For the square column shown below:

$(1)$ Derive the theoretical flexural buckling load, P$\_($cr$),$ and corresponding theoretical effective length factor, K$,$ using the principles shown in class $($assume the column buckles in the $+$y direction$).$ Cut the column directly above the fixed end $($x$=0)$ for the FBD $($free$-$body diagram$)$ where the deflected shape is concave down $($increasing slope in the $+$y direction$).$

$(2)$ Draw the buckled shape and clearly illustrate the equivalent pin$-$pin column $($hint: draw the moment diagram$).$

Be sure to draw clean and orderly FBDs$.$ Hint: This column buckles in an anti$-$symmetric form. Therefore, use statics about the

whole buckled column to compute all reaction forces at the supports. There will be horizontal reactions.