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.

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 P_cr required to buckle the pinned-pinned column is the Euler Buckling Load:

The Buckling Strength s_cr 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.

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 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: