© 2019 Proceedings of the 3rd International Conference on Uncertainty Quantification in Computational Sciences and Engineering, UNCECOMP 2019. All rights reserved. The paper deals with the modelling, response quantification and vibration control of rigid-plastic blocks in presence of stochastic forcing with indicative application to seismic engineering. The full dynamic interaction between a rigid-plastic block and a linear base-isolation system is considered and efficient piecewise numerical solutions are derived for analysing the true nonlinear response, in comparison with the base-fixed counterpart. Stochastic forcing is modelled as stationary filtered white noise, characterised by a modified version of the Kanai-Tajimi power spectrum suggested by Clough and Penzien, commonly used in earthquake engineering applications. A statistical linearisation approach is adopted in view of approximating the strongly nonlinear systems during the sliding motion regime, which conveniently permits quantification of the steady-state, stationary response statistics. The accuracy of the linearisation approximation is investigated, and the effectiveness of the base isolation in suppressing the extreme forcing delivered to the block is assessed. The work delivers insights into the determination and understanding of the probabilistic characteristics of the response of dynamically driven base-fixed and base-isolated rigid-plastic systems, further encouraging investigations on other types of structures, isolation systems and hazard scenarios.