Fred B. Oswald and Erwin V. Zaretsky

Glenn Research Center, Cleveland, Ohio

Joseph V. Poplawski 

J.V. Poplawski & Associates, Bethlehem, Pennsylvania

Interference-Fit Life Factors for Ball Bearings

NASA/TM—2010-216913

November 2010
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Fred B. Oswald and Erwin V. Zaretsky

Glenn Research Center, Cleveland, Ohio

Joseph V. Poplawski 

J.V. Poplawski & Associates, Bethlehem, Pennsylvania

Interference-Fit Life Factors for Ball Bearings

NASA/TM—2010-216913

November 2010

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NASA/TM—2010-216913 1

Interference-Fit Life Factors for Ball Bearings

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

The effect of hoop stresses on the rolling-element fatigue life 

of angular-contact and deep-groove ball bearings was determined for common inner-ring interference fits at the ABEC-5 

tolerance level. The analysis was applied to over 1150 bearing 

configurations and load cases. Hoop stresses were superimposed on the Hertzian principal stresses created by the applied 

bearing load to calculate the inner-race maximum shearing 

stress. The resulting fatigue life of the bearing was recalculated 

through a series of equations. The reduction in the fatigue life is 

presented as life factors that are applied to the unfactored 

bearing life. The life factors found in this study ranged from 

1.00 (no life reduction)—where there was no net interface 

pressure—to a worst case of 0.38 (a 62-percent life reduction).

For a given interference fit, the reduction in life is different for 

angular-contact and deep-groove ball bearings. Interference fits 

also affect the maximum Hertz stress-life relation. Experimental 

data of Czyzewski, showing the effect of interference fit on 

rolling-element fatigue life, were reanalyzed to determine the 

shear stress-life exponent. The Czyzewski data shear stress-life 

exponent c equals 8.77, compared with the assumed value of 9.

Results are presented as tables and charts of life factors for 

angular-contact and deep-groove ball bearings with light, 

normal, and heavy loads and interference fits ranging from 

extremely light to extremely heavy. 

Introduction

Rolling-element bearings often utilize a tight interference fit 

between the bore of the bearing inner ring and shaft and/or 

between the outer diameter of the bearing ring and housing bore 

to prevent fretting damage at the respective interfaces. American 

National Standards Institute/American Bearing Manufacturers 

Association (ANSI/ABMA) standards (ANSI/ABMA-7 and -20

(Refs. 1 to 2)) as well as catalogs of bearing manufacturers 

specify suggested fits for various operating conditions. Any fit

must be based on the most severe operating conditions expected, 

including the highest speeds and highest vibration levels.

A tight fit of the bearing inner ring over the shaft reduces 

internal bearing clearance. Initial internal clearance can be 

added to the bearing to compensate for the clearance change. In 

addition, the interference fit of the inner ring over the shaft adds 

a hoop stress on the bearing inner ring. Czyzewski (Ref. 3) first 

showed that tensile hoop stresses induced in the bearing inner 

ring can negatively affect rolling-element fatigue life.

Coe and Zaretsky (Ref. 4) analyzed the effect of these hoop 

stresses on rolling-element fatigue life. Their work was based 

on the analysis of Hertzian principal stresses from Jones (Ref. 5)

and the Lundberg-Palmgren bearing life theory (Lundberg and 

Palmgren (Ref. 6)). Coe and Zaretsky (Ref. 4) superimposed the 

hoop stresses on the Hertzian principal stresses, whereby the 

shearing stresses in the stressed volume below the contact 

between the rolling element (ball or roller) and inner race of the 

bearing increase. The increased maximum shearing stress at a 

depth z below the contacting surface due to hoop stress is 

( ) max max 2

h

h

σ

τ =τ − (1a) 

where τmax (a negative quantity) is the maximum shearing stress, 

σh (a positive quantity for the inner ring) is the hoop stress at the 

depth of the maximum shearing stress, and (τmax)h is the maximum shearing stress including the effect of the hoop stress. 

Where bearing inner rings are manufactured from casecarburized steel, there are compressive residual stresses induced 

in the raceway. There may also be compressive residual stresses 

induced by the heat-treating process in through-hardened steels. 

The presence of these residual stresses may offset the tensile 

stresses created by the interference fit. Equation (1a) can be 

modified to account for residual stress as follows:

( ) max max

1 ( )

2 h h r τ =τ − σ +σ (1b) 

where σr is a negative quantity for compressive stress. As a 

result, the detrimental effect of an interference fit on bearing life 

can be mitigated by the presence of compressive residual 

stresses in the bearing ring (Zaretsky (Ref. 7)).

The life of the race is inversely proportional to the maximum 

shearing stress including (tensile) hoop stress and residual stress 

to an exponent c: 

max

1 ~ c

h

L

τ

(2) 

Zaretsky (Ref. 7) and Zaretsky et al. (Ref. 8) developed a 

procedure (Zaretsky’s Rule) for separating the lives of the 
NASA/TM—2010-216913 2

rolling element set from the lives the of bearing races. This 

provides an improvement on the prior Coe and Zaretsky 

analysis (Ref. 4), that had, for simplicity, assumed that all 

bearing components are equally affected by the inner-race 

interference fit. By using Zaretsky’s Rule, the analysis applies 

the reduction in life only to the inner race without modifying 

the computed lives of the outer race or the rolling elements.

Zaretsky et al. (Ref. 9) performed a finite element analysis on 

the inner ring of an angular-contact ball bearing to determine 

the radius of an equivalent cylindrical race. They then analyzed 

45- and 120-mm angular-contact ball bearings as though the 

inner rings were cylindrical to calculate the hoop stress. By 

applying Zaretsky’s Rule, they calculated the life adjusted for 

heavy interference fits and inertial effects for bearings used in 

high-speed jet engines.

Oswald et al. (Ref. 10) applied this method to cylindrical 

roller bearings to calculate life factors for interference fit in 

roller bearings and to show the effect of interference fit on 

the relationship between life and the maximum Hertz stress. 

Heavy interference fits on the inner ring of a cylindrical 

roller bearing were found to significantly reduce bearing 

fatigue life. For example, the tightest fit at the high end of 

the tolerance band produced a life reduction of approximately 60 percent for an inner-race maximum Hertz stress of 

1200 MPa (175 ksi) in comparison to the life of a bearing 

without interference fit.

In view of the aforementioned, the objectives of this work 

were to expand the analysis of Oswald et al. (Ref. 10) and 

Zaretsky et al. (Ref. 9) to (1) calculate the reduction in fatigue 

life due to the interference fit of the inner ring for several 

classes of radially loaded deep-groove ball and thrust-loaded 

angular-contact ball bearings; (2) independently determine the 

lives of the inner races, outer races, and ball sets subject to 

inner-ring interference fit; and (3) develop life factors for 

interference fits that can be applied to the bearing life calculation according to the ANSI/ABMA standards for shaft fitting

(Refs. 1 to 2). 

Enabling Equations

Subsurface Shearing Stresses

A representative deep-groove ball bearing is shown in Figure

1(a) and (b). The bearing comprises an inner and outer ring and 

a plurality of balls interspersed between the two rings and 

positioned by a cage or separator. The angular-contact ball 

bearing shown in cross section in Figure 1(c) is similar to the 

deep-groove bearing except that the race shoulders are not 

symmetrical; a shoulder on one or both of the rings is generally 

relieved (removed) on one side.
NASA/TM—2010-216913 3

Figure 2 is a schematic of the contact profile of a ball on a 

race. Figure 3(a) shows the surface (Hertz) stress distribution 

under the ball and the principle stresses at z, a critical location 

below the surface. Figure 3(b) shows the stress distribution 

below 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. For the 

analysis reported herein, only the maximum shearing stress is 

considered. The maximum shearing stress is one-half the 

maximum difference between the principal stresses:

max 2

σ −σ z x τ = (3)

Note that with the sign convention used here τmax is negative.

From Jones (Ref. 5), the maximum subsurface shearing 

stress τmax below a rolling element is proportional to the 

maximum Hertz stress Smax: 

max 1 max τ =−k S (4)

where k1 is a conversion constant and Smax is positive. 

Oswald et al. (Ref. 10), based on earlier work by Coe and 

Zaretsky (Ref. 4), showed that in a cylindrical roller bearing 

the maximum subsurface shear stress due to Hertzian loading 

is –0.300 Smax and that this stress occurs at a nondimensional 

depth u below the surface, where u = z/b = 0.786 and b is the 

semiwidth in the direction of rolling of the rectangular contact 

zone beneath the most heavily loaded roller. This result is 

consistent with Figure 4 (from Jones (Ref. 5)) for an inverseellipticity ratio b/a = 0.

The principal stresses in the normal and tangential direction 

for a ball bearing and the effect of the added hoop stress are 

illustrated in Figure 5, where Sn is the normal stress, St is the 

tangential stress in the direction of rolling due only to Hertzian 

loading, and t S′ is the hoop stress superimposed on St. The 

maximum shear stress is one-half the difference between Sn

and St. 

In a ball bearing, the contact area is elliptical. The solution 

for the location and magnitude of the maximum subsurface 

shear stress involves calculating the size of both axes of the 

contact ellipse, which requires evaluation of elliptic integrals.
NASA/TM—2010-216913 4

Jones (Ref. 5) provides a procedure for estimating the size 

of the contact ellipse and the location and magnitude of the 

maximum subsurface shear stress by looking up parameters in 

charts such as Figure 4. 

Brewe and Hamrock (Ref. 11) and Hamrock and Brewe

(Ref. 12) provide curve fits for estimating the contact ellipse. 

Antoine et al. (Ref. 13) provide a more accurate, but slightly 

more complicated, procedure. Bearing analysis software 

typically evaluates these parameters through iteration.

Zaretsky (Ref. 7) and Oswald et al. (Ref. 10) give a simplified procedure for finding the effect due to hoop stress from 

inner-ring interference fits on a cylindrical roller bearing. This 

simplified procedure, when used to calculate the resulting life 

of a roller bearing, gives results within 1 percent of the value 

found by iterating for the actual location of the maximum 

shear stress, even with a very heavy fit. With a few modifications, this procedure can also be applied to ball bearings.

Zaretsky’s procedure (Ref. 7) requires the contact pressure 

pi between the inner ring and the shaft. For the case of a 

bearing ring shrunk on a solid shaft where both components 

have the same material properties, pi can be found from 

Equation (5) (adapted from Juvinall (Ref. 14)), 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 outside diameter of the inner ring.

( ) 2 2

eff

2

eff 2

S

i

S

ED D

p

D D

∆ −

= (5)

The effective inner-ring outside diameter for a ball bearing 

is 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 as shown in Figure 6(a) and (b). For the typical 

shoulder height of 20 percent of the ball diameter d, the sector 

angle θ = 1.854 radians (106.24°), as shown in Figure 6(c).

1 0.3 / 2 cos 1.854

0.5

d

d

−   θ =   ⇒ θ=   (6)

The area removed in grinding the ball track is 

( )

2

2

sector sin 0.1118

2

R A d = θ− θ = (7)

where R is the radius of curvature, and the effective diameter 

for a deep-groove ball bearing is

sector

eff 0.2 IR

A DD d

W

=+ − (8a)

where DIR is the diameter of the inner race and W is the width 

of the ball bearing ring.

On an angular-contact ball bearing, the shoulder is generally relieved (removed) on one side (see (Fig. 6(b)); hence the

 

second and third terms on the right-hand side of Equation (8a) 

are divided by 2. 

sector

eff 0.1

2 IR

A DD d

W

=+ − (8b)

However, if the inner race shoulder is not relieved, Equation 

(8a) should be used.
NASA/TM—2010-216913 5

Determining Hoop Stress at the Location 

of Maximum Subsurface Shearing Stress

Equation (9), adapted from Juvinall (Ref. 14), gives the 

tangential (hoop) stress in a bearing ring shrunk on a solid 

shaft. This equation is based on the stress in an internally

pressurized, thick-wall cylinder where the shaft and ring are 

made from the same material— 

2 2

eff

2 2

eff

1 i S

h

S

pD D

D D D

   

σ= +         −    

(9) 

where σh is the tangential (hoop) stress, pi is the internal 

pressure from Equation (5), DS is the common diameter of the 

shaft and bearing bore, Deff is the effective thick-wall cylinder 

diameter of the inner ring from Equations (8a) or (8b), and D

is the diameter at the location of the maximum shear stress 

beneath the surface of the inner race.

Strict Series Reliability and Zaretsky’s Rule

Lundberg and Palmgren (Ref. 6) first derived the relationship between individual rolling-element bearing component 

lives and system life. A bearing is a system of multiple components, each with a different life. As a result, the life of the 

system is different from the life of an individual component in 

the system. The fatigue lives of each of the bearing components are combined to calculate the system L10 life using strictseries system reliability (Lundberg and Palmgren (Ref. 6)) and 

the two-parameter Weibull distribution function (Weibull 

(Refs. 15 to 17)) for the bearing components comprising the 

system. The resultant life of the bearing cannot be longer than 

the life of the bearing component with the shortest life. Lundberg and Palmgren (Ref. 6) express the bearing system fatigue

life at a given reliability, S, as follows:

11 1

ee e LL L IR OR

= + (10) 

where L is life and LIR and LOR are the lives of the inner and 

outer races, respectively.

Zaretsky et al. (Ref. 8) note that the life of the rollingelement set is implicitly included in the inner- and outer-race 

lives in Equation (10). If the life of the balls LB (taken as a set)

is separated from the race lives, Equation (10) can be rewritten

as

1111

ee ee LL LL IR adj B OR adj − −

= ++ (11) 

where adj in the subscript indicates adjusted race lives that are

greater than the corresponding lives in Equation (10). Zaretsky 

(Ref. 7) and Zaretsky et al. (Ref. 8) observe that LOR is generally greater than LIR. 

For a deep-groove bearing, LB is equal to or greater than 

LOR. Herein, it is assumed that, for a deep-groove bearing, 

LB = LOR; thus, Equation (11) becomes 

11 2

ee e LL L IR adj OR adj − −

= + (12a) 

For an angular-contact ball bearing, LB is equal to or greater 

than LIR. Herein, it is assumed that, for an angular-contact 

bearing, LB = LIR; thus, Equation (11) becomes

12 1

ee e LL L IR adj OR adj − −

= + (12b) 

The ratio of the lives of the outer and inner races is defined as

OR

IR

L

X

L = (13) 

If the life ratio X does not change when the ball set life is 

separated from the race lives, Equation (12a) for deep-groove 

bearings becomes
NASA/TM—2010-216913 6

( )

11 2

e e e IR adj IR adj

L L − X L −

= +

⋅

 (14a) 

Likewise, for angular-contact bearings, Equation (12b) 

becomes

( )

12 1

e e e IR adj IR adj

L L − X L −

= +

⋅

 (14b) 

Bearing Life Factor for Interference Fit

Coe and Zaretsky (Ref. 4) assumed that LR, the life ratio for 

the hoop stress on the inner race, is the 9th power of the ratio

of τmax (from Eq. (4)) to (τmax)h. 

( )

( )

9

max

max

h

h

L

LR

L

  τ = =  

τ    

(15) 

For roller bearings, (τmax)h can be computed from the simplified procedure of Zaretsky (Ref. 7); and for ball bearings, 

(τmax)h can be computed from a modified version of the simplified procedure.

Equation (15) is based on earlier work by Lundberg and 

Palmgren (Ref. 6) that uses life exponents for shear-stress that 

range from 6.9 to 9.3. An exponent of 9 is assumed for the

current work. For further discussion of life exponents, see 

Poplawski et al. (Ref. 18) and the discussion later in this 

article. 

Coe and Zaretsky (Ref. 4) applied the life ratio to the life of 

the entire bearing, which produces an overly conservative 

estimate for the life of the bearing. Herein, the life ratio is

applied only to the inner race. This new value for the inner-race

life LR ⋅ LIR-adj is used in the first term on the right-hand side of 

Equation (14a) to calculate the life of a deep-groove bearing

(L)h, including the effects of Hertzian loading and hoop stress. 

Equation (16a) is identical to the equation developed for the life 

of a cylindrical roller bearing in Oswald et al. (Ref. 10). 

( ) ( ) ( )

11 2

e ee L LR L X L h IR adj − − IR adj

= + ⋅ ⋅ (16a) 

Likewise, for angular-contact bearings, Equation (14b) is 

modified to include the effects of Hertzian loading and hoop 

stress: 

( ) ( )

( )

1 1

1 1

e e

h IR adj

e e

IR adj IR adj

L LR L

L X L

−

− −

=

⋅

+ +

⋅

 (16b) 

Finally, the life factor for hoop stress (LF)h is computed as the 

ratio of (L)h divided by the original life of the bearing L: 

( ) ( )h

h

L

LF

L = (17) 

Determining Maximum Hertz Stress on the Basis 

of Static Radial Load Capacity 

Rolling-element bearing static load capacity was first defined in terms of deformation by Palmgren (Ref. 19): “... the 

allowable permanent deformation of rolling element and 

bearing ring [race] at a contact as 0.0001 times the diameter of 

the rolling element....” For ball bearings, this corresponds to a 

maximum Hertz stress of 4000 MPa (580 ksi). From Hertz 

theory, Jones (Ref. 5) shows that the relationship between 

maximum Hertz stress and radial load P for a cylindrical roller 

bearing is

S P max ~ (18) 

For a ball bearing, the corresponding relationship is

3 S P max ~ (19) 

Nearly all bearing manufacturers’ catalogs provide the static 

radial load capacity Cor for any bearing size. Hence, to determine the appropriate stress at the applied radial load on a ball

bearing inner race, Equation (19) can be rewritten as follows:

1/3

max 2 k

or

P S

C

  =    

(20) 

where the conversion constant k2 = 4000 for SI units with Smax

expressed in megapascals, or k2 = 580 for English traditional 

units with Smax expressed in kips per square inch (ksi). Table 1, 

which uses bearing catalog data from The Timken Company 

(Ref. 20), gives the static radial load capacity Cor for the 

bearings discussed herein. Figure 7 shows Smax plotted as a 

function of static load capacity from Equation (20). The 

appropriate life factor can be interpolated from Tables 2 to 5 

for the various interference fits.

The value of k2 = 4000 MPa (580 ksi) is implicitly included 

in the static load ratings in the 1978 revision of ANSI/ABMA 

standard 9 (ANSI/AFBMA-Std 9 (Ref. 21)). In the most recent

(2000) revision (ANSI/ABMA-9 (Ref. 22)), this permissible 

stress has been increased by 5 percent to 4200 MPa 

(609 ksi).The corresponding ISO standard, (ISO 76 (Ref. 23)) 

also specifies the larger value, 4200 MPa. Herein k2 = 4000 

was used because it agrees better with the catalog values that 

were used.
NASA/TM—2010-216913 7

As an example of using this procedure, consider a 210-size 

deep-groove ball bearing with a radial load P = 1847 N 

(415.1 lbf). From Table 1, Cor = 23 162 N (5207 lbf). Using 

either Figure 7, with P/Cor = 0.0797, or Equation (20), with 

3 P C/ or = 0.43043, and the appropriate value for k2 yields 

Smax = 1722 MPa (249.8 ksi). The bearing analysis code (Poplawski et al. (Ref. 24)) calculated Smax= 1720 MPa (249.5 ksi). 

For most low-speed ball bearing applications (less than 

1 million DN, where DN is the inner-ring speed in rpm multiplied by the bearing bore diameter in millimeters), the determination of the appropriate life factor based on bearing size, 

radial load, and interference fit can be related to Cor without 

the need to perform extensive calculations.

Czyzewski’s Experiment

Czyzewski (Ref. 3) performed rolling-element fatigue tests 

on modified inner-races from NU 209 roller bearings. These 

tests show the effect of interference fits between the inner-race 

and the shaft of his test machine. For all the Czyzewski tests, 

the stressed volume calculated according to Lundberg and 

Palmgren (Ref. 6) and the maximum Hertz (contact) stress 

remained unchanged.

The test apparatus, shown in Figure 8 (redrawn from 

Czyzewski (Ref. 25)), employs four loading rollers through a 

compound lever mechanism. The inner-race test article has a 

55-mm outside diameter, 45-mm bore, and 19-mm length. The 

size of the loading rollers is not given; however from Figure 8, 

they apparently have the same diameter as the test races, 55 mm.

The test races were made from an unspecified bearing steel 

annealed to a Brinell hardness of 170 HB (Czyzewski (Ref. 25)) 

(approximately Rockwell B 86 or less than Rockwell C 15). 

(We assume that the races were made from 52100 steel tempered from Rockwell C 60.) The races had a transverse crown 

of 50-mm radius with 5-mm flat length. 

TABLE 1.—DEEP-GROOVE AND ANGULAR-CONTACT BALL BEARING PROPERTIES 

Bearing

size

Bore,

mm

Outside 

diameter,

mm

Number 

of 

balls

Ball diameter, 

d

Inner-race outside 

diameter, 

DIR

Static radial load 

capacity, 

Cor

Dynamic radial 

load capacity, 

Cr

mm in. mm in. N lbf N lbf

1906 30 47 14 4.7625 3/16 33.738 1.3282 5 009 1 126 7 242 1 628

206 62 9 9.525 3/8 36.475 1.4360 11 254 2 530 19 457 4 374

1910 50 72 17 6.35 1/4 54.65 2.1516 11 152 2 507 13 398 3 012

1010 80 12 9.525 3/8 55.475 2.1841 16 610 3 734 23 291 5 236

210 90 10 12.7 1/2 57.30 2.2559 23 162 5 207 35 083 7 887

310 110 8 19.05 3/4 60.95 2.3996 37 828 8 504 61 835 13 901

1915 75 105 17 12.7 1/2 80.289 3.1610 25 008 5 622 27 895 6 271

215 130 10 19.05 3/4 82.550 3.2500 44 762 10 063 62 119 13 965

1920 100 140 17 12.7 1/2 105.969 4.172 44 456 9 994 46 818 10 525

220 180 10 25.4 1 111.531 4.3910 92 639 20 826 122 161 27 463

Adapted from The Timken Company (Ref. 20).
NASA/TM—2010-216913 8

TABLE 2.—LIFE FACTORS FOR 30-MM-BORE DEEP-GROOVE AND

ANGULAR-CONTACT BALL BEARINGS WITH ABEC−5 TOLERANCES

ABMA

fit class

Clearance,

mm

Inner-race Hertz stress Life factor fit

deep-groove

Life factor fit

MPa ksi angular contact

j5-min +0.004 1200 175 1 1

1720 250 1 1

2240 325 1 1

j5 –0.0035 1200 175 1 1

1720 250 1 1

2240 325 1 1 

j5-max –0.011 1200 175 0.74 0.82

1720 250 .81 .87

2240 325 .85 .90

j6-min +0.004 1200 175 1 1

1720 250 1 1

2240 325 1 1

j6 –0.0055 1200 175 0.94 0.96

1720 250 .96 .97

2240 325 .97 .98

j6-max –0.015 1200 175 0.62 0.72

1720 250 .72 .80

2240 325 .77 .84

k5-min –0.002 1200 175 1 1 

1720 250 1 1

2240 325 1 1

k5 –0.0095 1200 175 0.79 0.85

1720 250 .85 .90

2240 325 .88 .92

k5-max –0.017 1200 175 0.56 0.67

1720 250 .67 .76

2240 325 .74 .82

m5-min –0.008 1200 175 0.84 0.89

1720 250 .89 .93

2240 325 .91 .94

m5 –0.0155 1200 175 0.60 0.70

1720 250 .70 .79

2240 325 .76 .84

m5-max –0.023 1200 175 0.43 0.54

1720 250 .55 .66

2240 325 .64 .73

Results averaged from size 1906 and 206 bearings.
NASA/TM—2010-216913 9

TABLE 3.—LIFE FACTORS FOR 50-MM-BORE DEEP-GROOVE AND 

ANGULAR-CONTACT BALL BEARINGS WITH ABEC−5 TOLERANCES 

ABMA

fit class

Clearance,

mm 

Inner-race Hertz stress Life factor 

deep-groove

Life factor 

MPa ksi angular-contact

j5-min +0.005 1200 175 1 1

1720 250 1 1

2240 325 1 1

j5 –0.0045 1200 175 0.99 0.99

1720 250 .99 .99

2240 325 .99 1

j5-max –0.014 1200 175 0.76 0.83

1720 250 .83 .88

2240 325 .86 .91

j6-min +0.005 1200 175 1 1

1720 250 1 1

2240 325 1 1 

j6 –0.007 1200 175 0.92 0.95

1720 250 .95 .96

2240 325 .96 .97

j6-max –0.019 1200 175 0.66 0.75

1720 250 .75 .83

2240 325 .80 .87

k5-min –0.002 1200 175 1 1

1720 250 1 1

2240 325 1 1

k5 –0.0115 1200 175 0.82 0.87

1720 250 .87 .91

2240 325 .90 .93

k5-max −0.021 1200 175 0.62 0.72

1720 250 .72 .80

2240 325 .78 .85

m5-min −0.009 1200 175 0.87 0.92

1720 250 .91 .94

2240 325 .93 .96

m5 −0.0185 1200 175 0.67 0.76

1720 250 .76 .83

2240 325 .81 .87

m5-max −0.028 1200 175 0.51 0.62

1720 250 .63 .73

2240 325 .70 .79

m6-min −.009 1200 175 0.87 0.92

1720 250 .91 .94

2240 325 .93 .96

m6 –0.021 1200 175 0.62 0.72

1720 250 .72 .80

2240 325 .78 .85

m6-max −0.033 1200 175 0.44 0.55

1720 250 .57 .67

2240 325 .65 .74

Results averaged from size 1910, 1010, 210, and 310 bearings.
NASA/TM—2010-216913 10

TABLE 4—LIFE FACTORS FOR 75-MM-BORE DEEP-GROOVE AND 

ANGULAR-CONTACT BALL BEARINGS WITH ABEC−5 TOLERANCES

ABMA

fit class

Clearance,

mm

Inner-race Hertz stress Life factor 

deep-groove

Life factor 

MPa ksi angular-contact

j5-min +0.007 1200 175 1 1

1720 250 1 1

2240 325 1 1

j5 –0.004 1200 175 1 1

1720 250 1 1

2240 325 1 1

j5-max –0.015 1200 175 0.80 0.86

1720 250 .86 .91

2240 325 .89 .93

j6-min +0.007 1200 175 1 1

1720 250 1 1

2240 325 1 1

j6 –0.007 1200 175 0.94 0.96

1720 250 .96 .97

2240 325 .97 .98

j6-max –0.021 1200 175 0.71 0.79

1720 250 .79 .85

2240 325 .83 .89

k5-min –0.002 1200 175 1 1

1720 250 1 1 

2240 325 1 1

k5 –0.013 1200 175 0.84 0.89

1720 250 .88 .92

2240 325 .91 .94

k5-max –0.024 1200 175 0.66 0.76

1720 250 .75 .83

2240 325 .80 .87

m5-min –0.011 1200 175 0.87 0.91

1720 250 .91 .94

2240 325 .93 .95

m5 –0.0122 1200 175 0.69 0.78

1720 250 .78 .85

2240 325 .82 .88

m5-max –0.033 1200 175 0.55 0.66

1720 250 .66 .75

2240 325 .73 .81

m6-min –0.011 1200 175 0.87 0.91

1720 250 .91 .94

2240 325 .93 .95

m6 –0.025 1200 175 0.65 0.75

1720 250 .74 .82

2240 325 .80 .86

m6-max –0.039 1200 175 0.48 0.59

1720 250 .60 .71

2240 325 .68 .77

n6-min –0.020 1200 175 0.72 0.80

1720 250 .80 .86

2240 325 .84 .89

n6 –0.034 1200 175 0.54 0.65

1720 250 .65 .75

2240 325 .72 .80

n6-max –0.048 1200 175 0.40 0.51

1720 250 .53 .64

2240 325 .61 .72

Results averaged from size 1915 and 215 bearings.
NASA/TM—2010-216913 11

TABLE 5.—LIFE FACTORS FOR 100-MM-BORE DEEP-GROOVE AND 

ANGULAR-CONTACT BALL BEARINGS WITH ABEC−5 TOLERANCES

ABMA

fit class

Clearance,

mm 

Inner-race Hertz stress Life factor

deep-groove

Life factor

MPa ksi angular-contact

j5-min +0.009 1200 175 1 1

1720 250 1 1

2240 325 1 1

j5 −0.0035 1200 175 1 1

1720 250 1 1

2240 325 1 1

j5-max −0.016 1200 175 0.84 0.89

1720 250 .89 .92

2240 325 .91 .94

j6-min +0.009 1200 175 1 1

1720 250 1 1

2240 325 1 1

j6 −0.007 1200 175 0.96 0.97

1720 250 .97 .98

2240 325 .98 .99

j6-max −0.023 1200 175 0.76 0.83

1720 250 .83 .88

2240 325 .87 .91

k5-min −0.003 1200 175 1 1

1720 250 1 1

2240 325 1 1

k5 −0.0155 1200 175 0.85 0.90

1720 250 .89 .93

2240 325 .92 .94

k5-max −0.028 1200 175 0.71 0.78

1720 250 .79 .85

2240 325 .83 .88

m5-min −0.013 1200 175 0.88 0.92

1720 250 .92 .94

2240 325 .93 .96

m5 −0.0255 1200 175 0.73 0.81

1720 250 .81 .86

2240 325 .85 .90

m5-max −0.038 1200 175 0.61 0.70

1720 250 .71 .79

2240 325 .77 .84

m6-min −.013 1200 175 0.88 0.92

1720 250 .92 .94

2240 325 .93 .96

m6 −0.029 1200 175 0.70 0.78

1720 250 .78 .84

2240 325 .83 .88

m6-max −0.045 1200 175 0.55 0.64

1720 250 .66 .74

2240 325 .73 .80

n6-min −0.023 1200 175 0.76 0.83

1720 250 .83 .88

2240 325 .87 .91

n6 −0.039 1200 175 0.60 0.69

1720 250 .70 .78

2240 325 .76 .83

n6-max −0.055 1200 175 .47 0.57

1720 250 .59 .69

2240 325 .67 .76

P6-min −0.037 1200 175 0.62 0.71

1720 250 .72 .79

2240 325 .78 .84

p6 −0.053 1200 175 0.48 0.58

1720 250 .61 .70

2240 325 .68 .76

p6-max −0.069 1200 175 0.38 0.47

1720 250 .51 .61

2240 325 .60 .69

Results averaged from size 1920 and 220 bearings.
NASA/TM—2010-216913 12

The loading rollers were cylindrical (no crown) and presumably made from the equivalent of much harder American Iron 

and Steel Institute (AISI) 52100 steel. The surface speed was 

14.25 m/s, which corresponds to 495 rpm or 27 225 DN (low 

speed). The lubricant was “highly refined pure paraffinic oil” 

(Czyzewski (Ref. 3)). 

Czyzewski (Ref. 3) reported inner-race life data in the form 

of Weibull plots for three test series: three levels of interference 

fits, 80, 40, and 5 MPa (11.6, 5.8, and 0.725 ksi), given in terms 

of the bore-shaft interface pressure. The inner-race lives scaled 

from Czyzewski’s plot are shown in Table 6 and are replotted

and reanalyzed in Figure 9. The three test series had L10 lives of 

3.80×104

, 3.07×105, and 3.02×106 cycles and L50 lives of 

1.54×105

, 1.78×106, and 9.44×106 cycles, respectively.

Czyzewski reported (Ref. 3) that all of the races of series 1 

burst (failed by fracture), three of ten races in series 2 fractured, 

and none of series 3 fractured. We suspect that incipient rollingelement fatigue spalls initiated these race fractures. The Czyzewski results suggest that there is a limit pressure above which 

inner ring fracture can occur.

Czyzewski’s interference fits (Ref. 3) are extremely tight, 

particularly for series 1. The Czyzewski bearing inner races 

are approximately the same size as those on the 210-size deepgroove ball bearing considered in this paper, for which the 

heaviest fit recommended is an m6 fit. At the tight end of the 

tolerance band, an m6 fit in this bearing produces an interface 

pressure of 16.7 MPa (2.43 ksi). The Czyzewski series-1 fit 

pressure (80 MPa) is 4.8 times higher than the pressure would 

be for the maximum m6 fit for this ball bearing. 

Czyzewski’s experiments (Ref. 3) may also be compared 

with results from 210-size roller bearings in Oswald et al. 

(Ref. 10), where at the tight end of the tolerance band, an m6 

fit produces an interface pressure of 14.8 MPa (2.15 ksi). The 

Czyzewski series-1 fit pressure is 5.4 times higher than the 

pressure would be for the tight end of the m6 fit for this roller 

bearing.

Czyzewski (Ref. 3) did not mention the radial load he employed for his tests or the resulting Hertz stress. He reported 

instead the maximum subsurface shear stress (including the 

effect of hoop stress) normalized to the maximum Hertz stress 

as values of 0.520, 0.406, 0.316, and 0.302 for interference 

pressures of 80, 40, 5, and 0 MPa respectively. 

TABLE 6—ROLLING-ELEMENT FATIGUE LIFE IN LOAD CYCLES FROM CZYZEWSKI 

(REF. 3) FOR THREE LEVELS OF ROLLER BEARING INTERFERENCE FIT PRESSURE: 

80, 40 AND 5 MPA (11.6, 5.8 AND 0.725 ksi) FOR TEST SERIES 1, 2 AND 3, RESPECTIVELY 

Point Series 1 Series 2 Series 3

Max shear stress, MPa (ksi) 390 (56.6) 305 (44.2) 237 (34.4)

L10 life, cycles 3.80×104 3.07×105 3.02×106

L50 life, cycles 1.54×105 1.78×106 9.44×106

Weibull slope, e 1.35 1.07 1.65

Failure index 10 / 10 10 / 10 10 / 10

1 a25 620 a290 000 1 915 000

2 a60 000 530 000 5 910 000

3 a112 800 a610 000 6 460 000

4 a123 800 a1 300 000 7 380 000

5 a136 700 1 550 000 9 000 000

6 a156 000 1 740 000 9 800 000

7 a169 900 2 040 000 10 900 000

8 a203 000 3 160 000 12 500 000

9 a304 000 3 950 000 16 100 000

10 a476 000 8 800 000 20 900 000

a

Race fractured at failure.
NASA/TM—2010-216913 13

From the given data, we calculated the hoop stress at z (the 

depth below the surface to the maximum shearing stress) to be 

325, 162, and 20.3 MPa (47, 24, and 2.9 ksi) for the three test 

series, respectively. From the hoop stress and the maximum 

shear stress, we estimated the maximum Hertz stress to be 

approximately 750 MPa (110 ksi). This is a very low stress 

value; however, the test inner race is made from annealed 

material that will have a much shorter life than rings made from 

typical Rockwell C 60 hardness material. 

From our estimate of 750 MPa (110 ksi) maximum Hertz 

stress and from Czyzewski’s reported normalized maximum 

subsurface shear stress, we calculated the actual maximum 

subsurface shear stress to be 390, 305, and 237 MPa (56.6, 44.2, 

and 34.4 ksi) for series 1, 2, and 3, respectively. We calculated 

the subsurface shear stress with no interference fit to be 227 

MPa (33 ksi).

In Figure 10, we show the relationship between the normalized maximum subsurface shear stress and the inner-race life 

for Czyzewski’s data. The shear stress-life exponent c was 

found to be 8.77 for the L10 life and 8.40 for the L50 life. The L10

result compares reasonably well with the shear stress-life 

exponent c of 9 assumed by Coe and Zaretsky (Ref. 4) and by 

Oswald et al. (Ref. 10). 

Using a power-series curve fit, we calculated the L10 and L50

lives for no interference fit to be 4.24×106 and 1.73× 107 cycles. 

This suggests that inner-race L10 fatigue lives were reduced by 

99, 93, and 29 percent and that the L50 fatigue lives were reduced by 99, 90, and 45 percent by the interference fits of series 

1, 2, and 3, respectively. 

The series 3 fit, which reduced the race L10 life by 29 percent 

and which had an interface pressure of 5 MPa (0.725 ksi), is 

comparable to the pressure from the tight end of a j5 fit on a 

210-size roller bearing (5.1 MPa (0.74 ksi), which produces a 

life reduction of 20 percent. The other two Czyzewski fits (Ref. 

3) are much tighter than any of the recommended interference 

fits in ANSI/ABMA-7 (Ref. 1). 

It is commonly accepted that where tight interference fits are 

used and/or large hoop stresses are induced in bearing inner 

rings, case carburizing should be utilized to prevent ring fracture from incipient rolling-element fatigue spalls. The typical

explanation is that the soft ductile core of the case-carburized 

steel prevents crack propagation through the core. However, in 

the Czyzewski experiments (Ref. 3), the ring hardness is less 

than Rockwell C 15, which is significantly lower than the core 

of case-carburized steels such as M-50 NiL and AISI 9310

(Zaretsky (Ref. 7)). We theorize that it is the compressive 

residual stresses induced by heat treatment in case-carburized 

steels rather than the ductile (soft) core that offsets the effect of 

hoop stresses to prevent ring fracture.

Results and Discussion

The analysis described in the previous sections was applied 

to radially loaded deep-groove ball bearings and to thrustloaded angular-contact ball bearings made of AISI 52100 steel.

Table 1 lists the dimensions, static and dynamic load capacity of 

the bearings. Identical dimensions were chosen for the two 

bearing types except for the contact angle: 0° for the deepgroove bearings and 25° for the angular-contact bearings.

The analysis included four bore sizes and either two or four 

dimension series for each bore size. The dimension series are 

shown schematically in Figure 11. Each bearing was analyzed 

at three levels of inner-race Hertz stress. A commercial bearing 

analysis code (Poplawski et al. (Ref. 24)) was used to calculate 

the unfactored L10 lives for the inner and outer races operating 

without interference fit. Life factors were calculated for up to 

seven fit classes for each bearing, with each fit taken at the 

maximum, average, and minimum values within the fit class 

for ABEC–5 tolerances (ANSI/ABMA-20 (Ref. 2)). (ABEC, 

Annular Bearing Engineering Committee of the American 

Bearing Manufacturers Association, is a system for specifying 

bearing tolerances with an odd number between 1 and 9. 

ABEC–5 is a medium tolerance level.) Figure 12 shows a 

graphical representation of the shaft fits (adapted from 

ANSI/ABMA-7 (Ref. 1)). Harris (Ref. 26) discusses the effect 

of surface finish on interference fit as a result of the smoothing 

of asperities on the surface. He recommends reducing the 

calculated interference to account for asperity smoothing, 

depending on the quality of the finish. For very accurately 

ground surfaces, the reduction is 4 μm (2 μm for each surface: 

the inside diameter of the bore and the outside diameter of the 

shaft). In this work, the apparent interference was reduced by 

4 μm (160×10–6 in.) to account for surface finish effects. 
NASA/TM—2010-216913 14

Bearing dynamic load ratings were developed by Lundberg 

and Palmgren based on a geometry and material coefficient

designated fc (currently called fcm in the standards 

(ANSI/ABMA-9 (Ref. 22))) that was benchmarked to unreported experimental data (Lundberg and Palmgren (Ref. 6)).

Unfortunately, the interference fits used to generate the test 

data were also not reported. We have calculated life factors 

based on the assumption that the Lundberg and Palmgren 

bearing data did not include the effect of interference fit. 

For this article, all bearings were modeled at the zero internal operating clearance condition and low speed. In the case of 

inner-ring interference fit or thermal strain, this means that the 

bearings would have an appropriate initial (unmounted) 

clearance between the balls and the races. In addition, the 

analysis assumes that the bearing speed is less than one 

million DN. For high-speed bearings or unusual thermal 

environments, an appropriate analysis code should be used.

Over 1150 bearing configurations and load cases were analyzed in our study. Each bearing was analyzed for three values 

of inner-race maximum Hertz stress: 1200, 1720, and 

2240 MPa (175, 250, and 325 ksi). The load on each bearing 

was chosen to produce the desired stress value. These loads 

are designated light, normal, and heavy, respectively. The 

analysis code calculated the inner- and outer-race lives using 

the traditional Lundberg-Palmgren (Ref. 6) method. Therefore, 

these lives implicitly include the life of the ball set.

Equation (13) was used to find the ratio of the inner- and 

outer-race lives. The adjusted value of the inner-race life (with 

the ball set life separated) was calculated from Equation (14a) 
NASA/TM—2010-216913 15

for deep-groove bearings and from Equation (14b) for angular-contact bearings.

Data for limiting values of bearing bore sizes were taken 

from the table for tolerance class ABEC−5 (ANSI/ABMA-20

(Ref. 2)). Because this reference does not have information for 

shaft size limits, values for shaft diameter deviations were 

taken from the ANSI/ABMA shaft-fitting practice table

(ANSI/ABMA-7 (Ref. 1)), which gives tolerance limits for 

bearings with the ABEC−1 quality level. The practice of using 

bore tolerances from an ABEC−5 table and shaft tolerances 

from an ABEC−1 table is consistent with an example given by 

Harris (Ref. 26). 

The shaft diameter was subtracted from the bore diameter 

and then 0.004 mm (160×10–6 in.) was added to the difference 

to account for asperity smoothing. If the resulting fit was 

positive (indicating clearance), the interface pressure and, 

thus, the hoop stress were assumed to be zero. If the fit was 

negative, the resulting interference (as a positive number) was 

used to calculate the interface pressure due to the chosen 

interference fit in Equation (5). 

The simplified procedure described in the next section was 

used to find (τmax)h. Equation (15) was used to calculate the 

life ratio (hence the revised life) for the inner race, and finally, 

the life of the entire bearing was calculated from 

Equations (16a) or (16b) using the reduced life of the inner

race and the original lives of the ball set and outer race.

Ball bearings have an elliptical (point) contact between the 

ball and the inner race rather than the rectangular contact of 

roller bearings. Therefore, ball bearings have an inverseellipticity ratio b/a different from zero. (Elliptical contact 

dimensions a and b are illustrated in Fig. 2.) The bearings 

considered herein have inner-race conformity fi = 0.52. From 

the bearing analysis code, it was found that b/a = 0.11, 

τmax/Smax = 0.317, and u = 0.765. Similar results can be obtained from Figure 4 (Jones (Ref. 5)) using b/a = 0.11.

Curve fits were calculated for τmax/Smax and u in terms of b/a

from the data given in Figure 4 (Jones (Ref. 5)). These equations may be used to find parameters for the procedure in the 

following section to calculate the life factor for bearings with 

conformities different from 0.52, where the value of b/a

differs from 0.11.

3

max

max

2

0.1002

0.2431 0.1639 0.30

b

S a

b b

a a

τ   =    

  − ++    

 (21) 

3 2

0.3108 0.5131

0.1159 0.786

b b

u

a a

b

a

   = −    

  − +    

 (22) 

The bearing analysis code was used to verify that the shear 

stress ratio and the depth to maximum shear given in Equations (21) and (22) is correct for all the bearings analyzed 

here. In addition, one case of angular-contact bearings was 

checked that had different conformities: 0.515 for the inner

race and 0.53 for the outer race. For the inner race on these 

bearings, b/a = 0.139, τmax/Smax = 0.320, and u = 0.760. These 

changes have no effect (within the precision reported here) on 

the calculation of the life factor. (Note, however, that the 

different conformities alter the lives of the races and that the 

lives affect the life ratio.)

Analysis of 210-Size Deep-Groove 

Ball Bearing With m6 Fit 

As an example of the methods presented herein, consider a

210-size deep-groove ball bearing carrying a normal radial 

load of 1847 N (415.1 lbf) and a middle-of-the-tolerance-band

m6 inner-ring interference fit. From the analysis code or 

Equation (20), Smax = 1720 MPa (250 ksi). The analysis code 

predicted the L10 life of the bearing as 3054 million inner-race 

revolutions and the outer- and inner-race lives as 19 950 and 

3443 million revolutions: 

From Equation (13), the ratio of the outer-race life to innerrace life is X = 19 950/3443 = 5.79. The adjusted life of the 

inner race was found by solving Equation (14a): 

1/

10

2 1

e

IR adj e L L

X −

  = +     (23) 

The solution gives LIR–adj = 3827 million revolutions. (Although not needed for this calculation, the adjusted life of the 

outer race is 22 175 million revolutions.)

The ANSI/ABMA shaft-fitting practice table for ABEC−1 

bearings (ANSI/ABMA-7 (Ref. 1)) was used to find limiting 

diameters for a 50-mm (2-in.) shaft with an m6 fit. Figure 12 

illustrates the shaft fits schematically, showing deviations 

from the nominal bearing bore and shaft diameter of 

50.000 mm (1.9685 in.). The shaft deviation can range from 

0.009 to 0.025 mm (350 to 1000×10–6 in.) (shown as dimensions a and b in the figure). The bearing bore deviations were

found in the table for tolerance class ABEC−5 (ANSI/ABMA20 (Ref. 2)). The bore deviation can range from 0.000 to –

0.008 mm (0 to –300×10–6 in.) (shown as 0 and dimension c in 

Fig. 12.) 

The loosest within-tolerance m6 interference fit (0.009 mm, 

or 350×10–6 in., tight) occurs when the largest bore (50 mm) is

mounted on the smallest shaft (50.009 mm or 1.9689 in.) 

(before adjusting for surface finish), shown as dimension a in 

Figure 12. The tightest within-tolerance fit (0.033 mm, or 

1300×10–6 in., tight) occurs when the smallest bore 

(49.992 mm or 1.9682 in.) is mounted on the largest shaft 

(50.025 mm or 1.9695 in.), shown as dimension d in Figure

12. The average of these extremes is an interference fit of 

0.021 mm (830×10–6 in.) tight. This interference fit was 
NASA/TM—2010-216913 16

reduced by 0.004 mm (160×10–6 in.) to account for asperity 

smoothing, assuming smooth-ground surfaces. The resulting 

middle-of-the-tolerance-band m6 interference fit was 0.017 

mm (670×10–6 in.) tight.

For the example bearing, DS = 50 mm (1.9685 in.), 

DIR = 57.3 mm (2.2559 in.), ball diameter d = 12.7 mm (0.500

in.), race conformity fi = 0.52, bearing ring width W = 20 mm 

(0.7874 in.), Smax = 1720 MPa (250 ksi), E = 205 878 MPa

(29.86×103

 ksi), Poisson’s ratio ν = 0.3, and diametrical 

interference ∆ = 0.017 mm (670×10–6 in.). 

The life factor for the inner race due to the interference-fit 

stress was calculated by the following simplified procedure 

(adapted from Zaretsky (Ref. 7)) to calculate the maximum 

shear stress including the effect of hoop stress: 

(1) Determine the maximum shearing stress from Equation (4), where τmax = –(0.317) Smax = –545.2 MPa 

(–79.08 ksi). 

(2) Determine the effective outside diameter of the inner

race from Equation (8a), where Deff = 58.94 mm

(2.3205 in.).

(3) Determine the contact pressure from Equation (5), 

pi = 9.81 MPa (1.423 ksi).

(4) Determine R′, where R′ = DIR/d = 4.5118 (dimensionless). 

(5) Determine k3, where k3= E(R′ + 1)/[4(1 – ν2

)Smax] = 

181.25 (dimensionless). 

(6) Use the value for the nondimensional depth for an inner 

race conformity of fi = 0.52: u = 0.765 (dimensionless).

(7) Calculate the diameter to the maximum shear stress: 

D = DIR[1 – u/k3] = 57.058 mm (2.246 in.). 

(8) Calculate the hoop stress at D from Equation (9): 

σh = 52.065 MPa (7.55 ksi). 

(9) Substitute values for τmax and σh in Equations (1a) or 

(1b) to calculate (τmax)h = –571.27 MPa (–82.86 ksi). 

(10) Compute the life ratio for the inner race from Equation

(15): LR = [–545.2/(–571.27)]9 = 0.6572.

The inner-race life ratio from step (10) was used in Equation (16a) to calculate the life of the bearing, including the 

effect of the interference fit. The result was 2169 million 

inner-race revolutions:

( ) ( ) ( ) 1.11 1.11 1.11

1.11

11 2

0.6572 3827 5.79 3827

1

2169

h L

= +

⋅ ⋅

=

 (24) 

The life factor for hoop stress from Equation (17) is

( ) 2169 0.71

3054 h LF = = (25) 

Therefore, the middle-of-the-tolerance-band m6 interference 

fit will reduce the life of this bearing by 29 percent.

Analysis of 210-Size Angular-Contact 

Ball Bearing With m6 Fit 

Consider a thrust-loaded angular-contact ball bearing comparable to the radially loaded deep-groove example given in 

the previous section. For a thrust load of 4010 N (581.6 lbf), 

the analysis code gives Smax as 1720 MPa (250 ksi). Note that 

the applied thrust load to produce the same Smax (1720 MPa) is 

much higher than the radial load in the example in the previous section. This is because all balls carry the maximum 

load and maximum stress.

The analysis code predicts the L10 life of the bearing as 

550.3 million inner-race revolutions and the outer- and innerrace lives as 3447 and 624.2 million revolutions. The ratio of 

the outer-race life to the inner-race life X = 3447/624.2 = 5.52. 

Equation (14b) was solved in Equation (26) to find the adjusted life of the inner race: 1097 million revolutions. The 

adjusted life of the outer race was 6057 million revolutions: 

1/

10

1 2

e

IR adj e L L

X −

  = +     (26) 

As in the deep-groove bearing example in the previous section, a middle-of-the-tolerance-band m6 fit with smoothly

ground surfaces was used. This produced an effective interference of 0.017 mm (670×10–6 in.).

For the example bearing, DS = 50 mm (1.9685 in.), 

DIR = 57.3 mm (2.2559 in.), d = 12.7 mm (0.500 in.), 

fi = 0.52, W = 20 mm (0.7874 in.), Smax = 1720 MPa (250 ksi), 

E = 205 878 MPa (29.86×103

 ksi), ν = 0.3, ∆ = 0.017 mm 

(670×10–6 in.), and contact angle α = 25°. 

Next, the life factor of the inner race due to the added stress 

from the interference fit was calculated from the following 

simplified procedure (adapted from Zaretsky (Ref. 7)) to 

calculate the maximum shear stress including the effect of 

hoop stress: 

(1) Determine the maximum shearing stress from Equation

(4), where k1 = 0.317: τmax = –545.24 MPa (–79.08 ksi).

(2) Determine the effective outside diameter of the inner race 

from Equation (8b): Deff = 58.12 mm (2.288 in.). (We assume that the inner race shoulder is relieved.)

(3) Determine the contact pressure from Equation (5): 

pi = 9.10 MPa (1.320 ksi).

(4) Determine R′, where R′ = DIR/d = 4.5118 (dimensionless).

(5) Determine k3, where k3= E(R′ + 1)/[4(1 – ν2

)Smax] = 

181.25 (dimensionless).

(6) Use the value for the nondimensional depth for an innerrace conformity of fi = 0.52: u = 0.765 (dimensionless).

(7) Compute the z-component of the nondimensional depth, 

uz = ucos (α) = 0.693. 

(8) Calculate the diameter, D to the maximum shear stress

(note that a term was added because of the contact angle, 

see Fig. 13): D = DIR(1 – uz/k3) + d/2(1 – cos α) = 

57.696 mm (2.271 in.). 
NASA/TM—2010-216913 17

(9) Calculate the hoop stress at D from Equation (9): 

σh = 52.21 MPa (7.57 ksi). 

(10) Substitute values for τmax and σh in Equations (1a) or (1b) 

to calculate (τmax)h = –571.34 MPa (–82.87 ksi). 

(11) Compute the life ratio for the inner race from Equation

(15): LR = [–545.24/(–571.34)]9 = 0.6565. 

The inner-race life ratio from step (11) was used in Equation (16b) to calculate the life of the bearing, including the 

effect of the interference fit. The result was 441.5 million 

inner-race revolutions:

( ) ( )

( )

1.11 1.11

1.11 1.11 1.11

1 1

0.6565 1097

11 1

1097 5.52 1097 441.5

h L

=

⋅

++ =

⋅

(27)

From Equation (17), the life factor for the entire angularcontact bearing for hoop stress is

( ) 441.5 0.80

550.3 h LF = = (28)

Therefore, the middle-of-the-tolerance-band m6 interference fit will reduce the life of this bearing by 20 percent.

If the correction for the contact position of the ball on the 

race in steps (7) to (8) is neglected, the calculated value of the 

inner-race life ratio will change by less than 0.5 percent. This 

makes no difference in the calculated life ratio of the bearing 

to within the two decimal digits shown.

In this analysis, it is assumed that the contact angle of the 

bearing will remain at 25°, regardless of the thrust load. This 

means that the unloaded contact angle must be less than 25°.

For the normal-load example described, the unloaded contact

angle would be 20.4° to produce a loaded contact angle of 25°. 

With the smaller contact angle, the bearing life would be about

1 percent smaller at the same inner-race maximum Hertz stress

but there would be no difference in the calculated life factor. 

Interference-Fit Life Factors for 

ABEC−5 Ball Bearings

The analysis described in the previous sections was applied 

to over 1150 configurations and load cases of deep-groove and 

angular-contact ball bearings. It included four bore sizes, up to 

four dimension series (extremely light, extra light, light, and

medium), at three values of inner-race Hertz stress and up to 

seven inner-ring interference-fit classes. Each fit class was 

evaluated at the minimum, average, and maximum fit level for 

the ABEC−5 tolerance class. 

The shaft interference table from ANSI/ABMA-7 (Ref. 1) (as 

illustrated in Fig. 12) shows fit classes ranging from g6 (loose) 

to r7 (heavy interference). There is no effect on life for loose fits 

that produce no pressure at the bore, and no values are given for 

very heavy fits on small bearings because these are not recommended. Hence, life factors were calculated for four fit classes 

(j5 to m5) for 30-mm bearings (Table 2), five classes (j5 to m6) 

for 50-mm bearings (Table 3), six classes (j5 to n6) for 75-mm 

bearings (Table 4), and seven classes (j5 to p6)) for 100-mm 

bearings (Table 5). The k6 fit was not included in the analyses.

All interference fits were adjusted for the effect of asperity 

smoothing (assuming accurately ground surfaces) by adding 

0.004 mm (160×10–6 in.) to the clearance between the shaft and 

inner ring. If the resulting clearance value was negative (indicating interference), then the pressure and the resulting life factor 

were calculated. 

The life factors were lower (greater life reduction) for deepgroove ball bearings than for thrust-loaded angular-contact ball 

bearings. However, the deep-groove bearings had longer lives 

for the same value of maximum Hertz stress. Note that, in a 

deep-groove bearing, the maximum stress occurs on only one 

ball at a time, but in an angular-contact ball bearing, all the balls 

carry the maximum load continuously.

The various bearing dimension series within a particular bore 

size exhibited almost identical results for the interference-fit life 

factor for a particular fit, despite significant differences in the 

interface pressure at the bore required to produce that fit. For 

example, the four 50-mm deep-groove bearings analyzed (sizes

1910, 1010, 210, and 310) for an average m6 interference fit of 

0.021-mm (827×10–6 in.) (before adjusting for finish effect) had 

fit pressures of 6.64, 7.83, 9.81, and 13.13 MPa (0.963, 1.14, 

1.42, and 1.90 ksi), respectively. However, the resulting life 

factors were nearly identical (0.80, 0.77, 0.77, and 0.78 for the 

heavy load case). Therefore, Tables 2 to 5 combine these 

results, showing the average life factors for each bore size. 

Although the different dimension series within a particular 

bore size had similar life factors, the life factor was not the same 

for different bore sizes. In general, the larger bore sizes had a 

slightly lower life factor.

The life factors found in this study range from 1.00 (no effect)—where there was no net interface pressure—to a worst case 
NASA/TM—2010-216913 18

of 0.38 (62-percent life reduction) for the maximum p6 fit on a 

100-mm-bore deep-groove ball bearing at 1200 MPa (175 ksi) 

maximum Hertz stress. The corresponding angular-contact 

bearing had a life factor of 0.47 (53-percent life reduction) under 

the same conditions. However, the angular contact bearing, which 

carries a much higher load, had a much shorter life—about 9270 

million revolutions for the 220 size, reduced to 4495 million 

revolutions by the maximum p6 interference fit, in contrast to the

220-size deep-groove bearing, which had a life of 51 475 million 

revolutions, reduced to 18 810 million revolutions.

As expected, tighter fits produced smaller life factors (i.e., 

shorter lives). In addition, the life factor was smallest (greatest 

life reduction) for bearings running under a light load, where the 

added stress due to the fit has a greater impact on the life.

Table 3 shows the variation (minimum to maximum) in life 

factor for 50-mm-bore bearings operating under three load cases 

(three levels of Hertz stress) at the five fit classes considered (j5 

to m6). Figure 14 shows lives for the light and heavy load cases

averaged from size 1910, 1010, 210, and 310 bearings (Table 3 

data without the average fits). In the three loosest fit classes (j5 

to k5), the minimum fit will produce no interface pressure; 

hence, the life factor is 1.00. 

For example, “k5-min” in the seventh row of Table 3 shows a 

“clearance” of –0.002 mm (–79×10–6 in.). (This is actually a 

slight interference.) However, after adjusting for asperity 

smoothing, it becomes a clearance of 0.002 mm. There is no 

interface pressure for a clearance; thus, (LF)h = 1.0. In the bar 

chart of Figure 14(a), the nonshaded part of the third bar that 

represents this k5-min fit shows an LF of 1.0.

For the heavier fits, the life factor is much smaller, especially 

at the light load level of 1200 MPa (175 ksi) maximum Hertz 

stress, where life is higher. The smallest life factor found for 50-

mm deep-groove bearings was 0.44 (56-percent life reduction). 

Likewise, on a 50-mm angular-contact bearing, the smallest life 

factor was 0.55 (45-percent life reduction). These values are 

represented by the shaded part of the right-most pair of bars in 

Figure 14(a).

On both bearings, the interference for “m6-max” was 

0.033 mm, reduced to 0.029 mm (1300×10–6 in.) by asperity 

smoothing. This fit produced an interface pressure of 22.40 MPa 

(3.25 ksi) on the 310-size deep-groove bearing and 21.00 MPa 

(3.05 ksi) on the 310-size angular-contact bearing. (Interface 

pressures are not shown in the tables.) 

Figures 15 and 16 show the life factors at the tight end of the 

tolerance band for 30-, 50-, 75-, and 100-mm-bore deep-groove 

and angular-contact ball bearings at the three maximum Hertz 

stress levels. These plots can be used to estimate the interference-fit life factor at Hertz stress levels between the values 

analyzed in this article. The curves also show that interference fit 

has relatively less effect at higher loads, where the unfactored 

bearing life is shorter. Note that if the bearing or shaft has a 

different tolerance than the ABEC-5 tolerance considered in this 

article, the life factor values should be adjusted appropriately.

The life factors found here for deep-groove ball bearings are 

similar to, but slightly greater than, those reported for cylindrical 

roller bearings in Oswald et al. (Ref. 10). Note, however, that 

cylindrical roller bearings have a much higher radial load capacity than ball bearings. 

Application of the life factors presented here may result in 

conservative predictions of bearing life because bearing life 

calculations and published life values are based on tests that may 

have included some level of fit. However, the fits used for the 

tests underlying the life calculations are unreported and thus not 

known. We have not attempted to account for these unknown 

fits.

The authors do not recommend that tight interference fits be 

avoided where operating conditions (speed and vibration levels) 

demand such fits unless an alternate method for preventing 

fretting between the bearing race and shaft is provided. Such 

alternate methods may include splines or keyways.

Effect of Interference Fit on 

Stress-Life Exponent

From Lundberg and Palmgren (Ref. 6), the theoretical relationship between the maximum Hertz stress and life in a roller 

bearing (with line contact) is an inverse eighth power. For ball 

bearings (with point contact) the relationship is an inverse ninth 

power where n = 9 is the stress-life exponent for ball bearings:

max

1 ~ n L

S

(29)

Interference fits can affect this maximum Hertz stress-life 

relation. Six curves for maximum Hertz stress versus life are 

shown in Figure 17 (for five interference fits plus zero fit) on 

210-size ball bearings. For each interference fit, the Hertz stresslife exponent n was calculated. The values for n are also shown 

in Figure 17.

With no interference fit, the Hertz stress-life exponent n is 9, 

based on Lundberg and Palmgren (Ref. 6). However, with an 

average m6 interference fit, the calculation resulted in 

n = 8.63 for deep-groove ball bearings and 8.74 for angularcontact ball bearings. When the results were recalculated based 

on the tight end of the tolerance band for the m6 interference fit 

(not shown in Fig. 17), n was found to be 8.36 and 8.52 for deepgroove and angular-contact ball bearings, respectively. Similar 

results were obtained for other bearing sizes.

This result of a reduction in the Hertz stress-life exponent is 

similar to that reported in Oswald et al. (Ref. 10) for cylindrical 

roller bearings except that for roller bearings, the nominal Hertz 

stress-life exponent, n is 8 rather than 9.

This effect on the stress-life exponent can impact the results 

of accelerated testing on bearings that have a heavy interference 

fit. If such tests are performed with a heavy load (thus at high 

Hertz stress) and the test results are then extrapolated to lower 

stress levels using the usual Hertz stress-life exponent n = 9, the 

predicted value of life may be too high, giving a nonconservative 

design.
NASA/TM—2010-216913 19
NASA/TM—2010-216913 20
NASA/TM—2010-216913 21
NASA/TM—2010-216913 22

Summary of Results

The effect of hoop stresses in reducing deep-groove and 

angular-contact ball bearing fatigue life was determined for 

various classes of inner-ring interference fit. Calculations 

were performed for up to seven interference-fit classes for 

each bearing series. Each fit was taken at the maximum, 

average, and minimum values within the fit class for ABEC−5 

tolerances, thus requiring over 1150 separate analyses. 

The hoop stresses were superimposed on the Hertzian principal stresses created by light, normal, and heavy applied 

radial loads to determine ball bearing fatigue life. The results 

are presented as life factors for bearings loaded to maximum 

Hertz stress levels of 1200, 1720, and 2240 MPa (175, 250, 

and 325 ksi) in up to seven fit classes (very light to very 

heavy) and for bearing accuracy class ABEC−5. If the bearing 

or shaft has a different tolerance than considered herein, the 

life factor values should be adjusted appropriately.

The experimental data of Czyzewski (Ref. 3), showing the 

effect of interference fit on rolling-element fatigue life, were

reanalyzed to determine the shear stress-life exponent c. This 

was compared with the analysis reported in this paper. 

All calculations are for zero initial internal clearance conditions and assume low speed (below 1 million DN). Any 

reduction in internal bearing clearance due to the interference 

fit would be compensated for by increasing the initial (unmounted) clearance. 

Bearing life data as calculated from bearing standards or as 

found in manufacturers' catalogs are based on tests that 

presumably would have included interference fits on the rings.

However, these fits are unreported and unknown. Therefore, 

the life factor calculations reported herein are made using the 

conservative assumption that the bearing life calculations did 

not include the effect of interference fit.

The life factor for interference fit in low-speed ball bearings can be determined through charts or tables from the 

maximum Hertz stress, which is easily calculated from the 

applied radial load and the static-load capacity. The following 

results were obtained. 

(1) Interference fits on the inner bearing ring of a ball bearing can significantly reduce fatigue life. A heavy fit (maximum end of m6 tolerance band) on a 50-mm-bore deepgroove ball bearing reduced the fatigue life by 56, 43, and 

35 percent from the standard life at maximum Hertz stresses 

of 1200, 1720, and 2240 MPa (175, 250, and 325 ksi), respectively.

(2) The experimental data of Czyzewski exhibit a shear 

stress-life exponent c that equals 8.77, in comparison to an

assumed value of 9. 

(3) Life factors on thrust-loaded angular-contact ball bearings were higher than on radially loaded deep-groove ball 

bearings. However, the life of angular-contact bearings for a 

given maximum Hertz stress is lower than that of deep-groove 

bearings. 

(4) Tighter interference fits produced smaller life factors 

(i.e., shorter lives). Life factors due to hoop stresses found in 

this study ranged from 1.00 (no effect)—where there was no 

interface pressure—to as low as 0.38 (62-percent life reduction) for the maximum p6 fit on a 100-mm-bore deep-groove

ball bearing and a life factor of 0.47 (53-percent reduction) on 

a 100-mm angular contact ball bearing at 1200 MPa (175 ksi) 

maximum Hertz stress.

(5) In general, the life factor was smallest (greatest life 

reduction) for bearings running under light load where the 

unfactored life was highest. For any particular bearing size 

and interference fit, as the maximum Hertz stress on the innerrace was increased, the effect of the hoop stresses on life was 

reduced, thus increasing the resulting life factor.

(6) Interference fits affected the maximum Hertz stress-life 

relation. With no interference fit, a Hertz stress-life exponent 

of n = 9 was assumed. An m6 middle-of-the-tolerance-band

interference fit resulted in n = 8.63 for deep-groove ball 

bearings and n = 8.74 for angular-contact ball bearings.
NASA/TM—2010-216913 23

Appendix.—Nomenclature

Asector Area of bearing ring cross-section removed in grinding the ball track

a Semiwidth of major axis of Hertzian contact area, 

mm (in.)

b Semiwidth of minor axis of Hertzian contact area, 

mm (in.)

CD Dynamic load capacity, N (lbf)

Cor Static radial load capacity, N (lbf)

c Shear stress-life exponent

D Diameter at the location of the maximum shear stress 

beneath the surface of the inner race, mm (in.) 

DN Bearing speed parameter: bearing speed in rpm multiplied by the bearing bore diameter in millimeters

d Ball diameter, mm (in.)

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

e Weibull slope or modulus (taken to be 1.11 herein)

F Probability of failure (1 − S), fraction or percent

fc Bearing geometry and material coefficient used in 

Lundberg-Palmgren equation (Lundberg and 

Palmgren (Ref. 6)). (Standards currently designate 

this as fcm)

fi Inner race conformity Ri/d, where Ri is the raceway 

groove radius and d is the ball diameter

k1 Conversion constant for Equation (4) 

k2 Conversion constant for Equation (20) 

k3 Parameter used to calculate location of maximum 

shear stress (in the example analyses provided in this 

paper)

L Life, millions of inner-race revolutions or hours

L10 10-percent life, or life at which 90 percent of a 

population survives, millions of inner-race revolutions or hours

(LF)h Life factor for hoop stress

LR Life ratio for hoop stress on the inner ring defined in 

Equation (15) 

n Hertz stress-life exponent

P Radial load on bearing, N (lbf)

p Load-life exponent

pi Contact pressure between shaft and inner ring due to 

interference fit, MPa (psi)

R Radius of curvature, mm (in.)

R′ Ratio of DIR to d 

S Reliability or probability of survival (1 − F), fraction 

or percent

Smax Maximum Hertz stress, MPa (ksi)

Sn Normal stress, MPa (ksi)

St Tangential stress, MPa (ksi)

t S′ Tangential stress including hoop stress superimposed 

on Hertz stress, MPa (ksi)

u Dimensionless depth (z/b) below surface to location 

of maximum shear stress

W Width of bearing ring, mm (in)

X Ratio of the lives of the inner and outer races

z Distance below surface to maximum shear stress due 

to Hertzian load, mm (in.)

α Contact angle, deg

∆ Diametral interference, mm (in.) 

θ Sector angle, rad

ν Poisson’s ratio

σ Stress, MPa (ksi)

τ Shear stress, MPa (ksi)

(τmax)h Maximum shear stress modified by hoop stress, MPa 

(ksi)

Subscripts

adj Adjusted life

B Ball set

eff Effective, used to adjust the diameter of the inner 

ring in Equations (8a) and (8b) 

eq Equivalent

h Hoop stress (in tangential or x-direction)

IR Inner race of bearing

max Maximum

n Normal direction

OR Outer race of bearing

r residual stress

S Shaft and inner-ring bore

t Tangential direction

x Tangential direction (i.e., in rolling direction)

y Transverse direction

z Normal direction
NASA/TM—2010-216913 24

References

1. American Bearing Manufacturers Association (ABMA). (2001), 

“Shaft and Housing Fits for Metric Radial Ball and Roller Bearings (Except Tapered Roller Bearings) Conforming to Basic 

Boundary Plan,” ANSI/ABMA-7:1995 (R2001), The American 

Bearing Manufacturers Association, Washington, DC.

2. American Bearing Manufacturers Association (ABMA). (2008), 

“Radial Bearings of Ball, Cylindrical Roller and Spherical Roller Types—Metric Design,” ANSI/ABMA 20-1996 (R2008), The

American Bearing Manufacturers Association, Washington, DC.

3. Czyzewski, T. (1975), “Influence of a Tension Stress Field Introduced in the Elastohydrodynamic Contact Zone on Rolling 

Element Fatigue,” Wear, 34, 2, pp. 201–214.

4. Coe, H.H. and Zaretsky, E.V. (1986), “Effect of Interference 

Fits on Roller Bearing Fatigue Life,” ASLE Transactions, 30, 2, 

pp. 131–140.

5. Jones, A.B. (1946), Analysis of Stress and Deflections, Vol. 1, 

New Departure Div., General Motors Corp., Bristol, CT.

6. Lundberg, G. and Palmgren, A. (1947), “Dynamic Capacity of 

Rolling Bearings,” Acta Polytech, Mechanical Engineering Series, 1, 3, Stockholm, Sweden.

7. Zaretsky, E.V., ed. (1999), STLE Life Factors for Rolling Bearings, STLE SP–34, 2nd edition, Society of Tribologists and Lubrication Engineers, Park Ridge, IL. ISBN 10: 9993313599, 

ISBN 13: 9789993313595.

8. Zaretsky, E.V., Poplawski, J.V. and Root, L.E. (2007), “Reexamination of Ball-Race Conformity Effects on Ball Bearing 

Life,” Tribology Transactions, 50, 3, pp. 336–349.

9. Zaretsky E.V., Coe H.H. and August R. (1997), “Effect of Hoop 

Stress on Ball Bearing Life Prediction,” Tribology Transactions, 

40, 1, pp. 91–101.

10. Oswald, F.B., Zaretsky, E.V. and Poplawski, J.V. (2009), “Interference-Fit Life Factors for Roller Bearings,” Tribology Transactions, 52, 4, 415–426.

11. Brewe, D. E. and Hamrock, B. J. (1977), “Simplified Solution 

for Elliptical-Contact Deformation Between Two Elastic Solids,” ASME Journal of Lubrication Technology, 99, series F, 

pp. 485–487.

12. Hamrock, B.J. and Brewe, D. (1983), “Simplified Solution for 

Stresses and Deformations,” ASME Journal of Lubrication

Technology, 105, pp. 171–177.

13. Antoine J-F., Visa, C., Sauvey, C. and Abba, G. (2006), “Approximate Analytical Model for Hertzian Elliptical Contact 

Problems,” ASME Journal of Tribology, 128, 3, pp. 660–664. 

14. Juvinall, R.C. (1967), Engineering Considerations of Stress 

Strength and Strain, McGraw Hill, New York, NY. LCCN: 67-

011208.

15. Weibull, W. (1939), “A Statistical Theory of the Strength of 

Materials,” Ingeniorsvetenskapsakad, Handl., No. 151, Roy. 

Swed. Inst. Eng. Res, Stockholm, Sweden. 

16. Weibull, W. (1939), “Phenomenon of Rupture of Solids,” Ingeniorsvetenskapsakad Handl., No. 153, Roy. Swed. Inst. Eng. 

Res, Stockholm, Sweden.

17. Weibull, W. (1951), “A Statistical Distribution Function of 

Wide Applicability,” Journal of Applied Mechanics, ASME

Transactions, 18, 3, pp. 293−297.

18. Poplawski, J.V., Peters, S.M. and Zaretsky, E.V. (2001), “Effect 

of Roller Profile on Cylindrical Roller Bearing Life Prediction—

Part I Comparison of Bearing Life Theories,” Tribology Transactions, 44, 3, pp. 339–350.

19. Palmgren, A. (1945), Ball and Roller Bearing Engineering, 1st

Edition, translation by G. Palmgren and B. Ruley, SKF Industries, Philadelphia, PA.

20. The Timken Company (2004), “Timken Aerospace Design Guide 

for Precision Metric Ball and Cylindrical Roller Bearings,” 3M 04-

04-06 Order No. 5912, The Timken Company, 

http://www.timken.com/enus/products/bearings/productlist/aerospace/Documents/aerospace%

20product%20design%20guide.pdf, accessed April 23, 2010.

21. Anti-Friction Bearing Manufacturers Association (AFBMA), 

(1978), “American National Standard, AFBMA Standard—Load 

Ratings and Fatigue Life for Ball Bearings,” ANSI/AFBMA-Std 

9:1978, Anti-Friction Bearing Manufacturers Association, 

Arlington, VA.

22. American Bearing Manufacturers Association (ABMA). (2000), 

“Load Ratings and Fatigue Life for Ball Bearings,” 

ANSI/ABMA-9:1990 (R2000), American Bearing Manufacturers Association, Washington, DC.

23. International Organization for Standardization (ISO), (2006), 

“Rolling Bearings—Static Load Ratings,” ISO 76:2006, International Organization for Standardization, Geneva, Switzerland.

24. Poplawski, J. V., Rumbarger, J. H., Peters, S. M., Flower, R. and 

Galaitis, H. (2002) “Advanced Analysis Package for High Speed 

Multibearing Shaft Systems: COBRA–AHS,” Final report, 

NASA Contract NAS3–00018.

25. Czyzewski, T. (1975), “Changes in the Stress Field in the Elastohydrodynamic Contact Zone Due to Some Operating Factors 

and Their Possible Role in the Rolling Contact Fatigue of Cylindrical Surfaces,” Wear, 31, 1, pp. 119–140

26. Harris, T.A. (1991), Rolling Bearing Analysis, 3rd Edition, John 

Wiley & Sons, Inc., New York, NY, ISBN 0–471–51349–0.
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14. ABSTRACT

The effect of hoop stresses on the rolling-element fatigue life of angular-contact and deep-groove ball bearings was determined for common 

inner-ring interference fits at the ABEC-5 tolerance level. The analysis was applied to over 1150 bearing configurations and load cases. 

Hoop stresses were superimposed on the Hertzian principal stresses created by the applied bearing load to calculate the inner-race maximum 

shearing stress. The resulting fatigue life of the bearing was recalculated through a series of equations. The reduction in the fatigue life is 

presented as life factors that are applied to the unfactored bearing life. The life factors found in this study ranged from 1.00 (no life 

reduction)--where there was no net interface pressure--to a worst case of 0.38 (a 62-percent life reduction). For a given interference fit, the 

reduction in life is different for angular-contact and deep-groove ball bearings. Interference fits also affect the maximum Hertz stress-life 

relation. Experimental data of Czyzewski, showing the effect of interference fit on rolling-element fatigue life, were reanalyzed to determine 

the shear stress-life exponent. The Czyzewski data shear stress-life exponent c equals 8.77, compared with the assumed value of 9. Results 

are presented as tables and charts of life factors for angular-contact and deep-groove ball bearings with light, normal, and heavy loads and 

interference fits ranging from extremely light to extremely heavy.

15. SUBJECT TERMS

Interface fit; Ball bearings; Rolling bearings; Fatigue analysis; Contacts; Life prediction methods; Maintenance; Rolling-contact 

fatigue; Wear; Stress; Analysis 

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