Brian L. Vlcek

Georgia Southern University, Statesboro, Georgia

Erwin V. Zaretsky

Glenn Research Center, Cleveland, Ohio

Rolling-Element Fatigue Testing and

Data Analysis—A Tutorial

NASA/TM—2011-216098

March 2011
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Brian L. Vlcek

Georgia Southern University, Statesboro, Georgia

Erwin V. Zaretsky

Glenn Research Center, Cleveland, Ohio

Rolling-Element Fatigue Testing and

Data Analysis—A Tutorial

NASA/TM—2011-216098

March 2011

National Aeronautics and

Space Administration

Glenn Research Center

Cleveland, Ohio 44135

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NASA/TM—2011-216098 1 

Rolling-Element Fatigue Testing and Data Analysis—A Tutorial 

Brian L. Vlcek 

Georgia Southern University 

Statesboro, Georgia 30458 

Erwin V. Zaretsky*

National Aeronautics and Space Administration 

Glenn Research Center 

Cleveland, Ohio 44135 

Abstract 

In order to rank bearing materials, lubricants and other design variables using rolling-element bench 

type fatigue testing of bearing components and full-scale rolling-element bearing tests, the investigator 

needs to be cognizant of the variables that affect rolling-element fatigue life and be able to maintain and 

control them within an acceptable experimental tolerance. Once these variables are controlled, the number 

of tests and the test conditions must be specified to assure reasonable statistical certainty of the final 

results. There is a reasonable correlation between the results from elemental test rigs with those results 

obtained with full-scale bearings. Using the statistical methods of W. Weibull and L. Johnson, the 

minimum number of tests required can be determined. This paper brings together and discusses the 

technical aspects of rolling-element fatigue testing and data analysis as well as making recommendations 

to assure quality and reliable testing of rolling-element specimens and full-scale rolling-element bearings. 

Nomenclature 

A area due to plastic deformation and wear, m2, (in.2) 

C confidence number, fractional percent or percent 

e Weibull slope or Weibull modulus 

F probability of failure (1 – S), fractional percent 

H depth of stressed surface due to plastic deformation and wear, maximum distance from chord to 

contact surface, m, (in.) 

h' distance from chord to original surface, m, (in.) 

j sequential failure number where j = 1,2,3,...r

L life, revolutions, stress cycles or hours. 

L10 10-percent life or life at which 90-percent of population survives, revolutions, stress cycles or hours 

L50 median life or life at which fifty percent of a population fails, revolutions, stress cycles or hours 

Lβ characteristic life or life at which 63.2 percent of population fails, revolutions, stress cycles or hours 

L location parameter or life below which no failures will occur, revolutions, stress cycles or hours 

l 1/2 chord length, m, (in.) 

m total number of bearings or specimens in a subgroup, number of bearing testers 

n number of data sets, test series, or samples 

Pt thrust load, N, (lb) 

Pr radial load, N, (lb) 

R original radius or curvature, m, (in.) 

 

*

Distinguished Research Associate. 
NASA/TM—2011-216098 2 

Rp profile radius, m, (in.) 

r number of failures or number of subgroups 

S probability of survival (1 – F), fractional percent 

S90 90-percent probability of survival, fractional percent 

ω rotational speed, rpm 

Subscripts: 

ir inner race 

or outer race 

re rolling element 

x probability of survival 

Introduction 

While there can be multiple failure modes of rolling-element bearings, the ultimate failure mode 

limiting bearing life is contact (rolling-element) surface fatigue of one or more of the running tracks of 

the bearing components. Rolling-element fatigue is extremely variable but is statistically predictable 

depending on the material (steel) type, the processing, the manufacturing, and operating conditions 

(Ref. 1). Sadeghi, et al. (Ref. 2) provide an excellent review of this failure mode. Alley and Neu (Ref. 3) 

provide a recent attempt at modeling rolling-element fatigue. With improved manufacturing and material 

processing, the potential improvement in bearing life can be as much as 80 times that attainable in the late 

1950s or as much as 400 times that attainable in 1940 (Ref. 4). 

In 1896, R. Stribeck (Ref. 5) in Germany began fatigue testing full-scale rolling-element bearings. 

Rolling-element fatigue life analysis is based on the initiation or first evidence of fatigue spalling on a 

loaded, contacting surface of a bearing. This spalling phenomenon is load cycle dependent. Generally, the 

spall begins in the region of maximum shearing stresses, located below the contact surface, and 

propagates into a crack network. Failures other than that caused by classical rolling-element fatigue are 

considered avoidable if the component is designed, handled, and installed properly and is not overloaded 

(Ref. 1). However, under low elastohydrodynamic (EHD) lubricant film thickness conditions, rollingelement fatigue can be surface or near-surface initiated with the spall propagating into the region of 

maximum shearing stresses. C.A Moyer and E.V. Zaretsky discuss in detail “failure modes related to 

bearing life” in Reference 1. 

The database for ball and roller bearing fatigue testing is extensive. A concern that arises from these 

data and their analysis is the variation between life calculations and the actual endurance characteristics of 

these components. Experience has shown that endurance tests of groups of identical bearings under 

identical conditions can produce a variation in life at a 90-percent probability of survival (L10 life) from 

group to group. If a number of apparently identical bearings are tested to fatigue at a specific load, there is 

a wide dispersion of life among these bearings. For a group of 30 or more bearings, the ratio of the 

longest to the shortest life may be 20 or more (Ref. 1). This variation can exceed reasonable engineering 

expectations. 

Variables affecting rolling-element fatigue have been experimentally studied and documented by 

many investigators for over 100 years. In 1963, J.E. Moult (Ref. 6) discussed the then known variables 

that affect rolling-element fatigue life. He concluded that, “Qualitative comparisons of bearing 

performance require test bearing and test conditions to be statistically equivalent at all stages of 

processing, including the material, except for the variable being compared. Quantitative comparisons 

have the additional requirement that the results must be related to the performance of a control bearing 

sample in combination with a control lubricant.” A compilation of those variables that effect rollingelement fatigue life were codified in Reference 7 as ASME life modifying factors to the then existing ISO 

 
NASA/TM—2011-216098 3 

and ANSI/ABMA rolling bearing life prediction standards. Later these variables were expanded by the 

STLE to become more inclusive (Ref. 1). The life modifying factors were updated from those of 

Reference 7 to reflect the then existing data base (Ref. 1). 

In order to rank bearing materials, lubricants and design variables using rolling-element bench type 

fatigue testing and full-scale rolling-element bearing tests, the investigator needs to be aware of the 

variables that effect rolling-element fatigue life, and to be able to maintain and control them within an 

acceptable experimental tolerance. Once these variables are controlled, the number of tests and the test 

conditions need to be specified to assure reasonable statistical certainty of the final results. Using the 

statistical methods of W. Weibull (Refs. 8 and 9) and L. Johnson (Ref. 10), the minimum number of tests 

required can be determined. 

There is reasonable correlation between the results from elemental test rigs with those results 

obtained with full-scale bearings. It is the objective of the work reported herein to bring together and 

discuss the technical aspects of rolling-element fatigue testing as well as making recommendations to 

assure quality and reliable testing of rolling-element specimens and full-scale rolling-element bearings. 

Rolling-Element Fatigue Testing 

Bench-Type Fatigue Testers 

The determination of the rolling-element fatigue life of a bearing made from a particular steel or the 

effect of a specific lubricant on fatigue life is an expensive undertaking. Very high scatter or dispersion in 

fatigue life makes it necessary to test a large number of specimens. The cost is high both in money and 

time. This is borne out by the fact that a high-quality aircraft bearing can cost upwards of several 

thousand dollars each and run for several thousand hours. Bearing companies and various research 

laboratories have pioneered the use of bench type rolling-element fatigue testers which can simulate to 

various degrees the conditions found under full-scale bearing operation. Generally, these test rigs perform 

accelerated rolling-element fatigue testing of element specimens such as a ball or a roller at maximum 

Hertz (compressive) stress levels beyond 4.14 GPa (600 ksi). 

The results obtained with the bench type testers have been used to indicate trends and to rank 

materials and lubricants. The vast majority of published data have been obtained in bench type fatigue 

testers. The results from these testers qualitatively, although not necessarily quantitatively, compare with 

test results from full-scale bearing tests (Ref. 11). 

The best compendium of rolling-element fatigue testing and types of bench type rolling-element 

fatigue testers was that compiled by J.J.C. Hoo (Ref. 12) for the American Society for Testing Materials 

(ASTM). The more commonly used test rigs discussed in Reference 12 are (a) Two-disk Machine or 

Ring-to Ring Rolling-Contact Fatigue Tester, (b) Barwell Four-Ball Fatigue Tester, (c) NASA Spin Rig, 

(d) Pratt-Whitney One-Ball Fatigue Tester (Fig. 1), (e) General Electric R-C (Rolling-Contact) Tester 

(Fig. 2), (f) NASA Five-Ball Fatigue Tester (Fig. 3), (g) Unisteel Flat Washer Fatigue Tester, (h) FederalMogul Ball-Rod Rolling-Contact Fatigue (RCF) Tester, (i) NTN Cylinder-to-Ball Rolling-Contact 

Fatigue Tester, and (j) NTN Cylinder-to-Cylinder Rolling Contact Fatigue Tester. These test rigs employ 

an automatic failure detection and shutdown system. 

Lubrication mode for the test specimen depends on the type of test rig and the operating conditions. 

The lubrication mode can be by oil mist, oil jet (recirculating oil system), oil drip, or oil bath. Oncethrough, oil-mist lubrication for these types of tests appears to be the most efficient lubrication mode. 

After each test the oil should be discarded and not reused. However, a sample of oil from each test should 

be saved and cataloged for later laboratory analysis, if necessary. 
NASA/TM—2011-216098 4 
NASA/TM—2011-216098 5 

Test Procedures for Bench Type Testers 

When testing in a bench-type rig, various precautions must be taken in order to obtain valid 

comparison of the lubricant, material, and/or operating variables. The procedure as outlined is that 

followed by the authors and other investigators who have conducted a considerable number of rollingelement fatigue studies. It is mandatory, first of all, to have all the rolling-element specimens fabricated 

from a single heat of material and heat treated simultaneously to the same hardness. All specimens must 

have the same surface finish unless, of course, the surface finish is the parameter being studied. The steel 

microstructure should be reported together with the retained austenite and residual stress before and after 

testing. It is of prime importance that the difference in hardness between the contacting test specimens 

remains constant. 

Prior to test, all contacting elements should be cleaned thoroughly with a solvent and wiped dry with 

a clean cloth. Subsequently, the mating elements should be coated with the test lubricant and installed in 

the tester. 

Lubricant flow into the test assembly should be monitored. In addition, the specimens should be 

loaded prior to start-up. If no load is placed on the specimens prior to starting the test, damage can result 

due to skidding of the contacting surfaces. Speed and test temperature should be periodically monitored 

during the testing. Subsequent to testing, all specimens should be examined and their condition recorded. 

It has been the authors’ experience that 30 specimens should be run for a given series of tests anticipating 

that at least one-third to one-half of the specimens will fail. Care must be taken to make sure that, in any 

series of tests in which material parameters are compared, all test are conducted with the same lubricant 

formulation obtained from the same lubricant batch. The same lubricant obtained from different batches 

can give significantly different results. In testers where mating elements are not replaced subsequent to 

test, the elements must be examined to make sure that they have not been damaged during prior operation. 
NASA/TM—2011-216098 6 

Where elevated temperature tests are to be conducted it is a general procedure to heat the specimen 

housing prior to beginning a test. Before start-up, the lubricant flow is begun to make sure that there is a 

sufficient amount of lubricant in the test system to prevent wear at startup. 

It is generally a good idea, where material comparisons are to be made, that the lubricant has good 

storage stability. It has been the authors’ experience that super-refined mineral oils of the paraffinic or 

naphthenic type and polyalphaolefin (PAO) are stable over extremely long periods of time and can give 

consistent results, making material comparisons possible. However, when using ester based lubricants, 

long term stability of the lubricant must be considered. To avoid water absorption, the lubricant should 

not be exposed to the atmosphere for any length of time. During storage, the lubricant must be kept under 

an inert cover gas. 

For a given type fatigue tester where there are multiple rigs and/or multiple test heads being used, it is 

prudent to assure that each test rig and/or test head is providing the same test results. This assures that 

there are no sources of random variability or systematic differences between the test rigs and/or test heads 

(Ref. 13). Because wear out of the test apparatus is a random variable, these comparisons need to be made 

on a continuous basis.

Deformation Effects on Contact Geometry and Stress 

In order to obtain rolling-element fatigue data in a bench-type, rolling-element fatigue test rig within 

a reasonable time, the maximum Hertz test stresses that are used are generally at or above the static load 

capacity of the hardened bearing steel surface at which plastic deformation will occur. Besides plastic 

deformation, experience has shown that fracture of the contacting surfaces may occur at maximum Hertz 

stresses at or beyond 6.9 GPa (1000 ksi). For efficiency of testing in these element testers, it is 

recommended that maximum Hertz stresses between 4.83 to 5.52 GPa (700 to 800 ksi) be used. 

With maximum compressive stresses of less than 4.14 GPa (600 ksi), gross plastic deformation of 

rolling surfaces will generally not occur for normal hardened bearing steels (Ref. 14). However, 

Drutowski and Mikus (Ref. 15) reported that small amounts of plastic deformation can occur at maximum 

Hertz stresses as low as 1.1 GPa (160 ksi). Drutowski (Ref. 16) further reported that for AISI 52100 steel 

there was an inversion between hardness and the maximum Hertz stress necessary for the initiation of 

plastic deformation. For a Rockwell C hardness of 62, the onset of plastic deformation occurs at a 

maximum Hertz stress of 1.92 GPa (278 ksi). For a Rockwell C hardness of 58, the onset occurs at a 

maximum Hertz stress of 2.6 GPa (377 ksi). Drutowski (Ref. 16) concluded that the contact stress at 

which plastic deformation is initiated is a function of the steel structure, but is independent of the 

diameter of the rolling ball (roller). 

Usually, in normal bearing operation, gross plastic deformation is of no concern. However, in 

laboratory fatigue tests of rolling elements, stresses greater than 4.14 GPa (600 ksi) are the rule rather 

than the exception. Due to the isotropic or kinematic hardening the gross plastic deformation is 

accompanied by changes in the contact geometry, residual stresses and yield limit. As the amount of 

plastic deformation increases, the contact stress is reduced from that calculated using the Hertz formulas. 

It was found that the resultant Hertz stress is not only affected by the applied load, material hardness and 

structure but also by the elastohydrodynamic film formed (Ref. 14). 

Figure 4(a) is a schematic diagram of the transverse section of a ball surface or crowned roller. The 

line of the true sphere or crown and the profile after plastic deformation and wear are shown. Figure 4(b) 

is a schematic diagram of a surface trace of a transverse section with deviations from that of the true 

sphere highly magnified (Ref. 17). 

 
NASA/TM—2011-216098 7 

For many applications where gross plastic deformation occurs, it is necessary to calculate the 

effective Hertz stress or the contact stress after plastic deformation occurs (Ref. 18). This can be 

accomplished by deriving the radius of curvature Rp of the deformed rolling element as shown in 

Appendix A. Referring to Figure 5(a), for a convex surface such as a ball or a crowned roller, 





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

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

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













 



  





 







H

H

A 2 R R

H

A

R

1 2 2

2

2

p / (1)

The deformed radius Rp’ (not shown) in the plane perpendicular to the plane of the profile shown in 

Figure 5(a) is 

 Rp’ = R’ – H (2) 

Where R’ is the radius of the body in the perpendicular plane. Since H is extremely small relative to R; 

 Rp’ = R’ (3) 

Substituting Rp into the Hertz equations for contact stress, the effective Hertz stress after gross plastic 

deformation of the rolling–element surface can be calculated. 
NASA/TM—2011-216098 8 
NASA/TM—2011-216098 9 

For a race groove surface having a concave or negative radius R, as shown in Figure 5(b), 

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

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

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H

H

A 2 R R

H

A

R

1 2 2

2

2

p / (4) 

The area of deformation (and wear) A and the track depth H can be measured directly from a surface 

contour trace. Because plastic deformation can hardly change the roller diameter, the modified radius of 

curvature Rp as derived should only be applied to ball bearings, not to cylindrical roller bearing. 

The effect of deformation and wear on the profile radius and the resultant maximum Hertz stress is 

illustrated in Table 1 for the NASA Five-Ball Fatigue Tester. The radius of curvature across the running 

track of the upper test ball is increased. Only considering deformation of the upper test ball, the resultant 

maximum Hertz stress is decreased by approximately 5 percent. However, considering deformation of 

both the upper and lower balls, the maximum Hertz stress is decreased by approximately 10 percent 

(Ref. 18). At an initial maximum Hertz stress of approximately 4.14 GPa (600 ksi) no plastic deformation 

was measured. This stress reasonably correlates with those stresses reported in the ISO Standards 

(Ref. 19) of 4.2 (609 ksi) and 4.0 GPa (580 ksi) related to the static load capacity of ball and roller 

bearings, respectively. 

TABLE 1.—DEFORMATION AND WEAR AND THEIR EFFECT ON MAXIMUM HERTZ STRESS FOR AISI M–50 

STEEL BALLS IN NASA FIVE-BALL FATIGUE TESTER 

[Ball dia., 12.7×10–3 m (0.500 in.); Rockwell C hardness, 62; initial maximum Hertz stress, 5.52 GPa (800 ksi); 

contact angle, 10; number of stress cycles, 30×103

) (data from (Ref. 17)).] 

Lubricant 

type 

Lubricant 

designation 

Original profile 

Radius, R, 

m (in.) 

Deformation 

and wear 

area from 

surface trace, 

A (Fig. 4), 

m2

 (in.2) 

Depth from 

original profile 

to deformed 

profile, H, 

(Figs. 4 and 5(a)), 

m (in.) 

Calculated 

profile radius, 

Rp (Fig. 5(a)), 

(from Eq. (1)), 

m (in.) 

Effective maximum 

Hertz stress, 

GPa (ksi) 

No 

deformation 

and wear of 

support balls 

assumed 

Deformation 

and wear of 

support balls 

assumed equal 

to test ball 

Ester NA–XL–3 6.35×10–3 4.95×10–10 16.68×10–7 8.35×10–3 5.27 5.03 

 (0.250) (7.67×10–7) (6.567×10–5) (0.329) (765) (730) 

 NA–XL–8 6.35×10–3 5.16×10–10 16.75×10–7 8.18×10–3 5.29 5.06 

 (0.250) (8.00×10–7) (6.596×10–5) (0.322) (767) (734) 

Mineral oil NA–XL–4 6.35×10–3 6.13×10–10 20.19×10–7 8.79×10–3 5.23 4.95 

 (0.250) (9.50×10–7) (7.947×10–5) (0.346) (759) (718) 

 NA–XL–7 6.35×10–3 5.37×10–10 17.95×10–7 8.51×10–3 5.27 5.00 

 (0.250) (8.33×10–7) (7.067×10–5) (0.335) (763) (725) 

The contact model is presented as an introduction to the topic, and can be used with reasonable 

engineering certainty to determine the reduction in the contact stress resulting from gross plastic 

deformation. However, for greater accuracy, recent additional work in elasto-plastic point contacts was 

reported on by Chen et al. (Ref. 20). Nelias et al. (Refs. 21 and 22) have reported on elastic-plastic sliding 

contacts, and Wang et al. (Ref. 23) have modeled elastic-plastic contacts between a sphere and a flat. 
NASA/TM—2011-216098 10 

Full-Scale Bearing Testers 

As discussed above, the use of rolling-element fatigue testers qualitatively compare with results of fullscale bearing tests and/or from field data. However, the variables once defined from bench type tests must 

be subject to realistic operating conditions such as magnitude and type of load, speed, mode of lubrication 

and temperature such as those found in actual bearing applications (Ref. 24). In general, for fatigue testing 

full-scale bearings it is recommended that maximum resultant Hertz stresses be equal to or less than 

2.41 GPa (350 ksi). However, maximum Hertz stresses as high as 3.1 GPa (450 ksi) can be used for testing 

many types of bearings. These requirements were discussed by A.T. Galbato (Ref. 24). He appropriately 

stated that, “The combined influence of important environmental conditions which simulate as closely as 

possible the expected service environment and that a sufficient number of tests be performed to evaluate the 

scatter of fatigue life as characterized by the Weibull distribution (function) (Ref. 24).” 

From Galbato (Ref. 24), the various arrangements for arranging testing ball and roller bearings are 

illustrated in Figure 6. “The standard bearing test rigs include a stationary frame, a movable housing frame, 

and a single test shaft on which test bearings are mounted. The type and size of the bearing will determine 

the overall size of the test rig and drive system. Operating conditions include the magnitude and type of 

load, speed, mode of lubrication and temperature.” These variables should be continuously monitored. 
NASA/TM—2011-216098 11 

A four bearing configuration for radial only loading is shown in Figure 6(a) (Ref. 24). This 

arrangement permits simultaneous testing of four bearings. The applied radial load (2P) acts on the two 

inboard bearings. The reactionary forces (P) act on the two inboard bearings. 

A three-bearing configuration for combined loading is shown in Figure 6(b) (Ref. 24). This 

arrangement provides for the combined loading of the two outboard bearings. The applied load (2P) acts 

on the center load radial bearing (B). The thrust load (T) is applied axially on one outboard bearing and 

reacted to by the other. In this arrangement, the reactionary radial load (P), in combination with the 

applied and reactionary thrust load (T) acting on the outboard test bearings, provides a simultaneous 

combined loading on these two bearings (Ref. 24). 

A two bearing configuration for thrust loading only is shown in Figure 6(c). In this arrangement, the 

thrust load (T) is applied axially to one bearing and is reacted to by the second bearing. 

The lubrication mode for these type tests is usually recirculating oil jet lubrication. Oil filtration of 

10 μm or better should be provided and the filter replaced after each test. Other lubrication modes that are 

used are under- and through-the-race lubrication and oil mist. While oil bath lubrication can be used for 

low speed tests, it is not recommended. It is recommended that the oil condition be monitored and that the 

oil be changed after each test. The used oil should be discarded and not reused. A sample of oil from each 

test should be saved and cataloged for later analysis, if necessary. In either case, the same lubrication 

conditions have to be assured in the most loaded areas for all four bearings, especially when bath 

lubrication is used. 

Automatic shut-off systems are used for each test rig based on temperature rise and monitoring any 

vibration amplitude increase that indicates the occurrence of an incipient fatigue spall. The bearings’ oilin and out temperatures and the bearings’ outer- (and, if possible, inner-) ring temperatures are measured 

and monitored. Lubricant flow is measured and monitored. Chip detectors in the lubricating system 

should be critically placed to detect a metallic chip of a significant size emerging from a spall. Time of 

operation should be monitored for each test machine and each test bearing (Ref. 24).

Procedures for Testing Full-Scale Bearings 

It should again be emphasized that bench-type testing is used to observe comparative trends due to 

changes in lubricant, material type, material processing, operating parameters, etc. While great efforts are 

made to appropriately select and control testing technique and test parameters the only way to obtain 

quantitative data is to test full-scale bearings. However, the cost of testing bearings can be more than 100 

times greater than simplified rolling-element specimens. Similar caution and test procedures should be 

taken with full-scale bearings as with bench type testers discussed above. 

A common problem in testing full-scale bearings is misalignment of the bearing within the rig. Prior 

to test, the bearing should be measured for dimensional clearances and tolerances. In general, test stresses 

lower than 2.41 GPa (350 ksi) maximum Hertz are suggested. However, maximum Hertz stresses as high 

as 3.1 GPa (450 ksi) on the inner race can be used for testing many types of bearings. 

Another common problem associated with full-scale bearing testing is the failure of investigators to 

consider or report the interference fits between the bearing bore and shaft and the bearing outer diameter 

and housing. The interference fit between the bearing bore and the shaft will induce tensile hoop stresses 

in the bearing inner race. These tensile stresses can increase the magnitude of shearing stresses below the 

contacting surface between the rolling elements and the bearing inner race and reduce the bearing fatigue 

life (Refs. 25, 26, and 27). 

The interference fit between the bearing bore and shaft in combination with the interference fit 

between the bearing outer diameter and the housing will reduce the bearing’s internal clearance and affect 

the bearing fatigue life. Hence, the same bearing run under extensively the same operating conditions can 

produce significantly different life results because of interference fits. Accordingly, a “fit-up study” 

should be conducted and reported regarding the effect of interference fit and bearing internal clearance. 

Detailed findings and insights regarding interference fit can be found in Coe and Zaretsky (Ref. 25) and 

Oswald et al. (Refs. 26 and 27). 
NASA/TM—2011-216098 12 

Another problem to be alerted to is that the static load capacity of the bearing should not be exceeded. 

This is the maximum load that a bearing can be permitted to support when not rotating. It is arbitrarily 

defined as the load that will produce a permanent indentation of the race having a depth equal to 0.0001 

times the rolling-element diameter. Based upon ISO Standard ISO 76:2006 (Ref. 19) for through 

hardened bearing steel of hardness Rockwell C 58 and above, this load correlates to a maximum Hertz 

stress of approximately 4.0 GPa (580 ksi) for cylindrical roller bearings. For ball bearings this load 

correlates to a maximum Hertz stress of approximately 4.2 (609 ksi) (Ref. 19). When permanent 

deformation exceeds this value, bearing vibration and noise can noticeably increase when the bearing is 

subsequently rotated under lesser loads. 

Although it is not recommended, a bearing can be loaded above the static load capacity as long as the 

load is applied when the bearing is rotated. For an angular-contact ball bearing, the plastic deformation 

that occurs during rotation will be distributed evenly around the periphery of the races and will not be 

harmful until it becomes more extensive. However, for a radially loaded ball or roller bearing, when the 

maximum Hertz stress on the outer race exceeds 4.0 GPa (580 ksi), the plastic deformation will be 

concentrated at the position of the maximum loaded rolling element. In this event, exceeding the 

bearing’s static load capacity would be unacceptable. 

In many applications where marginal lubrication is a factor, such as at elevated temperature, highsurface tangential speeds are desirable in the bearings because of elastohydrodynamic effects. This can 

make a difference between the bearing operating for relatively short time periods with wear and surface 

distress being the failure mode or operating for extended time periods with rolling-element fatigue being 

the failure mode. Lower speeds will not necessarily give longer lives in terms of inner-race revolutions or 

conversely, higher speeds will not give shorter lives in terms of actual running time. Temperatures should 

be monitored on the inner race, the outer race and in the housing around the bearing. Lubricant sump and 

ambient temperatures should also be monitored. 

Post test examination of the failed test bearings should record those components, i.e., balls or rollers, 

inner and outer races, of each bearing that has failed and their respective failure mode. Based on the 

methods of Johnson (Ref. 10), it is possible to not only determine the statistical life and failure 

distribution of the bearing group but also the lives of the ball or roller set, the inner race and the outer 

race. 

Correlation of Bench-Type Tests With Full-Scale Bearing Results 

Bench-type fatigue tests can qualitatively rank material, lubricant, or operating variables. This is 

illustrated in Figure 7, which is a comparison of the relative dynamic load capacity obtained in the NASA 

Five-Ball Fatigue Tester with four different test lubricants at 149 C (300 F) to fatigue tests with 7208-

size deep-groove ball bearings with the same four test lubricants (Ref. 17). Each test lubricant for the 

bench tests and the full-scale bearing tests came from a single batch. While the operating conditions, 

bearing steels and contact geometries between the bench test rig configuration and the full-scale bearings 

are different, both type of tests ranked the fatigue lives with each of the four lubricants in the same order. 

These data are summarized in Table 2. 

The operating conditions of the bench-type tests typically differ from the in-service operating 

conditions for full-scale rolling-element bearings. Most bench-type tests are accelerated tests with 

extreme conditions (higher speeds, higher loads, and/or higher operating temperatures) as compared to 

typical bearing operating conditions. While there may be differences in the magnitude of the quantitative 

lives, the general trends are comparable, that is, there is agreement in relative rankings of materials, 

lubricants, operating conditions, etc. 

 
NASA/TM—2011-216098 13 

TABLE 2.—COMPARISON OF ROLLING-ELEMENT FATIGUE LIVES 

OBTAINED WITH FOUR TEST LUBRICANTS IN NASA FIVE-BALL FATIGUE 

TESTER AND 7208-SIZE ANGULAR-CONTACT BALL BEARINGS (REF. 17) 

Lubricant 

type 

Lubricant 

designation 

Five-ball fatigue testera

 7208-size angular-contact 

ball bearingsb

L10 life, 

millions of 

stress cycles

Ratio of L10 life 

to L10 life with 

NA–XL–7 

L10 life, 

millions of 

inner-race 

revolutions

Ratio of L10 life 

to L10 life with 

NA–XL–7 

Ester NA–XL–3 13.7 0.31 1.1 0.33 

 NA–XL–8 14.9 0.34 1.4 0.42 

Mineral oil NA–XL–4 42.5 0.96 2.5 0.76 

 NA–XL–7 44.5 1.00 3.3 1.00 

a

Temperature, 149 C (300 F); maximum Hertz stress, 5.52 GPa (800 ksi); material, 

vacuum processed (VP) AISI M–1 steel; steel Rockwell C hardness, 63; speed, 

10,000 rpm. b

Temperature, 149 C (300 F); thrust load, 20,461 N (4,600 lb); maximum Hertz stress, 

2.41 GPa (350 ksi); materials (assumed to be) air-melt processed (AM) 

AISI 52100 steel; hardness, not reported; speed, 3,450 rpm. 

Figure 8 compares the relative load capacity also obtained in the NASA Five-Ball Fatigue Tester to 

the load capacity of 207-size deep-groove ball bearings where ball hardness is the variable (Ref. 28). As 

can be seen for the lubricant and hardness variables, the correlation of the Five-Ball Fatigue Tester results 

with those of full-scale bearing tests on a relative basis is excellent. For both sizes of bearings, the 

maximum stress level was approximately 2.41 GPa (350 ksi) at the inner race-ball contact in comparison 

with 5.52 GPa (800 ksi) for the five-ball tester. 
NASA/TM—2011-216098 14 

A third example of the correlation between the NASA Five-Ball Fatigue Tester and full-scale 

bearings is shown in Figure 9 (Refs. 29 and 30). The relative L10 lives of bearings made from three steels 

are compared to rolling-element fatigue data from the NASA Five-Ball Fatigue Tester. These data show 

the effect of total percent weight of alloying element on bearing life. The bearings were thrust loaded 

120-mm bore angular-contact ball bearings made from AISI M-1, AISI M-50, and WB-49 steels 

(Ref. 30). The AISI M-42 steel run in the five-ball tester had a similar microstructure to WB-49. Both 
NASA/TM—2011-216098 15 

steels contained relative high percentages of cobalt. The bearings were run at a maximum Hertz stress at 

the inner race of 2.23 GPa (323 ksi) and a temperature at the outer ring of 316 C (600 F). The five-ball 

tests were run at a nominal temperature of 66 C (150 F). Again, there is excellent correlation between 

the NASA Five-Ball Fatigue Tester fatigue data with the full-scale bearing fatigue data. 

These comparisons show that bench-type fatigue testers can reliably identify qualitative effects of 

many variables on rolling-element fatigue life. By bench marking these life results to an already existing 

data base, it is possible to develop bearing life modifying factors with the Lundberg-Palmgren theory 

(Ref. 31) to predict bearing life with reasonable engineering certainty (Ref. 1). 

Testing Methodologies That Reduce Total Test Time 

Experimentally determining rolling-element bearing life is a complex, time- consuming, and costly 

task. In addition to the driving need to find methods to reduce testing time (and cost), not all bearing 

and/or specimens that are tested will or can be expected to fail in a reasonably, prescribed time limit. 

Thus, methods to reduce testing time are of great importance. An early explanation of three 

methodologies for reducing testing time can be found in the classic work of Leonard G. Johnson 

(Ref. 10). In the first method, more specimens are run simultaneously than are intended to fail. For 

example, for a Weibull slope of 1.0, the median time to fail 10 out of 20 samples is 76 percent less than 

the median time required to fail 10 out of 10 samples. This is assuming that there are no replacements of 

failed specimens, all testers are the same, and all specimens are run simultaneously. Since the width of the 

confidence band is determined by the number of items failed and not the total number of specimens in a 

test (Ref. 10), the number failed is 10 in both cases. As a result, the only difference is that the 10 items 

having the lowest lives in a lot of 20 are plotted on a Weibull plot instead of all 10 specimens in a lot of 

10 (Ref. 10). The first scenario will take 24 percent of the time it takes for the second, and yet the 

necessary life information can be obtained by the first case. 

Sequential analysis is a second methodology for shortening total test time described by Johnson 

(Ref. 10). Test specimens are failed sequentially one after the other. The total number of specimens 

needed to fail is not known in advance. The investigator decides after each failure whether or not 

additional testing is required. In this manner, the bare minimum number of runs needed to demonstrate an 

improvement or worsening of life is conducted. A method similar to this has been used extensively by 

E.V. Zaretsky and colleagues since the late 1950s. An estimate of the 50-percent life (L50) is made and 

test bearings are then run on identical testers until this life is reached. As samples fail or are suspended at 

the L50 life, they are removed, and new samples are mounted and evaluated to the estimated L50. In this 

manner, at least 50 percent of the samples are typically failed out of the entire available population. In 

some cases, the L50 target may have to be adjusted as dictated by the number of failures encountered. The 

experimenter decides after each failure whether or not additional testing is required. In this manner, the 

minimum number of runs needed to demonstrate an improvement or worsening of life is conducted. 

Sudden death testing (SDT) is the third method of reducing testing time described by Johnson 

(Ref. 10). The total accumulated test time is reduced by not running all specimens to failure. The total 

number of specimens to be evaluated n is divided into equal-sized subgroups according to the number of 

available experimental testers. Thus, there are m specimens in each equal-sized subgroup, and there are a 

total of r subgroups. The total number of specimens n equal m times r. The specimens in each subgroup 

are fatigue tested identically and simultaneously on different testers. The first subgroup of specimens is 

run until the first failure occurs. At this point, the surviving specimens are suspended and removed from 

testing. An equal set of new specimens numbering m samples is next tested until the first failure in that 

subpopulation. This process is repeated until one failure is generated for each of the subgroups. In the 

end, r failures are generated while (m1) r samples are suspended. Thus, the total accumulative test 

time is the time to fail r specimen times the number of samples concurrently tested m, not the time for n

failures. With proper corrections and analysis (Johnson (Ref. 8)), reliability of the life predicted by this 

sudden death methodology from r failures is comparable to that obtained when failing the entire 

population. A detailed description of sudden death testing (SDT) is given in Appendix B. 
NASA/TM—2011-216098 16 

Proving that any of these methodologies reduces total test time while predicting accurate bearing 

fatigue lives would require the generation of a significant and unreasonable quantity of experimental data. 

Vlcek, Hendricks, and Zaretsky (Ref. 32) have shown that computer modeling of bearing life based upon 

Weibull-Johnson Monte Carlo simulations results in reasonable engineering predictions of bearing life 

that are relatively easy to determine. Vlcek, et al. (Ref. 33) also performed Monte Carlo simulations 

combined with sudden death testing in order to compare resultant bearing lives to calculated bearing life 

and the cumulative test time and calendar time relative to sequential and censored sequential testing. 

Reductions up to 40 percent in bearing test time and calendar time can be achieved by testing to failure or 

the L50 life and terminating all testing when the last of the predetermined bearing failures has occurred. 

Vlcek, Hendricks, and Zaretsky found that sudden death testing is not a more efficient method to reduce 

bearing test time or calendar time when compared to censored sequential testing (Ref. 33). 

Fatigue Data Analysis 

Weibull Distribution Function 

In 1939, W. Weibull (Refs. 8 and 9) developed a method and an equation for statistically evaluating 

the fracture strength of materials based upon small population sizes. This method can be and has been 

applied to analyze, determine, and predict the cumulative statistical distribution of fatigue failure or any 

other phenomenon or physical characteristic that manifests a statistical distribution. The dispersion in life 

for a group of homogeneous test specimens can be expressed by 

ln where 0 0 1 1 lnln

μ

μ      

















 



L ; S

L L

L L

e

S (5)

where S is the probability of survival as a fraction (0  S  1); e is the slope of the Weibull plot; L is the 

life cycle (stress cycles); Lμ is the location parameter, or the time (cycles) below which no failure occurs; 

and L is the characteristic life (stress cycles). The characteristic life is that time at which 63.2 percent of 

a population will fail, or 36.8 percent will survive. 

The format of Equation (5) is referred to as a three-parameter Weibull analysis. For most—if not 

all—failure phenomenon, there is a finite time period under operating conditions when no failure will 

occur. In other words, there is zero probability of failure, or a 100-percent probability of survival, for a 

period of time during which the probability density function is nonnegative. This value is represented by 

the location parameter L. Without a significantly large data base, this value is difficult to determine with 

reasonable engineering or statistical certainty. As a result, L is usually assumed to be zero and 

Equation (5) can be written as 

ln where 0 ; 0 1 1 lnln      













 



L S

L

L

e

S (6)

This format is referred to as the two-parameter Weibull distribution function. The estimated values of the 

Weibull slope e and L for the two-parameter Weibull analysis may not be equal to those of the threeparameter analysis. As a result, for a given survivability value S, the corresponding value of life L will be 

similar but not necessarily the same in each analysis. 

By plotting the ordinate scale as ln ln(1/S) and the abscissa scale as ln L, a Weibull cumulative 

distribution will plot as a straight line, which is called a “Weibull plot.” Usually, the ordinate is graduated 

in statistical percent of specimens failed F where F = [(1 – S) × 100]. Figure 10(a) is a generic Weibull 

plot with some of the values of interest indicated. Figure 10(b) is a Weibull plot of actual bearing fatigue 

data. 
NASA/TM—2011-216098 17 

The Weibull plot can be used to evaluate any phenomenon that results in a statistical distribution. The 

tangent of the resulting plot, called the “Weibull slope” and designated by e, defines the statistical 

distribution. Weibull slopes of 1, 2, and 3.57 represent exponential, Rayleigh, and Gaussian (normal) 

distributions, respectively. 

The scatter in the data is inversely proportional to the Weibull slope; that is, the lower the value of the 

Weibull slope, the larger the scatter in the data, and vice versa. The Weibull slope is also liable to 

statistical variation depending on the sample size (data base) making up the distribution (Ref. 10). As the 

sample size becomes smaller, there is a greater statistical variation in the Weibull slope. 
NASA/TM—2011-216098 18 

Whether the data is generated experimentally or simulated analytically, the ultimate goal is to 

determine the life and characteristics of a larger population from a limited amount of data. There are 

various statistical methods for determining bearing life estimates from fatigue data that for practical 

engineering purposes, give similar results. These methods differ significantly; however, in their level of 

complexity to apply and limitations to their application. For its relative ease of use and engineering 

application with comparable reliability, we have selected the Weibull-based Leonard G. Johnson 

approach (Ref. 10) which utilizes a linear regression least-square-fit method over the significantly more 

complicated maximum likelihood estimation method (Ref. 34). Experience of the authors’ has been that 

least-square-fit method of Johnson (Ref. 10) gives similar results to that of the maximum likelihood 

estimation methods (Refs. 34 and 35). However, the maximum likelihood estimation method is more 

sensitive to early failures, biasing the Weibull slope to indicate more scatter than actually exists in the 

data. There are computer programs that are commercially available that can perform these Weibull 

analyses. 

Bearing Component Lives 

Where there are design, manufacturing, material and metallurgical variables among the inner race, 

outer race and rolling-element (ball or roller) set, it is important to distinguish the effects of these 

variables on the resultant life of a rolling-element bearing. Lundberg and Palmgren (Ref. 31) present 

bearing life as only a function of the lives of the inner and outer races as follows: 

e e e

L L L 















 













  















10 ir or

1 1 1 (7a) 

They do not directly calculate the life of the rolling-element (ball or roller) set of the bearing. However, 

through benchmarking of the equations with bearing life data by use of a material-geometry factor, the 

life of the rolling-element set is implicitly included in the bearing life calculation. The material-geometry 

factor does not differentiate the effects of these variables but empirically blends them together. 

Accordingly, Lundberg and Palmgren (Ref. 30) should have written their equation relating bearing life to 

the individual components lives based on the Weibull equation (Refs. 8 and 9) as follows: 

e e e e

L L L L 















 















 













  















10 ir re or

1 1 1 1 (7b) 

where the Weibull slope e is the same for each of the components as well as for the bearing as a system. 

The value of the L10 life is the same as that in Equation (7a). However, the values for the inner and outer 

race live will be greater than those of Equation (7a). Equations (7a) and (b) are based on strict series 

reliability derived in Appendix C. Equation (7b) can be rewritten as 

1 =

e

L

L 



 





ir

10 +

e

L

L 



 





re

10 +

e

L

L 



 





or

10 (7c)

While the experimental bearing L10 life may equal or greater than that calculated according to Lundberg 

and Palmgren (Ref. 31), the experimental values of the inner and outer race lives should always be higher 

than that calculated.

From Johnson (Ref. 10), the fraction of failures due to the failure of a bearing component is 
NASA/TM—2011-216098 19 

Fraction of inner-race failures 

e

L

L 



 



 

ir

10 (8a) 

 Fraction of rolling-element failures 

e

L

L 



 



 

re

10 (8b) 

 Fraction of outer-race failures 

e

L

L 



 



 

or

10 (8c) 

From Equations (8a) to (c), if the life of the bearing and the fractions of the total failures represented 

by the inner race, the outer race and the rolling-element set are known, the life of each of these 

components can be calculated. Hence, by observation, it is possible to determine the life of each of the 

bearing components with respect to the life of the bearing. 

Bearing Life Variation 

Vlcek et al. (Ref. 32) randomly assembled and tested 340 virtual bearing sets totaling 31,400 radially 

loaded and thrust-loaded rolling-element bearings. It was assumed that each bearing was assembled from 

three separate bins of components, with one bin containing 1000 inner rings; one with 1000 rolling 

element sets, and one with 1000 outer rings. The median ranks of the individual components were 

assigned and then virtual bearing assemblies were created using a Monte Carlo technique. 

Vlcek et al. (Ref. 32) determined the L10 maximum limit and L10 minimum limit for the number of 

bearings failed, r, using a Weibull-based Monte Carlo method. By fitting the resultant lives for different 

size populations of failed bearings (Fig. 11), equations were determined for both of these limits: 

 Maximum variation L10 life = calculated L10 life (1+6r

–0.6) (9a) 

 Minimum variation L10 life = calculated L10 life (1–1.5r

–0.33) where r  3 (9b) 

 Minimum L10 life = 0 where r  3 (9c) 

These curves compared favorably with the 90-percent confidence limits of Johnson (Ref. 10) at a Weibull 

slope of 1.5 (Fig. 12) (Ref. 32). 

Moult (Ref. 6) published a series of successive fatigue tests with cylindrical roller bearings made 

from a single heat of air melted (AM) and subsequently consumable-electrode vacuum remelted (CVM) 

AISI 8620 steel conducted over a five year period. These bearings were lubricated with a single batch of 

Mil-L-7808 diester lubricant and three individual types of SAE 30 mineral oil lubricants. The base stocks 

of these mineral oils were a super-refined naphthenic, a naphthenic and a paraffinic. Moult’s data base 

comprised approximately 106 roller bearing fatigue failures (Ref. 6). Individual test series ranged from 

10 to 14 bearings with as many as 39 bearings making up a composite group used for purposes of 

comparison. The variation in Moult’s data base varied in accordance with Equations (9a) to (c). For an 

example, the individual L10 lives for the air-melt (AM) AISI 8620 steel lubricated with the Mil-L-7808 

lubricant varied from 85 to 1900 percent from the composite bearing L10 life value with this lubricant. 

Further, when comparing the variation in bearing L10 lives for the air-melt (AM) AISI 8620 steel 

lubricated with the Mil-L-7808 lubricant with the composite L10 bearing life obtained with the air-melt 

(AM) AISI 8620 steel bearings lubricated with the SAE 30 mineral oil, the bearing L10 lives with the 

Mil-L-7808 lubricant ranged from 7 to 115 percent of that obtained with the SAE 30 mineral oil (Ref. 6). 
NASA/TM—2011-216098 20 

 
NASA/TM—2011-216098 21 

Rules can be implied from these results to compare and distinguish resultant lives of identical 

bearings either from two or more sources or made using different manufacturing methods. The following 

rules are suggested to determine if the bearings are acceptable for their intended application or if there are 

significant differences between the two groups of bearings. 

1. If the L10 lives of both bearing tests are between the maximum and minimum L10 life variations, there 

can be no conclusion that there is a significant difference between the two sets of bearings regardless 

of the ratio of the L10 lives. The bearing sets are acceptable for their intended application (Fig. 12(a)). 

2. If the L10 life of one set of bearings is greater than the maximum variation and the second set is less 

than the minimum value, there exists a significant difference between the bearing sets. Only one 

bearing set is acceptable for its intended application (Fig. 12(b)). 

3. If the L10 lives of both sets of bearings exceed the maximum variation, the bearing life differences 

may or may not be significant and should be evaluated based upon calculation of confidence numbers 

according to the method of Johnson (Ref. 10). Both sets of bearings are acceptable for their intended 

application (Fig. 12(c)). 

4. If the L10 lives of both sets of bearings are less than the minimum variation, the bearing life 

differences may or may not be significant. However, neither set of bearings is acceptable for its 

intended application (Fig. 12(d)). 

 
NASA/TM—2011-216098 22 

5. If the L10 life of one set of bearings exceeds the maximum variation and the other set is between the 

maximum and minimum variation, the bearing life differences may or may not be significant and 

should be evaluated based upon calculation of confidence numbers according to the method of 

Johnson (Ref. 10). Both sets of bearings are acceptable for their intended application (Fig. 12(e)). 

6. If the L10 life of one set of bearings is less than the minimum variation and the other set is between the 

maximum and minimum variation, there exists a significant difference between the bearing sets. Only 

one set of bearings is acceptable for its intended application (Fig. 12(f)). 

Weibull Slope Variation 

Johnson (Ref. 10) analyzed the probable variation of the Weibull slope as a function of the number of 

bearings tested to failure. Based on the Johnson analysis, in 90 percent of all possible cases the resultant 

Weibull slope will be within the limits shown in Figure 13 based upon a Weibull slope of 1.11. Based on 

Johnson, the approximate relation for the number of bearings failed r and the limits of the value of 

Weibull slope e equal to 1.11 are as follows: 

 Maximum Weibull slope = 1.11 + 1.31 r

–0.5 (10a) 

 Minimum Weibull slope = 1.11 – 1.31 r

–0.5 (10b) 

The results of the extremes in the Weibull slopes for each group of the 10 bearing trials of r bearings 

from the Monte Carlo simulation of Vlcek, et al. (Ref. 33) are compared with the Johnson analysis 

(Ref. 10) in Figure 13. 
NASA/TM—2011-216098 23 

Confidence Number 

It is important to know the confidence in the conclusions that can be derived from one series of tests 

relative to a standard test series. Johnson (Ref. 10) developed a statistical method for comparing two 

populations and/or variables that he termed “Confidence Numbers.” For purposes of example, Weibull 

plots of lives of bearings made from two hypothetical bearing steels are compared in Figure 14. Lot 1 is 

made from a standard bearing steel containing 7 failures. Lot 2 is made from an experimental bearing 

steel and contains 10 failures. The L10 for Lots 1 and 2 are 38 and 76 hr, respectively. Both lots have a 

Weibull slope of 1.7. The question that can be asked is. “How much confidence can be placed in the 

apparent superiority of the experimental steel population (Lot 2) over the standard steel population 

(Lot 1)?” 

In addition to the dependence upon the amount of separation between the two Weibull plots, or the 

ratio of the longer life to the shorter life at one level [F(x)], the answer to the question of confidence 

depends upon the number of failures (number of data points), r1 and r2 in each population of Lots 1 and 2, 

respectively, and also upon the Weibull slopes of their populations. An important point to be aware of is 

that the degree of confidence of one lot over the other need not be constant from one level [F(x)] to 

another. As an example, it is possible to have a significant improvement at the 50-percent life, L50, 

without any improvement at the L10 life and vice versa. Since improvement in early life is of prime 

importance and is used for comparison purposes, the confidence in the L10 life will be considered. An 

extensive treatment was presented by Johnson (Ref. 10). 

Johnson (Ref. 10) created a series of graphs shown in Figure 15 where 1/(1-C) is plotted against the 

L10 life ratio on logarithmic coordinates with the Weibull slope and [(r1 – 1)(r2 – 1)] held constant. The 

graphs shown in Figure 15 are straight lines (or very nearly so). For convenience, the ordinate is 

graduated as the confidence number. Johnson (Ref. 10) claims no theoretical basis for the linearity of the 

plots. 
NASA/TM—2011-216098 24
NASA/TM—2011-216098 25 

The quantity (r1 – 1) is known as the number of degrees of freedom for Lot 1. There are (r1 – 1) 

degrees of freedom because for a fixed value of failures r1, only (r1 – 1) can be chosen arbitrarily, the last 

one being determined from the remaining values. Similarly, (r2 – 1) is the number of degrees of freedom 

in Lot 2. The product of [(r1 – 1)(r2 – 1)] is the total degrees of freedom of the pair of lots. 

Referring to the Weibull plots of Figure 14, 

 r1 = number of failures of Lot 1 = 7 

 r2 = number of failures of Lot 2 = 10 

Thus, the total degrees of freedom = (7 – 1)(10 – 1) = 54. 

The Weibull slopes of both lots are approximately 1.7 and the L10 life ratio equals 76/38 or 2. 

Interpolating between Figures 15(d) and (e) for 54 degrees of freedom at Weibull slopes of 1.6 and 1.8, 

the confidence numbers are 75 and 78 percent, respectively. At a Weibull slope of 1.7 the average 

confidence number is 76.5 percent [(75 + 78)/2]. This indicates that there are 765 chances out of 1000 

that the L10 life in the population of Lot 2 is superior to the L10 life in the population of Lot 1. (A 

95-percent confidence number is equivalent to 2σ confidence band while a 68-percent confidence number 

is equivalent to a 1σ confidence band.) In general, it is desirable to have a confidence number of 90 or 

greater to assure a statistically significant difference between two populations. 

If Lots 1 and 2 have unequal Weibull slopes, the confidence number can be determined by treating 

the two lots as if both had the Weibull slope of Lot 1. Then obtain a second confidence number by 

treating both lots as if both had a Weibull slope of Lot 2. To arrive at the true confidence number the two 

results are averaged together (Ref. 10). 

To determine the confidence number for successive comparisons or successive tests, Johnson 

(Ref. 10) provides the following formula 

  

   

n

i 1

i n 1 C (11) 1 2 1 C

As an example, suppose there is a second set of tests to compare Lots 1 and 2, for the second set of 

tests, the resultant confidence number is 0.8. Then from Equation (11), the combined confidence number 

is 

 C = 1 – 2(1-C1)(1-C2) = 1 – 2(1 – 0.765)(1 – 0.8) = 0.906 (12) 

Assume that there is a third set of data that provides a confidence number of 0.65. 

The combined confidence number is 

 C = 1 – 2(3-1)(1-C1)(1-C2) (1-C3) = 

 1 – 4(1 – 0.765)(1 – 0.8)(1 – 0.65) = 0.934 (13) 

From the above, it may be concluded with reasonable engineering and statistical certainty that Lot 2 will 

provide a longer rolling-element fatigue life than Lot 1 approximately 93 percent of the time. However, 

there will be instances (approximately 7 percent of the time) where the population of Lot 1 will give 

longer lives. 
NASA/TM—2011-216098 26 

Determining Significance of Test Results 

If the data being compared is from either full-scale bearing tests or bench-type tests, confidence 

numbers can be used. The previously established rules based on variation in life, however, should only be 

applied to full-scale bearing testing where the lives of a bearing population with which the experimental 

or field data are to be compared are known and/or predicted. However, there are instances where there are 

test results where there are either insufficient test failures and/or the ratio of the life results are not 

sufficient to conclude with any reasonable engineering and/or statistical certainty that life differences in 

the variables being tested exist. This can occur where a single variable is being tested such as material 

chemistry, additive and lubricant chemistry, material hardness, etc. 

For purposes of example, let us assume that there appears to be a relation between the volume of an 

additive present in a lubricant and bearing fatigue life. In this example we will assume that in four test 

series life increases with increases in additive content where Test Series D with the highest percentage of 

additive gives the longest life and Test Series A with no additive present gives the lowest life. 

Test Series C has an additive content greater than Test Series B but less than Test Series D. The order of 

the test results from longest life to shortest life are D, C, B, and A. However, the confidence number 

between Test Series D and A is less than 90 percent. It may be possible to combine the confidence 

numbers as discussed in the previous section to obtain a confidence number greater than 90 percent 

showing an effect of additive volume on fatigue life. Alternatively, the following formula may be used. 

 Percent Probability of a Relation Existing = [1–1/(n!)]  100 (14) 

where n is the number of test series or groups. 

In the example above, n = 4 and factorial (n!) = 4321 = 24. From Equation (14) there is a 

96 percent probability that there exists a relation between additive volume percent and fatigue life. 

However, assume that there were only three test series whereby (n!) = 321 = 6. From Equation (14) 

there is an 83 percent probability that there exists a relation between additive volume percent and fatigue 

life. This percentage may be insufficient to conclude with any statistical certainty that a relation does 

exist. 

Summary 

In order to rank bearing materials, lubricants and design variables using rolling-element bench type 

fatigue testing and rolling-element bearing tests, the investigator needs to be cognizant of the variables 

that affect rolling-element fatigue life and be able to maintain and control them within an acceptable 

experimental tolerance. This paper brings together and discusses the technical aspects of rolling-element 

fatigue testing and data analysis as well as making recommendations to assure quality and reliable testing 

of rolling-element specimens and full-scale rolling-element bearings. These are as follows: 

1. When testing to obtain valid comparison of the lubricant, material, and/or operating variables in 

bench-type rolling-element fatigue testers, all the rolling-element specimens must fabricated from 

a single heat of material, heat treated simultaneously to the same hardness. All specimens must 

have the same surface finish unless the surface finish is the parameter being studied. The steel 

microstructure should be reported together with the retained austenite and residual stress before 

and after testing. It is of prime importance that the difference in hardness between the contacting 

test specimens remains constant. 

2. Where high reliability is desired for a given application it is preferable to test a group of bearings 

to determine their actual lives. In such cases, the same precautions should be taken as with a 

bench-type tester discussed above. In general, test stresses lower than 2.4 GPa (350 ksi) 
NASA/TM—2011-216098 27 

maximum Hertz are suggested. However, maximum Hertz stresses as high as 3.1 GPa (450 ksi) 

on the inner race can be used for testing many types of bearings. 

3. The interference fit between the bearing bore and the shaft needs to be controlled and reported. 

The interference fit can change the bearing’s internal diametrical clearance and will also induce 

tensile hoop stresses in the bearing inner race. These tensile stresses can increase the magnitude 

of shearing stresses below the contacting surface between the rolling elements and the bearing 

inner race and reduce the bearing fatigue life. It can also alter the Hertz stress-life relation. 

4. Comparisons show that bench-type fatigue testers can reliably identify qualitative effects of many 

variables on rolling-element fatigue life. By bench marking these life results to an already 

existing data base, it is possible to develop bearing life modifying factors with the LundbergPalmgren theory to predict bearing life with reasonable engineering certainty. 

5. Reductions up to 40 percent in bearing test time and calendar time can be achieved by testing half 

of the specimens to failure or to the L50 life and terminating all testing when the last of the 

predetermined bearing failures have occurred. Sudden death testing is not a more efficient method 

to reduce bearing test time or calendar time when compared to censored sequential testing. 

6. Once variables are controlled, the number of tests and the test conditions need to be specified to 

assure reasonable statistical certainty of the final results. Using the statistical methods of 

W. Weibull and L. Johnson, the minimum number of tests required can be determined. 

 

NASA/TM—2011-216098 29 

Appendix A.—Effect of Gross Plastic Deformation on Contact Radii 

For many applications where gross plastic deformation occurs, it is necessary to calculate the 

effective Hertz stress or the stress after plastic deformation (Ref. 17). This can be accomplished by 

deriving the radius of curvature Rp of the deformed rolling element as follows: 

Referring to Figure 5(a) let 

H  2l

2

1 A  (A1) 

approximately, then 

h' h  H or h  h'H (A2) 

Then 

 

2

p

2 2

p

2

2R h h

R R h

 

l    (A3) 

and 

2h

h R

2 2

p

  l (A4) 

If Rp = R and h = h' , Equation (A3) becomes 

 

  











  

 

1/2 2 2

2

h' R R x

and

l 2Rh' h'

 (A5) 

Substituting Equation (A5) into Equation (A2) and solving for h in terms of A, R, and H results in 

H

H

A h R R

1/2 2

2  



















 



    (A6) 

If H is substituted into Equation (A4) and a negative sign selected for the radical in Equation (A6), 





















 



















 



  





















 



















 



      







H

H

A 2 R R

H

H

A R R

H

A

R

1/2 2

2

2 1/2 2

2

2

p (A7) 
NASA/TM—2011-216098 30 

Since 

2 1/2 2

2 H

H

A R R





















 



















 



   is small relative to 

2

H

A 



 



 , it can be neglected in the numerator, then 





















 



















 



  





 







H

H

A 2 R R

H

A

R

1/2 2

2

2

p (A8) 

The deformed radius R'p (not shown) in the plane perpendicular to the plane of the profile shown in 

Figure 5(a) is 

R'p  R'H

where R' is the radius of the body in the perpendicular plane. Since H is extremely small relative to R' , 

R' R' p  (A10) 

Substituting Rp in the Hertzian equations will give the effective compressive Hertz stress after gross 

plastic deformation of the rolling-element surface. 

For a race groove surface having a concave or negative radius R, as shown in Figure 5(b), 





















 



















 



  





 







H

H

A 2 R R

H

A

R

1/2 2

2

2

p (A11) 

The area of deformation A and track depth H can be measured directly from a surface contour trace. 
NASA/TM—2011-216098 31 

Appendix B.—Sudden Death Testing Techniques 

As discussed by Vlcek, Hendricks, and Zaretsky (Ref. 33), in sudden death testing, the total number 

of specimens to be evaluated is divided into equal subgroups that can be evaluated simultaneously. The 

first subgroup of specimens is run simultaneously until the first failure occurs. The surviving tests are 

terminated (i.e., suspended), and new test specimens are mounted in the testers. This process is repeated 

until all of the test specimens in the population have been screened. A technique developed by Vlcek, 

Zaretsky, and Hendricks (Ref. 32) was used to generate virtual bearing sets that were analyzed as if they 

were sudden death tested. Total populations with 36, 72, and 144 deep-groove bearings were generated. 

The populations were then sequentially broken into subgroups representing all possible combinations of 

sudden death test series of (m  r), where m was the number of bearing testers used simultaneously and r

was the number of sets of m bearings necessary to achieve the total number of bearings n in the total 

population. The total number of bearings n equal m times r. For example, if 36 bearings were to be 

evaluated, and there were 4 bearing fatigue testers available, the value of m would be 4 while the value of 

r would be 9; whereby n equals 36 (49). For this example, the first four bearings were run 

simultaneously until the first failure occurred. The three surviving tests were suspended, and four new 

bearings were mounted. This process was repeated until nine failures occurred; one from each of the sets 

of 4, with (m – 1)  r or 27 suspensions. 

These nine failures and their corresponding median ranks, one for each subgroup of size m, were then 

plotted on a Weibull plot. This is illustrated as Step 1 in the schematic of a generic Weibull plot of the 

sudden death line (SDL) of Figure 16. This SDL represents the distribution of first failures in each 

subgroup. The SDL of each series was next shifted on its respective Weibull plot, Steps 2 and 3 of 

Figure 16, so that the curve represented the failures of the total population, not just that of one out of r

bearings. 

Various methods exist for shifting the SDL (Fig. 16) and finding the life of a total population n based 

upon sudden death testing data where only r failures are considered. The slope and characteristic life of 

the subpopulation generated during sudden death testing can be found from maximum likelihood 

estimators of Cohen (Ref. 35) that are obtained from an iterative process. This life for r samples must be 

corrected to represent the life of the original population containing n samples. One way to achieve this, as 

reported by McCool (Ref. 36) is to multiply the subpopulation life estimator by the number of samples 

(m) in each equally sized subpopulation raised to the inverse of the slope estimator. Confidence limits can 

be placed upon these values for a limited number of cases provided in tables in the open literature or by 
NASA/TM—2011-216098 32 

extensive Monte Carlo simulations (Ref. 36). Houpert (Ref. 37) also proposes a technique for determining 

the life of a larger population based upon a sub-population determined from sudden death testing. 

For its simplicity of application and relative engineering accuracy, we prefer a technique presented in 

Johnson (Ref. 10) for shifting the SDL so that the life and characteristics of the larger bearing population 

(n) can be projected. The generic Weibull plot of Figure 16 accompanies the following steps, and includes 

many of the elements mentioned. 

(Step 1) Plotting the subgroup of r failures on Weibull paper. To plot the failures, median ranks are 

assigned to the sequentially ordered lives. Median ranks were defined using 

 median rank = (j – 0.3)/(r + 0.4) (B1) 

where j = 1, 2, 3, ... r. For sudden death testing, the number of subsets r equal n. A discussion and 

comparison of median rank definitions is available in Houpert (Ref. 37) and Skinner et al. (Ref. 37). The 

median ranks along with their corresponding lives are next plotted on Weibull paper. The locus of points 

is fitted with a linear curve. From the Weibull plot the Weibull slope, the L10 life, L50 life and 

characteristic life, Lβ, are determined. This SDL represents the distribution of first failures at the median 

rank for one failure out of the number of samples (m) in each of the subgroups. 

(Step 2) Determining by how much the SDL must be shifted to accurately estimate the L10 life for the 

entire population. The median rank (Eq. (B1)) must be determined for the first failure out of the number 

of bearings simultaneously evaluated (m). In the above example, for a [(m = 4)  (r = 9)] test, the four 

simultaneous testers are stopped after the first test failure occurs, thus the mean order number j = 1 and 

the subgroup size m = 4 where it is assumed that in Equation (B1) m = r. The median rank for 1 out of 4, 

found using Equation (B2) below is 0.1591. This is the value to which the SDL must be shifted (Fig. 16). 

In general, 

 First failure median rank (FFMR) = (1 – 0.3)/(m + 0.4) (B2) 

(Step 3) Constructing the Total Population Line (TPL) by shifting the SDL. At the L50 intersection of 

the SDL, a vertical line is drawn down to the median rank value determined in Step 2. Through this point, 

a line is drawn parallel to the SDL created in Step 1. The slopes of both lines are assumed to be equal. 

Figure16 is a generic Weibull plot of the SDL and the shifted Total Population Line (TPL) representative 

of this technique. 

(Step 4) Determining lives from the shifted TPL. The Weibull slope and lives are read directly from 

the TPL generated in Step 3. Table 3 contains the Weibull slope, L10 life, L50 life, and Characteristic life, 

Lβ, for a typical [(m = 4)  (r = 9)] sudden death test of a virtual deep groove bearing. The values obtained 

from the SDL and the shifted TPL are provided. 

TABLE 3.—COMPARISON BETWEEN LIVES OBTAINED FROM 

MONTE CARLO SUDDEN DEATH TESTING AND THAT 

ADJUSTED FOR TOTAL POPULATION 

[Number of failures, 9; number of bearings tested, 36; number 

of test rigs, 4; assumed Weibull slope, 1.11; resultant Weibull 

slope, 1.033; bearing size and type, 50-mm bore deep-groove 

ball bearing (Ref. 33).] 

Bearing 

population size 

Bearing life, hr Weibull 

L10 L50 L slope, e 

Sudden death for 9 failures 2354 14569 20772 1.033 

Adjusted for total population 

from sudden death 9003 55729 79453 1.033 

Calculateda (actual) for total 

population 6912 37729 47729 1.11 

a

Life based on Zaretsky’s rule and lubricant life factor (Ref. 1).
NASA/TM—2011-216098 33 

Appendix C.—Derivation of Strict Series Reliability 

As discussed and presented in References 39 and 40, G. Lundberg and A. Palmgren (Ref. 31) in 1947, 

using the Weibull Equations (Refs. 8 and 9) for rolling-element bearing life analysis, first derived the 

relationship between individual bearing component lives and bearing system L10 life. The following 

derivation is based on but is not identical to the Lundberg-Palmgren analysis (Ref. 31) which did not 

include the rolling-element (ball or roller) set life. 

Referring to Figure 10(a) and Equation (6), the Weibull equation can be written as 

















 



 





L

L

e

S

ln 1 lnln

x

 (C1) 

where L is the number of cycles to failure at a probability of survival of Sx. The characteristic life Lβ is the 

life at a 36.8 percent probability of survival or a 63.2 [(1 – 0.368)  100 = 63.2] percent probability of 

failure. 

Figure 17 is a sketch of multiple Weibull plots where each Weibull plot represents a cumulative 

distribution of the bearing and each component of the bearing system. The Weibull plot of the bearing 

represents the combined Weibull plots of the (1) inner race, (2) rolling-element set, and (3) outer race. All 

plots are assumed to have the same Weibull slope e. The slope e can be defined as follows: 

10

x 90

ln ln

1 lnln 1 lnln

L L

S S

e 





 



  



 





 (C2a) 

or 

e

L

L

S

S 



 



 





 









 





10

90

x

1 ln

1 ln

 (C2b) 

From Equations (C1) and (C2b), 

e e

L

L

L

L

S S 















 



 









 



  



 





x 90 10 

1 ln 1 ln (C3) 

and 

e

L

L S















  



x exp (C4) 

where Equation (C4) is identical to Equation (C1). 

 
NASA/TM—2011-216098 34 

For a given time or life L, each component in a system will have a different reliability S. For a series 

reliability system the probability of survival of the system Sx at a given time or life L is the product of the 

probabilities of survival of each of the components making up the system where for a bearing 

Sx S1 S2 S3    (C5) 

Combining Equations (C4) and (C5) gives 



















 















  



















  

e e e

L

L

L

L

L

L

1 2

exp exp exp

e

L

L



















3

exp (C6a) 







































































  



















   

e e e e

L

L

L

L

L

L

L

L

1 2 3

exp exp (C6b) 

It is assumed that the Weibull slope e is the same for all components. From Equation (C6b) 







































































  



















e e e e

L

L

L

L

L

L

L

L

β β1 β2 β3

 (C7a) 

Factoring out L from Equation (C7a) gives 
NASA/TM—2011-216098 35 

e e e e

L L L L 





































































 1 2 3

1 1 1 1 (C7b) 

From Equation (C3) the characteristic lives L1, L2, L3, etc., can be replaced with the respective lives 

L1, L2, and L3, at S90 (or the lives of each component that have the same probability of survival S90) as 

follows: 

e e e e

S L S L S L S L 



 









 



  



 









 



  



 









 



  



 









 





90 10 90 1 90 2 90 3

1 1 ln 1 1 ln 1 1 ln 1 1 ln (C8) 

where, in general, from Equation (C3) 

e e

L S L 



 









 



 

















 90 10

1 1 ln 1 (C9a) 

and 

,etc.

1 1 ln 1

1 90 1

e e

L S L 



 









 



 



















 (C9b) 

Factoring out 



 





90

1 ln

S from Equation (C8) gives 

/ e e e e

L L L L

1

10 1 2 3

1 1 1 1





















 



  



 



  



 



  



 



 (C10) 

or rewriting Equation (C10) results in 

e e e e

L L L L 















 















 













  















10 ir re or

1 1 1 1 (C11) 

Equation (C11) is identical to Equation (7b) of the text. 

 
NASA/TM—2011-216098 36 

References 

1. Zaretsky, E.V., Ed., (1992), STLE Life Factors for Rolling Bearings, STLE SP–34, Society of 

Tribologists and Lubrication Engineers, Park Ridge, IL. 

2. Sadeghi, F.; Jalalahmadi, B., Slack, T.S., Raje, N., and Arakere, N.K., (2009), “A Review of Rolling 

Contact Fatigue,” Jour. of Tribology, ASME Trans., 131, 4, Art. No. 041403, OCT 2009. 

3. Alley, E.S., and Neu, R.W., (2010) “Microstructure-Sensitive Modeling of Rolling Contact Fatigue,” 

International Jour. of Fatigue, 32, 5: pp. 841–850. 

4. Zaretsky, E.V., Ed., (1997), Tribology for Aerospace Applications, STLE SP–37, Society of 

Tribologists and Lubrication Engineers, Park Ridge, IL. 

5. Stribeck, R., (1900), “Reports From the Central Laboratory for Scientific Investigation,” Translation 

by H. Hess, 1907, ASME Trans., 29, pp. 420–466. 

6. Moult, J.F., (1963) “Critical Aspects in Bearing Fatigue Testing,” J. ASLE (now STLE), (Dec. 1963), 

pp. 503–511. 

7. Bamberger, E.N., Harris, T.A., Kacmarsky, W.M., Moyer, C.A., Parker, R.J., Sherlock, J.J., and 

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Rolling-Element Fatigue Testing and Data Analysis--A Tutorial 

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Vlcek, Brian, L.; Zaretsky, Erwin, V. 

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14. ABSTRACT

In order to rank bearing materials, lubricants and other design variables using rolling-element bench type fatigue testing of bearing 

components and full-scale rolling-element bearing tests, the investigator needs to be cognizant of the variables that affect rolling-element 

fatigue life and be able to maintain and control them within an acceptable experimental tolerance. Once these variables are controlled, the 

number of tests and the test conditions must be specified to assure reasonable statistical certainty of the final results. There is a reasonable 

correlation between the results from elemental test rigs with those results obtained with full-scale bearings. Using the statistical methods of 

W. Weibull and L. Johnson, the minimum number of tests required can be determined. This paper brings together and discusses the

technical aspects of rolling-element fatigue testing and data analysis as well as making recommendations to assure quality and reliable 

testing of rolling-element specimens and full-scale rolling-element bearings.

15. SUBJECT TERMS

Bearings; Roller bearings; Ball bearings; Rolling-element fatigue life 

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