This paper presents a method to directly evaluate floor response spectra (FRS) from a specified ground response spectrum (GRS) without time history analysis. In general, floor response spectra are utilized to evaluate seismic design force of secondary systems such as non-structural components and equipment in buildings. The secondary system having small mass is mounted on the building idealized as a single degree of freedom linear oscillator. In this study, target fundamental natural period of the buildings are limited between 0.2 and 2 seconds, because dynamic amplification factors are known to be relatively stable for specified damping factors in this range. The damping factors range from 0.02 and 0.3 in order to cover wide variety of the buildings. In the contrast, the damping factors of the secondary systems are assumed to be less than 0.1.
Firstly, a solution by double convolution integral of impulse response functions is firstly discussed. By integration, a closed form representing the seismic response in the secondary system is decomposed into four parts in time domain. These are composed of the displacement and the velocity of the building and a secondary system virtually mounted on the ground. In the perfect tuning case, where the building and the secondary system have the same natural period, difference between these responses corresponds to the seismic response of the secondary system. In order to obtain the maximum response, a spectrum difference rule (SPD rule) is employed. Consequently, spectral acceleration corresponding both to the building and the second system is combined with a correlation factor of the complete quadratic combination (CQC) rule developed by Der Kiureghian. This rule is extended to a non-tuning case in the similar manner. This approach is confirmed to improve the accuracy of the dynamic amplification factor compared to foregoing studies through time history analysis. Variation of the smooth FRS based on the SPD rule is accurately traced in accordance with changes of both of the two damping factors.
Secondary, convenient evaluation method of the dynamic amplification factor in the tuning case is discussed. As a result of semi-empirically formulation, it is cleared that the amplification factor divided by a certain factor is critically governed by the average of the two damping factors of the building and the secondary systems. This factor is identified as a reduction factor for response spectra due to damping. As a result, an explicit formula of the amplification factor for specified damping even without GRS is proposed. The values by this formula correspond to perfectly the same values obtained by time history analysis using simulated ground motions with uniform random phase angles. This result holds for various kinds of combination of two damping factors. In the contrast, the estimated values give roughly mean values compared to the results when observed ground motions are used.
Finally, numerical examples are demonstrated for representative observed ground motions. It is confirmed that the FRS directly evaluated by the proposed method extremely resemble one obtained by time history analysis in shape.
