Material: $C\frac{30}{S}420$

Concrete cover: $d^{\text{'}}=40mm$

${}_{cu}=0\mathrm{.}003$

$f_{ck}=30$MPa

$f_{yk}=420$MPa

a$)$ The rectangular $RC$ beam in the figure is subjected to pure bending. Calculate the moment capacities and curvature values for

i$)$ the rebar yielding $M_y-\frac{O}{{?}_y},($Assume concrete stress distribution is linear.$)$

ii$)$ the ultimate strength conditions $M_u-\frac{O}{{?}_u},$

and draw the moment$-$curvature curve by using the calculated values. Determine the section ductility ratio of this section.

b$)$ For the same section, if the half amount of the tension reinforcement is added as compression reinforcement