The element shown below is a bar element $($subject to only axial deformation$)$ that

contains three nodes, also known as a higher$-$order element.

$($Note: This is not two elements!$)$:

Assuming the following function for the displaced shape of the element:

$u(x)=c_0+c_{1}x+c_2{x}^2$

Determine the three unknown constants $c_0,c_1,$ and $c_2$ using the following boundary

conditions:

$u(0)=u_1,u(\frac{L}{2})=u_2,u(L)=u_3$

Determine the $N$ and $B$ matrices for the element:

$u(x)=[N{]}_{13}\{d{\}}_{31},(x)=\frac{du}{dx}=[B{]}_{13}\{d{\}}_{31}$

Derive the stiffness matrix for the element:

$[K{]}_{33}={}_V[B{]}^T[C][B]dV=AE{\displaystyle \underset{x=0}{\overset{x=L}{}}}[B{]}_{31}^T[B{]}_{13}dx$