ISO 10300-3:2023

INTERNATIONAL ISO
STANDARD 10300-3
Third edition
2023-08
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 10300-3:2023(E)
© ISO 2023

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ISO 10300-3:2023(E)
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Published in Switzerland
ii
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ISO 10300-3:2023(E)
Contents Page
Foreword .iv
Introduction .v
1 Scope . 1
2 Normative references . 1
3 Terms and definitions . 2
4 Symbols, general subscripts and abbreviated terms . 2
5 General rating procedure . 6
6 Gear tooth rating formulae — Method B1 . 7
6.1 Tooth root stress formula . 7
6.2 Permissible tooth root stress . 8
6.3 Calculated safety factor . 9
6.4 Tooth root stress factors . 9
6.4.1 Tooth form factor, Y . 9
Fa
6.4.2 Stress correction factor for load at tooth tip, Y . 14
Sa
6.4.3 Contact ratio factor, Y . 14
ε
6.4.4 Bevel spiral angle factor, Y . 15
BS
6.4.5 Load sharing factor, Y . 16
LS
6.5 Permissible tooth root stress factors . 16
6.5.1 Relative surface condition factor, Y . 16
R,relT-B1
6.5.2 Relative notch sensitivity factor, Y . 17
δ,relT-B1
7 Gear tooth rating formulae — Method B2 .19
7.1 Tooth root stress formula . 19
7.2 Permissible tooth root stress . 19
7.3 Calculated safety factor . 20
7.4 Tooth root stress factors .20
7.4.1 General .20
7.4.2 Stress parabola according to Lewis . . 21
7.4.3 Basic formula of geometry factor, Y . 21
J
7.4.4 Geometry factor, Y , for bevel gears (for hypoid gears, see 7.4.5) .22
J
7.4.5 Geometry factor, Y , for hypoid gears . 26
J
7.4.6 Additional tooth strength parameters (for bevel and hypoid gears) . 35
7.4.7 Root stress adjustment factor, Y .38
A
7.5 Permissible tooth root stress factors .38
7.5.1 Relative surface condition factor, Y .38
R,relT-B2
7.5.2 Relative notch sensitivity factor, Y .38
δ,relT-B2
8 Factors for permissible tooth root stress common for method B1 and method B2 .39
8.1 Size factor, Y . 39
X
8.1.1 General .39
8.1.2 Structural and through hardened steels, spheroidal cast iron, perlitic
malleable cast iron.39
8.1.3 Case, flame, induction hardened steels, nitrided or nitro carburized steels .39
8.1.4 Grey cast iron and spheroidal cast iron (ferrit structure) .39
8.2 Life factor, Y .39
NT
8.2.1 General .39
8.2.2 Method A . . .40
8.2.3 Method B . .40
Bibliography .43
iii
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ISO 10300-3:2023(E)
Foreword
ISO (the International Organization for Standardization) is a worldwide federation of national standards
bodies (ISO member bodies). The work of preparing International Standards is normally carried out
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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 document should be noted. This document was drafted in accordance with the
editorial rules of the ISO/IEC Directives, Part 2 (see www.iso.org/directives).
ISO draws attention to the possibility that the implementation of this document may involve the use
of (a) patent(s). ISO takes no position concerning the evidence, validity or applicability of any claimed
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www.iso.org/iso/foreword.html.
This document was prepared by Technical Committee ISO/TC 60, Gears, Subcommittee SC 2, Gear
capacity calculation.
This third edition cancels and replaces the second edition (ISO 10300-3:2014), which has been
technically revised.
The main changes are as follows:
— Table 1 has been inserted;
— Table 2 has been inserted;
— Figure 4 — surface condition factor, Y , for permissible stress number relative to standard test
R,relT
gear dimensions has been removed;
— Figure 5 — relative notch sensitivity factor with respect to standard test gear dimensions has been
removed;
— new Figure 5 — life factor, Y (standard reference test gears) has been added;
NT
— Figure 7 — size factor, Y , for permissible tooth root stress has been removed.
X
A list of all parts in the ISO 10300 series can be found on the ISO website.
Any feedback or questions on this document should be directed to the user’s national standards body. A
complete listing of these bodies can be found at www.iso.org/members.html.
iv
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ISO 10300-3:2023(E)
Introduction
When ISO 10300:2001 (all parts) 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 document, 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 10300-1:2023, Figure 1.
NOTE ISO 10300 (all parts) gives no preferences in terms of when to use method B1 and when to use
method B2.
Failure of gear teeth by tooth root breakage can be brought about in many ways; severe instantaneous
overloads, excessive macropitting, case crushing and bending fatigue are a few. The strength ratings
determined by the formulae in this document 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 document). 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 document.
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 document 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
[1]
method C for the 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 root
F
breakage higher than the safety factor against damage due to macropitting is, therefore, generally to be
preferred (see ISO 10300-1).
v
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INTERNATIONAL STANDARD ISO 10300-3:2023(E)
Calculation of load capacity of bevel gears —
Part 3:
Calculation of tooth root strength
1 Scope
This document 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 they are the
mn
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 tooth root
F
stress σ . This document is not applicable in the assessment of tooth flank fracture.
FP
The formulae in this document 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 within the
vα
range of the applied factors as specified in ISO 10300-1. The bending strength formulae are applicable
to fractures at the tooth fillet, but not to those on the active flank surfaces, to failures of the gear rim or
of the gear blank through the web and hub.
3
This document does not apply to stress levels above those permitted for 10 cycles, as stresses in that
range can exceed the elastic limit of the gear tooth.
NOTE This document is not applicable to bevel gears which have an inadequate contact pattern under load.
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 facewidths b > 13 m , the
m1 m2 e mn
calculated results of this document should be confirmed by experience.
2 Normative references
The following documents are referred to in the text in such a way that some or all of their content
constitutes requirements of this document. For dated references, only the edition cited applies. For
undated references, the latest edition of the referenced document (including any amendments) applies.
ISO 701, International gear notation — Symbols for geometrical data
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 10300-1:2023, Calculation of load capacity of bevel gears — Part 1: Introduction and general influence
factors
ISO 10300-2:2023, Calculation of load capacity of bevel gears — Part 2: Calculation of surface durability
(macropitting)
ISO 17485, Bevel gears — ISO system of accuracy
ISO 23509:2016, Bevel and hypoid gear geometry
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ISO 10300-3:2023(E)
3 Terms and definitions
For the purposes of this document, the terms and definitions given in ISO 1122-1, ISO 23509 and the
following apply.
ISO and IEC maintain terminology databases for use in standardization at the following addresses:
— ISO Online browsing platform: available at https:// www .iso .org/ obp
— IEC Electropedia: available at https:// www .electropedia .org/
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 (bending)
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, general subscripts and abbreviated terms
For the purposes of this document, the symbols given in ISO 701, ISO 17485, ISO 23509, and the
following shall apply.
Table 1 — Symbols
Symbol Description or term Unit
a Auxiliary value —
BS
b Facewidth mm
b Developed length of one tooth as facewidth of the calculation model mm
a
b Auxiliary value —
BS
b Calculated effective facewidth mm
ce
b Mean facewidth mm
k
2
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ISO 10300-3:2023(E)
TTabablele 1 1 ((ccoonnttiinnueuedd))
Symbol Description or term Unit
b Facewidth of virtual cylindrical gears mm
v
b Relative facewidth of virtual cylindrical gear pair —
v,rel
C Conversion factor used in Formula (202), C = 25,4 mm mm
mm mm
c Auxiliary value —
BS
d Tip diameter of virtual cylindrical gear in normal section mm
van
d Base diameter of virtual cylindrical gear in normal section mm
vbn
E, G, H Auxiliary quantities for tooth form factor (method B1) —
F Nominal tangential force at mid-facewidth of the reference cone N
mt
F Nominal tangential force of virtual cylindrical gears N
vmt
g Relative length of action to point of load application (method B2) —
J
g Auxiliary term mm
f0
g Relative distance from blade edge to centre line —
rb
g Relative length of action in normal section —
vαn
g Relative length of action in normal section for hypoid gears —
vαn,hyp
g Auxiliary term —
xb
g Auxiliary term —
yb
g Auxiliary term —
za
g Auxiliary term —
zb
g Relative length of action within the contact ellipse —
η
g Auxiliary term —
0
g Intermediate value mm
1
h Tool addendum mm
a0
h Bending moment arm for tooth root stress (load application at tooth tip) mm
Fa
h Mean dedendum mm
fm
h Mean whole depth used for bevel spiral angle factor mm
m
h Relative load height from critical section (method B2) —
N
h Relative mean virtual dedendum —
vfm
h Auxiliary values —
1,2,3,4
K Application factor —
A
K Transverse load factor for bending stress —
Fα
K Face load factor for bending stress —
Fβ
K Dynamic factor —
v
k' Contact shift factor —
L Factor to calculate the stress correction factor according to Dolan and Broghamer —
L Auxiliary value —
a,D,C
l Part of the model’s facewidth covered by the contact line mm
bb
l Theoretical length of middle contact line mm
bm
M Factor to calculate the stress correction factor according to Dolan and Broghamer —
m Outer transverse module mm
et
m Mean normal module mm
mn
m Mean transverse module mm
mt
N Number of load cycles —
L
O Factor to calculate the stress correction factor according to Dolan and Broghamer —
q Notch parameter —
s
3
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ISO 10300-3:2023(E)
TTabablele 1 1 ((ccoonnttiinnueuedd))
Symbol Description or term Unit
R Relative radius from tool centre to critical pinion coast side fillet point —
CL2
R Relative radius from tool centre to critical pinion drive side fillet point —
DL2
R Mean cone distance mm
m
Rz Mean roughness μm
r Relative pinion radius to fillet point —
L1o
r Relative wheel radius to pinion fillet point —
L2o
r Relative tooth fillet radius at the root in mean section —
mf
r Mean pitch radius mm
mpt
r Mean transverse radius to point of load application (method B2) mm
my0
r Relative mean virtual tip radius —
va
r Relative mean virtual pitch radius —
vn
Relative distance from pitch circle to the pinion point of load application and the
Δr —
y0
wheel tooth centre line
S Safety factor for bending stress (against tooth root breakage) —
F
S Minimum safety factor for bending stress —
F,min
s Tooth root chord in calculation section mm
Fn
s Relative horizontal distance from centreline to critical fillet point (method B2) —
N
s Mean normal circular thickness mm
mn
s Amount of protuberance mm
pr
s Relative virtual tooth thickness —
vmn
W Wheel mean slot width mm
m2
x Profile shift coefficient —
hm
x Thickness modification coefficient —
sm
x Tooth strength factor (method B2) —
N
x Distance from mean section to point of load application mm
oo
x Relative horizontal distance from pitch circle to fillet point —
1
Y Root stress adjustment factor (method B2) —
A
Y Bevel spiral angle factor —
BS
Y Tooth form factor for load application at the tooth tip (method B1) —
Fa
Y Combined tooth form factor for generated gears —
FS
Y Stress concentration and stress correction factor (method B2) —
f
Y Bending strength geometry factor (method B2) —
J
Y Load sharing factor (bending) —
LS
Y Life factor (bending) —
NT
Y Combined geometry factor (method B2) —
P
Y Surface condition at the root fillet —
R
Y Surface condition at the test gear —
RT
Y Relative surface condition factor —
R,relT
Y Stress correction factor for load application at the tooth tip —
Sa
Y Stress correction factor for dimensions of the standard test gear —
ST
Y Size factor for tooth root stress —
X
Y Tooth form factor of pinion and wheel (method B2) —
1,2
Y Relative notch sensitivity factor —
δ,relT
Y Contact ratio factor for bending (method B1) —
ε
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ISO 10300-3:2023(E)
TTabablele 1 1 ((ccoonnttiinnueuedd))
Symbol Description or term Unit
Relative location of point of load application for maximum bending stress on path
y —
J
of action (method B2)
y Relative vertical distance from pitch circle to fillet point —
1
Relative distance from the beginning of the path of action to the point of load
y —
3
application on path of action for maximum root stress
Z Load sharing factor (method B1) —
LS
z Number of teeth of virtual cylindrical gear in normal section —
vn
α Coast flank pressure angle in wheel root coordinates °
Cnf
α Drive flank pressure angle in wheel root coordinates °
Dnf
α Load application angle at tooth tip of virtual cylindrical gear (method B1) °
Fan
α Normal pressure angle at point of load application (method B2) °
L
α Generated pressure angle at fillet point °
LN
α Adjusted pressure angle (method B2) °
a
α Normal pressure angle at tooth tip °
an
α Effective pressure angle for drive side/coast side °
eD,C
α Limit pressure angle in wheel root coordinates (method B2) °
f
α Auxiliary term °
h
α Limit pressure angle °
lim
α Generated pressure angle for drive side/coast side °
nD,C
β Intermediate angle °
a
β Helix angle of virtual gear (method B1), virtual spiral angle (method B2) °
v
β Helix angle at base circle of virtual cylindrical gear °
vb
γ Auxiliary angle for tooth form and tooth correction factor °
a
ε Load sharing ratio for bending (method B2) —
N
ε Lengthwise load sharing factor —
b
ε Profile load sharing factor —
f
ε Transverse contact ratio of virtual cylindrical gears —
vα
ε Transverse contact ratio of virtual cylindrical gears in normal section —
vαn
Transverse contact ratio of virtual cylindrical gears in normal section for hypoid
ε —
vαn,hyp
gears
ε Face contact ratio of virtual cylindrical gears —
vβ
ε Virtual contact ratio (method B1), modified contact ratio (method B2) —
vγ
θ Auxiliary angle for tooth form and tooth correction factors rad
θ Wheel angle from centreline to pinion tip on drive side rad
D1
Δθ Wheel angle between fillet points °
ϑ Auxiliary value —
µ Relative distance from centreline to tool critical fillet point —
ξ Assumed angle in locating weakest section rad
One half of angle subtended by normal circular tooth thickness at point of load
ξ rad
h
application
ρ Cutter edge radius mm
a0
ρ Fillet radius at point of contact of 30° tangent mm
F
2
σ Tensile strength (corresponds to R of ISO 6892-1) N/mm
B m
2
σ Allowable stress number (bending) N/mm
FE
2
σ Permissible tooth root stress N/mm
FP
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ISO 10300-3:2023(E)
TTabablele 1 1 ((ccoonnttiinnueuedd))
Symbol Description or term Unit
2
σ Nominal stress number (bending) N/mm
F,lim
ρ Fillet radius at point of contact of 30° tangent in normal section mm
Fn
ρ Root fillet radius of basic rack for cylindrical gears mm
fP
2
σ Yield stress (corresponds to R of ISO 6892-1) N/mm
S p
2
σ Proof stress (0,2 % permanent set) (corresponds to R of ISO 6892-1) N/mm
0,2 p0,2
ρ Radius of curvature change —
Δred
X -1
χ Relative stress drop in notch root mm
X -1
Relative stress drop in notch root of standardized test gear
mm
χ
T
Table 2 — General subscripts
Subscripts Description
0 Tool
1 Pinion
2 Wheel
A, B, B1, B2, C Value according to method A, B, B1, B2 or C
D Drive flank
C Coast flank
T Relative to standardized test gear dimensions
(1), (2) Trials of interpolation
Table 3 — Abbreviated terms in accordance with ISO 6336-5
Abbreviated term Material Type
St Wrought normalized low carbon steels
Normalized low carbon steels/cast steels
St (cast.) Cast steels
Black malleable cast iron
GTS (perl.)
(perlitic structure)
Cast iron materials Nodular cast iron
GGG (perl., bai., ferr.)
(perlitic, bainitic, ferritic structure)
GG Grey cast iron

V Through hardened wrought steels Carbon steels, alloy steels
V (cast) Through hardened cast steels Carbon steels, alloy steels
Eh Case-hardened wrought steels
Flame or induction hard
...