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 |
 |  |  |  |
