Unsymmetrical Bending and Bending Combined with Axial Loading of a Beam of Rectangular Cross Section into the Plastic Range

Summary

This note considers two cases of bending beyond the limit of proportionality which have not received a great measure of attention in the past. These are Case (i). A beam subjected to a pure bending moment acting in a plane other than one of symmetry. Case (ii). A beam subjected to a bending moment and an axial load.

A mathematical form is used for the stress–strain curve in order that the results shall be applicable to a large number of materials and to avoid the tedious arithmetical summations which would be necessary if actual stress-strain curves were used. All the results are presented in terms of form factors since this notation is convenient for design purposes and is consistent with current British practice. Although the method adopted in the analysis is applicable to a number of different types of beam cross section, only rectangular cross sections are considered in detail at the present time.

Curves are presented showing the variation of the form factor with respect to the angle of the plane of loading, for Case (i), and with respect to the ratio of bending moment to axial load for Case (ii). These curves are based upon a standardised form of the stress–strain curve which may be taken as representative of a range of aircraft materials. Consideration is given to the automatic satisfaction of present military proof loading requirements when these form factors are used for the estimation of maximum permissible bending moments and axial loads under ultimate loading conditions.

The recommendations of F. P. Cozzone, for dealing with these two problems are examined and comparison is made between the theoretical work of this note and a limited number of test results available for Case (i).