Abstract | Nonlinear static procedures, which relate the seismic demand of a structure to that of an equivalent single-degree-offreedom (SDOF) oscillator, are well-established tools in the performance-based earthquake engineering paradigm. Initially, such procedures made recourse to inelastic spectra derived for simple elastic-plastic 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, near-source (NS) pulse-like ground motions have been receiving increased attention, since they can induce a distinctive type of inelastic demand. Pulse-like 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 double-sided 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 aforementioned lines of research motivate the present study on the ductility demands imposed by pulse-like NS ground motions on oscillators that feature pinching hysteretic behavior with trilinear backbone curves. Incremental dynamic analysis (IDA) is used considering one hundred and thirty pulse-like-identified ground motions. Median, 16% and 84% fractile IDA curves are calculated and fitted by an analytical model. Leastsquares estimates are obtained for the model parameters, which importantly include pulse period Tp. The resulting equations effectively constitute an R – μ – T – T p relation for pulse-like NS motions. Potential applications of this result towards estimation of NS seismic demand are also briefly discussed.