Consideration of uncertainties in seismic analysis of non-classically damped coupled systems
Highlights
► We evaluate the design response of a non-classically damped coupled primary–secondary system. ► Uncertainties in modal properties of uncoupled systems have been considered. ► Gupta and Choi (2005) proposed the SRMS method for SDOF–SDOF type coupled system ► We study the applicability of SRMS method to MDOF–MDOF type coupled system. ► For modeling closely spaced frequenciesin MDOF systems, a joint density function is formulated.
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.
Section snippets
Coupled system analysis
The equation of motion for an N-DOF coupled primary–secondary system is given by:where M, C and K are the mass, damping and stiffness matrices of the coupled system; are the displacement, velocity and acceleration vectors of the coupled system relative to the fixed base of the primary system; Ub is the static displacement vector of the coupled system; and is the ground acceleration.
The uncoupled mode shapes [Φ] are given by
Consideration of uncertainties in coupled analysis framework
The response of secondary system is quite sensitive to the degree of tuning between the modes of uncoupled primary and secondary systems. The effect of non-classical damping is also significant in tuned or nearly tuned primary–secondary systems. A slight change in tuning due to a variation in the frequencies of uncoupled system can lead to large variation in secondary system response. A variation in the eigenvectors does not result in a significant variation in the secondary system response.
Limitations of originally proposed SRMS approach
Implementing the SRMS method involves conducting multiple analyses of the coupled system. This comprises of generating multiple sets of natural frequencies for uncoupled MDOF primary and MDOF secondary systems. Generating the random sample sets of natural frequencies of an MDOF structure requires knowledge of their probability density functions (pdfs). The SRMS method proposed by Gupta and Choi (2005) samples the frequencies of the uncoupled systems by treating each frequency as an independent
Formulation for joint density function
Let μ1 represent the mean frequency of a lower order mode and μ2 represent the mean frequency of the consecutive higher order mode and assuming the frequencies are uniformly distributed over the range of ±15%, the pdfs would overlap if 1.15μ1 > 0.85μ2 ≡ μ2 < 1.353μ1. This implies that if the mean frequencies are closer by 35.3% or less, the chance of incorrect sampling increases significantly with increasing closeness of the frequencies. Consequently, any sampling scheme should consider a definite
Numerical examples: verification of the SRMS approach
The interaction between the primary and secondary systems is significant when modes of the primary system are tuned with modes of the secondary system. Consequently, it is expected that the response of the secondary system is considerably influenced by uncertainties in frequencies in addition to uncertainties in ground motion. Gupta and Choi (2005) illustrated using SDOF primary–SDOF secondary systems that uncertainties in frequencies and modal damping ratios can result in much larger variation
Summary and conclusions
In the seismic analysis of primary–secondary coupled systems, the degree of tuning between the primary and secondary system modes can influence the response of a secondary system significantly. Uncertainties in frequencies can cause the modes of uncoupled primary and secondary systems to become tuned or detuned. The effect of non-classical damping can be significant in tuned or nearly tuned systems. This study investigates the effect of uncertainties in dynamic characteristics (frequencies and
Acknowledgements
This research was partially supported by the Center for Nuclear Power Plant Structures, Equipment and Piping at North Carolina State University. Resources for the Center come from sponsoring industrial organizations, National Science Foundation, the Civil Engineering Department, and College of Engineering in the University.
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