A circular shaft has a diameter $D=20mm$ and a length $L=1\mathrm{.}5m$ 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 ${}_y.$

b$)$ Calculate the maximum shear strain ymax.

c$)$ Determine the radius py to the elasticplastic boundary.

d$)$ Determine the torque T $.$ If the member has yielded and then unloaded.

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