Problem $3$

You are given a solid cylindrical torsion bar of cold drawn mild steel of length

${15}^{\text{'}\text{'}}$ and radius ${1\mathrm{.}5}^{\text{'}\text{'}},$ which is subjected to a steady torque M$.$

$($a$)$ If the solid bar is twisted elastically through a small angle $,$ as shown

above, derive the following expression for maximum shear strain:

${}_z=\frac{R_0}{2}\frac{}{L}$

State your assumptions.

$($b$)$ If the bar was made up of a composite of two materials, with a stainless

steel core surrounded by a steel case, derive an expression, in terms of $R_0,$

$R_i,L,$ and elastic modulus, for the torque $M^{\text{'}}$ required to twist the

composite bar through the sample angle $.$

$($c$)$ For the solid bar, determine the torque required to twist the bar elastically

through $=1.$

$($d$)$ For the composite bar, determine the maximum radius of the stainless

steel core to still retain $90\%$ of the torsional strength of the solid bar.