Perform a matrix analysis on the steel beam having a rectangular cross$-$section with a base of $9$ cm and height of $20$ cm and Young's modulus E$=210$GPa shown in Figure $1.$ The beam is fixed at one end and free at another end. The beam is supported by a roller at Node $2$ and hinge at the other two nodes. The load carried by the beam are P$1=1\mathrm{.}5$kN$,$P$2=1\mathrm{.}6$kN$,$ M$1=5$kN$-$mM$2=5$kN$-$m and q$=1\mathrm{.}5$kN$//$m$.$

$($a$)$ Identify number of nodes, elements and draw the FEM Model $($show displacement vectors$)$ for the beam.

$($b$)$ How many degrees of freedom does the structure have?

$($c$)$ Define element stiffness matrices.

$($d$)$ Assemble the stiffness matrix for the beam.

$($e$)$ Define the displacement, joint force vectors, and fixed end moment vectors for the beam.

$($f$)$ Formulate the system of equations and apply boundary conditions

$($g$)$ Derive the reduced system matrices.

$($h$)$ Find the unknown displacements and reactions.

$($i$)$ Find forces and moments in each element.

$($j$)$ Provide Analysis summary.