GNGTS 2021 - Atti del 39° Convegno Nazionale

GNGTS 2021 S essione 2.2 340 NONSTATIONARY STOCHASTIC SIMULATION OF EARTHQUAKE GROUND MOTIONS: APPLICATION TO THE ITALIAN DATABASE F. Sabetta 1 , G. Fiorentino 2 , G. Lanzano 3 , L. Luzi 3 , A. Pugliese 4 1 Department of Engineering, Roma Tre University, Rome, Italy 2 Department of Civil Engineering, University of Bristol, UK 3 Istituto Nazionale di Geofisica e Vulcanologia (INGV), Milano, Italy 4 Independent Researcher, Rome, Italy Introduction The selection of the input motion in the earthquake engineering practice has become progressively more important with the growing use of nonlinear dynamic analyses. Despite of the increasing availability of large strong motion databases, ground motion records are not always available for a given earthquake scenario and site condition, requiring the adoption of simulated time series. Among the different techniques for generating accelerograms (Douglas and Aochi, 2008), in this work we focused on the methods based on stochastic simulations, considering the time-frequency decomposition of the ground motion. In the proposed method (Sabetta et al. 2021) we updated the non-stationary stochastic model initially developed by Sabetta & Pugliese (1996) and later modified by Pousse et al. (2006) and Laurendeau et al. (2012). The model is based on the S-transform that implicitly considers both the amplitude and frequency modulation. Non-stationary stochastic ground motion model The simulation of non-stationary time series is based on the summation of Fourier series with random phases and time-dependent coefficients. The coefficients of the Fourier series are obtained from a frequency-time decomposition of the signal that, according to the definition given by Stockwell et al. (1996), is named S-transform. The S-transform of a signal x(t), such as a ground-motion record, is defined as: NONSTATIONARY STOCHASTIC SIMULATION OF EARTHQUAKE GROUND APPLICATION TO THE ITALIAN DATABASE. F. Sabetta 1 , G. Fiorentino 2 , G. Lanzano 3 , L. Luzi 3 , A. Pugliese 4 1 Department of Engineering, Roma Tre University, Rome, Italy. 2 Department of Civil Engineering, University of Bristol, UK. 3 Istituto Nazionale di Geofisica e Vulcanologia (INGV), Milano, Italy. 4 Independent Researcher, Rome, Italy. Introduction The selection of the input motion in the earthqu ke engineering practice has become progress important with the growi g use of nonlinear dynamic analyses. Despite of he increasi g availabil strong mo ion databases, ground motio records are n t always availabl f r given earthquake s site condition, requiring the adopti n of simulated time series. Among the different techniques for accelerograms (Douglas and Aochi, 2008), in this work we focused on the methods based on simulations, considering the time-frequency decomposition of the ground motion. In the propos (Sabetta et al. 2021) we updated the non-stationary stochastic model initially developed by Pugliese (1996) and later modified by Pousse et al. (2006) and Laurendeau et al. (2012). The mo on the S-transform that implicitly considers both the amplitude and frequency modulation. N -stationary stochastic ground motion model The simulation of non-stationary time series is based on the summation of Fourier series with ran and time-dependent coefficients. The coefficients of the Fourier series are obtained from a freq decomposition of the signal that, according to the definition given by Stockwell et al. (1996), i transform. The S-transform of a signal x(t), such as a ground-motion record, is defined as: S ( f , τ ) = ∫ − ∞ ∞ x ( t ) w ( f , ¿ τ − t ) e − i 2 πft dt ¿ ( in which S( f ,τ) is the S-transform of x(t), f and t are frequency and time, and τ is the center of window function w(f, τ – t). Figure 1 shows an application of the S-transform to an accelerogra during the Central Italy earthquake of 30 October 2016 at CSC (Cascia) station. It can be noted th frequencies decrease with increasing time. If we take the square of the S-transform, it becomes a natural extension of the power spectrum stationary case and it is constituted by a series of Power Spectral Densities (PSDs), calculated times. Xs ( t , f ) = | S ( t , f ) | 2 ( The PSDs can be fitted with a lognormal function defined through three parameters derived from of the spectral moments (Lai, 1982) extended to the non-stationary case: the average total power P integral over time corresponds to the Arias Intensity; the central frequency Fc(t), giving a measu the PSD is concentrated along the frequency axis; the frequency bandwidth Fb(t), correspon dispersion of PSD around the central frequency. To allow the correlation with earthquake (magnitude, distance, site condition), the time series simulation is done with a lognorm approximating Xs(t,f) : 2 2 (1) in which S( f ,τ) is the S-transform of x(t), f and t are frequency and time, and τ is the center of a Gaussian window function w( , τ – t). Figure 1 shows an application of the S-transform to an accelerogram recorded during the Central Italy e rthquake of 30 October 2016 at CSC (Cascia) station. It can be noted that t e high frequencies d crease with increasing time. If we take th square of the S-transform, it becomes a natural extension of the power spectrum to the non-stationary case and it is constituted by a se ies of Power Spectral Densities (PSDs), calculated at different times. NONSTATIONARY STOCHASTIC SIMULATION OF EARTHQUAKE GROUND APPLICATION TO THE ITALIAN DATABASE. F. Sabetta 1 , G. Fiorentino 2 , G. Lanzano 3 , L. Luzi 3 , A. Pugliese 4 1 Department of Engineering, Roma Tre University, Rome, Italy. 2 Department of Civil Engineering, University of Bristol, UK. 3 Istituto Nazionale di Geofisica e Vulcanologia (INGV), Milano, Italy. 4 Independent Researcher, Rome, Italy. Introduction The selection of the input motion in the earthquake engineering practice has become progress important with the growing use of nonlinear dynamic analyses. Despite of the increasing availabil strong motion databases, ground motion records are not always available for a given earthquake s site condition, requiring the adoption of simulated time series. Among the different techniques for ccelerograms (Douglas and Aochi, 2008), in this w rk we focused on he ethods based on simulatio s, considering the time-frequency decomp sition of the ground motion. In the propos tt et al. 2021) we updated the non-stationary stochastic model initially d veloped by Pugliese (1996) and later mo ified by Pousse et al. (2006) and Laure eau et al. (2012). The mo on the S-transform th t implicitly considers both the amplitude and frequency modulation. Non-stationary stochastic ground motion model The simulation of non-stationary time series is based on the summation of Fourier series with ran and ti e-dependent coefficients. The coefficients of the Fourier series are obtained from a freq decomposition of the signal that, according to the definition given by Stockwell et al. (1996), i transform. The S-transform of a signal x(t), such as a ground-motion record, is defined as: S ( f , τ ) = ∫ − ∞ ∞ x ( t ) w ( f , ¿ τ − t ) e − i 2 πft dt ¿ ( in which S( f ,τ) is the S-transform of x(t), f and t are frequency and time, and τ is the center of window function w(f, τ – t). Figure 1 shows an application of the S-transform to an accelerogra during the Central Italy earthquake of 30 October 2016 at CSC (Cascia) station. It can be noted th frequencies decrease with increasing time. If we take the square of the S-transfor , it becomes a natural ext nsion of the power spectrum tationary cas and it is constituted by a series of Power Sp ctral Densities (PS s), calculated time . Xs ( t , f ) = | S ( t , f ) | 2 ( The PSDs can be fitted with a lognormal function defined through three parameters derived from of the spectral moments (Lai, 1982) extended to the non-stationary case: the average total power P integral over ti e corresponds to the Arias Intensity; the central frequency Fc(t), giving a measu the PSD is concentrated along the frequency axis; the frequency bandwidth Fb(t), correspon dispersion of PSD around the central frequency. To allow the correlation with earthquake (magnitude, distance, site condition), the time series simulation is done ith a lognorm approximating Xs(t,f) : ~ X s ( t,f ) = Pa(t) e - [ lnf-ln β(t) ] 2 /2 δ 2 ( (2) The PSDs can be fitted with a lognormal function defined through three parameters derived from the theory of the spectral moments (Lai, 1982) extended to the non-stationary case: the average total power Pa(t), who’s integral over time corresponds to the Arias Intensity; the central frequency Fc(t), giving a measure of where the PSD is concentrated along the frequency axis; the frequency bandwidth Fb(t), corresponding to the dispersion of PSD around th central frequency. To allow the correlation with earthquake parameters (magnitude, distance, site condition), the time series simulation is done with a lognormal function approximating Xs(t,f) :