Improved procedure for equivalent linearization of bridges supported on hysteretic isolators M. Jara a,⇑ , J.M. Jara a , B.A. Olmos a , J.R. Casas b a Civil Engineering School, Universidad Michoacana de San Nicolás de Hidalgo, Morelia, Mexico b Construction Engineering Department, Technical University of Catalonia, Barcelona, Spain article info Article history: Received 5 May 2011 Revised 8 September 2011 Accepted 26 October 2011 Keywords: Equivalent linearization Isolated bridges Lead rubber bearings Displacement prediction abstract Maximum displacement demands can be obtained through non-linear time history analysis, however, many approximate methods have been proposed in recent codes to reduce the required computational time distinctive of non-linear approaches. Some of these methods are based on equivalent linearization of the system by using an effective lateral stiffness (k ef ) and equivalent damping ratio (n eq ). The dynamic characteristics of earthquake ground motions, ductility capacities, type of hysteretic relationships, and stiffness and strength degradation characteristics of the structure are aspects that strongly affect both, the energy dissipation capacity and the effective stiffness of the system; nevertheless, these conditions have not been adequately accounted for in the analyses. This work presents an improved expression, intended for bridges supported on hysteretic isolators, that takes into account some of these aspects. Since lead rubber bearings (LRB) are the most common isolators used on bridges, the expression is focused on the hysteretic behavior of this type of bearing. A simple and rational expression to evaluate the equivalent damping ratio n eq , tying the physical behavior of these systems, is proposed. In this way, the prediction capability of the linear equivalent model of bridges supported on LRB isolators is improved. The proposed equation predicts a displacement that is in good agreement with the one obtained through inelastic time history analysis. Furthermore, it can be easily incorporated into the displacement-based design framework, and into the code specifications. Ó 2011 Elsevier Ltd. All rights reserved. 1. Introduction The response spectrum provides some of the most important characteristics of earthquake motions and gives the maximum elas- tic deformation for structures over the entire range of periods. However, it is not able to predict damage level, in as much as dam- age involves inelastic deformations. Of course, maximum displace- ment demands can be obtained through non-linear time history analysis, but in most practical cases linear response spectra or uni- form hazard elastic response spectra are used. Hence, many approx- imate methods have been proposed to overcome this difficulty. Some of them are based on an equivalent linearization of the sys- tem by using an effective lateral stiffness (k ef ) and equivalent damping ratio (n eq ). Equivalent linear models have been incorpo- rated in the design of structures with passive energy dissipation systems in [1–4]. The main difference among the existing equiva- lent linear methods is the way in which n eq and k ef are determined. The expressions that have been proposed for computing these parameters are based on analytical formulations, empirical rela- tions and/or expressions derived from experimental tests. Initially, general models were proposed to deal with a great variety of energy dissipation systems in spite of the important differences that have been identified for different type of hysteretic cycles. Although there is considerable scatter in the data, a good approximation can be obtained on the average with the existing linearization parameters [5]. The important scatter found in other studies [6,7], can be explained for the intent in achieving generalized expressions obtained from particular conditions that fit a wide variety of inelas- tic systems. The equivalent viscous damping ratio n eq can be obtained by equating the energy dissipated in one cycle of a steady state response to harmonic excitation at maximum displacement to the corresponding linear viscous damping value, as it was proposed originally by Jacobsen in 1930 [8], and it is now adopted in several codes like the AASHTO [1]. This model is compatible with the assumption of structural characterization by stiffness and damping at peak response. The typical assumption for deriving the equivalent damping ratio depends on the energy dissipated during one cycle at peak response. Nevertheless, most of the time the displacement amplitudes are significantly less than the maximum response dur- ing an earthquake ground motion, and the energy dissipation can be overestimated. Chopra and Goel [5] have also reported displace- ment underestimation, with errors approaching 50%. Therefore, this model is not capable to appropriately incorporate the influence of 0141-0296/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.engstruct.2011.10.028 ⇑ Corresponding author. Address: Santiago Tapia 403, Col. Centro, Postal Code 58000, Morelia, Michoacan, Mexico. Tel.: +52 443 317 22 48, 443 322 35 00x4336; fax: +52 443 304 10 02. Engineering Structures 35 (2012) 99–106 Contents lists available at SciVerse ScienceDirect Engineering Structures journal homepage: www.elsevier.com/locate/engstruct