19.3:
Circular Shaft – Stresses in Linear Range
Consider a case when the torque applied to the circular shaft is within Hooke's law limit, so there is no permanent deformation.
Now, recall the expression for shearing strain. Multiplying it by the modulus of rigidity and using Hooke's Law for shearing stress and strain, an expression for shearing stress in a shaft can be determined.
Recall that the sum of the moments of the elementary forces exerted on any cross-section of the shaft must be equal to the magnitude of the torque exerted on the shaft.
Substituting for shearing stress and rearranging the terms gives an expression with an integral term, which represents the polar moment of inertia of the cross-section with respect to its center.
Further rearrangements and substitutions for maximum shearing stress give the elastic torsion formula for the shearing stress in a rigid uniform circular shaft.
However, for a hollow shaft with r1 and r2 as the inner and outer radii, the polar moment of inertia is expressed as a difference in the fourth power of two radii.
19.3:
Circular Shaft – Stresses in Linear Range
Consider a scenario where a circular shaft is subject to torque that remains within the boundaries of Hooke's Law, avoiding any permanent deformation. So, the formula for shearing strain is revisited. This formula is multiplied by the modulus of rigidity, and then Hooke's Law for the shearing stress and strain is applied. As a result, the equation for shearing stress in a shaft can be derived.
Furthermore, it is crucial to remember that the sum of the moments of the elementary forces acting on any cross-section of the shaft must be identical to the torque applied on that shaft. An integral term emerges when the equation is adjusted to substitute for shearing stress. This term signifies the polar moment of inertia of the cross-section concerning its center. After more adjustments and substitutions for the maximum shearing stress, the elastic torsion formula can be derived for the shearing stress in a uniformly rigid circular shaft.
However, the scenario differs for a hollow shaft where r1 and r2 are represented as the inner and outer radii. In this case, the polar moment of inertia is expressed as the difference in the fourth power of both radii.