The joint shown below is connected together using a $5$ mm diameter bolt. When the force $\backslash ($ P $\backslash )$ is applied, the bolt deforms. The bolt is made from a material having a shear stress$-$strain diagram that is approximated as shown. When a force P is applied, the bolt deforms by shear only. Note that stress$-$strain diagrams for shear stresses have the same features as those for normal stresses.

$($a$)$ Draw a diagram of the system showing the deformed bolt. Label shear stresses and strain.

$($b$)$ If a force $\backslash (\backslash$mathrm$\{$P$\}=17\mathrm{.}1\backslash$mathrm$\{$kN$\}\backslash )$ is applied, find the shear stress. Note that in this case you have two symmetric force couples in equilibrium. The sign $($sense$)$ of the strain is not important here.

$($c$)$ Find the shear modulus.

$($d$)$ Determine the permanent shear strain of the bolt when an applied force $\backslash ($ P$=17\mathrm{.}1\backslash$mathrm$\{$kN$\}\backslash )$ is removed. Again, the sign $($sense$)$ of the strain is not important.