ISO/TR 10657:1991
(Main)Explanatory notes on ISO 76
Explanatory notes on ISO 76
Notes explicatives sur l'ISO 76
Zapisek razlag k standardu ISO 76
General Information
Relations
Standards Content (Sample)
SLOVENSKI STANDARD
SIST ISO/TR 10657:2001
01-julij-2001
Zapisek razlag k standardu ISO 76
Explanatory notes on ISO 76
Notes explicatives sur l'ISO 76
Ta slovenski standard je istoveten z: ISO/TR 10657:1991
ICS:
21.100.20 Kotalni ležaji Rolling bearings
SIST ISO/TR 10657:2001 en
2003-01.Slovenski inštitut za standardizacijo. Razmnoževanje celote ali delov tega standarda ni dovoljeno.
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SIST ISO/TR 10657:2001
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SIST ISO/TR 10657:2001
TECHNICAL lSO/TR
REPORT 10657
First edition
1991-05-15
Explanatory notes on IS0 76
Notes explicatives SW /‘/SO 76
Reference number
ISO/TR 10657 : 1991 (E)
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SIST ISO/TR 10657:2001
ISO/TR 10657 : 1991 (E)
Contents
1 Scope. 1
2 BriefHistory . 1
2.1 ISO/R76-1958 . 1
2.2 ISO76-1978 . 2
2.3 ISO76-1987 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
............................................. 6
3 Basic Static Load Ratings
3.1 Basic static radial load rating &,-for radial ball bearings. . 10
3.1.1 Radial and angular contact groove ball bearings. . 10
3.1.2 Self-aligning ball bearings . 12
14
3.2 Basic static axial load rating C,, for thrust ball bearings. .
.............. 15
3.3 Basic static radial load rating &for radial roller bearings
.............. 16
3.4 Basic static axial load rating C,, for thrust roller bearings
................................................ 17
4 Static Equivalent Load
4.1 Theoretical static equivalent radial load P,, for radial bearings . 17
4.1 .I Single row radial bearings and radial contact groove ball bearings 17
.................................
4.1.2 Double row radial bearings 25
.......... 28
4.2 Theoretical static equivalent axial load P,, for thrust bearings
............................. 28
4.2.1 Single direction thrust bearings
4.2.2 Double direction thrust bearings. . 30
32
4.3 Approximate formulae for theoretical static equivalent load .
4.3.1 Radial bearings . 32
........................................... 34
4.3.2 Thrust bearings
0 IS0 1991
All rights reserved. No part of this publication may be reproduced or utilized in any form or by any
means, electronic or mechanical, including photocopying and microfilm, without permission in
writing from the publisher.
International Organization for Standardization
Case postale 56 l CH-1211 Geneve 20 l Switzerland
Printed in Switzerland
ii
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SIST ISO/TR 10657:2001
ISO/TR 10657 : 1991 (E)
Practical formulae of static equivalent load 35
4.4 .
35
4.4.1 Radial bearings .
.“. 40
4.4.2 Thrust bearings . .
Static radial load factor X, and static axial load factor Y,. . 42
4.5
42
4.5.1 Radial bearings
..............................................
42
4.5.1.1 Radial contact groove ball bearings .
44
4.5.1.2 Angular contact groove ball bearings. .
....... 47
4.5.1.3 Self-aligning ball bearings and radial roller bearings.
4.5.2 Thrust bearings 48
..............................................
Annexes
50
Valuesfory,xandE(x). .
AnnexA
53
AnnexB Symbols .
56
AnnexC References .
. . .
III
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SIST ISO/TR 10657:2001
ISO/TR 10657 : 1991 (E)
rewo rd
Fo
IS0 (the International Organization for Standardization) is a worldwide federation of
national standards bodies (IS0 member bodies). The work of preparing International
Standards is normally carried out through IS0 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, govern-
mental and non-governmental, in liaison with ISO, also take part in the work. IS0
collaborates closely with the International Electrotechnical Commission (IEC) on all
matters of electrotechnical standardization.
The main task of IS0 technical committees is to prepare International Standards. In
exceptional circumstances a technical committee may propose the publication of a
Technical Report of one of the following types:
-
type 1, when the required support cannot be obtained for the publication of an
International Standard, despite repeated efforts;
-
type 2, when the subject is still under technical development or where-for any
other reason there is the future but not immediate possibility of an agreement on an
International Standard;
-
type 3, when a technical committee has collected data of a different kind from
that which is normally published as an International Standard (“state of the art”, for
example).
Technical Reports of types 1 and 2 are subject to review within three years of publi-
cation, to decide whether they can be transformed into International Standards.
Technical Reports of type 3 do not necessarily have to be reviewed until the data they
provide are considered to be no longer valid or useful.
ISO/TR 10657, which is a Technical Report of type 3, was prepared by Technical Com-
mittee ISO/TC 4, Rolling bearings, Sub-Committee SC 8, Load ratings and life.
ISO/TR 10657 has been prepared for the guidance of users of IS0 76 : 1987, Rofing
bearings - Static load ratings. It is a purely scientific document intended for use by
specialists in this field, and it is not envisaged that it will become an International Stan-
dard.
Annexes A, B and C form an integral part of this Technical Report.
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SIST ISO/TR 10657:2001
TECHNICAL REPORT ISO/TR 10657 : 1991 (E)
Explanatory notes on IS0 76
1 Scope
This Technical Report gives supplementary background information
regarding the derivation of formulae and factors given in IS0 76,
Rolling bearings - Static load ratings.
2 Brief History
ISO/R 76 - 1958
2.1
The IS0 Recommendation R 76, Ball and Roller Bearings - Methods of
Evaluating Static Load Ratings, was drawn up by Technical Committee
ISO/TC 4, Ball and Holler Bearings, the Secretariat of which is held
-_.-
by the Sveriges
Standardiseringskownission (SIS).
This Recommendation is based on the studies[l]*, [Z] by A. Palmgren
and so on. It is defined in the Recommendation that the basic static
load ratings correspond to a total permanent deformation of rolling
elemenk and raceway at the most heavily stressed rolling element/
raceway contact ef 0,OOOl of the rolling element diameter. And then
the Standard values confined to the basic static load ratings for
special inner design rolling bearings are laid down,
*
Figures in brackets indicate literature references in annex C.
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SIST ISO/TR 10657:2001
ISO/TR 10657 : 1991 (E)
Technical Committee ISO/TC 4 discussed the questions dealt with by
the IS0 Recommendation at the following meetings:
the third meeting, held in Gsteborg, in September 1953,
the fourth meeting, held in Madrid, in May 1955,
the fifth meeting, held in Vienna, in September 1956.
At the third meeting of the Techrrical Committee, Working Group No.
3 was appointed to assist the ISO/TC 4 Secretariat in preparatory
work and in drawing up proposals.
The Working Group composed of
Germany, Sweden and USA held the following meetings:
the first meeting, held in Madrid, in May 1955,
the second meeting, held in Vienna, in September 1956.
On 4th June 1957, the Draft IS0 Recommendation was sent out to all
the IS0 Member Bodies and was approved by 28 (out of a total of 58)
Member Bodies.
The Draft IS0 Recommendation was then submitted by correspondence
to the IS0 Council, which decided, in December 1958, to accept it
as an IS0 Recommendation.
2.2 IS0 76
- 1978
The Working Group P(o. 3 was transformed into SC I at the 13th meeting
helci in Paris in May 1972. The: SC 8 Secretariat proposed to
include the revision of ISO/R 76 in the future work a.t the first
meeting held in London in November 1973, and SC 8 requested its
a Draft Proposal for an International Stan-
Secretariat to prepare
dard replacing ISO/R 76 and it was decided that this proposal should
be submitted to the SC 8 member Bodies for consideration prior to
1 October 1974 (SC 8 aESOLUTI0N 21, 21 London 1973).
2
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SIST ISO/TR 10657:2001
lSO/TR 10657 : 1991 (E)
The SC 8 Secretariat distributed & DRAFT PROPOSAL (Revision of
ISO/Ez 76) in July 1974.
The TC 4 decided to include the revision of ISO/ 76 in its pro-
gramme of work (TC 4 RESOLUTION 514, item No. 132) and SC 8 Sec-
retariat was requested to prepare a Draft Proposal (SC 8 HESOLUTION
38, 13 Mami Beach 1974). As a result, the Secretariat submitted
a DP [ 31 in January 1.976.
The Draft Pmprjsal DP 76 Was accepted by correspondence by 6 of the
9 Membws OS SC 8, Of the remaining two, Japan would prefer further
study and USA its counter proposal 9 document 418 N 64 [4] l The DIS
76 ~3s then st\bifri tted to the ISO Central Secretariat. After the DIS
had teen appmmd by the IS0 Mernbev brjdies, the IS0 Coun‘cil decided
i1-t June 1978 to accept it as an Intmmational Standard,
This Standwd adqked the SI unit ne&on and was revised in total,
but i4i ttmut: e55enti al changes af substance. However z values of x
0
0
and Y. for the naMna1 cantac,t angles 13 and 45 far angular COP
tact groove ball bearings were added ta the table whirh ShoWs the
values of Xo and Yo in the for-mulae to calculate the static equiva-
lent radial
lmds o-f radi ~1 ball bearings.
I!33 7b - 1’387
2.3
Duriflg the mtir,ion of 1533/R 76 - 1958, USA had in 1975 submitted a
4
counter proposal for the basic static load ratings based on a
c1
calm1 ated contact stress,
3
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SIST ISO/TR 10657:2001
ISO/TR 10657 : 1991 (E)
The Secmtwiat requested a vote an the wvi5ion of the static load
ratings based on a contact strr?ss level in January 1978 and after-
ward circulated the voted results in June 1978, and the item No.
of revision work had became No. 157 of the programme of work of
TC 4.
This Draft Proposal DP 76 was dealt with at the following TC 4
meetings:
15th meeting,
held in Moscow, in April 1977,
16th meeting,
held in London, in November 1979,
17 th meeting, held in Budapest, in May 1983,
and then, dealt with at the following SC 8 meetings:
the third meeting, held in London, in November 1979,
the fourth meeting, held in Budapest, in May 1983i
the fifth meeting, held in Arlington, in November 1984.
The following resolutions for the contents of the Standard were
adopted during the third meeting to the fifth meeting:
SC 8, considering the proposals made in the documents J/8 N 75 [5]
) as well as the comments made by TC 4 Members and
and 4 N 865 C6]
that several SC 8 Members expressed a need for updating IS0 76,
agreed to continue its study taking into account the possibility
of using either permanent deformation or stress level as a basis
for static load ratings (SC 8 RESOLUTION 45, 5 London 1979)) and
SC 8 requested its Secretariat to prepare a new draft. The new draft
should be prepared with the principles and formulae of the document
and to include levels of contact stress for various
418 N 75,
stated to be generally corresponding to
rolling element contact
4
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SIST ISO/TR 10657:2001
ISO/TR 10657 : 1991 (E)
-4
a permanent deformation of 10 D at the centre of the mast heavily
W
stressed rolling element/raceway contact. Far roller bearings a
stress level of 4000 MPa wa s agreed EC 8 REXILUTIUN 51, 4 Budapest
1%33> and then SC 8 agreed, by a majority vote, that static Ioad
ratiny~ should ccmxspand te calculated contact stresses of
4000 MPa for roller bearings,
4600 MPa for self-aligning ball bearings and
4200 MPa for all other ball bearings to which the
Standard applies (SC 8 RESOLUTION 56, 3 Arlington 1984). Moreover,
SC 8 recommended the document 4/8 N 121 [7] 9 amended in accordance
with SC 8 Resolution 56, as a revised IS0 76 (SC 8 i?ESOLUTION 57,
4 Arlington 1984).
For these calculated contact stresses, a total permanent deformation
occures at the centre of the most heavily stressed roliing element
/raceway contact, and its deformation is approximately 0,OOOl of
the rolling element diameter.
The n1s A5 has xk~~itted to the 193 Central Secretariat 1985, and
;If ter it had bmm approved by the ISG Members, the IS0 Ccxmcil de-
tided in-February VI?87 ta accept it as an Xnternatianal Standard.
Fw t hemore p SC 8 derided at its fifth meeting in April 1986 that
sLlpp1 emnt ary k3ckguaund i flf OUiUt i an, regarding the derivation of
fW3lUlX? Wd -fX3XX~ gi.ven in Ml X-3, skmuld be pub1 ished as a Tech-
I
nical Report EL r 8 FEXLlJTIt3N c- 71, 11 t-!angzhrjU 1%%d.
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SIST ISO/TR 10657:2001
ISO/TR 10657 : 1991 (E)
Basic Static Load RatinRs
3
(1) Basic equation for point contact
The relationship between
a calculated contact stress and a rolling element load within
an elliptical contact area is given as follows [8] 9
--.
36
1
(3-l)
u-
=27Tab
where
= calculated contact stress, MPa,
r
= major semi axis of the contact
a
ellipse, mm,
b = minor semi axis of the COntaCt
ellipse, mm,
= normal force between rolling element and
Q
raceway, M)
x = distanc.e in a-direction, mxn,
= distance in b-direction, mm .
Y
It is concluded that the maximum calculated contact
stress (Gax) occurs at the point of x = 0 and y = 0,
-. -
-. -.
36
2TTab
f
r
(3-2)
Q tT
max
=27Tab'
max l
3
According to the Hertz's theory,
2
l-
(2 lt2E(K) )1/3 3Q
VI
+
a=
( (3a3j
A E
c
*uJ
1
+ 1 - & v3 ,
3Q (1 - VI?
b _.(2S(K))l~3
(3-4)
E E
mt
2uJ )I
2
1
[
. ___.-. __ . .
-. --. - .--.
6
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SIST ISO/TR 10657:2001
lSO/TR 10657 : 1991 (E)
where
E(K) - complete .e.lliptic integral of the second kind
---
modulus of elasticity (Young's modulus), MPa,
By** = . . . .
._
. _
= Poisson% ratio, _. .
VW2
+p ) - . _. .
t/,=P+p +p
22
11 12 21
.--
2
N principal curvature of body 1 (ball),
P P 1-
D
11 9 21 l u
= principal curvature of body 2 (ring) tit the -point .*.
P
P
9
22
12
contrct l
Substituting equations \+T) and (3-4) into equation (5-2) for .the
. .
1= E2=
case of E E and VI= V2= y,
g-3 ~~(.~. *
s
(3-5)
1 9
Q
2
max
tP
3E
. 0
and
2
(W
1
- 'g-y E(XJ - 1) - WI - 0 n
(3-G
0
where
E
E 1111
?
0 2
1-v
5
E-2,07 @lOMPa,
v = 0,3
K(X) = complete elliptic integral of the flrat kind
?m
t
1. 42
2
o-8 n
2
4
31 1 - (1 ) i 41
0 u 2
I[
P - P
12 + p21 +22
11
Fo) =
P
11+ f)12+p21+p22 l
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SIST ISO/TR 10657:2001
ISO/TR 10657 : 1991 (E)
Consequently,
- 6,4762065 x lo
Q
(3-7)
(2) The relationship between
Basic equation for line contact
a calculated contact stress and arolling element load for s
._.-- _-. _ ,. ". .
line contact is given as follows [:9] 9
u - A;;ebF - if)*] l’* 9
(3-8)
where
= calculated contact stress, MPa )
r
b = semiwidth of the contact surface, mm I
L length of roller applicable to calculate load
we =
ratings, mm t
=
normal force between rollin+? element and raceway, N j
&
= distance in b-direction, mm .
Y
It is concluded that the maximum calculated contact stress
\Fmax) occurs at the line of y = 0,
_ 2Q
7r Lwob
CT
(3-Y)
- fALw,b ' Or ' - 2 &mx l
max
.
~
And also b is given by the following equation,
. ___.
_
2
l/2
48
v2
br
(
(3-w
m
AL~ew
)3
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SIST ISO/TR 10657:2001
ISO/TR 10657: 1991 (E)
where
c/II - q1 + P,, + n,, +
P 7
22
P m
- 2
P 1,
2 Y
11 u 9
12
+-- 1ry
9 $1
- 0 9 1322
= 0 '
D
we
W8
the upper sign applies to inner ring contact and
the lower to outer ring contact,
D = roller diameter applicable to calculate load
we
ratings, mm y
D~,coW
m
Y I
D
Pw
=
D pitch diameter of roller set, mm.
PW
Substituting equation (3-10) into equation (3.9j for the case of
=
E E *= E and Vl= v2= y,
1
L
we
Q - 27~
8
max E*J/) *
where
E
E
o=
2
1-v
E = *,07 x 105HPa2
u= 093 0
Consequently,
. -
L
Q = 2.7621732 x LOW5 -= Q-*
(3-11)
max l
ZP
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SIST ISO/TR 10657:2001
ISO/TR 10657 : 1991 (E)
0 Basic static radial load rating C for radial ball bearings
3 1
or
.- . .
Radial and angular contact groove ball bearings
3.1.1
The curvature sum Z/)and curvature difference F(p) of radial and an@=
contact groove ball bearings is given by the following equation,
2
1
Y
m
ZP
(3-W
d
2fqy2q&'
W
1
If
2
+
2f
l?:Y
c
it >
F(P) = (3-13)
2**- l .
2fi(e)
where _ .
the upper sign applies to inner ring contact and
-the lower to outer ring contact,
D II ball diameter, mm B
W
D cosa
w
Y
D
PW
D pw 05 pitch diameter of ball Bet, mm P
rr .
,
f IL )
i D
W
r
e
=-
f
t D.'
W
r
= inner ring groove radius, mm j
i
r
= outer ring groove radius, mm .
c
Substituting equation (3-12) into equation (3-T),
2
D
3
E(K)
LI 6,4762065 x lO-1o,($
(3-14)
Q )(7
max
Y
1
2*-w
lTy 2fi(e)
Substituting equations (3-12) and (3-14) into the following equation
9 furthermore exchanging Q for Qmax,
10
I: 1
1
E . ~08 a
C
a
(3-15)
raax
or S
IO
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SIST ISO/TR 10657:2001
ISO/TR 10657 : 1991 (E)
where
_
C = basic static radial load rating, Ni j
or
z = number of balls per row 2
= maximum normal force between rolling element
Q’
max
and raceway, N 9
S is a function of the loaded zone parameter &
If one half of the balls are loaded then S = 4,37 applies.
A common value used in general bearing calculations is
S
= 5, which leads to a rather conservative estimate of
the maximum ball load,
= nominal contact angle,',
a
= 0,22& cosa l (39 _.
C
.
max
or
_. -
_ . .
Consequently,
3
3
Q‘Paax
C = 0,2 x 6,4762065 x loolOx (4000) (-)u
or
4000
2
1
E(X) 2
x-
ZDwc
(' ofa
>
1
4 Y
2+
1
rY
2fi(e)
where the upper sign refers to the inner ring and the lower sign
refers to the outer ring. Therefore,
introducing the number of rows
i of balls,
2
C
* f iZD c00a
)
or
0 w (3-17 1
where
f
factor which depends on the geometry
OP
of the bearing components and on appli-
cable stress level
_---_.--. _
2
3
Finax E(K)
= 29072&j$ a
0
1
11
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SIST ISO/TR 10657:2001
ISO/TR 10657 : 1991 (E)
The formula applies to bearings with a cross-sectional raceway groove
radius not larger than 0,52 D, in radial and angular contact groove
ball bearing inner rings, and 0,53 D, in radial and angular contact
groove ball bearing outer rings and self-aligning ball bearing inner
rings.
The load-carrying ability of a bearing is not necessarily increased
by the use of a smaller groove radius, but is reduced by the use of
a correspondingly reduced
a larger groove radius. In the latter case,
value of f,
shall be used.
. _.
= 0,52,
For an inner ring with f
i
2
3
rmax
E(K)
f. = *9°72(4000/ Jt( 1 9 (3-19)
1
Y
. * + 1-y -
1,04
I. . . . . .-.- .-. _ I _.-. . . .
and for an outer ring with f
0,53,
e =
--_
E(K) 2
(3-20) 1
1 0
2 Y 1
YTji'1,66
values calculated from equations
'the smaller value between the f
0
(T-19) and (T-20) shall be adopted.
- 1987 are calculated
The values of factor f on table 1 in IS0 76
0
shown
from substituting the values for X, E(K) and y= D cos Q/D
W
PW
and r = 4200 MPa into the above equation.
in table A-1 I
max
Self-aligning ball bearings
3.12
The curvature sum r/,of self-aligning ball bearings is given by
the following equation for an outer ring,
1
4
(3-20
)
(
.v
=DVl+y l
12
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SIST ISO/TR 10657:2001
ISO/TR 10657 : 1991 (E)
Substituting equation (3-21) into equation (3-7),
. .
e = 6,4762065 x lo’loU [>(l + &(K)]2c;x 0 (3-22)
. . _ .-
In general, x= a/b =
1 for the case of contact between an outer
_-. .
ring raceway and balls of self-aligning ball bearings. Consequently,
_.
l/2
=,+ l
Therefore,
equation (3-22) yields
(3-23)
I 6,4762065 x 10°l'/(
Q
Substituting equation (3-23) into equation (3-16) and moreover
.
Q
exchanging Q for
max '
2
3,
Gnax
C
-) [$(I + y)]
= 2;072(4()()0 Z+a l
or
Introducing the mmber of rows ,i of balls,
2
C - f iZD ooera
9
(3-24)
or 0 w
where
(3-25)
The values of factor f
on table 1 in IS0 76 are calculated from
0
substituting rmax = 4600 MPa and values of b/= D cosQ/D
shown
W
PW
in the table of IS0 76 into equation (3-25).
13
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SIST ISO/TR 10657:2001
ISO/TR 10657: 1991 (E)
for thrust ball be.arings
0 Basic static axial load rating C _
3 2 .
.
. _
The curvature sum rp and curvature difference F(p) of thrust.ball .
bearings is given by the following equations,
. .
2 1
_.
-4-Y_
a-
(3-26 1
ZP (2 >
D @
-1sFy -z?
w
._ _ - --
. _ - . . _ . - - _ _ _.__ .--._-. -.
(3-27) -
F(P) =
9 .
1
-. - P
2t y
1TY 2f
where
r = groove radius of housing washer, mm.
Substitut$ng equation (T-26.) into equation.(+T), -
. -
D 2
E(U)
3
II 6,4762065 x 10°l'+
Q (3-W
0 0"
max
2 *L-h
-Y
Substituting equation (3-28) into the following equation (3-29),
C
t3ina B
= ZQ
(3-29)
oa max
where
C = basic static axial load rating, N )
oa
z = number of balls carrying load in one direction)
=
maximum normal force between rolling element
Q
max
and raceway, FJ .
Therefore,
2
3
E(K)
C ) ZD2*irlcr 0
Oa -‘ l&%2() K(
1 w
Y
2**pJi-=
_ .
._. -- --_ . -._. .---_.-- _
14
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SIST ISO/TR 10657:2001
ISO/TR 10657: 1991 (E)
The formula applies to bearings with a cross-sectional raceway groove
radius not larger than 0,54 D,,
The load-carrying ability of a bearing is not necessarily increased
by the use of a smaller groove radius, but is reduced by the use of
a larger groove radius. In the latter case, a correspondingly reduced
value of f, shall be used.
The smaller value C calculated from equation (3-30) shall be
oa
. .
adopted. For washers with f = 034, using the upper sign, _.
.
. --.
2
C =fZDsina )
(3-31)
08
0 w
where
3
f. = 10,362(E) )c( (3-32)
The values of factor f
on table 1 in IS0 76 are calculated from
. . . . -
0
. . . - .
-.
substituting the values for )dt E(K) and Y= D cosQ/D
shown in
pw .
W
table! A.2 r and r = 4200 MPa into equation (3-32).
max
Basic static radial load rating C for radial roller bearings
0
33
or
The curvature su~ll z/> for radial roller bearings is given by the
following equation,
2 1
(3-33)
m- -
zp
D welTY .
into equation (3-11) and adopting the
Substituting equation (3-33)
smaller Q,
2
.
Q = 1,3810867 II; 1005(1 - Y)LweDwe&
_ . .__ - . . . - . _ _ - _ .
15
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SIST ISO/TR 10657:2001
ISO/TR 10657 : 1991 (E)
Substituting equation (3-34) into the following @quatian,
C
ZQ
9
Or
maxcos a
(3-15)
_ - . _ ____
._ .
where
c = basic static radial load rating, N: b
-
or
- . .
Z = number of rollers per row )
= maximum normal force between rolling element
Q
max
and raceway* N )
S is a function of the loaded zone parameter &.
If one half of the rollers are loaded then S = 4,08 applies.
A common value used in general bearing calculations is
S = 5, which leads to a rather conservative estimate of
the maximum roller load.
= nominal contact angle, *)
a
2
C
or - 44,194774(s) (1 - )+L
D COB cy -.
we WC
Consequently, adopting ?max = 4000 MPa and introducing the number
. __ .
_ - . - . .
of rows i of roller8, .
. . . _ _ -
D
wccso~ a
C
or - 440 o D
)iZLw,Dwecos ct a
(3-35)
PW
Basic static axial load rating Coa
0 for thrust roller bearings
3 4
The curvature SUIII zp of thrust roller bearings is given by the
equation (3-33)
and Q is given by the equation (3-34).
Substituting equations (3-33) and (3-34) into equation (T-29),
2
C
*a - 220,97387(-
2;;)
(1 - Y)ZLweDweBlnCI .
Consequently, adopting a;nax = 4000 MPa,
Dw~cosa)ZLyeDyesinQ . (3-36)
C
* 220(1 -
oa
PW
16
---------------------- Page: 22 ----------------------
SIST ISO/TR 10657:2001
ISO/TR 10657: 1991 (E)
Static Equivalent Load
-4
Theoretical static equivalent radial load P fo'r radial
4.1
bearings
4.1.1 Single row radial bearings and radial contact groove ball
bearings (nominal contact angle w= 0)
Assuming both the bearing rings will yield a parallel displacement
when a radial and axial loads act simultaneously on a single row
radial bearings, the maximum rolling element load Qax (N) is given
.- _
by the following equation [ll]
P
F
r
a
Q
(4-1)
max 131c ZcosQJ
= ZainQJ *
T
a
where
F = radial load, N P
r
F
= axial load, N )
a
J = radial load integral '
r
J = axial load integral J
a
z
= number of rolling elements per row,
a = nominal contact angle,' l
17
---------------------- Page: 23 ----------------------
SIST ISO/TR 10657:2001
ISO/TR 10657 : 1991 (E)
The radial and axial load integrals are given by the following
equations
4 0
f
t
J = J,(E> = & - co3
1 - &\l
co+& 9
r
4
IL
(4-e)
r
t
= J,(E) =.& - &(I - toe $11 d$J )
J
4%
J
where
t = '3/2 for point contact
= 1.1 for line contact I
=
one half of the loaded arc.
+
0
=
a parameter indicating the width of the loaded
&
zone in the bearing .
Assuming the bearing has no radial internal clearance under mounting,
- F when the rings displace
the static equivalent radial load P
or- r
in the radial direction (&= 0,5). Consequently, since we can obtain
the following equation from equation (4-l)
P
OT
Q
max - 2c0sC1Jr(0,5) ’
the following relationship yields
.
F J
. (4-3) .
P
-$ 'qk3T '
P c0ta J
a
(4-4)
P
-,3-&J l
or
---------------------- Page: 24 ----------------------
SIST ISO/TR 10657:2001
ISO/TR 10657 : 1991 (E)
The values calculated from equations (4-3) and- (4-4) for a constant
contact angle a are given in table 4-l. In accordance with the
functional relationship given by this table, the static equivalent
.
radial load P for the given values of Fr 9 F and a-may be obtained.
or a
/Par and Facota/P also is shown by
The relationship between F
r or
figure 4-1.
Table 4-1
Values for F-/P or and F cotC(IP vs. F tans/F
a or r a
for single row radial bearings
roller bearings
ball bearia
F
t
&
Frtana/Fa
Frtan a/F FacotC(/Por F cota/P
F/P
F,/p
a or or a Or
1
c
0,794o 1
0,8225 1 I,2158 I,2595
1 0,5
0,7835 LO558 1,3475 0,7482 1,0469 113993
096
0,700o
0,7427 LO949 1,4743 l,o746 1,5353
! %7
,
,---~4.
1 0,8 l&83
195988 o,6484 1,67oY
0,6995
t
1,0711 1,8102
I,1255 1,723Y o,5917
o,6529
1 0,y
1
1 0,600o . I,1128 l,o286
0547 0,5238 1,963
E
I
1,0003
1 1,25 o,36oo
22043 o,8474 2,3541
o,4538
o,308o o,8165 2,6512
I 1,67 02333 o,6464 2,77o3
t
0,185o oJ372
.I 23 0,5852 3,1637 0,438~ 3,1948
I
f
c
0,0611 o,2218
o,o831 3,74oo
5 %3108 3,6317
.
0
0 0 0
co 4,37o6 4,o766
i-
19
---------------------- Page: 25 ----------------------
SIST ISO/TR 10657:2001
ISO/TR 10657 : 1991 (E)
ball
bearing
I
roller bearing
04
I
02
9
0 10
9 20
t
pt a/Par
Figure 4-1 Theore tical relationship between radhl and axial
loads vu, static equivalent load for single
row radial bearinga
---------------------- Page: 26 ----------------------
SIST ISO/TR 10657:2001
ISO/TR 10657 : 1991 (E)
Table 4-l and figure 4-l are calculated and plotted based on the
assumption of a constant contact angle. However, the above relation- .
ship is also approximately applicable to ball bearings (angular
contact groove ball bearings etc.), a contact angle of which varies
with the load, if cota'given by the following equation (4-5) is
substituted for cota[l2]
F
2/ji
CCBB cr’ a
=l+
0 . (4-5) :-
(
3 1
c08 al
zD’8inCT’
V
C in the above equation is a compression constant depending on the
elastic modulus and conformity 2r/Dw, where r is a curvature radius .
of a raceway cross-section and D w is a ball diameter (see table q-2):.
Table J-2 Values of c
and ,2T/Dy-l
--
unit N and mm
. .
2r/D,. LO325 1,035 LO375 1,06
c
4,4745 4,9s47
CXlO' 4 units 4,3377 4,3871
in
C
2r/D -1 N , mm 0,01323 0,03253 0,01193 o,oo8258
f
x
L98 2,Ol 2,05 2,27
cxlo 3 units
l in
C
2r/D -1 kgf,mm o,o6062 QO5743 0,05440 ho3783
W
i
For an example of 2r/D
w= 1,035, the values of cot& for eetch value
. , _ ._.__ . ._.
* .
for angular contact groove ball bearings with CT= 15'-45*
of F /ZD2
a w
are given in table 4-3.
Furthermore, for single and double row radial contact groove
ball bearings, table 4-4 may be obtained from equations
21
---------------------- Page: 27 ----------------------
SIST ISO/TR 10657:2001
ISO/TR 10657 : 1991 (E)
Table 4-3 Example of values of co
...
TECHNICAL lSO/TR
REPORT 10657
First edition
1991-05-15
Explanatory notes on IS0 76
Notes explicatives SW /‘/SO 76
Reference number
ISO/TR 10657 : 1991 (E)
---------------------- Page: 1 ----------------------
ISO/TR 10657 : 1991 (E)
Contents
1 Scope. 1
2 BriefHistory . 1
2.1 ISO/R76-1958 . 1
2.2 ISO76-1978 . 2
2.3 ISO76-1987 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
............................................. 6
3 Basic Static Load Ratings
3.1 Basic static radial load rating &,-for radial ball bearings. . 10
3.1.1 Radial and angular contact groove ball bearings. . 10
3.1.2 Self-aligning ball bearings . 12
14
3.2 Basic static axial load rating C,, for thrust ball bearings. .
.............. 15
3.3 Basic static radial load rating &for radial roller bearings
.............. 16
3.4 Basic static axial load rating C,, for thrust roller bearings
................................................ 17
4 Static Equivalent Load
4.1 Theoretical static equivalent radial load P,, for radial bearings . 17
4.1 .I Single row radial bearings and radial contact groove ball bearings 17
.................................
4.1.2 Double row radial bearings 25
.......... 28
4.2 Theoretical static equivalent axial load P,, for thrust bearings
............................. 28
4.2.1 Single direction thrust bearings
4.2.2 Double direction thrust bearings. . 30
32
4.3 Approximate formulae for theoretical static equivalent load .
4.3.1 Radial bearings . 32
........................................... 34
4.3.2 Thrust bearings
0 IS0 1991
All rights reserved. No part of this publication may be reproduced or utilized in any form or by any
means, electronic or mechanical, including photocopying and microfilm, without permission in
writing from the publisher.
International Organization for Standardization
Case postale 56 l CH-1211 Geneve 20 l Switzerland
Printed in Switzerland
ii
---------------------- Page: 2 ----------------------
ISO/TR 10657 : 1991 (E)
Practical formulae of static equivalent load 35
4.4 .
35
4.4.1 Radial bearings .
.“. 40
4.4.2 Thrust bearings . .
Static radial load factor X, and static axial load factor Y,. . 42
4.5
42
4.5.1 Radial bearings
..............................................
42
4.5.1.1 Radial contact groove ball bearings .
44
4.5.1.2 Angular contact groove ball bearings. .
....... 47
4.5.1.3 Self-aligning ball bearings and radial roller bearings.
4.5.2 Thrust bearings 48
..............................................
Annexes
50
Valuesfory,xandE(x). .
AnnexA
53
AnnexB Symbols .
56
AnnexC References .
. . .
III
---------------------- Page: 3 ----------------------
ISO/TR 10657 : 1991 (E)
rewo rd
Fo
IS0 (the International Organization for Standardization) is a worldwide federation of
national standards bodies (IS0 member bodies). The work of preparing International
Standards is normally carried out through IS0 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, govern-
mental and non-governmental, in liaison with ISO, also take part in the work. IS0
collaborates closely with the International Electrotechnical Commission (IEC) on all
matters of electrotechnical standardization.
The main task of IS0 technical committees is to prepare International Standards. In
exceptional circumstances a technical committee may propose the publication of a
Technical Report of one of the following types:
-
type 1, when the required support cannot be obtained for the publication of an
International Standard, despite repeated efforts;
-
type 2, when the subject is still under technical development or where-for any
other reason there is the future but not immediate possibility of an agreement on an
International Standard;
-
type 3, when a technical committee has collected data of a different kind from
that which is normally published as an International Standard (“state of the art”, for
example).
Technical Reports of types 1 and 2 are subject to review within three years of publi-
cation, to decide whether they can be transformed into International Standards.
Technical Reports of type 3 do not necessarily have to be reviewed until the data they
provide are considered to be no longer valid or useful.
ISO/TR 10657, which is a Technical Report of type 3, was prepared by Technical Com-
mittee ISO/TC 4, Rolling bearings, Sub-Committee SC 8, Load ratings and life.
ISO/TR 10657 has been prepared for the guidance of users of IS0 76 : 1987, Rofing
bearings - Static load ratings. It is a purely scientific document intended for use by
specialists in this field, and it is not envisaged that it will become an International Stan-
dard.
Annexes A, B and C form an integral part of this Technical Report.
---------------------- Page: 4 ----------------------
TECHNICAL REPORT ISO/TR 10657 : 1991 (E)
Explanatory notes on IS0 76
1 Scope
This Technical Report gives supplementary background information
regarding the derivation of formulae and factors given in IS0 76,
Rolling bearings - Static load ratings.
2 Brief History
ISO/R 76 - 1958
2.1
The IS0 Recommendation R 76, Ball and Roller Bearings - Methods of
Evaluating Static Load Ratings, was drawn up by Technical Committee
ISO/TC 4, Ball and Holler Bearings, the Secretariat of which is held
-_.-
by the Sveriges
Standardiseringskownission (SIS).
This Recommendation is based on the studies[l]*, [Z] by A. Palmgren
and so on. It is defined in the Recommendation that the basic static
load ratings correspond to a total permanent deformation of rolling
elemenk and raceway at the most heavily stressed rolling element/
raceway contact ef 0,OOOl of the rolling element diameter. And then
the Standard values confined to the basic static load ratings for
special inner design rolling bearings are laid down,
*
Figures in brackets indicate literature references in annex C.
---------------------- Page: 5 ----------------------
ISO/TR 10657 : 1991 (E)
Technical Committee ISO/TC 4 discussed the questions dealt with by
the IS0 Recommendation at the following meetings:
the third meeting, held in Gsteborg, in September 1953,
the fourth meeting, held in Madrid, in May 1955,
the fifth meeting, held in Vienna, in September 1956.
At the third meeting of the Techrrical Committee, Working Group No.
3 was appointed to assist the ISO/TC 4 Secretariat in preparatory
work and in drawing up proposals.
The Working Group composed of
Germany, Sweden and USA held the following meetings:
the first meeting, held in Madrid, in May 1955,
the second meeting, held in Vienna, in September 1956.
On 4th June 1957, the Draft IS0 Recommendation was sent out to all
the IS0 Member Bodies and was approved by 28 (out of a total of 58)
Member Bodies.
The Draft IS0 Recommendation was then submitted by correspondence
to the IS0 Council, which decided, in December 1958, to accept it
as an IS0 Recommendation.
2.2 IS0 76
- 1978
The Working Group P(o. 3 was transformed into SC I at the 13th meeting
helci in Paris in May 1972. The: SC 8 Secretariat proposed to
include the revision of ISO/R 76 in the future work a.t the first
meeting held in London in November 1973, and SC 8 requested its
a Draft Proposal for an International Stan-
Secretariat to prepare
dard replacing ISO/R 76 and it was decided that this proposal should
be submitted to the SC 8 member Bodies for consideration prior to
1 October 1974 (SC 8 aESOLUTI0N 21, 21 London 1973).
2
---------------------- Page: 6 ----------------------
lSO/TR 10657 : 1991 (E)
The SC 8 Secretariat distributed & DRAFT PROPOSAL (Revision of
ISO/Ez 76) in July 1974.
The TC 4 decided to include the revision of ISO/ 76 in its pro-
gramme of work (TC 4 RESOLUTION 514, item No. 132) and SC 8 Sec-
retariat was requested to prepare a Draft Proposal (SC 8 HESOLUTION
38, 13 Mami Beach 1974). As a result, the Secretariat submitted
a DP [ 31 in January 1.976.
The Draft Pmprjsal DP 76 Was accepted by correspondence by 6 of the
9 Membws OS SC 8, Of the remaining two, Japan would prefer further
study and USA its counter proposal 9 document 418 N 64 [4] l The DIS
76 ~3s then st\bifri tted to the ISO Central Secretariat. After the DIS
had teen appmmd by the IS0 Mernbev brjdies, the IS0 Coun‘cil decided
i1-t June 1978 to accept it as an Intmmational Standard,
This Standwd adqked the SI unit ne&on and was revised in total,
but i4i ttmut: e55enti al changes af substance. However z values of x
0
0
and Y. for the naMna1 cantac,t angles 13 and 45 far angular COP
tact groove ball bearings were added ta the table whirh ShoWs the
values of Xo and Yo in the for-mulae to calculate the static equiva-
lent radial
lmds o-f radi ~1 ball bearings.
I!33 7b - 1’387
2.3
Duriflg the mtir,ion of 1533/R 76 - 1958, USA had in 1975 submitted a
4
counter proposal for the basic static load ratings based on a
c1
calm1 ated contact stress,
3
---------------------- Page: 7 ----------------------
ISO/TR 10657 : 1991 (E)
The Secmtwiat requested a vote an the wvi5ion of the static load
ratings based on a contact strr?ss level in January 1978 and after-
ward circulated the voted results in June 1978, and the item No.
of revision work had became No. 157 of the programme of work of
TC 4.
This Draft Proposal DP 76 was dealt with at the following TC 4
meetings:
15th meeting,
held in Moscow, in April 1977,
16th meeting,
held in London, in November 1979,
17 th meeting, held in Budapest, in May 1983,
and then, dealt with at the following SC 8 meetings:
the third meeting, held in London, in November 1979,
the fourth meeting, held in Budapest, in May 1983i
the fifth meeting, held in Arlington, in November 1984.
The following resolutions for the contents of the Standard were
adopted during the third meeting to the fifth meeting:
SC 8, considering the proposals made in the documents J/8 N 75 [5]
) as well as the comments made by TC 4 Members and
and 4 N 865 C6]
that several SC 8 Members expressed a need for updating IS0 76,
agreed to continue its study taking into account the possibility
of using either permanent deformation or stress level as a basis
for static load ratings (SC 8 RESOLUTION 45, 5 London 1979)) and
SC 8 requested its Secretariat to prepare a new draft. The new draft
should be prepared with the principles and formulae of the document
and to include levels of contact stress for various
418 N 75,
stated to be generally corresponding to
rolling element contact
4
---------------------- Page: 8 ----------------------
ISO/TR 10657 : 1991 (E)
-4
a permanent deformation of 10 D at the centre of the mast heavily
W
stressed rolling element/raceway contact. Far roller bearings a
stress level of 4000 MPa wa s agreed EC 8 REXILUTIUN 51, 4 Budapest
1%33> and then SC 8 agreed, by a majority vote, that static Ioad
ratiny~ should ccmxspand te calculated contact stresses of
4000 MPa for roller bearings,
4600 MPa for self-aligning ball bearings and
4200 MPa for all other ball bearings to which the
Standard applies (SC 8 RESOLUTION 56, 3 Arlington 1984). Moreover,
SC 8 recommended the document 4/8 N 121 [7] 9 amended in accordance
with SC 8 Resolution 56, as a revised IS0 76 (SC 8 i?ESOLUTION 57,
4 Arlington 1984).
For these calculated contact stresses, a total permanent deformation
occures at the centre of the most heavily stressed roliing element
/raceway contact, and its deformation is approximately 0,OOOl of
the rolling element diameter.
The n1s A5 has xk~~itted to the 193 Central Secretariat 1985, and
;If ter it had bmm approved by the ISG Members, the IS0 Ccxmcil de-
tided in-February VI?87 ta accept it as an Xnternatianal Standard.
Fw t hemore p SC 8 derided at its fifth meeting in April 1986 that
sLlpp1 emnt ary k3ckguaund i flf OUiUt i an, regarding the derivation of
fW3lUlX? Wd -fX3XX~ gi.ven in Ml X-3, skmuld be pub1 ished as a Tech-
I
nical Report EL r 8 FEXLlJTIt3N c- 71, 11 t-!angzhrjU 1%%d.
---------------------- Page: 9 ----------------------
ISO/TR 10657 : 1991 (E)
Basic Static Load RatinRs
3
(1) Basic equation for point contact
The relationship between
a calculated contact stress and a rolling element load within
an elliptical contact area is given as follows [8] 9
--.
36
1
(3-l)
u-
=27Tab
where
= calculated contact stress, MPa,
r
= major semi axis of the contact
a
ellipse, mm,
b = minor semi axis of the COntaCt
ellipse, mm,
= normal force between rolling element and
Q
raceway, M)
x = distanc.e in a-direction, mxn,
= distance in b-direction, mm .
Y
It is concluded that the maximum calculated contact
stress (Gax) occurs at the point of x = 0 and y = 0,
-. -
-. -.
36
2TTab
f
r
(3-2)
Q tT
max
=27Tab'
max l
3
According to the Hertz's theory,
2
l-
(2 lt2E(K) )1/3 3Q
VI
+
a=
( (3a3j
A E
c
*uJ
1
+ 1 - & v3 ,
3Q (1 - VI?
b _.(2S(K))l~3
(3-4)
E E
mt
2uJ )I
2
1
[
. ___.-. __ . .
-. --. - .--.
6
---------------------- Page: 10 ----------------------
lSO/TR 10657 : 1991 (E)
where
E(K) - complete .e.lliptic integral of the second kind
---
modulus of elasticity (Young's modulus), MPa,
By** = . . . .
._
. _
= Poisson% ratio, _. .
VW2
+p ) - . _. .
t/,=P+p +p
22
11 12 21
.--
2
N principal curvature of body 1 (ball),
P P 1-
D
11 9 21 l u
= principal curvature of body 2 (ring) tit the -point .*.
P
P
9
22
12
contrct l
Substituting equations \+T) and (3-4) into equation (5-2) for .the
. .
1= E2=
case of E E and VI= V2= y,
g-3 ~~(.~. *
s
(3-5)
1 9
Q
2
max
tP
3E
. 0
and
2
(W
1
- 'g-y E(XJ - 1) - WI - 0 n
(3-G
0
where
E
E 1111
?
0 2
1-v
5
E-2,07 @lOMPa,
v = 0,3
K(X) = complete elliptic integral of the flrat kind
?m
t
1. 42
2
o-8 n
2
4
31 1 - (1 ) i 41
0 u 2
I[
P - P
12 + p21 +22
11
Fo) =
P
11+ f)12+p21+p22 l
---------------------- Page: 11 ----------------------
ISO/TR 10657 : 1991 (E)
Consequently,
- 6,4762065 x lo
Q
(3-7)
(2) The relationship between
Basic equation for line contact
a calculated contact stress and arolling element load for s
._.-- _-. _ ,. ". .
line contact is given as follows [:9] 9
u - A;;ebF - if)*] l’* 9
(3-8)
where
= calculated contact stress, MPa )
r
b = semiwidth of the contact surface, mm I
L length of roller applicable to calculate load
we =
ratings, mm t
=
normal force between rollin+? element and raceway, N j
&
= distance in b-direction, mm .
Y
It is concluded that the maximum calculated contact stress
\Fmax) occurs at the line of y = 0,
_ 2Q
7r Lwob
CT
(3-Y)
- fALw,b ' Or ' - 2 &mx l
max
.
~
And also b is given by the following equation,
. ___.
_
2
l/2
48
v2
br
(
(3-w
m
AL~ew
)3
---------------------- Page: 12 ----------------------
ISO/TR 10657: 1991 (E)
where
c/II - q1 + P,, + n,, +
P 7
22
P m
- 2
P 1,
2 Y
11 u 9
12
+-- 1ry
9 $1
- 0 9 1322
= 0 '
D
we
W8
the upper sign applies to inner ring contact and
the lower to outer ring contact,
D = roller diameter applicable to calculate load
we
ratings, mm y
D~,coW
m
Y I
D
Pw
=
D pitch diameter of roller set, mm.
PW
Substituting equation (3-10) into equation (3.9j for the case of
=
E E *= E and Vl= v2= y,
1
L
we
Q - 27~
8
max E*J/) *
where
E
E
o=
2
1-v
E = *,07 x 105HPa2
u= 093 0
Consequently,
. -
L
Q = 2.7621732 x LOW5 -= Q-*
(3-11)
max l
ZP
---------------------- Page: 13 ----------------------
ISO/TR 10657 : 1991 (E)
0 Basic static radial load rating C for radial ball bearings
3 1
or
.- . .
Radial and angular contact groove ball bearings
3.1.1
The curvature sum Z/)and curvature difference F(p) of radial and an@=
contact groove ball bearings is given by the following equation,
2
1
Y
m
ZP
(3-W
d
2fqy2q&'
W
1
If
2
+
2f
l?:Y
c
it >
F(P) = (3-13)
2**- l .
2fi(e)
where _ .
the upper sign applies to inner ring contact and
-the lower to outer ring contact,
D II ball diameter, mm B
W
D cosa
w
Y
D
PW
D pw 05 pitch diameter of ball Bet, mm P
rr .
,
f IL )
i D
W
r
e
=-
f
t D.'
W
r
= inner ring groove radius, mm j
i
r
= outer ring groove radius, mm .
c
Substituting equation (3-12) into equation (3-T),
2
D
3
E(K)
LI 6,4762065 x lO-1o,($
(3-14)
Q )(7
max
Y
1
2*-w
lTy 2fi(e)
Substituting equations (3-12) and (3-14) into the following equation
9 furthermore exchanging Q for Qmax,
10
I: 1
1
E . ~08 a
C
a
(3-15)
raax
or S
IO
---------------------- Page: 14 ----------------------
ISO/TR 10657 : 1991 (E)
where
_
C = basic static radial load rating, Ni j
or
z = number of balls per row 2
= maximum normal force between rolling element
Q’
max
and raceway, N 9
S is a function of the loaded zone parameter &
If one half of the balls are loaded then S = 4,37 applies.
A common value used in general bearing calculations is
S
= 5, which leads to a rather conservative estimate of
the maximum ball load,
= nominal contact angle,',
a
= 0,22& cosa l (39 _.
C
.
max
or
_. -
_ . .
Consequently,
3
3
Q‘Paax
C = 0,2 x 6,4762065 x loolOx (4000) (-)u
or
4000
2
1
E(X) 2
x-
ZDwc
(' ofa
>
1
4 Y
2+
1
rY
2fi(e)
where the upper sign refers to the inner ring and the lower sign
refers to the outer ring. Therefore,
introducing the number of rows
i of balls,
2
C
* f iZD c00a
)
or
0 w (3-17 1
where
f
factor which depends on the geometry
OP
of the bearing components and on appli-
cable stress level
_---_.--. _
2
3
Finax E(K)
= 29072&j$ a
0
1
11
---------------------- Page: 15 ----------------------
ISO/TR 10657 : 1991 (E)
The formula applies to bearings with a cross-sectional raceway groove
radius not larger than 0,52 D, in radial and angular contact groove
ball bearing inner rings, and 0,53 D, in radial and angular contact
groove ball bearing outer rings and self-aligning ball bearing inner
rings.
The load-carrying ability of a bearing is not necessarily increased
by the use of a smaller groove radius, but is reduced by the use of
a correspondingly reduced
a larger groove radius. In the latter case,
value of f,
shall be used.
. _.
= 0,52,
For an inner ring with f
i
2
3
rmax
E(K)
f. = *9°72(4000/ Jt( 1 9 (3-19)
1
Y
. * + 1-y -
1,04
I. . . . . .-.- .-. _ I _.-. . . .
and for an outer ring with f
0,53,
e =
--_
E(K) 2
(3-20) 1
1 0
2 Y 1
YTji'1,66
values calculated from equations
'the smaller value between the f
0
(T-19) and (T-20) shall be adopted.
- 1987 are calculated
The values of factor f on table 1 in IS0 76
0
shown
from substituting the values for X, E(K) and y= D cos Q/D
W
PW
and r = 4200 MPa into the above equation.
in table A-1 I
max
Self-aligning ball bearings
3.12
The curvature sum r/,of self-aligning ball bearings is given by
the following equation for an outer ring,
1
4
(3-20
)
(
.v
=DVl+y l
12
---------------------- Page: 16 ----------------------
ISO/TR 10657 : 1991 (E)
Substituting equation (3-21) into equation (3-7),
. .
e = 6,4762065 x lo’loU [>(l + &(K)]2c;x 0 (3-22)
. . _ .-
In general, x= a/b =
1 for the case of contact between an outer
_-. .
ring raceway and balls of self-aligning ball bearings. Consequently,
_.
l/2
=,+ l
Therefore,
equation (3-22) yields
(3-23)
I 6,4762065 x 10°l'/(
Q
Substituting equation (3-23) into equation (3-16) and moreover
.
Q
exchanging Q for
max '
2
3,
Gnax
C
-) [$(I + y)]
= 2;072(4()()0 Z+a l
or
Introducing the mmber of rows ,i of balls,
2
C - f iZD ooera
9
(3-24)
or 0 w
where
(3-25)
The values of factor f
on table 1 in IS0 76 are calculated from
0
substituting rmax = 4600 MPa and values of b/= D cosQ/D
shown
W
PW
in the table of IS0 76 into equation (3-25).
13
---------------------- Page: 17 ----------------------
ISO/TR 10657: 1991 (E)
for thrust ball be.arings
0 Basic static axial load rating C _
3 2 .
.
. _
The curvature sum rp and curvature difference F(p) of thrust.ball .
bearings is given by the following equations,
. .
2 1
_.
-4-Y_
a-
(3-26 1
ZP (2 >
D @
-1sFy -z?
w
._ _ - --
. _ - . . _ . - - _ _ _.__ .--._-. -.
(3-27) -
F(P) =
9 .
1
-. - P
2t y
1TY 2f
where
r = groove radius of housing washer, mm.
Substitut$ng equation (T-26.) into equation.(+T), -
. -
D 2
E(U)
3
II 6,4762065 x 10°l'+
Q (3-W
0 0"
max
2 *L-h
-Y
Substituting equation (3-28) into the following equation (3-29),
C
t3ina B
= ZQ
(3-29)
oa max
where
C = basic static axial load rating, N )
oa
z = number of balls carrying load in one direction)
=
maximum normal force between rolling element
Q
max
and raceway, FJ .
Therefore,
2
3
E(K)
C ) ZD2*irlcr 0
Oa -‘ l&%2() K(
1 w
Y
2**pJi-=
_ .
._. -- --_ . -._. .---_.-- _
14
---------------------- Page: 18 ----------------------
ISO/TR 10657: 1991 (E)
The formula applies to bearings with a cross-sectional raceway groove
radius not larger than 0,54 D,,
The load-carrying ability of a bearing is not necessarily increased
by the use of a smaller groove radius, but is reduced by the use of
a larger groove radius. In the latter case, a correspondingly reduced
value of f, shall be used.
The smaller value C calculated from equation (3-30) shall be
oa
. .
adopted. For washers with f = 034, using the upper sign, _.
.
. --.
2
C =fZDsina )
(3-31)
08
0 w
where
3
f. = 10,362(E) )c( (3-32)
The values of factor f
on table 1 in IS0 76 are calculated from
. . . . -
0
. . . - .
-.
substituting the values for )dt E(K) and Y= D cosQ/D
shown in
pw .
W
table! A.2 r and r = 4200 MPa into equation (3-32).
max
Basic static radial load rating C for radial roller bearings
0
33
or
The curvature su~ll z/> for radial roller bearings is given by the
following equation,
2 1
(3-33)
m- -
zp
D welTY .
into equation (3-11) and adopting the
Substituting equation (3-33)
smaller Q,
2
.
Q = 1,3810867 II; 1005(1 - Y)LweDwe&
_ . .__ - . . . - . _ _ - _ .
15
---------------------- Page: 19 ----------------------
ISO/TR 10657 : 1991 (E)
Substituting equation (3-34) into the following @quatian,
C
ZQ
9
Or
maxcos a
(3-15)
_ - . _ ____
._ .
where
c = basic static radial load rating, N: b
-
or
- . .
Z = number of rollers per row )
= maximum normal force between rolling element
Q
max
and raceway* N )
S is a function of the loaded zone parameter &.
If one half of the rollers are loaded then S = 4,08 applies.
A common value used in general bearing calculations is
S = 5, which leads to a rather conservative estimate of
the maximum roller load.
= nominal contact angle, *)
a
2
C
or - 44,194774(s) (1 - )+L
D COB cy -.
we WC
Consequently, adopting ?max = 4000 MPa and introducing the number
. __ .
_ - . - . .
of rows i of roller8, .
. . . _ _ -
D
wccso~ a
C
or - 440 o D
)iZLw,Dwecos ct a
(3-35)
PW
Basic static axial load rating Coa
0 for thrust roller bearings
3 4
The curvature SUIII zp of thrust roller bearings is given by the
equation (3-33)
and Q is given by the equation (3-34).
Substituting equations (3-33) and (3-34) into equation (T-29),
2
C
*a - 220,97387(-
2;;)
(1 - Y)ZLweDweBlnCI .
Consequently, adopting a;nax = 4000 MPa,
Dw~cosa)ZLyeDyesinQ . (3-36)
C
* 220(1 -
oa
PW
16
---------------------- Page: 20 ----------------------
ISO/TR 10657: 1991 (E)
Static Equivalent Load
-4
Theoretical static equivalent radial load P fo'r radial
4.1
bearings
4.1.1 Single row radial bearings and radial contact groove ball
bearings (nominal contact angle w= 0)
Assuming both the bearing rings will yield a parallel displacement
when a radial and axial loads act simultaneously on a single row
radial bearings, the maximum rolling element load Qax (N) is given
.- _
by the following equation [ll]
P
F
r
a
Q
(4-1)
max 131c ZcosQJ
= ZainQJ *
T
a
where
F = radial load, N P
r
F
= axial load, N )
a
J = radial load integral '
r
J = axial load integral J
a
z
= number of rolling elements per row,
a = nominal contact angle,' l
17
---------------------- Page: 21 ----------------------
ISO/TR 10657 : 1991 (E)
The radial and axial load integrals are given by the following
equations
4 0
f
t
J = J,(E> = & - co3
1 - &\l
co+& 9
r
4
IL
(4-e)
r
t
= J,(E) =.& - &(I - toe $11 d$J )
J
4%
J
where
t = '3/2 for point contact
= 1.1 for line contact I
=
one half of the loaded arc.
+
0
=
a parameter indicating the width of the loaded
&
zone in the bearing .
Assuming the bearing has no radial internal clearance under mounting,
- F when the rings displace
the static equivalent radial load P
or- r
in the radial direction (&= 0,5). Consequently, since we can obtain
the following equation from equation (4-l)
P
OT
Q
max - 2c0sC1Jr(0,5) ’
the following relationship yields
.
F J
. (4-3) .
P
-$ 'qk3T '
P c0ta J
a
(4-4)
P
-,3-&J l
or
---------------------- Page: 22 ----------------------
ISO/TR 10657 : 1991 (E)
The values calculated from equations (4-3) and- (4-4) for a constant
contact angle a are given in table 4-l. In accordance with the
functional relationship given by this table, the static equivalent
.
radial load P for the given values of Fr 9 F and a-may be obtained.
or a
/Par and Facota/P also is shown by
The relationship between F
r or
figure 4-1.
Table 4-1
Values for F-/P or and F cotC(IP vs. F tans/F
a or r a
for single row radial bearings
roller bearings
ball bearia
F
t
&
Frtana/Fa
Frtan a/F FacotC(/Por F cota/P
F/P
F,/p
a or or a Or
1
c
0,794o 1
0,8225 1 I,2158 I,2595
1 0,5
0,7835 LO558 1,3475 0,7482 1,0469 113993
096
0,700o
0,7427 LO949 1,4743 l,o746 1,5353
! %7
,
,---~4.
1 0,8 l&83
195988 o,6484 1,67oY
0,6995
t
1,0711 1,8102
I,1255 1,723Y o,5917
o,6529
1 0,y
1
1 0,600o . I,1128 l,o286
0547 0,5238 1,963
E
I
1,0003
1 1,25 o,36oo
22043 o,8474 2,3541
o,4538
o,308o o,8165 2,6512
I 1,67 02333 o,6464 2,77o3
t
0,185o oJ372
.I 23 0,5852 3,1637 0,438~ 3,1948
I
f
c
0,0611 o,2218
o,o831 3,74oo
5 %3108 3,6317
.
0
0 0 0
co 4,37o6 4,o766
i-
19
---------------------- Page: 23 ----------------------
ISO/TR 10657 : 1991 (E)
ball
bearing
I
roller bearing
04
I
02
9
0 10
9 20
t
pt a/Par
Figure 4-1 Theore tical relationship between radhl and axial
loads vu, static equivalent load for single
row radial bearinga
---------------------- Page: 24 ----------------------
ISO/TR 10657 : 1991 (E)
Table 4-l and figure 4-l are calculated and plotted based on the
assumption of a constant contact angle. However, the above relation- .
ship is also approximately applicable to ball bearings (angular
contact groove ball bearings etc.), a contact angle of which varies
with the load, if cota'given by the following equation (4-5) is
substituted for cota[l2]
F
2/ji
CCBB cr’ a
=l+
0 . (4-5) :-
(
3 1
c08 al
zD’8inCT’
V
C in the above equation is a compression constant depending on the
elastic modulus and conformity 2r/Dw, where r is a curvature radius .
of a raceway cross-section and D w is a ball diameter (see table q-2):.
Table J-2 Values of c
and ,2T/Dy-l
--
unit N and mm
. .
2r/D,. LO325 1,035 LO375 1,06
c
4,4745 4,9s47
CXlO' 4 units 4,3377 4,3871
in
C
2r/D -1 N , mm 0,01323 0,03253 0,01193 o,oo8258
f
x
L98 2,Ol 2,05 2,27
cxlo 3 units
l in
C
2r/D -1 kgf,mm o,o6062 QO5743 0,05440 ho3783
W
i
For an example of 2r/D
w= 1,035, the values of cot& for eetch value
. , _ ._.__ . ._.
* .
for angular contact groove ball bearings with CT= 15'-45*
of F /ZD2
a w
are given in table 4-3.
Furthermore, for single and double row radial contact groove
ball bearings, table 4-4 may be obtained from equations
21
---------------------- Page: 25 ----------------------
ISO/TR 10657 : 1991 (E)
Table 4-3 Example of values of cot@' for angular
contact groove ball bearings
.F /ZD2"
a w
t
095 1 2 5 10
a I
4 ._ . . . _
cota’
0
1,865 '
15 3,024 2,793 2,526 2,154
I
0
20
2,450 2,164 1,905 1,691
2,322
0
1,664 1,511
25 .1,997 1,929 1,834
0
1,444 1,337
1,651 1,613 1,552
30
f
0
1,171
1,317 1,248
35 1,381 1,356
0
1,018
1,122
1,146 1,072
40 1,163
0
0,879
0,975 0,969
45
2 2
Fa/ZDV = (Fa/Cor)focos&
ZDgosQ,
* unita in N and mm. Since Car = f
0
(4.3), (4-4) and the following equation (4-6) [12)
aina’- tana)L (
~*13’8cl - -&J3’*$ LZD2Y4
(4-G
a w
where
i = number of rows of balls 2
Z
= number of balls per row .
For given values of F ztnd F a a provisional value of dis found
* -
r
using equation (4-7). Next, table 4-4 is used to find& and F,/Por
or Facotcc’/POr and then PO, can be determined,
22
---------------------- Page: 26 ----------------------
ISO/TR 10657: 1991 (E)
Table 4-4 Values of F /P and FacotCIyP VS.
r Or
or
Frtan&/F for radial contact groove
a
ball bearings
FacO tapor
F,/P
Ol?
i
0
1
a3
I,0558 0,923a
1,1432
096
l,2oY6
l,og49
0,9o55
097
l&83 134231
0,7859
088
I,6051
I,1255
0,7013
O,Y
191128 l,7737
0~628
...
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