# Spatial variation of seismic ground motions: An overview

01 May 2002-Applied Mechanics Reviews (American Society of Mechanical Engineers)-Vol. 55, Iss: 3, pp 271-297

TL;DR: In this paper, the authors address the topic of the spatial variation of seismic ground motions as evaluated from data recorded at dense instrument arrays, focusing on spatial coherency and its interpretation.

Abstract: This study addresses the topic of the spatial variation of seismic ground motions as evaluated from data recorded at dense instrument arrays. It concentrates on the stochastic description of the spatial variation, and focuses on spatial coherency. The estimation of coherency from recorded data and its interpretation are presented. Some empirical and semi-empirical coherency models are described, and their validity and limitations in terms of physical causes discussed. An alternative approach that views the spatial variation of seismic motions as deviations in amplitudes and phases of the recorded data around a coherent approximation of the seismic motions is described. Simulation techniques for the generation of artificial spatially variable seismic ground motions are also presented and compared. The effect of coherency on the seismic response of extended structures is highlighted. This review article includes 133 references. @DOI: 10.1115/1.1458013#

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TL;DR: In this paper, the authors used ground motions observed during seven past earthquakes to estimate correlations between spatially distributed spectral accelerations at various spectral periods, which is required for the joint prediction of ground-motion intensities at multiple sites.

Abstract: Risk assessment of spatially distributed building portfolios or infrastructure systems requires quantification of the joint occurrence of ground-motion intensities at several sites, during the same earthquake. The ground-motion models that are used for site-specific hazard analysis do not provide information on the spatial correlation between ground-motion intensities, which is required for the joint prediction of intensities at multiple sites. Moreover, researchers who have previously computed these correlations using observed ground-motion recordings differ in their estimates of spatial correlation. In this paper, ground motions observed during seven past earthquakes are used to estimate correlations between spatially distributed spectral accelerations at various spectral periods. Geostatistical tools are used to quantify and express the observed correlations in a standard format. The estimated correlation model is also compared with previously published results, and apparent discrepancies among the previous results are explained. The analysis shows that the spatial correlation reduces with increasing separation between the sites of interest. The rate of decay of correlation typically decreases with increasing spectral acceleration period. At periods longer than 2 s, the correlations were similar for all the earthquake ground motions considered. At shorter periods, however, the correlations were found to be related to the local-site conditions (as indicated by site V s 30 values) at the ground-motion recording stations. The research work also investigates the assumption of isotropy used in developing the spatial correlation models. It is seen using the Northridge and Chi-Chi earthquake time histories that the isotropy assumption is reasonable at both long and short periods. Based on the factors identified as influencing the spatial correlation, a model is developed that can be used to select appropriate correlation estimates for use in practical risk assessment problems.

372 citations

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TL;DR: In this article, an approximate method to model and simulate spatially varying ground motions on the surface of an uneven site with non-uniform conditions at different locations in two steps is presented.

153 citations

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01 Jan 2014TL;DR: In this paper, the authors present a short but comprehensive review of the available seismic design methods, denoting the crucial issues and the problems that an engineer could face during the seismic analysis, and present the most recent developments on the evaluation of adequate fragility curves for shallow tunnels.

Abstract: Underground structures, tunnels, subways, metro stations and parking lots, are crucial components of the build environment and transportation networks. Considering their importance for life save and economy, appropriate seismic design is of prior significance. Their seismic performance during past earthquakes is generally better than aboveground structures. However several cases of severe damage to total collapse have been reported in the literature, with that of the Daikai metro station in Kobe during the Hyogoken-Nambu earthquake (1995) being one of the most characteristic. These recent damages revealed some important weaknesses in the current seismic design practices. The aim of this chapter is not to make another general presentation of the methods used for the seismic design of underground structures, but rather to discuss and highlight the most important needs for an improved seismic performance and design. In that respect it is important to consider that the specific geometric and conceptual features of underground structures make their seismic behavior and performance very distinct from the behavior of aboveground structures, as they are subjected to strong seismic ground deformations and distortions, rather than inertial loads. Several methods are available, from simplified analytical elastic solutions, to sophisticated and in principal more accurate, full dynamic numerical models. Most of them have noticeable weaknesses on the description of the physical phenomenon, the design assumptions and principles and the evaluation of the parameters they need. The chapter presents a short but comprehensive review of the available design methods, denoting the crucial issues and the problems that an engineer could face during the seismic analysis. The main issues discussed herein cover the following topics: (i) force based design against displacement based design, (ii) deformation modes of rectangular underground structures under seismic excitation, (iii) seismic earth pressures on underground structures, (iv) seismic shear stresses distribution on the perimeter of the structure, (v) appropriateness of the presently used impedance functions to model the inertial and the kinematic soil-structure interaction effects, (vi) design seismic input motion, accounting of the incoherence effects and the spatial variation of the motion and (vii) effect of the build environment (i.e. city-effects) on the seismic response of underground structures. The discussion is based on detailed numerical analysis of specific cases and recent experimental results in centrifuge tests. Other important issues like the design of submerged tunnels to liquefaction risk, or the complexity to evaluate the response of the joints of submerged tunnels are also shortly addressed. Finally we present the most recent developments on the evaluation of adequate fragility curves for shallow tunnels.

133 citations

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TL;DR: In this article, a semi-empirical estimation of the correlation coefficient, as a function of intersite separation distance, between residuals with respect to ground motion prediction equations (GMPEs) of horizontal peak ground acceleration (PGA) and peak ground velocity (PGV).

Abstract: Spatial modeling of ground motion intensity measures (IMs) is required for risk assessment of spatially distributed engineering systems. For example, when a lifeline system is of concern, classical site-specific hazard tools, which treat IMs at different locations independently, may not be adequate to accurately assess the seismic risk. In fact, in this case, modeling of ground motion as a random field is required; it basically consists of assigning a correlation structure to the IM of interest. This work focuses on semiempirical estimation of the correlation coefficient, as a function of intersite separation distance, between residuals with respect to ground motion prediction equations (GMPEs) of horizontal peak ground acceleration (PGA) and peak ground velocity (PGV). In particular, subsets of the European Strong-Motion Database (ESD) and the Italian Accelerometric Archive (ITACA) were employed to evaluate the intraevent residual correlation based on multiple earthquakes, considering different GMPEs fitted to the same records. The analyses were carried out through geostatistical tools, which enabled results to be found that are generally consistent between the two datasets. Correlation for PGV appears to attenuate more gradually with respect to PGA. In order to better understand the dependency of the results on the adopted estimation approach and dataset, some aspects related to the working hypotheses are critically discussed. Finally, estimated correlation models are used to develop illustrative applications of regional probabilistic seismic-hazard analysis.

112 citations

### Cites background or result from "Spatial variation of seismic ground..."

...This seems to be consistent with past studies of ground-motion coherency (Zerva and Zervas, 2002)....

[...]

...…coherency of ground-motion signals, which represents the similarity of ground motion in the frequency domain and describes the degree of positive or negative correlation between amplitudes and phase angles of two time histories at each of their component frequencies (e.g., Zerva and Zervas, 2002)....

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TL;DR: In this article, the seismic performance of a sea-crossing cable-stayed bridge is comprehensively evaluated based on the fragility function methodology, and the effect of seawater on the bridge seismic responses is modeled using the hydrodynamic added mass method.

95 citations

##### References

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TL;DR: In this paper, the authors used the representations of the noise currents given in Section 2.8 to derive some statistical properties of I(t) and its zeros and maxima.

Abstract: In this section we use the representations of the noise currents given in section 2.8 to derive some statistical properties of I(t). The first six sections are concerned with the probability distribution of I(t) and of its zeros and maxima. Sections 3.7 and 3.8 are concerned with the statistical properties of the envelope of I(t). Fluctuations of integrals involving I2(t) are discussed in section 3.9. The probability distribution of a sine wave plus a noise current is given in 3.10 and in 3.11 an alternative method of deriving the results of Part III is mentioned. Prof. Uhlenbeck has pointed out that much of the material in this Part is closely connected with the theory of Markoff processes. Also S. Chandrasekhar has written a review of a class of physical problems which is related, in a general way, to the present subject.22

5,806 citations

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01 Aug 1969TL;DR: In this article, a high-resolution frequency-wavenumber power spectral density estimation method was proposed, which employs a wavenumber window whose shape changes and is a function of the wave height at which an estimate is obtained.

Abstract: The output of an array of sansors is considered to be a homogeneous random field. In this case there is a spectral representation for this field, similar to that for stationary random processes, which consists of a superposition of traveling waves. The frequency-wavenumber power spectral density provides the mean-square value for the amplitudes of these waves and is of considerable importance in the analysis of propagating waves by means of an array of sensors. The conventional method of frequency-wavenumber power spectral density estimation uses a fixed-wavenumber window and its resolution is determined essentially by the beam pattern of the array of sensors. A high-resolution method of estimation is introduced which employs a wavenumber window whose shape changes and is a function of the wavenumber at which an estimate is obtained. It is shown that the wavenumber resolution of this method is considerably better than that of the conventional method. Application of these results is given to seismic data obtained from the large aperture seismic array located in eastern Montana. In addition, the application of the high-resolution method to other areas, such as radar, sonar, and radio astronomy, is indicated.

5,415 citations

01 Jan 1975

TL;DR: In this article, a single-degree-of-freedom (SDF) dynamic system is considered, and the effect of different degrees of freedom on the dynamics of the system is investigated.

Abstract: TABLE OF CONTENTS PREFACE 1 INTRODUCTION 1.1 Objectives of the Study of Structural Dynamics 1.2 Importance of Vibration Analysis 1.3 Nature of Exciting Forces 1.4 Mathematical Modeling of Dynamic Systems 1.5 Systems of Units 1.6 Organization of the Text PART I 2 FORMULATION OF THE EQUATIONS OF MOTION: SINGLE-DEGREE-OF-FREEDOM SYSTEMS 2.1 Introduction 2.2 Inertia Forces 2.3 Resultants of Inertia Forces on a Rigid Body 2.4 Spring Forces 2.5 Damping Forces 2.6 Principle of Virtual Displacement 2.7 Formulation of the Equations of Motion 2.8 Modeling of Multi Degree-of-Freedom Discrete Parameter System 2.9 Effect of Gravity Load 2.10 Axial Force Effect 2.11 Effect of Support Motion 3 FORMULATION OF THE EQUATIONS OF MOTION: MULTI-DEGREE-OF-FREEDOM SYSTEMS 3.1 Introduction 3.2 Principal Forces in Multi Degree-of-freedom Dynamic System 3.3 Formulation of the Equations of Motion 3.4 Transformation of Coordinates 3.5 Static Condensation of Stiffness matrix 3.6 Application of Ritz Method to Discrete Systems 4 PRINCIPLES OF ANALYTICAL MECHANICS 4.1 Introduction 4.2 Generalized coordinates 4.3 Constraints 4.4 Virtual Work 4.5 Generalized Forces 4.6 Conservative Forces and Potential Energy 4.7 Work Function 4.8 Lagrangian Multipliers 4.9 Virtual Work Equation For Dynamical Systems 4.10 Hamilton's Equation 4.11 Lagrange's Equation 4.12 Constraint Conditions and Lagrangian Multipliers 4.13 Lagrange's Equations for Discrete Multi-Degree-of-Freedom Systems 4.14 Rayleigh's Dissipation Function PART II 5 FREE VIBRATION RESPONSE: SINGLE-DEGREE-OF-FREEDOM SYSTEM 5.1 Introduction 5.2 Undamped Free Vibration 5.3 Free Vibrations with Viscous Damping 5.4 Damped Free vibration with Hysteretic Damping 5.5 Damped Free vibration with Coulomb Damping 6 FORCED HARMONIC VIBRATIONS: SINGLE-DEGREE-OF-FREEDOM SYSTEM 6.1 Introduction 6.2 Procedures for the Solution of Forced Vibration Equation 6.3 Undamped Harmonic Vibration 6.4 Resonant Response of an Undamped System 6.5 Damped Harmonic Vibration 6.6 Complex Frequency Response 6.7 Resonant Response of a Damped System 6.8 Rotating Unbalanced Force 6.9 Transmitted Motion due to Support Movement 6.10 Transmissibility and Vibration Isolation 6.11 Vibration Measuring Instruments 6.12 Energy Dissipated in Viscous Damping 6.13 Hysteretic Damping 6.14 Complex Stiffness 6.15 Coulomb Damping 6.16 Measurement of Damping 7 RESPONSE TO GENERAL DYNAMIC LOADING AND TRANSIENT RESPONSE 7.1 Introduction 7.2 Response to an Impulsive force 7.3 Response to General Dynamic Loading 7.4 Response to a Step Function Load 7.5 Response to a Ramp Function Load 7.6 Response to a Step Function Load With Rise Time 7.7 Response to Shock Loading 7.8 Response to a Ground Motion Pulse 7.9 Analysis of Response by the Phase Plane Diagram 8 ANALYSIS OF SINGLE-DEGREE-OF-FREEDOM SYSTEMS: APPROXIMATE AND NUMERICAL METHODS 8.1 Introduction 8.2 Conservation of Energy 8.3 Application of Rayleigh Method to Multi Degree of Freedom Systems 8.4 Improved Rayleigh Method 8.5 Selection of an Appropriate Vibration Shape 8.6 Systems with Distributed Mass and Stiffness: Analysis of Internal Forces 8.7 Numerical Evaluation of Duhamel's Integral 8.8 Direct Integration of the Equations of Motion 8.9 Integration Based on Piece-wise Linear Representation of the Excitation 8.10 Derivation of General Formulae 8.11 Constant Acceleration Method 8.12 Newmark's beta Method 8.13 Wilson-theta Method 8.14 Methods Based on Difference Expressions 8.15 Errors involved in Numerical Integration 8.16 Stability of the Integration Method 8.17 Selection of a Numerical Integration Method 8.18 Selection of Time Step 9 ANALYSIS OF RESPONSE IN THE FREQUENCY DOMAIN 9.1 Transform Methods of Analysis 9.2 Fourier Series Representation of a Periodic Function 9.3 Response to a Periodically Applied Load 9.4 Exponential Form of Fourier Series 9.5 Complex Frequency Response Function 9.6 Fourier Integral Representation of a Nonperiodic Load 9.7 Response to a Nonperiodic Load 9.8 Convolution Integral and Convolution Theorem 9.9 Discrete Fourier Transform 9.10 Discrete Convolution and Discrete Convolution Theorem 9.11 Comparison of Continuous and Discrete Fourier Transforms 9.12 Application of Discrete Inverse Transform 9.13 Comparison Between Continuous and Discrete Convolution 9.14 Discrete Convolution of an Infnite and a Finite duration Waveform 9.15 Corrective Response Superposition Methods 9.16 Exponential Window Method 9.17 The Fast Fourier Transform 9.18 Theoretical Background to Fast Fourier Transform 9.19 Computing Speed of FFT Convolution 9.16 Exponential Window Method 9.17 The Fast Fourier Transform 9.18 Theoretical Background to Fast Fourier Transform 9.19 Computing Speed of FFT Convolution PART III 10 FREE VIBRATION RESPONSE: MULTI-DEGREE-OF-FREEDOM SYSTEM 10.1 Introduction 10.2 Standard Eigenvalue Problem 10.3 Linearized Eigenvalue Problem and its Properties 10.4 Expansion Theorem 10.5 Rayleigh Quotient 10.6 Solution of the Undamped Free-Vibration Problem 10.7 Mode Superposition Analysis of Free-Vibration Response 10.8 Solution of the Damped Free-Vibration Problem 10.9 Additional Orthogonality Conditions 10.10 Damping Orthogonality 11 NUMERICAL SOLUTION OF THE EIGENPROBLEM 11.1 Introduction 11.2 Properties of Standard Eigenvalues and Eigenvectors 11.3 Transformation of a Linearized Eigenvalue Problem to the Standard Form 11.4 Transformation Methods 11.5 Iteration Methods 11.6 Determinant Search Method 11.7 Numerical Solution of Complex Eigenvalue Problem 11.8 Semi-definite or Unrestrained Systems 11.9 Selection of a Method for the Determination of Eigenvalues 12 FORCED DYNAMIC RESPONSE: MULTI-DEGREE-OF-FREEDOM SYSTEMS 12.1 Introduction 12.2 Normal Coordinate Transformation 12.3 Summary of Mode Superposition Method 12.4 Complex Frequency Response 12.5 Vibration Absorbers 12.6 Effect of Support Excitation 12.7 Forced Vibration of Unrestrained System 13 ANALYSIS OF MULTI-DEGREE-OF-FREEDOM SYSTEMS: APPROXIMATE AND NUMERICAL METHODS 13.1 Introduction 13.2 Rayleigh-Ritz Method 13.3 Application of Ritz Method to Forced Vibration Response 13.4 Direct Integration of the Equations of Motion 13.5 Analysis in the Frequency Domain PART IV 14 FORMULATION OF THE EQUATIONS OF MOTION: CONTINUOUS SYSTEMS 14.1 Introduction 14.2 Transverse Vibrations of a Beam 14.3 Transverse Vibrations of a Beam: Variational Formulation 14.4 Effect of Damping Resistance on Transverse Vibrations of a Beam 14.5 Effect of Shear Deformation and Rotatory Inertia on the Flexural Vibrations of a Beam 14.6 Axial Vibrations of a Bar 14.7 Torsional Vibrations of a Bar 14.8 Transverse Vibrations of a String 14.9 Transverse Vibration of a Shear Beam 14.10 Transverse Vibrations of a Beam Excited by Support Motion 14.11 Effect of Axial Force on Transverse Vibrations of a Beam 15 CONTINUOUS SYSTEMS: FREE VIBRATION RESPONSE 15.1 Introduction 15.2 Eigenvalue Problem for the Transverse Vibrations of a Beam 15.3 General Eigenvalue Problem for a Continuous System 15.4 Expansion Theorem 15.5 Frequencies and Mode Shapes for Lateral Vibrations of a Beam 15.6 Effect of Shear Deformation and Rotatory Inertia on the Frequencies of Flexural Vibrations 15.7 Frequencies and Mode Shapes for the Axial Vibrations of a Bar 15.8 Frequencies and Mode Shapes for the Transverse Vibration of a String 15.9 Boundary Conditions Containing the 15.10 Free-Vibration Response of a Continuous System 15.11 Undamped Free Transverse Vibrations of a Beam 15.12 Damped Free Transverse Vibrations of a Beam 16 CONTINUOUS SYSTEMS: FORCED-VIBRATION RESPONSE 16.1 Introduction 16.2 Normal Coordinate Transformation: General Case of an Undamped System 16.3 Forced Lateral Vibration of a Beam 16.4 Transverse Vibrations of a Beam Under Traveling Load 16.5 Forced Axial Vibrations of a Uniform Bar 16.6 Normal Coordinate Transformation, Damped Case 17 WAVE PROPAGATION ANALYSIS 17.1 Introduction 17.2 The Phenomenon of Wave Propagation 17.3 Harmonic Waves 17.4 One Dimensional Wave Equation and its Solution 17.5 Propagation of Waves in Systems of Finite Extent 17.6 Reection and Refraction of Waves at a Discontinuity in the System Properties 17.7 Characteristics of the Wave Equation 17.8 Wave Dispersion PART V 18 FINITE ELEMENT METHOD 18.1 Introduction 18.2 Formulation of the Finite Element Equations 18.3 Selection of Shape Functions 18.4 Advantages of the Finite Element Method 18.5 Element Shapes 18.6 One-dimensional Bar Element 18.7 Flexural Vibrations of a Beam 18.8 Stress-strain Relationship for a Continuum 18.9 Triangular Element in Plane Stress and Plane Strain 18.10 Natural Coordinates 19 COMPONENT MODE SYNTHESIS 19.1 Introduction 19.2 Fixed Interface Methods 19.3 Free Interface Method 19.4 Hybrid Method 20 ANALYSIS OF NONLINEAR RESPONSE 20.1 Introduction 20.2 Single-degree-of-freedom System 20.3 Errors involved in Numerical Integration of Nonlinear Systems 20.4 Multiple Degree-of-freedom System ANSWERS TO SELECTED PROBLEMS INDEX

5,044 citations

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01 Jan 1968

TL;DR: In this paper, Spectral Analysis and its Applications, the authors present a set of applications of spectral analysis and its application in the field of spectroscopy, including the following:

Abstract: (1970). Spectral Analysis and its Applications. Technometrics: Vol. 12, No. 1, pp. 174-175.

4,220 citations

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TL;DR: The purpose of this book is to bring together existing and new methodologies of random field theory and indicate how they can be applied to these diverse areas where a "deterministic treatment is inefficient and conventional statistics insufficient."

1,639 citations