SummaryNonlinear static procedures, which relate the seismic demand of a structure to that of an equivalent singleādegreeāofāfreedom oscillator, are wellāestablished tools in the performanceābased earthquake engineering paradigm. Initially, such procedures made recourse to inelastic spectra derived for simple elasticāplastic bilinear oscillators, but the request for demand estimates that delve deeper into the inelastic range, motivated investigating the seismic demand of oscillators with more complex backbone curves. Meanwhile, nearāsource (NS) pulseālike ground motions have been receiving increased attention, because they can induce a distinctive type of inelastic demand. Pulseālike NS ground motions are usually the result of rupture directivity, where seismic waves generated at different points along the rupture front arrive at a site at the same time, leading to a doubleāsided velocity pulse, which delivers most of the seismic energy. Recent research has led to a methodology for incorporating this NS effect in the implementation of nonlinear static procedures. Both of the previously mentioned lines of research motivate the present study on the ductility demands imposed by pulseālike NS ground motions on oscillators that feature pinching hysteretic behaviour with trilinear backbone curves. Incremental dynamic analysis is used considering 130 pulseālikeāidentified ground motions. Median, 16% and 84% fractile incremental dynamic analysis curves are calculated and fitted by an analytical model. Leastāsquares estimates are obtained for the model parameters, which importantly include pulse period Tp. The resulting equations effectively constitute an R ā Ī¼ ā T ā Tp relation for pulseālike NS motions. Potential applications of this result towards estimation of NS seismic demand are also briefly discussed. Copyright Ā© 2016 John Wiley & Sons, Ltd.