Comput Mech (2010) 45:431–441 DOI 10.1007/s00466-009-0451-y ORIGINAL PAPER Identification of an extended Bouc–Wen model with application to seismic protection through hysteretic devices Tudor Sireteanu · M. Giuclea · A. M. Mitu Received: 29 May 2009 / Accepted: 15 November 2009 / Published online: 10 January 2010 © Springer-Verlag 2010 Abstract In this paper is proposed an extended Bouc–Wen model for improving its capability to approximate experi- mental symmetric hysteretic loops. On the basis of the generalized equation there are defined integral and differ- ential conditions that describe the essential geometric prop- erties of a hysteretic curve. Next, a new method based on Genetic Algorithms is developed to identify the Bouc–Wen model parameters from experimental hysteretic loops obtained from periodic loading tests. The performance of presented approach is illustrated for two types of seismic pro- tection devices with hysteretic characteristics: elastomeric base isolators and buckling restrained dissipative braces. The applicability of proposed method is highlighted by using the derived models to analyse by numerical simulation the effi- ciency of these devices for reducing seismic response of a three stories civil structure. Keywords Mechanics · Vibration · Hysteresis · Inverse problem · Modeling · Seismic protection 1 Introduction The Bouc–Wen model, widely used in structural and mechan- ical engineering, gives an analytical description of a smooth hysteretic behaviour. It was introduced by Bouc [1] and extended by Wen [2], who demonstrated its versatility by producing a variety of hysteretic characteristics. The hyster- etic behaviour of materials, structural elements or vibration isolators is treated in a unified manner by a single nonlin- ear differential equation with no need to distinguish different T. Sireteanu (B ) · M. Giuclea · A. M. Mitu Institute of Solid Mechanics, Romanian Academy, 15 Constantin Mille Street, 010141 Bucharest, Romania e-mail: siretimsar@yahoo.com; siret@imsar.bu.edu.ro phases of the applied loading pattern. In practice, the Bouc–Wen model is mostly used within the following inverse problem approach: given a set of experimental input–output data, how to adjust the Bouc–Wen model parameters so that the output of the model matches the experimental data. Once an identification method has been applied to tune the Bouc–Wen model parameters, the resulting model is consid- ered as a “good” approximation of the true hysteresis when the error between the experimental data and the output of the model is small enough from practical point of view. Usually, the experimental data are obtained by imposing cyclic rel- ative motions between the mounting ends on the testing rig of a sample material, structural element or vibration control device and by recording the evolution of the developed force versus the imposed displacement. Once the hysteresis model was identified for a specific input, it should be validated for different types of inputs that can be applied on the testing rig, such as to simulate as close as possible the expected real inputs. Then this model can be used to study the dynamic behaviour of different systems containing the tested struc- tural elements or devices under different excitations. Various methods where developed to identify the model parameters from the experimental data of periodic vibration tests. A frequency domain method was employed to model the hysteretic behaviour of wire-cable isolators in [3] and iterative procedures were proposed for the parametric iden- tification of a smoothed hysteretic model with slip, see [4]. Another application, presented in [5], proposes to use a mod- ified Bouc–Wen model to portray the dynamic behaviour of magnetorheological dampers. Also, in [6], is given a gener- alized Bouc–Wen model for highly asymmetric hysteresis, successfully fitted to the experimental data obtained for flex- ible connectors. Closed analytical relationships were derived for standard Bouc–Wen model (n = 1, 2) such as the pre- dicted and experimental hysteresis loops to have exactly the 123