The double-impulse input is introduced as a substitute for the fling-step nearfault ground motion, and a closed-form solution for the approximate elastic-plastic response of a structure with viscous damping under the "critical double impulse" is derived. Since only the free vibration appears under such a double impulse, the energy approach plays an important role in the derivation of the closed-form solution for a complicated elastic-plastic response with viscous damping. The quadratic function

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... approximation for the damping force-deformation relationship is introduced. The validity and accuracy of the proposed theory are investigated through comparison of the results of the response analysis to the corresponding one-cycle sinusoidal input and actual recorded ground motions. This is an open access article under the terms of the Creative Commons Attribution-NonCommercial-NoDerivs License, which permits use and distribution in any medium, provided the original work is properly cited, the use is non-commercial and no modifications or adaptations are made.