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Top1. 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.
Many engineering systems involve the hysteresis. For instance, seismic protection devices, electronic systems, mechanical components, and others (Ismail, Ikhouane, & Rodellar, 2009). In this case, memory effect appears in the behaviour of the system. This means that the nonlinear restoring force cannot be expressed as a function of the instantaneous values of displacement and velocity. For this reason, many hysteretic models are based on a set of differential equations to be integrated over the entire event, in order to include the time dependency of the force. 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 nonlinear hysteretic systems is very complex. Moreover, practical applications require simple and effective models that can be managed and easily controlled. Thus, alternative models have been developed. They do not focus on the detailed description of the system. Instead, they view the system in terms of its input and output, without analysing its intrinsic behaviour. This kind of models are often known as ‘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.