The double impulse input is introduced as a substitute of the fling-step near-fault ground motion and a closed-form solution of the elastic–plastic response of a structure by the "critical double impulse input" is derived. Since only the free-vibration appears under such double impulse input, the energy approach plays an important role in the derivation of the closed-form solution of a complicated elastic–plastic response. It is shown that the maximum inelastic deformation can occur either after the first impulse or after the second impulse depending on the input level. The validity and accuracy of the proposed theory are investigated through the comparison with the response analysis to the corresponding one-cycle sinusoidal input as a representative of the fling-step near-fault ground motion. Since the critical input means the resonant case, the present theory dealing with the resonant response should be applied to buildings except very flexible ones.