This paper attempts to mathematically formulate the total input energy of secondary structures under earthquake, which stand on the primary structures. First, the total energy input of the primary structures is found to be simply expressed in terms of wave components of the seismic ground acceleration, neglecting the free vibration of the system and the influence of the phase angles between wave components. The acceleration response can be also expressed in terms of wave components of the ground acceleration, neglecting the free vibration and phase angles. Based on the acceleration response, the total energy input of the secondary structures can be formulated. As a result, the energy spectrum of the primary structure and the energy spectrum of the secondary structure can be directly related one another. Mathematical results are compared with numerical examples and are proved to be applicable to practical design purposes.
