Torsion in Shafts

When a shaft is subjected to a pure torsion load, it experiences twisting along its length. This twisting causes shear stresses to develop in the cross-section of the shaft. The design of such a shaft involves determining the appropriate diameter to ensure that the material can withstand the applied torsional load without failure.

Theories for Designing Shaft Diameter

There are several theories that can be used to design the diameter of a shaft subjected to pure torsion. These theories are based on different assumptions and considerations, and they may lead to different results. The theories commonly used for designing shaft diameter include:

1. Maximum Principal Stress Theory: This theory states that failure occurs when the maximum principal stress in the shaft reaches the maximum allowable stress for the material. The maximum principal stress theory assumes that failure is governed by normal stress.

2. Maximum Shear Stress Theory: According to this theory, failure occurs when the maximum shear stress in the shaft reaches the maximum allowable shear stress for the material. The maximum shear stress theory assumes that failure is governed by shear stress.

3. Strain Energy Theory: This theory states that failure occurs when the strain energy per unit volume in the shaft reaches the strain energy per unit volume at yield point for the material. The strain energy theory assumes that failure is governed by the total strain energy stored in the material.

Determining the Largest Diameter

Among these theories, the Maximum Shear Stress Theory gives the largest diameter for a shaft subjected to pure torsion. This is because the maximum shear stress theory assumes that failure occurs when the maximum shear stress reaches the maximum allowable shear stress for the material. Shear stress is directly related to the torque applied to the shaft and inversely related to the polar moment of inertia, which is a function of the shaft diameter. Therefore, by maximizing the shear stress, the diameter of the shaft can be increased to accommodate the applied torsional load.

On the other hand, the maximum principal stress theory and the strain energy theory do not directly consider shear stresses. They focus on normal stresses and strain energy, respectively. As a result, these theories may lead to smaller diameters compared to the maximum shear stress theory.

In summary, when designing a shaft subjected to pure torsion, the maximum shear stress theory should be used to determine the largest diameter for the shaft. This theory takes into account the shear stresses developed in the shaft and allows for a larger diameter to be selected to ensure the shaft can withstand the applied torsional load.