$4\mathrm{.}10$ Derive the shape functions for a beam finite element of length $L$ assuming a cubic polynomial in the form

$v(x)=a_0+a_{1}x+a_2{x}^2+a_3{x}^3$

by satisfying the boundary conditions $v(0)=0,v(L)=0,v^{\text{'}}(0)=0,$ and $v^{\text{'}}(L)=0.$