Question $1(40$ Marks$)$

For the beam Cross$-$Section shown in Figure $1,$ determine the $k_1,x,$ and $A$ s required to balance the Equation $(1).[$Hint: Use Excel Solver$].$ Submit a softcopy of the Excel sheet as part of your final submission. After solving for the values, determine the resulting cross$-$section$\text{'}$s ultimate moment of resistance $(M_4).$ If the cross$-$section is to carry a design moment of $1\mathrm{.}2M_4,$ comment on the type of cross$-$section $($under$-$reinforced, balanced or over$-$reinforced$)$ and support your comments with the correct equations or illustrations.

$k_l{f}_{cu}bx=A_4{f}_s-A_i{f}_s(1)$

Please make the necessary assumptions where necessary. Use fcu$=25\frac{N}{m}{m}^2$ and $fy=500\frac{N}{m}{m}^2$

Figure $1.$ Cross section of a rectangular beam $($All dimensions are in $mm)$

Question $2(30$ Marks$)$

Using the data in Table $4-6-1($Page $111$ of the Textbook$),$ develop a two$-$parameter relationship between $K,\frac{z}{d},$ and $\frac{x}{d}.$ Plot a $3-$D graph of the relationship and comment on it $($Observation$,$ Quantification, and Physical meaning$).$

Question $3(30$ Marks$)$

Superimpose the following curves: Hognestad et al$.($Fig$.4\mathrm{.}4-1),$ BS $8110$ stress block $($Fig$.4\mathrm{.}4-4),$ and Whitney's block for the values of $kl$ and $k2$ only. Develop regression equations for these curves. Comment on the graph $($Observation$,$ Quantification, and Physical meaning$).$ Structural engineering questions