Problem $4(25$ pts$).$ An elastic bar of Young's modulus $E,$ mass density $,$ and length L

is hanging from the ceiling of a room. Due to the acceleration of gravity $g,$ the rod stretches

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$)$:

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

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

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

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

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

$($b$)4pts$ Sketch the finite element discretization $($mesh$)$ of the bar using two linear ele$-$

ments of equal length.

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

are respectively

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

Assemble the global system of equations for the nodal displacements.

$($d$)[6$pts$]$ Impose the essential B$.$C$.$ and solve for the nodal displacements.