The simply$-$supported beam shown below has constant $EI$ throughout its length.

a$)$ Using the basic singularity functions provided in Table $5\mathrm{.}16$ of the textbook $($also

available in the Class $38$ notes$),$ show that the distributed loading on the beam $w(x)$ can

be expressed using singularity functions as

$w(x)=(12k\frac{N}{m})[($:$x-2(m)$:${)}^0-\frac{1}{3(m)}($:$x-2(m)$:${)}^1+($:$x-8(m)$:${)}^0+\frac{1}{3(m)}($:$x-8(m)$:${)}^1]$

b$)$ Determine the shear force and bending moment distributions in terms of singularity

functions.

c$)$ Determine the elastic curve $y(x)$ in terms of singularity functions.