SIST ISO 10300-3:2015
Calculation of load capacity of bevel gears - Part 3: Calculation of tooth root strength
Calculation of load capacity of bevel gears - Part 3: Calculation of tooth root strength
ISO 10300-3:2014 specifies the fundamental formulae for use in the tooth root stress calculation of straight and helical (skew), Zerol and spiral bevel gears including hypoid gears, with a minimum rim thickness under the root of 3,5 mmn. All load influences on tooth root stress are included, insofar as they are the result of load transmitted by the gearing and able to be evaluated quantitatively. Stresses, such as those caused by the shrink fitting of gear rims, which are superposed on stresses due to tooth loading, are intended to be considered in the calculation of the tooth root stress, σF, or the permissible tooth root stress σFP. ISO 10300-3:2014 is not applicable in the assessment of the so-called flank breakage, a tooth internal fatigue fracture (TIFF).
The formulae in ISO 10300-3:2014 are based on virtual cylindrical gears and restricted to bevel gears whose virtual cylindrical gears have transverse contact ratios of Ɛvα
Calcul de la capacité de charge des engrenages coniques - Partie 3: Calcul de la résistance du pied de dent
Izračun nosilnosti stožčastih zobnikov - 3. del: Izračun nosilnosti zobnega korena
Ta del standarda ISO 10300 določa osnovne formule, ki se uporabljajo pri izračunu obremenitve zobnega korena ravnih, valjastih (poševnih) Zerol in spiralnih stožčastih zobnikov, vključno s hipoidnimi zobniki, z minimalno debelino venca pod korenom 3,5 mmn. Vključeni so vsi vplivi obremenitve na obremenitev zobnega korena, če so posledica obremenitve, prenesene z gonilom, in se lahko količinsko ocenijo. Obremenitve, kot so tiste, ki jih povzroči tolerančno vpenjanje zobatih vencev, superponiranih pri obremenitvah, povezanih z obremenitvijo zob, je treba upoštevati pri izračunu obremenitev zobnega korena, σF, ali dovoljene obremenitve zobnega korena σFP. Ta del standarda ISO 10300 se ne uporablja za oceno tako imenovane lomljivosti zobnih bokov oziroma lomljivosti notranje utrujenosti zoba (TIFF).
Formule v tem delu standarda ISO 10300 temeljijo na umišljenih valjastih zobnikih in so omejene na stožčaste zobnike z umišljenimi valjastimi zobniki s profilno stopnjo prekrivanja εvα < 2. Rezultati so veljavni znotraj obsega uporabljenih faktorjev iz standarda ISO 10300-1 (glej tudi ISO 6336-3[1]). Poleg tega se navedena razmerja uporabljajo za stožčaste zobnike, pri katerih je vsota koeficientov profilnega premika zobatega kolesca in kolesa nič (glej ISO 23509).
Ta del standarda ISO 10300 se ne uporablja za stopnje odpornosti, ki presegajo stopnje, dovoljene za 103 cikle, ker bi lahko obremenitve v tem obsegu presegle elastično omejitev zobnikovih zob. Opozorilo – Uporabnika opozarjamo, da naj bi se pri uporabi formul za velike povprečne srednje spiralne kote (βm1 + βm2)/2 > 45°, za kote efektivnega tlaka αe > 30° in/ali za veliko širino zoba b > 13 mmn izračunan rezultat ISO 10300 (vsi deli) potrdil z izkušnjami.
General Information
Relations
Standards Content (Sample)
SLOVENSKI STANDARD
01-marec-2015
1DGRPHãþD
SIST ISO 10300-3:2008
,]UDþXQQRVLOQRVWLVWRåþDVWLK]REQLNRYGHO ,]UDþXQQRVLOQRVWL]REQHJDNRUHQD
Calculation of load capacity of bevel gears - Part 3: Calculation of tooth root strength
Calcul de la capacité de charge des engrenages coniques - Partie 3: Calcul de la
résistance du pied de dent
Ta slovenski standard je istoveten z: ISO 10300-3:2014
ICS:
21.200 Gonila Gears
2003-01.Slovenski inštitut za standardizacijo. Razmnoževanje celote ali delov tega standarda ni dovoljeno.
INTERNATIONAL ISO
STANDARD 10300-3
Second edition
2014-04-01
Calculation of load capacity of bevel
gears —
Part 3:
Calculation of tooth root strength
Calcul de la capacité de charge des engrenages coniques —
Partie 3: Calcul de la résistance du pied de dent
Reference number
©
ISO 2014
© ISO 2014
All rights reserved. Unless otherwise specified, no part of this publication may be reproduced or utilized otherwise in any form
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ii © ISO 2014 – All rights reserved
Contents Page
Foreword .iv
Introduction .v
1 Scope . 1
2 Normative references . 1
3 Terms and definitions . 1
4 Symbols, units and abbreviated terms . 2
5 General rating procedure . 2
6 Gear tooth rating formulae — Method B1 . 3
6.1 Tooth root stress formula . 3
6.2 Permissible tooth root stress . 4
6.3 Calculated safety factor . 5
6.4 Tooth root stress factors . 5
6.5 Permissible tooth root stress factors .13
7 Gear tooth rating formulae — Method B2 .17
7.1 Tooth root stress formula .17
7.2 Permissible tooth root stress .17
7.3 Calculated safety factor .18
7.4 Tooth root stress factors .18
7.5 Permissible tooth root stress factors .36
8 Factors for permissible tooth root stress common for method B1 and method B2 .36
8.1 Size factor, Y .36
X
8.2 Life factor, Y .38
NT
Bibliography .41
Foreword
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bodies (ISO member bodies). The work of preparing International Standards is normally carried out
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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. www.iso.org/directives
Attention is drawn to the possibility that some of the elements of this document may be the subject of
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For an explanation on the meaning of ISO specific terms and expressions related to conformity
assessment, as well as information about ISO’s adherence to the WTO principles in the Technical Barriers
to Trade (TBT) see the following URL: Foreword - Supplementary information
The committee responsible for this document is ISO/TC 60, Gears, Subcommittee SC 2, Gear capacity
calculation.
This second edition cancels and replaces the first edition (ISO 10300-3:2001), which has been technically
revised.
ISO 10300 consists of the following parts, under the general title Calculation of load capacity of bevel
gears:
— Part 1: Introduction and general influence factors
— Part 2: Calculation of surface durability (pitting)
— Part 3: Calculation of tooth root strength
iv © ISO 2014 – All rights reserved
Introduction
When ISO 10300:2001 (all parts, withdrawn) became due for (its first) revision, the opportunity was
taken to include hypoid gears, since previously the series only allowed for calculating the load capacity
of bevel gears without offset axes. The former structure is retained, i.e. three parts of the ISO 10300
series, together with ISO 6336-5, and it is intended to establish general principles and procedures for
rating of bevel gears. Moreover, ISO 10300 (all parts) is designed to facilitate the application of future
knowledge and developments, as well as the exchange of information gained from experience.
In view of the decision for ISO 10300 (all parts) to cover hypoid gears also, it was agreed to include a
separate clause: “Gear tooth rating formulae — Method B2” in this part of ISO 10300, while the former
methods B and B1 were combined into one method, i.e. method B1. So, it became necessary to present
a new, clearer structure of the three parts, which is illustrated in ISO 10300-1:2014, Figure 1. Note,
ISO 10300 (all parts) gives no preferences in terms of when to use method B1 and when method B2.
Failure of gear teeth by breakage can be brought about in many ways; severe instantaneous overloads,
excessive pitting, case crushing and bending fatigue are a few. The strength ratings determined by the
formulae in this part of ISO 10300 are based on cantilever projection theory modified to consider the
following:
— compressive stress at the tooth roots caused by the radial component of the tooth load;
— non-uniform moment distribution of the load, resulting from the inclined contact lines on the teeth
of spiral bevel gears;
— stress concentration at the tooth root fillet;
— load sharing between adjacent contacting teeth;
— lack of smoothness due to a low contact ratio.
The formulae are used to determine a load rating, which prevents tooth root fracture during the design
life of the bevel gear. Nevertheless, if there is insufficient material under the teeth (in the rim), a fracture
can occur from the root through the rim of the gear blank or to the bore (a type of failure not covered
by this part of ISO 10300). Moreover, it is possible that special applications require additional blank
material to support the load.
Surface distress (pitting or wear) can limit the strength rating, either due to stress concentration
around large sharp cornered pits, or due to wear steps on the tooth surface. Neither of these effects is
considered in this part of ISO 10300.
In most cases, the maximum tensile stress at the tooth root (arising from bending at the root when the
load is applied to the tooth flank) can be used as a determinant criterion for the assessment of the tooth
root strength. If the permissible stress number is exceeded, the teeth can break.
When calculating the tooth root stresses of straight bevel gears, this part of ISO 10300 starts from the
assumption that the load is applied at the tooth tip of the virtual cylindrical gear. The load is subsequently
converted to the outer point of single tooth contact. The procedure thus corresponds to method C for the
[1]
tooth root stress of cylindrical gears (see ISO 6336-3 ).
For spiral bevel and hypoid gears with a high face contact ratio of ε > 1 (method B1) or with a modified
vβ
contact ratio of ε > 2 (method B2), the midpoint in the zone of action is regarded as the critical point
vγ
of load application.
The breakage of a tooth generally means the end of a gear’s life. It is often the case that all gear teeth are
destroyed as a consequence of the breakage of a single tooth. A safety factor, S , against tooth breakage
F
higher than the safety factor against damage due to pitting is, therefore, generally to be preferred
(see ISO 10300-1).
INTERNATIONAL STANDARD ISO 10300-3:2014(E)
Calculation of load capacity of bevel gears —
Part 3:
Calculation of tooth root strength
1 Scope
This part of ISO 10300 specifies the fundamental formulae for use in the tooth root stress calculation
of straight and helical (skew), Zerol and spiral bevel gears including hypoid gears, with a minimum rim
thickness under the root of 3,5 m . All load influences on tooth root stress are included, insofar as
mn
they are the result of load transmitted by the gearing and able to be evaluated quantitatively. Stresses,
such as those caused by the shrink fitting of gear rims, which are superposed on stresses due to tooth
loading, are intended to be considered in the calculation of the tooth root stress, σ , or the permissible
F
tooth root stress σ . This part of ISO 10300 is not applicable in the assessment of the so-called flank
FP
breakage, a tooth internal fatigue fracture (TIFF).
The formulae in this part of ISO 10300 are based on virtual cylindrical gears and restricted to bevel
gears whose virtual cylindrical gears have transverse contact ratios of ε < 2. The results are valid
vα
[1]
within the range of the applied factors as specified in ISO 10300-1 (see also ISO 6336-3 ). Additionally,
the given relationships are valid for bevel gears, of which the sum of profile shift coefficients of pinion
and wheel is zero (see ISO 23509).
This part of ISO 10300 does not apply to stress levels above those permitted for 10 cycles, as stresses
in that range could exceed the elastic limit of the gear tooth.
Warning — The user is cautioned that when the formulae are used for large average mean spiral
angles (β + β )/2 > 45°, for effective pressure angles α > 30° and/or for large face widths
m1 m2 e
b > 13 m , the calculated results of ISO 10300 (all parts) should be confirmed by experience.
mn
2 Normative references
The following documents, in whole or in part, are normatively referenced in this document and are
indispensable to its application. For dated references, only the edition cited applies. For undated
references, the latest edition of the referenced document (including any amendments) applies.
ISO 1122-1, Vocabulary of gear terms — Part 1: Definitions related to geometry
ISO 6336-5, Calculation of load capacity of spur and helical gears — Part 5: Strength and quality of materials
ISO 10300-1:2014, Calculation of load capacity of bevel gears — Part 1: Introduction and general influence
factors
ISO 10300-2:2014, Calculation of load capacity of bevel gears — Part 2: Calculation of surface durability
(pitting)
ISO 23509:2006, Bevel and hypoid gear geometry
3 Terms and definitions
For the purposes of this document, the terms and definitions given in ISO 1122-1 and ISO 23509
(geometrical gear terms) and the following apply.
3.1
tooth root breakage
failure of gear teeth at the tooth root by static or dynamic overload
3.2
nominal tooth root stress
σ
F0
bending stress in the critical section of the tooth root calculated for the critical point of load application
for error-free gears loaded by a constant nominal torque
3.3
tooth root stress
σ
F
determinant bending stress in the critical section of the tooth root calculated for the critical point of
load application including the load factors which consider static and dynamic loads and load distribution
3.4
nominal stress number
σ
F,lim
maximum tooth root stress of standardized test gears and determined at standardized operating
conditions as specified in ISO 6336-5
3.5
allowable stress number
σ
FE
maximum bending stress of the un-notched test piece under the assumption that the material is fully
elastic
3.6
permissible tooth root stress
σ
FP
maximum tooth root stress of the evaluated gear set including all influence factors
4 Symbols, units and abbreviated terms
For the purposes of this document, the symbols and units given in Table 1 and Table 2 of ISO 10300-1:2014,
as well as the abbreviated terms given in Table 1 of ISO 10300-2:2014, apply (see ISO 6336-5).
5 General rating procedure
There are two main methods for determining tooth bending strength of bevel and hypoid gears:
method B1 and method B2. They are provided in Clauses 6 and 7, while Clause 8 contains those influence
factors which are equal for both methods. With method B1, the same set of formulae may be used for
bevel and hypoid gears; method B2 partly has different sets of formulae for bevel gears and for hypoid
gears (see 7.4.3 for general aspects).
With both methods, the capability of a gear tooth to resist tooth root stresses shall be determined by the
comparison of the following stress values:
— tooth root stress σ , based on the geometry of the tooth, the accuracy of its manufacture, the
F
rigidity of the gear blanks, bearings and housing, and the operating torque, expressed by the tooth
root stress formula (see 6.1 and 7.1);
— permissible tooth root stress σ , based on the bending stress number, σ , of a standard test
FP F,lim
gear and the effect of the operating conditions under which the gears operate, expressed by the
permissible tooth root stress formula (see 6.2 and 7.2).
2 © ISO 2014 – All rights reserved
NOTE In respect of the permissible tooth root stress, reference is made to a stress “number”, a designation
adopted because pure stress, as determined by laboratory testing, is not calculated by the formulae in this part
of ISO 10300. Instead, an arbitrary value is calculated and used in this part of ISO 10300, with accompanying
changes to the allowable stress number in order to maintain consistency for design comparison.
The ratio of the permissible root stress and the calculated root stress is the safety factor S . The value of
F
the minimum safety factor for tooth root stress, S , should be ≥ 1,3 for spiral bevel gears. For straight
F,min
bevel gears, or where β ≤ 5°, S should be ≥ 1,5.
m F,min
It is recommended that the gear designer and customer agree on the value of the minimum safety factor.
Tooth breakage usually ends transmission service life. The destruction of all gears in a transmission can
be a consequence of the breakage of one tooth, then, the drive train between input and output shafts
is interrupted. Therefore, the chosen value of the safety factor, S , against tooth breakage should be
F
carefully chosen to fulfil the application requirements (see ISO 10300-1 for general comments on the
choice of safety factor).
6 Gear tooth rating formulae — Method B1
6.1 Tooth root stress formula
The calculation of the tooth root stress is based on the maximum bending stress at the tooth root. It is
determined separately for pinion (suffix 1) and wheel (suffix 2); in the case of hypoid gears, additionally
for drive flank (suffix D) and coast flank (suffix C):
σσ=
F-B1 F0-B1A v Fβα F FP-B1
with the load factors K , K , K , K as specified in ISO 10300-1.
A v Fβ Fα
The nominal tooth root stress is defined as the maximum bending stress at the tooth root (30° tangent
to the root fillet):
F
vmt
σ = YY YY Y (2)
F0-B1 Fa Sa ε BS LS
bm
v mn
where
F is the nominal tangential force of the virtual cylindrical gear which should be in accordance
vmt
with Formula (2) of ISO 10300-1:2014;
b is the face width of the virtual cylindrical gear calculated for the active flank, drive or coast
v
side, as specified in ISO 10300-1:2014, Annex A;
Y is the tooth form factor (see 6.4.1), which accounts for the influence of the tooth form on the
Fa
nominal bending stress at the tooth root for load application at the tooth tip;
Y is the stress correction factor (see 6.4.2), which accounts for the stress increasing notch
Sa
effect in the root fillet, as well as for the radial component of the tooth load and the fact that
the stress condition in the critical root section is complex, but not for the influence of the
bending moment arm;
Y is the contact ratio factor (see 6.4.3), which accounts for the conversion of the root stress
ε
determined for the load application at the tooth tip to the determinant position;
Y is the bevel spiral angle factor, which accounts for smaller values for l compared to the
BS bm
total face width, b , and the inclined lines of contact (see 6.4.4);
v
Y is the load sharing factor, which accounts for load distribution between two or more pairs of
LS
teeth (see 6.4.5).
The determinant position of load application is:
a) the outer point of single tooth contact, if ε = 0;
vβ
b) the midpoint of the zone of action, if ε ≥ 1;
vβ
c) interpolation between a) and b), if 0 < ε < 1.
vβ
6.2 Permissible tooth root stress
The permissible tooth root stress, σ , shall be calculated separately for pinion and wheel. The values
FP
should preferably be evaluated on the basis of the strength of a standard test gear instead of a prismatic
specimen, which deviates too much with respect to similarity in geometry, course of movement and
manufacture.
σσ= YY YY (3)
FP-B1 FE NT δ,relT-B1R,relT-B1 X
4 © ISO 2014 – All rights reserved
σσ= Y YY YY (4)
FP-B1F,lim ST NT δ,relT-B1R,relT-B1 X
where
σ is the allowable stress number (tooth root);
FE
σ = σ Y , is the basic bending strength of the un-notched specimen under the
FE F,lim1,2 ST
assumption that the material (including heat treatment) is fully elastic;
σ is the nominal stress number (bending) of the test gear, which accounts for mate-
F,lim
rial, heat treatment, and surface influence at test gear dimensions as specified in
ISO 6336-5;
Y is the stress correction factor for the dimensions of the standard test gear, Y = 2,0;
ST ST
Y is the relative notch sensitivity factor for the permissible stress number, related to the
δ,relT-B1
conditions at the standard test gear (see 6.5.2), Y = Y /Y accounts for the notch
δ,relT δ δT
sensitivity of the material;
Y is the relative surface condition factor (see 6.5.1), Y = Y /Y accounts for the sur-
R,relT-B1 R,relT R RT
face condition at the root fillet, related to the conditions at the test gear;
Y is the size factor for tooth root strength, which accounts for the influence of the module
X
on the tooth root strength (see 8.1);
Y is the life factor, which accounts for the influence of required numbers of cycles of
NT
operation (see 8.2).
6.3 Calculated safety factor
The evaluated tooth root stress, σ , shall be ≤σ , which is the permissible tooth root stress. The
F FP
calculated safety factor against tooth breakage shall be determined separately for pinion and wheel:
σ
FP-B1
SS=> (5)
F-B1 F,min
σ
F-B1
NOTE This is the calculated safety factor with respect to the transmitted torque.
Considerations in reference to the safety factors and the risk (probability) of failure are given in
ISO 10300-1:2014, 5.2.
6.4 Tooth root stress factors
6.4.1 Tooth form factor, Y
Fa
6.4.1.1 General
The tooth form factor, Y , accounts for the influence of the tooth form on the nominal tooth root stress
Fa
in the case of load application at the tooth tip. It is determined separately for pinion and wheel. In doing
so, the possibility to manufacture bevel and hypoid gears with different pressure angles at drive and
coast side shall be considered (see Figure 1).
In the case of gears with tip and root relief, the actual bending moment arm is slightly smaller, but this
should be neglected and the calculation is on the safe side.
Bevel gears without offset generally have octoid teeth and tip and root relief. However, deviations from
an involute profile are small, especially in view of the tooth root cord and bending moment arm, and
thus both, tip and root relief, may be neglected when calculating the tooth form factor.
The distance between the contact points of the 30° tangents at the root fillets of the tooth profile of the
virtual cylindrical gear is taken as the critical section for calculation (see Figure 1).
By method B1 of ISO 10300, the tooth form factor, Y , and stress correction factor, Y , are determined
Fa Sa
for the nominal gear without deviations. The slight reduction in tooth thickness for backlash between
teeth may be neglected for the load capacity calculation. However, the size reduction shall be taken into
account when the outer tooth thickness allowance A > 0,05 m .
sne mn
Figure 1 — Tooth root chordal thickness s and bending moment arm h in normal section
Fn Fa
for load application, F , at the tooth tip of the virtual cylindrical gear
n
6.4.1.2 Tooth form factor for generated gears
6.4.1.2.1 General
The tooth form factor, Y , of generated bevel gears is calculated with parameters of the active flank of
Fa
the virtual cylindrical gear in normal section which includes the corresponding effective pressure angle
α or α (see Annex A of ISO 10300-1:2014). However, the direction of the normal force, F , in relation
eD eC n
to the tangential force, F , is given by the generated pressure angle α or α .
vmt nD nC
Attention — The tooth form factor, Y , and its parameters shall be determined for the pinion
F
(suffix 1) and the wheel (suffix 2) separately:
6 © ISO 2014 – All rights reserved
h
FaD,C
6 cosα
FanD,C
m
mn
Y = (6)
FaD,C
s
Fn
cosα
nD,C
m
mn
where
h and α see 6.4.1.2.5;
FaD,C FanD,C
α = α = generated pressure angle for drive side (specified in ISO 23509);
n nD
α = α = generated pressure angle for coast side (specified in ISO 23509).
n nC
[1]
See Figure 1 for the explanation of parameters; see ISO 6336-3 for more information about the tooth
form factor.
6.4.1.2.2 Auxiliary quantities
For the calculation of the tooth root chord, s , and the bending moment arm, h , firstly, the auxiliary
Fn Fa
quantities E, G, H and ϑ shall be determined.
The parameter, E, is calculated for the magnitudes of the active flank. For generated hypoid gears,
the effective pressure angle α = α for the drive side and α = α (see ISO 23509) for the coast side,
e eD e eC
respectively, are used in Formula (7). Note, the cutter edge radii ρ and ρ as well as the protuberance,
a0D a0C
s , might also be different, but not h , which is the tool addendum:
prD,C a0
ρα1− sin −s
()
π a0D,CeD,CprD,CC
Ex=− tmh−−anα (7)
D,Cs mm na0eD,C
4 cosα
eD,C
ρ
h
a0D,C
a0
G =− +x (8)
D,C hm
m m
mn mn
E
2 ππ
D,C
H =− − (9)
D,C
z 23m
vnD,C mn
2G
D,C
ϑϑ=−tan H (10)
D,C D,CD,C
z
vnD,C
For the solution of the transcendent Formula (10), ϑ = π/6 may be inserted as the initial value. A suggested
value for the difference (ϑ – ϑ) is 0,000 001. In most cases, the calculation already converges after a
new
few iterations.
6.4.1.2.3 Tooth root chordal thickness, s
Fn
The tooth root chords s and s are calculated for pinion and wheel, each with the corresponding
FnD FnC
geometry data for the drive flank and the coast flank:
G ρ
π
D,C a0D,C
sm=−zmsin ϑ +−3 (11)
FnD,CmnvnD,C D,Cmn
3 cosϑ m
D,C mmn
Then, the respective tooth root chord s for pinion or wheel results in:
Fn
ss=0,5 +0,5s (12)
Fn FnDFnC
6.4.1.2.4 Fillet radius, ρ , at contact point of 30° tangent
F
The fillet radius, ρ , is calculated with the corresponding geometry data for the drive flank and the coast
F
flank:
2Gm
D,Cmn
ρρ=+ (13)
F D,Ca0D,C
coscϑϑzGos² − 2
()
D,CvnD,C D,CD,C
6.4.1.2.5 Bending moment arm, h
Fa
The bending moment arm, h , is calculated with geometry data referring to the drive flank and to the
Fa
coast flank:
d G ρ
m
π
vanD,C D,C a0D,C
mn
h =−cosγγsin tanα −z cos −−ϑ + (14)
()
FaD,C aD,C aD,C FanD,C vnD,C D,C
2 m 3 cosϑ m
mn D,C mn
where
αα=−γ (15)
FanD,C anD,CaD,C
d
vbnD,C
α = arccos (16)
anD,C
d
vanD,C
1 π
γ =+ 2 xx tanαα+ +−invinv α (17)
()
aD,C hm eD,C sm eD,C anDD,C
z 2
vnD,C
where
α = α = generated pressure angle for drive side;
e eD
α = α = effective pressure angle of coast side (specified in ISO 23509).
e eC
Data of the virtual cylindrical gears (pinion and wheel) in normal section, d , d and z , are specified
van vbn vn
in ISO 10300-1:2014, A.3. Dimensions at the basic rack profile of the tooth are shown in Figure 2.
At the design stage, the tooth form factor Y for bevel gears without offset may be calculated for a basic
Fa
rack profile of the tool with the following data α = 20°, h /m = 1,25, and ρ /m = 0,25. Diagrams
n a0 mn a0 mn
[1]
for this and other basic rack profiles are given in ISO 6336-3.
6.4.1.3 Tooth form factor for non-generated gears
The tooth form factor, Y , for non-generated bevel gears should be considered separately. In this case of
Fa
form cutting the slot profile of the wheel is identical to the tool profile and so the tooth form factor can
directly be determined (see Figure 2).
The tooth form factor of the pinion, which is manufactured by a specific generating process, may be
approximated by the formulae according to 6.4.1.2. Hypoid gears are calculated with geometry data for
drive side and coast side.
8 © ISO 2014 – All rights reserved
Tooth root thickness of the wheel (suffix 2):
sm=−πρ c22E −°os 30 (18)
FnD,CmnD,C a0D,C
where
ρα1− sin −s
π ()
a0D,CnD,CprD,CC
Ex=− tmh−−anα (19)
D,Csmmna0nD,C
4 cosα
nD,C
The tooth root chord s is then calculated by:
Fn
ss=+05,,05s (20)
Fn FnDFnC
Fillet radius at contact point of 30° tangent:
ρ = ρ (21)
FD,C a0D,C
Bending moment arm:
ρ
π
a0D,C
hh=− +−mx+− tan ααm tan (22)
FaD,Ca0 mn sm nD,C mn nD,C
Tooth form factor of the wheel according to Formula (6) with α = α :
FanD,C nD,C
h
FaD,C
m
mn
Y = (23)
FaD,C
s
Fn
m
mn
a) With protuberance
b) Without protuberance
Figure 2 — Dimensions at the basic rack profile of the tooth
10 © ISO 2014 – All rights reserved
6.4.2 Stress correction factor, Y
Sa
The stress correction factor, Y , accounts for the stress increasing notch effect in the root fillet as well
Sa
[1]
as for other stress components which arise beside the tooth root stress (see ISO 6336-3 for additional
information).
12,,12 + 3 L
aD,C
YL=+12,,013 q (24)
()
SaD,CaD,CsD,C
s
Fn
L = (25)
aD,C
h
FaD,C
s
Fn
q = (26)
sD,C
2ρ
FD,C
where
s is calculated for generated or non-generated gears according to Formula (12) or For-
Fn
mula (20);
h is calculated for generated or non-generated gears according to Formula (14) or For-
Fa
mula (22);
ρ is calculated for generated or non-generated gears is according to Formula (13) or For-
F
mula (21).
[1]
The range of validity of Formula (26) is 1 ≤ q < 8 (see ISO 6336-3 for the influence of grinding notches).
s
6.4.3 Contact ratio factor, Y
ε
The contact ratio factor, Y , converts the load application at the tooth tip, where the tooth form factor,
ε
Y , and stress correction factor, Y , apply, to the determinant point of load application.
Fa Sa
There are three ranges for ε to calculate Y :
vβ ε
a) for ε = 0:
vβ
07, 5
Y =+02, 5 ≥0,625 (27a)
ε
ε
vα
b) for 0 < ε ≤ 1:
vβ
07,,50 75
Y =+02, 5 −−ε 0,,375 ≥ 0 625 (27b)
ε vβ
ε ε
vααv
c) for ε > 1:
vβ
Y = 0,625
ε
(27c)
6.4.4 Bevel spiral angle factor, Y
BS
The bevel spiral angle factor, Y , accounts for the non-uniform distribution of the tooth root stress
BS
along the face width. The stress distribution depends on the inclination of the contact lines due to the
spiral angle. With an increasing spiral angle the inclination angle also increases till the contact lines are
limited by tip and root of the teeth. Thus, the face width is not completely used to carry the load. This
leads to a higher stress maximum in the tooth root in the middle of the face width (see Figure 3), where
a tooth developed into a plane is replaced by a cantilever beam.
Figure 3 — Definition of geometric parameters of tooth model
Y is given by the following empirical formulae [i.e. Formula (28) to Formula (31)]:
BS
a l
BS bb
Y =−10, 51⋅b + (28)
BS BS
c b
BS a
b b
aa
a =−0,,018 20 + 473 60− ,32 (29)
BS
h h
b b
aa
b =−0,,003 20 + 05260+ ,712 (30)
BS
h h
b b
aa
c =−0,,005 00 + 085 00+ ,54 (31)
BS
h h
with auxiliary values a , b , c
BS BS BS
12 © ISO 2014 – All rights reserved
The developed length of one tooth as face width of the calculation model:
b = b /cos β (32)
a v v
Part of the model’s face width covered by the contact line:
cosβ
vb
ll= (33)
bb bm
cosβ
v
Average tooth depth:
hh= +/h 2 (34)
()
m1 m2
with mean whole tooth depth, h , as specified in ISO 23509.
m
6.4.5 Load sharing factor, Y
LS
The load sharing factor, Y , for bending accounts for load sharing between two or more pairs of teeth:
LS
YZ= (35)
LS LS
with load sharing factor, Z , as specified in 6.4.2 of ISO 10300-2:2014.
LS
6.5 Permissible tooth root stress factors
6.5.1 Relative surface condition factor, Y
R,relT-B1
The tooth root strength depends on the surface condition at the root predominantly on the roughness in
the root fillet. The surface condition factor, Y , accounts for this dependence related to standard test
R,relT
[1]
gear conditions with Rz = 10 µm (see ISO 6336-3 for general remarks) and is determined separately
for pinion (suffix 1) and wheel (suffix 2). If no surface condition factors determined according to method
A are available, method B described in 6.5.1 shall be used.
Warning — This method is only valid if there are no scratches or similar defects deeper than 2 Rz.
The relative surface condition factor, Y , determined by tests with test specimens, may be taken
R,RelT
from Figure 4 as a function of roughness Rz and material.
For calculation, Formulae (36) to (41) shall be used depending on two ranges of roughness.
Range Rz < 1 µm:
a) For through hardened and case hardened steels:
Y = 1,12
R,relT
(36)
b) For non-hardened steels:
Y = 1,07
R,relT
(37)
c) For grey cast iron, nitrided and nitro carburized steels:
Y = 1,025
R,relT
(38)
Range 1 µm ≤ Rz ≤ 40 µm:
a) For through hardened and case hardened steels:
Y
R
Y == 1,,674−+0 529 Rz 1 (39)
()
R,relT
Y
RT
b) For non-hardened steels:
Y
1 100
R
Y == 5,,306−+4 203 Rz 1 (40)
()
R,relT
Y
RT
c) For grey cast iron, nitrided and nitro carburized
Y
1 200
R
Y == 4,,299−+3 259 Rz 1 (41)
()
R,relT
Y
RT
Key
Rz surface roughness (µm)
Y surface condition factor (–)
R,RelT
Figure 4 — Surface condition factor, Y , for permissible stress number
R,relT
relative to standard test gear dimensions
6.5.2 Relative notch sensitivity factor, Y
δ,relT-B1
The dynamic notch sensitivity factor, Y , indicates the amount by which the theoretical stress peak
δ
exceeds the permissible stress number in the case of fatigue breakage. It is a function of the material and
relative stress drop. It is possible to calculate the notch sensitivity factor on the basis of strength values
determined at un-notched or notched specimens, or at test gears. If more exact test results (method A)
are not available, method B described in 6.5.2 shall be used.
The calculation of permissible tooth root stresses of bevel gears is based on bending strength values
determined for both, bevel and cylindrical test gears. Therefore, the relative notch sensitivity factor,
Y , is the ratio between the sensitivity factor of the gear to be calculated and the sensitivity factor of
d,relT
14 © ISO 2014 – All rights reserved
the standard test gear. Y = Y /Y may be taken directly from Figure 5 as a function of q (see 6.4.2)
δ,relT δ δT s
of the gear to be calculated and of the material.
In order to calculate the relative notch sensitivity factor, Y according to method B1,
δ,relT
Formulae (42) and (43), representing the curves in Figure 5, shall be used:
' X
1+ ρχ
1,2
Y = (42)
δ,rel T 1,2
' X
1+ ρχ
T
X
χ =+12 q (43)
()
1,2 s1,2
where
ρ′ shall be taken from Table 1 as a function of the material;
X
is applicable to module m = 5, with the size influence accounted for by Y (see 8.1);
mn X
χ
1,2
X
is calculated with q =25, according to Formula (43).
χ =12,
sT
T
Table 1 — Slip layer thickness ρ′
Slip layer thickness
No. Material
ρ′
1 GG σ = 150 N/mm 0,3124
B
2 GG, GGG (ferr.) σ = 300 N/mm 0,3095
B
NT (nitr.), NV (nitr.), NV
3 for all hardnesses 0,1005
(nitrocar.)
4 St σ = 300 N/mm 0,0833
S
5 St σ = 400 N/mm 0,0445
S
6 V, GTS, GGG (perl., bain.) σ = 500 N/mm 0,0281
0,2
7 V, GTS, GGG (perl., bain.) σ = 600 N/mm 0,0194
0,2
8 V, GTS, GGG (perl., bain.) σ = 800 N/mm 0,0064
0,2
9 V, GTS, GGG (perl., bain.) σ = 1000 N/mm 0,0014
0,2
10 Eh, IF (root) for all hardnesses 0,0030
Key
q notch parameter (–)
s
Y stress correction factor (–)
Sa
Y relative notch sensitivity factor (–)
δ,relT
a
Complete insensitivity to notches.
b
Complete sensitivity to notches.
Figure 5 — Relative notch sensitivity factor with respect to standard test gear dimensions
16 © ISO 2014 – All rights reserved
7 Gear tooth rating formulae — Method B2
7.1 Tooth root stress formula
The tooth root stress is determined separately for pinion (suffix 1) and wheel (suffix 2):
σσ=
F-B2 F0-B2A v Fβα F FP-B2
with load factors K , K , K and K , as specified in ISO 10300-1.
A v Fβ Fα
The tooth root stress σ is defined as the maximum tensile stress arising at the tooth root due to the
F0-B2
nominal torque when an error-free gear is loaded.
When applying method B2, the combined geometry factor Y replaces the factors Y , Y , Y , Y and Y
P Fa Sa ε BS LS
of method B1 in the tooth root stress equation:
F
mt1,2
σ = Y (45)
F0-B2 P1,2
bm
12, mn
The value of Y is determined by Formula (46):
P
Y mm⋅
A1,2 mt1,2mn
Y = (46)
P1,2
Y
m
J1,2
et2
Substitution in Formula (45):
F m Y
mt1,2 mt1,2 A1,2
σ =⋅ ⋅ (47)
F0-B2
b Y
m
1,2 J1,2
et2
where
F is the nominal tangential force of bevel gears in accordance with 6.1 of ISO 10300-1:2014;
mt
Y is the root stress adjustment factor for method B2 (see 7.4.7);
A
Y is the bending strength geometry factor for method B2 (see 7.4.3).
J
The bending strength geometry factor, Y , evaluates the shape of the tooth, the position at which the
J
most damaging load is applied, the stress concentration due to the geometric shape of the root fillet,
the sharing of load between adjacent pairs of teeth, the tooth thickness balance between the wheel and
mating pinion, the effective face width due to lengthwise crowning of the teeth, and the buttressing
effect of an extended face width on one member of the pair. Both the tangential (bending) and radial
(compressive) components of the tooth load are included.
7.2 Permissible tooth root stress
The permissible tooth root stress, σ , is determined separately for pinion and wheel. It should be
FP
calculated on the basis of the strength determined at an actual gear. In this way, the reference value for
geometrical similarity, course of movement and manufacture lies within the field of application:
σσ= YY YY (48)
FP -B2 FE NT δ,,relT-B2R relT-B2X
σσ= YY YY Y (49)
FP-B2F,lim ST NTrδ, elT-B2 R,relT-B2X
where
σ is the allowable stress number (bending);
FE
σ = σ Y , the basic bending strength of the un-notched specimen under the
FE F,lim1,2 ST
assumption that the material (including heat treatment) is fully elastic;
σ is the nominal stress number (bending) of the standard test gear, which accounts for
F,lim
material, heat treatment and surface influence at test gear dimensions, as specified in
ISO 6336-5;
Y is the stress correction factor for the dimensions of the standard test gear, Y = 2,0;
ST ST
Y is the relative notch sensitivity factor (see 7.5.2) for the bending stress number related
δ,relT-B2
to the conditions at the standard test gear (Y = Y /Y accounts for the notch sensi-
δ,relT δ δT
tivity of the material);
Y is the relative surface condition factor (see 7.5.1) (Y = Y /Y accounts for the sur-
R,relT-B2 R,relT R RT
face condition at the root fillet, related to the conditions at the test gear);
Y is the size factor for tooth root strength (see 8.1), which accounts for the influence of
X
the module on the tooth root strength;
Y is the life factor, which accounts for the influence of required numbers of cycles of
NT
operation (see 8.2).
7.3 Calculated safety factor
The determined tooth root stress, σ , shall be ≤σ , which is the permissible tooth root stress. The
F FP
calculated safety factor against tooth breakage shall be determined separately for pinion and wheel, on
the basis of the bending stress number determined for the standard test gear:
σ
FP-B2
SS=> (50)
F-B2 F min
σ
F-B2
NOTE This is the calculated safety factor with respect to the transmitted torque.
Considerations in reference to the safety factors and the risk (probability) of failure are given in
ISO 10300-1:2014, 5.2.
7.4 Tooth root stress factors
7.4.1 General
To calculate the bending strength geometry factor, Y , the formulae in 7.4.3 should be used. Because of
J
the complexity of the calculation, computerization is recommended.
[3]
ANSI/AGMA 2003-C10 contains graphs for the bevel geometry factor, Y , for straight, Zerol and spiral
J
bevel gears for a series of gear designs, based on the smaller of the face width to be chosen b = 0,3R or
e
[4]
b = 10m . Corresponding graphs for hypoid gears can be found in AGMA 932-A05. These may be used
et
whenever the tooth proportions and thickness, face widths, tool edge radii, pressure and spiral angles
of the design, and driving with the concave side, correspond to those in the graphs.
18 © ISO 2014 – All rights reserved
7.4.2 Stress parabola according to Lewis
The basis for method B2 is the Lewis formula applied to a virtual cylindrical gear, which has been
defined in transverse section as specified in Annex B of ISO 10300-1:2014, with the following additions
and modifications:
— the tooth strength is considered in the normal section rather than in the transverse section;
— the position of the point of load application is determined by taking into account theoretical lines of
contact, tooth bearing modifications and experimental evidence;
— the amount of load carried by one tooth is estimated based on tooth bearing modifica
...
INTERNATIONAL ISO
STANDARD 10300-3
Second edition
2014-04-01
Calculation of load capacity of bevel
gears —
Part 3:
Calculation of tooth root strength
Calcul de la capacité de charge des engrenages coniques —
Partie 3: Calcul de la résistance du pied de dent
Reference number
©
ISO 2014
© ISO 2014
All rights reserved. Unless otherwise specified, 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
Case postale 56 • CH-1211 Geneva 20
Tel. + 41 22 749 01 11
Fax + 41 22 749 09 47
E-mail copyright@iso.org
Web www.iso.org
Published in Switzerland
ii © ISO 2014 – All rights reserved
Contents Page
Foreword .iv
Introduction .v
1 Scope . 1
2 Normative references . 1
3 Terms and definitions . 1
4 Symbols, units and abbreviated terms . 2
5 General rating procedure . 2
6 Gear tooth rating formulae — Method B1 . 3
6.1 Tooth root stress formula . 3
6.2 Permissible tooth root stress . 4
6.3 Calculated safety factor . 5
6.4 Tooth root stress factors . 5
6.5 Permissible tooth root stress factors .13
7 Gear tooth rating formulae — Method B2 .17
7.1 Tooth root stress formula .17
7.2 Permissible tooth root stress .17
7.3 Calculated safety factor .18
7.4 Tooth root stress factors .18
7.5 Permissible tooth root stress factors .36
8 Factors for permissible tooth root stress common for method B1 and method B2 .36
8.1 Size factor, Y .36
X
8.2 Life factor, Y .38
NT
Bibliography .41
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. 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. 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 on the meaning of ISO specific terms and expressions related to conformity
assessment, as well as information about ISO’s adherence to the WTO principles in the Technical Barriers
to Trade (TBT) see the following URL: Foreword - Supplementary information
The committee responsible for this document is ISO/TC 60, Gears, Subcommittee SC 2, Gear capacity
calculation.
This second edition cancels and replaces the first edition (ISO 10300-3:2001), which has been technically
revised.
ISO 10300 consists of the following parts, under the general title Calculation of load capacity of bevel
gears:
— Part 1: Introduction and general influence factors
— Part 2: Calculation of surface durability (pitting)
— Part 3: Calculation of tooth root strength
iv © ISO 2014 – All rights reserved
Introduction
When ISO 10300:2001 (all parts, withdrawn) became due for (its first) revision, the opportunity was
taken to include hypoid gears, since previously the series only allowed for calculating the load capacity
of bevel gears without offset axes. The former structure is retained, i.e. three parts of the ISO 10300
series, together with ISO 6336-5, and it is intended to establish general principles and procedures for
rating of bevel gears. Moreover, ISO 10300 (all parts) is designed to facilitate the application of future
knowledge and developments, as well as the exchange of information gained from experience.
In view of the decision for ISO 10300 (all parts) to cover hypoid gears also, it was agreed to include a
separate clause: “Gear tooth rating formulae — Method B2” in this part of ISO 10300, while the former
methods B and B1 were combined into one method, i.e. method B1. So, it became necessary to present
a new, clearer structure of the three parts, which is illustrated in ISO 10300-1:2014, Figure 1. Note,
ISO 10300 (all parts) gives no preferences in terms of when to use method B1 and when method B2.
Failure of gear teeth by breakage can be brought about in many ways; severe instantaneous overloads,
excessive pitting, case crushing and bending fatigue are a few. The strength ratings determined by the
formulae in this part of ISO 10300 are based on cantilever projection theory modified to consider the
following:
— compressive stress at the tooth roots caused by the radial component of the tooth load;
— non-uniform moment distribution of the load, resulting from the inclined contact lines on the teeth
of spiral bevel gears;
— stress concentration at the tooth root fillet;
— load sharing between adjacent contacting teeth;
— lack of smoothness due to a low contact ratio.
The formulae are used to determine a load rating, which prevents tooth root fracture during the design
life of the bevel gear. Nevertheless, if there is insufficient material under the teeth (in the rim), a fracture
can occur from the root through the rim of the gear blank or to the bore (a type of failure not covered
by this part of ISO 10300). Moreover, it is possible that special applications require additional blank
material to support the load.
Surface distress (pitting or wear) can limit the strength rating, either due to stress concentration
around large sharp cornered pits, or due to wear steps on the tooth surface. Neither of these effects is
considered in this part of ISO 10300.
In most cases, the maximum tensile stress at the tooth root (arising from bending at the root when the
load is applied to the tooth flank) can be used as a determinant criterion for the assessment of the tooth
root strength. If the permissible stress number is exceeded, the teeth can break.
When calculating the tooth root stresses of straight bevel gears, this part of ISO 10300 starts from the
assumption that the load is applied at the tooth tip of the virtual cylindrical gear. The load is subsequently
converted to the outer point of single tooth contact. The procedure thus corresponds to method C for the
[1]
tooth root stress of cylindrical gears (see ISO 6336-3 ).
For spiral bevel and hypoid gears with a high face contact ratio of ε > 1 (method B1) or with a modified
vβ
contact ratio of ε > 2 (method B2), the midpoint in the zone of action is regarded as the critical point
vγ
of load application.
The breakage of a tooth generally means the end of a gear’s life. It is often the case that all gear teeth are
destroyed as a consequence of the breakage of a single tooth. A safety factor, S , against tooth breakage
F
higher than the safety factor against damage due to pitting is, therefore, generally to be preferred
(see ISO 10300-1).
INTERNATIONAL STANDARD ISO 10300-3:2014(E)
Calculation of load capacity of bevel gears —
Part 3:
Calculation of tooth root strength
1 Scope
This part of ISO 10300 specifies the fundamental formulae for use in the tooth root stress calculation
of straight and helical (skew), Zerol and spiral bevel gears including hypoid gears, with a minimum rim
thickness under the root of 3,5 m . All load influences on tooth root stress are included, insofar as
mn
they are the result of load transmitted by the gearing and able to be evaluated quantitatively. Stresses,
such as those caused by the shrink fitting of gear rims, which are superposed on stresses due to tooth
loading, are intended to be considered in the calculation of the tooth root stress, σ , or the permissible
F
tooth root stress σ . This part of ISO 10300 is not applicable in the assessment of the so-called flank
FP
breakage, a tooth internal fatigue fracture (TIFF).
The formulae in this part of ISO 10300 are based on virtual cylindrical gears and restricted to bevel
gears whose virtual cylindrical gears have transverse contact ratios of ε < 2. The results are valid
vα
[1]
within the range of the applied factors as specified in ISO 10300-1 (see also ISO 6336-3 ). Additionally,
the given relationships are valid for bevel gears, of which the sum of profile shift coefficients of pinion
and wheel is zero (see ISO 23509).
This part of ISO 10300 does not apply to stress levels above those permitted for 10 cycles, as stresses
in that range could exceed the elastic limit of the gear tooth.
Warning — The user is cautioned that when the formulae are used for large average mean spiral
angles (β + β )/2 > 45°, for effective pressure angles α > 30° and/or for large face widths
m1 m2 e
b > 13 m , the calculated results of ISO 10300 (all parts) should be confirmed by experience.
mn
2 Normative references
The following documents, in whole or in part, are normatively referenced in this document and are
indispensable to its application. For dated references, only the edition cited applies. For undated
references, the latest edition of the referenced document (including any amendments) applies.
ISO 1122-1, Vocabulary of gear terms — Part 1: Definitions related to geometry
ISO 6336-5, Calculation of load capacity of spur and helical gears — Part 5: Strength and quality of materials
ISO 10300-1:2014, Calculation of load capacity of bevel gears — Part 1: Introduction and general influence
factors
ISO 10300-2:2014, Calculation of load capacity of bevel gears — Part 2: Calculation of surface durability
(pitting)
ISO 23509:2006, Bevel and hypoid gear geometry
3 Terms and definitions
For the purposes of this document, the terms and definitions given in ISO 1122-1 and ISO 23509
(geometrical gear terms) and the following apply.
3.1
tooth root breakage
failure of gear teeth at the tooth root by static or dynamic overload
3.2
nominal tooth root stress
σ
F0
bending stress in the critical section of the tooth root calculated for the critical point of load application
for error-free gears loaded by a constant nominal torque
3.3
tooth root stress
σ
F
determinant bending stress in the critical section of the tooth root calculated for the critical point of
load application including the load factors which consider static and dynamic loads and load distribution
3.4
nominal stress number
σ
F,lim
maximum tooth root stress of standardized test gears and determined at standardized operating
conditions as specified in ISO 6336-5
3.5
allowable stress number
σ
FE
maximum bending stress of the un-notched test piece under the assumption that the material is fully
elastic
3.6
permissible tooth root stress
σ
FP
maximum tooth root stress of the evaluated gear set including all influence factors
4 Symbols, units and abbreviated terms
For the purposes of this document, the symbols and units given in Table 1 and Table 2 of ISO 10300-1:2014,
as well as the abbreviated terms given in Table 1 of ISO 10300-2:2014, apply (see ISO 6336-5).
5 General rating procedure
There are two main methods for determining tooth bending strength of bevel and hypoid gears:
method B1 and method B2. They are provided in Clauses 6 and 7, while Clause 8 contains those influence
factors which are equal for both methods. With method B1, the same set of formulae may be used for
bevel and hypoid gears; method B2 partly has different sets of formulae for bevel gears and for hypoid
gears (see 7.4.3 for general aspects).
With both methods, the capability of a gear tooth to resist tooth root stresses shall be determined by the
comparison of the following stress values:
— tooth root stress σ , based on the geometry of the tooth, the accuracy of its manufacture, the
F
rigidity of the gear blanks, bearings and housing, and the operating torque, expressed by the tooth
root stress formula (see 6.1 and 7.1);
— permissible tooth root stress σ , based on the bending stress number, σ , of a standard test
FP F,lim
gear and the effect of the operating conditions under which the gears operate, expressed by the
permissible tooth root stress formula (see 6.2 and 7.2).
2 © ISO 2014 – All rights reserved
NOTE In respect of the permissible tooth root stress, reference is made to a stress “number”, a designation
adopted because pure stress, as determined by laboratory testing, is not calculated by the formulae in this part
of ISO 10300. Instead, an arbitrary value is calculated and used in this part of ISO 10300, with accompanying
changes to the allowable stress number in order to maintain consistency for design comparison.
The ratio of the permissible root stress and the calculated root stress is the safety factor S . The value of
F
the minimum safety factor for tooth root stress, S , should be ≥ 1,3 for spiral bevel gears. For straight
F,min
bevel gears, or where β ≤ 5°, S should be ≥ 1,5.
m F,min
It is recommended that the gear designer and customer agree on the value of the minimum safety factor.
Tooth breakage usually ends transmission service life. The destruction of all gears in a transmission can
be a consequence of the breakage of one tooth, then, the drive train between input and output shafts
is interrupted. Therefore, the chosen value of the safety factor, S , against tooth breakage should be
F
carefully chosen to fulfil the application requirements (see ISO 10300-1 for general comments on the
choice of safety factor).
6 Gear tooth rating formulae — Method B1
6.1 Tooth root stress formula
The calculation of the tooth root stress is based on the maximum bending stress at the tooth root. It is
determined separately for pinion (suffix 1) and wheel (suffix 2); in the case of hypoid gears, additionally
for drive flank (suffix D) and coast flank (suffix C):
σσ=
F-B1 F0-B1A v Fβα F FP-B1
with the load factors K , K , K , K as specified in ISO 10300-1.
A v Fβ Fα
The nominal tooth root stress is defined as the maximum bending stress at the tooth root (30° tangent
to the root fillet):
F
vmt
σ = YY YY Y (2)
F0-B1 Fa Sa ε BS LS
bm
v mn
where
F is the nominal tangential force of the virtual cylindrical gear which should be in accordance
vmt
with Formula (2) of ISO 10300-1:2014;
b is the face width of the virtual cylindrical gear calculated for the active flank, drive or coast
v
side, as specified in ISO 10300-1:2014, Annex A;
Y is the tooth form factor (see 6.4.1), which accounts for the influence of the tooth form on the
Fa
nominal bending stress at the tooth root for load application at the tooth tip;
Y is the stress correction factor (see 6.4.2), which accounts for the stress increasing notch
Sa
effect in the root fillet, as well as for the radial component of the tooth load and the fact that
the stress condition in the critical root section is complex, but not for the influence of the
bending moment arm;
Y is the contact ratio factor (see 6.4.3), which accounts for the conversion of the root stress
ε
determined for the load application at the tooth tip to the determinant position;
Y is the bevel spiral angle factor, which accounts for smaller values for l compared to the
BS bm
total face width, b , and the inclined lines of contact (see 6.4.4);
v
Y is the load sharing factor, which accounts for load distribution between two or more pairs of
LS
teeth (see 6.4.5).
The determinant position of load application is:
a) the outer point of single tooth contact, if ε = 0;
vβ
b) the midpoint of the zone of action, if ε ≥ 1;
vβ
c) interpolation between a) and b), if 0 < ε < 1.
vβ
6.2 Permissible tooth root stress
The permissible tooth root stress, σ , shall be calculated separately for pinion and wheel. The values
FP
should preferably be evaluated on the basis of the strength of a standard test gear instead of a prismatic
specimen, which deviates too much with respect to similarity in geometry, course of movement and
manufacture.
σσ= YY YY (3)
FP-B1 FE NT δ,relT-B1R,relT-B1 X
4 © ISO 2014 – All rights reserved
σσ= Y YY YY (4)
FP-B1F,lim ST NT δ,relT-B1R,relT-B1 X
where
σ is the allowable stress number (tooth root);
FE
σ = σ Y , is the basic bending strength of the un-notched specimen under the
FE F,lim1,2 ST
assumption that the material (including heat treatment) is fully elastic;
σ is the nominal stress number (bending) of the test gear, which accounts for mate-
F,lim
rial, heat treatment, and surface influence at test gear dimensions as specified in
ISO 6336-5;
Y is the stress correction factor for the dimensions of the standard test gear, Y = 2,0;
ST ST
Y is the relative notch sensitivity factor for the permissible stress number, related to the
δ,relT-B1
conditions at the standard test gear (see 6.5.2), Y = Y /Y accounts for the notch
δ,relT δ δT
sensitivity of the material;
Y is the relative surface condition factor (see 6.5.1), Y = Y /Y accounts for the sur-
R,relT-B1 R,relT R RT
face condition at the root fillet, related to the conditions at the test gear;
Y is the size factor for tooth root strength, which accounts for the influence of the module
X
on the tooth root strength (see 8.1);
Y is the life factor, which accounts for the influence of required numbers of cycles of
NT
operation (see 8.2).
6.3 Calculated safety factor
The evaluated tooth root stress, σ , shall be ≤σ , which is the permissible tooth root stress. The
F FP
calculated safety factor against tooth breakage shall be determined separately for pinion and wheel:
σ
FP-B1
SS=> (5)
F-B1 F,min
σ
F-B1
NOTE This is the calculated safety factor with respect to the transmitted torque.
Considerations in reference to the safety factors and the risk (probability) of failure are given in
ISO 10300-1:2014, 5.2.
6.4 Tooth root stress factors
6.4.1 Tooth form factor, Y
Fa
6.4.1.1 General
The tooth form factor, Y , accounts for the influence of the tooth form on the nominal tooth root stress
Fa
in the case of load application at the tooth tip. It is determined separately for pinion and wheel. In doing
so, the possibility to manufacture bevel and hypoid gears with different pressure angles at drive and
coast side shall be considered (see Figure 1).
In the case of gears with tip and root relief, the actual bending moment arm is slightly smaller, but this
should be neglected and the calculation is on the safe side.
Bevel gears without offset generally have octoid teeth and tip and root relief. However, deviations from
an involute profile are small, especially in view of the tooth root cord and bending moment arm, and
thus both, tip and root relief, may be neglected when calculating the tooth form factor.
The distance between the contact points of the 30° tangents at the root fillets of the tooth profile of the
virtual cylindrical gear is taken as the critical section for calculation (see Figure 1).
By method B1 of ISO 10300, the tooth form factor, Y , and stress correction factor, Y , are determined
Fa Sa
for the nominal gear without deviations. The slight reduction in tooth thickness for backlash between
teeth may be neglected for the load capacity calculation. However, the size reduction shall be taken into
account when the outer tooth thickness allowance A > 0,05 m .
sne mn
Figure 1 — Tooth root chordal thickness s and bending moment arm h in normal section
Fn Fa
for load application, F , at the tooth tip of the virtual cylindrical gear
n
6.4.1.2 Tooth form factor for generated gears
6.4.1.2.1 General
The tooth form factor, Y , of generated bevel gears is calculated with parameters of the active flank of
Fa
the virtual cylindrical gear in normal section which includes the corresponding effective pressure angle
α or α (see Annex A of ISO 10300-1:2014). However, the direction of the normal force, F , in relation
eD eC n
to the tangential force, F , is given by the generated pressure angle α or α .
vmt nD nC
Attention — The tooth form factor, Y , and its parameters shall be determined for the pinion
F
(suffix 1) and the wheel (suffix 2) separately:
6 © ISO 2014 – All rights reserved
h
FaD,C
6 cosα
FanD,C
m
mn
Y = (6)
FaD,C
s
Fn
cosα
nD,C
m
mn
where
h and α see 6.4.1.2.5;
FaD,C FanD,C
α = α = generated pressure angle for drive side (specified in ISO 23509);
n nD
α = α = generated pressure angle for coast side (specified in ISO 23509).
n nC
[1]
See Figure 1 for the explanation of parameters; see ISO 6336-3 for more information about the tooth
form factor.
6.4.1.2.2 Auxiliary quantities
For the calculation of the tooth root chord, s , and the bending moment arm, h , firstly, the auxiliary
Fn Fa
quantities E, G, H and ϑ shall be determined.
The parameter, E, is calculated for the magnitudes of the active flank. For generated hypoid gears,
the effective pressure angle α = α for the drive side and α = α (see ISO 23509) for the coast side,
e eD e eC
respectively, are used in Formula (7). Note, the cutter edge radii ρ and ρ as well as the protuberance,
a0D a0C
s , might also be different, but not h , which is the tool addendum:
prD,C a0
ρα1− sin −s
()
π a0D,CeD,CprD,CC
Ex=− tmh−−anα (7)
D,Cs mm na0eD,C
4 cosα
eD,C
ρ
h
a0D,C
a0
G =− +x (8)
D,C hm
m m
mn mn
E
2 ππ
D,C
H =− − (9)
D,C
z 23m
vnD,C mn
2G
D,C
ϑϑ=−tan H (10)
D,C D,CD,C
z
vnD,C
For the solution of the transcendent Formula (10), ϑ = π/6 may be inserted as the initial value. A suggested
value for the difference (ϑ – ϑ) is 0,000 001. In most cases, the calculation already converges after a
new
few iterations.
6.4.1.2.3 Tooth root chordal thickness, s
Fn
The tooth root chords s and s are calculated for pinion and wheel, each with the corresponding
FnD FnC
geometry data for the drive flank and the coast flank:
G ρ
π
D,C a0D,C
sm=−zmsin ϑ +−3 (11)
FnD,CmnvnD,C D,Cmn
3 cosϑ m
D,C mmn
Then, the respective tooth root chord s for pinion or wheel results in:
Fn
ss=0,5 +0,5s (12)
Fn FnDFnC
6.4.1.2.4 Fillet radius, ρ , at contact point of 30° tangent
F
The fillet radius, ρ , is calculated with the corresponding geometry data for the drive flank and the coast
F
flank:
2Gm
D,Cmn
ρρ=+ (13)
F D,Ca0D,C
coscϑϑzGos² − 2
()
D,CvnD,C D,CD,C
6.4.1.2.5 Bending moment arm, h
Fa
The bending moment arm, h , is calculated with geometry data referring to the drive flank and to the
Fa
coast flank:
d G ρ
m
π
vanD,C D,C a0D,C
mn
h =−cosγγsin tanα −z cos −−ϑ + (14)
()
FaD,C aD,C aD,C FanD,C vnD,C D,C
2 m 3 cosϑ m
mn D,C mn
where
αα=−γ (15)
FanD,C anD,CaD,C
d
vbnD,C
α = arccos (16)
anD,C
d
vanD,C
1 π
γ =+ 2 xx tanαα+ +−invinv α (17)
()
aD,C hm eD,C sm eD,C anDD,C
z 2
vnD,C
where
α = α = generated pressure angle for drive side;
e eD
α = α = effective pressure angle of coast side (specified in ISO 23509).
e eC
Data of the virtual cylindrical gears (pinion and wheel) in normal section, d , d and z , are specified
van vbn vn
in ISO 10300-1:2014, A.3. Dimensions at the basic rack profile of the tooth are shown in Figure 2.
At the design stage, the tooth form factor Y for bevel gears without offset may be calculated for a basic
Fa
rack profile of the tool with the following data α = 20°, h /m = 1,25, and ρ /m = 0,25. Diagrams
n a0 mn a0 mn
[1]
for this and other basic rack profiles are given in ISO 6336-3.
6.4.1.3 Tooth form factor for non-generated gears
The tooth form factor, Y , for non-generated bevel gears should be considered separately. In this case of
Fa
form cutting the slot profile of the wheel is identical to the tool profile and so the tooth form factor can
directly be determined (see Figure 2).
The tooth form factor of the pinion, which is manufactured by a specific generating process, may be
approximated by the formulae according to 6.4.1.2. Hypoid gears are calculated with geometry data for
drive side and coast side.
8 © ISO 2014 – All rights reserved
Tooth root thickness of the wheel (suffix 2):
sm=−πρ c22E −°os 30 (18)
FnD,CmnD,C a0D,C
where
ρα1− sin −s
π ()
a0D,CnD,CprD,CC
Ex=− tmh−−anα (19)
D,Csmmna0nD,C
4 cosα
nD,C
The tooth root chord s is then calculated by:
Fn
ss=+05,,05s (20)
Fn FnDFnC
Fillet radius at contact point of 30° tangent:
ρ = ρ (21)
FD,C a0D,C
Bending moment arm:
ρ
π
a0D,C
hh=− +−mx+− tan ααm tan (22)
FaD,Ca0 mn sm nD,C mn nD,C
Tooth form factor of the wheel according to Formula (6) with α = α :
FanD,C nD,C
h
FaD,C
m
mn
Y = (23)
FaD,C
s
Fn
m
mn
a) With protuberance
b) Without protuberance
Figure 2 — Dimensions at the basic rack profile of the tooth
10 © ISO 2014 – All rights reserved
6.4.2 Stress correction factor, Y
Sa
The stress correction factor, Y , accounts for the stress increasing notch effect in the root fillet as well
Sa
[1]
as for other stress components which arise beside the tooth root stress (see ISO 6336-3 for additional
information).
12,,12 + 3 L
aD,C
YL=+12,,013 q (24)
()
SaD,CaD,CsD,C
s
Fn
L = (25)
aD,C
h
FaD,C
s
Fn
q = (26)
sD,C
2ρ
FD,C
where
s is calculated for generated or non-generated gears according to Formula (12) or For-
Fn
mula (20);
h is calculated for generated or non-generated gears according to Formula (14) or For-
Fa
mula (22);
ρ is calculated for generated or non-generated gears is according to Formula (13) or For-
F
mula (21).
[1]
The range of validity of Formula (26) is 1 ≤ q < 8 (see ISO 6336-3 for the influence of grinding notches).
s
6.4.3 Contact ratio factor, Y
ε
The contact ratio factor, Y , converts the load application at the tooth tip, where the tooth form factor,
ε
Y , and stress correction factor, Y , apply, to the determinant point of load application.
Fa Sa
There are three ranges for ε to calculate Y :
vβ ε
a) for ε = 0:
vβ
07, 5
Y =+02, 5 ≥0,625 (27a)
ε
ε
vα
b) for 0 < ε ≤ 1:
vβ
07,,50 75
Y =+02, 5 −−ε 0,,375 ≥ 0 625 (27b)
ε vβ
ε ε
vααv
c) for ε > 1:
vβ
Y = 0,625
ε
(27c)
6.4.4 Bevel spiral angle factor, Y
BS
The bevel spiral angle factor, Y , accounts for the non-uniform distribution of the tooth root stress
BS
along the face width. The stress distribution depends on the inclination of the contact lines due to the
spiral angle. With an increasing spiral angle the inclination angle also increases till the contact lines are
limited by tip and root of the teeth. Thus, the face width is not completely used to carry the load. This
leads to a higher stress maximum in the tooth root in the middle of the face width (see Figure 3), where
a tooth developed into a plane is replaced by a cantilever beam.
Figure 3 — Definition of geometric parameters of tooth model
Y is given by the following empirical formulae [i.e. Formula (28) to Formula (31)]:
BS
a l
BS bb
Y =−10, 51⋅b + (28)
BS BS
c b
BS a
b b
aa
a =−0,,018 20 + 473 60− ,32 (29)
BS
h h
b b
aa
b =−0,,003 20 + 05260+ ,712 (30)
BS
h h
b b
aa
c =−0,,005 00 + 085 00+ ,54 (31)
BS
h h
with auxiliary values a , b , c
BS BS BS
12 © ISO 2014 – All rights reserved
The developed length of one tooth as face width of the calculation model:
b = b /cos β (32)
a v v
Part of the model’s face width covered by the contact line:
cosβ
vb
ll= (33)
bb bm
cosβ
v
Average tooth depth:
hh= +/h 2 (34)
()
m1 m2
with mean whole tooth depth, h , as specified in ISO 23509.
m
6.4.5 Load sharing factor, Y
LS
The load sharing factor, Y , for bending accounts for load sharing between two or more pairs of teeth:
LS
YZ= (35)
LS LS
with load sharing factor, Z , as specified in 6.4.2 of ISO 10300-2:2014.
LS
6.5 Permissible tooth root stress factors
6.5.1 Relative surface condition factor, Y
R,relT-B1
The tooth root strength depends on the surface condition at the root predominantly on the roughness in
the root fillet. The surface condition factor, Y , accounts for this dependence related to standard test
R,relT
[1]
gear conditions with Rz = 10 µm (see ISO 6336-3 for general remarks) and is determined separately
for pinion (suffix 1) and wheel (suffix 2). If no surface condition factors determined according to method
A are available, method B described in 6.5.1 shall be used.
Warning — This method is only valid if there are no scratches or similar defects deeper than 2 Rz.
The relative surface condition factor, Y , determined by tests with test specimens, may be taken
R,RelT
from Figure 4 as a function of roughness Rz and material.
For calculation, Formulae (36) to (41) shall be used depending on two ranges of roughness.
Range Rz < 1 µm:
a) For through hardened and case hardened steels:
Y = 1,12
R,relT
(36)
b) For non-hardened steels:
Y = 1,07
R,relT
(37)
c) For grey cast iron, nitrided and nitro carburized steels:
Y = 1,025
R,relT
(38)
Range 1 µm ≤ Rz ≤ 40 µm:
a) For through hardened and case hardened steels:
Y
R
Y == 1,,674−+0 529 Rz 1 (39)
()
R,relT
Y
RT
b) For non-hardened steels:
Y
1 100
R
Y == 5,,306−+4 203 Rz 1 (40)
()
R,relT
Y
RT
c) For grey cast iron, nitrided and nitro carburized
Y
1 200
R
Y == 4,,299−+3 259 Rz 1 (41)
()
R,relT
Y
RT
Key
Rz surface roughness (µm)
Y surface condition factor (–)
R,RelT
Figure 4 — Surface condition factor, Y , for permissible stress number
R,relT
relative to standard test gear dimensions
6.5.2 Relative notch sensitivity factor, Y
δ,relT-B1
The dynamic notch sensitivity factor, Y , indicates the amount by which the theoretical stress peak
δ
exceeds the permissible stress number in the case of fatigue breakage. It is a function of the material and
relative stress drop. It is possible to calculate the notch sensitivity factor on the basis of strength values
determined at un-notched or notched specimens, or at test gears. If more exact test results (method A)
are not available, method B described in 6.5.2 shall be used.
The calculation of permissible tooth root stresses of bevel gears is based on bending strength values
determined for both, bevel and cylindrical test gears. Therefore, the relative notch sensitivity factor,
Y , is the ratio between the sensitivity factor of the gear to be calculated and the sensitivity factor of
d,relT
14 © ISO 2014 – All rights reserved
the standard test gear. Y = Y /Y may be taken directly from Figure 5 as a function of q (see 6.4.2)
δ,relT δ δT s
of the gear to be calculated and of the material.
In order to calculate the relative notch sensitivity factor, Y according to method B1,
δ,relT
Formulae (42) and (43), representing the curves in Figure 5, shall be used:
' X
1+ ρχ
1,2
Y = (42)
δ,rel T 1,2
' X
1+ ρχ
T
X
χ =+12 q (43)
()
1,2 s1,2
where
ρ′ shall be taken from Table 1 as a function of the material;
X
is applicable to module m = 5, with the size influence accounted for by Y (see 8.1);
mn X
χ
1,2
X
is calculated with q =25, according to Formula (43).
χ =12,
sT
T
Table 1 — Slip layer thickness ρ′
Slip layer thickness
No. Material
ρ′
1 GG σ = 150 N/mm 0,3124
B
2 GG, GGG (ferr.) σ = 300 N/mm 0,3095
B
NT (nitr.), NV (nitr.), NV
3 for all hardnesses 0,1005
(nitrocar.)
4 St σ = 300 N/mm 0,0833
S
5 St σ = 400 N/mm 0,0445
S
6 V, GTS, GGG (perl., bain.) σ = 500 N/mm 0,0281
0,2
7 V, GTS, GGG (perl., bain.) σ = 600 N/mm 0,0194
0,2
8 V, GTS, GGG (perl., bain.) σ = 800 N/mm 0,0064
0,2
9 V, GTS, GGG (perl., bain.) σ = 1000 N/mm 0,0014
0,2
10 Eh, IF (root) for all hardnesses 0,0030
Key
q notch parameter (–)
s
Y stress correction factor (–)
Sa
Y relative notch sensitivity factor (–)
δ,relT
a
Complete insensitivity to notches.
b
Complete sensitivity to notches.
Figure 5 — Relative notch sensitivity factor with respect to standard test gear dimensions
16 © ISO 2014 – All rights reserved
7 Gear tooth rating formulae — Method B2
7.1 Tooth root stress formula
The tooth root stress is determined separately for pinion (suffix 1) and wheel (suffix 2):
σσ=
F-B2 F0-B2A v Fβα F FP-B2
with load factors K , K , K and K , as specified in ISO 10300-1.
A v Fβ Fα
The tooth root stress σ is defined as the maximum tensile stress arising at the tooth root due to the
F0-B2
nominal torque when an error-free gear is loaded.
When applying method B2, the combined geometry factor Y replaces the factors Y , Y , Y , Y and Y
P Fa Sa ε BS LS
of method B1 in the tooth root stress equation:
F
mt1,2
σ = Y (45)
F0-B2 P1,2
bm
12, mn
The value of Y is determined by Formula (46):
P
Y mm⋅
A1,2 mt1,2mn
Y = (46)
P1,2
Y
m
J1,2
et2
Substitution in Formula (45):
F m Y
mt1,2 mt1,2 A1,2
σ =⋅ ⋅ (47)
F0-B2
b Y
m
1,2 J1,2
et2
where
F is the nominal tangential force of bevel gears in accordance with 6.1 of ISO 10300-1:2014;
mt
Y is the root stress adjustment factor for method B2 (see 7.4.7);
A
Y is the bending strength geometry factor for method B2 (see 7.4.3).
J
The bending strength geometry factor, Y , evaluates the shape of the tooth, the position at which the
J
most damaging load is applied, the stress concentration due to the geometric shape of the root fillet,
the sharing of load between adjacent pairs of teeth, the tooth thickness balance between the wheel and
mating pinion, the effective face width due to lengthwise crowning of the teeth, and the buttressing
effect of an extended face width on one member of the pair. Both the tangential (bending) and radial
(compressive) components of the tooth load are included.
7.2 Permissible tooth root stress
The permissible tooth root stress, σ , is determined separately for pinion and wheel. It should be
FP
calculated on the basis of the strength determined at an actual gear. In this way, the reference value for
geometrical similarity, course of movement and manufacture lies within the field of application:
σσ= YY YY (48)
FP -B2 FE NT δ,,relT-B2R relT-B2X
σσ= YY YY Y (49)
FP-B2F,lim ST NTrδ, elT-B2 R,relT-B2X
where
σ is the allowable stress number (bending);
FE
σ = σ Y , the basic bending strength of the un-notched specimen under the
FE F,lim1,2 ST
assumption that the material (including heat treatment) is fully elastic;
σ is the nominal stress number (bending) of the standard test gear, which accounts for
F,lim
material, heat treatment and surface influence at test gear dimensions, as specified in
ISO 6336-5;
Y is the stress correction factor for the dimensions of the standard test gear, Y = 2,0;
ST ST
Y is the relative notch sensitivity factor (see 7.5.2) for the bending stress number related
δ,relT-B2
to the conditions at the standard test gear (Y = Y /Y accounts for the notch sensi-
δ,relT δ δT
tivity of the material);
Y is the relative surface condition factor (see 7.5.1) (Y = Y /Y accounts for the sur-
R,relT-B2 R,relT R RT
face condition at the root fillet, related to the conditions at the test gear);
Y is the size factor for tooth root strength (see 8.1), which accounts for the influence of
X
the module on the tooth root strength;
Y is the life factor, which accounts for the influence of required numbers of cycles of
NT
operation (see 8.2).
7.3 Calculated safety factor
The determined tooth root stress, σ , shall be ≤σ , which is the permissible tooth root stress. The
F FP
calculated safety factor against tooth breakage shall be determined separately for pinion and wheel, on
the basis of the bending stress number determined for the standard test gear:
σ
FP-B2
SS=> (50)
F-B2 F min
σ
F-B2
NOTE This is the calculated safety factor with respect to the transmitted torque.
Considerations in reference to the safety factors and the risk (probability) of failure are given in
ISO 10300-1:2014, 5.2.
7.4 Tooth root stress factors
7.4.1 General
To calculate the bending strength geometry factor, Y , the formulae in 7.4.3 should be used. Because of
J
the complexity of the calculation, computerization is recommended.
[3]
ANSI/AGMA 2003-C10 contains graphs for the bevel geometry factor, Y , for straight, Zerol and spiral
J
bevel gears for a series of gear designs, based on the smaller of the face width to be chosen b = 0,3R or
e
[4]
b = 10m . Corresponding graphs for hypoid gears can be found in AGMA 932-A05. These may be used
et
whenever the tooth proportions and thickness, face widths, tool edge radii, pressure and spiral angles
of the design, and driving with the concave side, correspond to those in the graphs.
18 © ISO 2014 – All rights reserved
7.4.2 Stress parabola according to Lewis
The basis for method B2 is the Lewis formula applied to a virtual cylindrical gear, which has been
defined in transverse section as specified in Annex B of ISO 10300-1:2014, with the following additions
and modifications:
— the tooth strength is considered in the normal section rather than in the transverse section;
— the position of the point of load application is determined by taking into account theoretical lines of
contact, tooth bearing modifications and experimental evidence;
— the amount of load carried by one tooth is estimated based on tooth bearing modification and
contact ratio;
— the radial component of the normal load is considered;
— a stress concentration factor based on experimental data are applied;
— the concept of effective face width is used.
Bending stress shall be calculated assuming the tooth shaped beam is simulated by a parabola tangent
to the tooth profile at the most highly stressed section. Figure 6 shows a layout for the cases of: a) no
load sharing and b) load sharing.
7.4.3 Basic formula of geometry factor, Y
J
The parameters for calculating the geometry factor, Y are the same for bevel and hypoid gears. However,
J
the calculation procedures are different. See 7.4.4 for bevel gears without hypoid offset or 7.4.5 for
hypoid gears.
The bevel geometry factor, Y , is calculated using Formula (51):
J
r
Y b m
my01,2
1,2 ce1,2mt1,2
Y = ⋅⋅ ⋅ (51)
J1,2
YY⋅⋅ε r b m
f1,2 Ni mpt1,2 12, et2
where
Y is the tooth form factor of pinion and wheel (see 7.4.4.4 for bevel gears and 7.4.5.4 for
1,2
hypoid gears);
ε is the load sharing ratio (see 7.4.4.3 and 7.4.5.2, respectively);
N
r is the mean transverse radius to point of load application for pinion or wheel, in millime-
my01,2
tres (see 7.4.4.2 and 7.4.5.5);
r is the mean transverse pitch radius, in millimetres (see ISO 23509);
mpt1,2
Y is the stress concentration and correction factor (see 7.4.6.2);
f1,2
Y is the inertia factor for gears with a low
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