Received: 25 September 2018 Revised: 21 January 2019 Accepted: 27 January 2019 DOI: 10.1002/eqe.3157 RESEARCH ARTICLE Seismic response assessment of rocking systems using single degree-of-freedom oscillators Spyridon Diamantopoulos Michalis Fragiadakis School of Civil Engineering, National Technical University of Athens, Athens, Greece Correspondence Michalis Fragiadakis, School of Civil Engineering, National Technical University of Athens, Athens, Greece. Email: mfrag@mail.ntua.gr Present Address Michalis Fragiadakis, 9 Iroon Polytechneiou, Zografou 15780, Athens, Greece Summary A new modeling for the seismic response assessment of free-standing, rigid or flexible, pure rocking systems is presented. The proposed modeling is based on equivalent single degree-of-freedom (SDOF) oscillators that can be implemented with common engineering software or user-made structural analysis codes. The SDOF models adopted use beam elements that are connected to a nonlinear rotational spring with negative stiffness that describes the self-centering capac- ity of the rocking member. The loss of energy at impact is treated with an “event-based” approach consistent with Housner's theory. Different variations pertinent to rigid blocks are first presented, and then the concept is extended to the flexible case. The implementation of the method requires some minor pro- gramming skills, while thanks to the versatility of the finite element method, it is capable to handle a variety of rocking problems. This is demonstrated with two applications: (a) a vertically restrained block equipped with an elastic tendon and (b) a rigid block coupled with an elastic SDOF oscillator. The accuracy and the efficiency of the proposed modeling is demonstrated using simple wavelets and historical ground motion records. KEYWORDS direct stiffness method, flexible blocks, nonlinear response history analysis, restrained columns, rigid block, rocking, seismic loading 1 INTRODUCTION Free-standing slender blocks when subjected to an excitation of their base may slide, uplift, rock, or overturn. Omitting the uplift and assuming that there is no sliding, the most significant motion is rocking, ie, the partial uplift of a structure from its base when the center of rotation changes. Rocking during an earthquake is common for free-standing objects and also for many other engineering systems. The seismic response of a solitary rigid block that rocks on a rigid base was first studied more than a century ago by Milne, 1 while today the problem is typically addressed using the framework proposed by Housner. 2 Over the years, there have been many studies shedding light on the problem, eg, Yim and Chopra, 3 Ishiyama, 4 Zhang and Makris, 5 Politopoulos, 6 Dimitrakopoulos and DeJong, 7 and Mathey et al. 8 Rocking of rigid, or flexible, bodies/systems is a fundamental problem in earthquake engineering. The most common option for handling a 2D rocking problem is directly solving the equation of motion proposed by Housner. 2 Since structural engineers are more comfortable with elastically deforming structures and software, there have been attempts to develop “equivalent” solutions based on the dynamics of the single degree-of-freedom (SDOF) oscillator. Priestley et al 9 proposed an equivalent SDOF oscillator with a constant damping ratio whose period depends on the amplitude of the rocking Earthquake Engng Struct Dyn. 2019;1–20. wileyonlinelibrary.com/journal/eqe © 2019 John Wiley & Sons, Ltd. 1