This paper treats the optimal design of a vertical column that is built-in at the lower end. In addition to its own weight, the column is to carry an axial compressive load at its unsupported upper end. The column is to be designed as a thin-walled tube. The median line is to be the same for all cross sections; the wall thickness, though constant along the median line of any cross section, is allowed to vary along the length of the tube. Accordingly, the weight per unit length of the tube is proportional to the bending stiffness. For given length and total weight, the variation of the wall thickness along the column is to be determined to maximize the critical value of the compressive load at the upper end. The influence of a maximum allowable compressive stress on the design is also investigated.