$($a$)$ Using DoE procedure, design a concrete mix to yield a $28$ day characteristic strength of

$50\frac{N}{m}{m}^2$ assuming that quality control during production results in a standard deviation

of $4\frac{N}{m}{m}^2$ with a $2\mathrm{.}5\%$ defective rate.

The following data are given:

Cement: $,$ Ordinary Portland $($Class $42\mathrm{.}5)$

$($Density of cement particles may be taken as $3150k\frac{g}{m^3})$

Aggregates: $,$ Coarse $-$ uncrushed, maximum size $20mm$

Fine $-$ uncrushed with $80\%$ passing a $600m$ sieve.

$($Density of combined aggregate may be taken as $2600k\frac{g}{m^3})$

Required workability: $,10-30mm$ slump

Durability requirements:

Max. w$/$c ratio $-0\mathrm{.}55$

Min. cement content $-300k\frac{g}{m^3}$

Max. cement content $-450k\frac{g}{m^3}$

Determine: target mean strength, w$/$c ratio using DoE procedure, water content, cement

content, coarse and fine aggregate content.