Circular Shaft - Stresses in Linear Range

The elastic torsion formula determines shearing stress in a rigid uniform circular shaft by combining Hooke's Law with the polar moment of inertia. It derives from multiplying shearing strain by the modulus of rigidity, then ensuring the sum of moments on any cross-section equals the applied torque. This formula applies when torque remains within Hooke's law limit, preventing permanent deformation.

For a solid circular shaft, the polar moment of inertia is calculated from the fourth power of the radius. For a hollow shaft with inner radius r1 and outer radius r2, the polar moment of inertia is expressed as the difference in the fourth power of both radii. This difference accounts for the removed material in the hollow center.

The modulus of rigidity relates shearing stress to shearing strain through Hooke's Law. When multiplied by the shearing strain expression, it enables derivation of the shearing stress equation for the shaft. This material property is essential for connecting elastic deformation behavior to the applied torque.

The sum of moments of elementary forces exerted on any cross-section of the shaft must equal the magnitude of the applied torque. This equilibrium condition is fundamental to deriving the elastic torsion formula. Substituting this relationship into the stress equation produces the integral term representing polar moment of inertia.

The elastic torsion formula applies when torque remains within Hooke's law limit, ensuring no permanent deformation occurs. Under these conditions, the shaft exhibits linear elastic behavior, and shearing stress is directly proportional to shearing strain. This linear range assumption is critical for formula validity.

Maximum shearing stress is inversely proportional to the polar moment of inertia. After substituting for maximum shearing stress in the equilibrium equation, the polar moment of inertia emerges as a key geometric property. Larger polar moments of inertia reduce maximum stress for the same applied torque, which is why hollow shafts are often preferred in design of transmission shafts.

The shaft must be rigid and uniform with constant cross-section. Torque must remain within Hooke's law limit to prevent plastic deformation. The formula assumes linear elastic material behavior where shearing stress is proportional to shearing strain. These conditions ensure the derived relationships between torque, stress, and geometric properties remain valid.