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Consider a cylindrical shaft with a length denoted by L and a consistent cross-sectional radius referred to as r. This shaft undergoes a torque at the free end. The highest shearing strain within the shaft is directly proportional to the twist angle and the radial distance from the shaft axis. When the shaft behaves elastically, this shearing strain can be articulated using variables such as the applied torque, radial distance, the polar moment of inertia, and the modulus of rigidity. By setting these two equations equal to each other, it is possible to formulate an expression for the twist angle within the elastic range.

This equation applies to a uniform homogeneous shaft, where torque is only exerted at its extremities. However, if the shaft is exposed to torques at varying points or composed of different parts with diverse cross-sections or materials, the twist angle must be evaluated distinctly for each section. The sum of all individual values from each shaft segment determines the total twist angle. Alternatively, it can be calculated by integrating along the length of shafts with non-uniform cross-sections. This approach presents a comprehensive understanding of the behavior of shafts under varying conditions.

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