Question $-3$:

$($a$)$ Using Castigliano's method, determine the deflections of points $\backslash ($ A $\backslash )$ and $\backslash ($ B $\backslash )$ due to the force $\backslash ($ F $\backslash )$ applied at the end of the step shaft shown in Figure. The second area moments for sections $\backslash ($ A B $\backslash )$ and $\backslash ($ B C $\backslash )$ are $\backslash ($ l$\_\{1\}\backslash )$ and $\backslash (2$ l$\_\{1\}\backslash ),$ respectively.

$($b$)$ Using industrial codes, specifications and standards of the different metallic materials, please give your opinion about choosing the right material for this type shaft. You may wish to use Table A$-22-$ results of tensile tests of some metals.