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The Mexico Earthquake of September 19, 1985—Nonstationary Models of Seismic Ground Acceleration
Abstract
Characteristics of a nonstationary process obtained by modulating the amplitude and frequency of a stationary process differ from those of an oscillatory process. An accelerogram recorded in the soft soil of Mexico City during the 1985 earthquake serves to calibrate both nonstationary models. Response of linear and nonlinear single-degree-of-freedom systems indicate that the process with modulated amplitude and frequency is preferable for reliability studies. Coefficients of variation of ductility demands of systems excited with accelerograms generated by the model with modulated amplitude and frequency are close to those corresponding to actual accelerograms.
... A(t) is a two-varied stochastic process, that acts in x and y globaldirections of the finite element numerical model (FEM) (Figure 1). The non-stationary model described in [19,20] is used to generate samples a x (t) and a y (t) of A x (t) and A y (t), respectively. The probabilistic model used for A(t) is calibrated with actual time histories recorded at the school base during the October 30th, 2016 earthquake. ...
... The demand parameter D dy is obtained by a three-step Monte Carlo algorithm: (i) n s ground motion acceleration time series a n (t) of the process A n (t), n = x, y, are generated using the method proposed in [19]; (ii) linear time domain numerical dynamic structural analyses are used to obtain response samples x(t) of X(t) assuming proportional damping ratio ζ = 5%; (iii) n s samples of the random vector D dy = {D y } are computed using Equation (4). In this case study the interstorey drifts in y-direction at the m = 12 node positions shown in Figure 1 are selected as demand parameters. ...
... In particular, in this paper the following steps are considered 2. generation of n s = 500 independent samples a n,i (t), i = 1, . . . , n s of A n (t), n = x, y using the method reported in [19]; ...
Seismic performance of structural systems is assessed by performing fragility analysis. Generally, downstream of this analysis, fragility curves are estimated in order to obtain the structural system performance. Seismic fragilities are the probability that the structural response of a system overcomes specified limit states for given seismic intensity measure. The most widespread procedure to estimate seismic fragility curves is based on scaling seismic accelerograms by a reference intensity measure (e.g. single/multiple ordinates of the pseudo-acceleration response spectrum or peak ground acceleration). Recently, it was shown that this methodology gives limited if any information on the structural seismic performance when the dependence between the intensity measure and the system demand parameter of interest (e.g. max inter-story displacement) is weak. This paper presents a general algorithm to improve the accuracy in fragilities estimation when the dependence between the intensity measure and the demand parameter is weak and the widely used method in Performance-Based Earthquake Engineering does not give accurate results. The proposed algorithm is based on a linear transformation of samples of a given intensity measure, which improves the correlation with a set of demand parameters. Fragility curves are obtained using the transformed intensity measure samples and compared with those estimated with the standard approach. The effectiveness of the proposed algorithm is demonstrated for an actual multi-degree of freedom structural system.
... A non-stationary probabilistic model is calibrated using experimental records of wind velocity fluctuations during one of the several thunderstorm outflows experienced in the north-western italian coast [14,15]. This model is based on the one presented in [16] to describe seismic loads in the time-frequency domain using suitable time varying functions for both amplitude and frequency coefficients. Virtual samples of wind velocity thunderstorm time histories are then generated and used to perform https://doi.org/10.1016/j.probengmech.2020.103103 ...
... The non-stationary properties described for the time series ( ) can be described by the model presented in [16] for generating seismic acceleration time histories. Let ( ) = ( ) be a sample time history of the non-stationary process ...
... It can be demonstrated that the variance of ( ) at time has the expression [16] ...
Wind loads on structures are commonly described as stationary phenomena that occur in neutral atmospheric conditions at the synoptic scale with velocity profiles in equilibrium with the atmospheric boundary layer. Nevertheless, structural systems can be also affected by thunderstorm outflows, which are non-stationary local phenomena at the mesoscale that occur in convective conditions with totally different velocity profiles with respect to synoptic winds. This paper presents a non-stationary probabilistic model that describes the wind velocity fluctuations experienced during a thunderstorm in order to estimate its effects on the dynamic structural response. The model is first calibrated on a typical thunderstorm recorded in the north-west italian coast and it is used to generate virtual time histories in a Monte Carlo simulation approach. The potential influence of the wind load non-stationary features on the peak structural response is investigated using single degree of freedom parametric analysis and statistical estimation.
... Structural analyses are developed for each sample of the process ( ) = { ( ); ( )}, for simplicity ( ) is a stochastic process with independent components, that acts in and global-directions of the FE numerical model (Fig. 11). The non-stationary model described in [30] is used to generate samples ( ) and ( ) of ( ) and ( ), respectively. The probabilistic model used for ( ) is calibrated with actual time histories recorded at the school base during the October 30th, 2016 earthquake following the procedure described in [31]. ...
... The demand parameters , = , , are obtained by the following Monte Carlo algorithm: (i) ground motion acceleration time series ( ) of the process ( ), = , , are generated using the method reported in [30,31]; (ii) linear time domain numerical dynamic structural analyses are used to obtain response samples ( ) of ( ) assuming proportional damping ratio = 5%; (iii) samples of the random vector = { (1) , (2) , … , ( ) }, = , , are computed using Eq. (5). ...
Seismic fragilities, i.e., probabilities that structural systems exceed specified limit states, are used in Performance-Based Earthquake Engineering to characterize structural behavior during seismic events. Generally, seismic fragilities are constructed from structural responses to scaled accelerograms described by single/multiple ordinates of the pseudo-acceleration response spectrum, peak ground acceleration or other intensity measures. Recently, it was shown that the resulting fragilities provide limited if any information on the structural seismic performance if the dependence between intensity measures and demand parameters is weak. Yet, the method is used in Performance-Based Earthquake Engineering since the statistical uncertainty caused by the limited numbers of recorded ground accelerations is overcome by scaling these records.
This study develops a method for estimating the fragility of linear systems for cases in which intensity measures and demand parameters are weakly correlated, i.e., situations in which the current methodology is inaccurate. The method is based on a linear transformation of samples of a given intensity measure which is designed to improve the correlation between demand and intensity parameters. The effectiveness of the proposed method for linear system is demonstrated by an elementary oscillator, a multi-degree of freedom system and a real complex multi-degree of freedom structural system.
... A(t) is a stochastic process with independent components, that acts in x and y global-directions of the finite element numerical model (FEM) (Figure 3). The non-stationary model described in (Grigoriu et al., 1988;Ciano et al., 2018b;Ciano et al., 2020b) is used to generate samples a x (t) and a y (t) of A x (t) and A y (t), respectively. The probabilistic model used for A(t) is calibrated with actual time histories recorded at the school base during the October 30th, 2016 earthquake. ...
... The probabilistic model used for A(t) is calibrated with actual time histories recorded at the school base during the October 30th, 2016 earthquake. The demand parameter D dy is obtained by a three-step Monte Carlo algorithm: (i) n s ground motion acceleration time series a n (t) of the process A n (t), n = x, y, are generated using the method proposed in (Grigoriu et al., 1988); (ii) linear time domain numerical dynamic structural analyses are used to obtain response samples x(t) of X(t) assuming proportional damping ratio ζ = 5%; (iii) n s samples of the random vector D dy = {D In this case study the maximum absolute displacement in y-direction at the m = 12 node positions shown in Figure 3 are selected as demand parameters. The dependence between the random variables D (j) y , the two IM s in Equations (2) and (3) and their modified version are investigated. ...
Fragility curves are commonly computed to estimate structural system performance. Seismic fragilities are the probability that the structural response of a system overcomes prefixed limit states for given seismic intensity measure. The most widespread procedure to estimate seismic fragility curves is based on scaling seismic accelerograms by a reference intensity measure (e.g. single/multiple ordinates of the pseudo-acceleration response spectrum or peak ground acceleration).
Recently, it was shown that this methodology gives limited if any information on the structural seismic performance when the dependence between the intensity measure and the system demand parameter of interest (e.g. max inter-story displacement) is weak. However, the widespread method is currently applied in Performance-Based Earthquake Engineering because it is simple and it overcomes the problem of the limited number of natural recorded ground motions available for fragility analysis.
In this work a general approach to improve the accuracy in fragilities estimation when the
dependence between the intensity measure and the demand parameter is weak and the widely used method does not give accurate results is presented. The proposed algorithm is based on a linear transformation of samples of a given intensity measure, which improves the correlation with a set of demand parameters. Fragility curves are obtained using the transformed intensity measure samples and compared with those estimated with the standard approach. The effectiveness of the proposed algorithm is demonstrated for a linear elementary oscillator and complex multi-degree of freedom structural system.
... Si los registros sísmicos no son suficientes para caracterizar el peligro sísmico del sitio, se recurre a la simulación de movimientos sísmicos sintéticos a partir de la información disponible del sitio de estudio y lugares cercanos. Para la generación de sismos sintéticos existen distintas técnicas como la que propone Grigoriu et al. (1988) quienes plantean un modelo que divide en segmentos el movimiento del suelo y utilizan procesos unidimensionales con amplitud y frecuencia moduladas. Ordaz et al. (1995) presentan una metodología de estimación de movimientos sísmicos con base en la función de Green. ...
En el presente trabajo se propone un enfoque para estimar la confiabilidad estructural en términos de dos indicadores: 1) la tasa media anual de falla y 2) el factor de confianza. Los indicadores de confiabilidad se estiman con base en un índice, IDC, que normaliza a la capacidad y demanda estructural. Con base en lo anterior, se llega a nuevas expresiones cerradas para estimar los indicadores de confiabilidad. Las expresiones cerradas consideran incertidumbres epistémicas y aleatorias. El enfoque propuesto se compara con la formulación original propuesta por Cornell et al (2002) que considera a la capacidad y demanda como variables no normalizadas. Se estiman tasas de excedencia y factores de confianza para distintos estados limite que recomienda la normativa actual. Los indicadores de confiabilidad se estiman en un puente de concreto reforzado diseñado para desarrollar distorsión de 0.002 y 0.004. Las estructuras se ubican en la Ciudad de México en terreno de transición. En la obtención de la confiabilidad estructural se consideran las incertidumbres mecánicas y geométricas. Con base en los resultados se dan recomendaciones sobre la viabilidad de diseñar puentes para que desarrollen cierta distorsión bajo la tipología estructural mostrada.
... Liu and Hong (2013) incorporated the coherence in the stochastic point source model and finite fault model to simulate ground motions at multiple sites for scenario events. Heredia-Zavoni and Santa-Cruz (2000) used the concept of time transformation (i.e., frequency modulation) to transform stationary process in a timescale τ (i.e., auxiliary timescale) to nonstationary process in the timescale t (Grigoriu, Ruiz, and Rosenblueth 1988;Yeh and Wen 1990) for simulating nonstationary seismic ground motions at multiple sites. However, the simulation was carried out first in the τ-domain adopting a coherence function that is developed in the t-domain. ...
In earthquake engineering, seismic ground motions are most often modelled as a nonstationary Gaussian process. A few studies indicated that seismic ground motions should be treated as a nonstationary non-Gaussian process, by showing that the kurtosis coefficient of the historical ground motion records is much greater than three. These findings and conclusions are queried in the present study, which analyzes a large number of historical ground motion records. Our results indicate that while the mixture marginal distribution of the acceleration of the records is non-Gaussian with a heavy distribution tail, the mixture marginal distribution of the standardized record, defined by the time-varying record to its standard deviation, is only mildly non-Gaussian. We point out that the mixture marginal distribution of a nonstationary Gaussian process may not be Gaussian. The implication of these observations in simulating records is explained. The sampled nonstationary Gaussian/non-Gaussian records are used to compare the responses of single-degree-of-freedom systems. The results indicate that the error introduced by adopting the Gaussian assumption is small, suggesting that the ground motions could be assumed to be a nonstationary Gaussian process with sufficient accuracy, especially if structures that are not very stiff are considered.
... The EQ was one of the most devastating EQs of the nineteenth century with a human death toll of 800, nearly 300 buildings collapsed, and there were number of buildings which were in beyond repair condition. Dynamic characteristics of Mexico City clay were investigated and reported elsewhere (Celebi et al. 1987;Grigoriu et al. 1998). In fact, the seismic behavior of structures may vary in a vast region for a particular EQ due to variation in local subsoil deposit and topography effect which is also known as "Basin effect" (Sharma et al. 2017). ...
Site-specific dynamic soil properties, such as nonlinear stiffness and damping of subsoil, play a crucial role in assessment of vulnerability of structures incorporating seismic soil structure interaction (SSSI). The present study aims at the evaluation of dynamic parameters of subsoil samples collected from a case study railway construction site at Agartala, capital of Tripura, a North Eastern (NE) state of India, which is situated in an active seismic zone. Soil samples extracted from the site were mainly constituted of peat clay and very soft clay at shallow depth followed by silty fine sand layer at moderate to higher depth below existing ground level (EGL). A series of small strain and large strain element tests were conducted using bender element (BE) and cyclic triaxial (CTX) apparatus to obtain the shear modulus and damping ratio. Then, implications of dynamic properties of soil are studied through one-dimensional ground response analysis (1D GRA) using site-specific synthetic ground motions attributing scenario earthquakes. This study provides valuable inputs on importance of consideration of local site effect while performing the seismic design of structures embedded in soft alluvial deposits. Besides, the stiffness contrast of subsoil layers and ground motion are found to be key influential parameters. Finally, a case study analysis on seismic response of railway embankment also validates the importance of local site effect.
... In addition, STFT could provide good time resolution at high frequency or good frequency resolution at low frequency but not both, because it uses fixed window length. Grigoriu et al. (1988) and Yeh and Wen (1990) applied the time transformation to take into account potential time-varying frequency, resulting in the ground-motion record that is characterized by a time-frequency-dependent PSD function, which should not be treated as a time-and frequency-dependent amplitude modulation function in the context of SRM. This was illustrated in Hong (2016) for the amplitude-and frequency-modulated nonstationary process. ...
A probabilistic model of the time–frequency power spectral density (TFPSD) is presented. The model is developed, based on the time–frequency representation of records from strike-slip earthquakes, in which the time–frequency representation is obtained by applying the S-transform (ST). The model for the TFPSD implicitly considers the amplitude modulation and frequency modulation for the nonstationary ground motions; this differs from the commonly used evolutionary PSD model. Predicting models for the model parameters, based on seismic source and site characteristics, are developed. The use of the model to simulate ground motions for scenario seismic events is illustrated, in which the simulation is carried out using a recently developed model that is based on the discrete orthonormal ST and ST. The illustrative example highlights the simplicity of using the proposed model and the physical meaning of some of the model parameters. A model validation analysis is carried out by comparing the statistics of the pseudospectral acceleration obtained from the simulated records to those obtained using a few ground-motion models available in the literature and considered actual records. The comparison indicates the adequacy of the proposed model.
... The ground motions based on such a case can be simulated using the spectral representation method. 12,13 If the ground motions are modeled as the amplitude and frequency modulated evolutionary processes, 14,15 where the frequency modulation is achieved through nonlinear time transformation, it can be shown that the lagged coherence of the nonstationary processes modeled in such a manner 16,17 depends on time. Also, Conte and Peng 18 considered the ground motions at a site can be expressed as the superposition of several amplitude modulated subprocesses; in such a case, the lagged coherence between the ground motions at different sites could also be time-dependent. ...
There are several well‐known empirical lagged spatial coherence models for seismic ground motions proposed in the literature. The models are often developed based on the ordinary Fourier transform. None of the parametric models depend on time and frequency. The present study is focused on the development of the time‐frequency dependent (TF‐dependent) lagged coherence model for the seismic ground motions. The estimation of the TF‐dependent lagged coherence is carried out using the records obtained from dense arrays in Taiwan by applying the S‐transform—a TF‐dependent windowed Fourier transform. The spectral analysis results show that the TF‐dependent lagged coherence decreases with increasing separation or increasing frequency. Most importantly, it is shown that the TF‐dependent lagged coherence varies with the time‐varying intensity within the duration of the records; a higher normalized intensity corresponds to a higher lagged coherence. This feature is included in the developed empirical parametric TF‐dependent lagged coherence model, which is a function of the frequency, the separation between recording sites, and the normalized intensity. A numerical example illustrating its application to simulate nonstationary ground motions at multiple points is presented by using the time‐frequency spectral representation method that was developed based on the S‐transform and discrete orthonormal S‐transform.
In this article, several spectral models describing the stationary stochastic process of earthquake ground motion are explored and compared. The Hu-Zhou spectrum, which is regarded as an improved model of the Kanai-Tajimi spectrum, is concerned. It is proven that the earthquake-induced ground acceleration process described by the Hu-Zhou spectrum is a twice filtered white noise process in essence, and two filters for modifying low-frequency components and moderate- and high-frequency components respectively are investigated. A total of 1946 strong earthquake records at different sites were employed to determine the parameters of spectral models, including the Kanai-Tajimi spectrum, the Clough-Penzien spectrum and the Hu-Zhou spectrum. The results showed that the Hu-Zhou spectrum fits well with the actual observed ground motions over the whole frequency range, and that it is not only distinct in physical meaning and concise in mathematical expression, but also reasonable in practice.
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