The ratio of the theoretical critical buckling load for a column with fixed ends to that of another column with the same dimensions and material, but with pinned ends, is equal to

The effective length of a column of length $$L$$ fixed against rotation and translation at one end is

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