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National Advisory Committee for Aeronautics, Technical Notes - Inelastic Column Behavior

The significant findings of a theoretical study of column behavior
in the plastic stress range are presented. When the behavior of a
straight column is regarded as the limiting behavior of an imperfect
column as the initial imperfection (lack of straightness) approaches
zero, the departure from the straight configuration occurs at the
tangent-modulus load. Without such a concept of the behavior of a
straight column, one is led to the unrealistic conclusion that lateral
deflection of the column can begin at any load between the tangent-
modulus value and the Euler load, based on the original elastic modulus.

The behavior of a column with vanishing initial lack of straightness
at loads beyond the tangent-modulus load depends upon the stress-strain
curve for the material. A family of curves showing load against lateral
deflection is presented for idealized H-section columns of various
lengths and of various materials that have a systematic variation of
their stress-strain curves. These curves show that, for columns in which
the material stress-strain curves depart gradually from.the initial
elastic slope as is characteristic of stainless steels, the maximum column
loads may be significantly above the tangentemodulus load. If the depar-
ture from the elastic curve is more abrupt, such as for the high-strength
aluminum or magnesium alloys, the maximum.load is only slightly above the
tangent-modulus load.

Until recently, the double-modulus theory had generally been
accepted as the correct theory of column failure in the inelastic range
of stress. This theory, originally developed by Considere and Engesser
and later extended by Von Karman (see reference I for a discussion of
this development), predicts that the load at which bending starts and
the maximum load that a pin-ended column can support are the same and can
be obtained from the Euler equation.

By substitution of a reduced modulus for Young's modulus. (The symbols
used in this paper are presented in appendix A.) The reduced modulus
is obtained by assuming that, at the start of bending of an originally
straight column, the direction of straining of the elements on the con-
vex side of the column reverses. When such reversal of strain occurs

in the plastic range, increments of stress are related to increments of
strain by the original elastic modulus. The part of the cross section
over which the strains reverse is determined by the condition that there
shall be no change in load during the bending process.