What is the longest column possible?

How long can a column be if it is loaded just by self-weight?     This is a rather different problem to those solved by the well-known Euler buckling formula and its variants.  The difference lies in the fact that under self-weight, loading is uniformly distributed along the length of the column, rather than being a point load on the column end, as the Euler formula assumes. This makes calculating the maximum length more complex.

History

Greenhill solved the problem in 1881 in this paper, in which he obtained solutions by integrating the moment-curvature equation for columns of constant and varying cross-sections.  Timoshenko refered to Greenhill’s solutions in his classic Theory of Elastic Stability and also developed an energy-based approximate solution. More recently Wang discussed the problem in several papers, for example here.

The problem has always appeared to be a curiosity rather than having a practical use.  Greenhill suggested his solution could be used to determine the maximum possible height of trees but this ignores other factors like wind loading that will influence trees and is rather fanciful.  Risers in offshore engineering may have somewhat similar loading  but the link still seems tangential.

Are there any practical applications for this problem?  What are the effects of factors such as plasticity, eccentric loads and so on, matters that all affect the capacity of end-loaded columns substantially?  There is work to be done here, either by the intellectually curious or by someone who finds a practical application for the problem.

Mathematical Summary

Greenhill’s starting point was, as for the Euler formula, the moment-curvature equation.

\displaystyle M=EI\frac{d^2y}{dx^2}

But, whereas for the Euler probem {M} in the above equation is simply a force times a distance which leads to a linear differential equation with an analytical solution, with distrubted self-weight loading it is more complex.  Full details are in Greenhill’s paper but the key equations to compare are

\displaystyle Py=EI\frac{d^2y}{dx^2}

the normal Euler equation that leads to the well-known  solution P_{cr}=\frac{\pi^2 EI}{L_{eff}^2}, and

\displaystyle wA\int^{x}_{0}(y'-y)dx=EI\frac{d^2y}{dx^2}

the equation for a column under self-weight.  The LHS is considerably more complex than for the Euler case and leads to a differential equation requiring Bessel functions to obtain an approximate solution for maximum length of column under self-weight of

\displaystyle L_{cr}=1.986\left(\frac{EI}{wA}\right)^{1/3}

where w is the density and A the area of the column.

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