Euler collapse load

Column buckling was first investigated by Euler who established that the buckling strength is inversely proportional to the square of the slenderness ratio.

In 1757 Leonard Euler propounded a theory for calculating the strength of an axially loaded column pinned at both ends. The relationship he established was as follows:

P_e = pie²EI/L² (1)

where, P_e = Euler collapse load.

E = Modulus of elasticity of the material of construction of the column.

I = Moment of inertia of the column section.

L = Length of the pin-ended column.

The corresponding buckling stress is obtained simply by dividing the collapse load by the cross-sectional area of the column, A. Thus:

p = P_e/A = pie²EI/AL²

Since the moment of inertia (I) and the cross-sectional area (A) are both dependent solely on the geometry of the cross-section they can be combined into a single variable. This is defined as the radius of gyration r of the column section and is related to I and A as follows:

I = A.r² (2)

Substituting equation (2) in (1)

p_e = pie²E./(L/r)² (3)

This relationship can be represented graphically as a graph of buckling stress against (L/r). Clearly the value of (L/r) is of considerable importance in determining the ability of a column to carry load without buckling.

The relationship between buckling strength and slenderness ratio depends on the support conditions at the column ends.

The above expressions describing the theoretical buckling behaviour of columns are valid for pinned ends only. Similar expressions can be obtained for other end conditions. For instance the buckling stress of a fixed ended column is given by:

P_e = 4pie²E / (L/r)²

which can be written as

P_e = pie²E /(0.5 L/r)²

The influence of different end conditions can most conveniently be accounted for by using the concept of an effective length.

It can be seen that the buckling strength of a fixed ended column is the same as for a pin ended column of half the length. This introduces the concept of effective length which can be defined as the length of an equivalent pin ended column with the same buckling strength.

Thus to extend the relationship (3) to include columns with end conditions other than pinned ends the value for length L should be substituted by effective length L_E.

Thus:

P_e = pie²E.A/ (L_E/r)² (4)

The ratio L_E/r is called the slenderness ratio of the section and the larger the value of this slenderness ratio, the smaller is the value of the collapse load P_e of the column.

It is also worth noting that, because the slenderness ratio is squared, a relatively small increase in its value can cause a large reduction in P_e. Thus, in theory, the buckling load of a column with a slenderness ratio of 160, is just one quarter that of a column with a slenderness ratio of 80.

Because the slenderness ratio is dependent on effective length and radius of gyration, these two items both influence the Euler critical load. A column of 5m height with both ends pinned, will carry four times as much load as a similar sized column with fixed base and free top.

For stocky columns the buckling stress can exceed the material strength and the dominant failure mode is therefore 'squashing'.

In spite of the elegance of Euler's formula in explaining the parameters contributing to the strength of a compression member, it is only effective in predicting the collapse load of columns with large values of slenderness ratio. For stocky columns, Euler's collapse load P_e exceeds the yield stress of steel, and the column fails by crushing or 'squashing'.

For example, for a value of L_E/r=20, and taking the modulus of elasticity for steel E as 205000N/mm², the compressive stress obtained from Euler's formula would be as follows:

p_e = P_e/A = pie²E/(LE/r)²

pe = pie² x 20500 / 400 = 5120 N/mm²

This value of Euler collapse stress is much larger than the yield stress of steel (210 - 450N/mm²) specified in BS 5950. The failure of stocky columns is therefore determined by material yield stress, whilst for slender columns failure corresponds to the Euler stress. For columns with intermediate slenderness ratios both buckling and yielding contribute to collapse. Clearly both these conditions must be accounted for in any design rule for columns.