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” (P_cr) or the Euler Buckling Load (P_E) 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 P_E (or P_cr) is the Euler Buckling Load (in lb or kips).

VersusDiagram

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.

Euler Buckling Stress

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

Substitute   (r is defined as radius of gyration) in the above equation:

Dividing the numerator and denominator by r²:

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

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

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