Euler's Theory:

One of the previous mathematical explanations of the phenomenon of buckling was given by Euler (1759) who proposed the following formula.

f_E =P_E/A = k(π² EI/Al²) = k (π² E/(1/r)²) =k π² E/ λ ²

where f_E = Euler's critical stress,

P_E = Euler's critical load,

A = Area of cross section,

I = Moment of inertia of the section,

r == Radius of gyration,

λ= l_e/ r = Slenderness ratio of the element, and

k = A coefficient which depends upon the end condition of the column.

Also various empirical formulae, example for Rankine's, Johnson's and Perry's are available for estimating the buckling load of columns.