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.