A rectangular beam of span $7m($centre$-$to$-$centre of supports$),$ resting on $300mm$ wide simple supports, is to carry a uniformly distributed dead load $($excluding selfweight$)$ of $15k\frac{N}{m}$ and a live load of $20k\frac{N}{m}.$ Using Fe $500$ steel, design the beam section at midspan, based on first principles. Check the adequacy of the section for strength. Also perform a check for deflection control. Assume that the beam is subjected to moderate exposure conditions.

Design the beam section in Problem $1,$ given that the overall beam depth is restricted to $550mm.$