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 fail ing by inelastic hinges. His analysis is now extended to the size effect. F irst, the dependence of the bending strength on the postpeak softening slop e of an inelastic hinge on the beam depth is analyzed based on the energy p rinciples of fracture mechanics. Since exact analytical solutions for struc tures with many hinges softening simultaneously are very complicated, the p resent study focuses on the asymptotic case of a sufficiently large structu res, for which no two inelastic hinges are softening at the same time. Simp le size effect trends are identified for this asymptotic case. For the oppo site asymptotic case of a very small structures, classical plasticity appli es. For the intermediate situations, approximate formulae of asymptotic mat ching type are proposed. The size effect obtained is very different from tw o- or three-dimensional structures failing due to propagation of one domina nt crack of damage band.