Abstract The double impulse has extensively been used to evaluate the critical response of an elastoplastic structure against a pulse-type input, including near-fault earthquake ground motions. In this paper, we propose a robustness assessment method for elastoplastic single-degree-of-freedom structures subjected to the double impulse input. Uncertainties in the initial velocity of the input, as well as the natural frequency and the strength of the structure, are considered. As fundamental properties of the structural robustness, we show monotonicity of the robustness measure with respect to the natural frequency. In contrast, we show that robustness is not necessarily improved even if the structural strength is increased. Moreover, the robustness preference between two structures with different values of structural strength can possibly reverse when the performance requirement is changed.