Page 358 - Introduction to Continuum Mechanics

342 The Elastic Solid

5.57. Compare the twisting torque which can be transmitted by a shaft with an elliptical
cross-section having a major axis equal to twice the minor axis with a shaft of circular
cross-section having a diameter equal to the major axis of the elliptical shaft. Both shafts are
of the same material. Also compare the unit twist under the same twisting moment,
5.58. Repeat the previous problem, except that the circular shaft has a diameter equal to the
minor axis of the elliptical shaft.
5.59. (a) For an elliptic bar in torsion, show that the magnitude of the maximum shearing stress
varies linearly along radial lines x^ = kx$ and reaches a maximum on the outer boundary.
(b) Show that on the boundary the maximum shearing stress is given by

so that the greatest shearing stress does occur at the end of the minor axis.
5.60. Consider the torsion of a cylindrical bar with an equilateral triangular cross-section as in
Fig.P5.4.

(a) Show that a warping function <p = a (3*2 £3 - x-$) generates an equilibrium stress field.
(b) Determine the constant a in order to satisfy the traction-free lateral boundary condition.
Demonstrate that the entire lateral surface is traction-free.

(c) Write out explicitly the stress distribution generated by this warping function. Evaluate
the maximum shearing stress at the triangular corners and along the line #3 = 0 in a cross-sec-
tion. Along the line x-$ = 0 where does the greatest shearing stress occur?

Fig. P5.4

5.61. An alternate manner of formulating the problem of the torsion of a cylinder of noncircular
cross-section employs a stress function ty fa, x$) such that the stresses are given by