<<Back to the Previous Page <<Fundamentals <<Knowledgebase


Buckling of Compression Members

Euler Buckling Load

When a structural member is subjected to compressive stresses at certain levels, it deflects outward (similar to bending). This is called “buckling”. The load at which a compression member buckles is called the “critical load” (Pcr) or the Euler Buckling Load (PE) after Leonhard Euler, the Swiss mathematician, who computed it about three hundred years ago:



In this equation, π = 3.14, E is the modulus of elasticity (psi or ksi), Ι is the moment of inertia (in4) about which the column buckles, kl is the effective length of the column against buckling (ft or in.), and PE (or Pcr) is the Euler Buckling Load (in lb or kips).

k value diagram


The column effective length depends on its length, l, and the effective length factor, k.  k depends on the type of columns’ end conditions. If the member is pin-ended (it can freely rotate), k=1.0. This means that the entire length of the member is effective in buckling as it bends in one-direction. If one or both ends of a column are fixed, the effective length factor is less than 1.0 as shown below. This means that the member buckles at a larger load or it is more difficult to make it buckle.

buckling diagram

Euler Buckling Stress

To compute the Euler buckling stress, fE, we divide the Euler buckling load, PE, by the member’s cross-sectional area, A:



Substitute  I/A=r square (r is defined as radius of gyration) in the above equation:


Dividing the numerator and denominator by r2:


In this equation, kl/r is defined as slenderness ratio. It is clear that as a column becomes more slender (larger kl/r), fE becomes smaller, i.e. buckles at a smaller stress level.

The variation of fE versus kl/r is :

fe versus kl/r


It is clear that for very small kl/r ( short columns), buckling does not occur but the member crushes under the load. Therefore, the above diagram changes as:

fe versus kl/r







<<Back to the Previous Page <<Fundamentals <<Knowledgebase



© 2011 Virginia Polytechnic Institute and State University