Wikipedia
A column can buckle due to its own weight with no other direct forces acting on it, in a failure mode called self-buckling. In conventional column buckling problems, the self-weight is often neglected since it is assumed to be small when compared to the applied axial loads. However, when this assumption is not valid, it is important to take the self-buckling into account.
Elastic buckling of a "heavy" column i.e., column buckling under its own weight, was first investigated by Greenhill at 1881. He found that a free-standing, vertical column, with density ρ, Young's modulus E, and cross-sectional area A, will buckle under its own weight if its height exceeds a certain critical value:
$l_{max} = \left(7.8373\,\frac{EI }{\rho gA}\right)^{1/3}$
where g is the acceleration due to gravity, I is the second moment of area of the beam cross section.
One interesting example for the use of the equation was suggested by Greenhill in his paper. He estimated the maximal height of a pine tree, and found it cannot grow over 90 ft tall. This length sets the maximum height for trees on earth if we assume the trees to be prismatic and the branches are neglected.