$(1)-$ Derive from first principles the following equation for the ultimate

moment of resistance $(M_u)$ of a singly reinforced section in which $f_{cu}$ is the

characteristic cube strength of concrete, $b$ is the width of the beam and $d$ the

effective depth

$=0\mathrm{.}156f_{cu}b{d}^2$

$(2)$ Explain what you understand by the term 'under$-$reinforced' and why

concrete beams are normally designed in this way.

$(3)$ A simply supported rectangular beam of $7m$ span carries characteristic

dead $($including self$-$weight of beam$),gk,$ and imposed, $qk,$ loads of $12$

$kN{m}^{-1}$ and $8kN{m}^{-1}$ respectively. The beam dimensions are breadth, b$,275$

$mm$ and effective depth, d$,450mm.$ Assuming the following material

strengths, calculate the area of reinforcement required.

$f_{cu}=30Nm{m}^{-2},f_y=500Nm{m}^{-2}$