Direct scaling of residual displacements for bilinear and pinching oscillators

The estimation of residual displacements in a structure due to an anticipated earthquake event has increasingly become an important component of performance-based earthquake engineering because controlling these displacements plays an important role in ensuring cost-feasible or cost-effective repairs in a damaged structure after the event. An attempt is made in this study to obtain statistical estimates of constant-ductility residual displacement spectra for bilinear and pinching oscillators with 5% initial damping, directly in terms of easily available seismological, site, and model parameters. None of the available models for the bilinear and pinching oscillators are useful when design spectra for a seismic hazard at a site are not available. The statistical estimates of a residual displacement spectrum are proposed in terms of earthquake magnitude, epicentral distance, site geology parameter, and three model parameters for a given set of ductility demand and a hysteretic energy capacity coefficient in the case of bilinear and pinching models, as well as for a given set of pinching parameters for displacement and strength at the breakpoint in the case of pinching model alone. The proposed scaling model is applicable to horizontal ground motions in the western U.S. for earthquake magnitudes less than 7 or epicentral distances greater than 20 km.