Problem $1$

For the $4-$node bar element shown in Figure $1,$ derive the shape function and element stiffness

matrix. Draw out how the shape functions look like in the domain $($either by hand or using plot in

matlab$).$ How many Guass points would you recommend to evaluate the stiffness matrix? Why?

$($reminder: $N$ points is needed to exactly integrate a polynomial of order $2N-1$ or less, for example,

$1$ point for $y=ax,2$ point for $y=ax+b{x}^2+c{x}^3)$