg ηc (κ,dm,βcc) (7)

introducing equation 7 in equation 6 we get:

L10aa = A 1 – ηb (κ)brgc– ––

Puw e

C p

ηc (κ,dm,βcc) ------- ----- (8)P P

To simplify the use of the above equation,in 1989 SKF introduced a Stress Life Fac-tor aSKF for each specific bearing (brg)class: radial ball bearings, radial rollerbearings, thrust ball bearings, thrust rollerbearings. This factor can be calculated andplotted as shown in figure 1:

PuaSKF κ,ηc ------- =P brg

c– ––Pu

w e

= A 1 – ηb (κ)brg ηc(κ,dm,βcc)------P

(9)Furthermore curves of the functionηc (κ,dm, βcc) were also derived, asexplained by Bergling and Ioannides(1994) and Ioannides and his colleagues(1999). Examples of these parametriccurves are given in figure 2. With thisinformation, the life equation 8 can bewritten in a simple way:

Pu Cp

L10aa = aSKF κ,ηc ------- ----- (10)P brg P

This is the formulation of the life equationused in the SKF General Catalogue since1989. This equation, in conjunction withthe diagrams of figures 1 and 2, can beused to calculate the life of rolling bearingsin a straightforward manner that is similarto the previous life calculation,ISO 281:1990. However, equation 10 canaccount for specific lubrication and con-tamination conditions of the bearing andfor the increased life that is experienced incase of lightly loaded, clean and well lubri-cated bearing applications.

Experimental verification

To verify the accuracy of the SKF BearingLife Equation (equation 10), extensivebearing endurance tests were performed onmore than 8,000 bearings under a varietyof lubrication and contamination test con-ditions. An overview of the mean L10 mea-

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technology

Fig.3 Overview of experimental

L10 bearing lives from

endurance testing.

Experimental bearing lifeBasic rating life L10Adjusted rating life,a23=2.5Adjusted rating life,a23=0.1

0.1 0.2 0.3 0.4 0.5 0.6

10 000

1000

100

10

1

0.1

Bea

rin

g li

fe L

10[M

revs

]

Bearing load P/C

■

a23=2.5

a23=0.1

sured in these tests is shown in figure 3. A comparison was made between pre-dicted lives and the actual endurance testdata of figure 3. This was performed byplotting the ratio between the experimentalbearing life and the basic rating life of thebearing, i.e. aexp=L10exp/L10 versus thecorresponding stress factor parameter,ηc Pu/P.

In figure 4, a group of experimentalpoints of ball bearing operating with a fulloil film is shown. The data points of the testresults mostly overlap with the correspond-ing life-ratio curve, i.e. aSKF , calculatedaccording to equation 9. Figure 4 showsvery good agreement between the relativelife calculated with the SKF Life Equationand the endurance test results. A morecomprehensive evaluation of the experi-mental data also indicated that the totalbias in life calculation could be reduced byhalf using the SKF Life Equation in com-parison to the results of life calculationsbased only on bearing standard ratings.This improved accuracy provides a bettermatch between theory and test data, vali-dating the choice of the model constantsused and the present Life Equation. Fur-thermore, the asymptotic trend displayedby the function aSKF (κ, ηc Pu/P), figures1 and 4 as the stress factor parameterηc Pu/P tends towards lower stress con-ditions for the bearing, and is a further indi-cation of the postulated fatigue stress limitin rolling contact fatigue. The behaviour ofthe curve aSKF (relative stress vs. relativelife) is indeed similar to the familiar Wöh-ler (stress vs. number of stress cycles)curves used for plotting the survival proba-bility of specimens subjected to fatiguetesting at different stress level.

Conclusions

Rolling bearings have made it possible tosupport heavy loads at high rotational speedswith good reliability and a minimum offriction. This basic technology has allowedthe massive mechanisation process that hascharacterised the past century and has pro-foundly revolutionised our way of life.Practical tools for better selection and useof rolling bearings have a significant effecton the way machines operate, their effi-ciency and cost. This affects each of usthrough the energy consumption and run-ning costs of society at large. The SKF LifeEquation introduces a new and higher stan-dard in life calculations to help in the pre-

diction of bearing performance and torespond to the continual quest for betterways to design, select and use bearings.

The SKF Life Equation manages todescribe the complex tribo-system inwhich the bearing operates using a few keyparameters. This is an important feature of

the model that concentrates on the effect ofa few important factors that have signifi-cant consequences for the bearing life.Furthermore, the SKF Life Equationrecognises that the fatigue effect arisingfrom these factors cannot be written as alinear superposition of the risks inducedfrom individual stress components. Ratherthan attempting to derive independent lifemodifying factors for lubrication, contami-nation etc., a single multidimensional fac-tor, aSKF = f (κ, ηc Pu/P), is introduceddepending on those relevant factorsdescribing the tribo-physical state of thesystem. By this approach, the best use ismade of the available input data, to the ben-efit of both the designer and the user of amachine. The good agreement with theexperimental results and improved accu-racy achieved using the SKF Life Equationsupport the use of this approach for the cal-culation of the life expectancy of modernrolling bearings. ■■

* Ioannides E., Bergling G. and Gabelli A.“An analytical formulation for the life of

rolling bearings.” Acta Polytechnica Scandinavica, ME 137, Espoo (1999).

2 8 I E V O L U T I O N evolution.skf .com 1 /0 1

Summary Rolling bearing design has benefited from improved materials technology,

especially steel cleanliness and better understanding of the tribo-physical

conditions that prolong operational life.

Users of bearings have put an increasing emphasis on safe design coupled

with lighter,more competitively priced products.

This has put increasing pressure on bearing manufacturers to offer pre-

dictable performance in rolling bearings.In turn,this has encouraged

researchers to review and update standards for bearing life to take into

account the tremendous progress in engineering technology.As a conse-

quence,SKF has introduced the latest manifestation of its famous formula for

predicting rolling bearing life expectancy.

Fig.4 Comparison of the present SKF Life

Equation with actual endurance test results

at full oil film (κκ = 4).

Experimental relative life,aexp = L10exp / L10

New SKF Equation relative life,aSKF = L10aa / L10

0.0001 0.001 0.01 0.1

10

1

0.1

ηc Pu/P

Rel

ativ

e lif

e:a

exp

,aSK

F

■