$Q_1$:

Given the figure below derive the values of the following:

$55$ points

$(10$ points$)$: $\frac{x}{d}($in terms of ${}_{}$ and ${}_{nd})$

$(10$ points$)$: $\frac{d}{x}($in terms of ${}_{ccc}$ and ${}_{xc})$

$(15$ points$)$: If $x=0\mathrm{.}5d$ and $f_s=f_{sc}=0\mathrm{.}95f_y,$ derive an expression for $A_{st}$ and $A_{sc}.$

$(20$ points$)$: A singly reinforced rectangular $RC$ section with $b=200,d=560mm,f_{cu}=30\frac{N}{m}{m}^2$ and $A_{st}=750m{m}^2$ with $f_y=460\frac{N}{m}{m}^2.$ What is the mode of failure of this section? Does the section fall within the BS$8110$ code limits of reinforcement ratio? What is the ultimate moment of resistance $M_r$ of the section?

Q$2$:

$45$ points

$(25$ points$)$ Calculate the cracking moment for the section shown below. The characteristic strengths of materials are $f_{cu}=35\frac{N}{m}{m}^2$ for concrete and $f_y=460\frac{N}{m}{m}^2$ for steel. The modulus of rupture of concrete, $f_r=0\mathrm{.}7\sqrt[\phantom{2}]{f_{cu}}\frac{N}{m}{m}^2.$

$(20$ points$)$ Calculate the required reinforcement for the same section to resist a bending moment of $200$ kNm