# Radius of gyration, cross-sectional shape and buckling

The radius of gyration is a convenient parameter, providing a measure of the resistance of a cross-section to lateral buckling.

The radius of gyration is not a physical entity in itself. It is a relationship derived to make prediction of column behaviour easy. It is simply the square root of the second moment of area of the section divided by the cross-sectional area of the section. Thus,

r = Ö (I/A)

where r = the radius of gyration

I = the moment of inertia or the second moment of area of the section

A = the cross sectional area of the section

The radius of gyration is related to the size and shape of the cross-section.

The shape and size of the cross-section of the compression member determines the least radius of gyration of the section. If all the material of the section is concentrated to produce a solid section of small overall dimensions the section will have a smaller radius of gyration than a hollow member with the material distributed further away from the centre of the cross section. This can be illustrated by comparing solid cross-sections (a) with cross-sectional shapes in which the overall dimensions are much larger (b), but the cross-sectional area is the same. In each case cross-section (b) has a larger radius of gyration than (a) and the buckling strength is therefore increased.

Columns will buckle in the direction of least cross-sectional stiffness.

It should also be noted that when a column of rectangular cross-section is loaded it will buckle in the direction of the smaller dimension in cross-section. A column of square cross-section will be equally prone to buckling in x and y directions. This is because the cross section will offer equal resistance to buckling in the direction x and y. (In practice the presence of imperfections will cause the column to buckle preferentially in one direction.) For a rectangular cross-section, the tendency to buckle will be in the direction of the smaller dimension, that is perpendicular to the line yy.

A special range of sections (UCs) is manufactured principally for use as columns; these sections have similar resistance to lateral buckling about both axes of the cross-section.

Hot-rolled steel sections, such as Universal Beams (UBs), channels and joists, are deeper in one direction than the other. These sections are ideal for use as beams, where the main strength of the section is required in the direction of bending. If these sections are used as columns, their strength will be determined by their ability to resist buckling in the weaker direction. This suggests that rolled steel sections which are efficient in bending are less good at resisting lateral buckling as columns. For this reason, a different set of sections are produced with more comparable resistance to lateral buckling in xx and yy directions. These are called Universal Columns (UC) and are normally used for columns carrying primarily axial load. However, in practice, columns may have to resist lateral bending as well as axial load and, if this is significant, a Universal Beam section may prove to be more economical.

Other cross-section shapes are often used for struts, and hollow sections are very efficient in compression.

For small scale columns, joists or channel sections may be used, whilst lighter sections such as angles are more common for use as struts in trusses. The use of hollow sections is also becoming increasingly popular. Although connections involving such sections are more difficult, they are efficient in compression and are often preferred for aesthetic reasons.

Apart from standard rolled sections, it is possible to fabricate sections to suit any specific requirement although there is a cost penalty in doing so.

Although generally not justifiable on the grounds of cost, tapered columns can be efficient in resisting buckling.

In a compression member the resistance to buckling can be increased by shaping the member in such a way as to strengthen the point which is most affected by buckling. For example, in a column with both ends pinned, the point most susceptible to buckling deformation would be at the mid height. This reduces gradually towards the end of the column. Ideally the column could be shaped by making the mid point of the largest cross-section and gradually reducing the cross-sectional size towards the ends. Similarly, for a column fixed at both ends, the susceptible part is the central 70% of its column length. It is possible to strengthen this part to make a more stable column overall by making a tapered section or by using tension wires. However, the cost of such extra fabrication is likely to be very high and only justified on the basis of architectural expression.