Pseudo-spectral Velocity Research Articles

A prolegomenon to Design Input Energy Spectra for the Himalayan region

Energy-based design is becoming progressively significant as a new age of earthquake-resistant design unfolds. Thitherto, very little is known about energy-based design in India despite the fact that the Himalayan region is one of the most seismically active regions in the world. Therefore, the current work provides a preliminary analysis of the elastic input energy and equivalent velocity (Vea) spectra for the active regions in the Himalayas. In this regard, the ground motions of Himalayan databases are blended with NGA WEST2 to estimate the Vea spectra, an inertia-independent representation of the input energy spectra. Incipiently, an Artificial Neural Network-based ground motion model (GMM) is developed for Vea spectra based on the predictor variables as magnitude, distance, hypocentral depth and site conditions. Consequently, a Probabilistic Seismic Hazard analysis is performed to obtain the uniform hazard energy spectra for a 2475-year return period. In addition, a Design Input Energy Spectra is proposed for the active regions in the Indian Himalayas. In addition, the relationship between the Vea and pseudo-spectral velocity (PSV) is analyzed to develop a Vea-informed GMM for PSV. Furthermore, the PSA estimated from the Vea-PSV model is compared with the design spectra recommended by the Indian Standard Code of Practice (IS1893-1:2016).

Improvements in the direct displacement-based design procedure for mid-rise steel MRFs equipped with viscous dampers

This paper aims to propose straightforward modifications in the direct displacement-based design (DDBD) procedure to reach an optimal design for mid-rise steel moment-resisting frames (MRFs) equipped with linear fluid viscous dampers (FVDs). The modifications include using an alternative equivalent damping as well as applying the ratio of pseudo-spectral velocity to spectral velocity. To compare the proposed method with respect to the conventional DDBD procedure, the seismic performance of a set of low to mid-rise structures consisting of 3-, 6-, 9- and 12-story steel MRFs is investigated. This is carried out via nonlinear time-history analyses by using a set of design spectrum compatible ground motion records at two seismic hazard levels, i.e., the design earthquake (DE) and maximum considered earthquake (MCE). Furthermore, two different patterns, i.e., uniform and non-uniform, for distribution of damping coefficients at the height of the structures are used. Results show that the proposed method can meet the performance targets defined at the DE and MCE hazard levels. Moreover, the peak roof displacement of the structures that is estimated by the proposed method is acceptably the same as captured via the nonlinear analyses. Finally, implementing the proposed method leads to a reduction in the weight of the steel used in the MRFs in comparison with the structures designed by the conventional DDBD procedure and smaller damper forces, i.e., lighter damper constants, can also be achieved.

Effects of Earthquake Magnitude, Distance, and Site Conditions on Spectral and Pseudospectral Velocity Relationship

ABSTRACT The spectral velocity (SV) is necessary information in the seismic design of structures with supplemental velocity-dependent dampers, and it is conventionally approximated by the pseudospectral velocity (PSV), which is available in seismic codes. Because of the significant approximation error, it is important to clarify the relationship between the two spectra to establish a suitable formulation to relate SV to PSV. Recent studies have point out that this relationship is influenced not only by the oscillator period and damping ratio but also by earthquake characteristics (Papagiannopoulos et al., 2013; Samdaria and Gupta, 2018). To clarify the seismological effects, in this study, an approach to relate SV to PSV based on the random vibration theory is proposed, and it is verified by comparing its results with those of traditional time-series analysis. The effects of earthquake magnitude and distance as well as site conditions on the relationship between the two spectra are explored based on the proposed approach as well as statistical analysis of recorded seismic motions. It is found that the SV approaches the PSV with increasing magnitudes at long oscillator periods but performs oppositely at short oscillator periods. The demarcation range beyond which the opposite trend is observed varies from (0.07–0.24) to (0.12–0.87) s using the proposed approach and considering the regions of central and eastern North America. The range varies from (0.1–0.15) to (0.3–0.7) s based on the results obtained by the statistical analysis of seismic records in Japan. The observed phenomena were theoretically explained, and the seismological effects were found to be governed by the ground-motion frequency content.

An improvement of direct displacement-based design approach for steel moment-resisting frames controlled by fluid viscous dampers

Direct displacement-based design (DDBD) method is one of the most effective methods for performance-base design of structures that has been also employed to design structures controlled by fluid viscous damper (FVD). In previous studies, a modified DDBD has been developed to apply the higher mode effects as well as difference between spectral velocity and pseudo-spectral velocity on the design velocity of FVD. To this end, two constants were defined to correct the damping coefficient of FVD that these correction constants had been determined in a non-classical and iterative manner. In this study, a new classical method is proposed for determining these constant such that no iteration is required in DDBD. In order to be able to introduce this design approach as a reliable framework, its performance is validated under different sets of earthquake records and this design approach is also developed for structures controlled by nonlinear FVD. Steel moment-resisting frames with different numbers of stories have been designed using this method. For comparison, structures have been also designed based on DDBD proposed in previous researches. The results show that DDBD approach improved in this study is capable to achieve the design performance level under different sets of earthquake records and this design approach has more effective performance than previous design methods. Performance of steel moment-resisting frames equipped with nonlinear FVD also shows excellent performance of this design approach in achievement of desirable performance level. Therefore, DDBD approach proposed in this study can be introduced as a new classical and reliable framework because of its simplicity and excellent performance under different sets of earthquakes.

A note on spectral velocity approximation at shorter intermediate periods

It is common to use the design pseudo spectral acceleration (PSA) spectrum together with a modal combination rule to estimate the peak linear seismic response of a structural system. Some of the modal combination rules proposed so far require an accurate estimation of spectral velocity (SV) spectrum consistent with the given design PSA spectrum in order to account for modal correlations in a simple manner. The traditional pseudo spectral velocity (PSV) approximation of SV ordinates is reliable only for the velocity-sensitive region of the spectrum. More reliable approximations have therefore been developed in the recent years for the acceleration-sensitive and displacement-sensitive regions. In this work, the approximation developed for the acceleration-sensitive region is proposed to be applied on a greater range of periods, involving the shorter periods of the velocity-sensitive region, for a more accurate estimation of the SV spectrum. Regression equations are developed to predict the limiting period of the extended range by using the information available from the given PSA spectrum. A correction factor is also proposed to be applied to the SV estimates in order for those to have a smooth transition near the limiting period.

Quantifying local seismic amplification from regional charts and site specific numerical analyses: a case study

Three levels of seismic microzonation are considered in the Italian guidelines: 1st level allows to qualitatively recognise those areas characterised by the same seismic performance, while 2nd and 3rd levels quantify the modification of the reference signal due to the specific site conditions. The paper summarises and compares the results of 2nd and 3rd level seismic microzonation studies for the Dovadola urban area (Emilia-Romagna, Italy). The site was selected due to the availability of in situ and laboratory investigations for the definition of a detailed geotechnical seismic model of the subsoil. The 2nd level study was conducted adopting the regional charts for the quantification of the stratigraphic amplification phenomena and the regional procedures for assessing the influence of topography. The 3rd level analysis was developed through mono- and two-dimensional seismic site response analyses adopting a numerical non-linear approach. Amplification factors for the peak ground acceleration and for the velocity spectrum intensity (i.e. the integral of the pseudo-spectral velocity of a ground motion from 0.1–0.5 to 0.5–0.1 s) were determined along two geotechnical sections of the site, both characterised by a central large and surficial valley bordered by irregular topography. The study indicates a general satisfactory match between the results of the simplified regional approach and the advanced numerical tool, especially when one-dimensional seismic site conditions prevail. Further comparative studies are encouraged to validate the 2nd level microzonation as a reliable tool for seismic risk mitigation.

Laguerre–Hermite pseudo-spectral velocity formulation of gyrokinetics

First-principles simulations of tokamak turbulence have proven to be of great value in recent decades. We develop a pseudo-spectral velocity formulation of the turbulence equations that smoothly interpolates between the highly efficient but lower resolution three-dimensional (3-D) gyrofluid representation and the conventional but more expensive 5-D gyrokinetic representation. Our formulation is a projection of the nonlinear gyrokinetic equation onto a Laguerre–Hermite velocity-space basis. We discuss issues related to collisions, closures and entropy. While any collision operator can be used in the formulation, we highlight a model operator that has a particularly sparse Laguerre–Hermite representation, while satisfying conservation laws and the H theorem. Free streaming, magnetic drifts and nonlinear phase mixing each give rise to closure problems, which we discuss in relation to the instabilities of interest and to free energy conservation. We show that the model is capable of reproducing gyrokinetic results for linear instabilities and zonal flow dynamics. Thus the final model is appropriate for the study of instabilities, turbulence and transport in a wide range of geometries, including tokamaks and stellarators.

A new model for spectral velocity ordinates at long periods

SummaryThe estimation of peak linear response via elastic design (response) spectra continues to form the basis of earthquake‐resistant design of structural systems in various codes of practice all over the world. Many response spectrum‐based formulations of peak linear response require an additional input of the spectral velocity (SV) ordinates consistent with the specified seismic hazard. SV ordinates have been conventionally approximated by pseudo spectral velocity (PSV) ordinates, which are close to the SV ordinates only over the intermediate frequency range coinciding with the velocity‐sensitive region. At long periods, PSV ordinates underestimate the SV ordinates, and this study proposes a formulation of a correction factor (>1) that needs to be multiplied by the PSV ordinates in order to close the gap between the two sets of ordinates. A simple model is proposed in the form of a power function in oscillator period to estimate this factor in terms of two governing parameters which are in turn estimated from two single‐parameter scaling equations. The parameters considered for the scaling equations are (1) the period at which the PSV spectrum is maximized and (2) the rate of decay of the pseudo spectral acceleration (PSA) amplitudes at long periods. For a given damping ratio, four regression coefficients are determined for the scaling equations with the help of 205 ground motions recorded in western USA. A numerical study undertaken with the help of several design PSA spectra and ensembles of spectrum‐compatible ground motions illustrates the effectiveness of the proposed correction factor, together with the proposed scaling models, in comparison with the PSV approximation in a variety of design situations. Both the input parameters mentioned above can be easily obtained from the specified design spectrum, and thus the proposed model is convenient to use.

The 2011 Mw 5.2 Lorca earthquake as a case study to investigate the ground motion variability related to the source model

Near-field recordings are very sensitive to the spatiotemporal details of the rupture process while far-field signals show the signature of the overall “point-source” earthquake mechanism. Near- and far-field recording ranges are dependent on the event magnitude and modulate the variability of the ground motion. This study investigates the ground motion and the source-related near-field variability for the 2011 Lorca earthquake, a moderate seismic event (Mw = 5.2) that caused significant localized damage in the Region of Murcia, Spain. The low-frequency content (up to 1 Hz) is simulated by the wavenumber integration method assuming four different source models obtained by inversion of geodetic or seismological data. As a first result, we estimate the variability of the ground motion. We observe that the dispersion in the peak and spectral parameters is larger at LOR, the closest station to the source, and decreases as the source distance increases (more than 50 km far from the source) where the finite-fault effects become negligible. The variability of the pseudo spectral velocity at 2 s is within the ground motion prediction equation ±1 σ, apart from the very near-source station and those stations affected by forward directivity effects. These effects are also found in high-frequency seismograms obtained by the empirical Green’s functions approach.

On the Calculation of Peak Ground Velocity for Seismic Performance Assessment

Ratios of pseudo-spectral velocity (averaged over the period range of 0.5–2 seconds and at a period of 1 second) to peak ground velocity (PGV) were computed for hundreds of ground motion records selected from the PEER NGA Strong Motion Database. Two-stage regression analyses were performed to develop models for the ratios. The inter-event terms of the ratios trend to moderate decrease as moment magnitude increases; the intra-event residuals do not depend on distance. The results of this study provide the technical basis for a procedure presented in the American Technology Council's ATC-58 project, Guidelines for Seismic Performance Assessment of Buildings, for estimating PGV by scaling values of pseudo-spectral velocity.

A direct method for the Boltzmann equation based on a pseudo-spectral velocity space discretization

A deterministic method is proposed for solving the Boltzmann equation. The method employs a Galerkin discretization of the velocity space and adopts, as trial and test functions, the collocation ba...

Effect of soil conditions on predicted ground motion: Case study from Western Anatolia, Turkey

We present a site effect study for the city of Izmir, Western Anatolia, Turkey. Local amplification was evaluated using state-of-practice tools. Ten earthquakes recorded at 16 sites were analysed using spectral ratios relative to a reference site, horizontal-to-vertical spectral ratios, and an inversion scheme of the Fourier amplitude spectra of the recorded S-waves. Seismic noise records were also used to estimate site effects. The different estimates are in good agreement among them, although a basic uncertainty of a factor of 2 seems difficult to decrease. We used our site effect estimates to predict ground motion in Izmir for a possible M6.5 earthquake close to the city using stochastic modelling. Site effects have a large impact on PSV (pseudospectral velocity), where local amplification increases amplitudes by almost a factor of 9 at 1Hz relative to the firm ground condition. Our results allow identifying the neighbourhoods of Izmir where hazard mitigation measurements are a priority task and will also be useful for planning urban development.

A direct method for the Boltzmann equation based on a pseudo-spectral velocity space discretization

A deterministic method is proposed for solving the Boltzmann equation. The method employs a Galerkin discretization of the velocity space and adopts, as trial and test functions, the collocation basis functions based on weights and roots of a Gauss–Hermite quadrature. This is defined by means of half- and/or full-range Hermite polynomials depending whether or not the distribution function presents a discontinuity in the velocity space. The resulting semi-discrete Boltzmann equation is in the form of a system of hyperbolic partial differential equations whose solution can be obtained by standard numerical approaches. The spectral rate of convergence of the results in the velocity space is shown by solving the spatially uniform homogeneous relaxation to equilibrium of Maxwell molecules. As an application, the two-dimensional cavity flow of a gas composed of hard-sphere molecules is studied for different Knudsen and Mach numbers. Although computationally demanding, the proposed method turns out to be an effective tool for studying subsonic slightly rarefied gas flows.

A kinematic rupture model generator incorporating spatial interdependency of earthquake source parameters

S U M M A R Y Based on the results obtained by analysing a large number of dynamic strike slip and dipping fault ruptures we construct a kinematic rupture model generator that incorporates some of the key source parameters extracted from the dynamic rupture models. In this kinematic rupture model, the slip rate function includes the final slip, the rise time, the local rupture velocity and the peak time (a measure of the impulsive part of the slip rate function). A four-dimensional correlation matrix is used to describe the spatial interdependency between the four source parameters—each of which is modelled as correlated random field. Each source parameter has a different marginal distribution determined from the analysis of the dynamic models. The marginal distributions are allowed to change as a function of distance from the hypocentre. The autocorrelation of each parameter is modelled by a power spectrum that follows a power law. The value of the spectral decay of the power law for the different source parameters is based on the results obtained from the dynamic rupture models. Finally, the values of rise time and peak time are adjusted such that the moment rate function fits a Brune spectrum for a specified corner frequency. We validate the rupture model generator using observed strong motion near field recording for the 1994 Northridge earthquake and the 1989 Loma Prieta earthquake. We perform comparisons with the rupture model generator of Liu et al. We find that only considering the response spectral bias of a best model is not sufficient to validate a rupture model generator. Thus we test the predictive power of the two rupture model generators if multiple ruptures are computed. Overall, the new rupture model generator yields a better prediction, compared to Liu et al., for the two validation events, especially in predicting observed PGV values and pseudospectral velocity at low frequencies (<1Hz).

Simulation of Ground Motion in the Zagros Region of Iran Using the Specific Barrier Model and the Stochastic Method

Abstract Physically based ground‐motion prediction equations for soil and rock sites in the Zagros region have been developed based on the specific barrier model (SBM) used within the context of the stochastic model. Instead of direct time‐domain simulation, random vibration theory was used to estimate measures of peak motion in terms of the pseudospectral velocity of anelastic harmonic oscillator with 5% viscous damping. To avoid the uncertainties, calibration of the source model uses a database of carefully selected strong motion data without ambiguity about the site condition. Therefore, only rock sites are selected for determining source parameters. Also, to avoid any inconsistencies caused by magnitude conversion formulas, we restricted the dataset only to events with available moment magnitudes. Regression analysis is performed using the random effects model that considers both interevent and intraevent variabilities to effectively deal with the problem of an unequal number of records from different earthquakes. No sign of self‐similarity breakdown is observed between the source radius and its seismic moment. The local and global stress drops derived for the Zagros region (39 and 116 bars, respectively) are more consistent with the values obtained by other authors for an interplate regime than the values for an intraplate region. However, from the viewpoint of source heterogeneity (as the ratio of the stress drops is an indicator of the complexity of the source and heterogeneity of slip on the fault) the Zagros events, which have a stress‐drop ratio of about three are more homogeneous than other interplate events. Stochastic simulations are then implemented to predict peak ground motion and response spectra parameters for rock and soil site conditions. Online Material: Catalog tables of earthquakes and attenuation tables.

Quantification of ground-motion parameters and response spectra in the near-fault region

This study focuses on the characteristics of near-fault ground motions in the forward-direction and structural response associated with them. These ground motions are narrow-banded in nature and are characterized by a predominant period at which structures excited by them are severely affected. In this work, predominant period is defined as the undamped natural period of a single-degree-of-freedom (SDOF) oscillator at which its 5% damped linear elastic pseudo-spectral velocity (PSV) contains a clear and dominant peak. It is found that a linear relationship exists between predominant period and seismic moment. An empirical equation describing this relationship is presented by using a large set of accelerograms. Attenuation equations are developed to estimate peak ground velocity (PGV) as a function of earthquake magnitude and source-to-site distance. In addition, a predictive equation for spectral shapes of PSV (i.e., PSV normalized by PGV) is presented as a continuous function of the undamped natural period of SDOF oscillators. The model is independent of PGV, and can be used in conjunction with any available PGV attenuation relation applicable to near-fault ground motion exhibiting forward-directivity effects. Furthermore, viscous damping of the SDOF is included in the model as a continuous parameter, eliminating the use of so-called damping correction factors. Finally, simple equations relating force reduction factors and displacement ductility of elasto-plastic SDOF systems are presented.

Scaling of response spectrum and duration for aftershocks

Aftershocks have the potential to cause collapse of a structure that has been already damaged by the preceding main shock. Seismic safety of a structure should therefore be ascertained through a damage analysis using the anticipated main shock and few larger-aftershock motions. Simulation of aftershock motions needs characterization of the seismic hazard due to aftershocks, and therefore it will be useful to develop a conditional scaling model that can predict the response spectrum of an anticipated aftershock motion consistent with the design spectrum of the main shock motion anticipated at the same station. In this study an attempt is made to develop a conditional scaling model for the pseudo spectral velocity spectrum via linear regression analysis on the aftershock and main shock recordings for the 1999 Chi–Chi earthquake. It is shown that it may be possible to obtain a simpler and approximate version of the conditional model from an unconditional model. Damage-causing potential of a ground motion also depends on its strong motion duration (SMD) and therefore a conditional scaling model is developed for SMD of the aftershock motion in several narrow frequency-bands. The model is developed for the larger-aftershock motions and it is shown that a reasonable replacement of such a model may be obtainable directly from an unconditional model. Finally, a simple weighted averaging scheme is proposed to obtain the composite SMD from the SMDs for different frequency bands by using the pseudo spectral acceleration spectrum of the motion.

Dynamic Rupture Models for the Southern San Andreas Fault

Abstract Dynamic rupture, and resultant ground motions up to 0.25 Hz, are simulated for an M w 7.6 earthquake on the southern San Andreas fault. Spontaneous rupture is modeled with slip-weakening friction, and 3D viscoelastic wave solutions are computed with a support-operator numerical method. The initial traction model is derived from inversions of the M w 7.3 1992 Landers strong ground-motion records, and borrows heavily from that used for the TeraShake2 simulations by Olsen et al. (2008). Heterogeneity in the traction model leads to focusing of the rupture front, and the focusing produces cases of supershear rupture velocity in asperities (areas of high initial traction), as well as cases of high peak slip rate and cohesive zone contraction in antiasperities. Separate solutions are computed for version 3.0 and 4.0, respectively, of the Southern California Earthquake Center Community Velocity Model (SCEC-CVM). We also compare the case of a flat ground surface (a common simplification made for finite-difference simulations) to the case of the ground surface conformed to regional topography. The overall distribution of simulated ground motion intensity is consistent with that derived from the empirical model of Campbell and Bozorgnia (2008), in the sense that the bulk of simulated pseudospectral velocity (PSV) values are within the 68% confidence intervals of the empirical model. Simulated PSVs corresponding to low probability in the empirical model are principally associated with basin wave-guide and directivity effects. An important example, first identified by the TeraShake1 simulations (Olsen et al. , 2006), is the stronger than expected ground motions at the site of Montebello due to a basin wave-guide effect. We find that this effect is lessened for version 4.0 of the SCEC-CVM, relative to version 3.0, due to a shallower model for the Chino basin.

Ground-Motion Prediction Equation for SI Based on Spectral Acceleration Equations

Spectrum intensity (SI), defined as the integral of the pseudospectral velocity of a ground motion from 0.1 to 2.5 sec, has recently been shown to be an intensity measure that efficiently predicts the seismic response of both liquefiable and nonliquefiable soil deposits as well as the seismic demands on pile foundations embedded in such deposits. In order for such an intensity measure to be used in performance-based assessment and design, ground-motion prediction relations are required to develop ground-motion hazard curves in terms of SI for various sites. As such relationships developed specifically for SI are sparse, we propose the development of a relationship based on current ground-motion prediction relations for spectral acceleration, which are available in most regions of seismic activity. Comparison with a direct prediction equation for SI provides a validation of the proposed approach. It is illustrated that SI is an intensity measure with a good predictability, thereby further promoting its attractiveness for use in reliability-based seismic response analysis.

A new approximation for spectral velocity ordinates at short periods

AbstractResponse spectrum methods in earthquake‐resistant design sometimes require information on the spectral velocity (SV) for a given single‐degree‐of‐freedom oscillator and specified seismic hazard. SV has been conventionally approximated as pseudo spectral velocity (PSV) in the case of lightly damped structures that are not too flexible. This study shows that the PSV approximation may lead to large overestimation errors when the structure is stiffer to the ground motion and the ground motion is a long‐period motion. It is also shown that a new approximation requiring the use of peak ground acceleration of the motion may significantly reduce these errors. Copyright © 2008 John Wiley &amp; Sons, Ltd.