Probabilistic analysis of peak response of MDOF systems with uncertain PSD function

The use of uniform hazard spectra which have the same probability of exceedance at different frequencies has been proposed for the future version of the National Building Code of Canada. Commonly used combination rules to estimate the peak responses of multi-degree-of-freedom (MDOF) systems are the square root of sum of squares rule and the complete quadratic combination rule. However, the probability that the peak response of a MDOF system exceeds the one estimated by using these rules with the peak modal responses from the uniform hazard spectra cannot be inferred directly. The assessment of the probability of exceedance of the peak response of MDOF systems is presented by considering that the uncertainty in seismic excitation due to all potential earthquakes can be lumped in the power spectral density function of the ground acceleration with uncertain model parameters. This probability is evaluated based on the random vibration of linear systems and the first-order reliability method. It is found that the under- or over-estimations are less than about 5 or 10% if the modal contributions are not within 10-90% of, or not within 20-80% of, the absolute sum of the effective modal peak responses, respectively. Otherwise, severe under- or over-estimation could result.