$4\mathrm{.}3.$ Consider the bar shown.

$($a$)$ Solve for the exact displacement response of the structure.

$($b$)$ Show explicitly that the principle of virtual work is satisfied with the displacement patterns

$($i$)\frac{?}{\text{b}\text{a}\text{r}}(u)=ax$ and $($ii$)\frac{?}{\text{b}\text{a}\text{r}}(u)=a{x}^2.$

$($c$)$ Identify a stress ${}_{}$ for which the principle of virtual work is satisfied with pattern $($ii$)$ but

not with pattern $($i$).$