Problem $2-$Design$/$Check: $(10$ points$)$

A $6\mathrm{.}0$ m $-$long simply supported rectangular beam supports a uniform service dead load of $30k\frac{N}{m}($with

selfweight$)$ and a uniform service live load of $15k\frac{N}{m}.$ Use $f_{ic}^{\text{'}}=28$MPa&$f_2=420$MPa. Assume the cover to

center of the steel $=60mm.$ The beam width $b=400mm.$

$(1)$ Calculate the ultimate positive moment of the beam.

$(2)$ Design a singly reinforced rectangular cross$-$section to resist the maximum negative moment.

Use $=0\mathrm{.}015.$ Make all necessary checks.

$(3)$ Sketch the final designed cross$-$section with full details