Site dependent and spatially varying response spectra

The goal of this study is to provide a stochastic method to investigate the effects of the randomness of soil properties due to their natural spatial variability on the response spectra spatial variation at sites with varying conditions. For this purpose, Monte Carlo Simulations are used to include the variability of both incident ground motion and soil parameters in the response spectra by mean of an appropriate coherency loss function and a site-dependent transfer function, respectively. The approach is built on the assumption of vertical propagation of SH type waves in soil strata with uncertain parameters. The response spectra are obtained by numerical integration of the governing equation of a single-degree-of-freedom(SDOF) system under non-stationary site-dependent and spatially varying ground motion accelerations simulated with non-uniform spectral densities and coherency loss functions. Numerical examples showed that randomness of soil properties significantly affects the amplitudes of the response spectra, indicating that as the heterogeneity induced by the randomness of the parameters of the medium increases, the spectral ordinates attenuate.     

Estimating ground motion incoherence through finite source simulation: a case study of the 1980 El-Asnam Earthquake

Spatial variability of ground motions has significant influence on dynamic response of extended structures such as bridges and tunnels. In this study, the widely used finite-source ground motion simulation approach, the so-called Empirical Green’s Function (EGF) method, is extended to synthesize seismic motions across an array of stations located at bedrock in the epicentral region of the 1980 El-Asnam region (North-West Algeria). The target event being simulated is the October 10 1980 \( M_{s} = 7.2 \) Earthquake, and the EGF is obtained from the ground motion recorded at Sogedia Factory station during the 8 November 1980 \( M_{L} = 5.6 \) aftershock. Coherency functions are then estimated from the simulated ground accelerations. A parametric study investigating the influence of shear wave velocity, earthquake magnitude, and epicentral distance is conducted by simulating ground acceleration for different scenarios using the Hybrid Green’s Function method. The main finding of the study is that finite source effects can cause significant loss in coherency at bedrock in the near-field. In the far-field, the source effect alone does not seem to produce incoherent motion, which implies that scattering and local site effects could be dominating there. Furthermore, coherency functions are found to be more sensitive to inter-station separation in the near-field than in the far-field. Increasing shear wave velocity seems to increase coherency functions, and larger earthquakes seem to produce more incoherent motion than smaller ones. The simulation method presented here produces incoherent motion mainly due to the finite source effect, while path effects are partially accounted for through the EGF, and local site effects are not considered. In this sense, the estimated coherency functions represent that of plane waves. A parametric model of plane wave coherency is calibrated and presented based on the simulation results. The results indicate that the parametric model can be used as a first approximation, and at least an upper bound of lagged coherency in the near-field region of the El-Asnam Earthquake scenario. This model could be useful in random vibration analysis or generation of spatially variable ground motion for time history analysis of lifeline structures in the study area.     

Response of indeterminate two-span beam to spatially varying seismic excitation

In order to examine the effect of the spatial variation of ground motion on the response of an indeterminate structure, the stochastic responses of a two-span beam to spatially varying support excitations are analysed. A space-time earthquake ground motion model that accounts for both coherency decay and seismic wave propagation is used to specify the support motions, and the results are compared with those for various simplified excitations that are commonly used in practice. The response is computed through a linear random vibration approach with the structure being modelled by finite elements. The results of the study indicate that, even for moderate lengths, the effect of the spatial variation of ground motion can be significant. The assumption of fully coherent support motions (same excitations at all supports or delayed excitations allowing only for wave propagation) may be overconservative for some beams and unconservative for others.     

Soil–structure interaction analysis of cable-stayed bridges for spatially varying ground motion components

In this study, it is intended to determine the effects of soil–structure interaction (SSI) and spatially varying ground motion on the dynamic characteristics of cable-stayed bridges. For this purpose, ground motion time histories are simulated for spatially varying ground motions, depending on its components of incoherence, wave-passage and site-response effects. The substructure method, which partitions the total soil–structure system into the structural system and the soil system, is used to treat the soil–structure interaction problem. To emphasize the relative importance of the spatial variability effects of earthquake ground motion, bridge responses are determined for the fixed base bridge model, which neglects the soil–structure interaction (no SSI) and for the bridge model including the soil–structure interaction (SSI). This parametric study concerning the relative importance of the soil–structure interaction and spatially varying ground motion shows that these effects should be considered in the dynamic analyses of cable-stayed bridges.     

Response spectrum method for multi-support seismic excitations

A new response spectrum method is developed for seismic analysis of linear multi-degree-of-freedom, multiply supported structures subjected to spatially varying ground motions. Variations of the ground motion due to wave passage, loss of coherency with distance and variation of local soil conditions are included. The method is based on fundamental principles of random vibration theory and properly accounts for the effects of correlation between the support motions as well as between the modes of vibration of the structure.     

CHARACTERISTICS OF TORSIONAL GROUND MOTIONS

Due to the inherent difficulty in directly recording the rotational ground motions, torsional ground motions have to be estimated from the recorded spatially varying translational motions. In this paper, an empirical coherency function, which is based on the recorded motions at the SMART-1 array, is suggested to model the spatial variation of translational motions. Then, the torsional ground motion power spectral density function is derived. It depends on the translational motion power spectral density function and the coherency function. Both the empirical coherency function and the torsional motion power spectral density function are verified by the recorded motions at the SMART-1 array. The response spectra of the torsional motions are also estimated. Discussion on the relations between the torsional motion response spectrum and the corresponding translational motion response spectrum is made. Numerical results presented can be used to estimate the torsional ground motion power spectral density function and response spectrum.     

Influence of irregular topography and random soil properties on coherency loss of spatial seismic ground motions

Coherency functions are used to describe the spatial variation of seismic ground motions at multiple supports of long span structures. Many coherency function models have been proposed based on theoretical derivation or measured spatial ground motion time histories at dense seismographic arrays. Most of them are suitable for modelling spatial ground motions on flat‐lying alluvial sites. It has been found that these coherency functions are not appropriate for modelling spatial variations of ground motions at sites with irregular topography (Struct. Saf. 1991; 10 (1):1–13). This paper investigates the influence of layered irregular sites and random soil properties on coherency functions of spatial ground motions on ground surface. Ground motion time histories at different locations on ground surface of the irregular site are generated based on the combined spectral representation method and one‐dimensional wave propagation theory. Random soil properties, including shear modulus, density and damping ratio of each layer, are assumed to follow normal distributions, and are modelled by the independent one‐dimensional random fields in the vertical direction. Monte‐Carlo simulations are employed to model the effect of random variations of soil properties on the simulated surface ground motion time histories. The coherency function is estimated from the simulated ground motion time histories. Numerical examples are presented to illustrate the proposed method. Numerical results show that coherency function directly relates to the spectral ratio of two local sites, and the influence of randomly varying soil properties at a canyon site on coherency functions of spatial surface ground motions cannot be neglected. Copyright © 2010 John Wiley & Sons, Ltd.     

Stochastic dynamic response of dam–reservoir–foundation systems to spatially varying earthquake ground motions

In this paper, stochastic dynamic responses of dam–reservoir–foundation systems subjected to spatially varying earthquake ground motions are investigated using the displacement-based fluid finite elements. For this purpose, variable-number-node two-dimensional (2D) fluid finite elements based on the Lagrangian approach is programmed in FORTRAN language and incorporated into a computer program SVEM, which is used for stochastic dynamic analysis of solid systems subjected to spatially varying earthquake ground motion. The spatially varying earthquake ground motion model includes incoherence, wave-passage and site-response effects. The incoherence effect is examined by considering the Harichandran and Vanmarcke coherency model. The effect of the wave passage is investigated by using various wave velocities. Homogeneous medium and firm soil types are selected for considering the site-response effect where the foundation supports are constructed. The Sar?yar concrete gravity dam, constructed in Turkey is selected for numerical example. The ground motion is described by filtered white noise and applied to each support point of the 2D finite element model of the dam–reservoir–foundation system. The record of Kocaeli earthquake in 1999 is used in the analyses. Displacements, stresses and hydrodynamic pressures occurring on the upstream face of the dam are calculated for four cases. It is concluded that spatially varying earthquake ground motions have important effects on the stochastic dynamic response of dam–reservoir–foundation systems.     

Response of multiply supported rigid plate to spatially correlated seismic excitations

The ground motions produced by an earthquake at the multiple support points of large structures can differ considerably. In this study, the quasi-static, the dynamic and the total structural responses of a multiply supported rigid plate to spatially varying multiple ground motion inputs are analysed. The results, together with those obtained by neglecting ground motion phase shifts and those by neglecting ground motion coherency losses, are compared with the corresponding structural responses to a single input. The comparisons show that the single input method overestimates translational responses and underestimates rotational responses. They also show that the responses are sometimes overestimated and sometimes underestimated by neglecting the coherency loss effects between the multiple ground motion excitations.     

Simulation of multi-support depth-varying earthquake ground motions within heterogeneous onshore and offshore sites

This paper presents a novel approach to model and simulate the multi-support depth-varying seismic motions (MDSMs) within heterogeneous offshore and onshore sites. Based on 1D wave propagation theory, the three-dimensional ground motion transfer functions on the surface or within an offshore or onshore site are derived by considering the effects of seawater and porous soils on the propagation of seismic P waves. Moreover, the depth-varying and spatial variation properties of seismic ground motions are considered in the ground motion simulation. Using the obtained transfer functions at any locations within a site, the offshore or onshore depth-varying seismic motions are stochastically simulated based on the spectral representation method (SRM). The traditional approaches for simulating spatially varying ground motions are improved and extended to generate MDSMs within multiple offshore and onshore sites. The simulation results show that the PSD functions and coherency losses of the generated MDSMs are compatible with respective target values, which fully validates the effectiveness of the proposed simulation method. The synthesized MDSMs can provide strong support for the precise seismic response prediction and performance-based design of both offshore and onshore large-span engineering structures.     

Spatial variability effects of ground motions on cable-stayed bridges

In this paper, stochastic analysis of a cable-stayed bridge subjected to spatially varying ground motions is performed. While the ground motion is described by power spectral density (PSD) function, the spatial variability of ground motions is taken into account with the incoherence and the wave-passage effects. The incoherence effect is examined by taking into account two extensively used models. As the effect of the wave-passage effect is investigated by using various wave velocities, the effect of local soil conditions where the bridge supports are constructed is outlined by using homogeneous firm, medium and soft soil conditions. Solutions obtained for the spatially varying ground motions are compared with those of the specialised cases of the ground motion model. Stationary as well as the transient response analyses are performed for the considered bridge model. It is concluded that spatial variability and propagation effects of ground motions have important effects on the dynamic behaviour of the bridge and the variability of the ground motions should be included in the stochastic analysis of cable-stayed bridges.     

Response spectrum analysis for non-classically damped linear system with multiple-support excitations

A new response spectrum method, which is named complex multiple-support response spectrum (CMSRS) method in this article, is developed for seismic analysis of non-classically damped linear system subjected to spatially varying multiple-supported ground motion. The CMSRS method is based on fundamental principles of random vibration theory and properly accounts for the effect of correlation between the support motions as well as between the modal displacement and velocity responses of structure, and provides an reasonable and acceptable estimate of the peak response in term of peak seismic ground motions and response spectra at the support points and the coherency function. Meanwhile, three new cross-correlation coefficients or cross covariance especially for the non-classically damped linear structures with multiple-supports excitations are derived under the same assumptions of the MSRS method of classically damped system. The CMSRS method is examined and compared to the results of time history analyses in two numerical examples of non-classically damped structures in consideration of the coherences of spatially variable ground motion. The results show that for non-classically damped structure, the cross terms representing the cross covariance between the pseudo-static and dynamic component are also quite small just as same as classically damped system. In addition, it is found that the usual way of neglecting all the off-diagonal elements in transformed damping matrix in modal coordinates in order to make the concerned non-classically damped structure to become remaining proportional damping property will bring some errors in the case of subjected to spatially excited inhomogeneous ground motion.     

The effect of spatially varying earthquake ground motions on the stochastic response of bridges isolated with friction pendulum systems

In this paper, a comprehensive investigation of the effect of spatially varying earthquake ground motions on the stochastic response of bridges isolated with friction pendulum systems is performed. The spatially varying earthquake ground motions are considered with incoherence, wave-passage and site-response effects. The importance of the site-response effect, which arises from the difference in the local soil conditions at different support points of the isolated bridge, is investigated particularly. Mean of maximum and variance response values obtained from the spatially varying earthquake ground motions are compared with those of the specialised cases of the ground motion model. It is shown that site-response component of the spatially varying earthquake ground motion model has important effects on the stochastic response of the isolated bridges. Therefore, to be more realistic in calculating the isolated bridge responses, the spatially varying earthquake ground motions should be incorporated in the analysis.     

Physics‐based probabilistic seismic hazard and loss assessment in large urban areas: A simplified application to Istanbul

A set of 3D physics‐based numerical simulations (PBS) of possible earthquakes scenarios in Istanbul along the North Anatolian Fault (Turkey) is considered in this article to provide a comprehensive example of application of PBS to probabilistic seismic hazard (PSHA) and loss assessment in a large urban area. To cope with the high‐frequency (HF) limitations of PBS, numerical results are first postprocessed by a recently introduced technique based on Artificial Neural Networks (ANN), providing broadband waveforms with a proper correlation of HF and low‐frequency (LF) portions of ground motion as well as a proper spatial correlation of peak values also at HF, that is a key feature for the seismic risk application at urban scale. Second, before application to PSHA, a statistical analysis of residuals is carried out to ensure that simulated results provide a set of realizations with a realistic within‐ and between‐event variability of ground motion. PBS results are then applied in a PSHA framework, adopting both the “generalized attenuation function” (GAF) approach, and a novel “footprint” (FP)‐based approach aiming at a convenient and direct application of PBS into PSHA. PSHA results from both approaches are then compared with those obtained from a more standard application of PSHA with empirical ground motion models. Finally, the probabilistic loss assessment of an extended simplified portfolio of buildings is investigated, comparing the results obtained adopting the different approaches: (i) GMPE, (ii) GAF, and (iii) FP. Only FP turned out to have the capability to account for the specific features of source and propagation path, while preserving the proper physically based spatial correlation characteristics, as required for a reliable loss estimate on a building portfolio spatially distributed over a large urban area.     

A framework for validation of seismic response analyses using seismometer array recordings

A framework for the validation of computational models used to predict seismic response based on observations from seismometer arrays is presented. The framework explicitly accounts for the epistemic uncertainty related to the unknown characteristics of the ‘site’ (i.e. the problem under consideration) and constitutive model parameters. A mathematical framework which makes use of multiple prediction–observation pairs is used to improve the statistical significance of inferences regarding the accuracy and precision of the computational methodology and constitutive model. The benefits of such a formal validation framework include: (i) development of consistent methods for determination of constitutive model parameters; (ii) rigorous, objective, and unbiased assessment of the validity of various constitutive models and computational methodologies for various problem types and ground motion intensities; and (iii) an improved understanding of the uncertainties in computational model assumptions, constitutive models and their parameters, relative to other seismic response uncertainties such as ground motion variability. Details regarding the implementation of such a framework to achieve the aforementioned benefits are also addressed.     

Vertical coherency function model of spatial ground motion

Many studies have focused on horizontal ground motion, resulting in many coherency functions for horizontal ground motion while neglecting related problems arising from vertical ground motion. However, seismic events have demonstrated that the vertical components of ground motion sometimes govern the ultimate failure of structures. In this paper, a vertical coherency function model of spatial ground motion is proposed based on the Hao model and SMART 1 array records, and the validity of the model is demonstrated. The vertical coherency function model of spatial ground motion is also compared with the horizontal coherency function model, indicating that neither model exhibits isotropic characteristics. The value of the vertical coherency function has little correlation with that of the horizontal coherency function. However, the coherence of the vertical ground motion between a pair of stations decreases with their projection distance and the frequency of the ground motion. When the projection distance in the wave direction is greater than 800 meters, the coherency between the two points can be neglected.     

Coherency analysis of accelerograms recorded by the UPSAR array during the 2004 Parkfield earthquake

Spatial variability of near‐fault strong motions recorded by the US Geological Survey Parkfield Seismograph Array (UPSAR) during the 2004 Parkfield (California) earthquake is investigated. Behavior of the lagged coherency for two horizontal and the vertical components is analyzed by separately examining the decay of coherency with frequency and distance. Assumptions, approximations, and challenges that are involved in estimation of the coherency from recorded data are presented in detail. Comparison of the UPSAR coherency estimates with coherency models that are commonly used in engineering practice sheds light on the advantages and limitations of different approaches to modeling the coherency, as well as on similarities and differences in the spatial variability exhibited by seismic ground motion arrays at different sites. Copyright © 2013 John Wiley & Sons, Ltd.     

Simulation of spatially varying ground motions including incoherence,wave‐passage and differential site‐response effects

A method is presented for simulating arrays of spatially varying ground motions, incorporating the effects of incoherence, wave passage, and differential site response. Non‐stationarity is accounted for by considering the motions as consisting of stationary segments. Two approaches are developed. In the first, simulated motions are consistent with the power spectral densities of a segmented recorded motion and are characterized by uniform variability at all locations. Uniform variability in the array of ground motions is essential when synthetic motions are used for statistical analysis of the response of multiply‐supported structures. In the second approach, simulated motions are conditioned on the segmented record itself and exhibit increasing variance with distance from the site of the observation. For both approaches, example simulated motions are presented for an existing bridge model employing two alternatives for modeling the local soil response: i) idealizing each soil‐column as a single‐degree‐of‐freedom oscillator, and ii) employing the theory of vertical wave propagation in a single soil layer over bedrock. The selection of parameters in the simulation procedure and their effects on the characteristics of the generated motions are discussed. The method is validated by comparing statistical characteristics of the synthetic motions with target theoretical models. Response spectra of the simulated motions at each support are also examined. Copyright © 2011 John Wiley & Sons, Ltd.     

A parametric study of the required seating length for bridge decks during earthquake

During the recent major earthquakes, some bridges suffered severe damage due to the pull-off-and-drop collapse of their decks. This is due to the large differential movements of the adjacent spans of bridges during strong shaking compared to the seating lengths provided. The differential movements are primarily due to the different vibration properties of adjacent spans and non-uniform ground excitations at the bridge supports. This paper analyses the effects of various bridge and ground motion parameters on the required seating lengths for bridge decks to prevent the pull-off-and-drop collapse. The random vibration method is used in the analysis. A two-span bridge model with different span lengths and vibration frequencies and subjected to various spatially varying ground excitations is analysed. Non-uniform spatial ground motions are modelled by the filtered Tajimi–Kanai power spectral density function and an empirical coherency function. Ground motions with different intensities, different cross-correlations and different site conditions are considered in the study. The required seating lengths for bridge decks are calculated. Numerical results are presented and discussed with respect to different bridge vibration and ground motion properties. © 1998 John Wiley & Sons, Ltd.