$(1)$ A RC beam section is subjected to a service dead load $($including self$-$weight$)$ moment of

$60tf-m$ and a live load moment of $80tf-m.$ Concrete has $f_c^{\text{'}}=280kg\frac{f}{c}{m}^2$ and beam

longitudinal reinforcement has $f_y=5,000kg\frac{f}{c}{m}^2.$ The beam has a sectional width $b$ of

$45$ cm and a sectional height $h$ of $90$ cm $.$

$($a$)$ Before the compression reinforcement is added, calculate the maximum factored

moment the beam can take by setting the maximum tension reinforcement ratio to be

$0\mathrm{.}9{}_{\text{m}\text{a}\text{x}}.$ Assume the distance from the centroid of tension reinforcement to the bottom

of the section $d$ is $6$ cm $.$

$($b$)$ Calculate the required total tension and compression reinforcement areas for the beam

to take the required factored moment $(M_u).$ Assume the distance from the centroid of

compression reinforcement to the extreme fiber of the compression side $d^{\text{'}}=6cm.$

$($c$)$ Design the least amount of tension reinforcement and compression reinforcement to

satisfy the required moment strength $(M_u)$ considering a concrete cover of $4cm,$#$4U$

stirrup and a symmetrical arrangement. The maximum size of the aggregate is $2$ cm $.$

Use #$10$ bars. Based on your arrangement of tension and compression reinforcement,

check $M_n{M}_u$ considering contribution from compression reinforcement.