Abstract The critical excitation of a dynamical system is defined as the input excitation with the lowest energy that drives the system from one specified state to another within a given time span. Critical excitations play an important role in first passage problems because they are the most probable point in the first passage failure region of the standard Normal stochastic load space. They may also be used to provide efficient solution of other stochastic analysis problems by means of asymptotic approximations. Although the solution of critical excitation for linear systems can be obtained through unit impulse response functions, the case of nonlinear hysteretic systems is still under research. The latter has important relevance in the study of nonlinear response of structures under severe earthquake loads, where the characteristics of critical excitations may aid understanding the collapse potential of earthquakes. This paper investigates the critical excitation of single-degree-of-freedom (sdof) elasto-plastic systems. Through observations on dynamic characteristics, the critical excitation is parameterized in the time domain that allows for its efficient numerical solution. It is found that, in addition to resonance phenomenon that is observed in linear systems, a mechanism called ‘boundary criticality’ is responsible for driving elasto-plastic systems to its target by maximizing the capability of gaining momentum during elastic loading while avoiding opposing plastic deformations.