Sample Problem $12\mathrm{.}7$ The $3-$in$.-$diameter steel bar in Fig. $($a$)$ carries the bending moment $\backslash (\backslash$mathrm$\{$M$\}=2\mathrm{.}21\backslash$mathrm$\{$kip$\}\backslash$cdot $\backslash$mathrm$\{$ft$\}\backslash )$ and a torque T$.$ If the yield strength of steel is $40$ ksi $,$ determine the largest torque T that can be applied without causing yielding. Use $(1)$ the maximum shear stress theory; and $(2)$ the maximum distortion energy theory. Answer:$8\mathrm{.}56$kip$$ft$.\backslash (9\mathrm{.}88\backslash$mathrm$\{$kip$\}\backslash$cdot $\backslash$mathrm$\{$ft$\}\backslash ).$

$($a$)$