This assumption is not true for short columns that fail mainly by direct compression, nor is it true for columns of medium length. Euler's formula assumes that the failure of a column is due solely to the stresses induced by sidewise bending. The Euler's formula developed during 1759 by Leonard Euler (Swiss mathematician, 1707 to 1783) is used in product designs and also in designs using columns in molds and dies that processes plastic. These formulas include the popular Euler's formula, different eccentric formulas, cross-bend formulas, wood and timber column formulas, and general principle formulas. Different formulas involving the use of empirical coefficients have been deduced, however, and they give results that are consistent with the results of tests. There is no exact theoretical formula that gives the strength of a column of any length under an axial load. The extent to which strength is increased by fixing the ends depends on the length of column fixed ends have a greater effect on long columns than on short ones. Their relative values may be taken as intermediate between those represented by the condition at either end. Usually columns have one end fixed and one end hinged, or some other combination. Columns with hinged ends are equivalent to those with round ends in the plane in which they have movement columns with flat ends have a value intermediate between those with fixed ends and those with round ends. Tests prove that columns with fixed ends are stronger than columns with flat, hinged, or round ends, and that columns with round ends are weaker than any of the other types. (4) Columns with fixed ends the ends are rigidly secured, so that under any load the tangent to the elastic curve at the ends will be parallel to the axis in its original position. (3) Columns with flat ends the bearing surface is normal to the axis of the column and of sufficient area to give at least partial fix to the ends of the columns against lateral deflection. (2) Columns with hinged ends they have perfect freedom of motion at the ends in one plane, as in compression members in bridge trusses where loads are transmitted through endpins. (1) Columns with round ends the bearing at either end has perfect freedom of motion, as there would be with a ball-and-socket joint at each end. Various conditions may exist at the ends of columns that usually are divided into four classes. The slenderness ratio is the length l divided by the least radius of gyration k. The length of a column is the distance between points unsupported against lateral deflection. As in beams, the moment of inertia is an important factor in the ability of the column to resist bending, but for purposes of computation it is more convenient to use the radius of gyration. Unless otherwise mentioned, an axis through the center of gravity of the section is the axis considered. Where I is the moment of inertia and A is the sectional area.

Slender definition