Rolling bearings -- Explanatory notes on ISO 281

This document specifies supplementary background information regarding the derivation of mathematical expressions and factors given in ISOÂ 281:2007.

Roulements -- Notes explicatives sur l'ISO 281

Le présent document spécifie des informations de base supplémentaires sur la manière dont ont été définies les expressions mathématiques et les facteurs donnés dans l'ISO 281:2007.

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Publication Date
24-May-2021
Current Stage
5060 - Close of voting Proof returned by Secretariat
Start Date
15-Apr-2021
Completion Date
15-Apr-2021
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TECHNICAL ISO/TR
REPORT 1281-1
Second edition
2021-05
Rolling bearings — Explanatory notes
on ISO 281 —
Part 1:
Basic dynamic load rating and basic
rating life
Roulements — Notes explicatives sur l'ISO 281 —
Partie 1: Charges dynamiques de base et durée nominale de base
Reference number
ISO/TR 1281-1:2021(E)
ISO 2021
---------------------- Page: 1 ----------------------
ISO/TR 1281-1:2021(E)
COPYRIGHT PROTECTED DOCUMENT
© ISO 2021

All rights reserved. Unless otherwise specified, or required in the context of its implementation, no part of this publication may

be reproduced or utilized otherwise in any form or by any means, electronic or mechanical, including photocopying, or posting

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Email: copyright@iso.org
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Published in Switzerland
ii © ISO 2021 – All rights reserved
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ISO/TR 1281-1:2021(E)
Contents Page

Foreword ........................................................................................................................................................................................................................................iv

Introduction ..................................................................................................................................................................................................................................v

1 Scope ................................................................................................................................................................................................................................. 1

2 Normative references ...................................................................................................................................................................................... 1

3 Terms and definitions ..................................................................................................................................................................................... 1

4 Symbols .......................................................................................................................................................................................................................... 1

5 General ............................................................................................................................................................................................................................ 4

6 Basic dynamic load rating ........................................................................................................................................................................... 4

6.1 General ........................................................................................................................................................................................................... 4

6.2 Basic dynamic radial load rating, C , for radial ball bearings .......................................................................... 5

6.3 Basic dynamic axial load rating, C , for single row thrust ball bearings ................................................ 9

6.3.1 Thrust ball bearings with contact angle α ≠ 90° ................................................................................... 9

6.3.2 Thrust ball bearings with contact angle α = 90° ................................................................................... 9

6.4 Basic dynamic axial load rating, C , for thrust ball bearings with two or more rows

of balls .........................................................................................................................................................................................................10

6.5 Basic dynamic radial load rating, C , for radial roller bearings ..................................................................11

6.6 Basic dynamic axial load rating, C , for single row thrust roller bearings ........................................13

6.6.1 Thrust roller bearings with contact α ≠ 90° ..........................................................................................13

6.6.2 Thrust roller bearings with contact angle α = 90° ...........................................................................13

6.7 Basic dynamic axial load rating, C , for thrust roller bearings with two or more

rows of rollers .......................................................................................................................................................................................14

7 Dynamic equivalent load ..........................................................................................................................................................................16

7.1 Expressions for dynamic equivalent load .....................................................................................................................16

7.1.1 Theoretical dynamic equivalent radial load, P , for single row radial bearings .....16

7.1.2 Theoretical dynamic equivalent radial load, P , for double row radial bearings ...20

7.1.3 Theoretical dynamic equivalent radial load, P , for radial contact groove

ball bearings .....................................................................................................................................................................22

7.1.4 Practical expressions for dynamic equivalent radial load, P , for radial

bearings with constant contact angle .........................................................................................................22

7.1.5 Practical expressions for dynamic equivalent radial load, P , for radial ball

bearings ................................................................................................................................................................................25

7.1.6 Practical expressions for dynamic equivalent axial load, P , for thrust bearings .26

7.2 Factors X, Y and e ..............................................................................................................................................................................28

7.2.1 Radial ball bearings ....................................................................................................................................................28

7.2.2 Values of X, Y and e for each type of radial ball bearing ..............................................................29

7.2.3 Tabulation of factors X, Y and e for radial ball bearings ..............................................................35

7.2.4 Calculated values of Y and e different from standard ...................................................................37

7.2.5 Thrust ball bearings ...................................................................................................................................................37

7.2.6 Radial roller bearings ...............................................................................................................................................38

7.2.7 Thrust roller bearings ..............................................................................................................................................39

8 Basic rating life ....................................................................................................................................................................................................39

Bibliography .............................................................................................................................................................................................................................42

© ISO 2021 – All rights reserved iii
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ISO/TR 1281-1:2021(E)
Foreword

ISO (the International Organization for Standardization) is a worldwide federation of national standards

bodies (ISO member bodies). The work of preparing International Standards is normally carried out

through ISO technical committees. Each member body interested in a subject for which a technical

committee has been established has the right to be represented on that committee. International

organizations, governmental and non-governmental, in liaison with ISO, also take part in the work.

ISO collaborates closely with the International Electrotechnical Commission (IEC) on all matters of

electrotechnical standardization.

The procedures used to develop this document and those intended for its further maintenance are

described in the ISO/IEC Directives, Part 1. In particular, the different approval criteria needed for the

different types of ISO documents should be noted. This document was drafted in accordance with the

editorial rules of the ISO/IEC Directives, Part 2 (see www .iso .org/ directives).

Attention is drawn to the possibility that some of the elements of this document may be the subject of

patent rights. ISO shall not be held responsible for identifying any or all such patent rights. Details of

any patent rights identified during the development of the document will be in the Introduction and/or

on the ISO list of patent declarations received (see www .iso .org/ patents).

Any trade name used in this document is information given for the convenience of users and does not

constitute an endorsement.

For an explanation of the voluntary nature of standards, the meaning of ISO specific terms and

expressions related to conformity assessment, as well as information about ISO’s adherence to the

World Trade Organization (WTO) principles in the Technical Barriers to Trade (TBT), see www .iso .org/

iso/ foreword .html.

This document was prepared by Technical Committee ISO/TC 4, Rolling bearings, Subcommittee SC 8,

Load ratings and life.

This second edition cancels and replaces the Technical Corrigendum 1 (ISO/TR 1281-1:2008/

Cor 1:2009) and the first edition (ISO/TR 1281-1:2008), which has been technically revised.

The main changes compared to the previous edition are as follows:

— The old Clause 7 “Life adjustment factor for reliability” of ISO/TR 1281-1:2008 has been deleted, this

subject is covered in ISO/TR 1281-2 (see ISO/TR 1281-1:2008/Cor 1:2009).

— The derivation of the old Formulae (29) and (46) [Formulae (28) and (45) in this edition] has been

corrected.

— Typing errors have been corrected in Formulae (30) and (31) and in the derivation of the factor Y .

A list of all parts in the ISO/TR 1281 series can be found on the ISO website.

Any feedback or questions on this document should be directed to the user’s national standards body. A

complete listing of these bodies can be found at www .iso .org/ members .html.
iv © ISO 2021 – All rights reserved
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ISO/TR 1281-1:2021(E)
Introduction
I S O/ R 281: 1962

A first discussion on an international level of the question of standardizing calculation methods for

load ratings of rolling bearings took place at the 1934 conference of the International Federation of

the National Standardizing Associations (ISA). When ISA held its last conference in 1939, no progress

had been made. However, in its 1945 report on the state of rolling bearing standardization, the ISA 4

Secretariat included proposals for definition of concepts fundamental to load rating and life calculation

standards. The definitions it contained are in essence those given in ISO 281:2007 for the concepts

“life” and “basic dynamic load rating” (now divided into “basic dynamic radial load rating” and “basic

dynamic axial load rating”).

In 1946, on the initiative of the Anti-Friction Bearing Manufacturers Association (AFBMA), New York,

discussions of load rating and life calculation standards started between industries in the USA and

Sweden. Chiefly on the basis of the results appearing in Reference [5], an AFBMA standard, Method of

[3]

evaluating load ratings of annular ball bearings , was worked out and published in 1949. On the same

basis, the member body for Sweden presented, in February 1950, a first proposal to ISO, “Load rating of

ball bearings”.

In view of the results of both further research and a modification to the AFBMA standard in 1950, as

well as interest in roller bearing rating standards, in 1951, the member body for Sweden submitted a

modified proposal for rating of ball bearings as well as a proposal for rating of roller bearings.

Load rating and life calculation methods were then studied. Reference [6] was then of considerable use,

serving as a major basis for the sections regarding roller bearing rating.
ISO 281-1:1977

In 1964, in view of the development of improved bearing steels, the time had come to review ISO/R281

and submitted a proposal

In 1969, on the other hand, TC 4 followed a suggestion by the member body for Japan and reconstituted

its WG 3, giving it the task of revising ISO/R281. The AFBMA load rating working group had at this time

started revision work.

The major part of ISO 281-1:1977 constituted a re-publication of ISO/R281, the substance of which had

been only very slightly modified. However, based mainly on American investigations during the 1960s,

a new clause was added, dealing with adjustment of rating life for reliability other than 90 % and for

material and operating conditions.

Furthermore, supplementary background information regarding the derivation of mathematical

expressions and factors given in ISO 281-1:1977 was published as ISO/TR 8646:1985.

ISO 281:1990

ISO 281:1990 was published as “First edition” and entitled “Dynamic load ratings and rating life”. It is

referred to as the “technical revision” of ISO 281-1:1977. The new rating factor b for “contemporary,

normally used material and manufacturing quality, the value of which varies with bearing type and

design” was the introduction as a co-value to the basic dynamic load ratings.
ISO 281:2007 (second edition)

Since the publication of ISO 281:1990 additional knowledge regarding the influence on bearing life of

contamination, lubrication, internal stresses from mounting, stresses from hardening, fatigue load

limit of the material, has been gained. In ISO 281:1990/Amd.2:2000, a general method was presented to

consider such influences in the calculation of a modified rating life of a bearing. The said Amendment

was incorporated into the second edition, which also provides a practical method to consider the

influence on bearing life of lubrication conditions, contaminated lubricant and fatigue load of bearing

© ISO 2021 – All rights reserved v
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ISO/TR 1281-1:2021(E)

material. The life modification factors for reliability, a , have been slightly adjusted and extended to

99,95 % reliability.
vi © ISO 2021 – All rights reserved
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TECHNICAL REPORT ISO/TR 1281-1:2021(E)
Rolling bearings — Explanatory notes on ISO 281 —
Part 1:
Basic dynamic load rating and basic rating life
1 Scope

This document specifies supplementary background information regarding the derivation of

mathematical expressions and factors given in ISO 281:2007.
2 Normative references

The following documents are referred to in the text in such a way that some or all of their content

constitutes requirements of this document. For dated references, only the edition cited applies. For

undated references, the latest edition of the referenced document (including any amendments) applies.

ISO 281:2007, Rolling bearings — Dynamic load ratings and rating life
3 Terms and definitions

For the purposes of this document, the terms and definitions given in ISO 281:2007 apply.

ISO and IEC maintain terminological databases for use in standardization at the following addresses:

— ISO Online browsing platform: available at https:// www .iso .org/ obp
— IEC Electropedia: available at http:// www .electropedia .org/
4 Symbols
A constant of proportionality
A constant of proportionality determined experimentally
B constant of proportionality determined experimentally
C basic dynamic radial load rating of a rotating ring
C basic dynamic radial load rating of a stationary ring
C basic dynamic axial load rating for thrust ball or roller bearing

C basic dynamic axial load rating of the rotating ring of an entire thrust ball or roller bearing

C basic dynamic axial load rating of the stationary ring of an entire thrust ball or roller bearing

C basic dynamic axial load rating as a row k of an entire thrust ball or roller bearing

C basic dynamic axial load rating as a row k of the rotating ring of thrust ball or roller bearing

a1k

C basic dynamic axial load rating as a row k of the stationary ring of thrust ball or roller bearing

a2k
© ISO 2021 – All rights reserved 1
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ISO/TR 1281-1:2021(E)
C basic dynamic load rating for outer ring
C basic dynamic load rating for inner ring
C basic dynamic radial load rating for radial ball or roller bearing
D pitch diameter of ball or roller set
D ball diameter
D mean roller diameter
E modified modulus of elasticity
F axial load
F radial load
J factor relating mean equivalent load on a rotating ring to Q
1 max
J factor relating mean equivalent load on a stationary ring to Q
2 max
J axial load integral
J radial load integral
L bearing life
L basic rating life
L effective contact length of roller
L L per row k
wek we
N number of stress applications to a point on the raceway
P dynamic equivalent axial load for thrust bearing
P dynamic equivalent radial load for radial bearing
P dynamic equivalent radial load for the rotating ring
P dynamic equivalent radial load for the stationary ring
Q normal force between a rolling element and the raceways
Q rolling element load for the basic dynamic load rating of the bearing

Q rolling element load for the basic dynamic load rating of a ring rotating relative to the applied load

Q rolling element load for the basic dynamic load rating of a ring stationary relative to the applied

load
Q maximum rolling element load
max
S probability of survival, reliability
V volume representative of the stress concentration
V rotation factor
X radial load factor for radial bearing
2 © ISO 2021 – All rights reserved
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ISO/TR 1281-1:2021(E)
X radial load factor for thrust bearing
Y axial load factor for radial bearing
Y axial load factor for thrust bearing
Z number of balls or rollers per row
Z number of balls or rollers per row k
a semimajor axis of the projected contact ellipse
a life adjustment factor for reliability
b semiminor axis of the projected contact ellipse
c exponent determined experimentally
c compression constant
e measure of life scatter, i.e. Weibull slope determined experimentally

e limiting value of F / F for the applicability of different values of factors X and Y in the new

a r
edition

f factor which depends on the geometry of the bearing components, the accuracy to which the

various components are made, and the material
h exponent determined experimentally
i number of rows of balls or rollers
l circumference of the raceway
r cross-sectional raceway groove radius
r cross-sectional raceway groove radius of outer ring or housing washer
r cross-sectional raceway groove radius of inner ring or shaft washer
t auxiliary parameter
z depth of the maximum orthogonal subsurface shear stress
α nominal contact angle
α′ actual contact angle
γ D cos α/D for ball bearings with α ≠ 90°
w pw
D /D for ball bearings with α = 90°
w pw
D cos α/D for roller bearings with α ≠ 90°
we pw
D /D for roller bearings with α = 90°
we pw
ε parameter indicating the width of the loaded zone in the bearing
η reduction factor
λ reduction factor
© ISO 2021 – All rights reserved 3
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ISO/TR 1281-1:2021(E)
µ factor introduced by Hertz
ν factor introduced by Hertz, or adjustment factor for exponent variation
σ maximum contact stress
max
Σρ curvature sum
τ maximum orthogonal subsurface shear stress
φ one half of the loaded arc
5 General

The derivation of the basic dynamic load ratings is described in Formulae (1) to (46). The dynamic

equivalent load and the radial and axial load factors are covered in Formulae (47) to (82), while basic

rating life is described in Formulae (83) to (89).
6 Basic dynamic load rating
6.1 General

The background to basic dynamic load ratings of rolling bearings according to ISO 281 appears in

References [5] and [6].

The expressions for calculation of basic dynamic load ratings of rolling bearings develop from a power

formula that can be written as follows:
τ NV
ln ∝ (1)
where
S is the probability of survival;
τ is the maximum orthogonal subsurface shear stress;
N is the number of stress applications to a point on the raceway;
V is the volume representative of the stress concentration;
z is the depth of the maximum orthogonal subsurface shear stress;
c, h are experimentally determined exponents;

e is the measure of life scatter, i.e. the Weibull slope determined experimentally.

For “point” contact conditions (ball bearings) it is assumed that the volume, V, representative of the

stress concentration in Formula (1) is proportional to the major axis of the projected contact ellipse,

4 © ISO 2021 – All rights reserved
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ISO/TR 1281-1:2021(E)

2a, the circumference of the raceway, l, and the depth, z , of the maximum orthogonal subsurface shear

stress, τ
Va ∝ 2 zl (2)
Substituting Formula (2) into Formula (1):
τ Nal
ln ∝ (3)
h−1

“Line” contact was considered in References [5] and [6] to be approached under conditions where the

major axis of the calculated Hertz contact ellipse is 1,5 times the effective roller contact length:

21aL = ,5 (4)

In addition, b/a should be small enough to permit the introduction of the limit value of ab as b/a

approaches 0:
23Q
ab = (5)
π E ∑ρ
(for variable definitions, see 6.2).
6.2 Basic dynamic radial load rating, C , for radial ball bearings

From the theory of Hertz, the maximum orthogonal subsurface shear stress, τ , and the depth, z , can

o o

be expressed in terms of a radial load F , i.e. a maximum rolling element load, Q , or a maximum

r max

contact stress, σ , and dimensions for the contact area between a rolling element and the raceways.

max
The relationships are:
τσ = T
omax
zb = ζ
1/2
( 21t − )
21 tt( + )
ζ =
1/2
( tt+− )11 (2 )
1/3
 
a = μ
 
E ∑ρ
 
1/3
 
bv =
 
E ∑ρ
 
where
σ is the maximum contact stress;
max
t is the auxiliary parameter;
a is the semimajor axis of the projected contact ellipse;
© ISO 2021 – All rights reserved 5
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ISO/TR 1281-1:2021(E)
b is the semiminor axis of the projected contact ellipse;
Q is the normal force between a rolling element and the raceways;
E is the modified modulus of elasticity;
Σρ is the curvature sum;
µ, v are factors introduced by Hertz.

Consequently, for a given rolling bearing, τ , a, l and z can be expressed in terms of bearing geometry,

o o

load and revolutions. Formula (3) is changed to a formula by inserting a constant of proportionality.

Inserting a specific number of revolutions (e.g. 10 ) and a specific reliability (e.g. 0,9), the formula is

solved for a rolling element load for basic dynamic load rating which is designated to point contact

rolling bearings introducing a constant of proportionality, A :
04, 1
(1,591ch+−,415,82)/(ch−+2)
 
13, 2r (1γγ)
QA= ×
 
C 1
()22ch+− /(ch−+2) 32ec/()−+h 3e/(ch−+2)
2rD−
40,5 (1±γ)
 
(6)
3/(()ch−+2
 
(2ch+−52)/()ch−+ −−3/ec()h+2
 
cosα 
where
Q is the rolling element load for the basic dynamic load rating of the bearing;
D is the ball diameter;
γ is D cos α/D ;
w pw
in which
D is the pitch diameter of the ball set;
α is the nominal contact angle;
Z is the number of balls per row.
The basic dynamic radial load rating, C , of a rotating ring is given by:
CQ== Z cos αα 0,407 QZ cos (7)
11C C1
The basic dynamic radial load rating, C , of a stationary ring is given by:
CQ== Z cosc αα0,389 QZ os (8)
22C C 2
where

Q is the rolling element load for the basic dynamic load rating of a ring rotating relative

to the applied load;

Q is the rolling element load for the basic dynamic load rating of a ring stationary rel-

ative to the applied load;
6 © ISO 2021 – All rights reserved
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ISO/TR 1281-1:2021(E)

J = J (0,5) is the radial load integral for zero diametral clearance (see Table 3);

r r

J = J (0,5) is the factor relating mean equivalent load on a rotating ring to Q for zero diametral

1 1 max
clearance (see Table 3);

J = J (0,5) is the factor relating mean equivalent load on a stationary ring to Q for zero dia-

2 2 max
metral clearance (see Table 3).

The relationship between C for an entire radial ball bearing, and C and C , is expressed in terms of the

r 1 2
product law of probability as:
−−3/(2ch+ )
(2ch−+ )/3
 
 
 
CC=+1 (9)
 
r 1
 
 
 

Substituting Formulae (6), (7) and (8) into Formula (9), the basic dynamic radial load rating, C , for an

entire ball bearing is expressed as:
004, 1
(1,59 ch+− 1,41 5,82)/(ch−+2)
 
13, ()1−γ
3/(()ch−+2
CA=04, 1 γ ×
 
r 1
()22ch+− /(ch−+23)/ec()−+h 2 3/ec()−+h 2
2rD−
40 ,5 (1+γ )
 
−33/()ch−+2
()ch−+2 /3
 04, 1 
(1,59 c +++ 1,41 he 32−−5,82)/(ch+ )
 
r 2rD−
  
 1−γ 
 
 i ew 
11+ ,04 ×
    
 
 
r 2rD− 1+γ
 
 
  e iw  
 
 
 
()ch−−12/(ch−+ )(ch−−32ec+−)/()hc++22()hc−−5 /( hh+2)
( iZcos α ) D (10)
where
A is the experimentally determined proportionality constant;
r is the cross-sectional raceway groove radius of the inner ring;
r is the cross-sectional raceway groove radius of the outer ring;
i is the number of rows of balls.

Here, the contact angle, α, the number of rolling elements (balls), Z, and the ball diameter, D , depend

on bearing design. On the other hand, the ratios of raceway groove radii, r and r , to a half-diameter of

i e

a rolling element (ball), D /2 and γ = D cos α/D , are not dimensional, therefore it is convenient in

w w pw

practice that the value for the initial terms on the right-hand side of Formula (10) to be designated as a

factor, f :
(1ch−− )/()ch−+22(3ch−− ec+−2)/( hc++) (2 h−5/)/(ch−+2)
Cf= ( iZcos α ) D (11)
rc w

With radial ball bearings, the faults in bearings resulting from manufacturing need to be taken into

consideration, and a reduction factor, λ, is introduced to reduce the value for a basic dynamic radial load

rating for radial ball bearings from its theoretical value. It is convenient to include λ in the factor, fc. The

value of λ is determined experimentally.
© ISO 2021 – All rights reserved 7
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ISO/TR 1281-1:2021(E)
04, 1
(1,591ch+−,415,82)/(ch−+2)
 
13, ()1−γ
3//(ch−+2)
fA=04, 1 λ γ ×
 
c 1
()22ch+− /(ch−+23)/ec()−+h 2 32ec/( −+h )
2rD−
40,5 (1+γ )
 
−−3/(c hh+2)
()ch−+2 /3
 
04, 1
(1,59c+11,41he+−325,82)/(ch−+ )
 
r 2rD−
  
  1−γ  
 i ew 
11+ ,04 (12)
 
  
 
 
r 2rD− 1+γ
 
  
 e iw 
 
 
 

Based on References [5] and, [6] the following values were assigned to the experimental constants in

the load rating formulae for ball bearings:

Substituting the numerical values into Formula (11) gives the following, however, a sufficient number of

test results are only available for small balls, i.e. up to a diameter of 25,4 mm (1 inch), and these show

1,8

that the load rating may be taken as being proportional to D . In the case of larger balls, the load

1,4

rating appears to increase even more slowly in relation to the ball diameter, and D can be assumed

where D > 25,4 mm:
07,,2/3 18
Cf= (iZ cos )α D
rc w
for D ≤ 25,4 mm (13)
0,72/3 1,4
Cf= 3,647 (iZ cos )α D
rc w
for D > 25,4 mm (14)
04, 1
03,,139
 2r 
γγ()1−
fA= 0,,089 041 λ ×
 
1//3
2rD−
(1+γ )
 
−31/ 0 (15)
10/3
 04, 1 
17, 2
 
r  2rD− 
1−γ
   
 i ew 
11+ ,04
   
   
 
r 2rD− 1+γ
 
  
 e iw 
 
 
 

Values of f in ISO 281:2007, Table 2, are calculated by substituting raceway groove radii and reduction

factors given in Table 1 into Formula (15).
The value for 0,089A is 98,066 5 to calculate C in newtons.
1 r
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ISO/TR 1281-1:2021(E)
6.3 Basic dynamic axial load rating, C , for single row thrust ball bearings
6.3.1 Thrust ball bearings with contact angle α ≠ 90°
As in 6.2, for thrust ball bearings with contact angle α ≠ 90°:
(1ch−− )/(2ch−+ )(ch−−32ec+−)/(2hc++) (2 hhc−−5)/( h+2)
Cf= (cos )ααtan ZD (16)
ac w

For most thrust ball bearings, the theoretical value of a basic dynamic axial load rating has to be reduced

on the basis of unequal distribution of load among the rolling elements in addition to the reduction

factor, λ, which is introduced in to radial ball bearing load ratings. This reduction factor is designated

as η.
Consequently, the factor f is given by:
0,,41
(1,59 ch+− 1,41 5,82)/(ch−+2)
13,   ()1−γ
3/(cch−+2)
fA=λη γ ×
 
c 1
()22ch+− /(ch−+23)/ec()−
...

RAPPORT ISO/TR
TECHNIQUE 1281-1
Deuxième édition
2021-05
Roulements — Notes explicatives sur
l'ISO 281 —
Partie 1:
Charges dynamiques de base et durée
nominale de base
Rolling bearings — Explanatory notes on ISO 281 —
Part 1: Basic dynamic load rating and basic rating life
Numéro de référence
ISO/TR 1281-1:2021(F)
ISO 2021
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ISO/TR 1281-1:2021(F)
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ii © ISO 2021 – Tous droits réservés
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ISO/TR 1281-1:2021(F)
Sommaire Page

Avant-propos ..............................................................................................................................................................................................................................iv

Introduction ..................................................................................................................................................................................................................................v

1 Domaine d'application ................................................................................................................................................................................... 1

2 Références normatives ................................................................................................................................................................................... 1

3 Termes et définitions ....................................................................................................................................................................................... 1

4 Symboles ....................................................................................................................................................................................................................... 1

5 Généralités .................................................................................................................................................................................................................. 4

6 Charge dynamique de base ........................................................................................................................................................................ 4

6.1 Généralités .................................................................................................................................................................................................. 4

6.2 Charge radiale dynamique de base, C , des roulements radiaux à billes .............................................. 5

6.3 Charge axiale dynamique de base, C , des butées à billes à une rangée ................................................ 9

6.3.1 Butées à billes à angle de contact α ≠ 90° ................................................................................................... 9

6.3.2 Butées à billes à angle de contact α = 90° ................................................................................................... 9

6.4 Charge axiale dynamique de base, C , des butées à billes à deux ou plusieurs rangées .......10

6.5 Charge radiale dynamique de base, C , des roulements radiaux à rouleaux ...................................11

6.6 Charge axiale dynamique de base, C , des butées à rouleaux à une rangée ....................................13

6.6.1 Butées à rouleaux à angle de contact α ≠ 90° .......................................................................................13

6.6.2 Butées à rouleaux à angle de contact α = 90° .......................................................................................13

6.7 Charge axiale dynamique de base, C , des butées à deux ou plusieurs rangées

de rouleaux ..............................................................................................................................................................................................14

7 Charge dynamique équivalente..........................................................................................................................................................16

7.1 Expressions de la charge dynamique équivalente.................................................................................................16

7.1.1 Charge radiale dynamique équivalente théorique, P , des roulements

radiaux à une rangée .................................................................................................................................................16

7.1.2 Charge radiale dynamique équivalente théorique, P , des roulements

radiaux à deux rangées ............................................................................................................................................20

7.1.3 Charge radiale dynamique équivalente théorique, P , des roulements à

billes, à gorges, à contact radial .......................................................................................................................22

7.1.4 Expressions pratiques de la charge radiale dynamique équivalente, P , des

roulements radiaux à angle de contact constant ..............................................................................23

7.1.5 Expressions pratiques de la charge radiale dynamique équivalente, P , des

roulements radiaux à billes .................................................................................................................................25

7.1.6 Expressions pratiques de la charge axiale dynamique équivalente, P , des

butées .....................................................................................................................................................................................26

7.2 Facteurs X, Y et e ................................................................................................................................................................................. 28

7.2.1 Roulements radiaux à billes ................................................................................................................................28

7.2.2 Valeurs de X, Y et e pour chaque type de roulement (radial) à billes ..............................29

7.2.3 Tableau récapitulatif des facteurs X, Y et e pour les roulements radiaux à billes .34

7.2.4 Valeurs calculées de Y et e et leur écart par rapport à celles de la norme ..................36

7.2.5 Butées à billes ..................................................................................................................................................................36

7.2.6 Roulements radiaux à rouleaux .......................................................................................................................37

7.2.7 Butées à rouleaux .........................................................................................................................................................38

8 Durée nominale ..................................................................................................................................................................................................38

Bibliographie ...........................................................................................................................................................................................................................41

© ISO 2021 – Tous droits réservés iii
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ISO/TR 1281-1:2021(F)
Avant-propos

L'ISO (Organisation internationale de normalisation) est une fédération mondiale d'organismes

nationaux de normalisation (comités membres de l'ISO). L'élaboration des Normes internationales est

en général confiée aux comités techniques de l'ISO. Chaque comité membre intéressé par une étude

a le droit de faire partie du comité technique créé à cet effet. Les organisations internationales,

gouvernementales et non gouvernementales, en liaison avec l'ISO participent également aux travaux.

L'ISO collabore étroitement avec la Commission électrotechnique internationale (IEC) en ce qui

concerne la normalisation électrotechnique.

Les procédures utilisées pour élaborer le présent document et celles destinées à sa mise à jour sont

décrites dans les Directives ISO/IEC, Partie 1. Il convient, en particulier de prendre note des différents

critères d'approbation requis pour les différents types de documents ISO. Le présent document a été

rédigé conformément aux règles de rédaction données dans les Directives ISO/IEC, Partie 2 (voir www

.iso .org/ directives).

L'attention est attirée sur le fait que certains des éléments du présent document peuvent faire l'objet de

droits de propriété intellectuelle ou de droits analogues. L'ISO ne saurait être tenue pour responsable

de ne pas avoir identifié de tels droits de propriété et averti de leur existence. Les détails concernant

les références aux droits de propriété intellectuelle ou autres droits analogues identifiés lors de

l'élaboration du document sont indiqués dans l'Introduction et/ou dans la liste des déclarations de

brevets reçues par l'ISO (voir www .iso .org/ brevets).

Les appellations commerciales éventuellement mentionnées dans le présent document sont données

pour information, par souci de commodité, à l’intention des utilisateurs et ne sauraient constituer un

engagement.

Pour une explication de la nature volontaire des normes, la signification des termes et expressions

spécifiques de l'ISO liés à l'évaluation de la conformité, ou pour toute information au sujet de l'adhésion

de l'ISO aux principes de l’Organisation mondiale du commerce (OMC) concernant les obstacles

techniques au commerce (OTC), voir le lien suivant: www .iso .org/ iso/ fr/ avant -propos .html.

Le présent document a été élaboré par le comité technique ISO/TC 4, Roulements, sous-comité SC 8,

Charges de base et durée.

Cette deuxième édition annule et remplace le rectificatif technique 1 (ISO/TR 1281-1:2008/Cor 1:2009)

et la première édition (ISO/TR 1281-1:2008), qui a fait l’objet d’une révision technique.

Les principales modifications par rapport à l’édition précédente sont les suivantes:

— supression de l’ancien Article 7 «Facteur de réduction de la durée en fonction de la fiabilité»

de l’ISO/TR 1281-1:2008, ce sujet étant couvert dans l’ISO/TR 1281-2 (voir ISO/TR 1281-1:2008/

Cor 1:2009).

— correction de la manière dont ont été définies les anciennes Formules (29) et (46) [Formules (28) et

(45) dans cette édition].

— correction des erreurs de saisie dans les Formules (30) et (31) et de la manière dont a été défini le

facteur Y .

Une liste de toutes les parties de l’ISO/TR 1281-1 se trouve sur le site Web de l’ISO.

Il convient que l'utilisateur adresse tout retour d'information ou toute question concernant le présent

document à l'organisme national de normalisation de son pays. Une liste exhaustive desdits organismes

se trouve à l'adresse www .iso .org/ fr/ members .html.
iv © ISO 2021 – Tous droits réservés
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ISO/TR 1281-1:2021(F)
Introduction
I S O/ R 281: 1962

Une première discussion de niveau international portant sur la normalisation des méthodes de calcul

des charges de base des roulements eut lieu en 1934 lors de la conférence de la Fédération Internationale

des Associations Nationales de Normalisation (ISA). Lorsque l'ISA tint sa dernière réunion en 1939,

aucun progrès n'était encore intervenu. Pourtant, dans son rapport de 1945 sur l'état de la normalisation

dans le domaine des roulements, le Secrétariat de l'ISA 4 incluait des propositions de définitions de

concepts fondamentaux pour les normes de calcul de charges de base et de durée. Les définitions qu'il

contenait étant en substance celles que reprend l'ISO 281:2007 sous les termes de «durée» et de «charge

dynamique de base» (cette dernière étant maintenant séparée en «charge dynamique radiale de base»

et «charge dynamique axiale de base»).

Les discussions sur les normes de calcul de durée et de charges de base reprirent en 1946 entre les

spécialistes américains et suédois à l'initiative de l'AFBMA — Anti-Friction Bearing Manufacturers

Association (New York). Une norme AFBMA, publiée en 1949, intitulée «Method of evaluating load

[3]

ratings of annular ball bearings» fut élaborée sur la base principalement des résultats figurant dans

la Référence [5]. Partant de la même source, le Comité membre suédois soumit en février 1950 une

première proposition à l'ISO intitulée «Charges de base des roulements à billes».

Compte tenu des recherches nouvelles, de la révision de la norme AFBMA en 1950 et également de

l'intérêt pour les normes de calcul des roulements à rouleaux, le Comité membre Suédois présenta, en

1951, une proposition modifiée de calcul des roulements à billes, puis une proposition de calcul des

roulements à rouleaux.

Ces méthodes de calcul furent étudiées. La Référence [6] eut alors un retentissement considérable sur

l'élaboration des chapitres relatifs au calcul des roulements à rouleaux.
ISO 281-1:1977

En 1964, au vu de l'amélioration des aciers pour roulements, il était temps de réviser l'ISO/R281 et de

soumettre une proposition.

En 1969, cependant, le TC 4 suivit la suggestion du Comité membre Japonais de reconstituer le GT 3 et

de lui donner pour tâche de réviser l'ISO/R281. Le groupe AFBMA de calcul des charges de base avait

également à l'époque repris les travaux pour réviser la norme.

La majeure partie de l’ISO 281-1:1977 constitue une réédition de l’ISO/R281 dont le fond n'est que très

peu modifié. Un nouvel article a cependant été ajouté, résultat de recherches américaines des années

1960 et qui traite de la correction à apporter à la durée si la fiabilité est supérieure à 90 % ou pour tenir

compte des matériaux et des conditions de fonctionnement.

Des informations complémentaires relatives à la manière dont sont déterminés les expressions et

facteurs de l'ISO 281-1:1977 ont été publiées sous la référence ISO/TR 8646:1985.

ISO 281:1990

La «première édition» de l’ISO 281:1990 a été publiée et intitulée «Charges dynamiques de base

et durée nominale». Il y est fait référence en tant que «révision technique» de l’ISO 281-1:1977. Le

nouveau facteur de notation b pour «facteur de calcul pour un matériau et une fabrication modernes

et habituels. Sa valeur dépend du type et de la conception du roulement» a été l’introduction d’une co-

valeur des charges dynamiques de base.
ISO 281:2007 (deuxième édition)

Depuis la publication de l’ISO 281:1990, des connaissances supplémentaires concernant l’influence

de la contamination, de la lubrification, des contraintes internes dues au montage et contraintes dues

au durcissement, et de la limite de charge de fatigue du matériau sur la vie des roulements, ont été

acquises. Dans l’ISO 281:1990/Amd 2:2000, une méthode générale a été présentée pour tenir compte de

© ISO 2021 – Tous droits réservés v
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ISO/TR 1281-1:2021(F)

ces influences dans le calcul de la durée de vie nominale modifiée d'un roulement. Le dit amendement

a été incorporé à la deuxième édition, qui fournit également une méthode pratique pour prendre en

compte l'influence des conditions de lubrification, du lubrifiant contaminé et de la charge de fatigue du

matériau du roulement sur la vie du roulement. Les facteurs de modification de la durée de vie pour la

fiabilité, a , ont été légèrement ajustés et étendus à une fiabilité de 99,95 %.
vi © ISO 2021 – Tous droits réservés
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RAPPORT TECHNIQUE ISO/TR 1281-1:2021(F)
Roulements — Notes explicatives sur l'ISO 281 —
Partie 1:
Charges dynamiques de base et durée nominale de base
1 Domaine d'application

Le présent document spécifie des informations de base supplémentaires sur la manière dont ont été

définies les expressions mathématiques et les facteurs donnés dans l'ISO 281:2007.

2 Références normatives

Les documents suivants sont cités dans le texte de sorte qu'ils constituent, pour tout ou partie de leur

contenu, des exigences du présent document. Pour les références datées, seule l'édition citée s'applique.

Pour les références non datées, la dernière édition du document de référence s'applique (y compris les

éventuels amendements).
ISO 281:2007, Roulements — Charges dynamiques de base et durée nominale
3 Termes et définitions

Pour les besoins du présent document, les termes et définitions donnés dans l'ISO 281:2007 ainsi que

les suivants s'appliquent.

L'ISO et l'IEC tiennent à jour des bases de données terminologiques destinées à être utilisées en

normalisation, consultables aux adresses suivantes:

— ISO Online browsing platform: disponible à l'adresse https:// www .iso .org/ obp

— IEC Electropedia: disponible à l'adresse https:// www .electropedia .org/
4 Symboles
A constante de proportionnalité
A constante de proportionnalité déterminée expérimentalement
B constante de proportionnalité déterminée expérimentalement
C charge radiale dynamique de base d'une bague tournante
C charge radiale dynamique de base d'une bague fixe
C charge axiale dynamique de base d'une butée à bille ou à rouleaux

C charge axiale dynamique de base de la rondelle tournante d'une butée à billes ou à rou-

leaux

C charge axiale dynamique de base de la rondelle fixe d'une butée à billes ou à rouleaux

C charge axiale dynamique de base de la rangée k d'une butée à billes ou à rouleaux

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ISO/TR 1281-1:2021(F)

C charge axiale dynamique de base de la rangée k de la rondelle tournante d'une butée à

a1k
billes ou à rouleaux

C charge axiale dynamique de base de la rangée k de la rondelle fixe d'une butée à billes ou

a2k
à rouleaux
C charge dynamique de base d'une bague extérieure
C charge dynamique de base d'une bague intérieure
C charge radiale dynamique de base d'un roulement radial à billes ou à rouleaux
D diamètre primitif
D diamètre de bille
D diamètre moyen de rouleau
E module d'élasticité modifié
F charge axiale
F charge radiale
J facteur rapportant à Q la charge moyenne équivalente sur une bague tournante
1 max
(par rapport à la charge appliquée)

J facteur rapportant à Q la charge moyenne équivalente sur une bague fixe (par rapport

2 max
à la charge appliquée)
J intégrale de charge axiale
J intégrale de charge radiale
L durée du roulement
L durée nominale
L longueur effective de contact du rouleau
L L pour la rangée k
wek we
N nombre d'application pour la contrainte en un point du chemin de roulement
P charge axiale dynamique équivalente d'une butée
P charge radiale dynamique équivalente d'un roulement radial
P charge radiale dynamique équivalente de la bague tournante
P charge radiale dynamique équivalente de la bague fixe
Q force normale entre un élément roulant et les chemins de roulement

Q charge sur l'élément roulant correspondant à la charge dynamique de base du roulement

Q charge sur l'élément roulant correspondant à la charge dynamique de base d'une bague

tournante par rapport à la charge appliquée

Q charge sur l'élément roulant correspondant à la charge dynamique de base d'une bague

fixe par rapport à la charge appliquée
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ISO/TR 1281-1:2021(F)
Q charge maximale sur l'élément roulant
max
S probabilité de survie, fiabilité
V volume représentatif de la concentration des contraintes
V facteur de rotation
X facteur de charge radiale pour roulement radial
X facteur de charge radiale pour butée
Y facteur de charge axiale pour roulement radial
Y facteur de charge axiale pour butée
Z nombre de billes ou de rouleaux par rangée
Z nombre de billes ou de rouleaux par rangée k
a demi grand axe de l'ellipse de contact projetée
a facteur de réduction de la durée en fonction de la fiabilité
b demi petit axe de l'ellipse de contact projetée
c exposant déterminé expérimentalement
c constante de compression

e mesure de dispersion de durée, c'est-à-dire pente de la courbe de Weibull, déterminée

expérimentalement

e valeur limite de F / F pour l'applicabilité de différentes valeurs des facteurs X et Y dans

a r
la nouvelle édition

f facteur qui dépend de la géométrie des éléments du roulement, de leur précision d'exécu-

tion et des matériaux
h exposant déterminé expérimentalement
i nombre de rangées de billes ou de rouleaux
l circonférence du chemin du roulement
r rayon de courbure transversal d'un chemin de roulement
r rayon de courbure transversal d'un chemin de roulement de bague extérieure ou
de rondelle-logement
r rayon de courbure transversal d'un chemin de roulement de bague intérieure ou
de rondelle-arbre
t paramètre auxiliaire

z profondeur de la contrainte maximale de cisaillement orthogonale sous la surface

α angle nominal de contact
α ′ angle réel de contact
© ISO 2021 – Tous droits réservés 3
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ISO/TR 1281-1:2021(F)
γ D cos α/D pour roulements à billes avec α ≠ 90°
w pw
D /D pour roulements à billes avec α = 90°
w pw
D cos α/D pour roulements à billes avec α ≠ 90°
we pw
D /D pour roulements à billes avec α = 90°
we pw
ε paramètre caractéristique de la grandeur de la zone chargée dans le roulement
η facteur de réduction
λ facteur de réduction
µ facteur introduit par Hertz

ν facteur introduit par Hertz, ou facteur de réduction de la variation de l'exposant

σ contrainte maximale de contact
max
Σρ somme des courbures
τ contrainte maximale de cisaillement orthogonale sous la surface
φ moitié de l'arc chargé
5 Généralités

La manière dont sont définies les charges dynamiques de base est décrite dans les Formules (1) à

(46). La charge dynamique équivalente et les facteurs de charge radiale et axiale sont traités dans les

Formules (47) à (82), tandis que la durée de vie nominale de base est décrite dans les Formules (83)

à (89).
6 Charge dynamique de base
6.1 Généralités

Les calculs de charges dynamiques de base de l'ISO 281 sur les roulements sont fondés sur les

Références [5] et [6].

Les formules de calcul des charges dynamiques de base des roulements dérivent de la relation suivante:

τ NV
ln ∝ (1)
S est la probabilité de survie;

τ est la composante orthogonale de la contrainte maximale de cisaillement sous la surface;

N est le nombre d'applications de la contrainte en un point donné du chemin de roulement;

V est le volume représentatif de la concentration des contraintes;
4 © ISO 2021 – Tous droits réservés
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ISO/TR 1281-1:2021(F)

z est la profondeur de la composante orthogonale de la contrainte maximale de cisaillement

sous la surface;
c, h sont des exposants déterminés expérimentalement;

e est la mesure de la dispersion de la durée, c'est-à-dire la pente de la courbe de Weibull,

déterminée expérimentalement.

Dans les conditions de contact «ponctuel» (roulements à billes), on prend comme hypothèse que le

volume, V, représentatif de la concentration des contraintes dans la Formule (1) est proportionnel

au grand axe de l'ellipse de contact projetée, 2a, à la circonférence du chemin de roulement, l, et à la

profondeur, z , de la composante orthogonale de la contrainte maximale de cisaillement sous la surface,

τ .
Va ∝ 2 zl (2)
D'où, si l'on introduit Formule (2) dans la Formule (1):
τ Nal
ln ∝ (3)
h−1
S z

Dans les Références [5] et [6], il a été considéré qu'on pouvait admettre un contact «linéaire» lorsque

le grand axe de l'ellipse de contact calculée (ellipse de Hertz) était de 1,5 fois la longueur effective de

contact du rouleau:
21aL = ,5 (4)

Il convient, en outre, que b/a soit suffisamment petit pour permettre d'introduire la valeur-limite de ab

pour b/a tendant vers 0:
23Q
ab = (5)
π E ∑ρ
(pour les notations, se reporter à 6.2).
6.2 Charge radiale dynamique de base, C , des roulements radiaux à billes

D'après la théorie de Hertz, la composante orthogonale de la contrainte maximale de cisaillement sous

la surface, τ , et sa profondeur, z , peuvent se rattacher à une charge radiale, F , c'est-à-dire une charge

o o r

maximale sur l'élément roulant, Q , ou une contrainte maximale de contact, σ , et aux dimensions de

max max

la zone de contact entre un élément roulant et les chemins de roulement. Les relations correspondantes

s'expriment comme suit:
τσ = T
omax
zb = ζ
1/2
( 21t − )
21 tt( + )
ζ =
1/2
( tt+− )11 (2 )
1/3
 
a = μ
 
E ∑ρ
 
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ISO/TR 1281-1:2021(F)
1/3
 
bv =
 
E ∑ρ
 
σ est la contrainte maximale de contact;
max
t est le paramètre auxiliaire;
a est le demi grand axe de l'ellipse de contact projetée;
b est le demi petit axe de l'ellipse de contact projetée;
Q est la force normale entre l'élément roulant et les chemins de roulement;
E est le module d'élasticité;
Σρ est la somme des courbures;
µ, v sont des facteurs introduits par Hertz.

En conséquence, pour un roulement donné, τ , a, l et z peuvent s'exprimer en fonction de la géométrie

o o

du roulement, de la charge et du nombre de tours. La Formule (3) devient une formule si l'on y introduit

une constante de proportionnalité. En supposant un nombre déterminé de tours (par exemple 10 ) et

une fiabilité également déterminée (par exemple 0,9), la formule peut être résolue pour une charge

sur l'élément roulant correspondant à la charge dynamique de base sur le roulement. Pour un contact

ponctuel et en désignant par A la constante de proportionnalité, cette charge s'exprime par:

04, 1
(1,591ch+−,415,82)/(ch−+2)
 
13, 2r (1γγ)
QA= ×
 
C 1
()22ch+− /(ch−+2) 32ec/()−+h 3e/(ch−+2)
2rD−
40,5 (1±γ)
 
(6)
3/(()ch−+2
 
(2ch+−52)/()ch−+ −−3/ec()h+2
 
cosα 

Q est la charge sur l'élément roulant correspondant à la charge dynamique de base du

roulement;
D est le diamètre de bille;
γ est D cos α/D
w pw

D est le diamètre primitif,
α est l'angle nominal de contact;
Z est le nombre de billes par rangée.
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ISO/TR 1281-1:2021(F)

La charge radiale dynamique de base, C , d'une bague tournante s'obtient comme suit:

CQ== Z cos αα 0,407 QZ cos (7)
11C C1
La charge radiale dynamique de base, C , d'une bague fixe s'obtient comme suit:
CQ== Z cosc αα0,389 QZ os (8)
22C C 2

Q est la charge sur l'élément roulant correspondant à la charge dynamique de base

d'une bague tournante par rapport à la charge appliquée;

Q est la charge sur l'élément roulant correspondant à la charge dynamique de base

d'une bague fixe par rapport à la charge appliquée;
J = J (0,5) est l'intégrale de la charge radiale (voir Tableau 3);
r r

J = J (0,5) est le facteur rapportant à Q la charge moyenne équivalente sur une bague tour-

1 1 max
nante par rapport à la charge appliquée (voir Tableau 3);

J = J (0,5) est le facteur rapportant à Q la charge moyenne équivalente sur une bague fixe

2 2 max
par rapport à la charge appliquée (voir Tableau 3).

La relation entre C , pour un roulement radial à billes complet, et C et C s'exprime selon la loi du

r 1 2
produit des probabilités:
−−3/(2ch+ )
(2ch−+ )/3
 
 
 
CC=+1 (9)
 
r 1
 C 
 
 

Si l'on remplace les termes de la Formule (9) par leurs valeurs données dans les Formules (6), (7) et (8),

la charge radiale dynamique de base, C , d'un roulement à billes complet s'exprime de la façon suivante:

004, 1
(1,59 ch+− 1,41 5,82)/(ch−+2)
 
13, ()1−γ
3/(()ch−+2
CA=04, 1 γ ×
 
r 1
()22ch+− /(ch−+23)/ec()−+h 2 3/ec()−+h 2
2rD−
40 ,5 (1+γ )
 iw 
−33/()ch−+2
()ch−+2 /3
 04, 1 
(1,59 c +++ 1,41 he 32−−5,82)/(ch+ )
 
2rD−
r  
1−γ
   
 i ew 
11+ ,04 ×
    
 
 
r 2rD− 1+γ
 
    
e iw
 
 
 
()ch−−12/(ch−+ )(ch−−32ec+−)/()hc++22()hc−−5 /( hh+2)
( iZcos α ) D (10)
A est la constante de proportionnalité déterminée expérimentalement;

r est le rayon de courbure du chemin de roulement de la bague inférieure (en section transver-

sale);

r est le rayon de courbure du chemin de roulement de la bague extérieure (en section trans-

versale);
i est le nombre de rangées de billes.

Dans le cas considéré, l'angle de contact, α, le nombre d'éléments roulants (billes), Z, et le diamètre de

la bille, D , dépendent de la conception du roulement. Par ailleurs, le rapport des rayons de courbure, r

w i
© ISO 2021 – Tous droits réservés 7
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ISO/TR 1281-1:2021(F)

et r , au demi-diamètre de l'élément roulant (bille), D /2 et λ = D cos α/D , sont des grandeurs sans

e w w pw

dimension. Il est donc commode, dans la pratique, de remplacer les premiers termes du membre de

droite de la Formule (10) par un facteur f :
(1ch−− )/()ch−+22(3ch−− ec+−2)/( hc++) (2 h−5/)/(ch−+2)
Cf= ( iZcos α ) D (11)
rc w
Dans le cas de roulements radiaux à billes,
...

TECHNICAL ISO/TR
REPORT 1281-1
Second edition
Rolling bearings — Explanatory notes
on ISO 281 —
Part 1:
Basic dynamic load rating and basic
rating life
Roulements — Notes explicatives sur l'ISO 281 —
Partie 1: Charges dynamiques de base et durée nominale de base
PROOF/ÉPREUVE
Reference number
ISO/TR 1281-1:2021(E)
ISO 2021
---------------------- Page: 1 ----------------------
ISO/TR 1281-1:2021(E)
COPYRIGHT PROTECTED DOCUMENT
© ISO 2021

All rights reserved. Unless otherwise specified, or required in the context of its implementation, no part of this publication may

be reproduced or utilized otherwise in any form or by any means, electronic or mechanical, including photocopying, or posting

on the internet or an intranet, without prior written permission. Permission can be requested from either ISO at the address

below or ISO’s member body in the country of the requester.
ISO copyright office
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Phone: +41 22 749 01 11
Email: copyright@iso.org
Website: www.iso.org
Published in Switzerland
ii PROOF/ÉPREUVE © ISO 2021 – All rights reserved
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ISO/TR 1281-1:2021(E)
Contents Page

Foreword ........................................................................................................................................................................................................................................iv

Introduction ..................................................................................................................................................................................................................................v

1 Scope ................................................................................................................................................................................................................................. 1

2 Normative references ...................................................................................................................................................................................... 1

3 Terms and definitions ..................................................................................................................................................................................... 1

4 Symbols .......................................................................................................................................................................................................................... 1

5 General ............................................................................................................................................................................................................................ 4

6 Basic dynamic load rating ........................................................................................................................................................................... 4

6.1 General ........................................................................................................................................................................................................... 4

6.2 Basic dynamic radial load rating, C , for radial ball bearings .......................................................................... 5

6.3 Basic dynamic axial load rating, C , for single row thrust ball bearings ................................................ 9

6.3.1 Thrust ball bearings with contact angle α ≠ 90° ................................................................................... 9

6.3.2 Thrust ball bearings with contact angle α = 90° ................................................................................... 9

6.4 Basic dynamic axial load rating, C , for thrust ball bearings with two or more rows

of balls .........................................................................................................................................................................................................10

6.5 Basic dynamic radial load rating, C , for radial roller bearings ..................................................................11

6.6 Basic dynamic axial load rating, C , for single row thrust roller bearings ........................................13

6.6.1 Thrust roller bearings with contact α ≠ 90° ..........................................................................................13

6.6.2 Thrust roller bearings with contact angle α = 90° ...........................................................................13

6.7 Basic dynamic axial load rating, C , for thrust roller bearings with two or more

rows of rollers .......................................................................................................................................................................................14

7 Dynamic equivalent load ..........................................................................................................................................................................16

7.1 Expressions for dynamic equivalent load .....................................................................................................................16

7.1.1 Theoretical dynamic equivalent radial load, P , for single row radial bearings .....16

7.1.2 Theoretical dynamic equivalent radial load, P , for double row radial bearings ...20

7.1.3 Theoretical dynamic equivalent radial load, P , for radial contact groove

ball bearings .....................................................................................................................................................................21

7.1.4 Practical expressions for dynamic equivalent radial load, P , for radial

bearings with constant contact angle .........................................................................................................22

7.1.5 Practical expressions for dynamic equivalent radial load, P , for radial ball

bearings ................................................................................................................................................................................25

7.1.6 Practical expressions for dynamic equivalent axial load, P , for thrust bearings .26

7.2 Factors X, Y and e ................................................................................................................................................................................ 28

7.2.1 Radial ball bearings ....................................................................................................................................................28

7.2.2 Values of X, Y and e for each type of radial ball bearing ..............................................................29

7.2.3 Tabulation of factors X, Y and e for radial ball bearings ..............................................................34

7.2.4 Calculated values of Y and e different from standard ...................................................................36

7.2.5 Thrust ball bearings ...................................................................................................................................................36

7.2.6 Radial roller bearings ...............................................................................................................................................37

7.2.7 Thrust roller bearings ..............................................................................................................................................38

8 Basic rating life ....................................................................................................................................................................................................38

Bibliography .............................................................................................................................................................................................................................41

© ISO 2021 – All rights reserved PROOF/ÉPREUVE iii
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ISO/TR 1281-1:2021(E)
Foreword

ISO (the International Organization for Standardization) is a worldwide federation of national standards

bodies (ISO member bodies). The work of preparing International Standards is normally carried out

through ISO technical committees. Each member body interested in a subject for which a technical

committee has been established has the right to be represented on that committee. International

organizations, governmental and non-governmental, in liaison with ISO, also take part in the work.

ISO collaborates closely with the International Electrotechnical Commission (IEC) on all matters of

electrotechnical standardization.

The procedures used to develop this document and those intended for its further maintenance are

described in the ISO/IEC Directives, Part 1. In particular, the different approval criteria needed for the

different types of ISO documents should be noted. This document was drafted in accordance with the

editorial rules of the ISO/IEC Directives, Part 2 (see www .iso .org/ directives).

Attention is drawn to the possibility that some of the elements of this document may be the subject of

patent rights. ISO shall not be held responsible for identifying any or all such patent rights. Details of

any patent rights identified during the development of the document will be in the Introduction and/or

on the ISO list of patent declarations received (see www .iso .org/ patents).

Any trade name used in this document is information given for the convenience of users and does not

constitute an endorsement.

For an explanation of the voluntary nature of standards, the meaning of ISO specific terms and

expressions related to conformity assessment, as well as information about ISO’s adherence to the

World Trade Organization (WTO) principles in the Technical Barriers to Trade (TBT), see www .iso .org/

iso/ foreword .html.

This document was prepared by Technical Committee ISO/TC 4, Rolling bearings, Subcommittee SC 8,

Load ratings and life.

This second edition cancels and replaces the Technical Corrigendum 1 (ISO/TR 1281-1:2008/

Cor 1:2009) and the first edition (ISO/TR 1281-1:2008), which has been technically revised.

The main changes compared to the previous edition are as follows:

— The old Clause 7 “Life adjustment factor for reliability” of ISO/TR 1281-1:2008 has been deleted, this

subject is covered in ISO/TR 1281-2 (see ISO/TR 1281-1:2008/Cor 1:2009).

— The derivation of the old Formulae (29) and (46) [Formulae (28) and (45) in this edition] has been

corrected.

— Typing errors have been corrected in Formulae (30) and (31) and in the derivation of the factor Y .

A list of all parts in the ISO/TR 1281 series can be found on the ISO website.

Any feedback or questions on this document should be directed to the user’s national standards body. A

complete listing of these bodies can be found at www .iso .org/ members .html.
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ISO/TR 1281-1:2021(E)
Introduction
I S O/ R 281: 1962

A first discussion on an international level of the question of standardizing calculation methods for

load ratings of rolling bearings took place at the 1934 conference of the International Federation of

the National Standardizing Associations (ISA). When ISA held its last conference in 1939, no progress

had been made. However, in its 1945 report on the state of rolling bearing standardization, the ISA 4

Secretariat included proposals for definition of concepts fundamental to load rating and life calculation

standards. The definitions it contained are in essence those given in ISO 281:2007 for the concepts

“life” and “basic dynamic load rating” (now divided into “basic dynamic radial load rating” and “basic

dynamic axial load rating”).

In 1946, on the initiative of the Anti-Friction Bearing Manufacturers Association (AFBMA), New York,

discussions of load rating and life calculation standards started between industries in the USA and

Sweden. Chiefly on the basis of the results appearing in Reference [5], an AFBMA standard, Method of

[3]

evaluating load ratings of annular ball bearings , was worked out and published in 1949. On the same

basis, the member body for Sweden presented, in February 1950, a first proposal to ISO, “Load rating of

ball bearings”.

In view of the results of both further research and a modification to the AFBMA standard in 1950, as

well as interest in roller bearing rating standards, in 1951, the member body for Sweden submitted a

modified proposal for rating of ball bearings as well as a proposal for rating of roller bearings.

Load rating and life calculation methods were then studied. Reference [6] was then of considerable use,

serving as a major basis for the sections regarding roller bearing rating.
ISO 281-1:1977

In 1964, in view of the development of improved bearing steels, the time had come to review ISO/R281

and submitted a proposal

In 1969, on the other hand, TC 4 followed a suggestion by the member body for Japan and reconstituted

its WG 3, giving it the task of revising ISO/R281. The AFBMA load rating working group had at this time

started revision work.

The major part of ISO 281-1:1977 constituted a re-publication of ISO/R281, the substance of which had

been only very slightly modified. However, based mainly on American investigations during the 1960s,

a new clause was added, dealing with adjustment of rating life for reliability other than 90 % and for

material and operating conditions.

Furthermore, supplementary background information regarding the derivation of mathematical

expressions and factors given in ISO 281-1:1977 was published as ISO/TR 8646:1985.

ISO 281:1990

ISO 281:1990 was published as “First edition” and entitled “Dynamic load ratings and rating life”. It is

referred to as the “technical revision” of ISO 281-1:1977. The new rating factor b for “contemporary,

normally used material and manufacturing quality, the value of which varies with bearing type and

design” was the introduction as a co-value to the basic dynamic load ratings.
ISO 281:2007 (second edition)

Since the publication of ISO 281:1990 additional knowledge regarding the influence on bearing life of

contamination, lubrication, internal stresses from mounting, stresses from hardening, fatigue load

limit of the material, has been gained. In ISO 281:1990/Amd 2:2000, a general method was presented to

consider such influences in the calculation of a modified rating life of a bearing. The said Amendment

was incorporated into the second edition, which also provides a practical method to consider the

influence on bearing life of lubrication conditions, contaminated lubricant and fatigue load of bearing

© ISO 2021 – All rights reserved PROOF/ÉPREUVE v
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ISO/TR 1281-1:2021(E)

material. The life modification factors for reliability, a , have been slightly adjusted and extended to

99,95 % reliability.
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TECHNICAL REPORT ISO/TR 1281-1:2021(E)
Rolling bearings — Explanatory notes on ISO 281 —
Part 1:
Basic dynamic load rating and basic rating life
1 Scope

This document specifies supplementary background information regarding the derivation of

mathematical expressions and factors given in ISO 281:2007.
2 Normative references

The following documents are referred to in the text in such a way that some or all of their content

constitutes requirements of this document. For dated references, only the edition cited applies. For

undated references, the latest edition of the referenced document (including any amendments) applies.

ISO 281:2007, Rolling bearings — Dynamic load ratings and rating life
3 Terms and definitions

For the purposes of this document, the terms and definitions given in ISO 281:2007 apply.

ISO and IEC maintain terminological databases for use in standardization at the following addresses:

— ISO Online browsing platform: available at https:// www .iso .org/ obp
— IEC Electropedia: available at http:// www .electropedia .org/
4 Symbols
A constant of proportionality
A constant of proportionality determined experimentally
B constant of proportionality determined experimentally
C basic dynamic radial load rating of a rotating ring
C basic dynamic radial load rating of a stationary ring
C basic dynamic axial load rating for thrust ball or roller bearing

C basic dynamic axial load rating of the rotating ring of an entire thrust ball or roller bearing

C basic dynamic axial load rating of the stationary ring of an entire thrust ball or roller bearing

C basic dynamic axial load rating as a row k of an entire thrust ball or roller bearing

C basic dynamic axial load rating as a row k of the rotating ring of thrust ball or roller bearing

a1k

C basic dynamic axial load rating as a row k of the stationary ring of thrust ball or roller bearing

a2k
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ISO/TR 1281-1:2021(E)
C basic dynamic load rating for outer ring
C basic dynamic load rating for inner ring
C basic dynamic radial load rating for radial ball or roller bearing
D pitch diameter of ball or roller set
D ball diameter
D mean roller diameter
E modified modulus of elasticity
F axial load
F radial load
J factor relating mean equivalent load on a rotating ring to Q
1 max
J factor relating mean equivalent load on a stationary ring to Q
2 max
J axial load integral
J radial load integral
L bearing life
L basic rating life
L effective contact length of roller
L L per row k
wek we
N number of stress applications to a point on the raceway
P dynamic equivalent axial load for thrust bearing
P dynamic equivalent radial load for radial bearing
P dynamic equivalent radial load for the rotating ring
P dynamic equivalent radial load for the stationary ring
Q normal force between a rolling element and the raceways
Q rolling element load for the basic dynamic load rating of the bearing

Q rolling element load for the basic dynamic load rating of a ring rotating relative to the applied load

Q rolling element load for the basic dynamic load rating of a ring stationary relative to the applied load

Q maximum rolling element load
max
S probability of survival, reliability
V volume representative of the stress concentration
V rotation factor
X radial load factor for radial bearing
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ISO/TR 1281-1:2021(E)
X radial load factor for thrust bearing
Y axial load factor for radial bearing
Y axial load factor for thrust bearing
Z number of balls or rollers per row
Z number of balls or rollers per row k
a semimajor axis of the projected contact ellipse
a life adjustment factor for reliability
b semiminor axis of the projected contact ellipse
c exponent determined experimentally
c compression constant
e measure of life scatter, i.e. Weibull slope determined experimentally

e limiting value of F / F for the applicability of different values of factors X and Y in the new edition

a r

f factor which depends on the geometry of the bearing components, the accuracy to which the

various components are made, and the material
h exponent determined experimentally
i number of rows of balls or rollers
l circumference of the raceway
r cross-sectional raceway groove radius
r cross-sectional raceway groove radius of outer ring or housing washer
r cross-sectional raceway groove radius of inner ring or shaft washer
t auxiliary parameter
z depth of the maximum orthogonal subsurface shear stress
α nominal contact angle
α′ actual contact angle
γ D cos α/D for ball bearings with α ≠ 90°
w pw
D /D for ball bearings with α = 90°
w pw
D cos α/D for roller bearings with α ≠ 90°
we pw
D /D for roller bearings with α = 90°
we pw
ε parameter indicating the width of the loaded zone in the bearing
η reduction factor
λ reduction factor
µ factor introduced by Hertz
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ISO/TR 1281-1:2021(E)
ν factor introduced by Hertz, or adjustment factor for exponent variation
σ maximum contact stress
max
Σρ curvature sum
τ maximum orthogonal subsurface shear stress
φ one half of the loaded arc
5 General

The derivation of the basic dynamic load ratings is described in Formulae (1) to (46). The dynamic

equivalent load and the radial and axial load factors are covered in Formulae (47) to (82), while basic

rating life is described in Formulae (83) to (89).
6 Basic dynamic load rating
6.1 General

The background to basic dynamic load ratings of rolling bearings according to ISO 281 appears in

References [5] and [6].

The expressions for calculation of basic dynamic load ratings of rolling bearings develop from a power

formula that can be written as follows:
τ NV
ln ∝ (1)
where
S is the probability of survival;
τ is the maximum orthogonal subsurface shear stress;
N is the number of stress applications to a point on the raceway;
V is the volume representative of the stress concentration;
z is the depth of the maximum orthogonal subsurface shear stress;
c, h are experimentally determined exponents;

e is the measure of life scatter, i.e. the Weibull slope determined experimentally.

For “point” contact conditions (ball bearings) it is assumed that the volume, V, representative of the

stress concentration in Formula (1) is proportional to the major axis of the projected contact ellipse,

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ISO/TR 1281-1:2021(E)

2a, the circumference of the raceway, l, and the depth, z , of the maximum orthogonal subsurface shear

stress, τ
Va ∝ 2 zl (2)
Substituting Formula (2) into Formula (1):
τ Nal
ln ∝ (3)
h−1

“Line” contact was considered in References [5] and [6] to be approached under conditions where the

major axis of the calculated Hertz contact ellipse is 1,5 times the effective roller contact length:

21aL = ,5 (4)

In addition, b/a should be small enough to permit the introduction of the limit value of ab as b/a

approaches 0:
23Q
ab = (5)
π E ∑ρ
(for variable definitions, see 6.2).
6.2 Basic dynamic radial load rating, C , for radial ball bearings

From the theory of Hertz, the maximum orthogonal subsurface shear stress, τ , and the depth, z , can

o o

be expressed in terms of a radial load F , i.e. a maximum rolling element load, Q , or a maximum

r max

contact stress, σ , and dimensions for the contact area between a rolling element and the raceways.

max
The relationships are:
τσ = T
omax
zb = ζ
1/2
( 21t − )
21 tt( + )
ζ =
1/2
( tt+− )11 (2 )
1/3
 
a = μ
 
E ∑ρ
 
1/3
 
bv =
 
E ∑ρ
 
where
σ is the maximum contact stress;
max
t is the auxiliary parameter;
a is the semimajor axis of the projected contact ellipse;
© ISO 2021 – All rights reserved PROOF/ÉPREUVE 5
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ISO/TR 1281-1:2021(E)
b is the semiminor axis of the projected contact ellipse;
Q is the normal force between a rolling element and the raceways;
E is the modified modulus of elasticity;
Σρ is the curvature sum;
µ, v are factors introduced by Hertz.

Consequently, for a given rolling bearing, τ , a, l and z can be expressed in terms of bearing geometry,

o o

load and revolutions. Formula (3) is changed to a formula by inserting a constant of proportionality.

Inserting a specific number of revolutions (e.g. 10 ) and a specific reliability (e.g. 0,9), the formula is

solved for a rolling element load for basic dynamic load rating which is designated to point contact

rolling bearings introducing a constant of proportionality, A :
04, 1
(1,591ch+−,415,82)/(ch−+2)
 
13, 2r (1γγ)
QA= ×
 
C 1
()22ch+− /(ch−+2) 32ec/()−+h 3e/(ch−+2)
2rD−
40,5 (1±γ)
 
(6)
3/(()ch−+2
 
(2ch+−52)/()ch−+ −−3/ec()h+2
 
cosα 
where
Q is the rolling element load for the basic dynamic load rating of the bearing;
D is the ball diameter;
γ is D cos α/D ;
w pw
in which
D is the pitch diameter of the ball set;
α is the nominal contact angle;
Z is the number of balls per row.
The basic dynamic radial load rating, C , of a rotating ring is given by:
CQ== Z cos αα 0,407 QZ cos (7)
11C C1
The basic dynamic radial load rating, C , of a stationary ring is given by:
CQ== Z cosc αα0,389 QZ os (8)
22C C 2
where
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ISO/TR 1281-1:2021(E)

Q is the rolling element load for the basic dynamic load rating of a ring rotating relative

to the applied load;

Q is the rolling element load for the basic dynamic load rating of a ring stationary rel-

ative to the applied load;

J = J (0,5) is the radial load integral for zero diametral clearance (see Table 3);

r r

J = J (0,5) is the factor relating mean equivalent load on a rotating ring to Q for zero diametral

1 1 max
clearance (see Table 3);

J = J (0,5) is the factor relating mean equivalent load on a stationary ring to Q for zero dia-

2 2 max
metral clearance (see Table 3).

The relationship between C for an entire radial ball bearing, and C and C , is expressed in terms of the

r 1 2
product law of probability as:
−−3/(2ch+ )
(2ch−+ )/3
 
 
 
CC=+1 (9)
 
r 1
 
 
 

Substituting Formulae (6), (7) and (8) into Formula (9), the basic dynamic radial load rating, C , for an

entire ball bearing is expressed as:
004, 1
(1,59 ch+− 1,41 5,82)/(ch−+2)
 
13, ()1−γ
3/(()ch−+2
CA=04, 1 ×
 
r 1
()22ch+− /(ch−+23)/ec()−+h 2 3/ec()−+h 2
2rD−
40 ,5 (1+γ )
 
−33/()ch−+2
()ch−+2 /3
 
04, 1
(1,59 c +++ 1,41 he 32−−5,82)/(ch+ )
 
r 2rD−
  
  1−γ  
 i ew 
11+ ,04 ×
 
  
 
 
r 2rD− 1+γ
 
  
 e iw 
 
 
 
()ch−−12/(ch−+ )(ch−−32ec+−)/()hc++22()hc−−5 /( hh+2)
( iZcos α ) D (10)
where
A is the experimentally determined proportionality constant;
r is the cross-sectional raceway groove radius of the inner ring;
r is the cross-sectional raceway groove radius of the outer ring;
i is the number of rows of balls.

Here, the contact angle, α, the number of rolling elements (balls), Z, and the ball diameter, D , depend

on bearing design. On the other hand, the ratios of raceway groove radii, r and r , to a half-diameter of

i e

a rolling element (ball), D /2 and γ = D cos α/D , are not dimensional, therefore it is convenient in

w w pw

practice that the value for the initial terms on the right-hand side of Formula (10) to be designated as a

factor, f :
(1ch−− )/()ch−+22(3ch−− ec+−2)/( hc++) (2 h−5/)/(ch−+2)
Cf= ( iZcos α ) D (11)
rc w

With radial ball bearings, the faults in bearings resulting from manufacturing need to be taken into

consideration, and a reduction factor, λ, is introduced to reduce the value for a basic dynamic radial load

© ISO 2021 – All rights reserved PROOF/ÉPREUVE 7
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ISO/TR 1281-1:2021(E)

rating for radial ball bearings from its theoretical value. It is convenient to include λ in the factor, f . The

value of λ is determined experimentally.
04, 1
(1,591ch+−,415,82)/(ch−+2)
 
13, ()1−γ
3//(ch−+2)
fA=04, 1 λ γ ×
 
c 1
()22ch+− /(ch−+23)/ec()−+h 2 32ec/( −+h )
2rD−
40,5 (1+γ )
 
−−3/(c hh+2)
()ch−+2 /3
 04, 1 
(1,59c+11,41he+−325,82)/(ch−+ )
 
r 2rD−
  
 1−γ  
 i ew 
11+ ,04 (12)
    
 
 
r 2rD− 1+γ
 
 
  e iw  
 
 
 

Based on References [5] and, [6] the following values were assigned to the experimental constants in

the load rating formulae for ball bearings:

Substituting the numerical values into Formula (11) gives the following, however, a sufficient number of

test results are only available for small balls, i.e. up to a diameter of 25,4 mm (1 inch), and these show

1,8

that the load rating may be taken as being proportional to D . In the case of larger balls, the load

1,4

rating appears to increase even more slowly in relation to the ball diameter, and D can be assumed

where D > 25,4 mm:
07,,2/3 18
Cf= (iZ cos )α D
rc w
for D ≤ 25,4 mm (13)
0,72/3 1,4
Cf= 3,647 (iZ cos )α D
rc w
for D > 25,4 mm (14)
04, 1
03,,139
 2r 
γγ()1−
fA= 0,,089 041 λ ×
 
1//3
2rD−
(1+γ )
 
−31/ 0 (15)
10/3
 
04, 1
17, 2
 
r 2rD− 
  1−γ
   
 i ew 
11+ ,04
 
  
 
 
r 2rD− 1+γ
 
  
 e iw 
 
 
 

Values of f in ISO 281:2007, Table 2, are calculated by substituting raceway groove radii and reduction

factors given in Table 1 into Formula (15).
The value for 0,089A is 98,066 5 to calculate C in newtons.
1 r
8 PROOF/ÉPREUVE © ISO 2021 – All rights reserved
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ISO/TR 1281-1:2021(E)
6.3 Basic dynamic axial load rating, C , for single row thrust ball bearings
6.3.1 Thrust ball bearings with contact angle α ≠ 90°
As in 6.2, for thrust ball bearings with contact angle α ≠ 90°:
(1ch−− )/(2ch−+ )(ch−−32ec+−)/(2hc++) (2 hhc−−5)/( h+2)
Cf= (cos )ααtan ZD (16)
ac w

For most thrust ball bearings, the theoretical value of a basic dynamic axial load rating has to be reduced

on the basis of unequal distribution of load among the rolling elements in addition to the reduction factor,

λ, which is introduced in to radial ball bearing load ratings. This reduction fa
...

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