Yongxin Wu

Effects of Topographic Amplification Induced by a U‐Shaped Canyon on Seismic Waves

Abstract The series solution of wave functions for 2D scattering and diffraction of plane SH (shear horizontal) waves induced by a U‐shaped canyon is proposed herein to account for the topographic effect of such a canyon. The wave function expansion method has been frequently employed to study the topographic effect because it can reveal the physics of the wave scattering and can test the accuracy of other methods. Through a new domain decomposition strategy, the half‐space having a U‐shaped canyon is divided into three subregions. Hence, we defined three cylindrical coordinate systems. In each coordinate system, the wave field satisfying the Helmholtz equation was represented by means of the separation of variables method, in terms of the series of both Bessel functions and Hankel functions with unknown complex coefficients. Then three wave fields are all represented in the same coordinate system using the Graf addition theorem. The unknown coefficients are solved by satisfying the continuity conditions of the auxiliary boundary and the traction‐free boundary conditions on the bottom of the canyon. To show the effects of symmetrical and nonsymmetrical U‐shaped canyons on the surface ground motion, a parametric analysis is carried out in the frequency domain. Surface and subsurface transient responses in the time domain demonstrate the phenomenon of wave propagating and scattering. It is found that a zone of amplification can obviously take place at the bottom of a U‐shaped canyon with nearly vertical walls.

Stability Charts for 3D Failures of Homogeneous Slopes

AbstractA three-dimensional (3D) kinematically admissible rotational failure mechanism is extended from toe failure to include face failure and base failure for homogeneous slopes in both purely cohesive and frictional/cohesive soils. In the strict framework of limit analysis, an analytical approach is derived to obtain the upper bounds on slope stability and the corresponding type of the critical failure mechanism. Compared with the available results from the finite-element limit-analysis method, the 3D rotational failure mechanisms give the best estimate on the upper bound. A set of stability charts is presented in a wide range of parameters for 3D homogeneous slopes under both static and pseudostatic seismic loading conditions. This set is useful in assessing the safety of the 3D homogeneous slopes in practical applications.

Scattering of SH waves induced by a symmetrical V-shaped canyon: a unified analytical solution

This paper reports a series solution of wave functions for two-dimensional scattering and diffraction of plane SH waves induced by a symmetrical V-shaped canyon with different shape ratios. A half-space with a symmetrical V-shaped canyon is divided into two sub-regions by using a circular-arc auxiliary boundary. The two sub-regions are represented by global and local cylindrical coordinate systems, respectively. In each coordinate system, the wave field satisfying the Helmholtz equation is represented by the separation of variables method, in terms of the series of both Bessel functions and Hankel functions with unknown complex coefficients. Then, the two wave fields are described in the local coordinate system using the Graf addition theorem. Finally, the unknown coefficients are sought by satisfying the continuity conditions of the auxiliary boundary. To consider the phase characteristics of the wave scattering, a parametric analysis is carried out in the time domain by assuming an incident signal of the Ricker type. Surface and subsurface transient responses demonstrate the characteristics and mechanisms of wave propagating and scattering.

Comparison of the Spectral Representation Method to Simulate Spatially Variable Ground Motions

The spectral representation method (SRM) is widely used when simulating spatially variable ground motions. It has mainly two formulas, i.e., the random amplitudes and the random phases formulas. There exist three methods for decomposing the cross spectral density matrix: Cholesky decomposition, eigen decomposition, and root decomposition. Therefore, there are six forms with respect to the different combinations of the simulation formulas and the decomposition methods. To provide researchers and engineers with the guidance on choosing simulation method, the six forms are systematically investigated from five aspects: the power intensity, response spectra, and stochastic error of auto/cross spectral density, Fourier spectra, and difference indexes for Fourier amplitudes and phases. Finally, we give the following advice: the characteristics of the ground motions simulated by the random amplitudes formula are independent of the decomposition method, while the characteristics of the ground motions simulated by random phases formula are dependent of the decomposition method. Furthermore, the root decomposition is strongly recommended when utilizing the random phases formula.

Simulation of Spatially Varying Ground Motions in V-shaped Symmetric Canyons

A method of simulating spatially varying ground motions in V-shaped symmetric canyon is presented. The topographic amplification effect is taken into account by considering a 2-dimension wave propagation model. Two approaches are developed to simulate spatially varying ground motions. First, the spatially varying ground motions are simulated by using the power spectral density and an empirical coherency loss function. Second, the spatially varying ground motions are simulated by using given ground motions and an empirical coherency loss function. These two proposed methods are validated by comparing statistical characteristics of the synthetic motions with target theoretical models.

Simulation of Spatially Varying Non-Gaussian and Nonstationary Seismic Ground Motions by the Spectral Representation Method

AbstractSimulation of sample realizations of stochastic processes is the bedrock of the Monte Carlo method, and the accurate modeling of stochastic processes is crucial to determine realistic struc...

Scattering of Plane and Cylindrical SH Waves by a Horseshoe Shaped Cavity

An analytical solution for diffraction of both plane and cylindrical SH waves induced by a horseshoe shaped cavity with an inverted arch is presented in this paper. The geometry of the cavity is assumed to be composed of two circular arcs. By introducing an auxiliary boundary, the whole physical region is divided into two computational regions. The scattered wavefield in the open region and the standing wavefield in the enclosed region are presented by means of the wave function expansion method. Both of the wavefields are given in terms of the wave function series with unknown coefficients. By applying the Graf’s addition formula, two systems of equations for seeking the unknowns are derived by taking advantage of the boundary conditions based on the region-matching strategy. The problem of wave scattering is finally solved after seeking the solutions of the two systems of equations through standard matrix techniques. Then the effects of the excitation frequency, the cavity embedment depth and cavity geometry are discussed. The differences in terms of ground motions under different excitations and the influence of source location under cylindrical waves are also examined.

An Improved Method for the Generating of Spectrum-Compatible Time Series Using Wavelets

In dynamic analyses of important structures, seismic input may be defined in the form of time series. It is required that the response spectrum of this input time series be compatible with a specified target response spectrum. Time domain spectral matching, which is used to generate spectrum compatible acceleration time series, is investigated in some detail. First, a new, improved wavelet is presented, and the new adjustment wavelet can prevent drifts in the resulting velocity and displacement time series without applying a baseline correction. Next, the analytical solution of the matrix accounting for the cross correlation of each wavelet is given in order to ensure the speed of the matching procedure. Finally, some aspects, such as the reduction factors and the matching order, are discussed to ensure the stability and efficiency of the matching procedure. Accordingly, the characteristics of the matching procedure are illustrated by numerical examples.

Error Assessment for the Coherency Matrix-Based Spectral Representation Method in Multivariate Random Processes Simulation

AbstractMultivariate random processes are usually simulated by the spectral representation method (SRM). According to the matrix for decomposition, the SRM has two main types, that is, the SRM based on the decomposition of the power spectral density (PSD) matrix denoting the PSD matrix-based SRM, and the SRM based on the decomposition of the coherency matrix denoting the coherency matrix based-SRM. The stochastic errors of the PSD for the PSD matrix-based SRM have been given. This paper presents the stochastic errors of the PSD for the coherency matrix-based SRM, and makes a comparison of these errors for the PSD matrix-based SRM. For the random amplitudes formulas and random phase formula and Cholesky decomposition method, the stochastic errors of the PSDs for the PSD matrix-based SRM are the same as or the coherency matrix-based SRM, whereas for the random phases formula and eigendecomposition method and random phases formula and root decomposition method, they are different. However, the differences ar...

Approximation approach to the SRM based on root decomposition in the simulation of spatially varying ground motions

The spectral representation method (SRM) is widely used to simulate spatially varying ground motions. This study focuses on the approximation approach to the SRM based on root decomposition, which can improve the efficiency of the simulation. The accuracy of the approximation approach may be affected by three factors: matrix for decomposition, distribution of frequency interpolation nodes and elements for interpolation. The influence of these factors on the accuracy of this approach is examined and the following conclusions are drawn. The SRM based on the root decomposition of the lagged coherency matrix exhibits greater accuracy than the SRM based on the root decomposition of the cross spectral matrix. The equal energy distribution of frequency interpolation nodes proposed in this study is more effective than the counter pith with an equal spacing. Elements for interpolation do not have much of an effect on the accuracy, so interpolation of the elements of the decomposed matrix is recommended because it is less complicated from a computational efficiency perspective.