Problem $1$: A bending beam

Perform a matlab simulation of a bending beam with a load of $\backslash (\backslash$mathrm$\{$P$\}=44400\backslash$mathrm$\{$~N$\}\backslash )$ in the downward direction as shown in the diagram below. The Young's modulus is $\backslash ($ E$=2\backslash$times $10^\{11\}\backslash$mathrm$\{$~Pa$\}\backslash ),$ the total length is $\backslash ($ L$=10\backslash$mathrm$\{$~m$\}\backslash ),$ the width $($out of the plane in image below$)$ is $40$ cm $,$ and the thickness is $20$ cm $.$ The beam is fully supported on both sides, and supported from vertical deflection $($not rotation$)$ at the pivot.

$1.$ Provide the displacement vector $($with variables for unknowns$).$

$2.$ Provide the force vector $($with variables for unknowns$).$

$3.$ Provide the global stiffness matrix.

$4.$ Solve for the unknown nodal displacements and provide the full displacement vector.

$5.$ Solve for the unknown forces and moments and provide the full force vector.

$6.$ In matlab, use the shape functions of the beam element to plot the vertical deflection $\backslash ($ w$($x$)\backslash )$ along the entire length of the beam. Use a different line color for each of the element's.

$7.$ Use the shape functions to compute the strain along both the top and bottom surfaces of the beam. Provide matlab plots of the strain and place an X where the strain $($and thus stress$)$ is largest.