$(25$ pts$)$ Two equal length elements are used to model and analyze the beam structure shown below. The beam is fixed to the wall at the left end $($node $1)$ and is simply supported at the right end $($node $3).$ It supports a couple $C=50N-m$ at the center $($node $2)$ and a uniformly distributed load of intensity $p_o=60\frac{N}{m}$ between nodes $2$ and $3.$ The length of the beam is $3$ m with a flexural rigidity $EI=15N-m^2.$

a$)(4$ pts$)$ For FEA, what are the boundary conditions for this structure?

b$)(6$ pts$)$ Obtain the structural stiffness matrix for the beam system.

c$)(6$ pts $)$ Calculate the deflections and slope angles at nodes $2$ and $3.$

d$)(5$ pts$)$ Calculate the reaction force and moment at the fixed end of the beam.

e$)(4$ pts$)$ Determine the bending moment for the beam section located at $\frac{1}{2}m$ to the right of node $1.$