1. Built-In Column
The critical load for other column can be expressed in terms of the critical buckling load for a pin-ended column.
From symmetry conditions at the point of inflection occurs at ¼ L.
Therefore the middle half of the column can be taken out and treated as a pin-ended column of length LE=L/2
Yielding:

2. Therefore:

3. Note on Moment of Inertia
Since Pcrit is proportional to I, the column will buckle in the direction corresponding to the minimum value of I

4. Critical Column Stress
A column can either fail due to the material yielding, or because the column buckles, it is of interest to the engineer to determine when this point of transition occurs.
Consider the Euler buckling equation

5. Critical Buckling Load
Pcrit is the critical or maximum axial load on the column just before it begins to buckle
E youngs modulus of elasticity
I least moment of inertia for the columns cross sectional area.
L unsupported length of the column whose ends are pinned.