Fred B. Oswald and Erwin V. Zaretsky

Glenn Research Center, Cleveland, Ohio

Joseph V. Poplawski

J.V. Poplawski & Associates, Bethlehem, Pennsylvania

Relation Between Residual and Hoop Stresses and

Rolling Bearing Fatigue Life

NASA/TM—2015-218893

December 2015
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Fred B. Oswald and Erwin V. Zaretsky

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Joseph V. Poplawski

J.V. Poplawski & Associates, Bethlehem, Pennsylvania

Relation Between Residual and Hoop Stresses and

Rolling Bearing Fatigue Life

NASA/TM—2015-218893

December 2015

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NASA/TM—2015-218893 1 

Relation Between Residual and Hoop Stresses and 

Rolling Bearing Fatigue Life 

Fred B. Oswald and Erwin V. Zaretsky*

National Aeronautics and Space Administration 

Glenn Research Center 

Cleveland, Ohio 44135 

Joseph V. Poplawski 

J.V. Poplawski & Associates 

Bethlehem, Pennsylvania 18018 

Abstract 

Rolling-element bearings operated at high speed or high vibration may require a tight interference fit 

between the bore of the bearing and shaft to prevent rotation of the bearing bore around the shaft and 

fretting damage at the interfaces. Previous work showed that the hoop stresses resulting from tight 

interference fits can reduce bearing lives by as much as 65 percent. Where tight interference fits are 

required, case-carburized steel such as AISI 9310 or M50 NiL is often used because the compressive 

residual stresses inhibit subsurface crack formation and the ductile core inhibits inner-ring fracture. The 

presence of compressive residual stress and its combination with hoop stress also modifies the Hertz 

stress-life relation. This paper analyzes the beneficial effect of residual stresses on rolling-element bearing 

fatigue life in the presence of high hoop stresses for three bearing steels. These additional stresses were 

superimposed on Hertzian principal stresses to calculate the inner-race maximum shearing stress and the 

resulting fatigue life of the bearing. The load-life exponent p and Hertz stress-life exponent n increase in 

the presence of compressive residual stress, which yields increased life, particularly at lower stress levels. 

The Zaretsky life equation is described and is shown to predict longer bearing lives and greater load- and 

stress-life exponents, which better predicts observed life of bearings made from vacuum-processed steel. 

Introduction 

Classical rolling-element fatigue is the process by which repeated cycles of a concentrated 

compressive surface load initiate subsurface cracks in the zone of maximum shearing stresses that 

propagate into a crack network resulting in a spall on the surface of the running track. In some cases, 

cracks can propagate radially into the structure causing inner-ring fracture and a catastrophic failure. 

It is generally accepted that if a rolling-element bearing is properly designed, manufactured, installed, 

lubricated and maintained, “classical” rolling-element fatigue is the only failure phenomenon that limits 

bearing life (Ref. 1). Rolling-element fatigue is extremely variable but is statistically predictable 

depending on the steel type, processing and heat treatment, bearing manufacturing process and type, and 

operating conditions. This type of failure is a cycle-dependent phenomenon resulting from repeated stress 

under rolling-contact conditions and is considered high-cycle fatigue. Sadeghi et al. (Ref. 2) provide an 

excellent review of rolling-element fatigue failure. 

Bearing dynamic load ratings were developed by Lundberg and Palmgren (Refs. 3 and 4). The 

dynamic load capacity is defined as the radial load for a life of 1 million inner-race revolutions with a 

90-percent probability of survival. Load ratings were adopted in bearing load and life standards such as 

ANSI/ABMA-9:1990 (Ref. 5) and ANSI/ABMA–11:1990 (Ref. 6). 

Life computation in the rating standards is based on the effect of the Hertzian loading only. Any other 

effects, including accounting for improvements in bearing materials, manufacturing processes, 

 

*

Distinguished Research Associate 
NASA/TM—2015-218893 2 

factors is that they are presented as constants rather than being related to loading, clearance, or the 

interaction of stress due to loading with the residual stress in the material. 

The ISO 281 International Standards (Ref. 7) initially followed the ANSI/ABMA life standards 

(Refs. 5 and 6). However, in a 2000 amendment (Ref. 8) to the 1990 standard and continuing with the 

2007 revision (Ref. 7), ISO incorporated the concept of a fatigue limit, with the assumption that a bearing 

made from high-quality steel with good manufacturing quality and lubrication may have infinite life if 

contact stresses are kept below this limit. 

Compressive residual stress in the tangential (hoop) direction improves the life of rolling-element 

bearings because it reduces the effective shearing stress beneath the contacting surfaces of the bearing, 

inhibiting subsurface crack formation, thus delaying the onset of fatigue failure and extending bearing 

life. Most bearings are made from through-hardened steels, such as AISI 52100 or AISI M-50. While 

these give satisfactory service in many applications, the hardened core can be susceptible to fracture. 

Bamberger and Kroeger (Ref. 9) investigated a carburizing-grade modified version of AISI M-50 

steel for use as a race material for rolling-element bearings. This steel, designated M50 NiL (AMS 6278), 

has reduced carbon content (from 0.8 to 0.13 percent) to improve the fracture toughness of the core and 

3.5 percent nickel is added to stabilize the austenite and prevent the formation of excessive amounts of 

ferrite and retained austenite. The case (surface) hardness was increased from 38 to 60.6 HRC by 

carburizing to 0.81 percent carbon to an effective case depth of 1.9 mm (0.075 in.), which produces 

compressive residual stress to a depth of at least 1.5 mm (0.060 in.). The induced compressive residual 

stress significantly increased the rolling-element fatigue life of the M50 NiL steel over the conventional 

AISI M-50 steel. 

Parker and Zaretsky (Ref. 10) performed tests in a five-ball fatigue tester using 12.7-mm- (0.5-in.-) 

diameter vacuum-degassed AIS1 52100 balls. Their tests were run at four levels of maximum Hertz stress 

ranging from 4500 to 6000 MPa (650 to 875 ksi). They found that the L10 life (10-percent fatigue life) is 

inversely proportional to maximum Hertz stress raised to the power of 12, instead of the ninth power as 

assumed in ball bearing life standards. With the results corrected for temperature effects, the stress-life 

exponent was reduced slightly from 12 to 11.5. 

Jalalahmadi and Sadeghi (Ref. 11) conducted a study of the scatter in the critical shearing stress, 

depth to this stress and the Weibull modulus (slope) of rolling contacts, using Voronoi tessellation 

methods to simulate the random grain structure of bearing steels. Their results generally agree with the 

Lundberg-Palmgren theory. They also show that steels without initial flaws in the grain structure have 

greater Weibull modulus (slope) and thus longer lives, particularly at higher reliability levels. Jalalahmadi 

and Sadeghi (Ref. 11) did not investigate the effect of variations in Hertzian loading on rolling-element 

fatigue life. 

Kotzalas (Ref. 12) investigated changes in the residual stress during operation on the full subsurface 

stress field in rolling-element bearing fatigue life prediction, concluding that instantaneous values of the 

material constants in most stress-field-based life equations are incalculable and thus the pre-fatigue 

residual stress field should be used in fatigue life calculations. 

Rosado et al. (Ref. 13) reported fatigue test results with thrust-loaded, 208-size angular-contact 

hybrid ball bearings made from AISI M-50 and M50 NiL (AMS 6278) steel with silicon nitride (Si3N4) 

balls. Their maximum Hertz stress was reported to be 3100 MPa (450 ksi). The AISI M-50 bearing tests 

produced only 1 failure out of 12 tests. M50 NiL produced 4 failures in 16 tests. 

Rosado et al. (Ref. 13) obtained seven times higher life than predicted by STLE life analysis (Ref. 1) 

for the AISI M-50 steel but only 65 percent of the life predicted by STLE life analysis for the M50 NiL 

steel. However, they reported that a TiN coating from the M50 NiL cage-land surface had delaminated 

with hard particles from the coating causing raceway surface damage that reduced life of the M50 NiL 

raceways. 

Townsend and Bamberger (Ref. 14) reported results of fatigue tests on spur gears at 1710 MPa 

(248 ksi) and rolling-contact bars at a maximum Hertz stress level of 4830 MPa (700 ksi). The tests 

compared AISI 9310 and M50 NiL operating under line contact. These fatigue tests were obtained on a 

test machine that applied load to half of the gear face width in order to obtain a high surface fatigue load 
NASA/TM—2015-218893 3 

without generating tooth-bending failures. They obtained fatigue lives 4.5 times higher on the spur gear 

tests and 13.2 times higher for the rolling-contact tests for the M50 NiL steel compared to AISI 9310. 

Oswald et al. (Refs. 15 and 16) investigated the effect of hoop stress due to interference fits on the 

lives of roller and ball bearings, showing that these hoop stresses can reduce bearing lives by as much as 

65 percent. This analysis did not consider the beneficial effects of residual stress from case-carburized 

steels. 

The objective of the work reported herein is to examine the effect on fatigue life from combined 

residual stress and hoop stress over a range of applied Hertz stress on rolling-element bearings. An analysis 

was carried out at four levels of Hertz stress ranging from 1380 to 2215 MPa (200 to 350 ksi) to calculate 

the unfactored bearing lives for cylindrical roller bearings and angular-contact ball bearings. Then these 

lives were adjusted to account for material life factors and tensile hoop stress from a heavy interference fit 

of the inner ring and for the beneficial effect of compressive residual stress in carburized steel. 

Nomenclature 

b semi-width of Hertzian contact area in direction of rolling, mm (in.) 

c shear stress-life exponent 

D diameter at the location of the maximum shear stress beneath the surface of the 

 inner race, mm (in.) 

E Young’s modulus of elasticity, MPa (ksi) 

f ratio of ball bearing raceway groove radius to ball diameter (conformity) 

FM material life factor 

h exponent in Equations (1) and (B1) 

k constant in Equations (18) and (B1) to (B3), mmh/m (in.h/m) 

l roller length, mm (in.) 

L life, millions of inner-race revolutions or hr 

L10 10-percent life: life at which 90 percent of a population survives, millions of 

 inner-race revolutions or hours 

LF life factor: ratio of computed life to life at reference Hertz stress without hoop 

 stress or residual stress 

LR life ratio for one race of a bearing 

m Weibull slope 

n Hertz stress-life exponent 

p load-life exponent 

pi contact pressure between shaft and inner ring due to interference fit, MPa (psi) 

PN normal load 

RL relative life

S stress, MPa (ksi)

Smax maximum Hertz stress, MPa (ksi) 

(Smax)ref reference value of maximum Hertz stress, MPa (ksi) 

V stressed volume, mm3

 (in.3) 

y transverse direction 

z distance below surface to maximum shear stress due to Hertizan load, mm (in.) 

 diametral interference, mm (in.) 

ν Poisson’s ratio 
NASA/TM—2015-218893 4 

σ stress, MPa (ksi) 

τ shear stress, MPa (psi) 

τmax maximum shear stress, MPa (psi) 

τo maximum orthogonal shearing stress, MPa (psi) 

(τmax)ref reference value of maximum shear stress, MPa (psi) 

(τmax)rh maximum shear stress modified by residual and hoop stress, MPa (psi) 

Subscripts: 

adj adjusted life 

eff effective, used to adjust the outside diameter of the inner ring in Equations (7) and (8) 

eq equivalent 

h hoop stress (in tangential or x-direction) 

IR,OR inner or outer races of bearing 

LP refers to Lundberg-Palmgren life equation (Eq. (1)) 

max refers to maximum hertz stress or maximum shear stress 

o refers to maximum orthogonal shearing stress 

R roller 

r residual stress 

RE rolling-element set 

S shaft and inner-ring bore 

x tangential direction 

y transverse direction 

Z refers to Zaretsky life equation (Eq. (3)) 

z normal direction 

Enabling Equations 

A representative cylindrical roller bearing is shown in Figure 1(a). The bearing comprises an inner 

and outer ring and a set of rollers interspersed between the two rings and positioned by a cage or 

separator. Profiles for flat (uncrowned cylindrical) and aerospace roller profiles are shown in Figure 1(b) 

and typical stress profiles across the rollers in Figure 1(c). Flat rollers have no crowning except for a 

small edge break at the ends of the rollers. The aerospace profile is defined by a flat portion in the center 

of the roller and a crown radius at each end. 

Figure 2(a) is a schematic of the contact of a cylindrical roller on a race. Figure 2(b) shows the 

principal stresses at and beneath the surface. From these principal stresses the shearing stresses can be 

calculated. Three shearing stresses can be applied to bearing life analysis: the orthogonal, the octahedral, 

and the maximum shearing stress. 

In 1947, Lundberg and Palmgren (Ref. 3) related the fatigue life L of a radially loaded bearing to the 

critical shear stress τo, stressed volume V and depth to the critical shear stress z: 

 h m

c m m

o

z

V

L /

1/ /

1

τ

1 ~ 



 



 













 (1) 

where c, m, and h are exponents chosen to fit available experimental data. Zaretsky et al. (Ref. 17) show 

that Equation (1) is based on earlier work by Weibull but with the addition of the term involving z, the 
NASA/TM—2015-218893 5 

depth to critical shear stress. The rationale for the term involving z is that a significant portion of the 

fatigue life represents the time required for a crack to propagate to the surface and produce a fatigue spall. 

A more important reason is that by adding this term, Equation (1) better fits experimental bearing fatigue 

data available at that time. 

The Lundberg-Palmgren life equation can be summarized by Equation (2), where Smax is the 

maximum Hertz stress, exponent n = 8 for roller bearings with line contact, and n = 9 for ball bearings 

with point contact. 

n

S

L 















max

1 ~ (2) 

Zaretsky et al. (Ref. 17) modified the Lundberg-Palmgren life equation to better fit post-1960 life data for 

bearings made from vacuum-processed steel, which have much longer lives, particularly at light load. 

c m

V

L

1/

max

1 1 ~ 

































 (3) 

The Zaretsky life equation (Eq. (3)), which does not include the term involving the depth to the 

critical shearing stress, results in the value n = 10 for the Hertz stress-life exponent n in Equation (2) for 

roller bearings with line contact and approximately n = 11 to 12 for ball bearings with point contact. 

Zaretsky et al. (Ref. 17) suggest that these larger values are more appropriate for contemporary steels. 

Lundberg and Palmgren (Ref. 3) chose the orthogonal shearing stress as the critical stress in 

Equation (1). The orthogonal shearing stress is not affected by hoop or residual stress, therefore, 

Equation (1) is not appropriate where these additional stresses are present. The Zaretsky life equation 

employs the maximum shearing stress in Equation (3). For the analysis reported herein, the maximum 

shearing stress is considered. There is also a difference in the stressed volume between Equations (1) and 

(3) because the depth to the maximum shearing stress is greater than the depth to the orthogonal shearing 

stress. 

For a frictionless contact, the maximum in-plane shearing stress is one-half the maximum difference 

between principal stresses: 

2 max

z  x   (4) 

From Reference 1, the maximum shear stress due to Hertzian loading for a cylindrical roller bearing with 

line contact is 

max 300 max    0. S (5a) 

where Smax is the maximum Hertz stress. Although Smax is compressive, it is represented here as a positive 

number. For a ball bearing with point contact and with typical race conformity f = 0.52, the corresponding 

relation is (Ref. 16) 

max 317 max   0. S (5b) 

The fatigue life of a bearing race is inversely related to the magnitude of the maximum shearing stress, 

τmax created by the loading on the bearing to the exponent c. For calculation purposes, c can be taken as 9. 

The maximum shearing stress τmax may be modified by the presence of hoop stress from an interference 

fit and/or by residual stress from heat treatment of the bearing race (Refs. 1 and 16). The residual and 

hoop stresses both occur in the σx direction of Figure 2(a). We designate the shearing stress modified by 

hoop and residual stress as (τmax)rh. 
NASA/TM—2015-218893 6 

  ( ) 2

1 max rh max  r  h   (6) 

The pressure pi at the interface from a bearing shrunk on a solid shaft is given by Equation (7) (adapted 

from Juvinall (Ref. 18)) 

  2 2

eff

2

eff 2

S

i

S

E D D

p

D D

 

 (7) 

where E is Young’s modulus,  is the diametral interference, DS is the common diameter of the shaft and 

bearing bore, and Deff is the effective thick-wall cylinder diameter of the inner ring. 

For a bearing without a shoulder on the inner race (as is typical for a roller bearing), the effective 

inner-ring outside diameter Deff is simply the outside diameter of the inner race. For a ball bearing with an 

inner-race shoulder, Deff can be calculated by adding the area of the shoulders of the bearing race to the 

“shoulderless” inner race and dividing by the width of the ring. This procedure is described in 

Reference 16. 

The hoop or tangential stress σh in a bearing ring is calculated at the location of maximum shear stress 

D beneath the surface of the inner race. The hoop stress is given by Equation (8) (Ref. 16) as adapted 

from Reference 18 

2 2

eff

2 2

eff

1 i S

h

S

pD D

D D D

                  

 (8) 

In Equations (4) to (6), the maximum shear stress τmax is defined to be negative. The hoop stress σh is 

normally positive (tensile) and compressive residual stress σr is negative. The life L may be represented as 

an inverse function of τmax to the power c, where c, for convenience, can be taken to be 9 although other 

values can be used, such as 9.3 from Lundberg and Palmgren (Ref. 3) or 10.3 from Shimizu et al. 

(Ref. 19). 

 c L

max rh

1 ~



 (9) 

The life ratio for a bearing race can be taken as the ninth power of the ratio of τmax to (τmax)rh, where 

τmax and (τmax)rh are defined by Equations (5) and (6). The life ratio in Equation (10) becomes unlimited 

when τmax = 1⁄2(σr + σh) and thus (τmax)rh approaches zero. 

 

 

c

L

L LR



















  

max rh

rh max (10) 

Using Equations (5) to (10), we can calculate a life factor for a bearing race. The factor must be 

calculated separately for the inner and outer races because the loading conditions are not equal. In this 

article, we are using life results from a database (described below) taken at a reference value of maximum 

Hertz stress, (Smax)ref = 1710 MPa (248 ksi). From this value, the reference maximum shearing stress 

(τmax)ref is calculated from Equations (5a) or (5b). 

   

 

 

c

r

c

LF FM 

















  



















 

max ref

max ref

max rh

max rh

/ 2

 (11) 
NASA/TM—2015-218893 7 

where the material life factor FM is based on experimental fatigue data, the maximum shear stress τmax is 

calculated from Equations (5a) or (5b) and the maximum shearing stress modified by hoop and residual 

stress is calculated from Equation (6). 

Lundberg and Palmgren (Ref. 3) expressed the bearing system fatigue life in terms of the lives of the 

inner and outer races. However, the race lives LIR and LOR in Equation (12) implicitly include the life of 

the rolling-element set. 

m

OR mIR mL L L

1 1 1   (12) 

In order to consider the effect of residual stress and hoop stress on the various components of a 

bearing, the life of the rolling-element set must be separated from the race lives. This can be done by 

employing Zaretsky’s Rule (Refs. 1 and 20) to calculate adjusted race lives designated (LIR)adj and (LOR)adj. 

The adjusted lives in Equation (13) are greater than the corresponding race lives of Equation (12). 

Application of Zaretsky’s Rule is described in detail by Oswald et al. (Refs. 15 and 16) and in Appendix A. 

The life factor computed from Equation (11) must be calculated and applied separately for the inner 

and outer races and for the rolling elements. From Zaretsky’s Rule, (Appendix A), the life of the rollingelement set of a cylindrical roller bearing or deep groove ball bearing is assumed equal to the life of the 

outer race. Likewise, the life of the ball set of an angular-contact ball bearing is assumed equal to the life 

of the inner race (Refs. 20 and 21). The life L of the bearing is given by Equation (13). 

m

OR

m

RE

m

IR

mL L adj L L adj

1 1 1 1

 

   (13) 

where m is the Weibull modulus (slope) and LRE is the life of the rolling-element set (balls or rollers). 

Zaretsky’s Rule does not change the bearing life L; it simply redistributes the failure probabilities. By 

using Zaretsky’s Rule in conjunction with a life factor as defined by Equation (11), we can apply the 

effect of hoop stress and residual stress to bearing life calculated by the Lundberg-Palmgren life equation 

(Eq. (1)). 

Bearing Material Database 

Three bearing steels were considered in this study: through-hardened AISI M-50 and case-carburized 

AISI 9310 and M50 NiL (AMS 6278). All of these materials were VIM-VAR (double-vacuum) 

processed, where VIM designates vacuum induction melted and VAR means vacuum arc remelted. 

AISI M-50 is a molybdenum-based through-hardened tool steel that is often used for rolling-element 

bearings in turbine engines. The near-surface microstructure (Fig. 3(a)) shows course bands of white 

carbides against a dark martensitic matrix. The AISI M-50 used in the test gear database was processed to 

limit retained austenite to less than 1 percent. 

AISI 9310 is a nickel-chromium-molybdenum steel with finer carbides (Fig. 3(b)) and no banding. 

The carbide structure of M50 NiL (Fig. 3(c)) shows very fine carbides dispersed evenly within the 

microstructure. Case carburizing produces a beneficial compressive residual stress in bearings made from 

AISI 9310 and M50 NiL. 

NASA spur gear fatigue data from References 14 and 22 were used to develop material life factors for 

these three steels. Spur gears produce line contact conditions, similar to a cylindrical roller bearing. The 

tooth profile in the axial direction is crowned and is finished identical to that for bearing rollers. Because 

the test gear ratio is one to one, that is both gears in contact are the same size and identical, the Hertzian 

stressed volume for each gear tooth is always in contact with the same and opposite stressed volume on 

the second gear. This leads to a more consistent and uniform contact and less scatter in the fatigue results 

manifested in higher Weibull modulus or slope. 

The failure morphology for spall formation in these gears is identical to that of classical rollingelement fatigue in ball and roller bearings. For the gears, crack initiation begins at or just above the pitch 
NASA/TM—2015-218893 8 

line of the gear tooth, in the subsurface region of the maximum resolved shear stress. The largest value of 

the maximum Hertzian stress occurs at the pitch line of the gear teeth in contact. At this location, nearly 

pure rolling occurs. There are no significant traction forces present necessary to modify the subsurface 

critical shearing stresses. The crack or cracks form a network that propagates to the surface resulting in 

classical fatigue spalls (Ref. 1). 

Gear fatigue data for the three steels from a maximum Hertz stress level of 1710 MPa (248 ksi) are 

plotted in Figure 4 and summarized in Table 1. Using the method of Johnson (Ref. 23), the 90 percent

confidence band was determined for the L10 life of each steel. Based upon the Weibull slope m and the 

number of fatigue failures for each steel reported, statistically 90 percent of the time value of the L10 life 

will fall between the values calculated in Table 1. 

X-ray diffraction was used to determine the magnitude of compressive respective residual stress as a 

function of depth below the surface. Electro-polishing removed surface material to each depth value 

required for these measurements. The resulting measured residual stress below the surface for each steel 

is plotted in Figure 5. 

Weibull slopes for rolling-element bearings generally range from 1.0 to 2.0, while for gears; the 

Weibull slope is normally between 2.0 and 3.0. The Weibull slope is subject to statistical variability 

relating to the size of the database (the number of specimens failed) (Refs. 20 and 24). However, the data 

of Table 1 for the AISI M-50 steel reflects a higher than expected Weibull slope of 3.8. According to 

Jalahamadi and Sadeghi (Ref. 11), this could be reflective of steel, without initial flaws in the grain 

structure. However, it can result in quantitative differences at different levels of reliability. Accordingly, 

we have restricted our analysis and conclusions to a 90-percent probability of survival or L10 life. 

The material life factors from the gear fatigue experiments as shown in the right column of Table 1 

were calculated by dividing the L10 life for each steel by the reference L10 life for AISI M-50. These 

material factors FM were applied to the lives of the individual bearing components (inner and outer races 

and rolling elements). 

In calculating relative rolling-element bearing lives, adjustments were made to account for the effect 

of residual stress from case carburization of the steel and the effect of hoop stress due to an interference 

fit. In order to account for the effect of compressive residual stress, the reported gear life values were 

normalized to a maximum Hertz stress of 1710 MPa (248 ksi). 

Example Calculations 

COBRA AHS (Advanced High Speed), a commercial bearing analysis code (Ref. 25), was used to 

calculate the inner- and outer-race fatigue lives, maximum Hertz stresses, and depth below the surface to 

the maximum shearing stress for radially loaded 210-size cylindrical roller bearings and thrust-loaded 

210-size angular-contact ball bearings. Unfactored lives (without life adjustment factors) were used in this 

analysis. 

Roller bearings were modeled with zero internal clearance and an aerospace crown on the rollers 

(Fig. 1(b)) with 61.5 percent flat length and with the crown radius equal to 100 times the roller length. 

Angular-contact ball bearings were modeled with a 25° free contact angle, with race conformity f = 0.52, 

with shoulder height of 20 percent of the ball diameter on one side of the inner race and relieved 

shoulders on the other side of the race. 

Bearing tolerances are typically specified by an RBEC or ABEC number, which is an odd integer 

between 1 and 9. RBEC and ABEC stand for Roller (and Annular) Bearing Engineering Committee of the 

American Bearing Manufacturers Association (Ref. 26). In this paper, we assume RBEC or ABEC–5, a 

medium tolerance level. 
NASA/TM—2015-218893 9 

Four load cases were chosen to produce particular values of inner-race maximum Hertz stress, Smax: 

1380, 1710, 1900, and 2415 MPa (200.2, 248.0, 275.6, and 350.3 ksi). The 1710 MPa (248 ksi) maximum 

Hertz stress was designated the reference level of maximum Hertz stress, (Smax)ref because this was the 

stress level used for the gear fatigue tests in our database (Refs. 14 and 22). The reference value of 

maximum shear stress, (τmax)ref = –513 MPa (–74.4 ksi) was calculated from (Smax)ref using Equation (5a). 

The maximum shear stress τmax for ball bearings as calculated from the maximum Hertz stress Smax in 

Equation (5b) differs by about 6 percent from the value from Equation (5a) for roller bearings. We did not 

adjust (Smax)ref for ball bearings to account for this difference. The error in the calculated relative life from 

not making this adjustment for ball bearings is about 1 percent. 

The procedure can best be illustrated through examples. The following is similar to examples given in 

References 15 and 16 except the bearing radial load and fit are different and here we add the effect of 

residual stress. 

Life of 210-Size Roller Bearing at Reference Hertz Stress With 

M50 NiL Inner Race and m6 Fit 

Consider a 210-size cylindrical roller bearing, as shown in Figure 1, with bore diameter D = 50 mm 

(1.9685 in.). With no shoulders on the inner ring, the effective inner-ring outside diameter is equal to the 

inner-race diameter. Thus Deff = 57.65 mm (2.2697 in.). The elastic modulus E = 205,878 MPa 

(29.86×106

 psi) and Poisson’s ratio  = 0.3. For this example, we have chosen an m6 fit; the tightest fit 

recommended for a 50-mm-bore bearing. 

If the inner ring is made to RBEC 5 tolerance and mounted with an m6 fit at the tight end of the 

tolerance band (Ref. 15), it will have an interference of 0.033 mm (0.0013 in.). Assuming both the 

bearing bore and shaft have a fine ground finish, the interference will be reduced by 0.004 mm due to 

asperity smoothing (Ref. 15), which means the actual interference fit is 0.029 mm (0.0011 in.). 

The applied radial load of 15,770 N (3525 lbf) was chosen to produce an inner-race maximum Hertz 

stress of Smax = 1710 MPa (248 ksi). From Equation (5a), τmax = –0.3(1710) = –513 MPa (–74.4 ksi). Note 

that for this case, Smax = (Smax)ref and τmax = (τmax)ref. 

From the analysis code (Ref. 25), the depth to maximum shear stress z = 0.127 mm (0.0050 in.). 

Thus, the diameter to the maximum shear stress: D = Deff – 2z = 57.396 mm (2.2597 in.). From 

Equations (7) and (8), the fit produces an interface pressure pi = 14.794 MPa (2.146 ksi) and the hoop 

stress at diameter D is h = 90.22 MPa (13.09 ksi). We assumed that the bearings have initial internal 

clearance such that with the interference fit applied the operating internal clearance is zero. 

The analysis code (Ref. 25) gave the unfactored life of the bearing as L10 = 118.4 million rev with 

inner- and outer-race lives of LIR = 131.8 and LOR = 816.99 M rev. These lives implicitly include the effect 

of the life of the rolling-element set. Zaretsky’s Rule (Appendix A, Refs. 1, 15, and 16) was used to 

separate the life of the roller set from the lives of the races. The result, with a Weibull slope m = 1.125 is 

LIR-adj = 145.05 M rev, LOR-adj = LRE = 899.14 M rev. The L10 life of the bearing is not changed by this 

procedure. 

Next, we adapted the adjusted inner-race life to account for effects of residual and hoop stress. We 

assumed the inner ring of the bearing is made from case-carburized M50 NiL with a residual stress of 

σr = –400 MPa (–58 ksi) and a material life factor FM = 3.6. We assumed the outer ring and rolling 

elements are made from through-hardened AISI M-50 with no residual stress and with FM = 1.0. For the 

inner ring, using Equation (6), (τmax)rh = –513 –1⁄2(–400+90.22) = –358.1 MPa. 

Using Equation (11), we calculated the life factor for the inner race: 

    1.07

513

513 400 / 2

358.1

513 3.6

9 9

rh  



 







  





 







 LF  (14) 
NASA/TM—2015-218893 10 

Thus, the modified life of the inner race is (LIR)rh = 1.07(145.05) = 155.2 M rev. The lives of the outer 

race and roller set do not change. From Equation (13) by substituting (LIR)rh for LIR-adj, the life of the entire 

bearing is 

1.125 1.125 1.125 1.125 899.14

1

899.14

1

155.2

1 1   

L 

  125 10 rh L  M rev 

This bearing, if made with M50 NiL inner race with an m6 fit and AISI M-50 outer race and rollers 

and operating at the reference load that produces a maximum Hertz stress, Smax = 1710 MPa, has a relative 

life of 125.0/118.4 = 1.06. At this Hertz stress, the beneficial effect of the M50 NiL steel is slightly 

greater than the detrimental effect of hoop stress from the m6 fit of the inner race on its shaft. 

Without the effect of the material factor and residual stress, the fit alone produces relative life for the 

entire bearing of 0.52. In the absence of an interference fit, the use of M50 NiL steel in the inner race 

alone, with a residual stress of 400 MPa (58 ksi) reduces the effective inner-ring maximum shearing stress 

from 513 to 313 MPa (74.4 to 46.0 ksi). This combined with the material factor produces a relative life 

for the inner ring of 307 and for the entire bearing of 2.3. 

Modifying Life Results for a Different Value of Hertz Stress 

The example above can be modified for a different load (thus a different stress level) by either 

performing a new analysis with the bearing code or, as shown here, by using Equation (16), which is 

based on Equation (2), to re-compute the adjusted lives of the inner and outer races. From the LundbergPalmgren life theory (Refs. 3 and 4) for roller bearings with line contact, the exponent n = 8. 

n

S

S

L L 













 

2

1

max

max

2 1 (16) 

With a smaller applied load such that the inner-race maximum Hertz stress is reduced from Smax = 1710 to 

1380 MPa (200 ksi), the adjusted inner-race life will be LIR-adj = 145.05(1710/1380)8

 = 806.2 M rev. 

In performing the analysis for this paper, we have observed that the ratio LOR/LIR remains nearly constant 

at 6.20 for this bearing even with significant changes in the loading and stress. Therefore, LOR-adj = LRE = 

(6.20) 806.2 = 4998 M rev. 

The life of the bearing from Equation (13) is L10 = 658 M rev. The relative life of the bearing 

(compared to the life at the reference stress) is RL = 658/118.4 = 5.6. The L10 life is 3.5 percent greater 

than the value obtained through a new analysis with the code. Later we show that for the crowned rollers 

modeled here, the Hertz stress-life exponent, n = 8.2, rather than 8.0. With exponent n = 8.2, the error in 

the L10 life is –0.7 percent. 

The analysis code results included the depth below the surface to the maximum shear stress 

z = 0.1024 mm (0.00403 in.). From z we calculated the diameter to the maximum shear stress D = Deff – 

2z = 57.445 mm (2.262 in.). The interface pressure was unchanged from the previous example at 

pi = 14.79 MPa (2.146 ksi) and the hoop stress at diameter D is h = 90.14 MPa (13.07 ksi). 

To complete this example, in addition to the hoop stress, we assumed as above that the inner ring of 

the bearing is made from M50 NiL with a residual stress of σr = –400 MPa (–58 ksi) and a material life 

factor FM = 3.6. We also assume the outer ring and rolling elements are made from through-hardened 

AISI M-50 with no residual stress and with FM = 1.0. From Equation (6), (τmax)rh = –414.0 –1⁄2(–

400+90.14) = –259.1 MPa (–37.6 ksi). 

(15a) 

(15b) 
NASA/TM—2015-218893 11 

Using Equation (11), where (τmax)ref = 513 MPa (74.4 ksi), the life factor for the inner race is 

    2.87

513

513 400 / 2

259.1

414 3.6

9 9

rh  



 







  





 







 LF  (17) 

The adjusted life of the inner race is (LIR)rh = 2.87(806.2) = 2311 M rev. From Equation (13) by 

substituting (LIR)rh for LIR-adj, the life of the entire bearing is (L10)rh = 1344 M rev. This bearing made with 

M50 NiL inner race with an m6 fit and AISI M-50 outer race and rollers and operating at the load that 

produces a maximum Hertz stress Smax = 1380 MPa has a relative life RL = 1344/118.4 = 11.4. Note that 

the relative life is given with respect to the L10 life computed at the reference stress. 

Life Results With the Zaretsky Life Equation 

The computer code and the procedure illustrated above implicitly employ the Lundberg-Palmgren life 

equation (Eq. (1); Lundberg and Palmgren (Refs. 3 and 4)), which assumes a Hertz stress-life exponent 

n = 8 for line contact and n = 9 for point contact. The procedure can be modified for the Zaretsky life 

equation (Eq. (3)) by removing the term involving the depth to critical shear stress in Equation (1) and by 

adjusting for the difference in critical shear stress and in the stressed volume. 

In Appendix B, we derived Equation (18) to convert the adjusted inner- and outer-race roller bearing 

lives from the Lundberg-Palmgren life equation to the Zaretsky life equation, where LZ is the L10 life from 

the Zaretsky Equation and LLP is the life from the Lundberg-Palmgren equation. 

  

2.071

0.5

1 0.1254 













  b

LZ k LLP (18) 

The constant k incorporates the dimension of 1/b2.071, where b is the depth to the critical shear stress. 

The value of k is unknown. The constant k will vary for different units of b. We assumed that k is unity 

for b expressed in millimeters. Appendix B includes a similar expression for converting lives of ball 

bearings. 

For the inner race of our roller bearing example, the contact half width b = 0.1614 mm (from analysis 

code). Thus, (LIR-adj)Z = (1)(145.05)(0.1254) [1/(0.5 · 0.1614)]2.071 = 3340 M rev. For the outer race, the 

analysis code yields, b = 0.1945 mm. Thus, LOR-adj = LRE = 899.14 (0.1254)[1/(0.5 · 0.1945)] 2.071 = 14,074 

M rev. 

From Equation (13) the life of the entire bearing is (L10)rh = [1/(LIR-adj)Z

m + 2/(LIR-adj)Zm ]-(1/m) = 2482 M 

rev. This life is 21 times the value calculated by the analysis code, which implements the LundbergPalmgren life equation (Refs. 3 and 4). 

The process to adjust the life of the inner race for M50 NiL steel and for an m6 interference fit is 

similar to the procedure described above. The inner-race life factor is the same as calculated in the 

previous section: (LFIR)rh = 1.07, thus the life of the inner race becomes 1.07 (3340) = 3579 M rev. The 

roller-set and outer-race lives are unchanged at 14,074 M rev. The life of the bearing from Equation (13) 

becomes 2607 M rev. The relative life RL for the Zaretsky life equation (as compared to the analysis code 

unfactored life) is RL = 22. 

Modifying Zaretsky Life Equation Results for a Different Hertz Stress 

To convert the life results in the section above for a new value of maximum Hertz stress, we used 

Equation (16) but with exponent n = 10. LIR-adj = 3340 (1710/1380)10 = 28,503 M rev. Likewise, LOR-adj = 

LRE = 14,074 (1710/1380)10 = 120,115 M rev. The adjusted outer-race life can also be obtained directly 

from the inner-race life as above except that the ratio LOR/LIR for roller bearings with the Zaretsky 

equation is 4.21 instead of the 6.20 obtained above. The life of the bearing from Equation (13) is 
NASA/TM—2015-218893 12 

L10 = 21,182 M rev. The relative life (compared to the unfactored life at the reference stress) is 

RL = 21,182/118.4 = 179. 

The L10 life found here is 5.2 percent less than the value obtained by applying Equation (18) to the 

results of the analysis code for this new stress value. Later we show that for the crowned rollers modeled 

here, the stress-life exponent is 10.3, rather than 10.0. With an exponent of 10.3, the error in the L10 life is 

+1.3 percent. The more accurate value was used in the results below. 

To calculate the effect on bearing life from using M50 NiL in the inner race and with an m6 fit, we 

note that much of the data from the second example above does not change, including Smax, τmax, b, z, 

diameter D, and life factor (LFIR)rh. The inner-race life is (LIR)rh = 2.87(28,496) = 81,690 M rev. LOR-adj = 

LRE = 120,085 M rev (unchanged from just above). 

Using Equation (13), the life of the bearing from the Zaretsky life equation (Eq. (3)) with M50 NiL 

inner race and m6 fit and AISI M-50 outer race and rollers without residual or hoop stress is L10 = 39,013 

M rev. The relative life as compared to the analysis code unfactored life at the reference stress and 

without residual stress or hoop stress is RL = 39,013/118.4 = 330. We showed above that using the 

Lundberg-Palmgren life equation, RL = 11.4. 

Results and Discussion 

Effect on Hertz Stress-Life Relation 

The procedure illustrated above was applied to radially loaded, 210-size cylindrical roller bearings 

with zero operating internal clearance and to thrust-loaded angular-contact ball bearings with a free 

contact angle of 25°, race conformity f = 0.52 and 20 percent shoulder height on one side of the race. 

Properties for the bearings are summarized in Table 2. Three steel materials were compared: (1) AISI 

M-50 through-hardened steel; (2) AISI 9310 case-carburized steel; and (3) M50 NiL (AMS 6278) casecarburized steel. 

The relation between the relative life of the inner race alone and the maximum Hertz stress for a 

cylindrical roller bearing with uncrowned (flat) rollers is shown in Figure 6(a) for the three steels 

described in Table 1. Life was calculated, using the bearing analysis code, which is based on the 

Lundberg-Palmgren life equation and then adjusted by Equation (11) and shown relative to AISI M-50 

steel at the reference stress level (Smax)ref = 1710 MPa (248 ksi). The stress-life exponent n is the expected 

value 8.0 for AISI M-50 steel. For AISI 9310 steel, with 200 MPa compressive residual stress, exponent n

increases to 10.1. For M50 NiL (AMS 6278), with 400 MPa compressive residual stress, n becomes 13.5. 

The corresponding relative life of the inner race alone of angular-contact ball bearings is given in 

Figure 6(b). The stress-life exponent n for AISI M-50 is 9.0, increasing to 11.1 and 14.5 for AISI 9310 

and M50 NiL steels, respectively. 

The relative lives calculated using both the Lundberg-Palmgren and Zaretsky life equations for 

210-size cylindrical roller bearings and angular-contact ball bearings at the reference level of maximum 

Hertz stress (Smax)ref = 1710 MPA (248 ksi) are given in Table 3. Relative lives are shown for both the no 

interference fit case and for an m6 interference fit at the tight end of the RBEC/ABEC tolerance class. 

Because 1710 MPa equals the reference stress level (Smax)ref, and AISI M-50 is the reference material, 

the relative life for both ball and roller bearings from the Lundberg-Palmgren equation equals 1.0 by 

definition for AISI M-50 steel at zero hoop stress. One might expect that the no hoop stress relative life 

for AISI 9310 would equal the value of FM = 0.8. However, Smax at the outer race is less than (Smax)ref. 

Therefore, the life is greater at the outer race and thus the relative life of the entire bearing is slightly 

greater than 0.8. Likewise, for bearings made from M50 NiL, the relative life is slightly greater than the 

value of FM = 3.6 for this steel. 

The lives shown in Table 3 for the Zaretsky life equation are about 22 times as high as the 

corresponding values for the Lundberg-Palmgren equation for roller bearings and 15 times as high for ball 

bearings. The difference in lives is greater for lower stress levels and less for higher stress because of the 

change in the value of the stress-life exponent n. The addition of a heavy m6 fit significantly reduces the 
NASA/TM—2015-218893 13 

life for all of the materials, by as much as a factor of 5 for roller bearings and a factor of 3 for ball 

bearings. 

The specific life value for the Zaretsky life equation will depend on the value of the conversion 

constant k in Equation (18), for which we have arbitrarily assigned a value of unity. Test data are required 

to establish the actual value of this constant. With k = 1.00, the life results give approximately the values 

we expect. 

The relative lives of both radially loaded cylindrical roller bearings and thrust-loaded, angular-contact 

ball bearings as predicted by the Lundberg-Palmgren (designated “LP model”) and Zaretsky (designated 

“Zaretsky model”) life equations are plotted in Figures 7 and 8. The Lundberg and Palmgren (Refs. 3 and 

4) theory assumes a stress-life exponent n = 8 for roller bearings with line contact and n = 9 for ball 

bearings, with point contact. The curves marked “Zaretsky model” involve the conversion based on 

Equation (B1), which produces a stress-life exponent n = 10.3 for roller bearings with aerospace crown 

and 11.1 for ball bearings. With either model, the relationship is altered by the presence of any effects that 

change the subsurface shearing stress. 

Each plot includes results for both no hoop stress (solid lines) and for an m6 interference fit between 

the bearing bore and shaft (dashed lines). The dotted lines indicate a relative life of 1.0 for the reference 

stress (Smax)ref = 1710 MPa (248 ksi). The curves are located horizontally relative to the reference steel 

AISI M-50 by the material life factor, FM, which was taken from the database described above. 

For roller bearings with no interference fit, the Hertz stress-life exponent nLP = 8.2 for AISI M-50 

roller bearings, according to the Lundberg-Palmgren (LP) model shown in Figure 7(a). The exponent 

differs slightly from the expected value n = 8.0 because the rollers were modeled with crowning. If flat 

rollers are modeled (as in Fig. 6(a)), then the stress-life exponent for roller bearings will be n = 8.0. 

For roller bearings made from AISI M-50 the life with the Zaretsky (Z) model is 21 times the life for 

the Lundberg-Palmgren (LP) at the reference stress. In addition, the stress-life exponent nZ increased to 

10.3. With an m6 fit, the life is reduced by almost 50 percent with either model. The stress-life exponent 

is reduced to 7.6 for the LP model and to 9.7 for the Zaretsky model. 

Similar results were obtained for roller bearings made with AISI 9310 steel (Fig. 7(b)), which has a 

compressive residual stress σr = –200 MPa (–29 ksi). The relative life at the reference stress is slightly 

lower than for M-50 (RL = 0.86 for the LP model and RL = 19 for the Zaretsky model) while the exponent 

nLP increases to 10.3 for the LP life model and to nZ = 12.4 for the Zaretsky model. An m6 fit reduces life 

by nearly 60 percent with either model. 

The analysis predicts that bearings made from M50 NiL steel (AMS 6278), which has residual stress 

σr = –400 MPa (–58 ksi) (Fig. 7(c)) will have much higher life, roughly four times the life of M-50 at the 

reference stress level. In addition, with no interference fit, the stress-life exponent nLP increases to 13.8 

and to nZ to 15.9, which means that life at light loads increases much more. The curves are not straight: at 

lower Hertz stress the effect of the residual stress becomes more important. If the Hertz stress becomes 

small enough, the denominator of the first fraction in Equation (11) will approach zero and the predicted 

life will become unlimited. An m6 fit reduces life by nearly 70 percent; however, the life is still longer 

than with M-50 steel with no fit. 

For critical bearing applications where large interference fits are required between the inner ring and 

shaft, case carburized steels having fracture toughness and high compressive residual stresses should be 

considered, particularly for the inner ring. Figure 7(d) shows the behavior of roller bearings made from a 

combination of M50 NiL for the inner ring and AISI M-50 for the outer ring and rolling elements. The life 

and stress-life exponent lie between what was observed above for the AISI M-50 and M50 NiL used alone. 

Figure 8 shows similar results for thrust-loaded angular-contact ball bearings. For AISI M-50 steel, 

which does not have residual stress and without interference fit, nLP = 9.0 and nZ = 11.1 for the LP and 

Zaretsky models, respectively (Fig. 8(a)). With an interference fit, the stress-life exponents become 8.6 

and 10.7, respectively. 

The relative lives between bearings made from different steels is a function of a combination of the 

Hertz stress level, inner-ring interference fit and any compressive residual stress in the ring material. With 

the case-carburized steels, the life at light loads (thus low values of Smax) is much longer than with AISI 
NASA/TM—2015-218893 14 

M-50 through-hardened steel. This effect makes the stress-life exponent much greater than would 

otherwise be expected. 

To illustrate the effect of residual stress without hoop stress from an interference fit on a radiallyloaded roller bearing, we compare the effect of substituting case carburized M50 NiL steel for AISI M-50 

on the inner ring alone. The M50 NiL has a residual stress of 400 MPa (58 ksi), while AISI M-50 

generally has no residual stress. 

Consider first a lightly loaded bearing with inner-ring maximum Hertz stress, Smax = 1380 MPa 

(200 ksi). The residual stress reduces the maximum shearing stress, τmax from 414 to 214 MPa 

(60 to 31 ksi). The change in shearing stress in addition to the material life factor FM for M50 NiL 

(Table 1), increases the life of the inner race by a factor of 16 and increases the life of the entire bearing 

by 3.56 times. 

However, for a heavy load case, where Smax = 2415 MPa (350 ksi) the effect on life is different 

because of the greater stress-life exponent of M50 NiL. Although the residual stress reduces the 

maximum shearing stress, τmax from 724.5 to 524.5 MPa (105 to 76 ksi), the life of the inner race is lower 

than the life of an AISI M-50 race by 23 percent and the life of the entire bearing is lower by 19 percent. 

Effect on Load-Life Relation 

From Hertz theory as given by Jones (Ref. 27), the maximum contact (Hertz) stress in a rollingelement bearing can be expressed as a function of the load. For line contact in roller bearings, the 

relationship is given by Equation (19a) and for point contact in ball bearings, by Equation (19b). 

Smax ~ P (19a) 

3 Smax ~ P (19b) 

Combining Equation (2) with Equation (19) allows us to relate the life to the applied bearing load P to the 

power p, or 

p

P

L 



 



 1 ~ (20) 

Thus, for roller bearings with line contact p = n/2 and for ball bearings with point contact p = n/3. Using 

the Lundberg-Palmgren life equation (Eq. (2)), for roller bearings, where n = 8 results in 

p  n / 2  4 (21a) 

and for ball bearings, where n = 9 results in 

p  n /3  3 (21b) 

Using the Zaretsky life equation (Eq. (3)) for roller bearings, where n = 10 results in 

p  n /2  5 (22a) 

and for ball bearings, where n = 11.1 results in 

p  n /3  3.7  4 (22b) 

The load-life exponent p will be affected by factors such as residual and hoop stresses in the bearing 

steel and in roller bearings, by roller crowning. The analysis was extended to calculate the load life 

relation for ball and roller bearings. Results for the four material combinations with and without a heavy 

interference fit are given in Tables 4 and 5. 

For roller bearings without residual or hoop stress effects and with either the Lundberg-Palmgren or 

Zaretsky life equation, the load-life exponent p was found to equal n/2.3 rather than the expected value of 
NASA/TM—2015-218893 15 

n/2. Likewise, for angular-contact ball bearings p was found to equal n/3.2 rather than the expected value 

of n/3. 

In analyzing angular-contact ball bearings, we maintained the free contact angle constant at 25°, 

allowing the loaded contact angle to increase with the applied load, which is consistent with normal use of 

such bearings. For the load that produced the reference stress Smax= 1710 MPa (248 ksi) the loaded 

contact angle was 28.8°. 

We also tried modeling a constant loaded contact angle, rather than constant free contact angle. For 

the reference stress case, if the free contact angle is reduced from 25° to 20.5°, then the loaded contact 

angle will be 25°. We also reduced the applied thrust load so that Smax remained constant at 1710 MPa 

(248 ksi). With these two changes, the life did not change (within computational accuracy) and thus the 

stress-life exponent n did not change. However, since the load changed, the load-life exponent p is greater 

than what is shown in Table 5. For this constant loaded contact angle case, the load life exponent p = n/3 

= 9/3, which agrees with Equation (21b) and the Lundberg-Palmgren theory. 

Effect of Retained Austenite on Compressive Residual Stresses 

In general, for a given through-hardened bearing steel the amount of retained austenite increases with 

increasing material hardness. Experience has also shown that test rollers made from AISI 52100 of 

Rockwell C hardnesses greater than 63 will have sufficient austenite-to-martensite transformation during 

rolling contact to alter the surface waviness and cause early surface spalling (Ref. 28). 

In 1982 Johnston et al. (Ref. 29) studied the effect of the decomposition of retained austenite and the 

inducement of compressive residual stress as a result of bearing operation. What is unique about their data 

is that the magnitudes of the compressive residual stresses are directly proportional to the decomposition 

of retained austenite (Ref. 28). 

Voskamp and Mittemeijer (Ref. 30) in 1997 reported tests that they performed at a maximum Hertz 

stress of 3.3 GPa (479 ksi) with 6309-size deep-groove ball bearings made from electroslag remelted AISI 

52100 bearing steel with 10 to 15 percent retained austenite by volume. Their tests showed the 

accumulation of plastic deformation during cyclic stressing under rolling-contact loading conditions that 

resulted in a complex state of residual stress in the subsurface volume of the deep-groove bearing inner 

races. They also cite unpublished test results that showed evidence of residual stress development in 

bainitically hardened bearing rings with less than 1 percent retained austenite. Hence, they suggest that 

the build up of residual stresses are not solely caused by the transformation of retained austenite (Ref. 30). 

Changes in microstructure (phase transformations) have been reported to occur in the same areas as 

the maximum induced residual stress (Refs. 31 and 32). Under some conditions of extremely high contact 

stresses, nonmicrostructural alteration was apparent after significant residual stresses had been induced in 

a few cycles (Ref. 32). H. Muro and N. Tsushima (Ref. 33) proposed that the induced residual stresses 

and the microstructural alterations are independent phenomena (Ref. 28). 

Research performed by D. Zhu et al. (Ref. 34) in 1985 on carburized rollers suggested that the 

structural change in the zone of maximum resolved shearing stresses observed by Jones (Ref. 35) in 1947 

and later by Carter (Ref. 36) in 1960 as well as others is a manifestation of retained austenite transforming 

to martensite under cyclic Hertzian stress conditions. A combination of thermal and strain energy and 

time is believed to cause this change (Ref. 28).

In 1972, Parker and Zaretsky (Ref. 37) made residual stress measurements on several AISI 52100 

207-size, deep-groove ball bearings that were run for different times suitable for inducing significant 

compressive residual stresses in the inner-race–ball groove. Compressive residual stresses greater than 

0.7 GPa (100 ksi) were included in the maximum shear stress region of the bearing inner race, which was 

run for 25 hr at a maximum Hertz stress of 3.3 GPa (480 ksi) and a shaft speed of 2750 rpm. Twentyseven bearings were prestressed for 25 hr at this condition and fatigue tested at a maximum Hertz stress of 

2.4 GPa (350 ksi). The results of these tests were compared with the results of baseline tests without a 

prestress cycle at identical test conditions. The L10 life of the prestressed ball bearings was greater than 

twice that of the baseline bearings. Additionally, after 3000 to 4000 hr of testing the differences between 
NASA/TM—2015-218893 16 

the measured residual stresses in the prestressed and baseline bearings were small. This result would 

suggest that the early presence of compressive residual stresses is a requirement for longer lived bearings. 

However, the time dependence of residual stress buildup with different materials under varying load may 

be an additional factor affecting bearing life. This factor has not been studied. 

In 1982, Lorosch (Ref. 38) published results of fatigue tests on three groups of vacuum-degassed 

7205B-size AISI 52100 inner races at maximum Hertz stresses of 2.6, 2.8, and 3.5 GPa (370, 406, and 

500 ksi), respectively (Ref. 38). These were very highly loaded bearings. From these tests, Lorosch 

(Ref. 38) concluded that “Under low loads and with elastohydrodynamic lubrication, there is no material 

fatigue, thus indicating that under such conditions bearing life is practically unlimited.” 

Zwirlein and Schlicht (Ref. 39) in a companion paper published concurrently in 1982 with that of 

Lorosch (Ref. 38) and using the same 7205B-size bearing inner races, reported large amounts of 

compressive residual stress due to the transformation of retained austenite into martinsite (Ref. 39). 

Lorosch (Ref. 38) and Zwirlein and Schlicht (Ref. 39) failed to account for the significantly large 

presence of these induced compressive residual stresses in their bearing raceways. Instead they assumed 

that the large increases in life that they reported were due to a ‘fatigue limit.’ Zwirlein and Schlicht 

(Ref. 39) concluded that “contact pressures (maximum Hertz stresses) less than 2.6 GPa (370 ksi) do not 

lead to the formation of pitting within a foreseeable period (Zwirlein and Schlicht (Ref. 39)). This 

corresponds to ‘true endurance strength’ ...” However, their observation is not supported by rollingelement fatigue data in the open literature for maximum Hertz (contact) stress levels under 2.6 GPa 

(370 ksi). If Lorosch (Ref. 38) and Zwirlein and Schlicht (Ref. 39) were correct, virtually no bearing in 

rotating machinery applications would fail by classical rolling-element fatigue since maximum Hertz 

stress levels in the range of 1.2 GPa (175 ksi) to 1.9 GPa (275 ksi) are typical. 

Summary of Results 

An analysis was performed using the COBRA-AHS rolling-element bearing code, to calculate the 

unfactored bearing lives for 210-size cylindrical roller bearings and angular-contact ball bearings at four 

levels of maximum Hertz stress ranging from 1380 to 2215 MPa (200 to 350 ksi). These lives were 

adjusted to account for (1) material life factors; (2) the beneficial effect of compressive residual stress 

from heat treatment of carburized steel; and (3) the detrimental effect of tensile hoop stress from a heavy 

interference fit. Results are given for both the traditional Lundberg-Palmgren life equation and for the 

Zaretsky life equation. The following results were obtained: 

1. Relative lives between bearings made from different steels is a function of a combination of the Hertz 

stress level, inner-ring interference fit, and any compressive residual stress in the ring material. 

2. For critical bearing applications, particularly where heavy interference fits between the inner-ring 

bore and shaft are required, case carburized steels with high fracture toughness and high compressive 

residual stress should be considered as the material of choice for the inner ring. 

3. For both ball and roller bearings, the Hertz stress-life exponent n and the load-life exponent p are 

functions of the combination of the compressive residual stresses in the zone of resolved maximum 

shearing stresses below the Hertzian contact and the tensile hoop stress resulting from the interference 

fit between the shaft and bearing inner-ring bore. 

4. The Zaretsky life equation predicts much longer lives and greater stress- and load-life exponents than 

the Lundberg-Palmgren life equation without resorting to life adjusting factors. The predicted lives 

from the Zaretsky life equation better match experimental life data for bearings made from vacuumprocessed steel. 
NASA/TM—2015-218893 17 

Appendix A—Zaretsky’s Rule on Determining Rolling-Element System Life 

Lundberg and Palmgren (Refs. 3 and 4) did 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 fcm, the life of the rolling-element set is implicitly included in the life 

calculation where 

m

OR

m

IR

mL L L

1 1 1   (A1) 

Equation (A1) is identical to Equation (13) of the text. 

The rationale for not including the rolling-element set in Equation (A1) appears in the 1945 edition of 

Palmgren’s book (Ref. 40) wherein he states that, “...the fatigue phenomenon which determines the life 

(of the bearing) usually develops on the raceway of one ring or the other. Thus, the rolling elements are 

not the weakest parts of the bearing ...” 

The rolling bearing fatigue data that Palmgren used to benchmark his and later the LundbergPalmgren equations were obtained under radially loaded conditions. Under these conditions, the life of the 

rolling elements as a system (set) will be equal to or greater than that of the outer race. As a result, failure 

of the rolling elements in determining bearing life was not initially considered by Palmgren. Had it been, 

Equation (A1) would have been written as follows: 

1

L10





 





m

 1

LIR





 





m



1

LRE





 





m



1

LOR





 





m

 (A2) 

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

Equation (A2) is identical to Equation (13) of the text. 

Comparing Equation (A2) with Equation (A1), the value of the L10 bearing life will be the same. 

However, the values of the LIR and LOR between the two equations will not be the same, but the ratio of 

LOR/LIR will remain unchanged. 

The fraction of failures due to the failure of a bearing component is expressed by Johnson (Ref. 23) as 

 Fraction of inner-race failures 

m

LIR

L 



 



  10 (A3a) 

 Fraction of rolling-element failures 

m

LRE

L 



 



  10 (A3b) 

 Fraction of outer-race failures 

m

LOR

L 



 



  10 (A3c) 

From Equations (A3a) to (A3c), 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. 

Equations (A3a) to (A3c) were verified using radially loaded and thrust-loaded 50-mm-bore ball 

bearings. Three hundred and forty virtual bearing sets totaling 31,400 bearings were randomly assembled 

and tested by Monte Carlo (random) number generation (Ref. 41). From the Monte Carlo simulation, the 

percentage of each component failed was determined and compared to those predicted from 

Equations (A3a) to (A3c). These results are shown in Table A1. There is excellent agreement between 

these techniques (Ref. 41). 
NASA/TM—2015-218893 18 

TABLE A1.—COMPARISON OF BEARING FAILURE DISTRIBUTIONS BASED UPON WEIBULLBASED MONTE CARLO METHOD AND THOSE CALCULATED FROM EQUATIONS (A3a) TO (A3c) 

FOR 50-mm-BORE DEEP-GROOVE AND ANGULAR-CONTACT BALL BEARINGS (REF. 41) 

Ball bearing type Component Percent failure 

Weibull-based Monte 

Carlo results 

Results from 

Equations (A31) to (A3c) 

Deep groove 

Inner race 70.1 69.9 

Rolling element 14.8 15.0 

Outer race 15.1 15.0 

Angular contact 

Inner race 45.4 45.1 

Rolling element 45.2 45.1 

Outer race 9.4 9.7 

Figure A1.—Rolling-element fatigue lives of AISI 52100 204-size angular-contact ball 

bearings. Contact angle is 10°; outer-race temperature, 79 °C (175 °F); thrust load, 

1108 N (249 lb); inner-ring speed, 10,000 rpm; lubricant, MIL-L-7808; L10 life, 

20.5106 inner-ring revolutions (34.2 hr); and failure index, 7 out of 12 (Ref. 42). 

Figure A1 summarizes rolling-element fatigue life data for ABEC 7 204-size angular-contact ball 

bearings made from AISI 52100 steel (Ref. 42). The bearings had a free contact angle of 10°. Operating 

conditions were an inner-ring speed of 10,000 rpm, an outer-ring temperature of 79 °C (175 °F), and a 

thrust load of 1108 N (249 lb). The thrust load produced maximum Hertz stresses of 3172 MPa (460 ksi) 

on the inner race and 2613 MPa (379 ksi) on the outer race. From a Weibull analysis of the data, the 

bearing L10 life was 20.5 million inner-race revolutions or approximately 34.2 hr of operation (Ref. 42). 
NASA/TM—2015-218893 19 

Seven of the twelve bearings failed from rolling-element fatigue. Two of the failed bearings had 

fatigue spalls on a ball and an inner race. Two bearings had inner-race fatigue spalls. Two bearings had 

fatigue spalls on a ball, and one bearing had an outer-race fatigue spall. Counting each component that 

failed as an individual failure independent of the bearing, there were four inner-race failures, four ball 

failures, and one outer-race failure for a total of nine failed components. Inner-race failures were 

responsible for 44.4 percent of the failures; ball failures, 44.4 percent; and outer-race failures, 

11.2 percent. Using each of these percentages in Equations (A3a) to (A3c) together with the experimental 

L10 life, the lives of the inner and outer races and the ball set were calculated. For purposes of the 

calculation, the Weibull slope m was assumed to be 1.11, the same as Lundberg and Palmgren (Ref. 3). 

The resultant component L10 lives were 53 million inner-race revolutions (88.3 hr) for both the inner race 

and ball set and 183.3 million inner-race revolutions (305.5 hr) for the outer race. 

For nearly all rolling-element bearings the number of inner-race failures is greater than those of the 

outer race. Accordingly, from Equations (A3a) and (A3c), the life of the outer race will be greater than 

that of the inner race. Zaretsky (Ref. 1) noted that for radially loaded bearings (ball or roller), the 

percentage of failures of the rolling-element set was generally equal to and/or less than that of the outer 

race. For thrust-loaded ball or roller bearings, Zaretsky (Ref. 1) further noted that the percent for the 

rolling-element set was equal to or less than that for the inner race but more than for the outer race. In 

order to account for material and processing variations, Zaretsky developed what is now referred to as 

Zaretsky’s Rule (Ref. 1): 

For radially loaded ball and roller bearings, the life of the rolling-element set is equal to or greater 

than the life of the outer race. Let the life of the rolling-element set (as a system) be equal to that of the 

outer race. 

From Equation (A2) 

m

OR

m

IR

m

L L L 















 













  













 1 2 1 1

10

 (A4) 

where LRE = LOR. 

For thrust-loaded ball and roller bearings, the life of the rolling-element set is equal to or greater than 

the life of the inner race but less than that of the outer race. Let the life of the rolling-element set (as a 

system) be equal to that of the inner race. 

From Equation (A2), 

m

OR

m

IR

m

L L L 















 













  













 1 1 2 1

10

 (A5) 

where LRE = LIR. 

Examples of using Equations (A4) and (A5) are given in References 15 and 16. As previously stated, 

the resulting values for LIR and LOR from these equations are not the same as those from Equation (A1). 

They will be higher. 

NASA/TM—2015-218893 21 

Appendix B—On Converting Life Results From the Lundberg-Palmgren Life 

Equation to the Zaretsky Life Equation 

Zaretsky et al. (Ref. 17) modified the Lundberg-Palmgren life equation (Refs. 3 and 4) by making 

three changes: (1) replacing the orthogonal shear stress with the maximum shear stress; (2) eliminating 

the dependence on the Weibull slope in the first term (which involves the shear stress τ); and 

(3) removing the term involving the depth to critical shear stress. 

If we solve for the life LZ from the Zaretsky Equation (3) in terms of the Lundberg-Palmgren life LLP, 

Equation (1), we have 

 

h m

o

m

Z

LP

c

o Z LP V z

V L k L

1/ /

max

1



















































  (B1) 

Where k is a constant that incorporates the dimension of z

h//m, where z is the depth to the critical shear 

stress and h/m = 2.071. The value of k is unknown. It will take a series of life tests to establish the value 

of this constant. In this article, we assume that k is unity when z is expressed in millimeters. The value of 

k will vary with different units of z. For example, for z in inches and with line contact, the constant 

becomes k = (1/25.4)2.071 = 0.001232. 

For roller bearings with line contact, the maximum orthogonal shear stress τo = 0.25Smax, while the 

maximum shear stress τmax = 0.300Smax, where Smax is the maximum Hertz stress. Therefore τo/τmax = 

0.25/0.300 = 0.833. Likewise, for ball bearings with typical conformity of 0.52, τo = 0.249Smax and τmax = 

0.317Smax therefore τo/τmax = 0.249/0.317 = 0.785. 

The Lundberg-Palmgren life equation is semi-empirical. The exponents for the various terms were 

chosen to fit the experimental data available at the time. In their 1952 paper Lundberg and Palmgren 

(Ref. 4) show the Weibull slope for 10 roller bearing tests, with 30 bearings in each test. The test bearings 

included tapered, cylindrical, and spherical roller bearings. The exponents varied from 0.7 to 1.4. 

Lundberg and Palmgren (Ref. 4) adjusted their exponents in order to have an integer value for p, the 

load-life exponent, where p = (c-h+1)/(2m). For roller bearings with line contact, they chose m = 9/8, 

which makes p = 4. 

For the term involving the critical shearing stress in the LP life equation (Eq. (1); Lundberg and 

Palmgren (Refs. 3 and 4), chose exponent c = 10.33 for either ball bearings or roller bearings, that is, 

L~1/τ

c/m. For roller bearings with line contact, c/m = 10.33/1.125 = 9.182, and for ball bearings with point 

contact, c/m = 9.306. 

When he modified the life relation, removing the dependence on the Weibull slope, Zaretsky adjusted 

the exponent c so that it is equal to the published values of the Lundberg-Palmgren quotient c/m. But then 

Zaretsky will not vary the exponent if the Weibull exponent is changed from the nominal value. 

Therefore, for roller bearings, the term (τo/τmax)

c

 = (0.833)9.182 = 0.1875 and for ball bearings, (τo/τmax)

c

 = 

(0.785)9.306 = 0.1057. 

The stressed volume in either life equation is the product of the circumference of the rolling-element 

running track, times the width of the contact, times the depth to the critical shear stress. In changing from 

the orthogonal to maximum shear stress, the only parameter for the stressed volume that changes is the 

depth to the critical shear stress. 

The maximum shear stress occurs at greater depth than the maximum orthogonal shear stress; 

therefore, zmax is greater than zo. For cylindrical roller bearings, zo = 0.5b, while zmax = 0.786b, where 

b = the half-width of the Hertzian contact stress zone and the Weibull slope m = 1.125. Therefore, the 

term (VLP/VZ)

1/m = (zo/zmax)1/m = (0.50/0.786)1/1.125 = 0.6689. 

For ball bearings with typical conformity, f = 0.52, zo = 0.49b while zmax = 0.767b. Thus, (VLP/VZ)

1/m = 

(0.49/0.767)1/1.11 = 0.6679. 

To remove the dependence on the depth to the critical shear stress for the Zaretsky life equation, the 

LP life is divided by zo

h/m, where for line contact, the exponent h/m = 2.33/1.125 = 2.071 and for point 

contact, h/m = 2.33/1.11 = 2.099. 
NASA/TM—2015-218893 22 

For roller bearings with line contact, Equation (B1) simplifies to Equation (B2) 

       

2.071 2.071

0.5

1 0.1254

0.5

1 0.1875 0.6689 













  













  b

k L

b

LZ k LLP LP (B2) 

and for ball bearings with point contact, Equation (B1) simplifies to Equation (B3). 

       

2.0991 2.0991

0.49

1 0.07054

0.49

1 0.1056 0.6689 













  













  b

k L

b

LZ k LLP LP (B3) 
NASA/TM—2015-218893 23 

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358–379. 

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Ruley, SKF Industries, Philadelphia, PA. 

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NASA/TM—2015-218893 25 

TABLE 1.—GEAR FATIGUE LIFE DATA FOR 1710 MPa (248 ksi) MAXIMUM HERTZ STRESS 

Material Compressive residual 

stress, 

MPa (ksi) 

Gear fatigue 

L10 life 

million cycles 

Weibull 

slope 

Material life 

factor, FM

90 percent 

limits on true 

L10 lifea

million cycles 

VIM-VAR AISI M-50b 0 60 3.8 1.0 39 to 77 

VIM-VAR AISI 9310c 200 (29) 48 1.3 0.8 20 to 94 

VIM-VAR M50 NiL 

(AMS 6278)c 400 (58) 217 2.3 3.6 117 to 688 

a

 From Johnson (Ref. 23) 

b From Townsend et al. (Ref. 22) 

c

 From Townsend and Bamberger (Ref. 14) 

TABLE 2.—BEARING PROPERTIES 

Bearing description Bore, 

mm 

Outside 

diameter, 

mm 

Number of 

elements 

Element diameter, 

d

Roller length 

mm in. mm in. 

210-size cylindrical roller bearing 50 90 10 13 0.5118 13 0.5118 

210-size angular-contact ball bearing 50 90 14 12.7 1⁄2 n/a n/a 

TABLE 3.—RELATIVE L10 LIVES FOR LUNDBERG-PALMGREN AND ZARETSKY MODELS 

AT MAXIMUM HERTZ STRESS Smax = 1710 MPa 

Steel Roller bearing Angular-contact ball bearing 

Lundberg-Palmgren 

model 

Zaretsky model Lundberg-Palmgren 

model 

Zaretsky model 

No fit m6 fit No fit m6 fit No fit m6 fit No fit m6 fit 

AISI M-50 1 0.52 21 11 1 0.67 15 10 

AISI 9310 0.86 0.37 19 8 0.82 0.49 12 7 

M50 NiL (entire bearing) 4.18 1.29 94 30 3.75 1.87 56 28 

M50 NiL (inner race only) 2.3 1.06 42 22 1.48 1.08 21 16 

TABLE 4.—LOAD- AND STRESS-LIFE EXPONENTS FOR CYLINDRICAL ROLLER BEARING WITH RADIAL LOAD

Steel 

material 

Lundberg-Palmgren life equation Zaretsky life equation 

No fit m6 shaft fit No fit m6 shaft fit 

Stress-life 

exp., n

Load-life 

exp., p

Stress-life 

exp., n

Load-life 

exp., p

Stress-life 

exp., n

Load-life 

exp., p

Stress-life 

exp., n

Load-life 

exp., p

AISI M-50 8.2 3.6 7.6 3.3 10.3 4.5 9.7 4.2 

AISI 9310 10.3 4.5 9.3 4.1 12.4 5.4 11.4 5.0 

M50 NiL 13.8 6.0 11.9 5.2 15.9 6.9 14.0 6.1 

M50 NiL (IR only) 10.9 4.7 11.0 4.8 12.5 5.4 12.8 5.6 

TABLE 5.—LOAD- AND STRESS-LIFE EXPONENTS FOR ANGULAR-CONTACT BALL BEARING 

WITH THRUST LOAD 

Steel 

material 

Lundberg-Palmgren life equation Zaretsky life equation 

No fit m6 shaft fit No fit m6 shaft fit 

Stress-life 

exp., n

Load-life 

exp., p

Stress-life 

exp., n

Load-life 

exp., p

Stress-life 

exp., n

Load-life 

exp., p

Stress-life 

exp., n

Load-life 

exp., p

AISI M-50 9.0 2.8 8.6 2.7 11.1 3.4 10.7 3.3 

AISI 9310 10.9 3.4 10.3 3.2 13.0 4.0 12.4 3.8 

M50 NiL 14.0 4.3 12.8 4.0 16.1 5.0 14.9 4.6 

M50 NiL (IR only) 10.1 3.1 10.5 3.2 12.1 3.7 12.5 3.9 
NASA/TM—2015-218893 26 

Figure 1.—Cylindrical roller bearing. 

(a) Schematic. (b) Flat and 

aerospace roller profiles. (c) Typical 

stress distribution across rollers. 
NASA/TM—2015-218893 27 

Figure 2.—Subsurface stress field under line contact 

for a frictionless contact. (a) Hertz stress distribution 

for roller on raceway showing principal stresses at 

depth z below surface. (b) Distribution of principal 

and shearing stress as a function of depth z below 

surface. 
NASA/TM—2015-218893 28 

Figure 3.—Comparison of microstructures of three steels in bearing materials 

database discussed in this article. (a) AISI M-50 steel showing carbide banding. 

(b) AISI 9310 steel. (c) M50 NiL (AMS 6278). 
NASA/TM—2015-218893 29 

Figure 4.—Gear fatigue data from test gears made from three bearing steels. (a) AISI 9310 (Ref. 14). (b) AISI M-50 

(Ref. 22). (c) M50 NiL (AMS 6278 (Ref. 14). (d) Summary. 

 

Figure 5.—Residual stress in bearing steels used for gear fatigue 

tests (Ref. 1). Residual stresses were measured by x-ray 

diffraction. 
NASA/TM—2015-218893 30 

Figure 6.—Effect of material and residual stress on stress-life relationship for single race. 

(a) Roller bearing inner race. (b) Ball bearing inner race. 
NASA/TM—2015-218893 31 

Figure 7.—Relative- and stress-life relationship for 210-size cylindrical roller bearings with aerospace roller profile 

made from different steels for both the Lundberg-Palmgren (LP) and Zaretsky (Z) life models. (a) AISI M-50. 

(b) AISI 9310. (c) M50 NiL. (d) M50 Nil inner race, AISI M-50 outer race, and rolling elements. Results are shown 

with and without m6 interference fit. Reference life is for AISI M-50 material without fit. The dotted lines show the 

reference life of 1.0 with the reference value of maximum Hertz stress of 1710 MPa (248 ksi). The life equals 1.0 

at reference stress only for the LP model for bearings made from AISI M-50. 
NASA/TM—2015-218893 32 

 

Figure 8.—Relative- and stress-life relationship for 210-size angular-contact ball bearings made from different steels 

for both the Lundberg-Palmgren (LP) and Zaretsky (Z) life models. (a) AISI M-50. (b) AISI 9310. (c) M50 NiL. 

(d) M50 Nil inner race, AISI M-50 outer race, and rolling elements. Results are shown with and without m6 

interference fit. Reference life is for AISI M-50 material without fit.