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v 7 2m + ic + k is the complex frequency response function, and c(ā) = p(t) exp(āiāt) dt

is the Fourier transform of the input motion (i.e., the Fourier transform of the ground motion time history), which takes advantage of computational efficiency using the fast fourier transform (Clough and Penzien 1975).

For design purposes, it is often sufficient to know only the maximum amplitude of the response time history. If the natural period of the SDOF is varied across a spectrum of engineering interest (typically, for natural periods from 0.03 to 3 or more seconds, or frequencies of 0.3 to 30+ Hz), then the plot of these maximum amplitudes is termed a response spectrum. Figure 17.8 illustrates this process, resulting in Sd, the displacement response spectrum, while Figure 17.9 shows (a) the Sd, displacement response spectrum, (b) Sv, the velocity response spectrum (also denoted PSV, the pseudospectral velocity, pseudo to emphasize that this spectrum is not exactly the same as the relative velocity response spectrum; Hudson, 1979), and (c) Sa, the acceleration response spectrum. Note that

Response spectra form the basis for much modern earthquake engineering structural analysis and design. They are readily calculated if the ground motion is known. For design purposes, however, response spectra must be estimated ā this process is discussed in another chapter. Response spectra may be plotted in any of several ways, as shown in Figure 17.9 with arithmetic axes, and in Figure 17.10, where the velocity response spectrum is plotted on tripartite logarithmic axes, which equally enables reading of displacement and acceleration response. Response spectra are most normally presented for 5% of critical damping.

While actual response spectra are irregular in shape, they generally have a concave-down arch or trapezoidal shape, when plotted on tripartite log paper. Newmark observed that response spectra tend to be characterized by three regions: (1) a region of constant acceleration, in the high frequency portion of the spectra; (2) constant displacement, at low frequencies; and (3) constant velocity, at intermediate frequencies, as shown in Figure 17.11. If a spectrum amplification factor is defined as the ratio of the spectral parameter to the ground motion parameter (where parameter indicates acceleration, velocity, or displacement), then response spectra can be estimated from the data in Table 17.4, provided estimates of the ground motion parameters are available. An example spectrum using these data is given in Figure 17.11.

A standardized response spectrum is provided in the Uniform Building Code (UBC 1997). The spectrum is a smoothed average of a normalized 5% damped spectrum obtained from actual ground motion records grouped by subsurface soil conditions at the location of the recording instrument, and are applicable for earthquakes characteristic of those that occur in California (SEAOC 1988). This normalized shape may be employed to determine a response spectra, appropriate for the soil conditions. Note that the maximum amplification factor is 2.5, over a period range approximately 0.15 s to 0.4-0.9 s, depending on the soil conditions.

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