Closed-form stochastic response of linear building structures to spectrum-consistent seismic excitations

Abstract

Design codes typically define the seismic action in terms of pseudo-acceleration response spectra, encouraging the use of the modal superposition method for the evaluation of the structural response. For linear structural systems, provided that the ground shaking has been appropriately modelled, the full characterization of the response processes of interest can be achieved with the application of the random vibration theory. An analytical model for the power spectral density (PSD) functions consistent with seismic response spectra has been recently proposed. In this paper, taking advantage of this novel PSD model, closed-form approximate expressions of the spectral moments of the structural response are derived and numerically validated for single-and multi-degree of freedom systems. The proposed formulation is applied to the case of base isolated multi-story buildings, aimed at overcoming the difficulties associated with the non-classical nature of their damping. The paper shows how the proposed approach can be used for an accurate and computationally efficient evaluation of the probabilistic distribution of the structural response maxima.

An Efficient Stochastic Linearisation Procedure for the Seismic Optimal Design of Passive Control Devices

“…Moreover, it is assumed that earthquake acceleration timehistories may be modeled as finite time duration samples of a zero-mean Gaussian stationary stochastic process, completely defined by the knowledge of its Power Spectral Density (PSD) function. In this case, the vector r collects the parameters of the seismic input stochastic model, obtained by using recently proposed analytical expressions of PSDs consistent with assigned RS, that cover most of the international building codes (Barone et al, 2015(Barone et al, , 2019. The Appendix reports further details of this PSD model along with the closed-form relationship between the RS parameters and the consistent PSD function ones.…”

“…These parameters designate natural frequency, modal damping and damped frequency of the i-th modal oscillator (Igusa et al, 1984), respectively. Taking advantage by the analytical definition of the seismic input PSD function reported in Appendix, it is possible to perform a closed-form approximate evaluation of direct spectral moments in the modal space, without resorting to numerical integrations (Barone et al, 2019). Referring to the ith modal oscillator response process Q i (t), the expression of the j-th spectral moment λ j,Q i ,k is:…”

“…and the quantities e i , ω s and GÜ g (ω s ) are the parameters of the analytical model of the PSD function, whose meaning is detailed in the Appendix. Further details on the derivation and application of Equations (24) to (25) can be found in Barone et al (2019). Di Paola and Muscolino (1988) demonstrated that the crossspectral moments of any order λ j,Q i ,Q k , if they exist for a given PSD function of the excitation, may be obtained recursively as linear combinations of the direct spectral moments.…”

“…The same approach would also apply for other environmental hazards, i.e., wind or ocean waves, when a PSD model of the input is available. The issue of accuracy of the seismic PSD analytical model has been investigated in Barone et al (2019), to which the reader is referred for further details. In Figure 2, however, a comparison between first order closed-form spectral moments (m = 0, 1, 2) and their numerical counterparts is shown for several values of natural periods and viscous damping ratios.…”

“…In order to reduce the great computational burden involved in these problems, two novelty aspects have been here proposed and introduced. Firstly, assuming that earthquake time-histories can be modeled as samples of a stationary stochastic process, an analytical representation of earthquake load Power Spectral Density (PSD) functions consistent with seismic codes Response Spectra (RS) ( Barone et al, 2015Barone et al, , 2019 have been adopted. Secondly, an algorithm has been set up in order to obtain approximate analytical expressions of the response first spectral moments for not-classically damped Multi-degrees-of-freedom (MDOF) systems, without performing any numerical integration.…”