Problem $1(60$ pts $)$ Our team is designing the structure to support a corridor. We fix the beams at one end and put a roller on the other end. Compared to simply supported beam, the current design should reduce the peak internal forces, thus reducing the size of the members. We are asked to analyze the beam while carrying the full live load as shown below.

$(1)$ Use force method to determine the reactions for the beam subjected to the live load as shown above. $(30$ pts $)$

$(2)$ Construct the shear and moment diagrams. $(20$ pts$)$

$(3)$ Draw the deflection shape. $(10$ pts$)$