Fred B. Oswald and Erwin V. Zaretsky

Glenn Research Center, Cleveland, Ohio

Joseph V. Poplawski

J.V. Poplawski & Associates

Effect of Internal Clearance on Load

Distribution and Life of Radially 

Loaded Ball and Roller Bearings

NASA/TM—2012-217115

April 2012
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Fred B. Oswald and Erwin V. Zaretsky

Glenn Research Center, Cleveland, Ohio

Joseph V. Poplawski

J.V. Poplawski & Associates

Effect of Internal Clearance on Load

Distribution and Life of Radially 

Loaded Ball and Roller Bearings

NASA/TM—2012-217115

April 2012

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Glenn Research Center

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NASA/TM—2012-217115 1 

Effect of Internal Clearance on Load Distribution and 

Life of Radially Loaded Ball and 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 

Summary 

The effect of internal clearance on radially loaded deepgroove ball and cylindrical roller bearing load distribution and 

fatigue life was determined for four clearance groups defined 

in the bearing standards. The analysis was extended to negative clearance (interference) conditions to produce a curve of 

life factor versus internal clearance. Rolling-element loads can 

be optimized and bearing life maximized for a small negative 

operating clearance. Life declines gradually with positive 

clearance and rapidly with increasing negative clearance. 

Relationships were found between bearing life and internal 

clearance as a function of ball or roller diameter, adjusted for 

load. Results are presented as life factors for radially loaded 

bearings independent of bearing size or applied load. In addition, a modified Stribeck Equation is presented that relates the 

maximum rolling-element load to internal bearing clearance. 

Introduction 

When a ball or roller bearing is subjected to a radial load, 

the rolling elements making up the bearing assembly are not 

equally loaded. Usually, less than half of the rolling elements 

are loaded at any given time. The actual load on each rolling 

element depends on its position in the bearing (Ref. 1). In 

1900, Stribeck (Ref. 2) investigated the variation of loads on 

the elements of a radially loaded ball bearing. Stribeck found 

that the load on the most heavily loaded ball is 4.37 times the 

average ball load for zero internal clearance. Stribeck recommended rounding his constant up to 5.0 to account for nonzero 

clearance and other effects (Ref. 2). 

Palmgren (Ref. 3) states that the theoretical value of 

Stribeck’s constant for roller bearings at zero internal 

clearance is 4.08. Palmgren suggested using the Stribeck 

(Ref. 2) recommended value of 5.0 for the Stribeck constant 

for either ball or roller bearings having typical clearance. 

Radially loaded rolling-element bearings often require a small 

internal clearance to accommodate interference fits and thermal 

strain. In addition, clearance is introduced by wear in the bearing. The American National Standards Institute/American Bearing Manufacturers Association (ANSI/ABMA) Standard 20 

(Ref. 4), the corresponding ISO Standard (ISO 5753 (Ref. 5)), 

and catalogs of bearing manufacturers specify five groups of 

internal diametral clearances for bearings of different sizes. 

These clearance groups are designated 2, N (or “normal”), 3, 4, 

and 5. 

The load distribution among the rolling elements varies for 

different amounts of diametral clearance. Bearing internal 

clearances can significantly increase or decrease the rollingelement fatigue life of a radially loaded ball or roller bearing. 

The Lundberg-Palmgren life equations (Refs. 6 to 7) for 

radially loaded bearings are based on zero internal clearance 

and the Stribeck formula (Ref. 2). However, because of 

shaft/bore interference fits and thermal effects, the mounted 

operating diametral clearance of a bearing with an initial 

clearance of zero can be negative. 

Sadeghi et al. (Ref. 8) gives an overview of rolling-element 

fatigue modeling including a discussion of the physical 

mechanisms of fatigue and the limitations of various models. 

Elmidany et al. (Ref. 9) performed an analytical study of the 

effect of internal interference, reporting that moderate interference (negative clearance) improves bearing life and rigidity 

and reduces noise. However, it may be prudent engineering 

practice to operate a radially loaded bearing with a small positive diametral clearance to prevent a significant decrease in the 

bearing fatigue life or even seizure due to thermal effects. 

Harris (Ref. 10) and Harris and Kotzalas (Ref. 11) show that 

preloading through a moderate interference fit improves load 

distribution, prevents rolling-element skidding, and can 

increase fatigue life. Harris and Kotzalas (Ref. 11), based on 

earlier work by Sjövall (Ref. 12), Palmgren (Ref. 3), and 

Lundberg and Palmgren (Refs. 6 to 7), provide a table for the 

load-distribution integral Jr in terms of the load-zone 

parameter ε (which they call the load-distribution factor). The 

Stribeck constant should equal the reciprocal of Jr. At zero 

clearance, where the load zone is 180, ε = 0.5. Then 1/Jr = 

4.37 for ball bearings with point contact and 1/Jr = 4.08 for 

roller bearings with line contact. 

Jones (Refs. 13 and 14) and Houpert (Ref. 15) present an 

analysis for bearing rolling-element loads for bearings with a 

nonzero contact angle and with loads involving displacements 

in 5 degrees of freedom. 

 

*Distinguished Research Associate. 
NASA/TM—2012-217115 2 

Houpert (Ref. 16) found values of 1.074 and 1.078 for the 

load-deflection exponent for line contact on the inner and 

outer races, respectively, of a roller bearing rather than the 

accepted value of 10/9 = 1.1 or 1.11 (Lundberg and Palmgren, 

Ref. 7) or the earlier value of 1.0 (Lundberg and Palmgren, 

Ref. 6). 

From the preceding discussion, the objectives of the work 

reported herein are (1) to investigate the effect of internal 

bearing clearance on rolling-element loading and the resultant 

bearing fatigue life for deep-groove ball bearings and cylindrical roller bearings operating at low speed, and (2) to 

develop a fatigue life relation for these bearings that incorporates internal bearing clearance, static radial load capacity, 

rolling-element diameter, and applied radial load. 

Symbols are defined in Appendix A. 

Stribeck’s Load-Distribution Factor at 

Zero Internal Clearance 

Load Distribution and Maximum Load in 

Radially Loaded Ball Bearings 

A representative deep-groove ball bearing is shown in 

Figures 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 internal diametral 

clearance Gr is indicated on the figure. A cylindrical roller 

bearing is similar except that the rolling elements are cylinders 

and typically there are no shoulders on the inner race. 

In 1900, Stribeck (Ref. 2) investigated the variation of loads 

on the elements of a ball bearing to calculate the frictional 

forces. When a ball or roller bearing is subjected to a radial 

load, the rolling elements are not equally loaded. The load 

zone for a radially loaded bearing with internal clearance will 

be less than π/2 (180). 

Figure 2 (from Anderson, Ref. 1) shows part of a ball bearing with zero internal clearance that has z = 12 balls oriented 

such that the load path passes through ball 1. For this bearing 

oriented as shown, five balls carry the load: ball 1 plus two 

more pairs. The parameter n designates the number of pairs of 

balls carrying load in addition to ball 1; thus in Figure 2, n = 2. 

In general, n can be found from Equation (1), where INT, the 

integer function, means that any fractional part in Equation (1) 

is truncated, and z is the number of rolling elements. 

1

4

z n INT         (1) 

The angle β between the rolling elements is given by 

2

z

   (2) Figure 1.—Deep-groove ball bearing details. (a) Sketch. 

(b) Cross section showing internal diametral clearance. 

 
NASA/TM—2012-217115 3 

Figure 2.—Geometry to calculate distribution of ball load, Pr, 

in radially loaded ball bearing at zero clearance. Adapted 

from Anderson (Ref. 1). 

For static equilibrium of a radially loaded ball bearing with 

zero internal clearance, the ball loads are given by (Ref. 1): 

12 3 1 2 cos 2 cos 2 ... 2 cos PP P P P n r n        (3) 

where Pr is the applied radial load on the bearing and 

P1, P2, ... Pn+1 are loads on individual rolling elements. The 

normal-direction approach between two surfaces for a ball on 

a raceway (point contact) is proportional to Pr

2/3 (Ref. 3), thus 

the ratio of deflection/load between rolling element 1 and the 

race and rolling element 2 and the same race under load is 

2/3

2 2

1 1

P

P

  

    

 (4) 

or 

3/2

2 2

1 1

P

P

       

 (5) 

In addition, 

1 cos j   j (6) 

for j = 2, 3, ... n + 1, as long as the races remain circular, 

where j is the number of pairs of balls or rollers. Therefore, 

    3/2

1 cos ( 1) PP j j   (7) 

The loads on the rolling elements are found by combining 

Equations (1) to (7). Stribeck found that, for zero clearance, Q

(the quantity in brackets in Eq. (8a)) is very nearly the constant 4.37 divided by z, the number of balls (i.e., z/Q = 4.37). 

5/2 5/2 5/2

1 1 2(cos ) 2(cos2 ) ... 2(cos ) PP n r            (8a) 

Pr P Q1   (8b) 

Because P1 is the maximum ball load Pmax, Equation (8b) 

becomes 

P PQ r  max (9) 

and thus the maximum ball load for zero clearance on a 

radially loaded, deep-groove ball bearing is proportional to 

the ratio of the applied load to the number of balls in the bearing. The constant of proportionality is called the Stribeck 

Number St: 

max r t r P S P P

Q z   (10) 

where 

max

/ t

r

z P S

Q Pz   (11) 

Note that Pr/z in the denominator is the average ball or 

roller load. 

The Stribeck analysis was for zero diametral clearance and 

did not include the effects of internal fit up or inertia. The 

Stribeck Number will vary with internal clearance, fit up, race 

distortion, and load (Ref. 13). To account for these effects and 

possible noncircularity of the balls and the races, Stribeck 

(Ref. 2) suggested rounding his constant z/Q up to 5.0. 

Load Distribution and Maximum Load in Cylindrical Roller Bearings 

Palmgren (Ref. 3) stated that the theoretical value of Stribeck’s zero-clearance constant will be 4.08 and suggested 

using Stribeck’s rounded-up value for the Stribeck Number 

St = 5.0 (Ref. 2) for either ball or roller bearings having typical 

clearance. 

The derivation for the maximum loaded roller in a cylindrical roller bearing is similar to what is shown in the preceding section for ball bearings. The normal approach between 

two surfaces in line contact is proportional to Pr

0.9 (Ref. 3). 

Thus Equation (5) for roller bearings becomes 

1/0.9 1.11

22 2

11 1

P

P

           

 (12) 
NASA/TM—2012-217115 4 

TABLE I.—CALCULATION OF THE STRIBECK NUMBER, St = Z/Q, 

FOR BALL AND ROLLER BEARINGS AT ZERO CLEARANCE 

Number of 

balls or 

rollers, 

z

Ball or 

roller angle, 

β, 

deg 

Value of n

in 

Equation 

(1) 

Radial ball bearings Cylindrical roller bearings 

Load 

distribution 

factor, 

Q

Stribeck 

Number 

Error in St, 

percent 

Load 

distribution 

factor, 

Q

Stribeck 

Number 

Error in St, 

percent 

3 120 0 1.000 3.00 31.4 1.000 3.000 26.5 

4 90 0 1.000 4.00 8.5 1.000 4.000 2.1 

5 72 1 1.106 4.52 –3.4 1.168 4.281 –4.8 

6 60 1 1.354 4.43 –1.4 1.463 4.100 –.4 

7 51.43 1 1.614 4.34 .7 1.738 4.027 1.4 

8 45 1 1.841 4.35 .6 1.963 4.076 .2 

9 40 2 2.052 4.39 –.3 2.189 4.111 –.7 

10 36 2 2.284 4.36 –.2 2.447 4.087 –.1 

20 18 4 4.578 4.37 .0 4.898 4.084 .0 

30 12 7 6.865 4.37 .0 7.345 4.084 .0 

50 7.2 12 11.441 4.37 .0 12.242 4.084 .0 

Equation (7) for j = 2, 3, ... n + 1 becomes 

    1.11

1 cos ( 1) PP j j   (13) 

and Equation (8a) applied to roller bearings becomes 

2.11 2.11 2.11

1 1 2(cos ) 2(cos2 ) ... 2(cos ) PP n r           

(14a) 

Pr   PQ P Q 1 max   (14b) 

Note that the Q for roller bearings in Equation (14b) differs 

from the Q for ball bearings in Equation (8b). 

Table I shows several values of the calculated z/Q for 

various numbers of balls and rollers. The error (difference 

from Stribeck’s value of 4.37) for ball bearings is less than 

1 percent if there are at least seven balls in the bearing. 

Likewise, for roller bearings, z/Q converges to 4.08 with an 

error of less than 1 percent if there are at least eight rollers in 

the bearing. The value 4.08 is 6.5 percent less than the 

Stribeck Number for ball bearings. 

If the load-deflection exponent in Equation (12) is taken as 

1.0 (as in Lundberg and Palmgren’s 1947 paper (Ref. 6), 

rather than as 1.1 (as in their 1952 paper, Ref. 7), then z/Q for 

roller bearings converges to 4.0. However, later work such as 

Palmgren (Ref. 3) states that the deflection is proportional 

to P0.9, where there is line contact on both races. Thus, the 

load-deflection exponent  in Equation (12) should be 10/9 

(typically rounded to 1.11). 

Effect of Internal Clearance on RollingElement Load Distribution 

Computer Analysis 

The loads between the rolling elements and the inner and 

outer races of a bearing are shown schematically in Figure 3. 

The initial radial clearance is shown in Figure 4(a). The compression in the normal direction of element j due to inner-ring 

displacement in a radially loaded ball or roller bearing is 

sin cos 0

2

r nj x j y j

G        (15) 

where  is the displacement of the inner race and j is the 

position angle of element j. A more general treatment, including nonzero contact angle and loading in 5 degrees of freedom 

is presented by Jones (Refs. 13 and 14) and Houpert (Ref. 15). 

The total compression δnj at rolling element j is the sum of 

the deflections at the inner and outer race contacts: 

nj oj ij    (16) 

From the force equilibrium on an element, 

0 P CF P ij j oj    (17) 

where 

P K ij ij ij

   (18a) 

P K oj oj oj

   (18b) 

2

1 cage 2

p

j j

D

CF k m       

 (19) 
NASA/TM—2012-217115 5 

Figure 3.—Loads, P, and deflections between 

rolling elements and races in radially loaded 

ball or roller bearings. Dp, bearing pitch 

diameter; j, rolling-element number (index); 

CF, body force due to centripetal motion. 

Figure 4.—Radially loaded rolling-element bearing. Gr, internal diametral 

clearance; g, radial clearance; max, maximum race deformation. (a) Concentric arrangement. (b) Initial contact. (c) Interference. Adapted from Hamrock 

and Anderson (Ref. 17). 
NASA/TM—2012-217115 6 

and where K is the stiffness, k1 is the body-force conversion 

constant, Dp is the bearing pitch diameter, m is the mass, and 

 is the rotating speed. 

Exponent α = 10/9 ≈ 1.11 for line contact in a roller bearing 

and α = 3/2 for point contact in a ball bearing. For units of 

newtons, kilograms, and millimeters, the body-force conversion constant k1 = 10–6, and for units of pounds and inches, 

k1 = 0.00259. 

The roller-race stiffness in a roller bearing can be found 

from Equation (20a), where le is the equivalent roller length 

and E' is the effective elastic modulus. The ball-race stiffness 

in a radial ball bearing is given in Equations (20b) and (20c), 

where E() is the elliptic integral of the second kind, R is the 

effective radius of curvature, and (α) is the elliptic integral of 

the first kind. Appendixes B and C provide derivations for 

Equations (20a) to (20c). These expressions are valid for both 

metric units (MPa and mm) and English units (lb and in.). 

1.11 0.3555 K K El ij oj e    (20a) 

3

2 () ( ) 9 ()

i i ij i

i

R K kE 

    

E

 (20b) 

3

2 () ( ) 9 ()

o o oj o

o

R K kE 

    

E

 (20c) 

The geometry parameters in Equations (20b) and (20c) are 

defined in the next section. 

Combining Equations (16) and (18) to (20) with the equilibrium equation (Eq. (17)) gives 

  0 K K CF oj nj ij ij ij j

        (21) 

Equation (21) can be solved for δij by iteration for an 

assumed deflection δnj. Then δoj can be found from δij, and Pij

and Poj can be found from Equation (18). The equilibrium 

between the applied radial loads and the estimated bearing 

reactions can be checked for a given Δx and Δy from Equations (22a) and (22b), where Fx and Fy are the applied radial 

load in the x and y directions and  is the position angle from 

the maximum-loaded rolling element. 

1

sin 0

n

x ij j x

j

F P



      (22a) 

1

cos 0

n

y ij j y

j

F P



      (22b) 

These equations are nonlinear and statically indeterminate. 

An iterative method, such as Newton-Rhapson as applied by 

Jones (Ref. 14) provides a stable and rapid solution. 

The implementation of a computer solution, given the bearing geometry, internal clearance, speed, and set of applied 

forces is based on the following steps: 

(1) Assume a starting value for the inner-race displacements Δx and Δy 

(2) Compute the compression δnj in the normal direction for 

each element using Equation (15) 

(3) Solve Equation (21) by iteration for the compression δij

at the inner race contact for each element 

(4) Compute the inner-race loads resulting from the compression at each element using Equation (18a) 

(5) Check the equilibrium of forces using Equations (22a) 

and (22b) 

(6) If the residual forces ε'x and ε'y are larger than a tolerance value, repeat steps 1 to 5 with new estimates for 

the inner-race displacements Δx and Δy 

Once this iteration has completed, L10, the 10-percent 

fatigue life for the bearing, can be calculated using the 

classical Lundberg-Palmgren method (Refs. 6 and 7) from the 

rolling-element loads calculated above. 

Calculation of Maximum Rolling-Element Load 

by Approximate Method 

The geometry of a radially loaded ball or roller bearing with 

internal clearance is shown in Figure 4. Consider that the 

bearing is initially arranged symmetrically, with the clearance 

Gr divided equally around the outer race (Fig. 4(a)). 

Next, the clearance is taken up by an infinitesimal radial 

load that shifts the inner ring by a distance Gr /2, as shown in 

Figure 4(b). Opposite the point of application of the radial 

load, the position angle ψ = 0 and the clearance is zero. At 

any other (nonzero) angle ψ, there will be a radial clearance 

g = (1 – cos ψ)Gr /2. At ψ = 90, the clearance g will be the 

initial value of Gr/2. 

Additional load applied to the bearing causes elastic deformation at the contact between the rolling elements and the 

races as shown in Figure 4(c). At ψ = 0, the maximum 

deformation δmax adds to the clearance takeup Gr /2. The total 

race displacement δT = δmax + Gr /2. If the race maintains a 

circular shape, the elastic compression of the rolling elements 

is given by Equation (23). For Gr > –2δT, the limiting contact 

angle ψl is given by Equation (24) (Ref. 17, where ε = 1⁄2 –

Gr/(4δT). 

cos / 2   T r   G (23) 

 
NASA/TM—2012-217115 7 

  1 1 / 2 cos cos 1 2 r l

T

    G

       

 (24) 

Harris and Kotzalas (Ref. 11) provide an iterative procedure 

to calculate the maximum rolling-element load in ball and 

roller bearings using the load-distribution integral, Jr(ε). They 

provide a chart and table for calculating Jr from estimated 

values of the load-zone parameter ε. The following text and 

equations provide a modified version of this procedure, and 

examples are given in Appendix D. 

Figure 5 shows a modified version of the Harris and 

Kotzalas (Ref. 11) load-distribution integral chart augmented 

with additional data from Houpert (Ref. 15). The left part of 

Figure 5 shows Jr for bearings with internal clearance where 

ε ≤ 0.5 (zone of contact not more than 180). The right part 

shows Jr for bearings with interference, where ε > 0.5 (zone of 

contact greater than 180). 

The data of Figure 5 were fitted to mathematical functions 

that allow estimating the load-distribution integral Jr. These 

functions are plotted in Figure 5. For roller bearings with line 

contact, the load-distribution integral Jr (ε) can be computed 

from Equation (25): 

0.4023 0.3268 0.1 0.5 r J    (25a) 

0.2451 0.5 r J   (25b) 

4 32 0.0852 0.5703 1.3343

1.775 0.0771 0.5 2.0

r J      

  

 (25c) 

For ball bearings with point contact, the load-distribution 

integral Jr(ε) can be computed from Equation (26): 

0.4175 0.3076 0.1 0.5 r J    (26a) 

0.2290 0.5 r J    (26b) 

432 0.0191 0.2020 0.6327 0.6805

0.0213 0.5 2.0

r J      

 

 (26c) 

The Stribeck Number is computed from the reciprocal of the 

load-distribution integral, St = 1/Jr(ε). 

Estimating the load-distribution integral requires an expression for the bearing normal stiffness, which involves the combined deflection between the maximum-loaded rolling element 

and the two (inner and outer) races. The combined stiffness is 

given by 

1/ 1/ (1/ ) (1/ ) n n n KK K ni o



      (27) 

where n = 1.11 for roller bearings with line contact and n = 3/2 

for ball bearings with point contact. 

For a roller bearing, the inner and outer race stiffnesses are 

nearly equal and can be computed from Equation (20a). Then 

Equations (20a) and (27) can be combined to find the normal 

stiffness Kn: 

8/9

8/9

1.11

0.3555 ' 0.1647 '

2

e n e

E l K   E l (28) 

Figure 5.—Load distribution integral versus load distribution factor for ball bearings with point contact 

and roller bearings with line contact. Data from Harris and Kotzalas (Ref. 11) and Houpert (Ref. 15). 

(a) Jr for bearings with internal clearance where ε ≤ 0.5 and the zone of contact ≤180°. (b) Jr for 

bearings with interference, where ε > 0.5 and the zone of contact >180°. The rolling-element load 

zone is shown in a secondary scale. 
NASA/TM—2012-217115 8 

For a ball bearing, the inner and outer race stiffnesses 

needed for Equation (27) can be computed by Equations (20b) 

and (20c). These equations depend on several geometry 

parameters that must be calculated separately for the inner and 

outer races as shown in Equations (29) to (33). 

The effective radii of curvature, Ri and Ro for the inner and 

outer ball-bearing races are given by 

1

2 2 1

( )

p i i

p i

D f R

d D d fd

          

 (29a) 

1

2 2 1

( )

p o o

p o

D f R

dD d fd

       

  

 (29b) 

where d is the rolling-element diameter and f is the race conformity. 

Hamrock and Anderson (Ref. 17) give approximate expressions for the ellipticity parameter and elliptic integrals for a 

radially loaded ball bearing in terms of the radius ratios, αi and 

αo. Coordinate axes x and y must be chosen such that both 

radius ratios are greater than 1. 

2

( )(2 1)

i p

i

p i

f D

Ddf

    (30a) 

2

( )(2 1)

o p

o

p o

f D

Ddf

 

 

 (30b) 

Using the appropriate values of the radius ratio α for the 

inner and outer races yields the approximate ellipticity parameter k(α) and elliptic integrals (α) and E(α) (Ref. 17): 

 2/ k( )   (31) 

( ) 1 ln( ) 2 2

            (32) 

/2 1 () 1  

 



E (33) 

The radial load on either a ball or roller bearing is given by 

Equation (34). Rearranging gives the total deflection δT in 

Equation (35). 

  /2 ( ) n P zK G J r nT r r    (34) 

 

1/

2

n

r r T

n r

G P

zK J

          

 (35) 

The total deflection δT is used to calculate the load-zone 

parameter, ε = 1⁄2 – Gr/(4δT), which is then used for an updated 

estimate of the load-distribution integral Jr(ε) from Equation (25) or (26). 

The preceding equations can be used to find the Stribeck 

Number from the load-distribution integral. The equations are 

valid for either metric units (N and mm) or English traditional 

units (lb and in.). The method is valid for zero, positive, and 

negative internal clearances in the bearing within the limits 

shown for the load-zone parameter ε. 

Appendix D provides example calculations for a deepgroove ball bearing and a cylindrical roller bearing using the 

equations developed above to find the Stribeck Number from 

the load-distribution integral. The method is valid for zero, 

positive, and negative internal clearances in the bearing within 

the limits shown for the load-zone parameter . Either metric 

units (N and mm) or English traditional units (lb and in.) may 

be used. 

Appendix E shows a more direct alternative procedure 

based on Hamrock and Anderson (Ref. 17) that is valid for 

zero or positive clearance but not for negative clearance. 

Before these calculations are performed, the bearing internal 

clearance must be adjusted to account for thermal expansion, 

wear, and the effects of any interference fit between the bearing 

inner ring and shaft and between the outer ring and housing. 

Parametric Study—Effect of Internal 

Clearance on Bearing Load Distribution 

and Life 

A commercial bearing analysis code (Poplawski et al., 

Ref. 18) was used to determine the effect of internal clearance 

and interference on the distribution of load among the rolling 

elements and on the life of radially loaded deep-groove ball 

bearings and cylindrical roller bearings made from AISI 

52100 steel. The code uses the traditional Lundberg-Palmgren 

method (Lundberg and Palmgren, Ref. 6). Each bearing was 

initially analyzed for the zero internal-clearance condition at 

three values of inner-race maximum Hertz stress: 1200, 1720, 

and 2240 MPa (175, 250, and 325 ksi). These are designated 

light, normal, and heavy loads, respectively. 

The radial load on each bearing was chosen to produce the 

desired stress at zero clearance. The same load was maintained 

as the internal clearance was varied over a range of positive 

and negative diametral clearances to examine the effect of 

clearance on bearing life and on the maximum ball or roller 

load. These clearance values are for net operating clearance 
NASA/TM—2012-217115 9 

including any effects from interference fit, thermal effects, and 

wear. The analysis was restricted to low speed, where inertial 

effects are not significant. 

The properties (dimensions, static, and dynamic load 

capacity) of the deep-groove ball bearings analyzed in this 

article are given in Table II, and the properties of the cylindrical roller bearings are given in Table III. Ball bearings were 

modeled with both inner- and outer-raceway groove radius 

(conformity) of 0.52d, where d is the ball diameter. Roller 

bearing rollers were modeled with an aerospace crown, with a 

flat length of 61.5 percent of the roller length and with the 

crown radius 100 times the roller length. The clearance values 

used in this analysis are shown in Tables IV and V for ball and 

roller bearings, respectively. 

TABLE II.—DEEP-GROOVE BALL BEARING PROPERTIES 

[Adapted from The Timken Company, Ref. 19.] 

Bearing 

size 

Bore, 

mm 

Outside 

diameter, 

mm 

Number 

of balls, 

z

Ball diameter, 

d

Static radial load 

capacity, 

Cor

Dynamic radial load 

capacity, 

Cr

mm in. N lbf N lbf 

206 30 62 9 9.525 3/8 11 254 2 530 19 457 4 374 

1910 

50 

72 17 6.35 1/4 11 152 2 507 13 398 3 012 

110 80 12 9.525 3/8 16 610 3 734 23 291 5 236 

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

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

220 100 180 10 25.4 1 92 639 20 826 122 161 27 463 

1940 200 280 16 25.4 1 167 360 37 624 156 573 35 199 

TABLE III.—CYLINDRICAL ROLLER BEARING PROPERTIES 

[Adapted from The Timken Company, Ref. 19.] 

Bearing 

size 

Bore, 

mm 

Outside 

diameter, 

mm 

Number 

of 

rollers, 

z

Roller 

diameter, 

d

Roller 

length, 

L

Static radial 

load capacity, 

Cor

Dynamic radial 

load capacity, 

Cr

mm in. mm in. N lbf N lbf 

206 30 62 12 10 0.3937 10 0.3937 37 632 8 460 41 035 9 225 

1910 

50 

72 20 7 .2756 7 .2756 32 881 7 392 29 038 6 528 

110 80 18 9 .3543 9 .3543 49 446 11 116 44 860 10 085 

210 90 14 13 .5118 13 .5118 77 226 17 361 75 464 16 965 

310 110 10 19 .7480 19 .7480 111 010 24 956 117 273 26 364 

220 100 180 14 25 .9843 25 .9843 295 980 66 539 259 069 58 241 

1940 200 280 26 24 .9449 24 .9449 553 270 124 380 361 449 81 257 

TABLE IV.—RADIAL BALL BEARING DIAMETRAL CLEARANCES 

[Adapted from ABMA, Ref. 4, and ISO, Ref. 5.] 

Bearing 

bore 

size, 

mm 

Diametral clearance 

Group 2 Group N Group 3 Group 4 

Minimum, 

mm (in.) 

Maximum, 

mm (in.) 

Minimum, 

mm (in.) 

Maximum, 

mm (in.) 

Minimum, 

mm (in.) 

Maximum, 

mm (in.) 

Minimum, 

mm (in.) 

Maximum, 

mm (in.) 

30 0.001 (0.00005) 0.011 (0.00045) 0.005 (0.0002) 0.020 (0.0008) 0.013 (0.0005) 0.028 (0.0011) 0.023 (0.0009) 0.041 (0.0016) 

50 0.001 (0.00005) 0.011 (0.00045) 0.006 (0.00025) 0.023 (0.0009) 0.018 (0.0007) 0.036 (0.0014) 0.030 (0.0012) 0.051 (0.0020) 

75 0.001 (0.00005) 0.015 (0.0006) 0.010 (0.0004) 0.030 (0.0012) 0.025 (0.0010) 0.510 (0.0020) 0.046 (0.0018) 0.071 (0.0028) 

100 0.001 (0.00005) 0.018 (0.0007) 0.012 (0.00045) 0.036 (0.0014) 0.030 (0.0012) 0.058 (0.0023) 0.053 (0.0021) 0.084 (0.0033) 
NASA/TM—2012-217115 10 

TABLE V.—CYLINDRICAL ROLLER BEARING DIAMETRAL CLEARANCES 

[Adapted from ABMA, Ref. 4, and ISO, Ref. 5.] 

Bearing 

bore 

size, 

mm 

Diametral clearance 

Group 2 Group N Group 3 Group 4 

Minimum, 

mm (in.) 

Maximum, 

mm (in.) 

Minimum, 

mm (in.) 

Maximum, 

mm (in.) 

Minimum, 

mm (in.) 

Maximum, 

mm (in.) 

Minimum, 

mm (in.) 

Maximum, 

mm (in.) 

30 0 (0) 0.025 (0.0010) 0.020 (0.0008) 0.045 (0.0018) 0.035 (0.0014) 0.060 (0.0024) 0.050 (0.0020) 0.075 (0.0030) 

50 0.005 (0.0002) 0.035 (0.0014) 0.030 (0.0012) 0.060 (0.0024) 0.050 (0.0020) 0.080 (0.0031) 0.070 (0.0028) 0.100 (0.0039) 

75 0.010 (0.0004) 0.045 (0.0018) 0.040 (0.0016) 0.075 (0.0030) 0.065 (0.0026) 0.100 (0.0039) 0.090 (0.0035) 0.125 (0.0049) 

100 0.015 (0.0006) 0.050 (0.0020) 0.050 (0.0020) 0.085 (0.0033) 0.075 (0.0030) 0.110 (0.0043) 0.105 (0.0041) 0.140 (0.0055) 

Figure 6.—Comparison of the effect of internal diametral clearance on maximum ball load for the bearing analysis code 

results and the two approximate methods presented in this article. Results are for 210-size, deep-groove ball bearings radially loaded to produce 1720-MPa (250-ksi) inner-race maximum Hertz stress at zero clearance. Insets show 

bearing ball load distribution at five values of clearance. 

Variation of Rolling-Element Load Distribution 

With Internal Clearance 

Figure 6 shows the variation of maximum element (ball) 

load for zero, positive, and negative internal diametral clearance in a 210-size radially loaded deep-groove ball bearing at 

normal load (1720 MPa maximum Hertz stress). Results are 

shown for three methods: (1) as computed by the bearing analysis code, (2) the load integral method, and (3) an alternative 

procedure described in Appendix E. The load integral method 

results were within 3.1 percent of the analysis code. The alternative method, which is restricted to nonnegative clearance, 

gave results within 1.7 percent of the analysis code. Similar 

results were obtained for cylindrical roller bearings. 

Figure 6 includes small bearing diagrams to show the 

rolling-element (ball) load distribution at five clearance 

values. The ball loads shown were calculated by the analysis code 

(Ref. 18). The load distribution is discussed in the next section. 
NASA/TM—2012-217115 11 

Life Factor for Internal Clearance 

The ratio of the life L at any clearance condition, to the life 

at zero-clearance L0 is designated the life factor LF for that 

value of internal clearance of the bearing. By definition, the 

life factor equals 1.00 for zero clearance on any bearing. 

0

L LF

L  (36) 

The four groups of diametral clearance investigated for 

210-size ball bearings are shown graphically in Figure 7. The 

clearance ranges from 0.001 mm (group 2, minimum), where 

LF = 0.99, to 0.051 mm (group 4, maximum), where 

LF = 0.63. There is also a group 5 clearance that we did not 

consider for this article. 

Figure 7 shows the variation of life factor with internal 

diametral clearance for 210-size radially loaded deep-groove 

ball bearings at normal load, 1720 MPa (250 ksi) maximum 

Hertz stress. The curve shows that increasing clearance results 

in reduced life. A small negative clearance (interference) of 

–0.008 mm provides the greatest life factor found, 1.12. With 

greater interference, the life factor falls off rapidly, reaching 

0.36 at –0.025 clearance because the maximum load has 

increased considerably. 

The ball load diagrams (insets) of Figure 6 are also shown 

on Figure 7. Diagram 3 shows the ball loads at zero clearance, 

where five balls share the load and the life factor is defined to 

be 1.0. The greatest maximum ball load (diagram 1) occurs at 

the largest clearance (0.051 mm), where the load Pr is shared 

by only three balls. The greatest life factor found, 1.12, is 

shown in diagram 4, where seven balls carried the load. The 

smallest Pmax (minimum point in the curves of Fig. 6) is at a 

slightly greater interference (–0.013 mm) than the maximum 

life factor condition shown in diagram 4. At the greatest 

negative clearance (–0.025 mm) shown (diagram 5), all 10 

balls in the bearing share the load; however, the total load has 

increased significantly and the life factor is only 0.36. 

Note that the maximum life shown in Figure 7 occurs at a 

slightly smaller value of interference than the minimum load 

shown in Figure 6 (0.008 vs. 0.013 mm). This is due to the 

effect of the increased loads on balls other than ball 1 (the ball 

carrying Pmax). 

If the life factors for different loads or for different size 

bearings are plotted on the same axes, the curves will not 

match on the horizontal (clearance) axis. Figure 8 shows life 

factors for 50-mm deep-groove ball bearings in four envelope 

sizes at the normal load condition, where, at zero clearance, 

the maximum Hertz stress is 1720 MPa (250 ksi). The four 

bearing envelope sizes—designated “extremely light,” “extra 

light,” “light,” and “medium”—are depicted schematically in 

the figure. 

Internal bearing clearance values as specified in the 

ANSI/ABMA Standard 20 (Ref. 4) and ISO Standard, 5753 

(Ref. 5) are given relative to the bearing bore diameter without 

consideration of any other bearing dimensions. However, as 

Figure 7.—Effect of internal diametral clearance on life factor for 210-size, deep-groove ball bearings radially 

loaded to produce 1720-MPa (250-ksi) inner-race maximum Hertz stress at the zero-clearance condition. 

Insets show bearing ball load distribution at five values of clearance. 
NASA/TM—2012-217115 12 

Figure 8.—Life factor versus internal diametral clearance in four envelope sizes of 50-mm deep-groove 

ball bearings radially loaded to produce 1720-MPa (250-ksi) inner-race maximum Hertz stress at zero 

clearance (normal load condition). 

can be seen in Figure 8, a given clearance has much more 

impact on the life of a small envelope bearing, such as 

“extremely light,” than on the larger sizes, such as “medium.” 

With the clearance values in Figure 8 adjusted relative to 

the ball size, the curves closely match. Figure 9 shows the 

result of this adjustment, where the clearance is made dimensionless by dividing the actual clearance Gr by the ball diameter d to produce the nondimensional clearance G'r. If an 

adjustment is made relative to the bearing bore size or pitch 

diameter, rather than to the rolling-element diameter, the 

curves will not match. 

r r

G G

d   (37) 

Where initial (or unmounted) bearing clearance is needed to 

accommodate inner-ring expansion due to interference fits or 

thermal strain, it is correct to base such clearance on the bore 

diameter. However, ANSI/ABMA Standard 20 (Ref. 4) and 

ISO Standard, 5753 (Ref. 5), the bearing standards that relate 

recommended clearance on the bore diameter, must be used 

with care to avoid excessive operating clearance. 

In Figure 9, the bearings all carry the same normal load (as 

defined earlier). If the applied radial load varies as shown in 

Figure 10, the life factor LF curves will no longer match. 

However, applying a further adjustment, shown in Figure 11, 

will bring the curves back into alignment. 

The adjustment involves multiplying the nondimensional 

clearance G'r by the reciprocal of the maximum Hertz stress 

ratio squared: (S'max/Smax)

2

, where Smax is the maximum innerring Hertz stress at zero clearance and S'max is the reference 

inner-race maximum Hertz stress: 1720 MPa (250 ksi). This 

load-adjusted, nondimensional clearance is called the dimensionless internal diametral clearance number, Ĝr. 

2

max

max

ˆ r r

G S G

d S

       

 (38) 

Figure 12 shows the result of plotting LF versus Ĝr for 

seven different sizes of deep-groove ball bearings at normal 

load. There are also curves for light and heavy loads for the 

210-size bearings. The maximum LF = 1.12 for deep-groove 

ball bearings occurs with a small amount of negative clearance 

at Ĝr = –0.00099. 

Equation (38) requires solving for the maximum Hertz 

stress at zero clearance Smax. The following text and equations 

present a simple procedure for calculating Ĝr without calculating Smax. This procedure requires only the reference value of 

maximum Hertz stress S'max, the bearing clearance Gr, the 

rolling-element diameter d, the static radial load capacity Cor, 

and the radial load on the bearing Pr. 

The maximum Hertz stress at zero clearance Smax can be 

found through a simple relation involving the applied load and 

the static radial load capacity as given in a bearing catalog. 
NASA/TM—2012-217115 13 

Figure 9.—Life factor versus dimensionless internal diametral clearance in four envelope sizes of 

50-mm deep-groove ball bearings radially loaded to produce 1720-MPa (250-ksi) inner-race 

maximum Hertz stress at zero clearance (normal load condition). 

Figure 10.—Life factor versus dimensionless internal diametral clearance in 210-size deep-groove ball 

bearings at three loads and three corresponding levels of maximum Hertz stress, Smax. 

 
NASA/TM—2012-217115 14 

Figure 11.—Life factor versus dimensionless internal diametral clearance number in 210-size deep-groove 

ball bearings at three loads. 

Figure 12.—Life factor versus dimensionless internal diametral clearance number for seven sizes of deepgroove ball bearings at three loads and three corresponding levels of maximum Hertz stress, Smax. 
NASA/TM—2012-217115 15 

This simple relation is based on the static radial load capacity 

in terms of an acceptable level of deformation as defined by 

Palmgren (Ref. 3, p. 86): “... 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 

radially loaded ball or cylindrical roller bearings, this corresponds to an Smax of either 4000 or 4200 MPa (580 or 609 ksi). 

Oswald et al. (Ref. 20) used the Palmgren criteria (Ref. 3) to 

relate Smax to the applied radial load Pr for radial load in either 

ball or roller bearings: 

(1/ )

max 2

i

r

or

P S k

C

      

 (39) 

where the conversion constant k2 = 4000 MPa (580 ksi) is the 

stress corresponding to a radial load equal to the static radial 

load capacity, Cor is the static radial load capacity of the bearing, which can be found in a bearing catalog, and i = 3 for ball 

bearings and i = 2 for roller bearings. Equation (39) implicitly 

assumes zero internal bearing clearance. 

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

included in the static load ratings in the 1978 revision of 

ANSI/ABMA standard 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 used (Ref. 19). 

By substituting the value 1720 MPa for S'max and using 

Equation (39) for Smax and simplifying, the dimensionless 

internal diametral clearance number Ĝr can be found without 

solving for the actual stress in the bearing: 

2/3

ˆ 0.185 r or r

r

G C G

d P

      

 for ball bearings (40a) 

ˆ 0.185 r or r

r

G C G

d P

      

 for roller bearings (40b) 

where Gr is the internal diametral bearing clearance and d is 

the rolling-element diameter. 

Figure 13 shows the same life factor data for deep-groove 

ball bearings as is shown in Figure 12 plotted on semi-log 

scales. Positive values of Ĝr are shown in the right half of the 

figure, and negative values of Ĝr are shown on the left half. 

Since negative values cannot be used with a log scale, absolute 

values of Ĝr were plotted and the axis was reversed and 

labeled with minus signs. Zero clearance, where by definition, 

LF = 1.0, is not shown. 

The data of Figure 13 were fitted piecewise to mathematical 

functions that allow LF to be estimated for deep-groove ball 

bearings from the following equations. Curves of these functions are shown with the data in Figure 13. 

ˆ ˆ 600 1.7 0.003 0.00099 LF G G      r r (41a) 

  ˆ ˆ exp 110 0.00099 0.01 LF G G     r r (41b) 

The procedure for ball bearings described earlier in this section can also be applied to determine the effect of internal 

clearance on the life of cylindrical roller bearings. Figure 14 

shows life factor LF versus the dimensionless internal diametral clearance number Ĝr as calculated from Equation (40b) for 

seven different sizes of cylindrical roller bearings at normal 

Figure 13.—Life factor for deep-groove ball bearings versus dimensionless internal diametral clearance 

number on a log scale. (a) Negative Ĝr (absolute value plotted with axis reversed). (b) Positive Ĝr. The 

curves added to the figure represent curve fits applied to the data. 
NASA/TM—2012-217115 16 

Figure 14.—Life factor versus dimensionless internal diametral clearance number for seven sizes of cylindrical roller bearings at three load levels and three corresponding levels of maximum Hertz stress, Smax. 

Figure 15.—Life factor for cylindrical roller bearings versus dimensionless internal diametral clearance 

number on a log scale. (a) Negative Ĝr (absolute value plotted with axis reversed). (b) Positive Ĝr. 

The curves added to the figure represent curve fits applied to the data. 

load. Normal load is defined as the load that produces a maximum Hertz stress Smax = 1720 MPa (250 ksi) for a bearing 

with zero internal clearance. In addition, there are curves for 

light and heavy loads for the 210-size bearings. The maximum 

LF = 1.15 for cylindrical roller bearings occurs with a small 

amount of negative clearance at Ĝr = –0.0013. The clearance 

values for cylindrical roller bearings shown in Table IV are 

approximately twice the values specified for radial ball 

bearings in Table V at a given clearance group. Because of the 

larger clearances, roller bearings have smaller life factors than 

deep-groove ball bearings do, as can be seen by comparing 

Figures 12 and 14. 

Figure 15 shows the same data as in Figure 14 plotted on 

semi-log scales. Positive values of Ĝr are shown in the right 
NASA/TM—2012-217115 17 

half of the figure, and negative values of Ĝr are shown on the 

left half using the same procedure as for Figure 13. 

The data of Figure 15 were fitted piecewise to mathematical 

functions that allow the life factor LF for cylindrical roller 

bearings to be estimated from the following equations. Curves 

of these functions are also shown in Figure 15. 

ˆ ˆ 500 1.8 0.003 0.0013 LF G G      r r (42a) 

  ˆ ˆ exp 100 0.0013 0.01 LF G G     r r (42b) 

Stribeck Number and Maximum Rolling-Element 

Load as Affected by Internal Clearance 

In addition to the effect of clearance on life factor, we 

investigated the relation between the maximum rollingelement load and internal clearance. The Stribeck Number St, 

which relates the maximum element load on the bearing to the 

average rolling-element load, is defined in Equation (11). 

Figure 16 shows St versus Ĝr for seven different sizes of 

deep-groove ball bearings at normal load. The figure also 

includes St for the 210-size bearings at light and heavy loads. 

Note that the value of St found for zero internal clearance 

closely matches Stribeck’s value of 4.37. The recommended 

value of 5.0 would be a reasonable estimate for bearings with 

small values of clearance except at light load. The minimum 

Stribeck Number for deep-groove ball bearings, St = 3.8, 

occurs at Ĝr = –0.0012. Note that the minimum St occurs with 

a small negative clearance (interference). 

Figure 17 shows the same ball bearing Stribeck Number 

data as in Figure 16 plotted on semi-log scales. Positive values 

of Ĝr are shown in the right half of the figure. Negative values 

of Ĝr are shown on the left half. Since negative values cannot 

be used in a log scale, absolute values were plotted and the 

axis was reversed and labeled with minus signs. Zero clearance, where St approaches the asymptotic value of 4.37, is not 

shown. 

The data of Figure 17 were fitted piecewise to mathematical 

functions that allow St for deep-groove ball bearings to be 

estimated from the following equations: 

ˆ ˆ 2.668exp( 300 ) 0.003 0.0012 t rr S GG      (43a) 

ˆ ˆ 4.37 exp(110 ) 0.0012 0 tr r SG G    (43b) 

ˆ ˆ 4.37 exp(75 ) 0 0.01 t rr S GG   (43c) 

Figure 16.—Stribeck Number for deep-groove ball bearings versus dimensionless internal diametral clearance number. Curves show seven bearing sizes and include three load levels and three corresponding 

levels of maximum Hertz stress, Smax.
NASA/TM—2012-217115 18 

Figure 17.—Stribeck Number for deep-groove ball bearings versus dimensionless internal diametral clearance number on a log scale. (a) Negative Ĝr (absolute value plotted with axis reversed). (b) Positive Ĝr. 

The curves added to the figure represent curve fits applied to the data. 

Figure 18.—Stribeck Number versus dimensionless internal diametral clearance number for seven sizes of 

cylindrical roller bearings at three load levels and three corresponding levels of maximum Hertz stress, Smax. 

In Figure 18, the Stribeck Number for cylindrical roller 

bearings was plotted in the same manner as it was for ball 

bearings earlier. At zero internal clearance, St = 4.21, which is 

3.7 percent less than for ball bearings and 3.2 percent greater 

than Palmgren’s value of 4.08 for roller bearings. The 

minimum Stribeck Number for cylindrical roller bearings, 

St = 3.76, occurs at Ĝr = –0.0016 (negative clearance). 

The Stribeck Number analysis just described was applied to 

cylindrical roller bearings with crowned rollers. Because we 

believed that Palmgren’s analysis was for flat rollers (no 

 
NASA/TM—2012-217115 19 

Figure 19.—Stribeck Number for cylindrical roller bearings versus dimensionless internal diametral clearance number on a log scale. (a) Negative Ĝr (absolute value plotted with axis reversed). (b) Positive Ĝr. 

The curves added to the figure represent curve fits applied to the data. 

crown), we tried a single case for bearings with flat rollers. 

The result for zero clearance was St = 4.09, which is very close 

to the 4.08 value obtained analytically earlier and by Palmgren 

(Ref. 3). This suggests that the Palmgren analysis was for 

bearings with flat rollers. 

In Figure 19, the Stribeck Number data of Figure 18 were 

plotted on semi-log scales. Positive values of Ĝr are shown in 

the right half of the figure, and negative values of Ĝr are 

shown on the left half. The data of Figure 19 were fitted 

piecewise to mathematical functions that allow St for cylindrical roller bearings to be estimated from Equation (44): 

ˆ ˆ 2.8exp( 185 ) 0.003 0.0016 t rr S GG      (44a) 

ˆ ˆ 4.21exp(70 ) 0.0016 0 tr r SG G    (44b) 

ˆ ˆ 4.21exp(44 ) 0 0.016 t rr S GG   (44c) 

Examples 3 and 4 of Appendix F illustrate the use of the 

curve-fit functions for calculating the Stribeck Number and 

maximum rolling-element load as presented in this section. 

The Stribeck Number computed in the examples in the 

appendixes is significantly greater than the 5.0 value that 

Stribeck and Palmgren recommended, even though the clearance value was specified as the mid-range of “normal” clearance defined in the standards. Harris and Kotzalas (Ref. 11) 

warn that estimating the maximum rolling-element load with 

Equation (10) using the approximate constant 5.0 is inadequate for light loads or where the bearing rings are not rigid. 

In addition, we calculated the Stribeck Number for 

Examples 3 and 4 of Appendix F except at a very light load, 

such that Smax = 600 MPa (87.5 ksi). For this case, St is 7.5 and 

7.8 for a deep-groove ball bearing and a cylindrical roller 

bearing, respectively, with mid-range “normal” clearance. 

This result of the Stribeck Number becoming large for light 

loads can be anticipated if one considers the physical situation. 

As the applied radial load becomes very small, the elastic 

deflection at the contacts vanishes and all of the load is carried 

by a single rolling element. For this case, Pmax ≈ Pr and the 

Stribeck Number as defined by Equation (11) approaches z, 

the number of rolling elements. 

Summary of Results 

The effect of internal diametral clearance on the fatigue life 

and rolling-element load distribution was investigated for 

deep-groove ball bearings and cylindrical roller bearings. The 

load at the most heavily loaded rolling element was described 

in terms of the Stribeck Number St defined as the maximum 

element load divided by the average element load. The 

Stribeck Number can be used to find the load distribution 

among the rolling elements in a bearing. 

A procedure was developed for calculating the rollingelement loads for positive or negative clearance from the 

load-distribution integral as introduced by Sjövall (Ref. 12). 

Empirical equations also were developed for calculating the 

maximum rolling-element load and life factors in low-speed 

cylindrical roller bearings and deep-groove ball bearings. The 

following results were obtained:

1. The effect of a given value of operating internal clearance on the load distribution and fatigue life of rollingelement bearings varies with the rolling-element size 

and not the bore size or pitch diameter. 

 
NASA/TM—2012-217115 20 

2. The “dimensionless internal diametral clearance number” Ĝr can be calculated from the bearing internal 

clearance, rolling-element diameter, static load capacity, 

and applied load. The parameter Ĝr can be used to estimate the Stribeck Number St and the bearing life factor 

LF. 

3. The concept of a Stribeck Number, St = Pmax/(Pr/z), was 

derived to relate bearing internal clearance to the maximum rolling-element load for radially loaded ball and 

roller bearings, where Pmax is the maximum element

load, Pr is the applied radial load, and z is the number of 

rolling-elements. 

4. The analysis confirmed that St = 4.37 for ball bearings at 

zero internal clearance and St = 4.08 for roller bearings 

at zero clearance. The roller bearing number agrees with 

Palmgren’s value of 4.08. For roller bearings with a 

specified aerospace crown on the rollers, St = 4.21. For 

bearings with clearance, St approaches z as the applied 

radial load approaches zero. 
NASA/TM—2012-217115 21 

Appendix A.—Symbols 

a semimajor axis of contact, mm (in.) 

b semiminor axis of contact, mm (in.) 

CF body force due to centripetal motion, N (kips) 

Cor static radial load capacity, N (kips) 

Cr dynamic radial load capacity 

Dp bearing pitch diameter, mm (in.) 

d rolling-element (ball or roller) diameter, mm (in.) 

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

E' effective elastic modulus, where 

E' = 2[(1 – νa

2

)/Ea + [(1 – νb

2

)/Eb]

–1, MPa (ksi) 

E (α) elliptic integral of the second kind 

Fx applied radial load in the x direction, N (lb) 

Fy applied radial load in the y direction, N (lb) 

(α) elliptic integral of the first kind 

f race conformity, ratio of race groove radius to bearing ball diameter 

Gr internal diametral clearance, mm (in.) 

G'r nondimensional internal diametral clearance, Gr/d

Ĝr dimensionless internal diametral clearance number 

g radial clearance, mm (in.) 

INT integer function 

i stress-load exponent in Equation (39) 

Jr load-distribution integral 

j rolling-element number (index) 

K stiffness, N/m (lb/in.) 

k(α) ellipticity parameter, a/b 

k1 conversion constant for Equation (19) 

k2 conversion constant for Equation (39) 

L life of rolling-element bearing, hours or millions of 

inner-ring revolutions 

L0 life at zero clearance of rolling-element bearing, 

hours or millions of inner-ring revolutions 

L10 10-percent fatigue life for the bearing 

LF life factor for internal clearance 

le roller effective length, m (in.) 

m mass, g (lbm) 

n number of pairs of loaded elements in Equations (1), 

(8a), and (14a) 

P load, N (kips) 

Pj load on rolling-element j, N (kips) 

Pmax maximum rolling-element radial load, N (kips) 

Pr applied radial load on bearing, N (kips) 

Q factor in Equation (9) relating applied load to maximum element load 

R effective radius of curvature, where R = (Σ(1/r))–1, m (in.) 

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

S'max reference value of maximum Hertz stress, 1720 MPa 

(250 ksi) 

St Stribeck Number defined in Equation (11) 

z number of rolling elements 

α load/deformation exponent—For roller bearings, 

α = 1.11; for ball bearings, α = 1.5. 

β angle between the rolling elements, radians (deg) 

Δ displacement of inner race in Equation (15) 

δ deflection of rolling elements and races under load, 

mm (in.) 

δmax maximum elastic race deformation (under maximumloaded rolling element), mm (in.) 

δT total race deflection including elastic deformation 

and clearance takeup, mm (in.) 

ε load-zone parameter, where ε = 1⁄2 – Gr/(4δT) 

ε' residual force in Equations (22a) and (22b) (error 

term for iterative calculation), N (lbf) 

v elastic constant 

ψ position angle from maximum-loaded rolling element, radians (deg) 
NASA/TM—2012-217115 22 

ψl limiting value of ψ (limit of load zone), radians (deg) 

ω rotating speed, rad/s 

Subscripts 

a,b refers to body a or b 

i inner race of bearing 

j rolling-element number (index) 

n normal direction 

o outer race of bearing 

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

y transverse direction 
NASA/TM—2012-217115 23 

Appendix B.—Derivation of Equation (20a) for Roller-Race Stiffness 

in a Roller Bearing 

Anderson (Ref. 1), based on Palmgren (Ref. 3), shows that 

the approach (deflection) between the axis of a finite-length 

cylinder and an infinite plane can be expressed as 

0.9 0.9 0.8 0.39 ( ) /      ab e P l (B1) 

where P is the load, le is the effective cylinder (roller) length, 

and θ is defined in terms of the elastic constants E and ν as 

θ = 4(1 – ν

2

)/E, which need not be the same for bodies a and b. 

These elastic constants can be expressed in terms of the effective modulus E' as 

2 2 (1 ) (1 ) 8 ( )4 a b

a b E EE a b

             

 (B2) 

Substituting the effective modulus from Equation (B2) into 

Equation (B1), gives the deflections, δi and δo, at the inner and 

outer races: 

 

0.9

0.9 0.8

2.537 i o

e

P

E l

     (B3) 

Solving for the force P gives 

8/9 10/9

8/9 10/9

10/9

' 0.3555

2.537

e e

E l P El    (B4) 

Defining the stiffness K = P/δα

 (recall that for line contact, 

α = 10/9 ≈ 1.11), and with the outer and inner race stiffnesses 

nearly equal, gives 

1.11 0.3555 K K El ij oj e    (B5) 

If the roller and race are the same material, the effective 

modulus simplifies to E' = E/(1 – ν

2

). 

NASA/TM—2012-217115 25 

Appendix C.—Derivation of Equations (20b) and (20c) for Ball-Race Stiffness 

in a Radially Loaded Ball Bearing 

Hamrock and Anderson (Ref. 17) give the maximum ballrace deflection as 

1/3 2

max max

9 ( ) 2 () ()

P

R kE

                       E (C1) 

In Equation (C1), the deflection should be calculated 

separately for the inner and outer races from terms defined in 

Equations (29) to (33). The stiffnesses in Equations (18a) and 

(18b) are defined as K = P/δα

 (where, for point contact, 

α = 3/2). Thus, 

3

3/2 max max

9 ()

2 () ()

P

RkE

   

   E 

 (C2) 

3/2 3

2

9

P R K kE  

      

E( ) ( )

 ( )

 (C3) 

This article presents a simplified approximate method from 

Hamrock and Anderson (Ref. 17) to calculate the ellipticity 

parameters in Equations (C1) to (C3). Computer solutions 

(Ref. 18) typically employ an iterative procedure using Hertz 

contact theory as presented in Jones (Refs. 13 and 14). 

NASA/TM—2012-217115 27 

Appendix D.—Example Calculations to Determine the Stribeck Number and 

Maximum Rolling-Element Load From the Approximate Method 

This appendix provides two examples that illustrate the use 

of the load-distribution integral approximate method to determine maximum rolling-element loads for deep-groove ball 

bearings and cylindrical roller bearings that have either positive or negative clearance. 

Example 1—Stribeck Number and Maximum 

Ball Load in a Deep-Groove Ball Bearing 

Consider a 210-size deep-groove ball bearing, with midband group N clearance and carrying a light radial load 

Pr = 627 N (0.141 kips). This load was chosen such that at 

zero clearance it produces an inner-race maximum Hertz stress 

of 1200 MPa (175 ksi). The properties (Table II) include the 

following: z = 10 balls of diameter d = 12.7 mm (0.5 in.), the 

ratio of race groove radius to bearing ball diameter f = 0.52, 

and the pitch diameter Dp = 70.000 mm (2.7559 in.). 

From ANSI/ABMA–20 (Ref. 4) (Table IV), group N internal diametral clearance for this ball bearing ranges from 0.006 

to 0.023 mm (0.00025 to 0.00090 in.). Thus, Gr = 0.0145 mm 

(0.00057 in.). 

For steel rollers and races, E = 205 900 MPa (29 900 ksi), 

ν = 0.3, and the effective modulus E' = 226 240 MPa 

(32 800 ksi). 

For a ball bearing, several geometry parameters must be 

calculated: 

1. Compute the effective radii of curvature from 

Equations (29a) and (29b): 

1 2 70 2 0.52 1

12.7(70 12.7) 0.52 12.7

Ri

             = 5.039 mm (0.1984 in.) 

1 2 70 2 0.52 1

12.7(70 12.7) 0.52 12.7

Ro

             = 7.176 mm (0.2825 in.) 

2. Compute the radius ratios from Equations (30a) and 

(30b): 

2 0.52 70

(70 12.7)(2 0.52 1) i

      = 31.763 

 and 

2 0.52 70

(70 12.7)(2 0.52 1) o

   

 

 = 22.007 

3. Compute the ellipticity parameters from Equation (31): 

ki = (31.763)2/π

 = 9.040 and ko = (22.007)2/π = 7.157 

4. Compute the first and second elliptic integrals from 

Equations (32) and (33): 

i = π/2 + (π/2 – 1) ln(31.763) = 3.545

o = π/2 + (π/2 – 1) ln(22.007) = 3.335 

E i = 1 + (π/2 – 1)/31.763 = 1.018 

E o = 1 + (π/2 – 1)/22.007 = 1.026 

5. Compute the stiffnesses of the inner and outer races 

from Equations (20b) and (20c): 

3

2 1.018 5.039 9.04 226240

9 (3.545)

Ki

     

= 1.028106

 N/mm1.5 (29 579 kips/in.1.5) 

3

2 1.026 7.176 7.157 226 240

9 (3.335)

Ko

     

= 1.068106

 N/mm1.5 (30 737 kips/in.1.5) 

6. Compute the combined normal stiffness Kn from 

Equation (27): 

3/2 2/3 2/3 6 (1/1.028) (1/1.068) 10 Kn

     

= 3.704105

 N/mm1.5 (10 659 kips/in.1.5) 

7. For computing the maximum ball loads, use Stribeck’s 

(Ref. 2) suggested value for the Stribeck Number, St = 5, 

as a starting guess. 

8. Compute the initial estimate of the load-distribution 

integral: 

 Jr(ε) = 1/St = 1/5 = 0.200 

9. Compute the initial estimate of the total displacement δT

from Equation (35): 

2/3

6

0.0145 627

2 10 3.704 10 0.20 T

         

= 0.01620 mm (0.000661 in.) 
NASA/TM—2012-217115 28 

10. Compute the load-zone parameter: 

ε = 1⁄2 – Gr/(4δT) = 0.5 – (0.0145)/(4  0.0162) = 0.2762 

11. Recompute the load-distribution integral Jr(ε) from 

Equation (26a) since 0.1 ≤ ε < 0.5: 

0.4175 0.3076 (0.2762) r J  = 0.1805 

12. Compute St = 1/Jr = 5.539. This value for St differs 

significantly from the starting guess of 5.0, thus further 

iteration is needed. 

Steps 7 to 12 were repeated with St = 5.539, which yielded a 

new value of 5.494. On the third iteration, St = 5.494 yielded 

a new value of 5.499. Further iterations alternated between 

5.498 and 5.499. We chose St = 5.499. This value is 

2.5 percent greater than the value found from the analysis code 

for this 210-size deep-groove ball bearing.

Example 2—Stribeck Number and Maximum 

Ball Load in a Cylindrical Roller Bearing 

Consider a 210-size cylindrical roller bearing, analogous to 

Example 1 with mid-band group N clearance and carrying a 

light radial load, Pr = 6940 N (1.56 kips). This load was chosen such that, at zero clearance, it produces a inner-race maximum Hertz stress of 1200 MPa (175 ksi). Properties 

(Table III) include the following: z = 14 rollers of diameter 

d = 13 mm (0.512 in.) and length l = 13 mm (0.5118 in.). The 

pitch diameter Dp = 70.65 mm (2.7815 in.). 

From ANSI/ABMA–20 (Ref. 4) (Table V), group N internal 

diametral clearance for this roller bearing ranges from 0.030 to 

0.060 mm (0.0012 to 0.0024 in.). Thus, Gr = 0.045 mm 

(0.0018 in.). 

The elastic constants are the same as in Example 1, thus 

E' = 226 240 MPa (32 800 ksi). We assumed crowned rollers; 

hence we should adjust the roller length. Typically the effective roller length is 85 to 90 percent of l. For this lightly 

loaded example, we assumed 85 percent; hence le = 11.05 mm 

(0.4350 in.). 

1. Compute the combined normal stiffness from 

Equation (28): 

8/9 0.1647 226 240 (11.05) Kn  

= 3.153105

 N/mm1.11 (2579 kips/in.1.11) 

2. Begin iteration assuming Palmgren’s (Ref. 3) suggested 

value for the Stribeck Number, St = 5 

3. Compute the initial estimate of the load-distribution 

integral: 

 Jr(ε) = 1/St = 0.20 

4. Compute the initial estimate of the total deflection δT

from Equation (35): 

1/1.11

5

0.045 6940

2 14 3.153 10 0.20 T

         

= 0.03521 mm (0.0001385 in.) 

5. Compute the load-zone parameter: 

ε = 1⁄2 – Gr/(4δT) = 0.5 – (0.045)/(4·0.03521) = 0.1805 

6. Recompute the load-distribution integral Jr(ε) from 

Equation (25a): 

0.4023 0.3268 (0.1805) r J  = 0.1643 

7. Compute St = 1/Jr = 6.086. This value differs significantly 

from the starting guess of 5.0, thus further iteration is 

needed. 

We repeated steps 2 to 7 choosing St = 6.086, which yielded 

a new value of 5.831. After three further iterations, St converged to 5.876. This value differs by less than 0.01 percent of 

the value found from the analysis code for this 210-size roller 

bearing. 

For large values of negative clearance (interference), this 

method of iterating using the load-distribution integral may 

not give stable convergence with an “automatic” procedure 

that uses the prior estimate for the next iteration. If the clearance value in Example 2 is changed to Gr = –0.02, and the 

initial estimate of St = 5.0, the procedure produces a very large 

calculated value St = 8.5. One must override this estimate, 

choosing more moderate iteration steps to converge to the 

final value of St = 5.897 with residual <0.001. 
NASA/TM—2012-217115 29 

Appendix E.—Alternative Method to Determine the Stribeck Number 

and Maximum Element Load 

The following procedure can be used to calculate the 

Stribeck Number and maximum element load in deep-groove 

ball bearings and cylindrical roller bearings that have zero or 

positive internal clearance. It does not give good results for 

negative clearance. 

1. Make a starting guess for the Stribeck Number, using 

our knowledge of the value for zero clearance, where 

St = 4.37 for ball bearings or 4.08 for cylindrical roller 

bearings. Also consider the typical value for a small 

clearance at moderate load, St ≈ 5. 

2. Compute the maximum ball load from the initial guess 

for St from Equation (10). 

3. Compute the elastic deflections at both races, δi and δo, 

from Equation (B3) for roller bearings or from 

Equation (C1) for ball bearings. 

4. Compute the total deflection: 

δT = δi + δo + Gr/2 

5. Compute the load-zone parameter: 

ε = 1⁄2 – Gr/(4δT) 

6. Compute the maximum element load Pmax from δT from 

Equations (E1) and (E2) (based on Hamrock and 

Anderson (Ref. 17)). These equations are valid for zero 

or positive clearance, Gr ≥ 0, where ε ≤ 0.5. 

For a cylindrical roller bearing, 

max

4

(1 2 )sin

r

l l

P

z P

 

         

 (E1) 

where the limiting contact angle ψl is given by Equation (24). 

For a deep-groove ball bearing, 

3/2

max 2

(2 )

2 2.491 1 1

1.23

Pr

z P

 

                       

 (E2) 

7. Compute the Stribeck Number St from Equation (11). 

If the Stribeck Number from step 7 differs significantly 

from the initial guess, repeat steps 1 to 7 using a refined 

estimate for St. 

NASA/TM—2012-217115 31 

Appendix F.—Determining the Life Factor and Stribeck Number for Internal 

Clearance for Ball and Roller Bearings From Bearing Catalog Data

Two examples are presented to show the simple method for 

determining bearing life factor and Stribeck Number from the 

bearing clearance Gr, the rolling-element diameter d, the static 

radial load capacity Cor, the load on the bearing Pr, and the 

reference value of maximum Hertz stress, S'max = 1700 MPa 

(250 ksi). 

Example 3—Life Factor and Stribeck Number 

for Internal Clearance in a Deep-Groove Ball 

Bearing 

Consider a 210-size deep-groove ball bearing with middleof-the-tolerance-band, group N clearance that is carrying a 

light radial load Pr = 627 N (141 lb). This is the same bearing, 

with the same clearance and at the same load, as in Example 1 

in Appendix C. From Table II, there are z = 10 balls of diameter d = 12.7 mm (0.5 in.). From the bearing catalog, the static 

radial load capacity Cor = 23 162 N (5207 lb). 

From ANSI/ABMA–20 (Ref. 4) (Table IV), the group N 

clearance for this ball bearing ranges from 0.006 to 0.023 mm 

(0.0025 to 0.009 in.). With mid-band clearance, 

Gr = 0.0145 mm (0.00057 in.). 

1. Compute the dimensionless internal diametral clearance 

number from Equation (40a): 

2/3 0.0145 23 162 ˆ 0.185

12.7 627

Gr

       = 0.002343 

2. Compute the life factor from Equation (41b), which is 

valid for –0.000986 ≤ Ĝr > 0.01: 

 LF = exp (–110 · 0.002343) = 0.77 

3. Compute the Stribeck Number from Equation (43c), 

which is valid for 0 ≤ Ĝr > 0.01: 

 St = 4.37 · exp (75 · 0.002343) = 5.21 

The life factor is about 4 percent larger than the analysis 

code result, and the Stribeck Number is 3 percent less than the 

analysis code value. 

Example 4—Life Factor and Stribeck Number 

for Internal Clearance in a Cylindrical Roller 

Bearing 

Consider a 210-size cylindrical roller bearing with middleof-the-tolerance-band, group N clearance that is carrying a 

light radial load of 6940 N (1560 lb). This is the same bearing, 

with the same clearance and the same load as in Example 2 

in Appendix C. From Table III, there are z = 14 rollers of 

diameter d = 13 mm (0.512 in.). The static radial load capacity 

Cor = 77 226 N (17 361 lb). 

From ANSI/ABMA–20 (Ref. 4) (Table V), group N clearance for this bearing ranges from 0.030 to 0.060 mm (0.0012 

to 0.0024 in.). With mid-band clearance, Gr = 0.045 mm 

(0.00177 in.). 

1. Compute the dimensionless internal diametral clearance 

number from Equation (40b): 

0.045 77 226 ˆ 0.185

13 6940

Gr

      

 = 0.007126 

2. Compute the life factor from Equation (42b), which is 

valid for –0.00132 ≤ Ĝr > 0.01: 

 LF = exp (–100 · 0.007126) = 0.49 

3. Compute the Stribeck Number from Equation (44c), 

which is valid for 0 ≤ Ĝr > 0.016: 

 St = 4.21 · exp (44 · 0.007126) = 5.76 

The life factor is about 5 percent larger than the analysis 

code result, and the Stribeck Number is 2 percent less than the 

analysis code value. 

NASA/TM—2012-217115 33 

References 

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Bearing Technology, E.E. Bisson and W.J. Anderson, 

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From the Central Laboratory for Scientific Technical 

Investigation, Translation by K.W. Van Treuren et al., 

ASME Trans., vol. 29, 1907, pp. 420–463. 

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Radial Bearings of Ball, Cylindrical Roller and Spherical 

Roller Types—Metric Design. ANSI/ABMA 20–1996, 

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Rull-Lager Vid Givna Yttre Radial-Och Axialbelastningar

(The Load Distribution Within Ball and Roller Bearings Under Given External Radial and Axial Load), Teknisk Tidskrift (Technical Magazine), Mek., h. 9 (in Swedish), 1933. 

13. Jones, A.B.: New Departure—Analysis of Stresses and 

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Roller Bearing. J. Tribol., vol. 119, 1997, pp. 851–857. 

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aerospace/Documents/aerospace product design guide.pdf 

Accessed August 31, 2011. 

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Life for Ball Bearings. Anti-Friction Bearing Manufacturers Association, Washington, DC, 1978. 

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Life for Ball Bearings. American Bearing Manufacturers 

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Switzerland, 2006. 
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Effect of Internal Clearance on Load Distribution and Life of Radially Loaded Ball and Roller 

Bearings 

5a. CONTRACT NUMBER

5b. GRANT NUMBER

5c. PROGRAM ELEMENT NUMBER

6. AUTHOR(S)

Oswald, Fred, B.; Zaretsky, Erwin , V.; Poplawski, Joseph, V. 

5d. PROJECT NUMBER

5e. TASK NUMBER

5f. WORK UNIT NUMBER

WBS 877868.02.07.03.01.01.01 

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)

National Aeronautics and Space Administration 

John H. Glenn Research Center at Lewis Field 

Cleveland, Ohio 44135-3191 

8. PERFORMING ORGANIZATION

 REPORT NUMBER

E-17247-1 

9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES)

National Aeronautics and Space Administration 

Washington, DC 20546-0001 

10. SPONSORING/MONITOR'S

 ACRONYM(S)

NASA 

11. SPONSORING/MONITORING

 REPORT NUMBER

NASA/TM-2012-217115 

12. DISTRIBUTION/AVAILABILITY STATEMENT

Unclassified-Unlimited 

Subject Category: 37 

Available electronically at http://www.sti.nasa.gov 

This publication is available from the NASA Center for AeroSpace Information, 443-757-5802

13. SUPPLEMENTARY NOTES

14. ABSTRACT

The effect of internal clearance on radially loaded deep-groove ball and cylindrical roller bearing load distribution and fatigue life was 

determined for four clearance groups defined in the bearing standards. The analysis was extended to negative clearance (interference) 

conditions to produce a curve of life factor versus internal clearance. Rolling-element loads can be optimized and bearing life maximized for 

a small negative operating clearance. Life declines gradually with positive clearance and rapidly with increasing negative clearance. 

Relationships were found between bearing life and internal clearance as a function of ball or roller diameter, adjusted for load. Results are 

presented as life factors for radially loaded bearings independent of bearing size or applied load. In addition, a modified Stribeck Equation is 

presented that relates the maximum rolling-element load to internal bearing clearance.

15. SUBJECT TERMS

Bearings; Roller bearings; Ball bearings; Fatigue life; Contact loads; Clearance 

16. SECURITY CLASSIFICATION OF: 17. LIMITATION OF

 ABSTRACT

UU 

18. NUMBER

 OF 

 PAGES

39 

19a. NAME OF RESPONSIBLE PERSON

STI Help Desk (email:help@sti.nasa.gov) 

a. REPORT

U 

b. ABSTRACT

U 

c. THIS 

PAGE

U 

19b. TELEPHONE NUMBER (include area code)

443-757-5802 

Standard Form 298 (Rev. 8-98)

Prescribed by ANSI Std. Z39-18