Asymptotic matching analysis of scaling of structural failure due to softening hinges. I: Theory

Propagation of a crack from the tensile face of a beam causes postpeak softening, i.e., the bending moment decreases at increasing rotation of the hinge. Examples are unreinforced concrete beams and plates, foundations plinths, retaining walls, tunnel linings, or arch dams. Softening in an inelastic hinge is also caused by compression crushing of concrete. This happens in reinforced concrete beams that are prestressed, overreinforced, retrofitted by laminates, or subjected to a large enough axial compressive force, which is typical of columns as well as frames or arches with a large enough horizontal thrust. Hinge softening may also be caused by plastic buckling of flanges in deep thin-wall steel beams. An inevitable consequence of inelastic hinge softening in statically indeterminate structures requiring more than one inelastic hinge to fail is a size effect. Although finite element solutions are possible, general analytical formulas for the size effect in such structures do not exist, because of the complexity of response. The idea of this two-part paper is to exploit the technique of asymptotic matching in order to derive approximate formulas for the entire size range. Exact analytical solutions of the nominal strength of structure are derived for the large-size asymptotic case, for which the hinges soften one by one rather than simultaneously, and for the small-size asymptotic case, for which the classical plastic limit analysis applies. Matching these asymptotic solutions by a smooth formula then yields simple, yet general, size effect laws for the peaks and troughs of the load-deflection diagram through the entire size range. The size effect found is very different from the classical size effect in quasibrittle structures failing due to a single dominant crack. The theory is developed in the present Part I, and analysis of its implications is relegated to Part II.