Consideration of uncertainties in seismic analysis of non-classically damped coupled systems

This paper proposes a method to evaluate the design response of a non-classically damped coupled primary–secondary system by statistically incorporating the effects of uncertainties in modal properties of its constituent uncoupled systems. Within the framework for the coupled system seismic analysis, the uncertainties can be accounted for by modeling the uncoupled modal properties of primary and secondary systems as random variables. Gupta and Choi (2005) proposed the Square-Root-of-Mean-of-Squares (SRMS) method which employs a limited Monte Carlo simulation to evaluate the design response of the secondary system statistically. The SRMS approach was illustrated to work well with representative single degree of freedom (SDOF) primary–SDOF secondary systems. In this paper, we study the applicability of SRMS methodology to MDOF primary–MDOF secondary systems. In such systems, two or more modes are likely to have closely spaced frequencies. The individual probability density functions of the closely spaced frequencies overlap with each other. Simulation of such closely spaced frequencies as independent random can give incorrect set of frequencies in the sense that the frequencies do not remain as ordered sets. Rejection of these incorrect sets does not resolve the problem as the simulated density functions no longer maintain the originally assumed distribution. The simulation of ordered sets of natural frequencies of an MDOF structure can be achieved by using a joint density function that considers the necessary constraints. The SRMS method for MDOF primary–MDOF secondary coupled systems is modified by incorporating a closed-form formulation for the joint density function of closely spaced frequencies. The modified SRMS approach is validated for MDOF secondary systems that are both singly as well as multiply connected to the MDOF primary system.