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.
