Calculation of hoop stress on a gear tooth

Gear teeth experience stresses at the root due to contact force from the meshing tooth. The stresses are due to bending and shear on the tooth. The hoop stress model is not applicable to analyzing gear teeth.

 

Gear teeth are in rolling contact with a specified tooth face contact angle. That angle results in two force components, one a peripheral torque, and a smaller one tending to push the gears apart. To push the gears apart requires that the teeth are pressed onto their wheel which cancels some of the root stress due to bending. There appears to be no stress equivalent to hoop stress in spur gears.

The same is not true for internal or some epicyclic gear systems where the force of tooth contact tends to expand the internally toothed gear. In that situation the outside of the internally cut gear will have a peripheral stress. That is analogous to hoop stress, but it's magnitude is small, being only the torque attenuated by the sine of a typical 14° to 20° contact angle.