Buckling

In engineering buckling is a failure mode of a structure characterised by a failure to react the bending moment generated by a compressive load.

Buckling in columns

The ratio of the length of a column to the least radius of gyration of its cross section is called the slenderness ratio. This ratio affords a means of classifying columns. All the following are approximate values used for convenience. A short steel column is one whose slenderness ratio does not exceed 50; an intermediate length steel column has a slenderness ratio ranging from 50 to 200, while long steel columns may be assumed as one having a slenderness ratio greater than 200. A short concrete column is one having a ratio of unsupported length to least dimension of the cross section not greater than 10. If the ratio is greater than 10 it is a long column. Timber columns may be classed as short columns if the ratio of the length to least dimension of the cross section is equal to or less than 10. The dividing line between intermediate and long timber columns cannot be readily evaluated. One way of defining the lower limit of long timber columns would be to set it as the smallest value of the ratio of length to least cross sectional area that would just exceed a certain constant K of the material. Since K depends on the modulus of elasticity and the allowable compressive stress parallel to the grain it can be seen that this arbitrary limit would vary with the species of the timber. The value of K is given in most structural handbooks.

If the load on a column is applied through the center of gravity of its cross section it is called an axial load. A load at any other point in the cross section is known as an eccentric load. A short column under the action of an axial load will fail by direct compression but a long column loaded in the same manner will fail by buckling (bending), the buckling effect being so large that the effect of the direct load may be neglected. The intermediate length column will fail by a combination of direct stress and bending.

In the middle of the 18th century a mathematician named Euler derived a formula which gives the maximum axial load that a long, slender ideal column can carry without buckling. An ideal column is one which is perfectly straight, homogenous, and free from initial stress. The maximum load, sometimes called the critical load, causes the column to be in a state of unstable equilibrium, that is, any increase in the loads or the introduction of the slightest lateral force will cause the column to fail by buckling. The Euler formula for columns is:

P = (Kπ²EI)/l²

Where

P = maximum or critical load

E = modulus of elasticity

I = moment of inertia of cross sectional area

l = unsupported length of column

K = a constant whose value depends upon the conditions of end support of the column, For both ends free to turn K = 1; for both ends fixed K = 4; for one end free to turn and the other end fixed K = 2 approximately, and for one end fixed and the other end free to move laterally K = 1/4.

Examination of this formula reveals the following interesting facts with regard to the bearing power of columns. First, that elasticity and not compressive strength of the materials of the column determines the critical load. Secondly, the critical load is directly proportional to the moment of inertia of the cross-section. The strength of a column may therefore be increased by distributing the material so as to increase the moment of inertia. This can be done without increasing the weight of the column by distributing the material as far from the principal axes of the transverse section as possible consistent with keeping the material thick enough to prevent local buckling. This bears out the well-known fact that a tubular section is much superior to a solid section for column service. Another bit of information that may be gleaned from this equation is the effect of length upon critical load. For a given size column, doubling the unsupported length quarters the allowable load. The restraint offered by the end connections of a column also affects the critical load. If the connections are perfectly rigid, the critial load will be four times that for a similar column where there is no resistance to rotation (hinged at the ends).

Since the moment of inertia of a surface is its area multiplied by the square of a length called the radius of gyration, the above formula may be rearranged as follows. Using the Euler formula for hinged ends and substituting Ar² for I the following formula results:

σ = P/A = (π²E)/(l/r)²

where P/A is the allowable unit stress of the column and l/r is the slenderness ratio.

Since the structural column is generally an intermediate length column and it is impossible to obtain an ideal column, the Euler formula has little practical application for ordinary design. Consequently, a number of empirical column formulae have been developed to agree with test data, all of which embody the slenderness ratio. For design, appropriate factors of safety are introduced into these formulae.