A circular shaft has a diameter $\backslash (\backslash$mathrm$\{$D$\}=40\backslash$mathrm$\{$~mm$\}\backslash )$ and a length $\backslash (\backslash$mathrm$\{$L$\}=1\mathrm{.}5\backslash$mathrm$\{$~m$\}\backslash )$ and is made of structural steel with elastic perfectly plastic behavior $(\backslash (\backslash$mathrm$\{$G$\}=\backslash )95\mathrm{.}98$ GPa and $\backslash (\backslash$tau y$=120\backslash$mathrm$\{$MPa$\}\backslash )).$ 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 $\backslash (\backslash$varphi$=0\mathrm{.}2\backslash )$ rad.

a$)$ Calculate the yield shear strain $\backslash (\backslash$gamma y $\backslash ).$

b$)$ Calculate the maximum shear strain ymax.

c$)$ Determine the radius py to the elastic$-$ plastic boundary.

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

e$)$ Determine the residual shear stress at $\backslash (\backslash$rho$=5\backslash$mathrm$\{$~mm$\}\backslash ).$

$0\mathrm{.}0016774$ rad

$0\mathrm{.}00225$ rad

$0\mathrm{.}002667$ rad $0\mathrm{.}001333$ rad

$11$

$[$

then unloaded.

$12\mathrm{.}5026$ mm

$12\mathrm{.}5806$ mm

$12\mathrm{.}5026$ mm

$8\mathrm{.}3351$ mm

$16\mathrm{.}6701$ mm

$15$ mm

$11\mathrm{.}25$ mm

$2\mathrm{.}5171$ kN$.$m

$2\mathrm{.}0426$ kN$.$m

$0\mathrm{.}17138$ kN$.$m

$\backslash (\backslash$square $\backslash )\backslash (\backslash$square $\backslash )$

$106\mathrm{.}6667$ MPa

$86\mathrm{.}6667$ MPa

$53\mathrm{.}33$ MPa

$80$ MPa

$120$ MPa