Floor response spectrum is useful information in seismic design of nonstructural components such as ceiling and equipment. Its spectral peak, which occurs in resonance, is important in terms of deciding seismic design force of nonstructural components. The peak acceleration normalized by floor acceleration, which is independent of seismic intensity is called dynamic amplification ratio (DAR). The magnitude of DAR is affected by damping ratios of building and nonstructural component. In addition, it is empirically known that the DAR decrease as natural period of nonstructural component is longer, or duration of ground motion is shorter. However, this tendency has not been well clarified. The objective of this paper is to investigate the effect of duration on DAR and propose a closed form of expected DAF denoted by the characteristics of both the system and ground motion.
Both buildings and nonstructural component are modeled as an elastic single degree of freedom (SDOF) system, respectively. Firstly, two idealized excitation are considered to obtain upper bound and lower bound of DAR. One is a stationary response to white noise excitation, which give an upper bound of DAR. Another is a response to white noise with infinitesimal duration, which gives a lower bound. The latter corresponds to response to impulse input. Based on convolutional integral, a closed formula is derived for these conditions.
Subsequently, DAR is discussed in case of excitation with finite duration. Equivalent number of cycles for building response is introduced. The number of cycles is defined by ratio of the duration of ground motion to natural period. This value is incorporated with geometric mean of two damping ratio of system. The proposed variable give a degree of acceleration development associated with lapse of time. Stochastic time history analysis is conducted to verify the non-dimensional variable. It is clarified that this variable comprehensively explains the effect of short excitation on decrease of DAR.
Next, effectiveness of the explanatory variable is shown for historical earthquakes. A total of 1, 684 historical records are selected from the PEER ground motion database. The DARs calculated by time history analysis are normalized with the predetermined lower and upper bounds previously discussed. As a result, the DARs are mapped into a space with numeric between 0 and 1. Statistical relationship between the mapped DAR and the explanatory variable approximately trace a single curve for wide variety of structural characteristics and earthquakes. In order to express this curve, a regression formula is proposed for practical engineering. The proposed formula has four variables including natural period, two damping ratio, and significant duration of ground motion.
Finally, it is confirmed that the DARs evaluated from the formula are in good agreement with expected DARs. The formula also implies that DAR dramatically decreases over certain threshold period.
