A Note on Bending Beyond the Proportional Limit

Abstract

Most theories of bending in the plastic range are based either upon the assumption of linear strain variation across the depth of the beam or by fitting some equation to the stress-strain curve. In this paper the following problem is solved: Assuming the principle of minimum strain energy to be valid over the entire range of the stress-strain curve, find the exact stress distribution over the depth of a beam in pure bending (or combined bending and axial load, provided strain energy due to axial load and beam column effects are neglected) compatible with equilibrium and minimum strain energy. Mathematically, this problem leads directly to a rather simple isoperimetric problem with variable limits in the calculus of variations.* The solution leads to the construction of an auxiliary or generalized stress, function, intimately connected with the actual stress through certain simple characteristics of the stressstrain curve (moduli ratios) and reducing to the actual stress for all values below the proportional limit. I t is shown that the exact stress distribution, in the sense tha t it is compatible with equilibrium and minimum strain energy, is such that its associated generalized stress distribution is again linear across the depth of the cross section. Once the associated generalized stress (as a curve vs. stress) has been determined from a given stress-strain curve, it is a comparatively easy matter to calculate maximum resisting moments for a given section on the basis of any limiting maximum fiber stress. Examples of such calculations for stainless steel and comparisons are given in the test.