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

Abstract

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.

Introduction

The operation and safe shut down of any nuclear power plant relies on a variety of secondary systems such as pipelines, mechanical equipment and electrical systems which are supported on the primary structures such as the buildings. The importance of secondary systems in evaluating the seismic performance of a nuclear power plant is now well recognized by researchers and practicing engineers. Seismic response of a secondary system, in addition to its own dynamic properties also depends on its interaction with the primary system it is supported on. Tuning between primary and secondary system modes can affect the response of a secondary system significantly. Also, the damping characteristics of the primary and the secondary systems are generally different, thus making the coupled system non-classically damped. The effect of non-classical damping can be significant in tuned or nearly tuned primary–secondary systems. An uncertainty in frequencies can cause the modes of uncoupled primary and secondary systems to become tuned or detuned. Therefore, incorporation of these uncertainties during the seismic analysis of these structural systems is essential.

The response of the secondary system is sensitive to uncertainties in both dynamic characteristics as well as earthquake loading. In nuclear power plant systems, USNRC (1978) and ASME (2007) recommend a ±15% uncertainty in building or primary system frequencies that is uniformly distributed over the entire range. In a conventional or decoupled analysis, these uncertainties are accounted for by modifying the floor spectra (Liu et al., 1973, Singh, 1980, Igusa and Der Kiureghian, 1985, Chen, 1993, Reed et al., 1994). Practicing engineers often modify the floor spectra by either ‘Peak-Broadening’ (USNRC, 1978, ASME, 2007) or ‘Peak-shifting’ (ASME, 2007) methods. However, these methods cannot be used directly in a coupled analysis where the floor spectra are neither generated nor required.

Within the coupled analysis framework, Gupta and Choi (2005) considered uncertainties in dynamic properties (natural frequencies and damping ratios of uncoupled primary and secondary systems) and ground motion input for evaluating the design response of secondary systems using Square-Root-of-Mean-of-Squares (SRMS) method. The SRMS approach was illustrated to work well with representative simple 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. At first, this verification study might appear trivial. However, as illustrated in this paper, we encountered certain limitations in the application of originally proposed SRMS methodology. The key limitation relates to simulation of an MDOF structure's natural frequencies by characterizing them as uniformly distributed random variables within ±15% range as per USNRC recommendations. The paper illustrates that if two or more modes of a structure have closely spaced frequencies, the individual pdfs of the closely spaced frequencies can overlap with each other. In such a case, the simulated sets of natural frequencies do not remain as ordered sets. It is illustrated that rejection of unordered sample sets does not resolve the problem because the simulated pdfs no longer maintain uniform distribution over the entire ±15% range. In order to correctly simulate the closely spaced frequencies, a closed form formulation for the joint probability density function is incorporated into the SRMS approach. Several different MDOF primary–MDOF secondary systems are considered for verification of the modified SRMS approach.