M56

M56
M56(1990)(cont)(Rev.1
1994/
Corr.
1996)
(Rev.2
Oct 2013)(Rev.3 Oct
2015)(Rev.4
Feb 2021)(Corr.1
Oct 2021)(Corr. 2
Mar 2023)

Marine gears – load capacity of involute parallelaxis spur and helical gears
M56.1

Basic principles - introduction and general influence factors

M56.1.1 Introduction
The following definitions are mainly based on the ISO 6336 series standard (hereinafter
called “reference standard”) for the calculation of load capacity of spur and helical gears.
M56.1.2 Scope and field of application
These requirements apply to enclosed gears, both intended for main propulsion and foressential auxiliary services, which accumulate a large number of load cycles (several
millions), whose gear set is intended to transmit a maximum continuous power equal to, or
greater than:-

220 kW for gears intended for main propulsion
110 kW for gears intended for essential auxiliary services

These requirements, however, may be applied to the enclosed gears, whose gear set isintended to transmit a maximum continuous power less than those specified above at therequest of the individual society.

Notes:
1.

The requirements of UR M56 Rev.2 are to be uniformly implemented from 1 January
2015 by all IACS Societies to any marine gear subject to approval and to any TypeApproved marine gear from the date of the first renewal after 1 January 2015. For amarine gear approved prior to 1 January 2015 where no failure has occurred, and nochanges in design / scantlings of the gear meshes or materials or declared load
capacity data has taken place the requirements of UR M56 Rev.2 may be waived.

2.

The requirements of UR M56 Rev.3 are to be uniformly implemented from 1 January
2017 by all IACS Societies to any marine gear subject to approval and to any TypeApproved marine gear from the date of the first renewal after 1 January 2017. For amarine gear approved prior to 1 January 2017 where no failure has occurred, and nochanges in design / scantlings of the gear meshes or materials or declared load
capacity data has taken place the requirements of UR M56 Rev.3 may be waived.

3.

Rev.4 of this UR is to be uniformly implemented by IACS Societies from 1 July 2022 toany marine gear subject to approval and to any Type Approved marine gear from thedate of the first renewal after 1 July 2022. For a marine gear approved prior to 1 July
2022 where no failure has occurred, and no changes in design / scantlings of the gear
meshes or materials or declared load capacity data has taken place the requirements of
UR M56 Rev.4 may be waived.

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The following definitions deal with the determination of load capacity of external and internal
involute spur and helical gears, having parallel axis, with regard to surface durability (pitting)and tooth root bending strength and to this purpose the relevant basic equations are provided
in Parts 2 and 3.
The influence factors common to said equations are described in the present Part 1.
The others, introduced in connection with each basic equation, are described in the following
Parts 2 and 3.
All influence factors are defined regarding their physical interpretation. Some of the influence
factors are determined by the gear geometry or have been established by conventions.
These factors are to be calculated in accordance with the equations provided. Other factors,
which are approximations, can be calculated according to methods acceptable to the Society.
M56.1.3 Symbols and units
The main symbols used are listed below.
Other symbols introduced in connection with the definition of influence factors are describedin the appropriate sections.
SI units have been adopted.
acentre distancemmbcommon face width

mm

b1,2

face width of pinion, wheelmmdreference diameter

mm

d1,2

reference diameter of pinion, wheel

mm

da1,2

tip diameter of pinion, wheel

mm

db1,2

base diameter of pinion, wheel

mm

df1,2

root diameter of pinion, wheel

mm

dw1,2

working diameter of pinion, wheelmm

Ft

nominal tangential loadN

Fbt

nominal tangential load on base cylinder in thetransverse sectionNhtooth depthmmmnnormal modulemmmttransverse module

mm

n1,2

rotational speed of pinion, wheel

revs/min (rpm)Pmaximum continuous power transmitted by the gear set

kW

T1,2

torque in way of pinion, wheelNmugear ratiov

linear velocity at pitch diameter
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x1,2

addendum modification coefficient of pinion, wheelznumber of teeth

z1,2

number of teeth of pinion, wheelznvirtual number of teeth

αnnormal pressure angle at reference cylinder°αttransverse pressure angle at ref. cylinder°αtwtransverse pressure angle at working pitch cylinder°βhelix angle at reference cylinder°βbhelix angle at base cylinder°εαtransverse contact ratio

εβoverlap ratio

εγtotal contact ratio

M56.1.4 Geometrical definitions
For internal gearing z2, a, d2, da2, db2 and dw2 are negative. The pinion is defined as the gearwith the smaller number of teeth, therefore the absolute value of the gear ratio, defined asfollows, is always greater or equal to the unity:
u=z2/z1=dw2/dw1=d2/d1For external gears u is positive, for internal gears u is negative.
In the equation of surface durability b is the common face width on the pitch diameter.
In the equation of tooth root bending stress b1 or b2 are the face widths at the respective toothroots. In any case, b1 and b2 are not to be taken as greater than b by more than one module(mn) on either side.
The common face width b may be used also in the equation of teeth root bending stress if
significant crowning or end relief have been adopted.
tan α t =tan α ncos β

tan β b = tan β ⋅ cos α t

d1, 2 =

z1, 2 mncos βd b1, 2 = d1, 2 cos α t

2a 
u + 1  where2au d w2 =
u + 1
d w1 =

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z n1, 2 =z1, 2
cos β b ⋅ cos β
2

mn
cos β

mt =

invα = tan α −

πα
180

;

α [°]

invα tw = invα t + 2 tan α nεα =x1 + x 2
z1 + z2orcosα tw =mt ( z1 + z 2 )cosα t2a

0,5 d a21 − d b21 ± 0,5 d a22 − d b22 − a ⋅ sin α tw

π ⋅ mt ⋅ cos α t
the positive sign is used for external gears, the negative sign for internal gearsb ⋅ sin βεβ =π ⋅ mnfor double helix, b is to be taken as the width of one helix
εγ = εα + ε βv = π ⋅ d 1,2 n1,2 / 60 ⋅ 10 3

M56.1.5 Nominal tangential load, Ft
The nominal tangential load, Ft, tangential to the reference cylinder and perpendicular to therelevant axial plane, is calculated directly from the maximum continuous power transmitted by
the gear set by means of the following equations:

T1, 2 =

30 ⋅103 P
π ⋅ n1, 2

Ft = 2000 ⋅ T1,2 / d 1,2

M56.1.6 General influence factors
M56.1.6.1 Application factor, KA 1)
The application factor, KA, accounts for dynamic overloads from sources external to thegearing.
KA, for gears designed for infinite life is defined as the ratio between the maximum repetitive
cyclic torque applied to the gear set and the nominal rated torque.
The nominal rated torque is defined by the rated power and speed and is the torque used in
the rating calculations.
The factor mainly depends on:
characteristics of driving and driven machines;
ratio of masses;
type of couplings;
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operating conditions (overspeeds, changes in propeller load conditions, ...).

When operating near a critical speed of the drive system, a careful analysis of conditions
must be made.
The application factor, KA, should be determined by measurements or by system analysis
acceptable to the Society. Where a value determined in such a way cannot be supplied, thefollowing values can be considered.
a) Main propulsion
diesel engine with hydraulic or electromagnetic slip coupling
diesel engine with high elasticity coupling
diesel engine with other couplings
1): 1.00: 1.30: 1.50

Where the vessel, on which the reduction gear is being used, is receiving an Ice Class
notation, the Application Factor or the Nominal Tangential Force should be adjusted toreflect the ice load associated with the requested Ice Class, i.e. applying the design
approach in UR I3 when applicable.

b) Auxiliary gears
electric motor, diesel engine with hydraulic orelectromagnetic slip coupling
diesel engine with high elasticity coupling
diesel engine with other couplings: 1.00: 1.20: 1.40M56.1.6.2 Load sharing factor, Kγ
The load sharing factor, Kγ accounts for the maldistribution of load in multiple pathtransmissions (dual tandem, epicyclic, double helix, etc.)
Kγ is defined as the ratio between the maximum load through an actual path and the evenly
shared load. The factor mainly depends on accuracy and flexibility of the branches.
The load sharing factor, Kγ, should be determined by measurements or by system analysis.
Where a value determined in such a way cannot be supplied, the following values can beconsidered for epicyclic gears:-

up to 3 planetary gears
4 planetary gears
5 planetary gears
6 planetary gears and over: 1.00: 1.20: 1.30: 1.40M56.1.6.3 Internal dynamic factor, Kv
The internal dynamic factor, Kv, accounts for internally generated dynamic loads due tovibrations of pinion and wheel against each other.
Kv is defined as the ratio between the maximum load which dynamically acts on the toothflanks and the maximum externally applied load (FtKAKγ).
The factor mainly depends on:-transmission errors (depending on pitch and profile errors);
masses of pinion and wheel;
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gear mesh stiffness variation as the gear teeth pass through the meshing cycle;transmitted load including application factor;
pitch line velocity;
dynamic unbalance of gears and shaft;shaft and bearing stiffnesses;
damping characteristics of the gear system.

The dynamic factor, Kv, is to be calculated as follows:This method may be applied only to cases where all the following conditions are satisfied:- running velocity in the subcritical range, i.e.:vz1u2< 10 m/s
100 1 + u 2- spur gears (β = 0°) and helical gears with β ≤ 30 °- pinion with relatively low number of teeth, z1< 50- solid disc wheels or heavy steel gear rim

vz1
u2
This method may be applied to all types of gears if< 3 m/s , as well as to helical100 1 + u 2
gears where β > 30 °.
For gears other than the above, reference is to be made to Method B outlined in thereference standard ISO 6336-1:2019.
a) For spur gears and for helical gears with overlap ratio εβ ≥1



 K
 vz
u2
1
+ K 2  ⋅ 1 K3
Kv = 1 + 
1 + u2
 K Ft
 100
 A

b


If KAFt/b is less than 100 N/mm, this value is assumed to be equal to 100 N/mm.
Numerical values for the factor K1 are to be as specified in the Table 1.1

spur gears
helical gears

3
2.1
1.9

4
3.9
3.5

K1
ISO accuracy grades 2)
5
6
7.5
14.9
6.7
13.3

7
26.8
23.9

8
39.1
34.8

Table 1.1 Values of the factor K1 for the calculation of Kv
For all accuracy grades the factor K2 is to be in accordance with the following:- for spur gears, K2=0.0193- for helical gears, K2=0.0087
Factor K3 is to be in accordance with the following:
If

vz1
u2
≤ 0.2 then100 1 + u 2

K 3 = 2 .0

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If

(cont)

vz1
u2
> 0.2
100 1 + u 2then

K 3 = 2.071 − 0.357 ⋅vz1
u2
100 1 + u 2b) For helical gears with overlap ratio εβ<1 the value Kv is determined by linear interpolation
between values determined for spur gears (Kvα) and helical gears (Kvβ) in accordance with:

K v = K vα − ε β (K vα − K vβ )Where:
Kvα is the Kv value for spur gears, in accordance with a);
Kvβ is the Kv value for helical gears, in accordance with a).
2)

ISO accuracy grades according to ISO 1328-2:2020ISO 1328-1:2013. In case of mating
gears with different accuracy grades, the grade corresponding to the lower accuracy
should be used.

M56.1.6.4 Face load distribution factors, KHβ and KFβ
The face load distribution factors, KHβ for contact stress, KFβ for tooth root bending stress,
account for the effects of non-uniform distribution of load across the face width.
KHβ is defined as follows:
K Hβ =

maximum load per unit face widthmean load per unit face width

KFβ is defined as follows:
K Fβ =

maximum bending stress at tooth root per unit face widthmean bending stress at tooth roo per unit face width

The mean bending stress at tooth root relates to the considered face width b1 resp. b2.
KFβ can be expressed as a function of the factor KHβ.
The factors KHβ and KFβ mainly depend on:
gear tooth manufacturing accuracy;
errors in mounting due to bore errors;
bearing clearances;wheel and pinion shaft alignment errors;
elastic deflections of gear elements, shafts, bearings, housing and foundations which
support the gear elements;
thermal expansion and distortion due to operating temperature;compensating design elements (tooth crowning, end relief, etc.).
The face load distribution factors, KHβ for contact stress, and KFβ for tooth root bending stress,
are to be determined according to the Method C outlined in the reference standard ISO 63361:2019.
Alternative methods acceptable to the Society may be applied.

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a)

(cont)

In case the hardest contact is at the end of the face width KFβ is given by the following
equations:
K Fβ = K HNβN=(b / h )22
1 + (b / h ) + (b / h )(b/h) = face width/tooth height ratio, the minimum of b1/h1 or b2/h2.
For double helical gears, the face width of only one helix is to be used.
When b/h<3 the value b/h=3 is to be used.
b)

In case of gears where the ends of the face width are lightly loaded or unloaded (endrelief or crowning):
K Fβ = K Hβ

M56.1.6.5 Transverse load distribution factors, KHα and KFα
The transverse load distribution factors, KHα for contact stress and KFα for tooth root bending
stress, account for the effects of pitch and profile errors on the transversal load distribution
between two or more pairs of teeth in mesh.
The factors KHα and KFα mainly depend on:
total mesh stiffness;
total tangential load Ft, KA, Kγ, Kv, KHβ;base pitch error;
tip relief;running-in allowances.
The transverse load distribution factors, KHα for contact stress and KFα for tooth root bending
stress, are to be determined according to Method B outlined in the reference standard ISO
6336-1:2019.
M56.2

Surface durability (pitting)M56.2.1 Scope and general remarks
The criterion for surface durability is based on the Hertz pressure on the operating pitch point
or at the inner point of single pair contact. The contact stress σH must be equal to or less thanthe permissible contact stress σHP.
M56.2.2 Basic equations
M56.2.2.1 Contact stressσ H = σ H 0 K A ⋅ K γ ⋅ K v ⋅ K Hα ⋅ K Hβ ≤ σ HP
where:

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σH0

= basic value of contact stress for pinion and wheel(cont)

σ H 0 = ZB ⋅ ZH ⋅ ZE ⋅ Zε ⋅ Z β

Ft (u + 1)
d1 ⋅ b u

for pinion

σ H 0 = ZD ⋅ ZH ⋅ ZE ⋅ Zε ⋅ Z β

Ft (u + 1)
d1 ⋅ b u

for wheel

where:
ZB= single pair tooth contact factor for pinion(see clause 2.3)

ZD

= single pair tooth contact factor for wheel(see clause 2.3)

ZH

= zone factor(see clause 2.4)ZE
Zε= elasticity factor= contact ratio factor(see clause 2.5)
(see clause 2.6)Zβ= helix angle factor(see clause 2.7)

Ft

= nominal tangential load at reference cylinder in thetransverse section(see Part 1)b= common face widthd1= reference diameter of pinionu= gear ratio(for external gears u is positive, for internal gears u is negative)

Regarding factors KA, Kγ, Kv, KHα and KHβ, see Part 1.
M56.2.2.2 Permissible contact stress
The permissible contact stress σHP is to be evaluated separately for pinion and wheel:

σ HP =

σ H lim ⋅ Z N
SH⋅ Z L ⋅ Z v ⋅ Z R ⋅ ZW ⋅ Z X

where:
σHlim = endurance limit for contact stress(see clause 2.8)

ZN

= life factor for contact stress(see clause 2.9)

ZL

= lubrication factor(see clause 2.10)

Zv= velocity factor(see clause 2.10)

ZR= roughness factor(see clause 2.10)

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ZW= hardness ratio factor(see clause 2.11)

ZX= size factor for contact stress(see clause 2.12)

SH= safety factor for contact stress(see clause 2.13)

M56.2.3 Single pair tooth contact factors, ZB and ZD
The single pair tooth contact factors, ZB for pinion and ZD for wheel, account for the influenceof the tooth flank curvature on contact stresses at the inner point of single pair contact inrelation to ZH.
The factors transform the contact stresses determined at the pitch point to contact stresses
considering the flank curvature at the inner point of single pair contact.
The single pair tooth contact factors, ZB for pinions and ZD for wheels, are to be determined
as follows:
For spur gears, εβ=0
ZB = M1 or 1 whichever is the larger value
ZD = M2 or 1 whichever is the larger value

M1 =

M2 =

tan α tw
2


 d2
 a1 − 1 − 2π  d a 2 − 1 − (ε α − 1) 2π 22
 d b1z1  d b 2
z 2 
tan α tw
2


 d2
 a 2 − 1 − 2π  d a1 − 1 − (ε α − 1) 2π 
 d b22z 2  d b21
z1 For helical gears when εβ ≥1
ZB = 1
ZD = 1For helical gears when εβ <1 the values of ZB and ZD are determined by linear interpolation
between ZB and ZD for spur gears and ZB and ZD for helical gears having εβ ≥1.
Thus:

Z B = M1 − ε β (M1 − 1) and Z B ≥ 1
ZD = M 2 − ε β (M 2 − 1) and ZD ≥ 1

For internal gears, ZD shall be taken as equal to 1.

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M56.2.4 Zone factor, ZH
The zone factor, ZH, accounts for the influence on the Hertzian pressure of tooth flankcurvature at pitch point and transforms the tangential load at the reference cylinder to thenormal load at the pitch cylinder.
The zone factor, ZH, is to be calculated as follows:

ZH =2 cos β bcos 2 α t tan α tw

M56.2.5 Elasticity factor, ZE
The elasticity factor, ZE, accounts for the influence of the material properties E (modulus ofelasticity) and ν (Poisson’s ratio) on the contact stress.
The elasticity factor, ZE, for steel gears (E= 206 000 N/mm2, ν= 0.3) is equal to:
ZE = 189.8

N/mm 2

In other cases, reference is to be made to the reference standard ISO 6336-2:2019.
M56.2.6 Contact ratio factor, Zε
The contact ratio factor, Zε, accounts for the influence of the transverse contact ratio and theoverlap ratio on the specific surface load of gears.
The contact ratio factor, Zε, is to be calculated as follows:
Spur gears:Zε =4 − εα3Helical gears:- for εβ <1Zε =εβ4 − εα(1− ε β )+3εα- for εβ ≥1
Zε =

1

εα

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M56.2.7 Helix angle factor, Zβ
The helix angle factor, Zβ, accounts for the influence of helix angle on surface durability,
allowing for such variables as the distribution of load along the lines of contact. Zβ isdependent only on the helix angle.
The helix angle factor, Zβ, is to be calculated as follows:

Zβ =

1
cos β

Where β is the reference helix angle.
M56.2.8 Endurance limit for contact stress, σHlim
For a given material, σHlim is the limit of repeated contact stress which can be permanently
endured. The value of σHlim can be regarded as the level of contact stress which the material
will endure without pitting for at least 5x107 load cycles.
For this purpose, pitting is defined by:- for not surface hardened gears:
pitted area > 2% of total active flank area
- for surface hardened gears:
pitted area > 0,5% of total active flank area, or> 4% of one particular tooth flank area.
The σHlim values are to correspond to a failure probability of 1% or less.
The endurance limit mainly depends on:material composition, cleanliness and defects;
mechanical properties;
residual stresses;
hardening process, depth of hardened zone, hardness gradient;material structure (forged, rolled bar, cast).
The endurance limit for contact stress σHlim, is to be determined, in general, making reference
to values indicated in the standard ISO 6336-5:2016, for material quality MQ.
M56.2.9 Life factor, ZN
The life factor ZN, accounts for the higher permissible contact stress in case a limited life
(number of cycles) is required.
The factor mainly depends on:material and heat treatment;
number of cycles;
influence factors (ZR, Zv, ZL, ZW, ZX).
The life factor, ZN , is to be determined according to Method B outlined in the reference
standard ISO 6336-2:2019.

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M56.2.10 Influence factors of lubrication film on contact stress, ZL, Zv and ZR
The lubricant factor, ZL, accounts for the influence of the type of lubricant and its viscosity.
The velocity factor, Zv, accounts for the influence of the pitch line velocity. The roughness
factor, ZR, accounts for the influence of the surface roughness on the surface endurance
capacity.
The factors may be determined for the softer material where gear pairs are of different
hardness.
The factors mainly depend on:
viscosity of lubricant in the contact zone;
the sum of the instantaneous velocities of the tooth surfaces;
load;
relative radius of curvature at the pitch point;
surface roughness of teeth flanks;
hardness of pinion and gear.
The lubricant factor, ZL, the velocity factor, Zv, and the roughness factor ZR are to be
calculated as follows:a)

Lubricant factor, ZL

The factor, ZL, is to be calculated from the following equation:
Z L = CZL +4(1 − CZL )
134 1.2 +
ν 40 
2In the range 850 N/mm2 ≤ σ H lim ≤ 1200 N/mm2, CZL is to be calculated as follows:σ− 850 
CZL =  0.08 H lim + 0.83
350

If σHlim < 850 N/mm2, take CZL = 0.83
If σHlim > 1200 N/mm2, take CZL = 0.91
Where:
ν40 = nominal kinematic viscosity of the oil at 40°C, mm2/sb)Velocity factor, Zv

The velocity factor, Zv, is to be calculated from the following equations:
Zv = C ZV +2(1 − C ZV )0 .8 +

32v

In the range 850 N/mm2 ≤ σHlim ≤ 1200 N/mm2, CZV is to be calculated as follows:

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CZV = CZL + 0.02
c)

Roughness factor, ZR

The roughness factor, ZR, is to be calculated from the following equations:

Where:

 3 

Z R = 
 R z10 

Rz =

CZR

R z1 + R z 2
2

The peak-to-valley roughness determined for the pinion Rz1 and for the wheel Rz2 are meanvalues for the peak-to-valley roughness Rz measured on several tooth flanks (Rz as defined in
the reference standard ISO 6336-2:2019).
R z10 = R Z 3

10

ρ red

relative radius of curvature:ρ red =

ρ1 ⋅ ρ 2
ρ1 + ρ 2

Wherein:ρ 1,2= 0.5 ⋅ d b1,2 ⋅ tan α tw(also for internal gears, db negative sign)
If the roughness stated is an arithmetic mean roughness, i.e. Ra value (=CLA value) (=AAvalue) the following approximate relationship can be applied:R a = CLA = AA = R z 6

In the range 850 N/mm2 ≤ σHlim ≤ 1200 N/mm2, CZR is to be calculated as follows:
CZR = 0.32 − 0.0002 ⋅ σ H lim

If σHlim < 850 N/mm2, take CZR = 0.150
If σHlim > 1200 N/mm2, take CZR = 0.080
M56.2.11 Hardness ratio factor, ZW
The hardness ratio factor, ZW, accounts for the increase of surface durability of a soft steel
gear meshing with a significantly harder gear with a smooth surface in the following cases:a) Surface-hardened pinion with through-hardened wheel

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(cont)

0.15

If HB< 130

 3 

ZW = 1.2 ⋅ R
 zH 

If 130 ≤ HB ≤ 470

HB − 130   3  ⋅ 
ZW = 1.2 −1700
R  zH 

If HB >470

 3 

ZW = 
R
 zH 

0.15

0.15

Where:
HB = Brinell hardness of the tooth flanks of the softer gear of the pair
RzH = equivalent roughness, μm

RzH =Rz1 ⋅ (10 / ρ red ) ⋅ (Rz1 / Rz 2 )(v ⋅ν 40 / 1500)0.33
0.33

0.66

ρred = relative radius of curvature (see clause 2.10 c)b) Through-hardened pinion and wheel
When the pinion is substantially harder than the wheel, the work hardening effect increases
the load capacity of the wheel flanks. ZW applies to the wheel only, not to the pinion.
If HB1/HB2 < 1.2
If 1.2≤ HB1/HB2≤1.7
If HB1/HB2>1.7

ZW = 1
HB1
ZW = 1 +  0.00898− 0.00829  ⋅ (u − 1)
HB2

ZW = 1 + 0.00698 ⋅ (u − 1)

If gear ratio u>20 then the value u=20 is to be used.
In any case, if calculated ZW <1 then the value ZW = 1.0 is to be used.
M56.2.12 Size factor, ZX
The size factor, ZX, accounts for the influence of tooth dimensions on permissible contact
stress and reflects the non-uniformity of material properties.
The factor mainly depends on:material and heat treatment;
tooth and gear dimensions;ratio of case depth to tooth size;
ratio of case depth to equivalent radius of curvature.
For through-hardened gears and for surface-hardened gears with adequate casedepth
relative to tooth size and radius of relative curvature ZX = 1. When the casedepth is relatively
shallow then a smaller value of ZX should be chosen.
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M56.2.13 Safety factor for contact stress, SH
The safety factor for contact stress, SH, can be assumed by the Society taking into accountthe type of application.
The following guidance values can be adopted:-

Main propulsion gears:
Auxiliary gears:

1.20 to 1.40
1.15 to 1.20

For gearing of duplicated independent propulsion or auxiliary machinery, duplicated beyond
that required for class, a reduced value can be assumed at the discretion of the Society.
M56.3

Tooth root bending strength

M56.3.1 Scope and general remarks
The criterion for tooth root bending strength is the permissible limit of local tensile strength in
the root fillet. The root stress σF and the permissible root stress σFP shall be calculated
separately for the pinion and the wheel.
σF must not exceed σFP.
The following formulae and definitions apply to gears having rim thickness greater than
3.5mn.
The result of rating calculations made by following this method are acceptable for normalpressure angles up to 25° and reference helix angles up to 30°.
For larger pressure angles and large helix angles, the calculated results should be confirmedby experience as by Method A of the reference standard ISO 6336-3:2019.
M56.3.2 Basic equations
M56.3.2.1 Tooth root bending stress for pinion and wheel

σF =

Ft
YF YS Yβ YBYDT K A K γ K v K Fα K Fβ ≤ σ FP
bmn

where:
YF= tooth form factor
YS = stress correction factor
Yβ= helix angle factor
YB = rim thickness factor
YDT = deep tooth factor
Ft , K A , K γ , K v , K Fα , K Fβ(see clause 3.3)
(see clause 3.4)
(see clause 3.5)
(see clause 3.6)
(see clause 3.7)
(see Part 1)bmn(see Part 1, clause 1.4)
(see Part 1, clause 1.3)

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M56.3.2.2 Permissible tooth root bending stress for pinion and wheel(cont)σ FP =σ FEYd YNSFYδrelT YRrelT YX

where:
σFE =
Yd
=
YN =
YδrelT =YRrelT =YX =SF=

bending endurance limit
design factor
life factor
relative notch sensitivity factor
relative surface factor
size factor
safety factor for tooth root bending stress

M56.3.3 Tooth form factor, YF
The tooth form factor, YF, represents the influence on nominal bending stress of the toothform with load applied at the outer point of single pair tooth contact. YF shall be determined
separately for the pinion and the wheel. In the case of helical gears, the form factors forgearing shall be determined in the normal section, i.e. for the virtual spur gear with virtual
number of teeth Zn.
The tooth form factor, YF, is to be calculated as follows:h
6 F cos α Fen
mn
YF =
2
 s Fn  cos α n mn 
Where:= bending moment arm for tooth root bending stressfor application of load at the outer point of single tooth
pair contact

mm= tooth root normal chord in the critical sectionmm

αFen = pressure angle at the outer point of single tooth paircontact in the normal section

°

hF

sFn

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Fig. 3.1 Dimensions of hF, sFn and αFen for external gear
For the calculation of hF, sFn and αFen, the procedure outlined in the reference standard ISO
6336-3:2019 (Method B) is to be used.
M56.3.4 Stress correction factor, YS
The stress correction factor YS, is used to convert the nominal bending stress to the local
tooth root stress, taking into account that not only bending stresses arise at the root.
YS applies to the load application at the outer point of single tooth pair contact.
YS shall be determined separately for the pinion and for the wheel.
The stress correction factor, YS, is to be determined with the following equation (having rangeof validity: 1 ≤ qs ≤ 8 ):


1


 1.21+ 2.3 L 
1.2 + 0.13L qs

YS = (

Where:
qs =

)

sFn
2ρF

qs

= notch parameter,ρF= root fillet radius in the critical section, mm

L= sFn /hF

For hF and sFn see clause 3.1
For the calculation of ρF the procedure outlined in the reference standard ISO 6336-3:2019 is
to be used.
M56.3.5 Helix angle factor, Yβ
The helix angle factor, Yβ, converts the stress calculated for a point loaded cantilever beam
representing the substitute gear tooth to the stress induced by a load along an oblique load
line into a cantilever plate which represents a helical gear tooth.

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The helix angle factor, Yβ is to be calculated as follows:(cont)

Yβ = 1 − ε ββ120where:β= reference helix angle in degrees.

The value 1.0 is substituted for εβ when εβ > 1.0, and 30° is substituted for β > 30°.
M56.3.6 Rim thickness factor, YB
The rim thickness factor, YB, is a simplified factor used to de-rate thin rimmed gears. For
critically loaded applications, this method should be replaced by a more comprehensive
analysis. Factor YB is to be determined as follows:a) for external gears:
if s R / h ≥ 1.2
if 0.5 < s R / h < 1.2
where:YB = 1h 
YB = 1.6 ⋅ ln 2.242
s R 
sR = rim thickness of external gears, mm
h = tooth height, mm
The case s R / h ≤ 0.5 is to be avoided.
b) for internal gears:
if s R / mn ≥ 3.5
if 1.75 < s R / mn < 3.5
where:YB = 1m 
YB = 1.15 ⋅ ln 8.324 n sR sR = rim thickness of internal gears, mm
The case s R / mn ≤ 1.75 is to be avoided.
M56.3.7 Deep tooth factor, YDTThe deep tooth factor, YDT, adjusts the tooth root stress to take into account high precision
gears and contact ratios within the range of virtual contact ratio 2.05 ≤ ε αn ≤ 2.5 , where:

ε αn =

εαcos 2 β b

Factor YDT is to be determined as follows:
if ISO accuracy grade ≤ 4 and ε αn > 2.5
if ISO accuracy grade ≤4 and 2.05 < ε αn ≤ 2.5

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YDT = 0.7
YDT = 2.366 − 0.666 ⋅ ε αn

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in all other cases

YDT = 1.0

M56.3.8 Bending endurance limit, σFE
For a given material, σFE is the local tooth root stress which can be permanently endured.
According to the reference standard ISO 6336-5:2016 the number of 3x106 cycles is regarded
as the beginning of the endurance limit.
σFE is defined as the unidirectional pulsating stress with a minimum stress of zero
(disregarding residual stresses due to heat treatment). Other conditions such as alternatingstress or prestressing etc. are covered by the design factor Yd.
The σFE values are to correspond to a failure probability 1% or less.
The endurance limit mainly depends on:material composition, cleanliness and defects;
mechanical properties;
residual stresses;
hardening process, depth of hardened zone, hardness gradient;material structure (forged, rolled bar, cast).
The bending endurance limit, σFE is to be determined, in general, making reference to values
indicated in the reference standard ISO 6336-5:2016, for material quality MQ.
M56.3.9 Design factor, Yd
The design factor, Yd, takes into account the influence of load reversing and shrinkfit
prestressing on the tooth root strength, relative to the tooth root strength with unidirectionalload as defined for σFE.
The design factor, Yd, for load reversing, is to be determined as follows:
Yd = 1.0
in general;
Yd = 0.9
for gears with occasional part load in reversed direction, such as main
wheel in reversing gearboxes;
Yd = 0.7
for idler gears
M56.3.10 Life factor, YN
The life factor, YN, accounts for the higher tooth root bending stress permissible in case alimited life (number of cycles) is required.
The factor mainly depends on:material and heat treatment;
number of load cycles (service life);
influence factors (YδrelT ,YRrelT , YX).
The life factor, YN, is to be determined according to Method B outlined in the reference
standard ISO 6336-3:2019.

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M56.3.11 Relative notch sensitivity factor, YδrelT
The relative notch sensitivity factor, YδrelT, indicates the extent to which the theoreticallyconcentrated stress lies above the fatigue endurance limit. The factor mainly depends onmaterial and relative stress gradient.
The relative notch sensitivity factor, YδrelT, is to be determined as follows:YδrelT =1 + 0.2 ρ ' (1 + 2q s )1 + 1 .2 ρ 'where:
qs = notch parameter (see clause 3.4)
ρ’ = slip-layer thickness, mm, from the following table
Material
ρ’, mmcase hardened steels, flame or induction hardened steels
0.0030500 N/mm²
0.0281
600
N/mm²0.0194through-hardened steels1), yield point Re=
800 N/mm²0.00641000 N/mm²
0.0014nitrided steels
0.1005
1)
The given values of ρ’ can be interpolated for values of Re not stated above
M56.3.12 Relative surface factor, YRrelT
The relative surface factor, YRrelT, takes into account the dependence of the root strength on
the surface condition in the tooth root fillet, mainly the dependence on the peak to valley
surface roughness.
The relative surface factor, YRrelT is to be determined as follows:
Rz < 1

1 ≤ R z ≤ 40

1.120

1.674 − 0.529(Rz + 1)

1.070

5.306 − 4.203(Rz + 1)

1.025

4.299 − 3.259(Rz + 1)

0.1

0.01

0.0058Material
case hardened steels, through - hardened steels(σ B ≥ 800 N/mm 2 )normalised steels(σ B < 800 N/mm 2 )nitrided steels

Where:
Rz= mean peak-to-valley roughness of tooth root fillets, μm

σB= tensile strength, N/mm2

The method applied here is only valid when scratches or similar defects deeper than 2Rz are
not present.

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If the roughness stated is an arithmetic mean roughness, i.e. Ra value (=CLA value) (=AAvalue) the following approximate relationship can be applied:R a = CLA = AA = RZ 6

M56.3.13 Size factor, YX
The size factor, YX, takes into account the decrease of the strength with increasing size.
The factor mainly depends on:material and heat treatment;
tooth and gear dimensions;ratio of case depth to tooth size.
The size factor, YX, is to be determined as follows:
YX = 1.00

for mn ≤ 5

YX = 1.03 - 0.06 mn

for 5 < mn < 30

YX = 0.85

for mn ≥ 30

YX = 1.05 - 0.010 mn

for 5 < mn < 25

YX = 0.80

for mn ≥ 25

generallynormalised and through-hardened steels
surface hardened steels

M56.3.14 Safety factor for tooth root bending stress, SF
The safety factor for tooth root bending stress, SF, can be assumed by the Society taking into
account the type of application.
The following guidance values can be adopted:
Main propulsion gears:
1.55 to 2.00
Auxiliary gears:
1.40 to 1.45
For gearing of duplicated independent propulsion or auxiliary machinery, duplicated beyond
that required for class, a reduced value can be assumed at the discretion of the Society.

End ofDocument

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