A solid $63-$mm$-$diameter cold$-$rolled brass $[$G $=39\mathrm{.}6$ GPa$]$ shaft that is $0\mathrm{.}96$ m long extends through and is completely bonded to a

hollow aluminum $[G=26\mathrm{.}5$GPa tube. Aluminum tube $(1)$ has an outside diameter of $92$ mm $,$ an inside diameter of $63$ mm $,$ and a length

of $0\mathrm{.}51$ m $.$ Both the brass shaft and the aluminum tube are securely attached to the wall support at $A.$ Assume $L_1=L_2=0\mathrm{.}51m,L_3=0\mathrm{.}45$

$m,T_B=23kN-m,$ and $T_C=8kN-m.$ When the two torques shown are applied to the composite shaft, determine:

$($a$)$ the maximum shear stress magnitude ${}_1$ in aluminum tube $(1).$

$($b$)$ the maximum shear stress magnitude ${}_2$ in brass shaft segment $(2).$

$($c$)$ the maximum shear stress magnitude ${}_3$ in brass shaft segment $(3).$

$($d$)$ the rotation angle ${}_B$ of joint $B.$

$($e$)$ the rotation angle ${}_C$ of joint $C.$

Answers:

$($a$){}_1=$

MPa.

$($b$){}_2=$

i

MPa.

$($c$){}_3=$

MPa.

$($d$){}_B=$

$$

$($e$){}_C=$