Torsion in Circular Shafts: Angle of Twist, Shear Strain and Stress

This video explains torsion in circular bars, showing how applied torque twists a shaft while cross sections remain axisymmetric. It introduces the angle of twist φ, which increases linearly from zero at the fixed end to φ at the free end, and presents a governing relation that involves the bar length, the applied torque, the shear modulus, and the polar moment of inertia. It also discusses how the cross section shape affects warping, the difference between hollow and solid shafts in resisting torsional loads, and how internal torque diagrams are used when multiple torques act on a shaft. The talk concludes with how ductile and brittle materials fail under pure torsion and an illustration based on Moore's circle.

Introduction

Torsion is the twisting of an object caused by torques applied about its longitudinal axis. For circular bars, cross sections rotate as a rigid body while remaining axisymmetric, so there is no distortion of the cross section itself. This section sets up the basic problem of a shaft fixed at one end and loaded by torque at the other, leading to a twist along the length.

Angle of Twist and Governing Parameters

The angle of twist varies linearly along the bar from zero at the fixed end to a total twist φ at the free end. The angle of twist depends on the bar length L, the applied torque T, the shear modulus G of the material, and the polar moment of inertia J of the cross section. These four parameters form the core relationship used to predict how much twisting occurs for a given shaft configuration.

Polar Moment of Inertia and Cross Sections

The polar moment of inertia J quantifies a cross section's resistance to torsional deformation based solely on its shape. For a hollow circular bar with outer radius RO and inner radius RI, J is given by a specific formula; setting RI to zero recovers the solid-bar expression. This shows why hollow bars can carry torsional loads more efficiently for the same outer radius.

Torsional Shear Strain and Stress

Shear strain on the surface is related to the geometry of deformation and increases with radial distance from the center. The shear strain inside the bar varies linearly with radius. The shear stress is also proportional to the radius, reaching its maximum on the outer surface. The fundamental equations link torque, radius, and J to the internal stresses and strains, enabling design checks for torsional failure.

Single versus Multiple Torque Loadings

Shafts in machines are often loaded by several torques. In such cases an internal torque diagram is built by free-body analysis and equilibrium, revealing how the internal torque varies along the shaft and identifying sections with the highest shear stress.

Failure under Pure Torsion: Ductile vs Brittle

Under pure torsion, ductile materials tend to fracture along planes of maximum shear stress, while brittle materials fail along planes of maximum tensile stress. Moore's circle illustrates the relationship between maximum shear and normal stresses for a given stress element, clarifying why different materials fail in distinct ways when subjected to torsion.

Practical Takeaways

Understanding torsion in circular bars informs shaft design, material selection, and safety margins for rotational power transmission in engines, wind turbines, and industrial machinery.