## 5 Conclusions  
Based on considerations of stability and minimum kinetic energy a simple theoretical two-dimensional bi-modal model was derived to determine the safe-unsafe boundaries in the acceleration-period domain for rigid blocks under seismic single pulses.  
UDEC numerical simulations have substantiated the proposed two-dimensional model showing bi-modal behavior and the existence of a transition period in agreement with the derived model. The transition between the response modes in the numerical model takes place at a period that is close to the theoretically calculated characteristic period of the rigid block.  
The time/scale similitude rule established in the theoretical derivation is verified numerically showing that the critical period to cause instability of a 1/N reduced block scales with 1/N\({}^{0.5}\).  
The pulse energy similitude rule established in the theoretical derivation is verified numerically showing that for the same pulse energy the safe-unsafe boundaries in the acceleration-period domain are similar for different shape regular pulses.  
The theoretical model and the numerical simulation results are in good agreement with the experimental results from shaking table tests on a cylindrical small sized block, showing good predictive capabilities. The discrepancy between the experimental data and the theoretical and numerical results is attributed to imperfections in the shaking table model, eccentricities, contact surface roughness, and 3D effects.  
The model provides a first order approximation to determine the stability or overturning of rigid blocks under critical pulses. The time/scale and pulse energy similitude rules verified theoretically and numerically between various scales provide a robust method to determine the behavior under earthquake loads of full-sized block structures from experiments on scaled down models.  
The model shows good agreement with the published body of work on the overturning acceleration-frequency spectrum of rocking blocks under single pulses and other waveforms and presents a practical lower bound for realistic block sizes and impulse periods.  
Further development work is required to extend the proposed approach to the analysis of earthquake histories for critical pulses or critical trains of pulses that can drive a rocking system to overturn. The results presented in this work can be applied to investigate the response of rigid block structures such as obelisks, columns, colonnades, walls, precariously balanced rocks, and various block shaped equipment and furniture to complex seismic events. The model can also be used to estimate the critical parameters of near-fault earthquake pulses that have resulted in the historical destruction of some of these structures contributing to in archaeoseismology.  
Conceptualized the problem and conducted the experimental program, G.K.; carried out the numerical modeling, M.T. All authors have read and agreed to the published version of the manuscript.  
This research received no external funding.  
The authors wish to acknowledge the valuable contribution of anonymous reviewers whose judicious comments have resulted in substantial improvements to this paper.  
The authors declare no conflict of interest.