Problem $4(25$ pts $).$ A column of Young's modulus $E,$ mass density $,$ and length L is

fixed at its top and bottom ends. Due to the acceleration of gravity $g,$ the rod stretches

vertically under its own weight.

The axial deformation of the rod is measured by the vertical displacement $u(x),$ which is

governed by the following boundary value problem $($BVP$)$:

$\frac{d}{dx}+g=0$ PDE

$(x)=E\frac{du}{dx}$ Hooke's law

$u(L)=0B.C.$

$u(0)=0B.C.$

$($a$)[8$pts$]$ Find the analytical solution of the BVP $(8).$

$($b$)[4$pts$]$ Now consider the Finite Element $($FE$)$ solution of the BVP$.$ Sketch the FE

discretization $($mesh$)$ of the column using two linear elements of equal length. In your

sketch, highlight elements, nodes, and specify the nodal degrees of freedom $($nodal

displacements$).$

$($c$)7pts$ The two elements have identical stiffness matrix $K$ and force vector $F,$ which

are respectively

$K=\frac{2E}{L}\left[\begin{array}{cc}1& -1\\ -1& 1\end{array}\right],$ and $,F=\frac{gL}{4}\left[\begin{array}{c}1\\ 1\end{array}\right]$

Assemble the global system of equations for the nodal displacements.

$($d$)[6$pts$]$ Impose the boundary conditions and solve for the nodal displacements.