Non-stationary Ground Motion Research Articles

Stochastic Dynamic Response Analysis and Probability Evaluation of Subway Station Considering Subjected to Stochastic Earthquake Excitation

As a relatively new means of transportation, the subway has become an important tool for the sustainable development of many cities. Being buried deep in soil under the weight of vital infrastructure, subway stations can be vulnerable to seismic excitations. Considering the high randomness of ground motions, it is important to research the failure probability and seismic performance of the subway station based on stochastic dynamic analysis. In this paper, a probability density evolution method (PDEM) coupled with a spectral representation random function is used to analyze the stochastic dynamic response and seismic probability of a subway station. First, according to the improved power spectral density model and the seismic design code of urban rail transit structures in China (GB 50909-2014), a set of nonstationary ground motions consistent with the code spectrum are obtained. Then, a great deal of deterministic dynamic calculations for Daikai subway station considering soil–structure interaction based on elastic–plastic methods are performed. In addition, the nonlinear stochastic response analysis and the dynamic probability analysis are obtained for the subway station by solving the PDEM equation. Finally, the probability density function (PDF) and cumulative distribution function (CDF) of the subway station under stochastic earthquake excitations are obtained based on three performance indices, including drift angle in the middle column, relative vertical displacement between floor and roof, and damage area ratio (DAR). The results show that the stochastic dynamic analysis and the probability density evolution method can analyze seismic response and evaluate seismic performance of subway stations effectively. The proposed method will serve as an effective tool for the seismic design of underground structures.

Community detection in spatial correlation graphs: Application to non-stationary ground motion modeling

In this paper, we propose a community detection method to find regions in spatial data with higher correlations. We construct a ‘correlation deviation graph’ that considers the influence of global correlation caused by the node's geographical location. We compare the performance of the correlation deviation graph to conventional correlation graph construction approaches for community detection, using synthetic data, and show that the proposed method has the best performance in the presence of a global correlation structure. We also compare the performance of signed spectral clustering and the signed Louvain algorithm for community detection on the correlation graphs, and find that the signed spectral clustering algorithm performs best. We apply our algorithms to simulated earthquake ground motion data in the Los Angeles region. The results suggest that communities of high correlation in ground shaking tend to be associated with common geological conditions and relative location along the rupture strike direction.

Spectral Representation-Based Multidimensional Nonstationary Ground Motion Model for Seismic Reliability Analysis of Frame Structures

A framework for a multidimensional nonstationary ground motion model based on spectral representation theory is proposed in this paper. The multidimensional nonstationary ground motion model is built from a local target to fit the multidimensional response spectrum. A four-stage modulation function takes into account the multidimensional intensity correlation and the modified Clough–Penzien (C-P) power spectrum with parameter correlation, which represent the two main aspects, the modulation function and the power spectrum of constructing the multidimensional nonstationary ground motion model. A multidimensional response spectrum constructed according to the standardizing response spectrum is used as the fitting target response spectrum. Samples of random ground motion for random seismic response and dynamic reliability study are finally obtained. The random seismic responses are then combined with the probability density evolution method (PDEM) to carry out the seismic reliability analysis of a randomly base-excited moment-resisting frame structure. In the numerical analysis, the nonlinear seismic responses and reliability of a 10-story reinforced concrete frame structure are carefully investigated in accordance with the Egyptian seismic code. As a result, the effectiveness of the proposed method is fully demonstrated.

Simulation of non-stationary stochastic ground motions based on recent Italian earthquakes

This work presents an up-to-date model for the simulation of non-stationary ground motions, including several novelties compared to the original study of Sabetta and Pugliese (Bull Seism Soc Am 86:337–352, 1996). The selection of the input motion in the framework of earthquake engineering has become progressively more important with the growing use of nonlinear dynamic analyses. Regardless 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 the generation of ground motion records, we focused on the methods based on stochastic simulations, considering the time- frequency decomposition of the seismic ground motion. We updated the non-stationary stochastic model initially developed in Sabetta and Pugliese (Bull Seism Soc Am 86:337–352, 1996) and later modified by Pousse et al. (Bull Seism Soc Am 96:2103–2117, 2006) and Laurendeau et al. (Nonstationary stochastic simulation of strong ground-motion time histories: application to the Japanese database. 15 WCEE Lisbon, 2012). The model is based on the S-transform that implicitly considers both the amplitude and frequency modulation. The four model parameters required for the simulation are: Arias intensity, significant duration, central frequency, and frequency bandwidth. They were obtained from an empirical ground motion model calibrated using the accelerometric records included in the updated Italian strong-motion database ITACA. The simulated accelerograms show a good match with the ground motion model prediction of several amplitude and frequency measures, such as Arias intensity, peak acceleration, peak velocity, Fourier spectra, and response spectra.

Dynamic reliability assessment of nonlinear structures using extreme value distribution based on L-moments

In this paper, an efficient method using the linear moments (L-moments) is proposed for dynamic reliability assessment of nonlinear structures subjected to non-stationary ground motions. The proposed method comprises three main steps. First, the extrema of structural responses are obtained from the random function-spectral representation model (RFSRM) and nonlinear time history analysis, in which the principal points are first combined with the RFSRM and a sampling strategy is suggested to determine the final samples of extrema by deleting close values. Second, after the samples of extrema are obtained, the first four L-moments can be estimated from their definitions. Finally, using the first four L-moments of extrema of structural responses, the probability distribution of extrema is approximated by a cubic normal distribution based on the L-moments and the corresponding failure probability of a nonlinear structure is calculated. Three numerical examples are presented to demonstrate the efficiency, accuracy, and usefulness of the proposed method for the dynamic reliability assessment of nonlinear structures, and the conventional spectrum representation method and subset simulation method are used for comparison.

A new energy‐compatible nonstationary stochastic ground‐motion simulation method

AbstractA new time‐frequency modulating function with a novel equivalent average‐intensity envelope based on the instantaneous energy is proposed to simulate fully nonstationary ground motions. The new model utilizes simple, uncoupled, and statistically well‐studied temporal and spectral nonstationary parameters. Then, a period‐by‐period simulation method based on the proposed stochastic model is developed to match multiple targets. A numerical comparison shows that the new method replicates well both the target temporal and spectral nonstationarities and it is efficient and easy to use in engineering practice. This study also demonstrates that for obtaining site‐specific seismic input by means of site response analysis: (1) the frequency contents of bedrock motion, including the energy concentration in terms of frequency and frequency decaying with respect to time, have a significant impact on the site response with moderate and large PGA levels; (2) using the proposed instantaneous‐frequency‐related time modulating function to approximate the real ground motion leads to results sufficiently close to the record for nonlinear site response regardless PGA levels and site condition. Therefore, attention should be paid to the change of nonstationary characteristics of ground‐motion frequency according to the source, path, and site condition when simulating ground motion on bedrock. As the proposed methods can approximately control various ground‐motion characters, different targets of ground‐motion simulation can be matched for different engineering purpose such as site response analysis, time‐history analysis of special structures, etc.

A Time–Frequency Representation Model for Seismic Ground Motions

ABSTRACTA 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.

A time‐frequency dependent coherence model for seismic ground motions

AbstractThere 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.

Time-dependent pounding probability analysis between adjacent decks of bridges under non-stationary stochastic seismic excitations

This paper proposes a time-dependent probability method to predict the time-variant pounding risk, owing to this reason that the required separation gap width fails to accommodate the relative movement between two adjacent decks, of bridge systems subjected to non-stationary stochastic seismic excitations. Firstly the pseudo excitation method in random theory is utilized to transform the motion equation of system under the non-stationary stochastic seismic excitations into the transient response equation under deterministic external loads. And then the high precision direct integration method (HPDIM) is also employed to reduce the large number of transient analysis into two times for overcoming the challenges of strenuous computing effort. The time-dependent pounding system probability of bridges is derived by the combination of extreme value distribution theory and matrix-based system probability theory. Finally based on the proposed time-dependent probability method the numerical analysis of a real two span simply-supported beam bridge subjected to spatial and non-stationary ground motions is conducted herein for the pounding risk assessment. The results of this study provides in-depth insight into the time-dependent pounding probability of bridge systems subjected to spatially varying and non-stationary ground motions.

Efficient seismic reliability analysis of non-linear structures under non-stationary ground motions

The seismic reliability assessment (SRA) of complex nonlinear structures which possess structural and seismic uncertainty is an open challenge for academics and practical engineers. This paper proposes an improved maximum entropy method (IMEM) for the SRA of nonlinear structures. Non-stationary ground motions are first generated via a random-function-based spectral representation method. High-dimensional random variables in the ground motion simulation are then converted into two basic random variables as the high-dimensional probability space is reduced to a low-dimensional probability space. A two-step IMEM constrained with fractional moments is deployed to estimate the extreme value distribution (EVD) of the nonlinear structural dynamic system. An improved Latin hypercube sampling approach is also established to calculate the fractional moments of the complex nonlinear dynamic system. The first-passage probability of the structural nonlinear seismic response can be computed conveniently by a numerical integration using the EVD. Two numerical examples, 1) a three-story nonlinear shear frame structure and 2) a real-world high-pier continuous rigid frame bridge model, both of which exhibit strong nonlinearity under seismic action, are analyzed to show that the proposed method is efficient and accurate.

A novel framework for conditional simulation of fully nonstationary spatially varying ground motion field

SummaryExisting procedures for conditionally generating the spatially varying ground motion (SVGM) through the coherency‐based methods can be broadly categorized into two approaches: (i) cross‐spectral density (CSD)‐based procedures and (ii) auto‐spectral density (ASD)‐based procedures (reported by the authors elsewhere). CSD‐based procedures consider only the phase variability, whereas ASD‐based procedures account for phase as well as the amplitude variability. However, the ASD‐based framework and most of the CSD‐based frameworks simulate the stationary SVGM, and nonstationarity is imparted through the uniform modulation, ignoring the temporal variation of the frequency content. This paper proposes an evolutionary power spectral density (EPSD)‐based framework for the simulation of nonstationary SVGM. This framework is developed in two steps. First, a new methodology based on Hilbert transform is proposed to compute the nonstationary ground motion given the EPSD, modulating functions, and phase. Second, the ASD‐based framework reported by authors elsewhere is extended to account for the EPSD, and the associated nonstationary ground motion is simulated using the proposed methodology. Nonstationary ground motions thus simulated are assessed against SMART1 array recordings. The close resemblance between the simulated accelerogram and the recorded data indicates that the proposed EPSD based framework accounts not only for the phase and amplitude variabilities but also for the nonstationarity.

The smooth transition GARCH model for simulation of highly nonstationary earthquake ground motions

The aim of this paper is to extend a stochastic model for simulation of highly nonstationary ground motions (GMs). In this model, the dual-tree complex discrete wavelet transform (DT-CDWT) is applied to a recorded ground motion to extract its wavelet coefficients and then the Smooth Transition Generalized Autoregressive Conditional Heteroscedastic (ST-GARCH) model is used to simulate these coefficients. This model simulates nonstationary features of real GMs greatly, because the conditional variance estimated by this model changes compatibly with the amplitude nonstationary features of real GMs. Because of having asymmetric structure, this model also estimates sudden changes in the amplitude of ground motions. Some of the special capabilities of this model are the simulation of multiple increasing–decreasing cycles in the temporal amplitude of GMs, prediction of several peaks in the frequency spectrum of GMs, estimation of steps of their energy curve, and simultaneous simulation of all shocks of a GM sequence. Also, this model simulates the time–frequency energy distribution of both wide-frequency and narrow-frequency bandwidth GMs.

Analysis and modeling of ground motion coherency at uniform site conditions

In this work, nonstationary ground motion coherency analysis and modeling are performed using wavelet analysis and relevance vector machine regression. Earthquake ground motion data from four events recorded at dense seismograph SMART-1 array in north-south and east-west horizontal directions are used to investigate the lagged coherency behavior. Wavelet transform is used to compute the nonstationary lagged coherency that characterizes the space-time variation of seismic ground motion. Lagged coherency behavior with frequency, distance, and time is examined. It is shown that the lagged coherency is a time-varying quantity. The results implied that the lagged coherency on uniform soil is relatively insensitive to the earthquake events, and depends mainly on time, frequency, and separation distance. Relevance Vector Machines (RVM) regression is used to develop a model for nonstationary lagged coherency that characterizes the space-time variation of seismic ground motion. Earthquake ground motion in the in the north-south and the first principal directions is used to determine the lagged coherency model. The proposed model does not require a fixed parametric functional form. The developed model can estimate the lagged coherency for different separation distances at different time instants and frequencies.

Use of Discrete Orthonormal S-Transform to Simulate Earthquake Ground Motions

ABSTRACTThe S-transform and discrete orthonormal S-transform (DOST) produce time–frequency representations, which is in contrast to the wavelet transformations. Similar in the Fourier transform, the use of the S-transform and DOST provides frequency-dependent resolution with absolutely referenced phase information. Although the decomposed signal using DOST is expressed as a sum of orthonormal basis function, this is not the case if the S-transform is used. In the present study, a procedure to simulate nonstationary ground motions based on DOST is proposed based on a seed record or given a target amplitude of DOST coefficients. It is shown that the model has zero mean, and its variance equals the assigned target. Using five real records, each from a larger earthquake, the application of the DOST and S-transform to the records is carried out. Although the time–frequency resolution obtained from DOST is coarse as compared to that obtained using the S-transform, its use identifies clearly time–frequency characteristics. Samples of ground motions are simulated using the proposed method based on the amplitude of the DOST coefficients of a seed record or on the average amplitude of the DOST coefficients of a set of actual records. The comparison of the time–frequency resolution, Fourier spectrum, time-varying power distribution, and response spectrum of the simulated and seed records indicates that the proposed simulation model is a useful and practical tool to simulate nonstationary ground motions.

Stochastic assessment of slope failure run-out triggered by earthquake ground motion

Analysis of the run-out of landslides is essential and vital for disaster mitigation. However, accurate run-out analysis is difficult because of the uncertainty of earthquake ground motion and variability of soil properties. To solve this problem, a new run-out assessment framework that combines the methods of probability density evolution and smoothed particle hydrodynamics is proposed. This novel framework can consider multiple stochastic factors and different slope failure models of changing sliding surfaces. We used a homogeneous 2D slope as an example and generated stochastic seismic loading samples with an intensity-frequency non-stationary ground motion model. Soil property parameters (cohesion and internal friction angle) were assumed to obey logarithmic normal distribution, and run-out parameters were evolved. Moreover, based on an equivalent extreme event, the distributions of final run-out parameters were obtained. In an example with slope height of 100 m and angle of 45°, the probability that the run-out distance is < 150 m is 90%. The probability of flow depth more than 24.5 m was about 50%. The new framework has the potential to provide references for coordinating relief efforts after landslides by predicting the extent and scope of the earthquake-induced hazard; thereby minimizing casualties and property losses caused by geological disasters triggered by earthquakes.

A Statistical Model for Phase Difference Spectrum of Ground-Motion and Its Application in Generating Non-Stationary Seismic Waves

The intensity non-stationarity is one of the most important features of earthquake records. Modeling of this feature is significant to the generation of arti- ficial earthquake waves. Based on the theory of phase difference spectrum, an intensity non-stationary envelope function with log-normal form is proposed. Through a tremendous amount of earthquake records downloaded on Kik-net, a parameter fitting procedure using the genetic algorithm is conducted to obtain the value of model parameters under different magnitudes, epicenter distances and site conditions. A numerical example is presented to describe the procedure of generating fully non-stationary ground motions via spectral representation, and the mean EPSD (evolutionary power spectral density) of simulated waves is proved to agree well with the target EPSD. The results show that the proposed model is capable of describing the intensity non-stationary features of ground motions, and it can be used in structural anti-seismic analysis and ground motion simulation.

Conditional Simulation of Spatially Varying Multicomponent Nonstationary Ground Motions: Bias and Ill Condition

AbstractThe application of the spectral representation method to the conditional simulation of ground motions can break down because of nonconvergence in inverting the conditioning covariance matri...

Gaussian mixture–based equivalent linearization method (GM‐ELM) for fragility analysis of structures under nonstationary excitations

SummaryGaussian mixture–based equivalent linearization method (GM‐ELM) is a recently developed stochastic dynamic analysis approach which approximates the random response of a nonlinear structure by collective responses of equivalent linear oscillators. The Gaussian mixture model is employed to achieve an equivalence in terms of the probability density function (PDF) through the superposition of the response PDFs of the equivalent linear system. This new concept of linearization helps achieve a high level of estimation accuracy for nonlinear responses, but has revealed some limitations: (1) dependency of the equivalent linear systems on ground motion intensity and (2) requirements for stationary condition. To overcome these technical challenges and promote applications of GM‐ELM to earthquake engineering practice, an efficient GM‐ELM‐based fragility analysis method is proposed for nonstationary excitations. To this end, this paper develops the concept of universal equivalent linear system that can estimate the stochastic responses for a range of seismic intensities through an intensity‐augmented version of GM‐ELM. Moreover, the GM‐ELM framework is extended to identify equivalent linear oscillators that could capture the temporal average behavior of nonstationary responses. The proposed extensions generalize expressions and philosophies of the existing response combination formulations of GM‐ELM to facilitate efficient fragility analysis for nonstationary excitations. The proposed methods are demonstrated by numerical examples using realistic ground motions, including design code–conforming nonstationary ground motions.

Probability density evolution analysis of stochastic nonlinear structure under non-stationary ground motions

In the present paper, a framework of dimension-reduction modeling method is developed for a dual stochastic dynamic structural system of spectrum-compatible non-stationary stochastic ground motion processes and stochastic structures. With the aid of the proposed method, the random variables used to describe the stochastic characteristics of the non-stationary ground motion processes and the structural parameters are respectively represented by the one-elementary-random-variable functions, contributing to an entire stochastic dynamic structural system readily described by merely two elementary random variables. Owing to the fact that the number of the elementary random variables needed is extremely small, the set of the representative points associated with the elementary random variables can thus be selected by the widely-used number theoretical method, and then the probability density evolution method can be conveniently combined to conduct the dynamic response analysis and dynamic reliability evaluation of nonlinear stochastic structures. In the numerical examples, the probability density evolution analysis of an eight-storey reinforced concrete frame structure with random parameters subjected to spectrum-compatible non-stationary stochastic ground motion processes is investigated as a case study. Numerical results fully demonstrated the effectiveness and robustness of the proposed method.

Stochastic dynamic response analysis and reliability assessment of non-linear structures under fully non-stationary ground motions

In this paper, stochastic dynamic response analysis and reliability assessment of nonlinear structures under fully non-stationary ground motions is investigated based on the probability density evolution method (PDEM). To handle this problem, a new method is proposed for points selection in high-dimensional random-variate space for PDEM, which is of paramount importance to the tradeoff of accuracy and efficiency. First, the original high-dimensional space is decomposed into several two-dimensional orthogonal subspaces. Then, each subspace is partitioned via the Voronoi cells and the representative points and assigned probabilities in each subspace are specified accordingly. Finally, the representative points and assigned probabilities in each subspace are randomly paired to formulate the representative points and assigned probabilities in high-dimension. A numerical example is studied to validate the proposed method, which indicates that the proposed method is of efficiency and accuracy for high-dimensional stochastic dynamic problem of structures. The problem, which needs to be further investigated, is also pointed out.