10a. Buckling  (SSES Ch. 10.1-10.3)
What is BucklingEuler Buckling FormulaRadius of Gyration
Slenderness RatioEffective Length
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What is Buckling?

When a slender member is subjected to an axial compressive load, it may fail by a condition called buckling. Buckling is not a failure of the material itself (as is yielding and fracture), but is due to geometric instability of the system.

Buckling animation.

Euler Buckling Formula

Consider a column of length L, cross-sectional moment of inertia I, and Young's modulus E. Both ends are pinned so they can freely rotate and cannot resist a moment. The critical load Pcr required to buckle the pinned-pinned column is the Euler Buckling Load:

The Buckling Strength scr is the Euler Buckling Load divided by the column's cross-sectional area:

The buckling strength is a new condition we need to check for columns in compression. Note that buckling is not dependent on material strength.

Buckled shape of a
pinned-pinned column under compressive force

Radius of Gyration

If all of the cross-sectional area A were massed a distance r away from the bending axis, the idealized lumped-area cross-section would have the same moment of inertia I as the actual cross-section if:

I = Ar2

Distance r is the radius of gyration. There generally two bending axes to consider, and thus two radii of gyration:

Slenderness Ratio

Slenderness ratio is a measure of how long the column is compared to its cross-section's effective width (resistance to bending or buckling). The slenderness ratio is the column's length divided by the radius of gyration.

Including the slenderness ratio and the radius of gyration reduces the Buckling Strength to:

Effective Length

How a column is supported governs its buckling strength. The effective length Le accounts for differences in the end supports. The effective length is the length the column would be if it were to buckle as a pinned-pinned column. The buckling formula for any column is therefore:

Common end-conditions are given here:
Effective Lengths for Columns with Various End Conditions
End Condition Pinned-Pinned Fixed-Free Fixed-Fixed Fixed-Pinned

The effective length is equal to the distance between points in the column where moment = 0 (between "pins"). This occurs when the curvature of the column changes.

The fixed-free column is "mirrored" through the fixed end to visualize Le=2L.

Effective Length, Le L 2L 0.5L 0.7L
Relative Buckling Strength
    (~ 1/ Le2) for same L
1 0.25 4 2

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Updated: 05/22/09 DJD