Scaling of failure of beams, frames and plates with softening hinges

Between 1970 and 1986, Giulio Maier published a series of pioneering papers on equilibrium path bifurcations and instabilities in beam structures failing by inelastic hinges. His analysis is now extended to the size effect. First, the dependence of the bending strength on the postpeak softening slope of an inelastic hinge on the beam depth is analyzed based on the energy principles of fracture mechanics. Since exact analytical solutions for structures with many hinges softening simultaneously are very complicated, the present study focuses on the asymptotic case of sufficiently large structures, for which no two inelastic hinges are softening at the same time. Simple size effect trends are identified for this asymptotic case. For the opposite asymptotic case of very small structures, classical plasticity applies. For the intermediate situations, approximate formulae of asymptotic matching type are proposed. The size effect obtained is very different from two-or three-dimensional structures failing due to propagation of one dominant crack or damage band.