1. Show that the stress function ?= k (r2- a2 )
 is applicable to the solution of a
   solid bar of circular cross-section of radius a. Determine (a) the
stress distribution
 in the bar in terms of the applied torque, (b) the rate of twist and (c) the
warping of the cross-section.

2. Show that the stress function
?=-G*d?/dz [1/2*(x^2+y+y^2)-(1/(2*a))*(x^3-3*x*y^2)-(2*a/27)]
is the correct solution for the bar having a cross-section in the form
of the equilateral
triangle shown in the figure. Determine (a) the shear stress distribution,
(b) the rate of twist, (c) the warping of the cross-section, and (d)
the position and
magnitude of the maximum shear stress.

3. Using the membrane analogy, determine the maximum
shear stress and the rate of twist in terms of the applied
torque for the section of constant thickness t shown in the
figure.

4. Two tubular members, one circular and the other square,
are subjected to a torque T. If the thickness t and the total area of the region
occupied by the material are the same, compare the maximum shear stress and
the rates of twist. Neglect stress concentration and assume t<<b.

5. Compare the maximum stress in
two tubes with equal thickness and
the same interior cavity areas, one
of which is circular in cross-section
and the other elliptic with b/a=2.
Assume t<<b