Problem $2(30$ pts$)$

For the following planar frame in Figure $2,$ the properties are $E=200$GPa,$I=200{10}^{6}m{m}^4,A=$

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

i$.$ Calculate the stiffness coefficient $k_{3y3}$ with direct stiffness method.

ii$.$ Find the element level stiffness matrix and equivalent fixed$-$end loads for element $(3)$ at both

local and global reference system. You may use Python or Excel for help, but you only need to

show the results.

iii. With matrix stiffness method, study and complete the Python code hw$2\_$q$2\_$incomplete.py and

find all the unknown displacements. Submit your completed hw$2\_$q$2\_$completed.py file.

iv$.$ Use SAP$2000$ to verify your results and compare with $($iii$).$ Besides displacements, show the

reaction forces. Also, use SAP$2000$ to output the axial force, the shear force, and the bending

moment diagrams of the frame.

Figure $220$ Frome of Problem $2$