A tubular steel shaft has outer radius and inner radius

PHYSICS · Multi-part · Fermi University Physics · Fluids & Continuum Physics · Solid continuum mechanics and complex fluids · Solid continuum mechanics and complex fluids

Bending, Ringing, Breaking and Flowing, Ch Torsion and bending of rods and beams

Problem

A tubular steel shaft has outer radius

a_o = 30\,\mathrm{mm}

and inner radius

a_i = 24\,\mathrm{mm}

, and it carries a torque

T = 1.0\,\mathrm{kN\,m}

. The shaft is long enough that end effects can be ignored (Saint-Venant's principle) and the material is linear elastic with shear modulus

G

. ((a)) Starting from the assumption that plane cross-sections rotate rigidly about the shaft axis by an angle proportional to distance along the shaft, derive the shear stress distribution

\tau(r) = Gr\theta'

and show that the torque carried by a tube of outer radius

a_o

and inner radius

a_i

T = GJ\theta'

with

J = \dfrac{\pi}{2}(a_o^4 - a_i^4)

. ((b)) A solid shaft of the same steel, with the same cross-sectional area as the tube, carries the same torque

T

. Find its radius

a

and its polar second moment of area

J

. ((c)) Compute the maximum shear stress in the tube and in the solid shaft, and give their ratio. State which section carries this torque more efficiently for the same mass per unit length.

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