Question

Determine the relationship between the torque N and the torsion angle ϕ for

(a) the tube whose wall thickness Δr is considerably less than the tube radius;

(b) for the solid rod of circular cross-section. Their length l, radius r, and shear modulus G are supposed to be known.

Answer 1

(a) Consider a hollow cylinder of length l, outer radius r + Δr inner radius r, fixed at one end and twisted at the other by means of a couple of moment N. The angular displacement φ, at a distance l from the fixed end, is proportional to both l and N. Consider an element of length dx at the twisted end. It is moved by an angle φ as shown. A vertical section is also shown and the twisting of the parallelopipe of length l and area Δr dx under the action of the twisting couple can be discussed by elementary means. If f is the tangential force generated then shearing stress is f/Δr dx and this must equal

(b) For a solid cylinder we must integrate over r. Thus