Question $2(50$ points$)$

A propped cantilever beam AC of length $2$ L is loaded by a concentrated load P as illustrated in the figure. The beam is of rectangular cross$-$section with the corresponding moment of inertia I and Young's modulus E$.$ The beam is supported at B by a linear elastic spring with stiffness $\backslash (\backslash$mathrm$\{$k$\}=6\backslash$mathrm$\{$EI$\}/\backslash$mathrm$\{$L$\}^\{3\}\backslash ).$ The spring exerts only transverse force reaction on the beam. Employing Castigliano's $\backslash (2^\{\backslash$text $\{$nd $\}\}\backslash )$ theorem, please determine the reactions at $\backslash ($ A $\backslash )$ and $\backslash ($ B $\backslash )$ in terms of the known parameters P$,$L$,$E$,$I. Neglect any transverse shear strain effect, in elastic strain energy expression.