Dimensional Analysis of Rigid-Plastic and Elastoplastic Structures under Pulse-Type Excitations

In this paper, the inelastic response of rigid-plastic and elastic-plastic systems subjected to pulse-type excitations is revisited with dimensional analysis. Starting from Newmark’s result on the maximum displacement of a sliding mass resting on a base that is subjected to a rectangular acceleration pulse, the paper introduces an energetic length scale of the excitation and the relevant dimensionless Π-products that govern the response of yielding structures. The introduction of Buckingham’s Π-theorem reduces the number of variables that govern the response of the elastic-plastic system from five (5) to three (3). The proposed dimensionless Π-products are liberated from the associated elastic system response and are consistent with the incremental evolution from the rigid-plastic to the elastic-plastic system. When the response is presented in terms of the dimensionless Π-terms remarkable order emerges. It is shown that for a given value of dimensionless yield displacement the response curves (maximum relative dimensionless displacements) become self-similar and follow a single master curve. The self-similar solutions show clearly how the inelastic response amplifies as the normalized yield displacement increases and that an increase in strength may lead to an increase in inelastic displacements. The main advantage of the analysis presented in this paper is that it brings forward the concept of self-similarity—an invariance with respect to changes in scale or size—which is a decisive symmetry that shapes nonlinear behavior.