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Table Of Integrals Series And Products

Theoretical predictions of seismic motions as a function of source strength are often expressed as frequency-domain scaling models. The observations of interest to strong-motion seismology, however, are usually in the time domain (e.g., various peak motions, including magnitude). The method of simulation presented here makes use of both domains; its essence is to filter a suite of windowed, stochastic time series so that the amplitude spectra are equal, on the average, to the specified spectra. Because of its success in predicting peak and rms accelerations (Hanks and McGuire, 1981), an ω -squared spectrum with a high-frequency cutoff ( f m), in addition to the usual whole-path anelastic attenuation, and with a constant stress parameter (Δ σ ) has been used in the applications of the simulation method. With these assumptions, the model is particularly simple: the scaling with source size depends on only one parameter—seismic moment or, equivalently, moment magnitude. Besides peak acceleration, the model gives a good fit to a number of ground motion amplitude measures derived from previous analyses of hundreds of recordings from earthquakes in western North America, ranging from a moment magnitude of 5.0 to 7.7. These measures of ground motion include peak velocity, Wood-Anderson instrument response, and response spectra. The model also fits peak velocities and peak accelerations for South African earthquakes with moment magnitudes of 0.4 to 2.4 (with f m = 400 Hz and Δ σ = 50 bars, compared to f m = 15 Hz and Δ σ = 100 bars for the western North America data). Remarkably, the model seems to fit all essential aspects of high-frequency ground motions for earthquakes over a very large magnitude range .

Although the simulation method is useful for applications requiring one or more time series, a simpler, less costly method based on various formulas from random vibration theory will often suffice for applications requiring only peak motions. Hanks and McGuire (1981) used such an approach in their prediction of peak acceleration. This paper contains a generalization of their approach; the formulas used depend on the moments (in the statistical sense) of the squared amplitude spectra, and therefore can be applied to any time series having a stochastic character, including ground acceleration, velocity, and the oscillator outputs on which response spectra and magnitude are based .

Stochastic simulation of high-frequency ground motions based on seismological models of the radiated spectra

Paleoseismological data for the Wasatch and San Andreas fault zones have led to the formulation of the characteristic earthquake model, which postulates that individual faults and fault segments tend to generate essentially same size or characteristic earthquakes having a relatively narrow range of magnitudes near the maximum. Analysis of scarp-derived colluvium in trench exposures across the Wasatch fault provides estimates of the timing and displacement associated with individual surface faulting earthquakes. At all of the sites studied, the displacement per event has been consistently large; measured values range from 1.6 to 2.6 m, and the average is about 2 m. On the basis of variability in the timing of individual events as well as changes in scarp morphology and fault geometry, six major segments are recognized along the Wasatch fault. On the basis of the most likely number of surface faulting events (18) that have occurred on segments of the Wasatch fault zone during the past 8000 years, an average recurrence interval of 400–666 years with a preferred average of 444 years is calculated for the entire zone. Geologic data on the distribution of slip associated with prehistoric earthquakes and slip rates along the south-central segment of the San Andreas fault suggest that the M 8 1857 earthquake is a characteristic earthquake for this segment. Comparisons of earthquake recurrence relationships on both the Wasatch and San Andreas faults based on historical seismicity data and geologic data show that a linear (constant b value) extrapolation of the cumulative recurrence curve from the smaller magnitudes leads to gross underestimates of the frequency of occurrence of the large or characteristic earthquakes. Only by assuming a low b value in the moderate magnitude range can the seismicity data on small earthquakes be reconciled with geologic data on large earthquakes. The characteristic earthquake appears to be a fundamental aspect of the behavior of the Wasatch and San Andreas faults and may apply to many other faults as well.

Fault behavior and characteristic earthquakes: Examples from the Wasatch and San Andreas Fault Zones

A simple and powerful method for simulating ground motions is to combine parametric or functional descriptions of the ground motion’s amplitude spectrum with a random phase spectrum modified such that the motion is distributed over a duration related to the earthquake magnitude and to the distance from the source. This method of simulating ground motions often goes by the name “the stochastic method.” It is particularly useful for simulating the higher-frequency ground motions of most interest to engineers (generally, f > 0.1 Hz), and it is widely used to predict ground motions for regions of the world in which recordings of motion from potentially damaging earthquakes are not available. This simple method has been successful in matching a variety of ground-motion measures for earthquakes with seismic moments spanning more than 12 orders of magnitude and in diverse tectonic environments. One of the essential characteristics of the method is that it distills what is known about the various factors affecting ground motions (source, path, and site) into simple functional forms. This provides a means by which the results of the rigorous studies reported in other papers in this volume can be incorporated into practical predictions of ground motion.

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Simulation of Ground Motion Using the Stochastic Method

Recent advances in nonlinear passive vibration isolators

A numerical procedure to obtain the dynamic Green's functions for layered viscoelastic media is presented. The procedure is based on numerical evaluation of certain Hankel-type integrals which appear in an integral representation derived previously by the authors. Comparisons illustrating the accuracy and flexibility of the approach are made with a number of solutions obtained by other methods.

On the Green's functions for a layered half-space. Part II

The optimal values for the distribution of passive dampers interconnecting two adjacent structures of different heights are determined. The dampers are selected to minimize the seismic response in the first and second modes of the taller of the two structures. For simplicity, the structures are represented as uniform damped shear beams subjected to a common ground motion. Under certain conditions, apparent damping ratios as high as 12 and 15 per cent can be achieved in the first and second modes of lightly damped structures by the introduction of interconnection dampers. The largest reduction of the response in the first mode is achieved when the taller structure is about twice the height of the second structure. © 1998 John Wiley & Sons, Ltd.

Optimal damping between two adjacent elastic structures

A study is made of the dynamic interaction, through the soil, between two parallel infinite shear walls placed on rigid foundations. The steady-state response of both structures for a vertically incident SH wave is obtained and compared with the corresponding values resulting from consideration of only one structure. 

It is found that the additional interaction effects caused by the presence of a second structure are especially important at low frequencies and in the neighborhood of the fixed-base natural frequencies of the second structure. For high frequencies it is sufficient to consider the interaction between each structure and the soil, ignoring the presence of other structures.

Dynamic structure-soil-structure interaction

Three functional forms for earthquake occurrence relations are compared, and slip rate constraints on each are derived. Constrained occurrence relations referred to total fault area, to rupture area of the maximum magnitude earthquake, and to a particular site along the fault and are presented and discussed. The slip rate constraints provide a means to estimate M max from occurrence rates of small magnitude earthquakes, or a means to estimate the threshold of observations in a qualitative earthquake catalog, as well as a means to estimate occurrence rates when M max is obtained from other considerations. This paper shows examples of these three types of applications. The relations considered are: N 1 ( M ) is such that the cumulative occurrence rate of earthquakes with magnitude greater than M is an exponential function truncated at M max ; N 2 ( M ) is such that the corresponding incremental occurrence rate is a truncated exponential function; and N 3 ( M ) is such that the corresponding incremental occurrence rate is an exponential function plus a constant which causes the rate to go to zero at M max . A speculative model for the magnitudes of the earthquakes at Pallett Creek observed by Sieh (1978a) suggests that N 2 ( M ) is more appropriate than N 1 ( M ) or N 3 ( M ) for these earthquakes.

Consequences of slip rate constraints on earthquake occurrence relations

A five-parameter discrete model that approximates the dynamic force4isplacement relationship for rigid foundations undergoing vertical vibrations on a uniform elastic half-space is presented. The model involves a combination of two springs, two viscous dampers and a mass. Values of the parameters for circular, square and rectangular foundations placed on the surface or embedded in an elastic half-space are listed. The parameters are obtained by minimizing the discrepancy between the force4isplacement relation for the model and that obtained by solution of the mixed boundary-value problem of the rigid foundation on an elastic half-space. The definition of an appropriate input motion to represent wave excitation is also discussed. The input motion to the discrete model differs from the input motion that should be used in a continuum model.

J. Enrique Luco

Papers

On the Green's functions for a layered half-space. Part II

Optimal damping between two adjacent elastic structures

Dynamic structure-soil-structure interaction

Consequences of slip rate constraints on earthquake occurrence relations

Discrete models for vertical vibrations of surface and embedded foundations