Question $2(20$ marks$)$

A solid $72-$mm diameter cold$-$rolled brass $G=39$GPa shaft that is $1\mathrm{.}6m(L_1+L_2)$ long

extends through $($tube $2$ and tube $3)$ and is completely bonded to a hollow aluminum

$[G=28$GPa $]$ tube, as shown in Figure $2.$ Aluminum tube $(1)$ has an outside diameter of

$96mm,$ an inside diameter of $72mm,$ and a length of $L_1$ of $0\mathrm{.}9m.$ Both the brass shaft and

the aluminum are securely attached to the wall at $A.$ When the two torques $T_1=26kNm$

and $T_2=12kNm$ are applied as shown to the composite shaft, determine:

$($i$)$ The maximum shear stress magnitude in aluminum tube $(1).$

$(4$marks$)$

$($ii$)$ The maximum shear stress magnitude in brass shaft tube $(2).$

$(4$marks$)$

$($iii$)$ The maximum shear stress magnitude in brass shaft tube $(3).$

$(4$marks$)$

$($iv$)$ The rotation angle of joint $B.$

$(4$marks$)$

$($v$)$ The rotation angle of joint $C.$

$(4$marks$)$

Figure $2$