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AGMA Gear Tooth Bending Stress Formula and Calculator

Gear Engineering and Design

AGMA Gear Tooth Bending Stress Formula and Calculator

Per standard ANSI/AGMA D04 Fundamental Rating Factors and Calculation Methods for Involute Spur and Helical Gear Teeth:

Preview: AGMA Gear Tooth Bending Stress Calculator

The fundamental formula for bending stress number in a gear tooth is:

Eq. 1
st = Wt · Ko · Kv · Ks · Pd / F · Km · KB / J

Eq. 2
= π / ( p x · tan ψs ) = Pnd · cos ψs for helical gears

Pnd = normal diametral pitch, in-1;
px = axial pitch, in;
ψs = helix angle at standard pitch diameter.

Eq. 3
= arcsin [ π / ( px / ( px · Pnd ) ]

Transmitted tangential load, Wt
Eq. 4
Wt = 33,000 P / vt = 2 T / d = 396,000 P / ( π np d )


P = transmitted power, hp;
T = transmitted pinion torque, lb in;
vt = pitch line velocity at operating pitch diameter, ft/min.


Eq. 5
vt = π np d / 12

d = operating pitch diameter of pinion, in.
for external gears
Eq. 6
d = 2C / ( mG + 1 )


Rim thickness factor, KB - Where the rim thickness is not sufficient to provide full support for the tooth root, the location of bending fatigue failure may be through the gear rim, rather than at the root fillet. Published data suggest the use of a stress modifying factor, KB, in this case. The rim thickness factor, KB, is not sufficiently conservative for components with hoop stresses, notches or keyways. This data is based on external gears with smooth bores and no notches or keyways.

The rim thickness factor, KB, adjusts the calculated bending stress number for thin rimmed gears. It is a function of the backup ratio, mB,

Eq. 2
mB = tR · ht

tR = gear rim thickness below the tooth root, in;
ht = gear tooth whole depth, in.

The effects of webs and stiffeners can be an improvement

The effect of tapered rims has not been investigated. When previous experience or detailed analysis justifies, lower values of KB may be used.

KB is applied in addition to the 0.70 reverse loading factor where it is applicable

The geometry factor, I, evaluates the radii of curvature of the contacting tooth profiles based on tooth geometry. These radii are used to evaluate the Hertzian contact stress in the tooth flank. Effects of modified tooth proportions and load sharing are considered.

The geometry factor, J, evaluates the shape of the tooth, the position at which the most damaging load is applied, and the sharing of the load between oblique lines of contact in helical gears. Both the tangential (bending) and radial (compressive) components of the tooth load are included.


ANSI/AGMA American Nation Standards Institutue . American Gear Manufacturers Association