a) The critical loads for columns with various kinds of support conditions can be determined from the differential equation of the deflection curve by following the same procedure as above

Step 1 With the column assumed to be in the buckled state, we obtain an expression for the bending moment in the column.

Step 2 Set up the differential equation of the deflection curve, using the bending-moment equation

Step 3 Solve the equation and obtain its general solution, which contains two constants of integration plus any other unknown quantities.

Step 4 Apply boundary conditions pertaining to the deflection v and the slope v^’ and obtain a set of simultaneous equations.

Step 5 Obtain the equation of deflection curve for buckled column

Step 6 Solve those equations to obtain the critical load          

	Fixed-Free column	Fixed-fixed Column	Fixed-pinned column
Step 1
Step 2
Step 3
Step 4
Step 5	Applying third boundary condition	Applying third boundary condition	Applying third boundary condition
Step 6	Lowest critical load	Lowest critical load	Lowest critical load