Adapting the CQC3 Rule for Three Seismic Components with Different Spectra

The complete quadratic combination with three components (CQC3) formula is adapted for determining the critical response of elastic structures to two horizontal and the vertical seismic components with arbitrary spectra. The critical response, defined as the largest value of response for all possible incident angles of the two horizontal seismic components, can be estimated by the CQC3 rule assuming a period-independent value for the spectral ratio for these components. The CQC3 estimate is compared with the critical response that considers different spectra for each component, using spectra obtained from a preliminary study. The accuracy of the CQC3 estimate is evaluated for two values of the spectral ratio; $\overline{\gamma }{}_1$ is the average of the maximum and minimum value of the spectral ratio, and $\overline{\gamma }{}_2$ is the average of the two spectral ratios corresponding to the significant modes. Parametric analyses of one-story buildings and numerical analyses of multistory buildings indicate that the CQC3 rule predicts results within 5% of the critical value. For design purposes these differences are small and the CQC3 rule using $\overline{\gamma }{}_1$ or $\overline{\gamma }{}_2$ can be used to estimate the critical response of structures.