Statistical analysis of variability in floor response spectra using random vibration theory

The floor response spectrum is the maximum response of a single degree of freedom oscillator (secondary structure) with varying frequency attached to the degree of freedom (floor) of interest in the primary structure. In general, the two available approaches for calculating floor response spectra, given the ground spectra (or an ensemble of base excitations), are: A) time history (TH) and b) floor spectrum generation (FSG). In the time history approach, one uses a set of base excitations corresponding to a given ground motion spectrum (or uses a given ensemble of base excitations) in a series of time history analyses to find a set of floor response spectra. To find the final floor response spectrum one then averages over this set to obtain a smooth spectrum. The floor spectrum generation approach on the other hand is an efficient method which avoids the time history analysis of the composite system while accounting for the various effects such as tuning, interaction as well as the non-classical damping in the composite system. In this paper we study the variability in the floor response spectra generated using TH approach, both theoretically and numerically.We do this for a special case in which oscillator and structure comprise a perfectly tuned primary-secondary (PS) system. To this end we use Gaussian approximations along with some previously derived expressions for covariances of modal responses (Sinha & Igusa 1995) to derive closed form expressions for the variance of sample correlation coefficient between modal responses. These expressions involve damping and frequency characteristics of the composite system as well as its tuning and the duration of the input excitation.We then combine these expressions with other results on mean square response of structures to random excitations (Der Kiureghian 1980, Sinha & Igusa 1995) to estimate the variability in the floor response spectra.