A beam has length L$,$ cross$-$sectional area A$,$ moment of inertia I, and polar moment of inertia J$.$ The beam is constructed from a material with Young$$s Modulus E and Shear Modulus G$.$ The beam is supported at the left end by cantilevered $($fixed$)$ constraint, at the right end by a roller, and is subjected to a uniformly distributed load w$.$ There is a point load applied downward on the roller support. Find an expression for the reaction forces and moments. Use Castigliano$$s theorem. Neglect shear.