A circular shaft has a diameter D $=40$ mm and a length L $=1\mathrm{.}5$ m and is made of structural steel with elastic perfectly plastic behavior $($G $=95\mathrm{.}98$ GPa and y $=160$ MPa$).$ The shaft fixed at one end and subjected to a torque T at the other end. The angle of twist is measured between the ends as $=0\mathrm{.}2$ rad.

a$)$ Calculate the yield shear strain y$.0\mathrm{.}001667$rad

b$)$ Calculate the maximum shear strain max$.0\mathrm{.}002667$rad

c$)$ Determine the radius y to the elastic$-$ plastic boundary.$12\mathrm{.}5026$mm

d$)$ Determine the torque T$.$ If the member has yielded and then unloaded.$2\mathrm{.}5171$kN$.$m

e$)$ Determine the residual shear stress at $=5$ mm$.$Blank $5$ Question $3$

f$)$ Determine the radius where the residual shear stress is zero.