# Angle of Twist - Elastic Range

JoVE Core
Mechanical Engineering
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JoVE Core Mechanical Engineering
Angle of Twist – Elastic Range

### Nächstes Video19.5: Angle of Twist: Problem Solving

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.

## Angle of Twist - Elastic Range

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.