Buckling of Slender Struts
A type of failure that is sometimes overlooked for a body subject to compressive loading, is that due to instability, called buckling. The longer and more slender the column is, the lower the safe compressive stress that it can stand. The slenderness of a column is measured by the slenderness ratio, L/k, where L is the length of the column and (lower case) k = (I/A)0.5 the radius of gyration of the cross sectional area about the centroidal axis. The minimum radius of gyration is the one to be considered. This corresponds to the minimum value of I, the second moment of area of the section. A is the cross section area.
A link to a 'buckling load calculator' based on the thoeory below, is given at the bottom of this page.
Euler analysis applies to slender columns, the formula, for the critical axial concentric load that causes the column to be on the point of collapse for frictionless pinned ends (no bending moment at the ends) is given below.
Click here for a derivation of the above equation.
For a particular column cross section and length, the load capacity Fc depends only upon the modulus of elasticity E. Since there is little variation in E among diferent grades of steel, there is no advantage in using an expensive, high strength alloy steel instead of structural steel for columns with L/k greatcr than about l20.
Eulers equation as written can be applied to a column with ends fixed in any manner if the length is taken as that between sections of zero bending moment. This length is called the effective length, Le and is equal to KL (upper case) K, where L is the actual length and K is a constant dependent upon the end fixings. Theoretical values for different types of column ends are shown below. It should be noted that design codes issued by some organisations often recommend values that are somewhat different to these theoretical values, see right hand column.
|End Fixings||Theoretical K value||Practical K value|
|pinned frictionless ends:||K=1||K=1|
|fixed - pinned and guided:||K=0.7||K=0.8|
|fixed - free:||K=2||K=2.1|
A typical factor of safety, or design factor, for Euler structural columns is between 2 and 3.5, but this is based on the critical load, not on the yield or ultimate strength of the material.
If the long column remains straight and the load concentric, the average stress in the column at the point of collapse is sc = Fc/A and it is local buckling at some point where the stress is below the yield stress of the material that leads to failure.
Short and Intermediate Columns
If Le/k is below a certain value for a particular material, the Euler formula gives a critical load which causes a stress greater than the yield stress of the material. Collapse in these cases is probably due to a combination of buckling and plastic action. For very short columns the yield stress (with appropriate design factor) can be used. For columns that are not short, but where the Euler formula gives stress above the yield stress, empirical methods of design are used. One popular equation in use since the early 1900s is the Johnson formula which can be used for columns with slenderness ratios below a transition slenderness ratio or column constant, Cc. Click here for further explanation.
The value of Le/k that indicates the transition slenderness ratio is given by: and when Le/k is below Cc, the Johnson formula can be applied:
As a very rough guide, for steel, the Euler buckling formula is only applicable for columns with Le/k exceeding about 100, depending upon the yield stress.
Link to Buckling Load Calculator and Interactive Graph (Netscape 4.5 or IE 4 or later).
Link to Buckling Load Calculator
Return to Module Introduction
Further reading: 'Design of Machine Elements', by V M Faires, Collier Macmillan, 1965, chapter 7.
It should be noted that sheets and plates may suffer buckling. Where a fabricated 'I' section beam has an insufficiently thick web, this can also suffer from buckling. Standard sections are sized so this is unlikely to be a problem.
David J Grieve, Amended, 1st March 2004, original dated 27th August 1999.