Designing a transmission shaft requires a thorough understanding of the stresses induced by bending moments and torques, especially in systems where power is transferred through gears. These forces create force-couple systems at the centers of the shaft's cross-sections, leading to both transverse and torsional loading. Although shearing stresses from transverse loads are typically smaller than those from torques and are often overlooked, the significant normal stresses from these loads contribute to the maximum shearing stress.

Calculating the normal stress exerted on a shaft section involves examining the shaft's cross-section at a specific point, considering both the torque and bending couples. The maximum normal stress typically occurs at the ends of the diameter perpendicular to the resultant of the bending moment couple. Determining the cross-section's minimum allowable polar moment ratio J/c is essential to ensure the shaft's integrity and functionality.

This ratio is vital as it accounts for both the maximum resultant bending moment and torque on the shaft, in addition to the allowable shearing stress. This analytical approach is fundamental in designing both solid and hollow circular shafts, enabling the creation of optimized shaft configurations that can reliably withstand operational stresses. Such methodology ensures that the designed shafts are robust, efficient, and tailored to their specific applications, guaranteeing mechanical systems' safety and functionality.