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Journal Article•DOI•

A Computationally Efficient Ground-Motion Selection Algorithm for Matching a Target Response Spectrum Mean and Variance

Nirmal Jayaram1, Ting Lin1, Jack W. Baker1•
Stanford University1
01 Aug 2011-Earthquake Spectra (EARTHQUAKE SPECTRA)-Vol. 27, Iss: 3, pp 797-815
TL;DR: In this article, the authors proposed a computationally efficient and theoretically consistent algorithm to select ground motions that match the target response spectrum mean and variance, which is used to estimate structural response estimates.
Abstract: Dynamic structural analysis often requires the selection of input ground motions with a target mean response spectrum. The variance of the target response spectrum is usually ignored or accounted for in an ad hoc manner, which can bias the structural response estimates. This manuscript proposes a computationally efficient and theoretically consistent algorithm to select ground motions that match the target response spectrum mean and variance. The selection algorithm probabilistically generates multiple response spectra from a target distribution, and then selects recorded ground motions whose response spectra individually match the simulated response spectra. A greedy optimization technique further improves the match between the target and the sample means and variances. The proposed algorithm is used to select ground motions for the analysis of sample structures in order to assess the impact of considering groundmotion variance on the structural response estimates. The implications for codebased design and performance-based earthquake engineering are discussed. [DOI: 10.1193/1.3608002]

Summary (4 min read)

Jump to: [INTRODUCTION] – [GROUND-MOTION SELECTION ALGORITHM] – [ILLUSTRATIVE GROUND-MOTION SELECTION] – [PARAMETERIZATION OF THE TARGET RESPONSE SPECTRUM DISTRIBUTION] – [RESPONSE SPECTRUM SIMULATION] – [SELECTION OF GROUND MOTIONS TO MATCH SIMULATED SPECTRA] – [GREEDY OPTIMIZATION TECHNIQUE] – [SELECTION OF A SMALLER NUMBER OF GROUND MOTIONS] – [GROUND-MOTION SELECTION] – [STRUCTURAL RESPONSE] – [Description of structural systems] – [Response of SDOF systems] – [Response of MDOF systems] – [IMPLICATIONS] and [CONCLUSIONS]

INTRODUCTION

  • Dynamic structural analysis is commonly used in performance-based earthquake engineering to predict the response of a structure subjected to earthquake ground motions.
  • One commonly used approach is to select recorded or simulated ground motions whose response spectra match a target mean response spectrum (e.g., Beyer and Bommer, 2007; Shantz, 2006; WatsonLamprey and Abrahamson, 2006).
  • No further variance need be applied to the UHS, as varying the spectral values is equivalent to varying the associated exceedance rate of the spectral accelerations from period to period (an operation which is unlikely to produce meaningful results).
  • The selection algorithm first uses Monte Carlo simulation to probabilistically generate multiple response spectra from a distribution parameterized by the target means and variances.
  • The proposed algorithm is then used to select ground motions for estimating the seismic response of sample single-degree-of-freedom (SDOF) and multiple-degree-of-freedom (MDOF) structures, in order to assess the impact of considering ground-motion variance on Jayaram – 4 the structural response estimates.

GROUND-MOTION SELECTION ALGORITHM

  • The objective of the proposed algorithm is to select a suite of ground motions whose response spectra have a specified mean and variance.
  • The first step in this algorithm is to parameterize the multivariate normal distribution of lnSa’s at multiple periods.
  • The mean and the variance of the simulated response spectra will approximately match the corresponding target values because they were sampled from the desired distribution.
  • Therefore, a ‘greedy’ optimization technique is used to further improve the match between the sample and the target means and variances.
  • Each ground motion selected previously is replaced one at a time with a ground motion from the database that causes the best improvement in the match between the target and the sample means and variances.

ILLUSTRATIVE GROUND-MOTION SELECTION

  • This section describes the application of the proposed algorithm for selecting structurespecific ground motions that have a specified spectral acceleration at the structure’s fundamental period.
  • The target response spectrum mean and covariance matrices are obtained using the conditional mean spectrum (CMS) method (Baker, 2010), which provides the mean and variance (and correlations) of the response spectrum conditioned on the specified spectral acceleration.
  • It is to be noted that while this example uses the targets from the CMS method, the proposed algorithm can be used with any arbitrary target mean and covariance (e.g., Jayaram and Baker, 2010).

PARAMETERIZATION OF THE TARGET RESPONSE SPECTRUM DISTRIBUTION

  • As described in the previous section, the first step in the algorithm is to parameterize the multivariate normal distribution of the lnSa’s using the means and the variances of the spectral accelerations (chosen to equal the target mean and the target variance respectively) Jayaram – 8 and the correlations between the spectral accelerations at two different periods.
  • The steps involved in parameterizing the distribution using the CMS method are listed below.
  • Tj of interest, compute the unconditional mean and the unconditional standard deviation of the response spectrum, given M and R. (Since lnSa’s at multiple periods follow a Jayaram – 10 multivariate normal distribution, the exponential of the mean lnSa equals the median spectral acceleration.

RESPONSE SPECTRUM SIMULATION

  • Forty response spectra are simulated (using Monte Carlo simulation) by sampling from a multivariate normal distribution with the mean and covariance matrices defined by Equations 6 and 9 for the target scenario described above.
  • The response spectra are simulated at 20 periods logarithmically spaced between 0.05s and 10.0s, and are shown in Figure 2a.
  • A large period range is used to ensure a good match in the entire response spectrum that covers regions of higher modes and nonlinearity.
  • It can be seen that the mean values agree reasonably well.
  • Figure 1b shows a reasonable agreement between the standard deviation of the simulated lnSa values and the target standard deviation.

SELECTION OF GROUND MOTIONS TO MATCH SIMULATED SPECTRA

  • Forty ground motions are selected from the Next Generation Attenuation (NGA) database (Chiou et al., 2008) that individually match the forty response spectra simulated in the previous step.
  • No constraints on, for example, the magnitudes and distances of the selected recordings are used, but such constraints are easily accommodated by simply Jayaram – 11 restricting the set of ground motions considered for selection.
  • The sample and the target means and standard deviations are shown in Figure 1.
  • This computational efficiency allows for the algorithm to be optionally applied multiple times if one wants several candidate sets to choose from.
  • While selecting the ground motions shown in Figure 2, the authors applied the algorithm multiple times (twenty times, in particular) to obtain multiple candidate ground-motion sets and chose the set with the minimum value of SSE.

GREEDY OPTIMIZATION TECHNIQUE

  • The greedy optimization technique is used to modify the ground-motion suite selected in the previous step.
  • The spectra of the selected ground motions are shown in Figure 2c.
  • The means and the standard deviations of the set are shown in Figure 1, and have a near perfect match with the target means and standard deviations.
  • The mean absolute error between the sample and the target correlations is 0.15.
  • In total, the computational time required to select the set of 40 ground motions from the 7102 available ground motions is about 180 seconds using a MATLAB implementation on an 8GB RAM 2.33GHz quad core processor.

SELECTION OF A SMALLER NUMBER OF GROUND MOTIONS

  • The response spectra of the selected records are shown in Figure 3a.
  • The set means and standard deviations are compared to the target means and standard deviations in Figure 3b-c.
  • It can be seen that the matches are good, illustrating the effectiveness of the algorithm in selecting small sets of ground motions.
  • The computational time required to select the set of 10 ground motions is about 25 seconds using a MATLAB implementation on an 8GB RAM 2.33GHz quad core processor.
  • Figure 3. (a) Response spectra of 10 selected ground motions (b) Response spectrum mean (c) Response spectrum standard deviation.

GROUND-MOTION SELECTION

  • The ground motions used for evaluating structural response are selected using the method described in the previous section for a target scenario with magnitude = 7, distance to rupture = 10km, Vs30 = 400m/s, and a strike-slip mechanism.
  • The Campbell and Bozorgnia (2008) ground-motion model is used to estimate the mean and variance of the response spectrum.
  • The values of ε and period T* are varied to obtain multiple test scenarios.
  • The structures considered in this work have periods (T*) ranging between 0.5s and 2.63s.

STRUCTURAL RESPONSE

  • This section describes the response of sample nonlinear single-degree-of-freedom (SDOF) structures and multiple-degree-of-freedom (MDOF) buildings designed according to modern building codes.
  • The authors consider only maximum displacement for the SDOF structures and maximum interstory drift ratio (MIDR) for the MDOF structures.

Description of structural systems

  • The SDOF structures considered in this work follow a non-deteriorating, bilinear forcedisplacement relationship (Chopra, 2007).
  • The R factor is controlled by varying the yield displacements of the SDOF structures relative to the *( )aS T value obtained from the target spectrum.
  • The MDOF structures used in this study were designed per modern building codes and modeled utilizing the Open System for Earthquake Engineering Simulation (2007) by Haselton and Deierlein (2007).
  • The structural models consider strength and stiffness deterioration (Ibarra et al., 2005) unlike in the SDOF case.
  • They have also been used for previous extensive ground-motion studies (Haselton et al., 2009).

Response of SDOF systems

  • Table 1 shows the mean, median and dispersion (dispersion refers to logarithmic standard deviation) of ductility ratios (spectral displacement divided by the yield displacement) of the SDOF structures under the different ground-motion scenarios described earlier.
  • The ductility statistics are estimated using the two sets of 40 ground motions selected using Method 1 (ground motions selected by matching only the target response spectrum mean) and Method 2 (ground motions selected by matching the target response spectrum mean and variance).
  • The CDF obtained using Method 2 is flatter with heavier tails as a result of the larger dispersion observed in this case.
  • As seen from Figure 5a, the upper tails of the CDFs are heavier than the lower tails.
  • Analytically, if the responses were to follow a lognormal distribution (a common assumption in performance-based earthquake engineering), the properties of the lognormal distribution will imply that a larger dispersion results in a larger mean for a fixed median, which also explains the larger means observed in Method 2.

Response of MDOF systems

  • Table 2 summarizes the maximum interstory drift ratio (MIDR) estimates for the MDOF structures considered in this study under various ground-motion scenarios, estimated using Jayaram – 16 Methods 1 and 2.
  • The distributions of responses are summarized using the probability of collapse (i.e., counted fraction of responses indicating collapse) and the median and the dispersion of the non-collapse responses.
  • As seen in the SDOF case, the CDF obtained using Method 2 is flatter and has heavier tails on account of larger dispersion.
  • Figure 6. Distribution of the structural response of the 20 story moment frame building corresponding to ε(T*) = 2: (a) Linear scale (b) Logarithmic scale.
  • The increased dispersion can result in more extreme responses, which can lead to a larger probability of structural collapse.

IMPLICATIONS

  • Code-based design is often concerned with the average response of the structure (e.g., ASCE, 2005).
  • The average response is typically interpreted as the mean response, although sometimes it is interpreted as the median.
  • The increase in the dispersion leads to higher and lower extremes of structural response and the associated damage states and losses.
  • PBEE calculations will thus almost certainly be affected by this issue.
  • In summary, the example analyses presented above and engineering intuition suggest that the target response spectrum variance used when selecting ground motions has an impact on the distribution of structural responses obtained from resulting dynamic analysis.

CONCLUSIONS

  • A computationally efficient, theoretically consistent ground-motion selection algorithm was proposed to enable selection of a suite of ground motions whose response spectra have a target mean and a target variance.
  • It was shown empirically that this selection algorithm selects ground motions whose response spectra have the target mean and variance.
  • It was seen that considering the response spectrum variance does not significantly affect the resulting median response, but slightly increases the mean response and considerably increases the dispersion (logarithmic standard deviation) of the response.
  • The increase in the mean and the dispersion is larger for more non-linear SDOF structures.
  • Two code-compliant MDOF structures with heights of 4 and 20 stories were also analyzed using the selected ground motions.

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Citations
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Journal Article•DOI•
Dynamics of structures

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L. G. Jaeger
01 Aug 1990-Canadian Journal of Civil Engineering

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Conditional spectrum-based ground motion selection. Part I: Hazard consistency for risk-based assessments

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Ting Lin1, Curt B. Haselton2, Jack W. Baker1•
Stanford University1, California State University, Chico2
10 Oct 2013-Earthquake Engineering & Structural Dynamics
TL;DR: In this article, the conditional spectrum (CS) with mean and variability is used to estimate the annual rate of exceeding a specified structural response amplitude for a 20-story reinforced concrete frame structure.
Abstract: SUMMARY The conditional spectrum (CS, with mean and variability) is a target response spectrum that links nonlinear dynamic analysis back to probabilistic seismic hazard analysis for ground motion selection. The CS is computed on the basis of a specified conditioning period, whereas structures under consideration may be sensitive to response spectral amplitudes at multiple periods of excitation. Questions remain regarding the appropriate choice of conditioning period when utilizing the CS as the target spectrum. This paper focuses on risk-based assessments, which estimate the annual rate of exceeding a specified structural response amplitude. Seismic hazard analysis, ground motion selection, and nonlinear dynamic analysis are performed, using the conditional spectra with varying conditioning periods, to assess the performance of a 20-story reinforced concrete frame structure. It is shown here that risk-based assessments are relatively insensitive to the choice of conditioning period when the ground motions are carefully selected to ensure hazard consistency. This observed insensitivity to the conditioning period comes from the fact that, when CS-based ground motion selection is used, the distributions of response spectra of the selected ground motions are consistent with the site ground motion hazard curves at all relevant periods; this consistency with the site hazard curves is independent of the conditioning period. The importance of an exact CS (which incorporates multiple causal earthquakes and ground motion prediction models) to achieve the appropriate spectral variability at periods away from the conditioning period is also highlighted. The findings of this paper are expected theoretically but have not been empirically demonstrated previously. Copyright © 2013 John Wiley & Sons, Ltd.

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TL;DR: The National Institute of Standards and Technology (NIST) funded a project to improve guidance to the earthquake engineering profession for selecting and scaling earthquake ground motions for the purpose of performing nonlinear response-history analysis.
Abstract: The National Institute of Standards and Technology (NIST) funded a project to improve guidance to the earthquake engineering profession for selecting and scaling earthquake ground motions for the purpose of performing nonlinear response-history analysis. The project supported problem-focused studies related to defining target spectra for seismic design and performance assessment, response-spectrum matching, and near-fault ground motions. Recommendations are presented for target spectra, selection of seed ground motions, and scaling of motions to be consistent with different target spectra. Minimum numbers of sets of motions are recommended for computing mean component and systems responses, and distributions of responses. Guidance is provided on selection and scaling of ground motions per ASCE/SEI 7-10.

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Journal Article•DOI•
Conditional Spectrum Computation Incorporating Multiple Causal Earthquakes and Ground-Motion Prediction Models

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Ting Lin1, Stephen C. Harmsen2, Jack W. Baker1, Nicolas Luco2•
Stanford University1, United States Geological Survey2
01 Apr 2013-Bulletin of the Seismological Society of America
TL;DR: The conditional spectrum (CS) is a target spectrum (with conditional mean and conditional standard deviation) that links seismic hazard information with ground-motion selection for nonlinear dynamic analysis.
Abstract: The conditional spectrum (CS) is a target spectrum (with conditional mean and conditional standard deviation) that links seismic hazard information with ground-motion selection for nonlinear dynamic analysis. Probabilistic seismic hazard analysis (PSHA) estimates the ground-motion hazard by incorporating the aleatory uncertainties in all earthquake scenarios and resulting ground motions, as well as the epistemic uncertainties in ground-motion prediction models (GMPMs) and seismic source models. Typical CS calculations to date are produced for a single earthquake scenario using a single GMPM, but more precise use requires consideration of at least multiple causal earthquakes and multiple GMPMs that are often considered in a PSHA computation. This paper presents the mathematics underlying these more precise CS calculations. Despite requiring more effort to compute than approximate calculations using a single causal earthquake and GMPM, the proposed approach produces an exact output that has a theoretical basis. To demonstrate the results of this approach and compare the exact and approximate calculations, several example calculations are per- formed for real sites in the western United States. The results also provide some in- sights regarding the circumstances under which approximate results are likely to closely match more exact results. To facilitate these more precise calculations for real applications, the exact CS calculations can now be performed for real sites in the United States using new deaggregation features in the U.S. Geological Survey hazard mapping tools. Details regarding this implementation are discussed in this paper.

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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 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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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Applied Multivariate Statistical Analysis

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Charles E. Heckler
01 Nov 2005-Technometrics
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Abstract: (2005). Applied Multivariate Statistical Analysis. Technometrics: Vol. 47, No. 4, pp. 517-517.

3,932 citations

Additional excerpts

  • ...…follows: R1 ¼ qðT1; T ÞrlnSaðT1ÞrlnSaðT Þ qðT2; T ÞrlnSaðT2ÞrlnSaðT Þ : : qðTn; T ÞrlnSaðTnÞrlnSaðT Þ 2 66664 3 77775 (8) The covariance matrix of ln SaðT1Þ; ln SaðT2Þ; :::; ln SaðTnÞð Þ conditioned on ln SaðT Þ can be computed as follows (e.g., Johnson and Wichern 2007): R ¼ R0 1 r2lnSaðT Þ R1R 0…...

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Book•DOI•
Minimum Design Loads for Buildings and Other Structures

[...]

null null
13 Jan 2003
TL;DR: Minimum Design Loads for Buildings and Other Structures as mentioned in this paper gives the latest consensus requirements for dead, live, soil, flood, wind, snow, rain, ice, and earthquake loads.
Abstract: Minimum Design Loads for Buildings and Other Structures gives the latest consensus requirements for dead, live, soil, flood, wind, snow, rain, ice, and earthquake loads, as well as their combinatio...

3,751 citations

Journal Article•DOI•
Dynamics of structures

[...]

L. G. Jaeger
01 Aug 1990-Canadian Journal of Civil Engineering

1,604 citations

"A Computationally Efficient Ground-..." refers background in this paper

  • ...The SDOF structures considered in this work follow a non-deteriorating, bilinear forcedisplacement relationship (Chopra 2007)....

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Frequently Asked Questions (10)
Q1. What contributions have the authors mentioned in the paper "A computationally efficient ground-motion selection algorithm for matching a target response spectrum mean and variance" ?

In this paper, the authors proposed a method to select ground motions that match the least variance from the target response spectrum. 

The computational time required for selecting the set of 10 ground motions without using the greedy optimization technique is 4 seconds. 

In total, the computational time required to select the set of 40 ground motions from the 7102 available ground motions is about 180 seconds using a MATLAB implementation on an 8GB RAM 2.33GHz quad core processor. 

SDOF structures with ‘R factors’ (the ratio of the target spectral acceleration at the period of the structure, *( )aS T , to the yield spectral acceleration = ω2 * yield displacement, where ω is the structure’s fundamental circular frequency) of 1, 4 and 8 are considered to study varying levels of non-linear behavior. 

A greedy optimization technique then further improves the match between the target and the sample means and variances by replacing one previously selected ground motion at a time with a record from the ground-motion database that causes the best improvement in the match. 

A MATLAB implementation of the proposed ground-motion selection algorithm can be downloaded from http://www.stanford.edu/~bakerjw/gm_selection.html.Jayaram – 12To test the effectiveness of the algorithm in sampling smaller ground motion sets, it is repeated to select a set of 10 ground motions for the scenario described earlier (magnitude = 7, distance to rupture = 10km, T* = 2.63s and ε(T*) = 2). 

Since the Monte Carlo simulated response spectra have the desired mean and variance, the response spectra of the selected recorded ground motions will also have the desired mean and variance. 

While selecting the ground motions shown in Figure 2, the authors applied the algorithm multiple times (twenty times, in particular) to obtain multiple candidate ground-motion sets and chose the set with the minimum value of SSE. 

Since the simulated response spectra have approximately the desired mean and variance, the response spectra selected using this approach will also have approximately the desired mean and variance. 

One commonly used approach is to select recorded or simulated ground motions whose response spectra match a target mean response spectrum (e.g., Beyer and Bommer, 2007; Shantz, 2006; WatsonLamprey and Abrahamson, 2006).