__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 (in^{4}) 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 r^{2}:
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.
The variation of *f*_{E} versus is :
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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