$3\mathrm{.}8-18$ Two pipes and $($:$L_2=1\mathrm{.}5(m)\}$ are joined at $B$ by flange plates $($thickness $t_f=14mm)$ with five bolts $)=(13(mm)$ arranged in a circular pattern $($see figure$).$ Also, each pipe segment is attached to a wall $($at A and $C,$ see figure$)$ using a base plate $)=(15(mm)$ and four bolts $)=(16(mm).$ All bolts are tightened until just snug. Assume $E_1=110$GPa,$E_2=73$GPa,$v_1=0\mathrm{.}33,$ and $v_2=0\mathrm{.}25.$ Neglect the self$-$weight of the pipes, and assume the pipes are in a stress$-$free state initially. The crosssectional areas of the pipes are $A_1=1500m{m}^2$ and $A_2=(\frac{3}{5})A_1.$ The outer diameter of Pipe $1$ is $60mm.$ The outer diameter of Pipe $2$ is equal to the inner diameter of Pipe $1.$ The bolt radius $r=64mm$ for both base and flange plates.

$($a$)$ If torque $T$ is applied at $x=L_1,$ find an expression for reactive torques $R_1$ and $R_2$ in terms of $T.$

$($b$)$ Find the maximum load variable $T($i$.$e$.,T_{\text{m}\text{a}\text{x}})$ if allowable torsional stress in the two pipes is ${}_{\text{a}\text{l}\text{l}\text{o}\text{w}}=65$MPa.

$($c$)$ Draw torsional moment $($TMD$)$ and torsional displacement $($TDD$)$ diagrams. Label all key ordinates. What is ${}_{\text{m}\text{a}\text{x}}?$

$($d$)$ Find $T_{\text{m}\text{a}\text{x}}$ if allowable shear and bearing stresses in the base plate and flange bolts cannot be exceeded. Assume allowable stresses in shear and bearing for all bolts are ${}_{\text{a}\text{l}\text{l}\text{o}\text{w}}=45$MPa and ${}_{\text{a}\text{l}\text{l}\text{o}\text{w}}=90$MPa.