19.4 : 扭转角度 - 弹性范围

Consider a circular shaft of length L and a uniform cross-section of radius r subjected to a torque at its free end.

The maximum shearing strain in the shaft is proportional to both the angle of twist and the radial distance from the shaft's axis.

In the elastic range, this shearing strain can be expressed in terms of the applied torque, radial distance, polar moment of inertia, and the modulus of rigidity.

By equating these two equations, we derive an expression for the angle of twist within the elastic range. This equation applies to a homogeneous shaft with a uniform cross-section when torque is only applied at one of its ends.

If the shaft is subjected to torques at different locations, or if it consists of various parts with different cross sections or materials, the angle of twist must be considered separately for each part.

The total angle of twist is calculated by summing all individual values from each part of the shaft or by integrating along the length for shafts with non-uniform cross-sections.