The double impulse is introduced as a substitute of the fling-step near-fault ground motion and a critical elastic–plastic response of a two-degree-of-freedom (2DOF) building structure under the “critical double impulse” is evaluated. Since only the free-vibration appears under such double impulse, the energy balance approach plays an important and essential role in the derivation of the solution of a complicated elastic–plastic critical response. It is shown that the critical timing of the double impulse is characterized by the timing of the second impulse at the zero story shear force in the first story. This timing guarantees the maximum energy input by the second impulse that causes the maximum plastic deformation after the second impulse. Because the response of 2DOF elastic–plastic building structures is quite complicated due to the phase difference between two masses compared to single-degree-of-freedom models for which a closed-form critical response can be derived, the upper bound of the critical response is introduced by using the convex model.
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