The strength of a compression member (column) depends on its geometry (slenderness ratio L_eff / r) and its material properties (stiffness and strength).

The Euler formula describes the critical load for elastic buckling and is valid only for long columns. The ultimate compression strength of the column material is not geometry-related and is valid only for short columns.

In between, for a column with intermediate length, buckling occurs after the stress in the column exceeds the proportional limit of the column material and before the stress reaches the ultimate strength. This kind of situation is called inelastic buckling.

This section discusses some commonly used inelastic buckling theories that fill the gap between short and long columns.

Suppose that the critical stress s_t in an intermediate column exceeds the proportional limit of the material s_pl. The Young's modulus at that particular stress-strain point is no longer E. Instead, the Young's modulus decreases to the local tangent value, E_t.

Replacing the Young's modulus E in the Euler's formula with the tangent modulus E_t, the critical load becomes,

The corresponding critical stress is,

Note:	1.	The proportional limit spl, rather than the yield stress sy, is used in the formula. Although these two are often arbitrarily interchangeable, the yield stress is about equal to or slightly larger than the proportional limit for common engineering materials. However, when the forming process is taken into account, the residual stresses caused by processing can not be neglected and the proportional limit may drop up to 50% with respect to the yield stress in some wide-flange sections.
	2.	The tangent-modulus theory tends to underestimate the strength of the column, since it uses the tangent modulus once the stress on the concave side exceeds the proportional limit while the convex side is still below the elastic limit.
	3.	The tangent-modulus theory oversimplifies the inelastic buckling by using only one tangent modulus. In reality,the tangent modulus depends on the stress, which is a function of the bending moment that varies with the displacement w.

For a column with rectangular cross section, the reduced modulus is defined by,

where E is the value of Young's modulus below the proportional limit. Replacing E in Euler's formula with the reduced modulus E_r, the critical load becomes,

The corresponding critical stress is,

Note:	1.	The reduced-modulus theory tends to overestimate the strength of the column, since it is based on stiffness reversal on the convex side of the column.
	2.	The reduced-modulus theory oversimplifies the inelastic buckling by using only one tangent modulus. In reality, the tangent modulus depends on the stress which is a function of the bending moment that varies with the displacement w.

Both tangent-modulus theory and reduced-modulus theory were accepted theories of inelastic buckling until F. R. Shanley published his logically correct paper in 1946. The critical load of inelastic buckling is in fact a function of the transverse displacement w. According to Shanley's theory, the critical load is located between the critical load predicted by the tangent-modulus theory (the lower bound) and the reduced-modulus theory (the upper bound / asymptotic limit).

However, the difference between Shanley's theory and the tangent-modulus theory are not significant enough to justify a much more complicated formula in practical applications, especially when manufacturing defects in mass production and geometric inaccuracies in assembly are taken into account.

Finally, if one must make and error in the design, engineers would much rather miss on the safe side. This is the reason why many design formulas are based on the overly-conservative tangent-modulus theory.