Please answer all parts$($a$)$ Sketch and clearly label an idealised stress distribution diagram for a doubly

reinforced rectangular concrete section subjected to a bending moment $M_{Ed},$ i$.$e$.$

a section requiring compression zone reinforcement.

Define the following non$-$dimensional terms used: $F_{ec},F_s^{\text{'}}$ and $f_{ck}.$

$($b$)$ Assuming the compression zone reinforcement has yielded, use the diagram

from $($a$)$ to derive the following expression for its total cross$-$sectional area, $A_x^{\text{'}}$ :

$A_s^{\text{'}}=\frac{(K-0\mathrm{.}167)f_{ck}b{d}^2}{0\mathrm{.}87f_{yk}(d-d^{\text{'}})}$

$($c$)$ The beam in Fig.O$3$ has a rectangular cross$-$section. Determine suitable

reinforcement arrangements to resist bending at locations A and $B.$

$(b=300(mm)$;$d=500(mm)$;$d^{\text{'}}=50(mm)$;$f_{ck}=30\frac{N}{m}{m}^2$;$f_{yk}=500\frac{N}{m}{m}^2)$

Sketch the beam cross$-$sections at A and B clearly showing the locations of the

reinforcement designed.

$($d$)$ Develop a mathematical inequality in terms of dimensions $d$ and $d^{\text{'}}$ to validate

the assumption above that the compression zone reinforcement has yielded at the

design moment $M_{Ed}.$

$({}_{cu}=0\mathrm{.}0035$;${}_{s,y}=0\mathrm{.}00217)$

Hence show that this is the case for the beam designed in $($c$).$