| Statistics of peak inelastic seismic displacement or ductility demand are needed for assessing seismic risk for a spatially distributed group of structures. To simplify the structural analyses while maintaining the essential dynamic characteristics of the buildings, simplified equivalent nonlinear structural systems with one- to three-degree-of-freedom (1DOF to 3DOF) systems under uni- or multi-directional excitations could be employed for estimating such statistics. The obtained statistics can also be used as the basis to develop simple rules for estimating the inelastic seismic displacement demand that could aid the seismic risk assessment as well as the calibration of seismic provisions in design codes. While the bilinear model (or elastoplastic model) is often considered for simplified structural systems, it cannot cope with the effects of the degradation of strength or stiffness degradation or the pinching behaviour. To overcome this, the phenomenological hysteretic model, known as the Bouc-Wen model can be used.;Samples of the peak inelastic seismic displacement or ductility demand are evaluated by considering different combinations of structural dynamic characteristics, hysteretic behaviour for 1DOF to 3DOF systems. The statistics obtained from the samples are employed to develop empirical equations for estimating the mean ductility demand, and to assign probabilistic models. Furthermore, statistics of dissipated hysteretic energy are also given. In some cases, simple rules are provided for estimating the ductility demand under bidirectional excitations. The statistics of torsional effect on the inelastic seismic displacement demand indicate that such an effect is sensitive to the torsional frequency ratio, the translational frequency ratio, the degree of torsional restraint and the strength and stiffness degradations for stiff structures. It is also concluded that in general, the ductility demand under bidirectional seismic excitations is higher than that under unidirectional excitations, and the ductility demand under bidirectional excitations can be modeled by a lognormal or Frechet variate depending on the vibration period, the normalized yield strength and the model parameter controlling the surface at yield. The coefficient of variation of the displacement ductility demand is similar for systems under unidirectional or bidirectional excitations.;By adopting the Bouc-Wen model, assessment of inelastic seismic displacement demand is carried out for the simplified equivalent nonlinear structural systems under uni- or bi-directional seismic excitations, which are defined by 381 records from 31 California seismic events. For the analysis, we have implemented the Bouc-Wen model in a standing alone program, and in a finite element software package. |
