Plastic bending is an extension of the Euler-Bernoulli beam equation that allows plastic deformation. It makes use of the fact that peak bending stresses only occur at the outside fibres of a beam, and therefore the entire cross-section will not yield simultaneously. Rather, outside regions will yield, redistributing stress and delaying rupture beyond what would be predicted by elementary E-B theory. Plastic bending theory makes identical predictions as E-B theory if all stresses are below the yield strength of the material.

Elementary E-B theory requires that bending stress varies linearly with distance from the neutral axis, but this extension allows a more complicated stress profile. The yielded portions at the outside of the beam are assumed to continue to carry the yield stress, even while deforming freely. In the unyielded core of the beam, the bending stress varies linearly with distance from zero at the neutral axis to the yield stress at the start of the yielded region. Rupture is assumed not to occur until the entire cross-section is yielded. This is under-conservative, severely so for efficient cross-sections such as I-beams.

As in the basic E-B theory, the moment at any section is equal to an area integral of bending stress across the cross-section. From this and the above additional assumptions, predictions of deflections and rupture strength are made.

Plastic bending has been tested for rectangular cross-sections, and found to be slightly underconservative. It should be used with caution, however, since it is known to be heavily underconservative for skinny sections at risk of web buckling and efficient sections such as I-beams.

Continuum mechanics | Building engineering | Structural engineering