# Deformation in a Circular Shaft

JoVE Core
Mechanical Engineering
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JoVE Core Mechanical Engineering
Deformation in a Circular Shaft

### Próximo Vídeo19.3: Circular Shaft – Stresses in Linear Range

A unique property of circular shafts is that under torsion, every cross-section remains plane and undistorted, rotating as a solid rigid slab.

To determine the distribution of shearing stress, consider a cylindrical section inside a circular shaft of length L and radius R, fixed at one end. The radius of the cylindrical section is r.

Now, consider the small square element formed by two adjacent circles and straight lines on the surface of the cylindrical section before any load is applied.

As the torsional load is applied to the shaft, the square element deforms into a rhombus. Since the two sides of the rhombus are fixed, the shearing strain is equal to the angle between lines AB and A'B.

Using a small angle approximation and suitable geometry, it can be shown that shearing strain at any given point of a shaft in torsion is proportional to the angle of twist and the distance r from the axis of the shaft. It is maximum at the surface of the shaft.

## Deformation in a Circular Shaft

One of the distinctive characteristics of circular shafts is their ability to maintain their cross-sectional integrity under torsion. In other words, each cross-section continues to exist as a flat, unaltered entity, simply rotating like a solid, rigid slab. To understand the distribution of shearing stress within such a shaft, consider a cylindrical section inside this circular shaft. This section has a length of L and a radius of R, with one end fixed. The radius of the cylindrical section is denoted as r.

Before any load is applied, consider a small square element on the surface of the cylindrical section. This element is formed by two neighboring circles and straight lines. This square element morphs into a rhombus shape upon applying a torsional load to the shaft. Given that the two sides of the rhombus are anchored, the shearing strain equals the angle between the vertical line AB drawn on the walls of the cylinder section and the inclined line A'B drawn along a side of the rhombus. By applying a small angle approximation and appropriate geometry, it is possible to demonstrate that the shearing strain at any specific point of a shaft undergoing torsion is directly proportional to the angle of twist and the distance r from the shaft's axis. This strain reaches its maximum at the shaft's surface.