$4.$ Design elements for a $2-$storey laboratory building. Refer to the plans and section below. $[42$ marks$]$

a$.$ Roof

i$.$ Select the lightest deck thickness suitable for the roof. Advise how many continuous spans will be needed for this to be acceptable. $[4$ marks$]$

ii$.$ Select the most economical joist for the roof. Indicate the depth of joist the architect will need to accommodate. Show all your checks. $[4$ marks$]$

iii. Select the most economical W$-$section for the Beam $1$ girder. Show all your checks. $[6$ marks$]$

The architect wants a $76$ mm roof deck on joists spanning the entire $16$ m roof. As designer, work with the following assumptions:

$-$ You decided to space your joists at $2\mathrm{.}5$ m $.$

$-$ The architects have given their intended finishes.

$-$ You estimate a total roof dead load of $1\mathrm{.}4$ kPa and a snow load of $2\mathrm{.}2$ kPa $.$

$-$ Considering serviceability, the roof must be limited to a deflection of L$/240$ for live load and $\backslash (\backslash$mathrm$\{$L$\}/180\backslash )$ for total loads