A $75$ mm by $200$ mm deep sawn timber beam in a domestic residence has a clear span of $2\mathrm{.}75$ m and a bearing length restricted to $100$ mm at each end. It is of strength class C$24,$ functions in service class $2$ conditions and is laterally restrained against lateral buckling along its length. Use the following medium term loads;

$G_k=1\mathrm{.}3k\frac{N}{m}($characteristic dead load$)$

$Q_k=1\mathrm{.}5k\frac{N}{m}($characteristic imposed load$)$

$P=1\mathrm{.}0kN($characteristic point dead load at mid $-$ span$)$