$475$ Solve Problem $458$ using Castigliano$$s theorem.

$4-58$ The figure illustrates a stepped torsion$-$bar spring $\backslash ($ O A $\backslash )$ with an actuating cantilever $\backslash ($ A B $\backslash ).$ Both parts are of carbon steel. Use superposition and find the spring rate $\backslash ($ k $\backslash )$ corresponding to a force $\backslash ($ F $\backslash )$ acting at $\backslash ($ B $\backslash ).$