Problem #$1(25$ pts$)$:

A cantilever beam is modeled as one bar element using the finite element method. If the bar has Young's Modulus $E,$ cross sectional area $A,$ density $,$ is length $L,$ and has a force applied at node $1(F_1)$ and node $2(F_2),$ then find the following: The unknowns are the reaction force $R$ at node $1$ and the displacement $u_2(t)$ at node $2.$

a$)$ Find the nodal displacement $u_2(t)$ if $F_1$ and $F_2$ are step inputs $($constant$).$ Remember $u_1(t)=0.($Boundary condition$)$

b$)$ Solve for the reaction force $R(t).$

c$)$ Solve for the finite element displacement solution $u(x,t)$ for the entire bar element.