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 Roller Bearings

NASA/TM—2009-215285

November 2009
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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 Roller Bearings

NASA/TM—2009-215285

November 2009

National Aeronautics and

Space Administration

Glenn Research Center

Cleveland, Ohio 44135

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Level of Review: This material has been technically reviewed by technical management. 
NASA/TM—2009-215285 1 

Interference-Fit Life Factors for Roller 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 in reducing cylindrical roller bearing fatigue life was determined for 

various classes of inner-ring interference fit. Calculations were performed for up to 7 fit classes for each 

of 10 bearing sizes. The hoop stresses were superimposed on the Hertzian principal stresses created by the 

applied radial load to calculate roller bearing fatigue life. A method was developed through a series of 

equations to calculate the life reduction for cylindrical roller bearings. All calculated lives are for zero 

initial internal clearance. Any reduction in bearing clearance due to interference fit would be compensated 

by increasing the initial (unmounted) clearance. Results are presented as tables and charts of life factors 

for bearings with light, moderate, and heavy loads and interference fits ranging from extremely light to 

extremely heavy for bearing accuracy class RBEC−5 (ISO class 5). Interference fits on the inner ring of a 

cylindrical roller bearing can significantly reduce bearing fatigue life. In general, life factors are smaller 

(lower life) for bearings running under light load where the unfactored life is highest. The various bearing 

series within a particular bore size had almost identical interference-fit life factors for a particular fit. The 

tightest fit at the high end of the tolerance band produces a life factor of approximately 0.40 for an innerrace maximum Hertz stress of 1200 MPa (175 ksi) and a life factor of 0.60 for an inner-race maximum 

Hertz stress of 2200 MPa (320 ksi). Interference fits also impact the maximum Hertz stress-life relation. 

Introduction 

Rolling-element bearings often utilize a tight interference fit between the bearing inner ring and shaft 

or between the outer ring and housing bore to prevent fretting damage at the interface. American National 

Standards Institute/American Bearing Manufacturers Association (ANSI/ABMA) standards (Refs. 1 

and 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 highest 

speeds and highest vibration levels. 

A tight fit of the bearing inner ring over the shaft reduces internal bearing clearance. The clearance 

change can be compensated for by adding initial internal clearance to the bearing. However, the 

interference fit of the inner ring over the shaft also adds a hoop stress on the bearing inner ring. 

Czyzewski (Ref. 3) showed that the hoop stresses are generally tensile (designated by a plus (+) sign) and 

can negatively affect fatigue life. 

Coe and Zaretsky (Ref. 4) analyzed the effect of hoop stresses on rolling-element fatigue. 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 of the 

contact between the rolling element (ball or roller) and inner race of the bearing are increased. The 

increased maximum shearing stress at a depth below the contacting surface due to hoop stress is 
NASA/TM—2009-215285 2 

 ( ) 2 max max

h

h

σ

τ =τ + (1) 

where τmax is the maximum shearing stress, (τmax)h is the maximum shearing stress including the effect of 

the hoop stress, and σh is the hoop stress. 

Coe and Zaretsky (Ref. 4) applied Equation (1) to modify the Lundberg-Palmgren life equation as follows: 

 ( )

p

eq

D

c e

h P

C L

































τ

τ =

/

max

max 10 (2) 

where c is a stress-life exponent, e is the Weibull slope or modulus, CD is the bearing dynamic load 

capacity, Peq is the equivalent bearing load, and p is a load-life exponent. 

The Coe-Zaretsky analysis assumed for simplicity that all components (inner race, rollers, and outer 

race) are affected equally by the inner-race interference fit. This assumption results in a conservative 

prediction of bearing life. A more rigorous analysis should apply the life reduction due to hoop stress to 

only the inner race without modifying the lives of the outer race and rolling-element set. 

Subsequent to the Coe-Zaretsky analysis, Zaretsky (Ref. 7) and (Zaretsky, Poplawski, and Root 

(Ref. 8)) developed a procedure (Zaretsky’s Rule) for separating the lives of bearing races from the lives 

of the rolling elements (considered as a set). This procedure allows for calculating the reduction of 

bearing life from an inner-ring interference fit without affecting the rolling-element set and outer race 

lives, thus providing a more accurate bearing life analysis. 

In view of the aforementioned, the objectives of this work were to (a) independently determine the 

lives of the inner races, outer races, and roller sets for several classes of radially loaded, cylindrical roller 

bearings subject to inner-ring interference fit; (b) calculate the reduction in cylindrical roller bearing 

fatigue life due to interference fit of the inner ring; and (c) develop life factors applied to the bearing life 

calculation for interference fits according to the ANSI/ABMA standards for shaft-fitting practice. 

Nomenclature 

b semiwidth of Hertzian contact area, m (in.) 

CD dynamic load capacity, N (lbf) 

C0 static load capacity, N (lbf) 

c stress-life exponent 

D diameter, mm (in.)

E elastic (Young’s) modulus, MPa (psi) 

e Weibull slope 

k conversion constant in Equation (17)

L life, millions of inner-race revolutions or h 

L10 10-percent life or life at which 90 percent of a population survives, millions of inner-race 

revolutions or h 

(LF)h life factor for hoop stress 

LR life ratio defined in Equation (13) 
NASA/TM—2009-215285 3 

P radial load on bearing, N (lbf) 

Pi pressure between shaft and inner race due to interference fit, MPa (psi) 

p load-life exponent 

R radius of curvature, mm (in.) 

S stress, MPa (ksi) 

IR

Smax maximum Hertz stress on inner race, MPa (ksi) 

OR Smax maximum Hertz stress on outer race, MPa (ksi) 

St

′ tangential stress including hoop stress superimposed on Hertz stress, MPa (ksi) 

u dimensionless depth below surface to maximum shear stress (= z/b) 

z distance below surface to maximum shear stress due to Hertzian load, m (in.) 

Δ diametral interference, mm (in.) 

ν Poisson’s ratio 

σ stress, MPa (ksi) 

τ shear stress, MPa (psi) 

(τmax)h maximum shear stress modified by hoop stress, MPa (psi)

Subscripts 

adj adjusted life 

eq equivalent 

h hoop stress (in tangential or x-direction) 

IR,OR inner or outer race of bearing 

n or z normal direction 

R roller set 

S shaft and inner-ring bore 

t or x tangential direction 

Enabling Equations 

Subsurface Shearing Stresses 

A representative cylindrical roller bearing is shown in Figure 1. The bearing comprises an inner and 

outer ring and plurality of rollers interspersed between the two rings and positioned by a cage or separator. 

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

the von Mises, and the maximum. For the analysis reported herein, only the maximum shearing stresses 

are considered. 

 
NASA/TM—2009-215285 4 

The maximum shearing stress is one-half the maximum difference between principal stresses: 

 2 max

σ z σ x τ − = (3) 

Coe and Zaretsky (Ref. 4) showed that the subsurface shear stress in a cylindrical roller bearing due 

to Hertzian loading as a function of z, the depth below the surface, can be expressed in terms of the 

maximum Hertz stress Smax and the non-dimensional depth below the surface u: 

 















+

τ = + − − 2

2 max

1

1 1

u

S u u (4) 

where u = z/b and b is the semiwidth of the Hertzian contact area (Fig. 2(a)). By setting the derivative of 

Equation (4) with respect to u equal to zero and solving by iteration, the maximum shearing stress from 

Hertzian loading occurs at u = 0.786152. Substituting u into Equation (4) gives the maximum shear stress 

due to Hertzian loading: 

 max max τ = −0.30028 S (5) 

Coe and Zaretsky (Ref. 4) considered effects due to the hoop stress from an interference fit of the 

inner ring on the shaft and the effects due to rotation superposed on the Hertz stress. Their analysis 

showed that the additional effects cause little change in the location of the maximum shear stress on the 

inner-race surface; the variation in u due to the added hoop stress is only 0.074 percent. 

The principal stresses in the tangential direction and the effect of the added hoop stress are illustrated 

in Figure 3 where Sn is the normal stress, St is the tangential stress due only to Hertzian loading, and St

′ is 

the tangential stress including hoop stress superimposed on St. The maximum shear stress is one-half the 

difference between Sn and St

′ . 

 
NASA/TM—2009-215285 5 

 
NASA/TM—2009-215285 6 

Zaretsky (Ref. 7) gives a simplified procedure for finding the effect due to hoop stress from inner-ring 

interference fits and inner-ring rotation, assuming the value u = 0.78667. 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. 

Zaretsky’s procedure requires the contact pressure Pi between the inner race and the shaft. For the 

case where both components have the same material properties, Pi is given as p in Equation (6) below 

from Juvinall (Ref. 9), where E is Young’s modulus, δ is the radial interference, a is the inside radius of 

the shaft, b is the radius of the inner ring to shaft interface, and c is the outside radius of the inner ring: 

 ( )( )

( ) 3 2 2

2 2 2 2

2b c a

E b a c b

p

−

δ − − = (6) 

For a bearing race shrunk on a solid shaft, dimensions δ = Δ/2, a = 0, b = DS/2, c = DIR/2, and 

Equation (6) becomes 

 ( )

2

2 2

2 S IR

IR S

i

D D

EΔ D − D

P = (7) 

Strict Series Reliability and Zaretsky’s Rule 

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

bearing component life 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 strict-series system reliability (Ref. 6) and the two-parameter Weibull distribution function (10–12) 

for the bearing components comprising the system. Lundberg and Palmgren (Ref. 6) expressed the 

bearing system fatigue life as follows: 

 e

OR e

IR eL L L

1 1 1 = + (8) 
NASA/TM—2009-215285 7 

Zaretsky, Poplawski, and Root (Ref. 8) note that the life of the rolling element set is implicitly 

included in the inner and outer race lives above. If the life of the rollers LR (taken as a set) is separated 

from the race lives, Equation (8) can be rewritten as 

 e

OR adj e

R e

IR adj eL L L L − −

= + +

1 1 1 1 (9) 

where adj in the subscript indicates adjusted lives for the races that will be greater than the corresponding 

lives in Equation (8). Zaretsky (Ref. 7) and Zaretsky, Poplawski, and Root (Ref. 8) observe that the life of 

the outer race LOR is generally greater than the life of the inner race LIR and the life of the roller set LR is 

equal to or greater than the life of the outer race. In this report, we assume that LR = LOR; thus, 

Equation (9) becomes 

 e

OR adj eIR adj eL L L − −

= +

1 1 2 (10) 

 

We define the ratio of lives between outer and inner races as 

 

IR

OR

L

L X = (11) 

If we assume that the life ratio X does not change when the roller life is separated from the race lives, 

Equation (10) becomes 

 ( )

e

IR adj

e

IR adj e X L L L − − ⋅

= +

1 1 2 (12) 

Bearing Life Factor for Interference Fit 

Coe and Zaretsky (Ref. 4) show that the life ratio for hoop stress LR is the ninth power of the ratio of 

maximum shear stress τmax from Hertz loading alone (from Eq. (5)) to the maximum shearing stress 

including both Hertizan loading and hoop stress (τmax)h, which can be computed from the simplified 

procedure of Zaretsky (Ref. 7): 

 ( )

( )

9

max

max

















τ

τ = =

h

h

L

L

LR (13) 

Equation (13) 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, Peters, and Zaretsky (Ref. 13). 

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 will be 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 (12) to calculate the life of the bearing (L)h, including the effects of Hertzian loading and 

hoop stress: 
NASA/TM—2009-215285 8 

 ( ) ( ) ( )e IR adj e IR adj e

h L LR L − X L − ⋅ + ⋅ = 1 1 2 (14) 

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

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

 ( ) ( )

L

L

LF h

h = (15) 

Determining Life Factor Based on Load and Fit 

For most low-speed roller 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 roller bearing size, radial load, and interference fit can be related to the bearing static 

load capacity C0 without the need to perform extensive calculations. 

Rolling element bearing static load capacity was first defined in terms of deformation by Palmgren 

(Ref. 14): “... 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 roller 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 

 Smax ~ P (16) 

Nearly all bearing manufacturers’ catalogs provide the static load capacity C0 for any bearing size. 

Hence, to determine the appropriate stress at the applied radial load on the roller bearing inner race, 

Equation (16) can be rewritten as follows: 

 

0 max C

P S =k (17) 

where the conversion constant k = 4000 for SI units with Smax expressed in megapascals or k = 580 for 

English traditional units with Smax expressed in kips per square inch. Table I (using data from The Timken 

Company (Ref. 15)) gives the static load capacity for the bearings discussed herein. 

TABLE I.—CYLINDRICAL ROLLER BEARING PROPERTIES 

[From The Timken Company (Ref. 15).] 

ABMA 

number 

Bore, 

mm 

Outside 

diameter, 

mm 

Number 

of 

rollers 

Roller 

diameter, 

mm 

Roller 

length, 

mm 

Inner-race outside 

diameter, 

mm (in) 

Static load capacity, 

C0, 

kN (lbf) 

1906 30 47 18 5 5 33.75 (1.3287) 14.737 (3 313) 

0206 62 12 10 10 37.02 (1.4575) 37.632 (8 460) 

1910 50 72 20 7 7 54.00 (2.126) 32.881 (7 392) 

1010 80 18 9 9 56.45 (2.2224) 49.446 (11 116) 

0210 90 14 13 13 57.65 (2.2697) 77.226 (17 361) 

0310 110 10 19 19 61.95 (2.439) 111.010 (24 956) 

1915 75 105 24 9 9 81.00 (3.189) 68.796 (15 466) 

0215 130 14 18 18 84.50 (3.3268) 149.420 (33 591) 

1920 100 140 24 12 12 108.00 (4.252) 123.278 (27 714) 

0220 180 14 25 25 115.00 (4.5275) 295.980 (66 539) 

 
NASA/TM—2009-215285 9 

The maximum Hertz stress Smax value as a function of the applied radial load and static load capacity 

C0 is plotted in Figure 4. The appropriate life factor can be interpolated from Tables II to V for the 

various interference fits. 

As an example of using this procedure, consider a 0210-size bearing with a radial load P = 6.95 kN 

(1562 lbf). From Table I, the static load capacity C0= 77.226 kN (17 361 lbf). Using either Figure 4, 

with P/C0 = 0.09, or Equation (17), with 0 P/C = 0.300, and the appropriate value for k yields 

Smax= 1200 MPa or 175 ksi. 

TABLE II.—LIFE FACTORS FOR 30-mm-BORE CYLINDRICAL ROLLER BEARING 

WITH RBEC−5 TOLERANCES 

[Results averaged from ABMA 1906 and 0206 bearings.]

ABMA 

fit class 

Clearance, 

mm 

Inner-race 

Hertz stress, 

MPa (ksi) 

Life 

factor 

ABMA 

fit class 

Clearance,

mm 

Inner-race 

Hertz stress, 

MPa (ksi) 

Life 

factor 

j5-min 0.004 1200 (175) 1 k5-min –0.002 1200 (175) 1 

1700 (250) 1 1700 (250) 1 

2200 (320) 1 2200 (320) 1 

j5 –0.0035 1200 (175) 1 k5 –0.0095 1200 (175) 0.77 

1700 (250) 1 1700 (250) .83 

2200 (320) 1 2200 (320) .87 

j5-max –0.011 1200 (175) 0.72 k5-max –0.017 1200 (175) 0.54 

1700 (250) .79 1700 (250) .65 

2200 (320) .84 2200 (320) .71 

j6-min 0.004 1200 (175) 1 m5-min –0.008 1200 (175) 0.83 

1700 (250) 1 1700 (250) .88 

2200 (320) 1 2200 (320) .90 

j6 –0.0055 1200 (175) 0.93 m5 –0.0155 1200 (175) 0.58 

1700 (250) .95 1700 (250) .68 

2200 (320) .96 2200 (320) .74 

j6-max –0.015 1200 (175) 0.59 m5-max –0.023 1200 (175) 0.40 

1700 (250) .69 1700 (250) .52 

2200 (320) .75 2200 (320) .61 
NASA/TM—2009-215285 10 

TABLE III.—LIFE FACTORS FOR 50-mm-BORE 

CYLINDRICAL ROLLER BEARING 

WITH RBEC−5 TOLERANCES 

[Results averaged from ABMA 1910, 

1010, 0210, 0310 bearings.]

ABMA 

fit class 

Clearance, 

mm 

Inner-race 

Hertz stress, 

MPa (ksi) 

Life 

factor 

j5-min 0.005 1200 (175) 1 

1700 (250) 1 

2200 (320) 1 

j5 –0.0045 1200 (175) 0.99 

1700 (250) .99 

2200 (320) .99 

j5-max –0.014 1200 (175) 0.74 

1700 (250) .81 

2200 (320) .85 

j6-min 0.005 1200 (175) 1 

1700 (250) 1 

2200 (320) 1 

j6 –0.007 1200 (175) 0.92 

1700 (250) .94 

2200 (320) .95 

j6-max –0.019 1200 (175) 0.64 

1700 (250) .73 

2200 (320) .78 

k5-min –0.002 1200 (175) 1 

1700 (250) 1 

2200 (320) 1 

k5 –0.0115 1200 (175) 0.80 

1700 (250) .85 

2200 (320) .89 

k5-max −0.021 1200 (175) 0.60 

1700 (250) .70 

2200 (320) .76 

m5-min −0.009 1200 (175) 0.86 

1700 (250) .90 

2200 (320) .92 

m5 −0.0185 1200 (175) 0.65 

1700 (250) .74 

2200 (320) .79 

m5-max −0.028 1200 (175) 0.48 

1700 (250) .60 

2200 (320) .67 

m6-min −.009 1200 (175) 0.86 

1700 (250) .90 

2200 (320) .92 

m6 –0.021 1200 (175) 0.60 

1700 (250) .70 

2200 (320) .76 

m6-max −0.033 1200 (175) 0.41 

1700 (250) .53 

2200 (320) .62 

TABLE IV.—LIFE FACTORS FOR 75-mm-BORE 

CYLINDRICAL ROLLER BEARING 

WITH RBEC−5 TOLERANCES 

[Results averaged from ABMA 

1915 and 0215 bearings.]

ABMA 

fit class 

Clearance, 

mm 

Inner-race 

Hertz stress, 

MPa (ksi) 

Life 

factor 

j5-min 0.007 1200 (175) 1

1700 (250) 1

2200 (320) 1

j5 –0.004 1200 (175) 1

1700 (250) 1

2200 (320) 1

j5-max –0.015 1200 (175) 0.80

1700 (250) .86

2200 (320) .89

j6-min 0.007 1200 (175) 1

1700 (250) 1

2200 (320) 1

j6 –0.007 1200 (175) 0.94

1700 (250) .96

2200 (320) .97

j6-max –0.021 1200 (175) 0.71

1700 (250) .79

2200 (320) .83

k5-min –0.002 1200 (175) 1

1700 (250) 1

2200 (320) 1

k5 –0.013 1200 (175) 0.84

1700 (250) .88

2200 (320) .91

k5-max –0.024 1200 (175) 0.67

1700 (250) .75

2200 (320) .80

m5-min –0.011 1200 (175) 0.87

1700 (250) .91

2200 (320) .93

m5 –0.0122 1200 (175) 0.70

1700 (250) .78

2200 (320) .82

m5-max –0.033 1200 (175) 0.55

1700 (250) .66

2200 (320) .73

m6-min –.011 1200 (175) 0.87

1700 (250) .91

2200 (320) .93

m6 –0.025 1200 (175) 0.65

1700 (250) .74

2200 (320) .80

m6-max –0.039 1200 (175) 0.48

1700 (250) .60

2200 (320) .68

n6-min –0.020 1200 (175) 0.73

1700 (250) .80

2200 (320) .84

n6 –0.034 1200 (175) 0.54

1700 (250) .65

2200 (320) .72

n6-max –0.048 1200 (175) 0.40

1700 (250) .53

2200 (320) .61
NASA/TM—2009-215285 11 

TABLE V.—LIFE FACTORS FOR 100-mm-BORE CYLINDRICAL ROLLER BEARING 

WITH RBEC−5 TOLERANCES 

[Results averaged from ABMA 1920 and 0220 bearings.]

ABMA 

fit class 

Clearance, 

mm 

Inner-race 

Hertz stress, 

MPa (ksi) 

Life 

factor 

ABMA 

fit class 

Clearance,

mm 

Inner-race 

Hertz stress, 

MPa (ksi) 

Life 

factor 

j5-min +0.009 1200 (175) 1 m5-max −0.038 1200 (175) 0.60 

1700 (250) 1 1700 (250) .70 

2200 (320) 1 2200 (320) .76 

j5 −0.0035 1200 (175) 1 m6-min −.013 1200 (175) 0.88 

1700 (250) 1 1700 (250) .91 

2200 (320) 1 2200 (320) .93 

j5-max −0.016 1200 (175) 0.84 m6 −0.029 1200 (175) 0.69 

1700 (250) .88 1700 (250) .77 

2200 (320) .91 2200 (320) .82 

j6-min +0.009 1200 (175) 1 m6-max −0.045 1200 (175) 0.54 

1700 (250) 1 1700 (250) .65 

2200 (320) 1 2200 (320) .72 

j6 −0.007 1200 (175) 0.96 n6-min −0.023 1200 (175) 0.76 

1700 (250) .97 1700 (250) .82 

2200 (320) .98 2200 (320) .86 

j6-max −0.023 1200 (175) 0.76 n6 −0.039 1200 (175) 0.59 

1700 (250) .82 1700 (250) .69 

2200 (320) .86 2200 (320) .75 

k5-min −0.003 1200 (175) 1 n6-max −0.055 1200 (175) 0.46 

1700 (250) 1 1700 (250) .58 

2200 (320) 1 2200 (320) .66 

k5 −0.0155 1200 (175) 0.85 P6-min −0.037 1200 (175) 0.61 

1700 (250) .89 1700 (250) .71 

2200 (320) .91 2200 (320) .77 

k5-max −0.028 1200 (175) 0.70 p6 −0.053 1200 (175) 0.47 

1700 (250) .78 1700 (250) .59 

2200 (320) .83 2200 (320) .67 

m5-min −0.013 1200 (175) 0.88 p6-max −0.069 1200 (175) 0.36 

1700 (250) .91 1700 (250) .49 

2200 (320) .93 2200 (320) .58 

m5 −0.0255 1200 (175) 0.73 

1700 (250) .80 

2200 (320) .84 

Results and Discussion 

We applied the analysis described in the previous section to radially loaded cylindrical roller bearings 

from four bore sizes at either two or four dimension series for each bore size. Each bearing was analyzed 

at three levels of inner-ring Hertz stress. The dimension series are shown schematically in Figure 5. All 

bearings are made from AISI 52100 steel and have a “square” cross section, with the roller length equal to 

the diameter. Properties for the bearings analyzed are listed in Table I. 

The calculations were repeated for up to seven fit classes for each bearing, with each fit taken at the 

tightest, average, and loosest values within the fit class for RBEC−5 tolerance. This required 486 separate 

analyses. A graphical representation of shaft fits is shown schematically in Figure 6, which was adapted 

from ABMA (Ref. 1). 

Harris (Ref. 16) 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 recommended 

reduction is 4 μm (2 μm for each surface, bore I.D. and shaft O.D.). In this work, we have reduced the 

apparent interference by 4 μm (160 μin.) to account for surface finish effects. 

A commercial bearing analysis code (Ref. 17) was used to calculate the unfactored L10 lives for the 

inner and outer races operating without interference fit. The rollers were modeled with an aerospace 

crown (Poplawski, Peters and Zaretsky (Ref. 18)) chosen to minimize the effect of stress concentrations at 
NASA/TM—2009-215285 12 

the ends of the rollers. The crown has a flat length of 61 percent of the actual roller length and a 

crown radius approximately 100 times the roller length. 

All bearings were modeled at the zero internal operating clearance condition. In the case of inner-ring 

interference fit, this means the bearings would have an appropriate initial (unmounted) clearance between 

the rollers and the races. 

Each bearing was analyzed for three values of inner-race maximum Hertz stress: 1200, 1700, and 2200 

MPa (175, 250, and 320 ksi). The radial load for each bearing was chosen to give the desired stress value. 

The analysis code estimated the inner and outer race lives using the traditional Lundberg-Palmgren method 

(Lundberg and Palmgren (Ref. 6)). Therefore, these lives implicitly include the life of the roller set. 

 
NASA/TM—2009-215285 13 

Equation (11) was used to find the ratio of the inner and outer race lives for the bearing, and 

Equation (12) was used to calculate the adjusted value of the inner-race life (with the roller set life 

separated), assuming that X, the ratio of the race lives, does not change when the roller set life is separated 

from the race lives. 

For this study, data for bearing bore sizes were taken from the table for tolerance class ABEC−5, 

RBEC−5 (ABMA (Ref. 2)). However, 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 

(Ref. 1), which gives tolerance limits for ABEC−1, RBEC−1 quality level for bearing bore sizes from 3 to 

1250 mm (0.1181 to 49.2126 in.). 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. 16). 

The shaft diameter was subtracted from the bore diameter and then 0.004 mm (160 μin.) was added to 

the difference to account for asperity smoothing. If the resulting fit was positive (indicating clearance), 

then the interface pressure and thus the hoop stress were assumed to be zero. It 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 (7). 

The simplified procedure described in the next section was used to find the maximum shearing stress 

(τmax)h, including the effect of Hertzian loading and the effect from hoop stress due to the interference fit 

on the inner ring. Equation (13) 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 Equation (14) using the reduced life of 

the inner race and the original lives of the rollers and outer race. 

Analysis of 0210-Size Cylindrical Roller Bearing With m6 Fit 

As an example of the methods presented in this report, consider a 0210-size cylindrical roller bearing 

carrying a moderate load of 6.95 kN (1562 lbf) and a middle-of-the-tolerance-band m6 inner-ring 

interference fit. From Equation (17), the inner-race maximum Hertz stress Smax is 1200 MPa (175 ksi). 

The analysis code gave lives of the outer and inner races of 14 240 and 2303 million inner-race 

revolutions. From Equation (8), assuming a Weibull slope of 1.125, the L10 life of the bearing is 2068 

million revolutions: 

 1.125 1.125 1.125 1.125 2068

1

14 240

1

2303

1 1 = + = L

 (18) 

The ratio X of outer race life to inner-race life from Equation (11) is 14 240/2303 = 6.18. The adjusted life 

of the inner race was found using Equation (12): 

 ( ) 1.125 1.125 1.125 6.18

1 2

2068

1

adj IR adj IR L − L −

= + (19) 

The solution gives LIR-adj = 2535 million revolutions. (Although not needed for this calculation, the 

adjusted life of the outer race is 15 675 million revolutions.) 

The ANSI/ABMA shaft-fitting practice table for ABEC−1, RBEC−1 bearings (ABMA (Ref. 1)) was 

used to find limiting diameters for a 50-mm (2-in.) shaft with an m6 fit. Shaft fits are illustrated 

schematically in Figure 6. This figure shows 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 μin.) 

(shown as dimensions a and b in the figure). The bearing bore deviations were found in the table for 

tolerance class ABEC−5, RBEC−5 (ABMA (Ref. 2)). The bore deviation can range from 0.000 to -0.008 

mm (0 to –300 μin.) (shown as 0 and dimension c in the figure). 
NASA/TM—2009-215285 14 

The loosest m6 fit occurs when the largest bore bearing (50 mm) is mounted on the smallest shaft 

(50.009 mm or 1.9689 in.), producing a fit of 0.009 mm (350 μin.) tight (before adjusting for surface 

finish). This fit is illustrated as dimension a in Figure 6. The tightest fit is from the smallest bore 

(49.992 mm or 1.9682 in.) on the largest shaft (50.025 mm or 1.9695 in.), or 0.033 mm (1300 μin.) tight, 

shown as dimension d in Figure 6. The average of these extremes is 0.021 mm (830 μin.) tight. This 

interference fit was reduced by 0.004 mm (160 μin.) to account for asperity smoothing, assuming smoothground surfaces. The resulting middle-of-the-tolerance-band m6 fit is 0.017 mm (670 μin.) tight. 

For our example bearing, DS = 50 mm (1.9685 in.); DIR = 57.65 mm (2.2697 in.); roller diameter 

DR = 13 mm (0.5118 in.); Smax = 1200 MPa (175 ksi); E = 205 878 MPa (29.86×106

 psi); Poisson’s 

ratio ν = 0.3; and Δ = 0.017 mm (670 μin.). From Equation (7), the interference-fit contact pressure 

Pi = 8.67 MPa (1.257 ksi). 

Next, the life factor of the inner ring due to the interference-fit stress was calculated by the following 

the 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 τmax from Equation (5), where τmax = –(0.3)Smax 

= –360 MPa (–52.2 ksi). 

(2) Determine the geometry constant B, where B = DS/DIR = 0.867303 (dimensionless). 

(3) Determine m, where m = PiB2

/(1 – B2) = 26.3269 MPa (3.818 ksi). 

(4) Determine R′, where R′ = DIR/d = 4.4346 (dimensionless). 

(5) Determine K2, where K2 = E(R′ + 1)/(4(1 – ν

2

)Smax) = 256.151 (dimensionless). 

(6) Assume the value for u to be 0.78667 (dimensionless). 

(7) Determine y, where y = 1 – u/K2 = 0.996929 (dimensionless). 

(8) Substitute values for Smax, m, and y in Equation (20) below to calculate (τmax)h = –386.489 MPa 

(–56.056 ksi): 

 ( ) max max 2

y

m

h τ = τ − (20) 

(9) Compute the life ratio for the inner ring from Equation (13): LR = (–360/(–386.489))9 = 0.5278. 

Using the life ratio from step 9 above in Equation (14), the life of the bearing, including the effect of 

the interference fit, was calculated to be 1205 million inner-race revolutions: 

 ( ) ( ) ( ) 1.125 1.125 1.125 1.125 1205

1

6.18 2535

2

0.5278 2535

1 1 = ⋅ + ⋅ =

h L

 (21) 

The life factor for hoop stress (LF)h is 

 ( ) 0.58

2068

1205 = = h LF (22) 

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

42 percent. 

Interference-Fit Life Factors for RBEC−5 Roller Bearings 

The analysis described in the previous section was applied to 486 separate bearing configurations. It 

included four bore sizes, up to four dimension series (ranging from extremely light to medium), at three 

values of inner-ring Hertz stress, and up to seven inner-ring interference-fit classes. Each fit class was 

evaluated at the minimum, maximum, and average fit level for the RBEC−5 tolerance class. 
NASA/TM—2009-215285 15 

The shaft interference table from ABMA (Ref. 1) (as illustrated in Fig. 6) shows fit classes ranging 

from g6 (loose) to r7 (heavy interference). There is no effect on life for the looser fits that produce no 

pressure at the bore and no values are given for the heavier fit classes for small bearings. Hence, for 30-mm 

bearings, we calculated life factors for only four fit classes (j5 to m5); for 50-mm bearings, five classes (j5 

to m6); for 75-mm bearings, six classes (j5 to n6) and for 100-mm bearings, seven classes (j5 to p6). 

All interference fits were adjusted for the effect of asperity smoothing (assuming accurately ground 

surfaces) by adding 0.004 mm (160 μin.) to the clearance between the shaft and inner ring. If the 

clearance value was negative (indicating interference), then the resulting pressure was calculated. The 

results are shown in Tables II to V. 

The life factors found in this study range from 1.00 (no effect) where there is no interface pressure to 

a worst case of 0.36 (64-percent life reduction) for the tightest p6 fit on a 100-mm (220-size) bearing at 

1200 MPa (175 ksi) maximum Hertz stress. As should be expected, tighter fits produce smaller life 

factors (i.e., shorter lives). In general, the life factor is smallest (greatest life reduction) for bearings 

running under a light load where the unfactored life is highest. 

Figure 7 and Table III show the variation in life factor for 50-mm-bore bearings operating under three 

levels of Hertz stress at the five fit classes considered (j5 to m6). In the three lightest fits shown (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 III shows a “clearance” of –0.002 mm (–79 μin.). (This is actually a slight 
NASA/TM—2009-215285 16 

 
NASA/TM—2009-215285 17 

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 7(a), the 

nonshaded part of the third bar that represents this k5-min fit shows a LF of 1.0.

For the heavier fits, the life factor is less, ranging to a low value of 0.41 (59-percent life reduction) for 

“m6-max”, an interference of 0.033 mm (1300 μin.). This fit produces an interface pressure of 14.79 MPa 

(2.15 ksi) (not shown in the table). For a bearing running at 1200 MPa (175 ksi) maximum Hertz stress, 

the resulting (LF)h = 0.41. This value is represented by the shaded part of the right-most bar of 

Figure 7(a). 

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 bearings analyzed (nos. 

1910, 1010, 0210, and 0310) for the average m6 fit of 0.021-mm (827 μin.) interference (before adjusting 

for finish effect) have fit pressures of 5.00, 7.55, 8.67, and 12.21 MPa (0.725, 1.10, 1.26, and 1.77 ksi), 

respectively. However, the resulting life factors are nearly identical (0.60, 0.60, 0.58, and 0.60). 

Therefore, we have combined these results, showing average life factor values for each bore size in 

Tables II to V. 

Interestingly, at a given Hertz stress level, the tightest fits defined in the ANSI/ABMA tables 

produced very similar life factors, even on different bore sizes and for different fit designations. For 

example, at the 1200-MPa (175-ksi) maximum Hertz stress level, the tightest (m5) fit on a 30-mm-bore 

bearing has a life factor of 0.40 whereas the tightest (p6) fit on a 100-mm-bore bearing has a life factor of 

0.36. Likewise, for the 2200-MPa (320 ksi) maximum Hertz stress, the m5 fit on a 30-mm-bore bearing 

has a life factor of 0.61 whereas the p6 fit on a 100-mm-bore bearing has a life factor of 0.58. 

Figure 8 shows the middle of the tolerance band life factor for 30-, 50-, 75-, and 100-mm-bore 

cylindrical roller bearings at the three maximum Hertz stress levels. This plot can be used to estimate the 

life factor for Hertz stress levels between the values analyzed in this report. However, for conservative 

design, life factors should be chosen based on the tight end of the tolerance band rather than based on 

midband values. 

Effect of Interference Fit on Stress-Life Exponent 

Poplawski, Peters, and Zaretsky (Ref. 13) state that the theoretical relation between maximum Hertz 

stress and life in a roller bearing (with line contact) is an inverse eighth power: 

 

max

1 ~ n S

L (23) 

where L is bearing life, Smax is the maximum Hertz stress, and n = 8 is the stress-life exponent. 

Interference fits can affect this maximum Hertz stress-life relation. The curves for maximum Hertz 

stress versus life are shown in Figure 9 for five interference fits on 210-size cylindrical roller bearings. 

For each interference fit, the Hertz stress-life exponent n was calculated. These values are also shown in 

Figure 9. 

With no interference fit, our calculation resulted in a Hertz stress-life exponent n = 8.1, which is 

close to the expected value n = 8. However, with an interference fit at the middle of the m6 tolerance 

range, the exponent n = 7.7. If the results are recalculated based on the tight end of the tolerance range for 

the m6 interference fit (not shown in Fig. 9), the Hertz stress-life exponent becomes n = 7.4. Similar 

results were obtained for other bearing sizes. 

This effect can impact the results of accelerated testing on bearings with a heavy interference fit. If 

such tests are performed at a high load (thus at high Hertz stress) and then the test results are extrapolated 

to lower stress levels using the usual stress-life exponent n = 8, the predicted value of life may be too 

high, thus giving a nonconservative design. 

 
NASA/TM—2009-215285 18 

It is conjectured that the variation in the Hertz stress-life exponent with interference fit may help to 

explain the load-life exponent p = 10/3 reported and used by Lundberg and Palmgren (Ref. 19) for 

cylindrical roller bearings. Where p = 10/3, n = 6.66. Unfortunately, they did not report the interference 

fit that they used in their test bearings. 

Summary of Results 

The effect of hoop stresses in reducing roller 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 tightest, average, and loosest values within the fit class for 

RBEC−5 tolerances, thus requiring 486 separate analyses. 

The hoop stresses were superimposed on the Hertzian principal stresses created by light, moderate, 

and heavy applied radial loads to determine roller bearing fatigue life. The results are presented as life 

factors for bearings loaded to maximum Hertz stress levels of 1200, 1700, and 2200 MPa (175, 250, and 

320 ksi) in up to seven fit classes (extremely light to extremely heavy) and for bearing accuracy class 

RBEC−5 (ISO class 5). All calculations are for zero initial internal clearance conditions. Any reduction in 

internal bearing clearance due to the interference fit would be compensated by increasing the initial 

(unmounted) clearance. 

The life factor for interference fit in low-speed roller 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 cylindrical roller bearing can significantly reduce 

bearing fatigue life. A heavy fit (tight end of m6 tolerance band) on a 210-size roller bearing reduced the 

fatigue life by 59, 47, and 38 percent from the standard life at maximum Hertz stresses of 1200, 1700, and 

2200 MPa (175, 250, and 320 ksi), respectively. 

(2) 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 is no interface pressure, to as low as 

0.36 (64-percent life reduction) for the tightest p6 fit on a 100-mm bore (220-size) bearing with a 

1200-MPa (175-ksi) maximum Hertz stress. 

(3) The various bearing series within a particular bore size had almost identical interference-fit life 

factors for a particular fit, despite significant differences in the interface pressure at the bore. Four series 

(1910-, 110-, 210- and 310-size) of 50-mm-bore cylindrical roller bearings having an average m6 fit of 

0.021 mm (827 μin.) interference (before adjusting for finish effect) and producing interference-fit 

pressures of 5.00, 7.55, 8.67, and 12.21 MPa (0.73, 1.10, 1.26, 1.77 ksi) had resulting life factors of 0.60, 

0.60, 0.58, and 0.60, respectively. 
NASA/TM—2009-215285 19 

(4) In general, the life factor was smallest (greatest life reduction) for bearings running under light 

load where the unfactored life is highest. For any particular bearing size and interference fit, as the 

maximum Hertz stress on the inner race was increased, the effect of the hoop stresses on life was reduced, 

thus increasing the resulting life factor. 

(5) The tightest fit at the high end of the RBEC−5 tolerance band defined in ANSI/ABMA shaft-fit 

tables produced a life factor of approximately 0.40 for an inner-race maximum Hertz stress of 1200 MPa 

(175 ksi) and a life factor of 0.60 for an inner-race maximum Hertz stress of 2200 MPa (320 ksi). 

(6) Interference fits affected the maximum Hertz stress-life relation. With no interference fit, a Hertz 

stress-life exponent of n = 8.1 was found, which is close to the accepted value of n = 8. With an m6 

middle-of-the-tolerance-range interference fit, the exponent was n = 7.7; with an interference fit at the 

tight end of the range for the m6, the Hertz stress-life exponent became n = 7.4. 

References 

 1. American Bearing Manufacturer 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 Manufacturer Association (ABMA) (2001), “Radial Bearings of Ball, Cylindrical 

Roller and Spherical Roller Types—Metric Design,” ANSI/ABMA-20:1996, 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. 

 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, second edition, 

Society of Tribologists and Lubrication Engineers, Park Ridge, IL. 

 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, June 2007, pp. 336–349. 

 9. Juvinall, R.C. (1967), “Engineering Considerations of Stress Strength and Strain,” McGraw Hill, NY. 

10. Weibull, W. (1939), “A Statistical Theory of the Strength of Materials,” Ingeniorsvetenskapsakad, 

Handl., no. 151. 

11. Weibull, W. (1939), “Phenomenon of Rupture of Solids,” Ingeniorsvetenskapsakad Handl., no. 153. 

12. Weibull, W. (1951), “A Statistical Distribution Function of Wide Applicability,” J. Appl. Mech. 

Trans. ASME, 18, 3, pp. 293−297. 

13. 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. 

14. Palmgren, A. (1945), “Ball and Roller Bearing Engineering,” 1st ed., Translation by G. Palmgren and 

B. Ruley, SKF Industries, Philadelphia, PA. 

15. 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/en-us/solutions/aerospace/ 

Documents/aerospace%20product%20design%20guide.pdf, Accessed 10/15/2008. 

16. Harris, T.A. (1991), “Rolling Bearing Analysis,” Third Edition, John Wiley & Sons, Inc. , ISBN 0-

471-51349-0. 
NASA/TM—2009-215285 20 

17. 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. 

18. Poplawski, J.V., Peters, S.M. and Zaretsky, E.V. (2001), “Effect of Roller Profile on Cylindrical 

Roller Bearing Life Prediction—Part II Comparison of Roller Profiles,” Tribology Transactions, 44, 

3, pp. 417–427. 

19. Lundberg, G.; and Palmgren, A. (1950), “Dynamic Capacity of Roller Bearings,” Acta Polytechnica, 

Mechanical Engineering Series, 2, 4, Stockholm, Sweden. 
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Interference-Fit Life Factors for Roller Bearings 

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13. SUPPLEMENTARY NOTES

14. ABSTRACT

The effect of hoop stresses in reducing cylindrical roller bearing fatigue life was determined for various classes of inner-ring interference fit. 

Calculations were performed for up to 7 fit classes for each of 10 bearing sizes. The hoop stresses were superimposed on the Hertzian 

principal stresses created by the applied radial load to calculate roller bearing fatigue life. A method was developed through a series of 

equations to calculate the life reduction for cylindrical roller bearings. All calculated lives are for zero initial internal clearance. Any 

reduction in bearing clearance due to interference fit would be compensated by increasing the initial (unmounted) clearance. Results are 

presented as tables and charts of life factors for bearings with light, moderate, and heavy loads and interference fits ranging from extremely 

light to extremely heavy for bearing accuracy class RBEC-5 (ISO class 5). Interference fits on the inner ring of a cylindrical roller bearing 

can significantly reduce bearing fatigue life. In general, life factors are smaller (lower life) for bearings running under light load where the 

unfactored life is highest. The various bearing series within a particular bore size had almost identical interference-fit life factors for a 

particular fit. The tightest fit at the high end of the tolerance band produces a life factor of approximately 0.40 for an inner-race maximum 

Hertz stress of 1200 MPa (175 ksi) and a life factor of 0.60 for an inner-race maximum Hertz stress of 2200 MPa (320 ksi). Interference fits 

also impact the maximum Hertz stress-life relation. 

15. SUBJECT TERMS

Rolling element bearings; Roller bearings; Interference fit; Bearing life prediction; Stress 

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