$(9$ points$)$ Light$-$grade steel channel was used as a purlin of a truss. The top chord of the truss is inclined $1V$:$3H$ and distance between trusses is equal to $7m.$ The purlin has a weight of $80\frac{N}{m}$ and spaced at $1\mathrm{.}5$ on centers. The dead load of load including the roof materials $731Pa,$ live load of $1120Pa$ and wind load of $1450Pa.$ Coefficient of pressure at leeward are $0\mathrm{.}6$ and $0\mathrm{.}25$ respectively. Assume all loads are acting on top of the purlin. And the allowable bending stress in major and minor axis is $210$MPa.

For $D+L+W,$ a $\frac{1}{3}$ increase in allowable stresses is allowed.

Properties of $C20077mm$

$Sx=6\mathrm{.}21{10}^{4}m{m}^3$;$,Sy=1\mathrm{.}40{10}^{{?}^{?}}4m{m}^3$

;$W=81\frac{N}{m}$

a$.)$ Calculate the max. ratio of actual to the allowable bending stress for load combination $0\mathrm{.}75(D+L+WL)$ at the windward side. b$.)$ Calculate the max. ratio of actual to the allowable bending stress for load combination $0\mathrm{.}75(D+L+WL)$ at the leeward side.

c$.)$ Calculate the bending stress for dead and live load combination $(D+L)$ if the dead load was increase by $20\%.$