$(20$ points$)$ A bridge is supported by several piers, and the geometry and loads of a typical pier are

as shown below. Note that the pier consists of a linearly decreasing, and then linearly increasing,

width.

The top surface load of $20k\frac{N}{m^2}$ represents the weight of the bridge and traffic on the bridge. The

concrete weighs $30k\frac{N}{m^3}$ and the modulus is $E=28{10}^{6}k\frac{N}{m^2}.$ We will model this pier as a $1-$D

structure, using a $3-$node, $2-$element discretization as shown in the figure. Recall that for each

element, the finite element matrix equation is of the form:

$[K^e]*[u^e]=[f^e]+[Q^e]$

We are using the linear Lagrange interpolation functions in this finite element model.

For this finite element discretization, derive $K_{11}^1$ and $f_1^1.$

i$.$e$.,$ for Element $1,$ derive the $K_{11}$ entry of the stiffness matrix and the $f_1$ entry of the RHS force

vector.