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A sufficiently slender column will fail under compression below yield stress by bowing, releasing it's axial elastic strain. This is called buckling. Depending on the slenderness of the column, the buckling may be fully elastic, or may involve some plastic deformation powered by the released elastic strain energy. If the column is too stubby, it will not buckle at all before reaching the yield stress.

One such column, which is ductile, yields in compression. What is the expected ultimate failure mode? Is it a fully plastic version of buckling, or something else?

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When a column is stressed beyond yield, the strain increases rapidly with small changes in stress, at this stage, the stress-strain curve is no longer linear, and the tangent modulus decreases. The tangent modulus is the slope of the stress-strain curve; in the linear elastic range, it is called "elastic modulus, $E$"; beyond yield, it is called "tangent modulus of elasticity, $E_t$".

As the tangent modulus decreases, the buckling stress will fall below the yield stress, so, while the tension side of the column continues to yield, the compressive side column will buckle and cause the column to fail.

enter image description here

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  • $\begingroup$ This answer assumes the top of the column is free to move in any direction, but I think OP is referring the standard scenario for buckling, where the top of the column moves only vertically. $\endgroup$
    – nanoman
    Nov 27, 2021 at 20:34
  • $\begingroup$ @nanoman Same phenomenon for failures in bending and axial compression. Consider this: when column material been pressed axially beyond yield strength, the tangent modulus changes from "E", to "E_t", which would lead to non-ductile failure (buckling) - the applied stress (fy) is much higher than the permissible stress calculated (fcr) by using E-t, at least mathematically. $\endgroup$
    – r13
    Nov 27, 2021 at 21:29
  • $\begingroup$ The concept present here is the influence of the "change in stiffness" of material after reaching/beyond yield strength, which is typically ignored in normal engineering practice but has some significance in the more advanced topics, such as cyclic dynamic loadgs. $\endgroup$
    – r13
    Nov 27, 2021 at 21:53
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American Institute of Steel Construction has this graph, (AISC Spec E2), for columns that are not crooked and do not have residual manufacturing stress, and are doubly symmetrical. There are safety factors that take into consideration a variety of imperfections and apply according to the specific load combinations.

The vertical axis is $F_{cr}/F_y \ $ and the horizontal axis is $\lambda_c=\frac{KL}{r\pi}\sqrt{\frac{F_y}{E}} \ $

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The chart uses two equations noted there that meet tangentialy at coordinates, $\lambda=1.5. \ $ , $\frac{F_{cr}}{F_y}=0.39$

It assumes yielded short columns Crush under the load, and gradually as the column slenderness or its length increases the may buckle under plastic, elastic-plastic, and finally elastic buckling.

cloumn chart

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