Problem $2$: Beam Bending Analysis

The architect James Law designed a residential tower where each residence flat has a personal swimming pool as the balcony, complete with glass containment walls.

As a structural engineer for this project, your task is to design the overhang beam that is used to support the swimming pool. The beam, denoted by the bold rectangular dashed lines, is hung from the nearest support column $($not seen in the above figure$)$ to support the area of the pool as shown above with thin dotted lines. Due to the overall design of the swimming pool as has just been illustrated, each beam supports more water at the hanging end of the beam and thus we can model the loading on the beam by the pool as a linear distributed load as shown below. The beam is designed using roller and hinge support, rather than a fixed support. The area supported by the beam between the roller and the hinge is actually inside the apartment $($ie$.$ the area between the nearest and the core support columns$).$

The specification given by James $($the architect$)$:

The beam should be able to support the swimming pool without yielding $($remember yielding is fundamentally started with atomic planes slipping against each other and these slips are activated by shear stress$).$

As the structural engineer, you approach the beam design analysis by answering the following steps:

a$)$ Draw the Free Body Diagram of the beam

b$)$ Draw the Shear Force and Bending Moment Diagram of the beam

c$)$ What and where is the maximum bending stress?

d$)$ What is the shear stress $($due to shear force$)$ at that location if we assume that shear stress $($due to shear force$)$ is uniform across the cross$-$sectional area?