A singly$-$reinforced rectangular concrete beam has an effective depth to reinforcement of $400mm$ and a width of $200mm.$ It is desired that both concrete and reinforcement materials are utilized optimally such that, under long$-$term service load conditions, the maximum bending stresses are at their elastic limit values in both materials simultaneously.

a$)$ Neatly sketch the elastic strain and stress distribution diagrams for the section, indicating relative magnitudes for concrete and reinforcement $($assume cracked section$).$

b$)$ Use the diagrams in $($a$)$ above to determine the long$-$term neutral axis position for the stated stress conditions. $[$Ans$.x=121\mathrm{.}7mm]$

c$)$ Calculate the optimum reinforcement area required to achieve the desired objective.

$[$Ans$.A_s=303m{m}^2]$

d$)$ Calculate the long$-$term maximum elastic moment that the section can sustain. $[$Ans$.M=36\mathrm{.}3kNm]$

Take: For concrete, $f_{cu}=25\frac{N}{m}{m}^2$; service stress limit for bending compression $p_{cb}=\frac{f_{cu}}{3}$; $E_c=25k\frac{N}{m}{m}^2$; creep coefficient $=1\mathrm{.}2.$

For reinforcement, $f_y=500\frac{N}{m}{m}^2$; service stress limit for bending tension $p_{st}=2\frac{f_y}{3}$; $E_s=200k\frac{N}{m}{m}^2$