When a structural beam is subjected to compressive loads, it can experience a sudden lateral deformation and loss of stiffness which is termed as buckling failure. This failure is produced by large deformation, but this doesn’t necessarily mean that there is material yielding or fracture. But buckling is still considered a failure mode, as there is no load carrying capacity for a buckled beam. A real-life example where buckling can happen is the support beams in a bed frame which are subjected to compressive loads.

There are two types of buckling modes: Local and global. In global (general) buckling, the structure buckles along the longitudinal axis of the member. The overall lateral stability of the structure is compromised due to compressive stresses in global buckling. Whereas in local buckling, the axis of the member remains undistorted, but the cross-section of the beam is buckled significantly reducing the load carrying capacity of the affected region. In this case, the overall stability of the structure may not be compromised.

## Critical Buckling Load

The critical buckling load depends on the material properties and geometry of the structure. It is calculated using Euler’s formula shown below.

where E is the young’s modulus, I is the moment of inertia, L is the column length and K is the effective length factor.

The critical bending stress is calculated as,

where A is the area of cross section of the beam.

The formula can be simplified for circular cross section with radius ‘r’ as,

We can see that in the above Euler’s formula, effective length factor (K) plays a crucial role in determining the critical buckling behavior of the beam. Physically this determines the distance between the inflection points on the deformed buckling shape of the beam. Inflection point is where the curvature of the beam changes from concave left to concave right or vice versa.

The location of the inflection points is determined by the support conditions at the ends of beam. Based on these support conditions only a certain length of the beam undergoes buckling. So, the length factor is a correction needed to find the effective length of the beam from its full length. For a 3D structural beam, there will be 3 rotational and 3 translational degrees of freedom (DOF) at each of its end points. There are 4 possible support conditions from these DOF. The boundary conditions and their pictorial representations are shown in the figure below.

Now, let’s see how the column buckles for different support conditions and the location of their inflection points in the figure below.

The critical buckling load is inversely proportional to the effective length of the beam. The higher effective length of the beam will lower the critical load. Since buckling is caused by lateral deformation of the columns, allowing the free translational DOF, or free rotational DOF or both will lead to low critical buckling loads by increasing the effective length.

## Higher Buckling Modes

The buckling phenomenon can be seen as analogous to vibrational mode shapes in a structure. Similar to how we have higher natural modes of vibration, buckling also has higher modes of deformation.

The general solution for a buckling problem is given as,

In the above equation, n is an integer value ranging from 1 to infinity. Substituting n=1, gives Euler critical load formula discussed above. This will require the least bending of the column. Higher buckling modes are obtained for n= 2,3 and so on as shown in the figure below. The critical load will increase in factors of squares of the integers (4,9,16, …). The higher buckling modes require extensive deformations in the column.

## Final Thoughts

Hopefully this article has provided some background theoretical information about buckling failure in structural columns and beams. This failure occurs when the lateral deformation in a column caused by compressive force leads to a sudden loss of stiffness. The critical load that causes buckling depends on the material and geometry of the structural column. In the next article, we will explore how to model buckling using FEA using linear and nonlinear buckling analysis.

If you have any questions about buckling problems, or would like help in solving them, don’t hesitate to get in touch with our expert engineering simulation team today!