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

Abstract

The objective of this study is to investigate the effect of uncertainties in modal properties of uncoupled primary and secondary systems in the seismic analysis of non-classically damped Multi Degree of Freedom (MDOF) primary – MDOF secondary coupled systems by response spectrum method.

The design response of the secondary system is evaluated by conducting multiple analyses of coupled system with randomly sampled sets of uncoupled natural frequencies of primary and secondary systems. Generating the random samples of frequencies require knowledge of their probability density functions. When the individual probability density functions overlap, generating sample sets by assuming the frequencies to be independent random variables can lead to incorrect sets of frequencies in the sense that the frequencies do not remain as ordered pairs or ordered sets. The frequencies of any uncoupled primary or secondary system should all be in an ascending order. This necessitates the need for considering correlations between such closely spaced frequencies. However, determination of correlations between the natural frequencies of a structure is either not possible or impractical. So, any sampling scheme should consider the constraint that any randomly sampled set of uncoupled frequencies be an ordered set. Moreover, rejecting the incorrect samples result in individual density functions that are significantly different from the distributions initially assumed for sampling of each natural frequency. A formulation for a joint probability density function for the frequencies is developed using fundamental probability approaches. Sampling the frequencies by using the joint density function ensures that each sampled set remains ordered while maintaining the individual density functions.

Development of formulation for the joint density function of frequencies enabled the application of the Square-root-of-mean-of-squares (SRMS) method proposed by Gupta and Choi (2005) to MDOF primary – MDOF secondary systems. Two types of MDOF primary – MDOF secondary system configurations considered in this validation and verification study are (i) singly-connected secondary systems and (ii) multiply-connected secondary systems. Various degrees of tuning between primary and secondary systems were considered. The modified SRMS method is validated for both types of configurations.