Problem #$1$:

A torsion$-$bar spring consists of a prismatic bar, usually of round cross section, that is twisted at one end and held fast at the other to form a stiff spring. An engineer needs a stiffer one than usual and so considers building in both ends and applying the torque somewhere in the central portion of the span, as shown in the figure. This effectively creates two springs in parallel. If the bar is uniform in diameter, that is$,$ if $d=d1=d2,($a$)$ determine how the spring rate and the end reactions depend on the location x at which the torque is applied, $($b$)$ determine the spring rate, the end reactions, and the maximum shear stress, if $d=12\mathrm{.}700mm,x=127\mathrm{.}000mm,l=254\mathrm{.}000mm,T=262\mathrm{.}680N*mm,$ and $G=79\mathrm{.}290$GPa.