Ductility, Snapback, Size Effect, and Redistribution in Softening Beams or Frames

Abstract

A layered finite element model with strain-softening material properties, whose applicability to reinforced concrete was corroborated by comparisons with experimental data in the preceding paper, is used in a parametric study aimed at the effect of several factors: structure size, finite element size, downward sl9pe of strain-softening stress-strain relation, length of the plastic yield plateau before the onset of strain softening (if any), and end-restraint stiffness. T() quantify t~e response, several new response characteristics are introduced: the ductile stren~h ening factor, characterizing how strain softening reduces the maximum load compared to the plastic limit load; the redistribution ratio, characterizing the degree of bending moment redistribution in comparison to that in plastic limit analysis; the energy safety factor, describing the energy to deform the structure to the peak load; and the ductility factor, characterizing the deflection increase at maximum load relative to the deflection from elastic analysis. The condition of snapback instability, which determines the ductility factor, is derived analytically for an elastically restrained beam. Finally, it is shown that strain-softening segments in beams cannot be modeled as softening hinges except for sufficiently slender beams.

From discrete to nonlocal continuum damage mechanics: Analysis of a lattice system in bending using a continualized approach

“…The only difference with respect to the previous model is the identification of the rotation discontinuity, which is here expressed with respect to the slope in the continuous perspective. One recognizes in this last case a typical damage hinge law, which can be viewed in this case as a kind of rotation-based damage cohesive law (see Bazˇant, 2005;Bazˇant et al, 1987;Florez-Lopez, 1995). Equation ( 33) is still valid and then the cohesive law of Figure 5 remains valid, the only difference is the rotation expression Á 0 ð Þ.…”

Scaling of structural failure

“…This inconvenience was later eliminated by the formulation of Pijaudier-Cabot and Baiant (1987) (see also Baiant and Pijaudier-Cabot, 11988), in which the main idea was that only the damage, considered in the sense of continuum damage mechanics (and later also yield limit degradation, Baiant and Lin l988), should be nonlocal and the elastic response should be local. The subsequent nonlocal continuum models with an averaging type integral were various variants on this idea.…”

Efficient finite element formulation for the analysis of localized failure in beam structures

“…This approach contrasts with those formulations, including a plastic hinge, in which strain softening occurs in zones with a finite length, see e.g. Darvall [6], Darvall and Mendis [7], Bažant et al [8,9], Sanjayan and Darvall [10].…”