b$)$ The transverse support member within the same platform can be considered to be a simply supported beam and is subjected to the loads given below. You have been tasked with amending the design and making weight savings wherever possible.

To allow your work to be checked by a colleague you are asked to produce a fully labelled sketch of the shear force distribution along the length of the beam.

As part of this analysis you have been asked to derive the expressions for the bending moments in the beam, and hence produce a graph showing the bending moment distribution along the length of the beam.

Given that the maximum allowable tensile stress in the beam is $350$ MPa $,$ determine the lightest possible section to use for this structure, using a universal beam to BS $4$ part $1$:$2005.$ Ensure that the relevant section of the steel stock table is presented with your answer, clearly indicating and justifying the beam you have selected.