A helical gear with 20° pressure angle and 30° helix angle mounted at the midspan of a shaft that is supported between two bearings at the ends. The nature of the stresses induced in the shaft is

The correct option is: normal stress due to bending in two planes and axial loading: shear stress due to torsion.

To understand why this is correct, let us analyze the forces acting on a helical gear mounted on a shaft and how they induce different types of stresses in the shaft.

1. Force Analysis of a Helical Gear:
When a helical gear transmits power, the tooth load is three-dimensional due to the helix angle (^β) and the pressure angle (^α). The total force acting on the gear tooth can be resolved into three mutually perpendicular components:
• Tangential Force (F_t): This force acts tangentially to the pitch circle of the gear. It transmits the torque and induces a torsional moment on the shaft, which results in shear stress due to torsion. Additionally, because it acts perpendicular to the shaft axis in the vertical plane, it causes bending of the shaft in that vertical plane.
• Radial Force (F_r): This force acts radially toward the center of the shaft. It causes bending of the shaft in the horizontal plane (perpendicular to the bending caused by the tangential force). Thus, the tangential and radial forces together cause bending in two perpendicular planes, leading to normal stress due to bending in two planes.
• Axial/Thrust Force (F_a): Due to the helix angle, an axial force is generated parallel to the axis of the shaft. This axial force acts along the shaft line, producing an axial tensile or compressive load on the shaft, which induces direct normal axial stress.

2. Stresses Induced in the Shaft:
Combining the effects of all three force components:
• The tangential force (F_t) and radial force (F_r) act in perpendicular directions, both causing bending moments. This results in bending stresses (which are normal stresses) acting in two planes.
• The axial force (F_a) acts along the axis of the shaft, producing a direct axial normal stress (axial loading).
• The torque transmitted by the tangential force (F_t) induces a twisting moment, which results in torsional shear stress.

Therefore, the nature of the stresses induced in the shaft is normal stress due to bending in two planes and axial loading, along with shear stress due to torsion.