This article presents an empirical ground-motion prediction model (attenuation relation) for inelastic (as opposed to elastic) spectral displacement (Sdi) for ground motions without forward directivity effects. It is a function of two earthquake parameters, moment magnitude (Mw) and the closest distance to rupture (Rrup), and two bilinear oscillator parameters, an undamped elastic period (T) and a yield displacement (dy). The dy is introduced via the predicted median strength-reduction factor (), a proxy for the ratio of elastic spectral displacement (Sde) to dy, which is identical with the familiar strength-reduction factor (R). The proxy recognizes that R can only be estimated indirectly because it implicitly contains the random variable, Sde, which cannot be known a priori; therefore, the median estimate or predicted median (Ŝde) from a conventional (elastic) ground-motion prediction model is used instead to calculate = Ŝde/dy. For enhanced generality, the inelastic spectral displacement prediction model here is based on a ratio concept, that is, the total model is a (any) conventional elastic prediction model coupled with a new inelastic displacement ratio prediction model, with proper statistical correlation between the two. We empirically consider the dependence of this ratio on source and path effects (i.e., Mw and Rrup), and find that Mw is significant, but Rrup is not. The resulting prediction model can easily be added to existing probabilistic seismic-hazard analysis (psha) software packages with only one extra structure-specific parameter, dy of the oscillator. In practical engineering applications, this will likely have been estimated from the conventional static pushover analysis of the multi-degree-of-freedom (mdof) structure under consideration.
The resulting psha product is a hazard curve for Sdi, the inelastic spectral displacement of a nonlinear oscillator. Such a curve can provide a more direct hazard- based target displacement for nonlinear static procedures (Federal Emergency Management Agency [fema] 356, 2000) and/or a basic input function for new probabilistic seismic-demand analyses that is based on Sdi (as opposed to Sde) as an efficient and sufficient intensity measure. This new attenuation relationship will be particularly useful in evaluating the performance of existing structures and specified designs with known lateral strength. In particular, unlike most past studies, it does not pre-fix the ductility level.