For a $3-$node $1-$d bar element, derive the shape functions using the following threepolynomial expansions. $(9)$

$$Apply displacement field corresponding to rigid body deformations $($u$1=$u$2=$u$3=$u$)$and examine if the strain inside the element is zero or not. Third node is at mid$-$length of the element

$$Apply displacement field corresponding to constant strain conditions $($u$1=0,$ u$2=$u$,$u$3=0\mathrm{.}5$u$)$ and examine if the strain in the element is really constant or not.

$$Critically comment on the results obtained with respect to the polynomialexpansion used.