$($a$)$ A horizontal beam $4\mathrm{.}2m$ long, resting on two simply supports $3\mathrm{.}0m$ apart,

carries a uniformly distributed load of $25k\frac{N}{m}$ between the supports with

concentrated loads of $15kN$ and $20kN$ at the ends, as shown in Figure $1$

below.

Assume the weight of the beam is negligible, calculate the following:

$($i$)$ the reactions $R1$ and $R2$ at the supports.

$($ii$)$ Sketch the diagrams of shear force and bending moments.

$($iii$)$ State the greatest bending moment and shear force.

$($iv$)$ The beam has a uniform rectangular cross$-$section with a width of $50$

$mm$ and $75\mathrm{.}6mm$ deep. Determine the maximum bending stress.

Use the bending moment equation below to aid your calculation.

$\frac{M}{\text{I}}=\frac{}{y}=\frac{E}{R}$