Degrading Bouc–Wen Model Parameters Identification Under Cyclic Load

Degrading Bouc–Wen Model Parameters Identification Under Cyclic Load

G. C. Marano (College of Civil Engineering Fuzhou University, Fuzhou, China), M. Pelliciari (‘Enzo Ferrari' Engineering Department, University of Modena and Reggio Emilia, Modena, Italy), T. Cuoghi (‘Enzo Ferrari' Engineering Department, University of Modena and Reggio Emilia, Modena, Italy), B. Briseghella (College of Civil Engineering Fuzhou University, Fuzhou, China), D. Lavorato (Department of Architecture, Roma Tre University, Rome, Italy) and A. M. Tarantino (‘Enzo Ferrari' Engineering Department, University of Modena and Reggio Emilia, Modena, Italy)
Copyright: © 2017 |Pages: 22
DOI: 10.4018/IJGEE.2017070104


The purpose of this article is to describe the Bouc–Wen model of hysteresis for structural engineering which is used to describe a wide range of nonlinear hysteretic systems, as a consequence of its capability to produce a variety of hysteretic patterns. This article focuses on the application of the Bouc–Wen model to predict the hysteretic behaviour of reinforced concrete bridge piers. The purpose is to identify the optimal values of the parameters so that the output of the model matches as well as possible the experimental data. Two repaired, retrofitted and reinforced concrete bridge pier specimens (in a 1:6 scale of a real bridge pier) are tested in a laboratory and used for experiments in this article. An identification of Bouc–Wen model's parameters is performed using the force–displacement experimental data obtained after cyclic loading tests on these two specimens. The original model involves many parameters and complex pinching and degrading functions. This makes the identification solution unmanageable and with numerical problems. Furthermore, from a computational point of view, the identification takes too much time. The novelty of this work is the proposal of a simplification of the model allowed by simpler pinching and degrading functions and the reduction of the number of parameters. The latter innovation is effective in reducing computational efforts and is performed after a deep study of the mechanical effects of each parameter on the pier response. This simplified model is implemented in a MATLAB code and the numerical results are well fit to the experimental results and are reliable in terms of manageability, stability, and computational time.
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1. Introduction

Reinforced concrete or steel structural systems show nonlinear behaviour of parts of the structures (plastic hinge) or devices (isolator, dissipative braces, etc.) applied on some structures to mitigate their seismic response when severe excitations occur during strong earthquakes. In that condition the restoring force becomes highly nonlinear, showing significant hysteresis. The modelling of this hysteretic behaviour is much important to analyse the structural seismic response and design properly structural details and mitigation devices.

This nonlinear behaviour is encountered in a wide variety of processes such as seismic engineering, biology, optics, electronics, ferroelectricity, magnetism, mechanics, among other areas (Ismail, Ikhouane, & Rodellar, 2009) in which the input–output dynamic relations between variables involve memory effects. This hereditary nature of nonlinear restoring force cannot be described as a function of the instantaneous displacement and velocity. Thus, many hysteretic restoring force models were developed to include the time dependent nature using a set of differential equations. In seismic engineering, the structural hysteresis depends on the natural mechanism of materials which produce restoring forces in function of the instantaneous deformation and the history of the deformation (memory material nature) when are subjected to great inelastic deformations.

The detailed modelling of these systems using the laws of Physics is an arduous task, and the obtained models are often too complex to be used for practical applications involving characterization of systems, identification, or control. For this reason, alternative models of these complex systems have been proposed. They do not come, in general, from the detailed analysis of the physical behaviour of the systems with hysteresis. Instead, they combine some physical understanding of the hysteretic system along with some kind of black–box modelling. For this reason, they are called ‘semi–physical’ models.

When complex loading patterns happened, such as those produced by earthquakes, the availability of smooth continuous mathematical models, able to describe realistically the time evolution of hysteresis properties, is crucial. In this way, it is possible to perform analyses about the structural capability of systems subjected to seismic actions with modest computational efforts.

Limiting our attention in this field, in recent papers, Ray et al. developed smooth hysteretic models with nonlinear kinematic hardening for sliding seismic isolators (Ray, Sarlis, Reinhorn, & Constantinou, 2013), enhanced a one–dimensional smooth model in order to portray the degradation of elastic and inelastic stiffness (Ray & Reinhorn, 2012), and showed that by introducing damping in form of supplemental devices, it is possible to control structural deformations during severe seismic events (Ray, Reinhorn, & Nagarajaiah, 2013). Algebraic smooth relations, describing the nonlinear stress–strain material behaviour for monotonic loading, such is the well–known Ramberg–Osgood (RO) model (Ramberg & Osgood, 1943), have been used as skeleton curves to obtain smooth algebraic hysteretic behaviour by applying the Masing’s Hypothesis (Jennings, 1964; 1965). The smooth algebraic hysteresis models do not violate Drucker’s stability postulate, which implies that the hysteresis model should not produce loops (i.e. negative energy), when unloading–reloading process occurs without load reversal. This property is not generally true in the case of smooth differential models, which can exhibit local violation of Drucker’s stability postulate (Bažant, 1978). The RO model has been used in a great extent to portray the hysteresis behaviour of materials, structures, and devices involved in seismic engineering analysis and design (Jennings, 1964; 1965; Elgamal, 1991; Faccioli & Ramirez, 1976; Bratosin & Sireteanu, 2002). Due to the arrangement of the variables in the RO equation, it is not a simple and straightforward task to obtain the values of model parameters which give a sufficiently accurate fit of experimental data (Mostanghel & Byrd, 2002). Computational procedures, based on experimental data of shear modulus and damping ratio at various shear strains, were developed for obtaining the value of these parameters (Ueng & Chen, 1992). Many applications of RO model have addressed the prediction of structures response to strong motion earthquakes (Kaldjian & Fan, 1967; Ganev, Yamazaki, Ishizaki, & Kitazawa,1998; Phocas & Pocanschi, 2003; Feng et al., 2000).

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