Intermediate Columns  
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 geometryrelated 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.  
TangentModulus Theory  
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 stressstrain 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,

ReducedModulus Theory  
The Reduced Modulus theory defines a reduced Young's modulus E_{r} to compensate for the underestimation given by the tangentmodulus theory.
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,

Intermediate Column Design 
Both tangentmodulus theory and reducedmodulus 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 tangentmodulus theory (the lower bound) and the reducedmodulus theory (the upper bound / asymptotic limit). However, the difference between Shanley's theory and the tangentmodulus 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 overlyconservative tangentmodulus theory. 