Using Bézier curves to model gradual stiffness transitions in nonlinear elements: Application to self‐centering systems

Abstract A number of techniques are available for modelling nonlinear elements, but most available hysteretic rules do not capture the gradual stiffness changes that are typical of physical systems. In particular, there has not previously been a hysteretic rule with rounded hysteretic corners that could be used to model self‐centering elements, where multiple stiffness changes occur within one loading cycle. This paper presents a new hysteretic rule that allows the gradual stiffness transitions that occur in real systems to be modelled. In this paper, the rule is formulated for flag‐shaped hystereses, but it is shown that the same model also produces hystereses that can be used to model systems that are not self‐centering. The same technique could also be applied to round the corners of different backbone hystereses. A previous study has shown how abrupt stiffness changes can cause very large acceleration spikes, particularly in self‐centering systems. This paper shows that acceleration spikes due to stiffness changes may be reduced by designing systems to change stiffness more gradually, and that this typically has little effect on other aspects of the seismic response. When modelling structural systems, especially if they are self‐centering, sharp‐cornered hysteretic models may be used for initial analysis, but round‐cornered hysteretic models should be considered when using nonlinear rotational springs or when accelerations are of particular importance. Copyright © 2011 John Wiley & Sons, Ltd.