A long structural column (length $$-L$$) with both ends hinged is acted upon by an axial compressive load $$P.$$ The differential equation governing the bending of column is given by: $$$EI{{{d^2}y} \over {d{x^2}}} = - Py$$$

Where $$y$$ is the structural lateral deflection and $$EI$$ is the flexural rigidity. The first critical load on column responsible for its buckling is given by

Four column of the same material having identical geometric properties are supported in different ways as shown below

It is required to order these four beams in the increasing order of their respective first buckling loads. The correct order is given by

When a column is fixed at both ends corresponding to Euler's criterion load is

The kern area (core) of a solid circular section column of diameter, $$D$$ is a concentric circle of diameter, $$'d'$$ is equal to