A simpli®ed analysis for the evaluation of stochastic response of elasto-plastic oscillators S. Caddemi * , P. Colajanni Dipartimento di Ingegneria Strutturale e Geotecnica, Universita Á degli Studi di Palermo, Viale delle Scienze, 1-90128 Palermo, Italy Received 1 February 1996; received in revised form 1 December 1997; accepted 1 February 1998 Abstract The paper deals with dynamic hysteretic oscillators without post-yielding hardening, called ideal elasto-plastic oscillators, subjected to white noise. They are characterized by the fact that they do not reach stationarity even though excited by stationary stochastic processes. A simpli®ed solution procedure to capture this behaviour is presented in this paper. It is based on modelling the accumulated plastic deformations as a homogeneous compound Poisson process. In particular, two aspects are addressed in the paper: (1) evaluation of the probabilistic parameters of the accumulated plastic deformation process; and (2) evaluation of the second-order cumulants of the response by means of closed form expressions. Although the presented results are not rigorous and rely on an empirical basis, the aim is a very handy and suf®ciently accurate procedure to obtain the evaluation of the second-order probabilistic parameters of elasto-plastic oscillators. Moreover, by testing this procedure against Monte Carlo simulations, a parametric study has been conducted in order to assess the range of validity of the homogeneous compound Poisson process model. The presented procedure can be easily extended to the case of non-normal delta correlated input processes. q 1999 Elsevier Science Ltd. All rights reserved. 1. Introduction Particular interest of several researchers has been devoted to the study of dynamic elasto-plastic oscillators showing hysteretic behaviour in view of several engineering applica- tions. Such kind of oscillators are in fact able to represent the elasto-plastic constitutive behaviour of real materials, and also particular Coulomb friction dissipative devices as isolation systems and friction dampers. The hysteretic non-linearity is quite complex and, further- more, it has to be often combined with random analysis since in nature the feature of most of the loading forces cannot be described by means of deterministic parameters only. Several solution procedures have been proposed to analyse elasto-plastic oscillators subjected to random processes [1±6]. Most of them are based on Gaussian and non-Gaussian closure techniques but lead to accurate results only whenever residual stiffness is shown in the post-elastic state. If no residual stiffness is shown in the post-elastic state, this hysteretic oscillator (also called ideal elasto-plastic oscillator) forced beyond its elastic limit, is subjected to accumulation of plastic displacements without any incre- ment of the restoring force. In this case it has been proved [5] that the hysteretic response does not reach stationarity even though excited by a stationary white noise. For this reason response analysis of such oscillators requires parti- cular attention. An appealing approach suggested by Vanmarke et al. [7] is based on the characterization of the accumulated plastic displacement process by means of a homogeneous compound renewal process (marked renewal point process). However, they also noted that plastic events are rare and with short duration as the level u of the elasticity limit approaches in®nity and in this case the accumulated plastic displacement process can be modelled as a homogeneous compound Poisson process. Within this context, Ditlevsen [8] proposed a probabilistic characterization of the plastic displacement process on the basis of the hypothesis that the plastic events are so rare that between two of them the response has time to renormalize to a linear elastic response (renormalization assumption). The renormalization assumption adopted by Ditlevsen [8] seems to be quite restrictive. In fact, the Poisson process assumption for plastic displacements could be adopted even though the renormalization assumption fails. The renorma- lization assumption was subsequently removed by Ditlevsen and Bognar [9] who provided a complete and satisfactory Probabilistic Engineering Mechanics 14 (1999) 269±280 0266-8920/99/$ - see front matter q 1999 Elsevier Science Ltd. All rights reserved. PII: S0266-8920(98)00006-X * Corresponding author. Tel: 1 39-91-6568444; Fax: 1 39-91- 6568407; e-mail: caddemi@stru.diseg.unipa.it