A propeller shaft carries a steady thrust and a steady torque together

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

Bending, Ringing, Breaking and Flowing, Ch Strain and stress in a deformed solid

Problem

A propeller shaft carries a steady thrust and a steady torque together. At a point on the outer surface of the shaft, with

x

along the shaft axis and

y

tangential to the surface, the resulting plane state of stress is

\begin{aligned}\sigma_{xx}&=80\,\mathrm{MPa}\ (\text{axial tension}),\qquad \sigma_{yy}=0,\\ \sigma_{xy}&=45\,\mathrm{MPa}\ (\text{torsional shear}).\end{aligned}

((a)) Find the two principal stresses and the angle, measured from the shaft axis, to the principal axis carrying the larger (tensile) principal stress. ((b)) Give the centre and radius of Mohr's circle for this state, and hence the maximum shear stress

\tau_{\max}

and the angle from the shaft axis of the plane on which it acts. ((c)) Under tension alone the principal axis of greatest tension lies along the shaft (

0^\circ

) and the maximum-shear directions at

\pm45^\circ

; under torsion alone the principal axis lies at

45^\circ

and the maximum-shear directions at

0^\circ

and

90^\circ

. By how many degrees, and in which sense, have these directions turned here from their tension-alone positions, and how far do they lie from their torsion-alone positions?

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