$(1)$ The rigid bar A D is supported as shown by pin A and two steel wires $($E$\_$s$=30\backslash$times $10^6$psi$)$ at B and C$.$ A downward load $=580$ lb is applied at D and the cross sectional area of each wire is $0\mathrm{.}03$ in$^2.$ Determine $($a$)$ the normal stresses in the two wires, $($b$)$ the corresponding deflection of point D$.$ A n s: $($a$)\backslash$sigma $\_$B E$=18$ ksi, $\backslash$sigma $\_$C F$=20$ ksi $($b$)\backslash$Delta $\_$D$=0\mathrm{.}018$ in $(2)$ A downward load $=36$ kN is applied to the tip D of the horizontal rigid bar A D that is pinned at support A and connected to two vertical links B F and C E as shown. Both links are made of aluminum $($E$=75$ GPa$)$ with a $12\backslash$times $20$ mm rectangular cross section. Determine $($a$)$ the normal stress in link B F and $($b$)$ the displacement of point D$.$ A n s: $($a$)-37\mathrm{.}5$ MPa $($b$)2$mm$(3)$ The brass shell $(\backslash$alpha $\_$b$=20\mathrm{.}9\backslash$times $10^-6/$ C$)$ is fully bonded as shown to the steel core $(\backslash$alpha $\_$s$=11\mathrm{.}7\backslash$times $10^-6/$ C$).$ Determine the largest allowable increase in temperature $\backslash$Delta T if the allowable normal stress in the steel core is $55$ MPa. Ans: $75\mathrm{.}4^$C

$($a$,40pt)$ A thin walled tube is subjected to equal magnitude torques $T$ at both ends. The tube has an outer radius of $r,$ wall thickness $t,$ and shear modulus $G.$ Determine $($i$)$ the maximum and minimum shear stresses in the tube; $($ii$)$ maximum and minimal shear strains; $($iii$)($relative$)$ angle of twist; $($iv$)$ rate of twist. In total $6$ quantities.

$($b$,20$ bonus pt$)$ The same tube is slit lengthwise as shown in picture. Under the same torques $T,$ determine its $($i$)$ maximum shear stress and $($ii$)$ angle of twist. How do they compare to the original tube?

Hint: this is not a circular cross$-$section now due to the slit. Refer to textbook.