The study of the tallest standing column is basic in the design of tall structures, especially antennae. Greenhill [1] first found the maximum height of the uniform column that does not buckle under its own weight. The height can be extended if the column tapers towards the top. The optimum shape of the tallest nonuniform column is more difficult to determine. Keller and Niordson [2], and Cox and McCarthy [3] discussed its asymptotic properties, and Hu and Kirmser [4] integrated the shape numerically. It was found that the optimum column shrinks very fast at the base, then more slowly, and becomes very sharp at the apex. However, such an optimal column is very difficult to manufacture due to the vastly nonuniform radius variations. A more practical solution is to weld two uniform rods of different radii together. The purpose of the present paper is to find the optimum dimensions (lengths and diameters) of a compound column such that for a given total weight, the maximum height can be attained.
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