## Column

A column or strut is a bar or structural member under axial com-

pression, which has an unbraced length greater than about eight or ten times the least dimension of its cross secdon. Because of its length, it is impossible to hold a column in a straight line under a load. A slight sidewise bending always occurs, causing flexural stresses in addition to the compressive stresses induced directly by die load. The lateral deflection will be in a direction perpendicular to that axis of the cross section about which the moment of inertia is the least. Thus in a complex shape such an H-column will bend in a direction perpendicular to its major axis. In a square shape it will bend perpendicular to its two major axes. With a tubular shape it is likely to bend in any direction.

The radius of gyradon of a column secdon with respect to a given axis is equal to the square root of the quodent of the moment of inertia with respect to that axis, divided by the area of the section, that is:

where I is the moment of inertia and A is the sectional area. Unless otherwise mentioned, an axis through the center of gravity of the section is the axis considered. 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. The length of a column is the distance between points unsupported against lateral deflection. The slenderness ratio is the length / divided by the least radius of gyration k.

Various conditions may exist at the ends of columns that usually are divided into four classes. (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. (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. (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. (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.

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. 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. Usually columns have

one end fixed and one end hinged, or some other combination. Their relative values may be taken as intermediate between those represented by the condition at either end. 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.

There is no exact theoretical formula that gives the strength of a column of any length under an axial load. Different formulas involving the use of empirical coefficients have been deduced, however, and they give results that are consistent with the results of tests. These formulas include the popular Euler's formula, different eccentric formulas, cross-bend formulas, wood and timber column formulas, and general principle formulas.

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