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response spectrum analysis, modal combination, spatial combination, modal superposition, 30% rule
Abstract:
Response Spectrum Analysis (RSA), one of the most popular methods to carry out the seismic design of multi-degree-offreedom (MDOF) structures, is based on the concept of modal superposition, by which the uncoupled equations of motion that represent each mode of vibration of the system can be solved independently and the resulting responses superimposed by assuming linear elastic behaviour. Each mode is represented by a single-degree-of-freedom (SDOF) system, whose peak response is retrieved from response spectra deemed suitable for design. However, while modal superposition allows for the total response of a MDOF system to be determined by simple addition of the individual modal responses at each time step, combination of spectral values needs to take into account the fact that peak modal responses do not necessarily occur at the same time or along the same horizontal directions. These considerations give rise to the use of modal and spatial combination rules that aim to calculate the likely peak response of a MDOF system instead of conservatively carrying out an algebraic sum of maxima. Current design codes
prescribe methodologies that were defined in the 1970s and 1980s, such as the Complete Quadratic Combination (CQC) [1], its three-dimensional extension CQC3 [2], the Square Root of the Sum of the Squares (SRSS) [3], or the 30% rules [4], based mostly on random vibration theory. However, access to large numbers of ground motion records at the present time allow us to revisit these approaches from a data-driven perspective, and investigate the relationship across the peaks of SDOF responses to seismic excitation at different orientations and at different points in time, with the ultimate goal of characterising this relationship in a fully probabilistic way. This paper presents results of a study of SDOF demands obtained considering 1,218 accelerograms from the RESORCE database [5], whose two horizontal perpendicular components were rotated around all non-redundant angles every 2° and applied to SDOF systems with periods of vibration of 0.2, 1.0 and 3.0 seconds, and sets of secondary systems with periods ranging from 0.5 through 0.95 times the three aforementioned periods. The concept of peak response was extended to include all peaks with amplitudes above two alternative thresholds of 80% and 95% of the maximum absolute response. Two main kinds of parameters were studied and are presented: (i) time differences between peaks of the same component and across perpendicular components, and (ii) ratios of instantaneous displacement demands between perpendicular components and the same component for different oscillator periods, as one of the components reaches a peak in the oscillator’s response. While results for the latter resemble the idea of the 0.3 coefficient from the 30% rule in average terms, the dispersion associated with all these parameters is large and should not be neglected.
