» 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 so much a failure of the material (as is yielding and fracture), but an instability caused by system geometry.

» Euler Buckling Formula
The critical load, P_cr, required to buckle the pinned-pinned column is given by the EULER BUCKLING FORMULA. Consider a column of length, L, cross-sectional Moment of Inertia, I, having Young's Modulus, E. Both ends are pinned, meaning they can freely rotate and can not resist a moment. The critical compression force at which the column buckles is given by:

Pcr = p2EI L2

The Buckling Strength, s_cr, is the Euler Buckling Load divided by the columns cross-sectional area:

scr = p2EI AL2

» Radius of Gyration
If all of the cross-sectional area of a column were massed a distance r away from the neutral axis, the lumped-area cross-section would have the same Moment of Inertia as the real cross-section. r is termed the RADIUS OF GYRATION and is given by:

rx = Ixx A 1/2   and   ry = Iyy A 1/2

» 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, s, is simply the column's Length divided by the Radius of Gyration. Applying the Slenderness Ratio and the Radius of Gyration reduces the Euler Buckling Formula to:

Pcr = p2EI L2 = p2E r L 2 = p2E s2

» Effective Length
How a column is supported governs its buckling strength. The EFFECTIVE LENGTH, L_e, accounts for differences in the end supports. The Effective Length is the length the column would have to be if it were to buckle as a pinned-pinned column. The Buckling Formula for any column is thus:

Pcr = p2EI Le2

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