Problem $1(25$ pts$)$

For the beam shown in Figure $1,$ the properties are $E=200$GPa,$I=200{10}^{6}m{m}^4,A=$

$2{10}^{4}m{m}^2,$

i$.$ The externally applied $10kN$ on node $2$ is redundant. Why?

ii$.$ Derive the stiffness matrix using the direct stiffness method for the active DOFs. You should

obtain a $33$ matrix,

iii. Find the right$-$hand$-$side force vector and solve for the DOFs. Find the reaction forces at node $1$

and node $4.$

iv$.$ Use SAP$2000$ to verify your results in $($iii$)$ and plot the deformed shape.